# 4847.Apel N. - Approaches to the Description of Anisotropic Material Behaviour at Finite Elastic and Plastic Deformations. Theory and Numerics (2004).pdf

код для вставкиСкачатьApproaches to the Description of Anisotropic Material Behaviour at Finite Elastic and Plastic Deformations — Theory and Numerics — Nikolas Apel PSfrag replacements ϕ X x B S F = ∇ϕ C, Gp g, cp f (x, Ξ(x)) = f (Q ? x, Ξ(Q ? x)) ∀Q ∈ G Bericht Nr.: I-12 (2004) Institut für Mechanik (Bauwesen), Lehrstuhl I Professor Dr.-Ing. C. Miehe Stuttgart 2004 Approaches to the Description of Anisotropic Material Behaviour at Finite Elastic and Plastic Deformations — Theory and Numerics — Von der Fakultät Bau- und Umweltingenieurwissenschaften der Universität Stuttgart zur Erlangung der Würde eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung vorgelegt von Nikolas Apel aus Ulm Hauptreferat : Prof. Dr.-Ing. Christian Miehe Korreferat : Prof. Dr. rer.nat. Bob Svendsen Tag der mündlichen Prüfung: 28. November 2003 Institut für Mechanik (Bauwesen) der Universität Stuttgart 2004 Herausgeber: Prof. Dr.-Ing. habil. C. Miehe Organisation und Verwaltung: Institut für Mechanik (Bauwesen) Lehrstuhl I Universität Stuttgart Pfaffenwaldring 7 70550 Stuttgart Tel.: ++49–(0)711/685–6378 Fax : ++49–(0)711/685–6347 c Nikolas Apel ° Institut für Mechanik (Bauwesen) Lehrstuhl I Universität Stuttgart Pfaffenwaldring 7 70550 Stuttgart Tel.: ++49–(0)711/685–6326 Fax : ++49–(0)711/685–6347 Alle Rechte, insbesondere das der Übersetzung in fremde Sprachen, vorbehalten. Ohne Genehmigung des Autors ist es nicht gestattet, dieses Heft ganz oder teilweise auf fotomechanischem Wege (Fotokopie, Mikrokopie) zu vervielfältigen. ISBN 3-937859-00-4 (D 93 Stuttgart) Zusammenfassung Die vorliegende Arbeit befaßt sich mit rein makroskopischen Beschreibungen richtungsabhängigen Materialverhaltens. Zentrale neue Entwicklungen liegen auf dem Gebiet der Theorie und Numerik anisotroper finiter Plastizität. Nach einer Diskussion der grundlegenden Konzepte zur Klassifizierung von Materialien anhand von materiellen Symmetriegruppen sowie der Zusammenstellung der Konzepte zur Formulierung isotroper Tensorfunktionen und -polynome werden alternative makroskopische Formulierungen finiter Plastizität diskutiert. Formulierungen auf der Basis der multiplikativen Zerlegung des Deformationsgradienten in einen elastischen und plastischen Anteil führen auf neundimensionale Fließregeln und erlauben die Abbildung der plastischen Rotation. Im Gegensatz dazu steht die Beschreibung der plastischen Deformation mittels einer plastischen Metrik. Für letzteres führt die Wahl logarithmischer Verzerrungen und die additive Zerlegung der totalen Verzerrung in elastische und plastische Anteile auf eine Klasse von Materialgesetzen im logarithmischen Verzerrungsraum. Sie zeichnet sich durch einen modularen Aufbau und Strukturen und Algorithmen ähnlich zu denen der geometrisch linearen Theorie aus. Auf der numerischen Seite werden implizite und explizite Integrations- und Spannungsaufdatierungsalgorithmen für anisotrope Plastizität bereit gestellt. Eine sorgfältige Konstruktion dieser Algorithmen ist von entscheidender Bedeutung für die Effizienz der numerischen Simulationen. Besonderes Augenmerk wird auf Algorithmen für Variationsformulierungen gelegt. Bedingt durch die inhärente (inkrementelle) Potentialstruktur arbeiten diese mit symmetrischen Größen und benötigen daher weniger Speicherplatz und Löserkapazität als klassische, unsymmetrische Verfahren. Abstract The present work deals with purely macroscopic descriptions of anisotropic material behaviour. Key aspects are new developments in the theory and numerics of anisotropic plasticity. After a short discussion of the classification of solids by symmetry transformations a survey about representation theory of isotropic tensor functions and tensor polynomials is given. Next alternative macroscopic approaches to finite plasticity are discussed. When considering a multiplicative decomposition of the deformation gradient into an elastic part and a plastic part, a nine dimensional flow rule is obtained that allows the modeling of plastic rotation. An alternative approach bases on the introduction of a metric-like internal variable, the so-called plastic metric, that accounts for the plastic deformation of the material. In this context, a new class of constitutive models is obtained for the choice of logarithmic strains and an additive decomposition of the total strain measure into elastic and plastic parts. The attractiveness of this class of models is due to their modular structure as well as the affinity of the constitutive model and the algorithms inside the logarithmic strain space to models from geometric linear theory. On the numerical side, implicit and explicit integration algorithms and stress update algorithms for anisotropic plasticity are developed. Their numerical efficiency crucially bases on their careful construction. Special focus is put on algorithms that are suitable for variational formulations. Due to their (incremental) potential property, the corresponding algorithms can be formulated in terms of symmetric quantities. A reduced storage effort and less required solver capacity are key advantages compared to their standard counterparts. Acknowledgements The work presented in this thesis was carried out in the years between 1999 and 2003, when I was a co-worker at the Institute of Applied Mechanics (Chair I) at the University of Stuttgart. At the end of this period I feel grateful to a lot of people who accompanied me in these five years. First of all, I want to thank my academic teacher Professor Christian Miehe for his scientific support and for the fruitful discussions we had. Without his research in the field of Applied Mechanics this thesis could not have achieved these results. My special thanks also go to Professor Bob Svendsen for his interest in my research and for acting as the external examiner of this thesis. Next I want to thank my fellow workers at the institute, who were mainly responsible for the friendly atmosphere within the institute. Especially, I would like to express my gratitude to my room mate Matthias Lambrecht for the support he gave me and the many discussions we had. Parts of this thesis are based on scientific results he was mainly involved with. Furthermore, I would like to thank Andreas Koch for the good collaboration and the many interesting discussions about a lot of mechanical topics. I am also very grateful to Grieta Himpel for her incessant interest in mechanical and computational problems throughout the years which is also true to Sonja Baumberger, whom I supervised during her diploma thesis, too. Their interest on high-end topics of theoretical and computational mechanics led to many interesting and helpful discussions that helped me to see some things more clearly. I want to thank my wife Heidi for giving me moral support in all these years. She kept my options open from many everyday-life problems and thus had a great share in the accomplishment of my research activities. Last but not least I should like to thank my father Karlheinz for the great help he gave me at the proof-reading stage as well as Dominik Zimmermann for his support regarding organizational matters. Stuttgart, January 2004 Nikolas Apel I Contents Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Fundamentals of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1. Finite Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1. Representation, Transformation and Rotation of Tensorial Objects . 7 2.1.2. Motion of a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2. Notion of Stresses and Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1. Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2. Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3. Balance Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1. Balance of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2. Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.3. Balance of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.4. Balance of Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.5. Balance of Entropy and Second Law of Thermodynamics . . . . . . . . 19 2.4. Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.1. Principle of Material Objectivity . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.2. Material Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3. Material Symmetries — Classification of Solids . . . . . . . . . . . . . . . . . . 23 3.1. Construction of a Space Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2. Symmetry Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.1. Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.2. Rotation-Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.3. Tensor Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.4. Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3. The 14 Bravais Lattices and 7 Crystal Systems . . . . . . . . . . . . . . . . . . . . 26 3.3.1. Triclinic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.2. Monoclinic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.3. Orthorhombic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.4. Tetragonal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.5. Cubic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.6. Trigonal and Hexagonal Symmetry . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4. The 32 Crystal Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1. The Motif — Inner Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 32 II Contents 3.4.2. Notation of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5. Icosahedral, Cylindrical and Spherical Symmetry Classes . . . . . . . . . . . . . 32 3.6. Classification into 14 Types of Anisotropy . . . . . . . . . . . . . . . . . . . . . . . 35 4. Representations of Anisotropic Tensor Functions . . . . . . . . . . . . . . . . . 37 4.1. Definitions and Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2. Isotropic Extension of Anisotropic Tensor Functions . . . . . . . . . . . . . . . . 38 4.3. Isotropic Functions of First- and Second-Order Tensors . . . . . . . . . . . . . . 39 4.3.1. Wang’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3.2. Smith’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.3. Comparison of the Results Obtained by Wang and Smith . . . . . . . 46 4.4. Isotropic Polynomials of First- and Second-Order Tensors . . . . . . . . . . . . 46 4.4.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4.2. Integrity Basis for Sets of First-Order Tensors . . . . . . . . . . . . . . . 47 4.4.3. Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4.4. Integrity Bases for Sets of First-Order and Second-Order Tensors . . 49 4.5. Irreducibility of Integrity Bases and Functional Bases . . . . . . . . . . . . . . . 51 4.6. Quadratic Functions of a Symmetric Second-Order Tensor . . . . . . . . . . . . 52 4.6.1. Triclinic Functions — Symmetry Group Ci . . . . . . . . . . . . . . . . . . 53 4.6.2. Monoclinic Functions — Symmetry Group C2h . . . . . . . . . . . . . . . 55 4.6.3. Orthorhombic Functions — Symmetry Group D2h . . . . . . . . . . . . . 57 4.6.4. Tetragonal Functions — Symmetry Group C4h . . . . . . . . . . . . . . . 58 4.6.5. Tetragonal Functions — Symmetry Group D4h . . . . . . . . . . . . . . . 60 4.6.6. Trigonal Functions — Symmetry Group S6 . . . . . . . . . . . . . . . . . . 61 4.6.7. Trigonal Functions — Symmetry Group D3d . . . . . . . . . . . . . . . . . 63 4.6.8. Hexagonal Functions — Symmetry Group C6h . . . . . . . . . . . . . . . 65 4.6.9. Hexagonal Functions — Symmetry Group D6h . . . . . . . . . . . . . . . 66 4.6.10. Cubic Functions — Symmetry Group Oh . . . . . . . . . . . . . . . . . . . 68 4.6.11. Cubic Functions — Symmetry Group Th . . . . . . . . . . . . . . . . . . . 69 4.6.12. Transversely Isotropic Functions — Symmetry Group C∞h . . . . . . . 70 4.6.13. Transversely Isotropic Functions — Symmetry Group D∞h . . . . . . 71 4.6.14. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5. Anisotropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1. General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2. Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.1. Fiber-Reinforced Technical Rubber . . . . . . . . . . . . . . . . . . . . . . . 76 5.3. Numerical Example: Tension Test of a Fiber-Reinforced Bar . . . . . . . . . . 77 Contents III 5.4. Numerical Example: Inflation of a Fiber-Reinforced Sheet . . . . . . . . . . . . 79 6. Approaches to Anisotropic Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.1. Kinematic Approach in Terms of a Plastic Map . . . . . . . . . . . . . . . . . . . 81 6.1.1. Geometry of Multiplicative Plasticity, Stress Tensors . . . . . . . . . . . 81 6.2. Constitutive Model for Plastic-Map Plasticity . . . . . . . . . . . . . . . . . . . . 82 6.2.1. Energy Storage and Elastic Stress Response . . . . . . . . . . . . . . . . . 82 6.2.2. Dissipation and Plastic Flow Response . . . . . . . . . . . . . . . . . . . . . 83 6.2.3. Decoupling of the Constitutive Functions . . . . . . . . . . . . . . . . . . . 84 6.2.4. Continuous Tangent Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3. Algorithmic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3.1. Outline of the Standard Stress Update Algorithms . . . . . . . . . . . . 86 6.3.2. Implicit Stress Update Algorithm (U1) . . . . . . . . . . . . . . . . . . . . 88 6.3.3. Explicit Stress Update Algorithm (U2) . . . . . . . . . . . . . . . . . . . . 90 6.3.4. Algorithmic Tangent Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.4. Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.4.1. Standard Formulation of Rate-Independent Plasticity . . . . . . . . . . 95 6.4.2. Incremental Variational Formulation . . . . . . . . . . . . . . . . . . . . . . 96 6.4.3. Specification to Multi-Surface Plasticity . . . . . . . . . . . . . . . . . . . . 97 6.4.4. Implicit Discrete Variational Formulation (V1) . . . . . . . . . . . . . . . 99 6.4.5. Algorithmic Solution of the Discrete Variational Formulation (V1) . 101 6.4.6. Stresses and Algorithmic Tangent Moduli (V1) . . . . . . . . . . . . . . . 102 6.4.7. Application of the Algorithm (V1) to the Model Problem . . . . . . . 102 6.4.8. Explicit Discrete Variational Formulation (V2) . . . . . . . . . . . . . . . 104 6.4.9. Algorithmic Solution of the Discrete Variational Formulation (V2) . 104 6.4.10. Stresses and Algorithmic Tangent Moduli (V2) . . . . . . . . . . . . . . . 105 6.4.11. Application to Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.5. Model Problem: Double Slip Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.5.1. Numerical Example: Rotation of the Slip Systems . . . . . . . . . . . . . 108 6.5.2. Numerical Example: Drawing of a Flange . . . . . . . . . . . . . . . . . . 110 6.6. Model Problem: Plasticity based on Quadratic Functions . . . . . . . . . . . . 111 6.6.1. Elastic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.6.2. Plastic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.6.3. D2h -, Oh -, D∞h and O(3)-invariant Fourth-Order Tensors . . . . . . . 112 6.6.4. Comparison of the Stress Update Algorithms . . . . . . . . . . . . . . . . 114 6.6.5. Kelvin-Mode Decomposition of Fourth-Order Tensors . . . . . . . . . . 116 6.7. Constitutive Model for Plastic-Metric Plasticity . . . . . . . . . . . . . . . . . . . 117 6.7.1. Energy Storage and Elastic Stress Response . . . . . . . . . . . . . . . . . 117 IV Contents 6.7.2. Dissipation and Plastic Flow Response . . . . . . . . . . . . . . . . . . . . . 118 6.7.3. Decoupling of the Constitutive Functions . . . . . . . . . . . . . . . . . . . 118 6.7.4. Continuous Tangent Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.8. Algorithmic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.8.1. Implicit Stress Update Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 119 6.8.2. Algorithmic Tangent Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7. Additive Plasticity in the Logarithmic Strain Space . . . . . . . . . . . . . . .123 7.1. Kinematic Approach in Terms of a Plastic Metric . . . . . . . . . . . . . . . . . . 123 7.1.1. Current Metric, Plastic Metric and Stresses . . . . . . . . . . . . . . . . . 123 7.2. Constitutive Model in the Logarithmic Strain Space . . . . . . . . . . . . . . . . 125 7.2.1. Energy Storage and Elastic Stress Response . . . . . . . . . . . . . . . . . 125 7.2.2. Dissipation and Plastic Flow Response . . . . . . . . . . . . . . . . . . . . . 126 7.2.3. Considered Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2.4. Continuous Elastic-Plastic Tangent Moduli . . . . . . . . . . . . . . . . . 128 7.3. Algorithmic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.3.1. Stress Update Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.3.2. Algorithmic Tangent Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.4. Variational Formulation in the Logarithmic Strain Space . . . . . . . . . . . . . 132 7.4.1. Standard Formulation of Inelasticity . . . . . . . . . . . . . . . . . . . . . . 133 7.4.2. Incremental Variational Formulation . . . . . . . . . . . . . . . . . . . . . . 133 7.4.3. Specification to Multi-Surface Models of Elasto-Plasticity . . . . . . . 134 7.4.4. Algorithmic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.4.5. Stresses and Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.4.6. Application to Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8. Finite Shell Element Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . .139 8.1. Parameterization of the Shell-Like Continuum . . . . . . . . . . . . . . . . . . . . 139 8.2. Finite Element Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8.2.1. Compatible Displacement Approach . . . . . . . . . . . . . . . . . . . . . . 140 8.2.2. Assumed Strain Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.2.3. Enhanced Strain Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.3. Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.4. Gradient-type Interface to Constitutive Models . . . . . . . . . . . . . . . . . . . 145 9. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147 9.1. Necking of an Isotropic Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.2. Necking of an Isotropic Rectangular Strip . . . . . . . . . . . . . . . . . . . . . . . 149 Contents V 9.3. Drawing of a Circular Flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.3.1. Comparison of Additive and Multiplicative Plasticity . . . . . . . . . . 154 9.3.2. Comparison of the Stress Update Algorithms . . . . . . . . . . . . . . . . 156 9.3.3. Comparison of Additive and Multiplicative Shell Element Design . . 160 9.4. Deep Drawing of Cubic and Orthotropic Sheets . . . . . . . . . . . . . . . . . . . 160 9.4.1. Comparison of Additive and Multiplicative Plasticity . . . . . . . . . . 161 9.4.2. Comparison of the Stress Update Algorithms . . . . . . . . . . . . . . . . 164 9.4.3. Comparison of Multiplicative and Additive Shell Element Design . . 164 10. Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169 A. On Matrix Representations of Symmetric Tensors . . . . . . . . . . . . . . . .179 A.1. Coordinate Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 A.2. Spectral Decomposition of Symmetric Fourth-Order Tensors . . . . . . . . . . 180 B. Incremental Variational Formulation — Derivatives . . . . . . . . . . . . . . .181 C. Positively Homogenous Functions of Degree One . . . . . . . . . . . . . . . . .182 Introduction 1 1. Introduction The modeling of anisotropic material behaviour is an area of research of great importance in material science. During the last years, the activity of research has been continuously increasing and this trend still continues. In particular, this concerns the modeling of anisotropic inelastic material response within the framework of elasto-plasticity which is part of this thesis. Symmetry theory provides a valuable set of concepts for the specification of structure, particularly important for crystalline materials but also useful for describing non-crystalline materials. An important class of materials are crystalline solids. In the ideal state their matter is arranged periodically in space and therefore the material properties depend on the spatial orientation of the solid. Based on the symmetry of their atomic structure they are divided into the 32 crystal classes. Ideal crystals possess 1-, 2-, 3-, 4- and 6-fold rotational symmetries and the corresponding rotoinversions obtained by combining these five rotations with a central inversion. But not all materials belong to these classes. In the eighties quasi-crystals were discovered. Their arrangement of matter violates the rotational symmetries for a periodic structure, 5- and 10-fold symmetry axes were observed. A key characteristic of the above mentioned crystalline materials is that their inherent symmetry can be described by discrete rotations and rotoinversions. The structure of many non-crystalline materials can be characterized by continuous symmetry transformations. Typical examples are engineering materials like composites consisting for example of fabrics embedded in a matrix material or bio-materials like softtissues possessing a fibrous structure. Homogenized material behaviour and overall behaviour of materials are often invariant with respect to continuous symmetry transformations, too. A comprehensive introduction to classical crystallography is the textbook by Voigt [142]. An illustrative introduction to symmetry transformations can be found in Kennon [60]. Recent publications are Whittaker [152] Borchardt-Ott [24], Kleber, Bautsch & Bohm [61] or Allen & Thomas [1]. A comprehensive work is the “International Tables for Crystallography” edited by Hahn [45]. Chapter 3 gives an overview of the classification of solids based on their inherent symmetries. Complex constitutive behaviour of materials is usually mathematically described by scalar-valued and tensor-valued tensor functions. The principle of material frame invariance and the principle of material symmetry restrict the form of the tensor functions. While the first restriction must hold for all constitutive equations the second one depends on the concrete material that is to be described and on the constitutive model that is used. In this work we do not differentiate between material symmetries and physical symmetries, cf. Zheng & Boehler [163]. The set of transformations that leave the constitutive response unchanged is denoted as material symmetry group. It belongs to one of the 32 crystal classes, the infinitely many classes of non-crystallographic point groups, the five cylindrical classes or the two spherical classes. The theory of isotropic tensor functions of arbitrary sets of vectors and second-order tensors is well-known. Representations for polynomials were derived by many authors, cf. for example the review article by Rychlewski & Zhang [111]. A comprehensive discussion of the theory is the contribution by Spencer [132]. Complete representations for tensor 2 Introduction functions go back to the competitive articles by Wang [144, 145, 146, 147] and Smith [129]. Later, the representations obtained by both authors with different approaches were compared and unified by Boehler [20]. The papers of Lokhin & Sedov [67], Boehler [21] and Liu [66] discuss, how the framework of isotropic tensor functions can be used to describe anisotropic behaviour. Therefore so-called structural tensors that characterize the material symmetry group are introduced as additional arguments into the constitutive function. Unfortunately, only a few classes of symmetry groups can be characterized by vectors or second-order tensors. Zheng [162] lists a single structural tensor of first-order up to fourth-order for almost all of the classes above mentioned. Representations for tensor functions of higher order argument tensors, however, are still topic of research and available only in some particular cases, see for example Zheng & Betten [160] and Betten & Helisch [14, 15]. Recently, Xiao [156, 157] extended the structural tensor approach to obtain complete representations for all of the 32 crystal classes and all noncrystal classes by introducing structural tensor functions as additional arguments instead of constant structural tensors. Chapter 4 gives a survey of the theory of isotropic tensor polynomials and isotropic tensor functions. The isotropic extension with structural functions is discussed and applied to the important case of quadratic potential functions for all classes of symmetry. Representations of the second derivatives are derived, which govern the well-known coordinate representations that are usually obtained by direct application of symmetry transformations onto the fourth-order second-derivative tensor. Before applying the theory of isotropic tensor functions to complex constitutive models for inelastic materials, we shortly illustrate its application in chapter 5 for elasticity. An important goal of this thesis is to pick up and discuss existing alternative approaches to finite strain elasto-plasticity and to present further developments which are the result of recent research. Essentially two different approaches exist as for the local constitutive modeling of finite elasto-plastic material behaviour. They can be traced back to the classical works of Green & Naghdi [43], Lee [65], Rice [107] and Mandel [71] and are controversially discussed in literature, cf. the review article of Naghdi [95]. The key ingredient of plastic-map plasticity is the multiplicative decomposition of the deformation gradient into an elastic and plastic part F = F e F p . In recent years, the microstructurebased theory for the description of finite elasto-plastic deformations in ideal single crystals, often denoted as the continuum slip theory, has achieved a degree of common acceptance. A common approach is the multiplicative composition of the plastic map and the total deformation measure to the Hencky-type elastic strain variable e Ē := 1 2 [ln[F p−T CF p−1 ] − ln[1]] (1.1) that enters a constitutive function for the energy storage of the material. The plastic map F p ∈ GL(3)+ is considered to be an internal variable that may describe the flow of matter through the crystal by multiple shearings on crystallographic slip planes. The reversible distortion of the lattice including a rigid body rotation is described by the elastic map F e . This is the classical approach to finite crystal plasticity proposed by Rice [107], Kröner & Teodosiu [64] and Mandel [71]. Recent works on multiplicative finite plasticity are Simó [122], Cuitiño & Ortiz [34], Anand & Kothari [2], Miehe [79, 80], Miehe, Schröder & Schotte [93] and Ortiz & Stainier [98]. These formulations take into account elastic and plastic anisotropies associated with certain characteristics of the crystals. Svendsen [136] and Svendsen, Arndt, Klingbeil & Sievert [138] 3 Introduction consider the incorporation of non-linear isotropic and kinematic hardening effects into the constitutive equations. The modeling of anisotropic inelastic material behaviour by phenomenological approaches using this multiplicative composition can be found e.g. in Svendsen [137] and Menzel [75]. The plastic rotational part of F p is well defined in crystal plasticity, cf. Boyce, Weber & Parks [25], but is often controversially discussed whenever the multiplicative ansatz (1.1) is used in purely phenomenological theories of finite plasticity, see for example Casey & Naghdi [30], Dafalias [36] and Naghdi [95]. Phenomenological assumptions for the plastic rotation often appear somewhat artificial, for example by simply setting the plastic spin to zero as considered e.g. in Moran, Ortiz & Shih [94], Miehe [78] and Simó [122]. It is convenient to formulate constitutive equations with respect to the reference configuration for isotropic and anisotropic purely phenomenological models of anisotropic plasticity with preferred structural directors deforming with the material. A framework of plastic-metric plasticity can be motivated by the example of a transversal isotropic material consisting of plastic fibers embedded into an elastic matrix material. Therefore it can be shown that the plastic map enters the stored energy functions only through the socalled plastic metric Gp := F pT F p . The dependence of the stored energy on C := F T F and Gp can alternatively be derived by an invariance principle, cf. Casey & Naghdi [30]. There is no need to introduce the multiplicative split of the deformation gradient to motivate the framework of plastic-metric plasticity. In fact the existence of a plastic metric that accounts for the plastic deformation of a material can be postulated a priori. This circumvents the restrictions that are apparent in the plastic-map framework. Considering the plastic metric to be an internal variable as is done in Miehe [82, 83] is consistent with the classical approach to finite plasticity proposed by Green & Naghdi [43], see also Casey & Naghdi [30] and Simó & Ortiz [126]. Reese & Svendsen [106] and Reese [105] model the material behaviour of a fiber reinforced composite using the plastic metric as an internal variable for the description of the inelastic deformation. A specific framework denoted here as additive plasticity is provided by the additive Hencky-type elastic strain measure E e := 1 2 [ln[C] − ln[Gp ]] . (1.2) Observe that (1.1) and (1.2) coincide for the special case of coaxial total and plastic deformations where C and Gp commute and the plastic map can be identified as F p := Gp1/2 . In this sense both formulations (1.1) and (1.2) are considered to be “close”. Formulations and numerical implementations of additive finite plasticity consistent with (1.2) have recently been outlined by Papadopoulos & Lu [99, 100] and Miehe, Apel & Lambrecht [87]. The additive decomposition of the total strains into elastic and plastic parts is a typical feature of the geometrically linear theory of plasticity. Hence, ansatz (1.2) provides a natural basis for a material-independent extension of constitutive structures from the geometrically linear to the non-linear theory at finite strains. A similar method for the extension of geometrically linear models to the finite strain range is considered in the incremental algorithmic setting of multiplicative plasticity (1.1) by Cuitiño & Ortiz [35], it is restricted, however, to certain isotropy properties. The convenience of logarithmic 4 Introduction strains has also been exploited in many works on computational multiplicative plasticity in the last decade. We refer to the treatments of isotropic finite plasticity by Eterovic & Bathe [39], Perić, Owen & Honnor [102], Simó [121, 122] and Miehe [83]. Chapter 6 discusses constitutive approaches to anisotropic plasticity. The main part deals with the plastic-map approach. A key point is the construction of one-step time integration algorithms for the evolution equations of the internal variables. New symmetric implicit and explicit algorithms are presented and compared with unsymmetric standard implicit and explicit return mapping schemes. Both above-mentioned symmetric formulations are related to variational formulations of finite plasticity that were recently developed by Miehe [85, 90] in the context of small strain elasto-plasticity and large strain crystal plasticity, respectively. The behaviour of the algorithms is compared in numerical examples. The framework of additive plasticity is presented in chapter 7. Two different approaches to the numerical treatment are presented, the classical general return scheme and a variational formulation derived in Miehe, Apel & Lambrecht [87]. On the side of computational analysis of shell-like structures we focus on a rotation-free shell formulation in chapter 8. It is in line with recently developed brick-type mixed finite element designs outlined in Miehe [84], Miehe & Schröder [92], Seifert [120] and Klinkel, Gruttmann & Wagner [62]. For a comprehensive overview of different shell element formulations and associated finite element technologies we refer to the textbook by Belytschko, Liu & Moran [9]. The brick-type finite shell element parameterizes — as originally proposed by Schoop [115, 116] — the deformation field in terms of displacements of material points at the top and bottom surfaces of the shell identical to the eight-node brick element of threedimensional continuum analysis. It is well known that this element yields very poor performance within the thin shell limit. This can be traced back to locking effects due to parasitic transverse shear stresses, which appear as a result of the low-order interpolation of the brick element, and the poor constant interpolation of the membrane strains and in particular the thickness normal strain that contradicts plate-type bending modes. These two locking effects can be circumvented by applying assumed strain and enhanced strain modifications on the pure displacement approach on the basis of mixed variational formulations. Both the enhanced as well as the assumed strain modifications are treated here in unified matrix notations with respect to the parameter chart of the shell. The enhanced strain formulation can be approached in a multiplicative and an additive format. The multiplicative approach defines the enhanced current metric with respect to the parameter space of the shell by ˜ := j̃ T g j̃ C̄ with j̃ := j C + j E (1.3) in terms of a modification j E of the compatible deformation gradient j C of the shell continuum. This multiplicative enhancing is conceptually in line with works on continuum elements by Simó & Armero [123] and Simó, Armero & Taylor [124] and has been applied to brick-type finite shell element formulations by Miehe [84] and Miehe & Schröder [92]. In this thesis we concentrate on an additive approach that defines the enhanced current metric in the parameter space by ˜ = C̄ ˜ + C̄ ˜ C̄ C E ˜ = j T gj with C̄ C C (1.4) Introduction 5 ˜ of the metric C̄ computed with the compatible deformain terms of a modification C̄ E C tion gradient j C . This additive enhancing is analogous to works on continuum elements by Simó & Rifai [127] and has been applied to brick-type finite shell element formulations by Seifert [120] and Klinkel, Gruttmann & Wagner [62]. As done in Miehe ˜ in (1.4) is assumed to cover two types of enhanced strain [84] for j E the modification C̄ E modes. Firstly, we enhance the normal strain in the thickness direction in order to incorporate a linear dependence in terms of the curvilinear parameter in thickness direction as conceptually suggested by Büchter & Ramm [28] and Büchter, Ramm & Roehl [29]. Secondly, we enhance the membrane-bending response of the shell element in a manner similar to formulations in plane problems outlined by Simó & Rifai [127] and Simó & Armero [123]. In order to avoid the shear locking phenomenon we apply the classical assumed strain interpolation of the transverse shear strains proposed by Dvorkin & Bathe [37]. A locking effect due to the poor interpolation of the director field is avoided by the introduction of the assumed strain interpolation of the thickness strains as proposed by Betsch & Stein [11] and Bischoff & Ramm [18]. The development results in an mixed eight-node brick-type shell element with an underlying trilinear displacement interpolation, five internal degrees for enhanced strain interpolations of the thickness and membrane strains, and eight collocation points for the assumed strain interpolation of the thickness and transverse shear strains. An interface to deformation-gradient-driven constitutive stress update algorithms of anisotropic finite plasticity is developed by using an assumption with respect to the enhanced assumed local rotation as suggested by Dvorkin, Pantuso & Repetto [38]. The strain-like interface to the three-dimensional constitutive box at a material point of the shell continuum is formulated in terms of curvilinear coordinate charts relative to a local parameter chart of the shell. In chapter 9 representative numerical examples are discussed. The four proposed stress algorithms for plastic-map plasticity are compared and evaluated for complex anisotropic phenomenological constitutive models of elasto-plasticity. This is done by considering driver tests as well as simulations of drawing processes. As already mentioned, the plastic map and the plastic-metric approaches are subject of intensive research. One purpose of this thesis is to compare and to evaluate both approaches. On the computational side this is done by means of complex boundary value problems. The multiplicatively enhanced shell element design published by Miehe [84] serves as a reference for the proposed additively enhanced shell formulation. The results obtained with both approaches are compared. 7 Fundamentals of Continuum Mechanics 2. Fundamentals of Continuum Mechanics The aim of this introductory chapter is to give a short and compact survey about the continuum mechanical background of this work and to introduce the notation that is used. We refer to the textbooks of Truesdell & Noll [140], Malvern [70] and Marsden & Hughes [72]. Recent publications are Chadwick [31] Başar & Weichert [4], Haupt [49] and Holzapfel [53] among many others. 2.1. Finite Kinematics The outline of this section is as follows. We discuss coordinate representations of tensors and their transformations and rotations in bases systems. In the first part, Cartesian bases are considered because they provide a simple lead-in to that topic. Then co-variant and contra-variant bases are taken into account. Here especially convected bases allow an illustrative geometric interpretation of the motion of material bodies. 2.1.1. Representation, Transformation and Rotation of Tensorial Objects 2.1.1.1. Cartesian Bases. In this paragraph, we restrict our attention to a right-handed orthonormal basis {ei }i=1,2,3 that is characterized by ei · ej = δij and ei × ej = ²ijk ek . (2.1) A second orthonormal frame is denoted by {ēi }i=1,2,3 . The frames are assumed to be related via an orthogonal transformation Q ∈ SO(3). Here SO(3) denotes the set of all second-order tensors Q with det[Q] = 1 and QT = Q−1 and is referred to as the proper orthogonal group. The rotation tensor Q maps the basis vectors onto each other ei = Qēi ⇔ ēi = QT ei . (2.2) For an illustration of this setting see Figure 1. Q is a two-point tensor with the typical E3 v, A PSfrag replacements P QT {ei } Q {ēi } Figure 1: Two Cartesian bases are linked via an orthogonal transformation tensor, e i = Qēi . Tensorial objects v and A at a point P of the Euclidean space E3 can be represented in any frame, however their characteristic properties are frame independent. structure consisting of the sum of dyadic products of corresponding basis vectors Q = ei ⊗ ēi ⇔ QT = ēi ⊗ ei , (2.3) 8 Fundamentals of Continuum Mechanics where two identical indices imply the summation due to Einstein’s convention. In the context of continuum mechanics, tensors describe geometrical objects which can be represented in any coordinate system. The coordinates with respect to a certain frame are obtained by a contraction of the tensor with the basis vectors of that frame. So the coordinates of Q with respect to {ei } are Qij = (Qej ) · ei = ((ek ⊗ ēk )ej ) · ei = δik ēk · ej = ēi · ej . (2.4) In literature one often finds transformations of basis vectors with rotation matrices. Such a representation is obtained from (2.4) by multiplication with ēi Qij ēi = (ēi · ej )ēi = (ēi ⊗ ēi )ej = ej and we obtain as analogous expressions to (2.2) the linear-combinations ei = Qji ēj ⇔ ēi = Qij ej . (2.5) Inserting (2.5) into the definition (2.3) leads to the coordinate representation of Q in the barred frame. The result is Q = Qji ēj ⊗ ēi , stating that the coordinates of Q are identical in both frames {ei } and {ēi }. For a given second-order tensor A, we have two one-point coordinate representations when considering the above introduced frames {ei } and {ēi }, see also figure 1. They are related, because the basis vectors of the frames are connected with the mapping (2.2) and we get A = Aij ei ⊗ ej = Qik Akl Qjl ēi ⊗ ēj = Āij ēi ⊗ ēj = Qki Ākl Qlj ei ⊗ ej (2.6) A comparison of the representations in both frames yields the transformation laws for the coordinates due to a change of basis, i.e. Āij = Qik Akl Qjl and Aij = Qki Ākl Qlj . (2.7) Remark 2.1: It is important to notice that the change of basis (2.7) is different from the rotation of A, QAQT = (ei ⊗ ēi )(Ākl ēk ⊗ ēl )(ēj ⊗ ej ) = Āij ei ⊗ ej 6= A . (2.8) The rotated tensor QAQT has the coordinate scheme of the barred setting in the basis {ei }. In general, the coordinates Aij and Āij are different and then QAQT is a new object. An exception is the rotation tensor itself, obviously it is QQQ T = Q. 2.1.1.2. Co-variant and Contra-variant Bases. For an extension of the discussion of tensor representations and transformations, we drop the assumption of orthonormal bases and consider an arbitrary co-variant coordinate frame {g i }i=1,2,3 spanned by three linear independent vectors of arbitrary length and orientation and g 1 · (g 2 × g 3 ) > 0. The metric coefficients are defined by the scalar products gij := g i · g j and g ij := g i · g j . (2.9) 9 Fundamentals of Continuum Mechanics Obviously we have gij 6= δij . The basis vectors are normalized and oriented with respect to a dual contra-variant basis {g i }i=1,2,3 so that g i · g j = δi j and g i × g j = ²ijk g k . (2.10) For a visualization see figure 2. In analogy to the linear combinations (2.5) the co-variant E3 v, A PSfrag replacements P g {g i } g −1 {g i } Figure 2: Dual bases are linked via metric tensors. basis vectors can be decomposed into the contra-variant system and vice versa which is denoted as index lowering and raising, g i = gij g j and g i = g ij g j . (2.11) In the context of dual bases the identity tensor has four different representations g i ⊗ g i = gij g i ⊗ g j = g ij g i ⊗ g j = g i ⊗ g i . (2.12) According to their mapping properties, they are formally distinguished with different symbols and names. The mixed variant representations i := g i ⊗ g i and i−1 := g i ⊗ g i . (2.13) are referred to as identity tensors. They map within the co-variant or contra-variant basis. The mappings in between the dual frames are referred to as co-variant and contra-variant metric tensors, g := gij g i ⊗ g j and g −1 := g ij g i ⊗ g j . (2.14) For a second-order tensor A the following co-variant, contra-variant and mixed-variant representations exist A = Aij g i ⊗ g j = Aij g i ⊗ g j = Ai j g i ⊗ g j = Ai j g i ⊗ g j . Here no difference between a multiplication with metric tensors and a coordinate transformation with (2.11) is observed. Both operations yield a change of basis gAg T = (gij g i ⊗ g j )(Aij g i ⊗ g j )(gkl g l ⊗ g k ) = gia Aab gkb g i ⊗ g k = Aik g i ⊗ g k . (2.15) 10 Fundamentals of Continuum Mechanics v, A E 3 P PSfrag replacements F {Gi } {g i } F −1 F −T {Gi } {g i } FT Figure 3: Two arbitrary dual bases are linked via F ∈ GL(3)+ . Tensorial objects v and A at a point P of the Euclidean space E3 can be represented in any frame, their characteristic properties, however, are frame independent. 2.1.1.3. Arbitrary Dual Bases. Consider two arbitrary dual frames, denoted by {G i }, {Gi } and {g i }, {g i } respectively, for i = 1, 2, 3. Assume that the co-variant bases are mapped onto each other with a map of the general linear group F ∈ GL(3)+ , i.e. g i = F Gi , Gi = F −1 g i , g i = F −T Gi , Gi = F g i , (2.16) see also figure 3. The transformation tensor F is a two-point tensor with the typical structure consisting of the sum of dyadic products of corresponding basis vectors F = g i ⊗ Gi , F −1 = Gi ⊗ g i , F −T = g i ⊗ Gi , F T = Gi ⊗ g i . (2.17) The coordinates of F are obtained by contractions with the corresponding basis vectors, e.g. F i j = (F Gj ) · Gi = Gi · g j . Insertion into (2.16) give the corresponding linear combinations g i = F a i Ga , Gi = F −1 a i g a , g i = F −1 i a Ga , Gi = F i a g a . (2.18) For a given second-order tensor A we discuss the two co-variant one-point forms in place of all possible representations, i.e. A = Aij g i ⊗ g j = F −1a i Aab F −1b j Gi ⊗ Gj = Āij Gi ⊗ Gj = F a i Āab F b j g i ⊗ g j . (2.19) When comparing the representations in both frames one obtains the transformation laws for a change of basis, i.e. Āij = F −1a i Aab F −1b j and Aij = F a i Āab F b j . (2.20) 11 Fundamentals of Continuum Mechanics Remark 2.2: Observe that the change of basis (2.20) differs from a generalized rotation with F of the tensor A, which is a new object F −T AF −1 = (g i ⊗ Gi )(Āab Ga ⊗ Gb )(Gj ⊗ g j ) = Āij g i ⊗ g j 6= A . (2.21) Operations of this type are denoted as push-forward and pull-back operations. They yield new tensors with identical coordinates but exchanged bases. 2.1.2. Motion of a Body In this thesis, we consider so called material bodies S, consisting of a continuous set of material points P which are in a one-to-one relation to the points of the three-dimensional Euclidean space E3 . The placement of the body in space changes with time, i.e. χ : S × R → E3 , (S, t) 7→ χ(S, t) . (2.22) Every point of E3 is described by means of a vector x ∈ E3 consisting of the coordinates of that point with respect to a global Cartesian basis {E i }i=1,2,3 . At frozen time t = t̄, the set of points only depends on S. This is emphasized by the notation χt (S) := χ(S, t = t̄). The image x = χt (S) is called actual configuration of the material body and is denoted by St . The motion is often described with regard to a fixed, so called reference configuration, associated with time t = t0 and denoted by B. It is not necessary but often convenient to choose the reference configuration identical to the position of the body at time t 0 so that χ0 := χ(t = t0 ) and B := St=t0 . For a visualization see figure 4. The relative motion is PSfrag replacements E3 X x2 x1 B St 1 St 2 χ0 {S} χt1 χt2 Figure 4: Reference configuration B and actual configurations St1 and St2 are images of a continuous set S of material points in the Euclidean space E3 at fixed times t1 and t2 . The motion of the body is described by the family of configurations parameterized in the time t. then described by a non-linear point map ϕ(X, t) : B → S , ϕ(X, t) = χt (χ−1 0 (X)) (2.23) where the points of B are defined by X := χ(S, t0 ). The map (2.23) is the basis for the description of the motion of the body. In what follows, we consider two configurations. The reference configuration B is denoted as Lagrangian or material configuration, the 12 Fundamentals of Continuum Mechanics actual configuration S is also referred to as Eulerian configuration. Both configurations are in a one-to-one relation via (2.23). We introduce a global Cartesian frame {E i } with coordinates θ i . The position vectors have the representations X = X i (θ1 , θ2 , θ3 )E i and x = xi (θ1 , θ2 , θ3 , t)E i . At any point X the images of the global coordinate lines are denoted as material lines which deform with the body θ 1 (α, t) = ϕ(Θ1 (α), t) Θ1 (α) = X(θ 1 + α, θ2 , θ3 ) Θ2 (α) = X(θ 1 , θ2 + α, θ3 ) θ 2 (α, t) = ϕ(Θ2 (α), t) (2.24) and 3 3 3 1 2 3 Θ (α) = X(θ , θ , θ + α) θ (α, t) = ϕ(Θ (α), t) . The tangent vectors to these lines constitute a natural co-variant basis {G i } and {g i } for the so-called tangent space TX B and Tx S, respectively. They are obtained by PSfrag replacements Gi := ∂α Θi = ∂θi X j E j , g i := ∂α θ i = ∇Θi ϕ ∂α Θi = ∇Θi ϕ Gi . (2.25) The dual bases {Gi } and {g i } span the contra-variant co-tangent spaces, denoted by TX? B and Tx? S, respectively. Tangent and co-tangent spaces are connected with the Lagrangian and Eulerian metric tensors ¾ ¾ g = gij g i ⊗ g j G = Gij Gi ⊗ Gj (2.26) and g −1 = g ij g i ⊗ g j , G−1 = Gij Gi ⊗ Gj respectively. Figure 5 visualizes the geometric setting. G, G−1 g, g −1 ϕ G3 G1 Θ g2 θ1 g1 g3 G2 θ3 Θ1 2 Θ3 x X θ2 S B {E i } Figure 5: Reference and actual configuration B and S of the body under consideration. The non-linear point-map ϕ maps material points X ∈ B to the current configuration x ∈ S. Tangent vectors to material lines θ i form the co-variant convective basis and constitute the metric tensors G−1 = Gij Gi ⊗ Gj and g −1 = g ij g i ⊗ g j . To the opinion of the author the introduction of the natural convected basis is essential for the geometric understanding of finite kinematics. In this context the deformation gradient is defined in accordance with (2.25) by F : TX B → Tx S , F = ∇X ϕ(X, t) . (2.27) It connects the co-variant basis vectors of the Lagrangian and Eulerian configurations and the results of section 2.1.1.3 can be applied to the context of finite kinematics. 13 Fundamentals of Continuum Mechanics The deformation gradient F maps tangent vectors to material lines in the Lagrangian PSfrag replacements configuration to tangent vectors to material lines in the Eulerian configuration. Due to property (2.10)2 , vectors in the co-tangent space are denoted as normals. They are mapped by F −T from the Lagrangian to the Eulerian configuration, see figure 6. On the i I F X x Tx S TX B g G F −T X TX? B I −1 x i −1 Tx? S Figure 6: Mappings between the co-variant and contra-variant tangent spaces of a point X and its Eulerian counterpart x = ϕ(X). I and I −1 are defined in analogy to (2.13). basis of these pictures, three fundamental geometric mappings are of high relevance. (i) Map of tangents. An infinitesimal line element of the tangent space dX ∈ T X B is mapped with the deformation gradient from the Lagrangian to the Eulerian configuration, F : TX B → Tx S , dx = F dX . (2.28) (ii) Map of area elements. Two infinitesimal line elements dX and dX̄ span an area element dA = dX × dX̄ which is in TX? B. Its Eulerian counterpart is obtained with the co-factor cof[F ] := det[F ]F −T and known as Nanson’s formula cof[F ] : TX? B → Tx? S , da = cof[F ]dA . (2.29) (iii) Map of volume elements. The volume of a parallelepiped spanned by dX, dX̄ and ¯ ]. A transformation with the Jacobi¯ is given by the box product dV = [dX, dX̄, dX̄ dX̄ determinant J := det[F ] yields the Eulerian counterpart J : R→R, dv = JdV . (2.30) So far we have dealt with the geometry of finite deformations in terms of convective curvilinear coordinates. In this formulation, the actual dual bases {g i } and {g i } are the images of their Lagrangian counterparts {Gi } and {Gi }. The coordinates of tensors do not change while performing a push-forward or pull-back operation, see (2.21). This is a consequence of the fact that the information about the deformation is exclusively stored in the basis vectors. 14 Fundamentals of Continuum Mechanics In several cases kinematics is formulated with respect to a single global Cartesian frame without introducing convected bases explicitly. In order to keep things clear, we distinguish between the tangent and co-tangent spaces of the material and actual configuration by formally introducing four Cartesian frames instead of only one which are denoted by {E A }, {E A } and {ea }, {ea }, respectively. Then the metric tensors take the forms ¾ ¾ g = δab ea ⊗ eb G = δAB E A ⊗ E B (2.31) and g −1 = δ ab ea ⊗ eb G−1 = δ AB E A ⊗ E B and they do no more contain any information about the actual deformation but only serve as maps between tangent and co-tangent spaces. Mappings between the Lagrangian and Eulerian configuration are performed on the basis of the deformation gradient. In analogy to (2.17) one then has to consider the two basic deformation maps F = F a A ea ⊗ E A , F −1 = F −1A a E A ⊗ ea . (2.32) 2.2. Notion of Stresses and Heat Flux In order define the notion of stresses and heat flux, consider a body S, from which we PSfragtoreplacements cut out an arbitrary part Sp . The effects of the outer part onto the cut-out part have t x x q S n h Sp Figure 7: Cut-off part Sp from the body S. The action of the part S\Sp is replaced by surface tractions t on ∂Sp and a heat flux normal through the surface of Sp . to be replaced by phenomenological quantities. In this work we consider mechanical and thermal effects. 2.2.1. Stresses The mechanical effects of the outer part S\Sp are replaced by surface tractions defined as the limit value of a force ∆f acting on an area element ∆a at points x ∈ ∂Sp of the surface, i.e. ∆f df = . ∆a→0 ∆a da t(x, t) := lim (2.33) According to Cauchy’s theorem, the traction vector in a point x ∈ ∂Sp is a linear function of the normal n of the area element da in x t(x, t, n) = σ(x, t)n (2.34) where σ denotes the Cauchy stress tensor. Within the context of finite deformations σ relates the actual force to the actual deformed area element. Therefore the Cauchy PSfrag replacements Fundamentals of Continuum Mechanics 15 F X x Tx S TX B S P F −T X TX? B τ, σ x Tx? S Figure 8: Definition of stress tensors. Cauchy stresses or true stresses σ, Kirchhoff stresses τ = Jσ, first Piola stresses or nominal stresses P = τ F −T , second Piola-Kirchhoff stresses S = F −1 τ F −T . stresses are also denoted as true stresses. σ is a co-variant tensor mapping normals of the co-tangent space Tx? into the tangent space Tx , see figure 8. The stress tensor τ = Jσ is called weighted Cauchy or Kirchhoff stress tensor. Relating the actual force to the undeformed area element dA yields the definition of the nominal stress tensor which is also denoted as first Piola-Kirchhoff stress tensor P . From N . Insertion into (2.34) gives the alternative Cauchy theorem (2.29) we get n = JF −T dA da T = PN with P = τ F −T (2.35) da and T = dA t. Finally, the second Piola-Kirchhoff stress tensor S is used. It is the Lagrangian counterpart of τ and is obtained by the pull-back operation S := F −1 τ F −T . (2.36) This tensor is a purely geometrical construct, with almost no physical interpretation but is convenient for the formulation of constitutive equations. 2.2.2. Heat Flux The thermal effects of the outer part S\Sp are represented by a scalar-field h on the surface ∂Sp . This field describes the heat-flux h through the surface in outer normal direction. According to Stoke’s heat-flux theorem, h is a linear function of the normal vector of the area element in a point x ∈ ∂Sp h(x, t, n) = q(x, t) · n . (2.37) Here q is the contra-variant Cauchy heat-flux vector. Its Lagrangian counterpart is obtained from the demand Z Z q · n da = Q · N dA . (2.38) ∂Sp ∂Bp With Nanson’s formula (2.29) the nominal heat flux vector is defined as Q := det[F ]F −1 q. 16 Fundamentals of Continuum Mechanics 2.3. Balance Principles Balance principles and conservation laws for physical quantities constitute the physical basis of continuum mechanics. They are at first formulated as integral equations for a cut-out part Sp of a body subjected to surface tractions and thermal loading representing the action of the cut-off part S\Sp , see figure 7. Application of the localization theorem yields the corresponding local or strong form of the balance law, valid for any x ∈ S. ˙ In the subsequent development we denote with d/dt[·] = [·] the material time derivative. This is the temporal change of the quantity [·] at an arbitrary but then fixed point of the reference configuration, i.e. X = constant. For Lagrangian fields F (X, t) the material time derivative equals the partial derivative Ḟ = ∂F/∂t whereas for Eulerian fields f (x(X, t), t) it consists of two parts, f˙(x, t) = ∂f /∂t + grad[f ] · ẋ, the local part and the convective part, respectively. Furthermore we introduce the following operators grad[·] := [·] ⊗ ∇x , Grad[·] := [·] ⊗ ∇X , div[·] := [·] · ∇x , Div[·] := [·] · ∇X (2.39) ∂ in terms of the differential operators ∇x := ∂x∂ i ei and ∇X := ∂X i ei with respect to a Cartesian basis {ei }. An important integral equation for Eulerian scalar-fields is d dt Z f (x, t) dv = Sp Z f˙ + f div ẋ dv . (2.40) Sp Note that in the Eulerian setting differentiation and integration do not commute due to the time dependence of the integral limits. 2.3.1. Balance of Mass For an arbitrary cut-out part of a body Bp and its deformed configuration Sp the density of mass is defined by ρ0 := dm/dV and ρ := dm/dv, respectively. Within this work we consider only processes, where the mass of any part of the body remains constant during the deformation process, i.e. Z Z ρ0 dV . (2.41) ρ dv = m= Sp Bp Corresponding to the two configurations, two local forms are obtained ) ρ̇ + ρ div[ẋ] = 0 ∀ x ∈ S ρ0 − ρJ = 0 ∀X∈B. (2.42) The first one follows directly from (2.40) when considering ṁ = 0. The second one is obtained with the transformation (2.30) of the volume elements. 2.3.2. Balance of Linear Momentum The linear momentum of a part of a body is defined by Z Z ρ0 ẋ dV . ρẋ dv = I := Sp Bp (2.43) 17 Fundamentals of Continuum Mechanics Any body remains in uniform motion as long as no resultant forces act on it. Otherwise the temporal change of momentum equals the resulting volume and surface forces Z Z Z Z İ = t̄ da = ργ̄ dv + ρ0 γ̄ dV + T̄ dA (2.44) ∂Sp Sp ∂Bp Bp which are prescribed by an acceleration field γ̄(x, t) and the surface tractions t̄(x, t) and T̄ (X, t) in the actual and reference setting, respectively. Using the Gauss theorem, the surface integral can be recast into a volume integral and one obtains the local equilibrium conditions related to the actual and reference unit-volume ) div[σ] + ρ(γ̄ − ẍ) = 0 ∀x∈S . (2.45) Div[P ] + ρ0 (γ̄ − ẍ) = 0 ∀ X ∈ B 2.3.3. Balance of Angular Momentum The angular momentum of a body S is defined with respect to the origin of the coordinate system “o” by Z Z x × ρ0 ẋ dV . (2.46) x × ρẋ dv = D o := Bp Sp The temporal change of the angular momentum equals the sum of the applied moments in consequence of body forces and surface tractions Z Z Z Z x × T̄ dA . (2.47) x × ρ0 γ̄ dV + x × t̄ da = x × ργ̄ dv + Ḋ o = Sp ∂Spt ∂Bpt Bp The corresponding local forms follow after a somewhat extensive derivation. They state the symmetries of the stress tensors σ = σT and S = S T . (2.48) This essential property does not transmit to the first Piola-Kirchhoff stress tensor. 2.3.4. Balance of Total Energy The total energy of a part of a body is defined by Z Z ρ0 e dV ρe dv = E := Sp (2.49) Bp where e denotes the specific energy density per unit mass. In the thermo-mechanical context, a temporal change of E may be caused by the power of mechanical tractions and body forces P and the thermal power Q, i.e. Z Z Z Z P := ρẋ · γ̄ dv = ρ0 ẋ · γ̄ dV ẋ · t̄ da + ẋ · T̄ dA + SpZ BpZ Z∂Bp Z∂Sp (2.50) ρ0 r dV −Q · N dA + ρr dv = −q · n da + Q := ∂Sp Sp ∂Bp Bp 18 Fundamentals of Continuum Mechanics respectively. r is the heat supply per unit mass and unit time. With these definitions, the balance of total energy reads Ė = P + Q . (2.51) The corresponding local equations related to the unit-volume of the actual and reference configuration run as follows ) ρė = div[ẋ · σ − q] + ρẋ · γ̄ + ρr ∀x∈S (2.52) ρ0 ė = Div[ẋ · P − Q] + ρ0 x · γ̄ + ρ0 r ∀ X ∈ B . The total energy can be additively split up into the kinetic energy and an internal energy, which is discussed in the next two subsections. 2.3.4.1. Balance of Kinetic Energy. The kinetic energy of a part of the body is defined by the integrals Z Z 1 1 K := ρẋ · ẋ dv = ρ ẋ · ẋ dV . (2.53) 2 2 0 Sp Bp Balance of kinetic energy is equivalent to the equilibrium condition. Multiplying the local Eulerian form (2.45)1 or the local Lagrangian form (2.45)2 with the velocity ẋ and integrating over the volume, one obtains K̇ = P − S . (2.54) Here P is defined in (2.50)1 and S denotes the so-called stress-power Z Z grad[ẋ] : (gσ) dv = Ḟ : (gP ) dV S := (2.55) Bp Sp with respect to unit-volume of the actual and reference configuration, respectively. 2.3.4.2. Balance of Internal Energy (First Law of Thermodynamics). The total energy can be additively decomposed into the kinetic part and a remaining part U := E − K, denoted as internal energy. The latter is related to the specific internal energy density per unit mass u according to Z Z ρ0 u dV . (2.56) ρu dv = U := Sp Bp The balance of the internal energy is then U̇ = Ė − K̇ = Q + S (2.57) and leads to the two alternative local forms ρu̇ = ρr − div[q] + grad[ẋ] : (gσ) ∀x ∈ S ρ0 u̇ = ρ0 r − Div[Q] + Ḟ : (gP ) ∀X ∈ B . ) (2.58) 19 Fundamentals of Continuum Mechanics 2.3.5. Balance of Entropy and Second Law of Thermodynamics Entropy is a state variable for a thermo-mechanical system which measures microscopic randomness and disorder and determines the direction of the thermodynamical process.. Its physical definition is part of Statistical Physics. The entropy H possessed by a part of a body is defined in terms of the specific entropy per unit mass η as Z Z ρ0 η dV . (2.59) ρη dv = H := Sp Bp A temporal change of the entropy of a body can be caused by (i ) a production of entropy ργ, (ii ) a source of entropy inside the body ρr/θ due to evolution of temperature and (iii ) a supply of entropy through the surface due to heat flux −q · n/θ. Here θ ≥ 0 denotes the absolute temperature. The balance of entropy takes the form Z Z Z Z r 1 1 r ρ + ργ dv − Ḣ = ρ0 + ρ0 γ dV − q · n da = Q · N dA (2.60) θ Sp θ ∂Sp θ Bp ∂Bp θ The second law of thermo-dynamics states that the production of entropy is always positive, i.e. γ ≥ 0. Solving the local forms corresponding to (2.60) for γ yields the local forms of the so-called Clausius-Duhem inequality r 1 1 ργ = ρη̇ − ρ + div[q] − 2 q · grad[θ] ≥0 ∀x ∈ S θ θ θ (2.61) 1 r 1 ∀X ∈ B . ρ0 γ = ρ0 η̇ − ρ0 + div[Q] − 2 Q · Grad[θ] ≥ 0 θ θ θ Insertion of (2.58) and introducing the Helmholtz free energy per unit mass through a Legendre transformation Ψ = u − θη yields the alternative forms 1 ∀x ∈ S θργ = (gσ) : grad[ẋ] − ρΨ̇ − ρθ̇η − q · grad[θ] ≥ 0 θ (2.62) 1 θρ0 γ = (gP ) : Ḟ − ρ0 Ψ̇ − ρ0 θ̇η − Q · Grad[θ] ≥0 ∀X ∈ B . θ A purely mechanical theory is obtained assuming an isothermal process which is characterized by constant temperature, i.e. θ = constant. In this case, the Clausius-Duhem inequality reduces to ) ρD = (gσ) : grad[ẋ] − ρΨ̇ ≥ 0 ∀x ∈ S (2.63) ρ0 D = (gP ) : Ḟ − ρ0 Ψ̇ ≥0 ∀X ∈ B where D := θγ is the mechanical dissipation per unit-mass of the process. 2.4. Constitutive Equations Within this thesis, the class of so-called standard dissipative materials is considered. Their constitutive behaviour is governed by two scalar-valued tensor functions. The locally stored energy is described by the free energy function, and the evolution of the internal variables is governed for instance by the dissipation function, the level-set function or the classical yield-criterion function. The form of these functions is not arbitrary but 20 Fundamentals of Continuum Mechanics restricted by fundamental principles. For a detailed discussion see e.g. Truesdell & Noll [140], Malvern [70] or Haupt [49]. The most important principle is that of frame invariance or material objectivity which is discussed in the first subsection. The inherent symmetries of a material poses restrictions on the constitutive functions. The material symmetry group introduced in the second part of this section provides a basis for the classification of materials and the construction of appropriate constitutive functions. 2.4.1. Principle of Material Objectivity The locally stored energy in a material that is completely reversible is characterized by a free energy function. Here we consider elastic material behaviour and a functional dependence on the deformation gradient F . The principle of material objectivity demands that the free energy function ψ = ρ0 Ψ is independent of the choice of reference frame. For two arbitrary Cartesian frames linked with orthogonal rotation tensor Q, it reads ψ(F ) = ψ(QF ) . (2.64) Thus the free energy function has to be invariant with respect to superimposed rigid body motions. A common way to satisfy this restriction in a Lagrangian setting is to assume a functional dependence on the right Cauchy-Green tensor C := F T gF . It is the pullback of the current metric and is objective because (QF )T g(QF ) = F T (QT gQ)F = C. Functions satisfying the principle of material objectivity a priori are denoted as reduced forms. They are often formulated in terms of a strain tensor of the Seth-Hill family ½ 1 m/2 [C − G] for m 6= 0 (m) m E := (2.65) 1 ln[C] for m = 0 2 in the form ψ = ψ(E (m) ). 2.4.2. Material Symmetry In order to motivate the classification of material according to their material symmetry, we consider a free energy function depending on the deformation gradient. In general, the amount of stored energy will not only depend on the applied deformation but also on the orientation of the material. As an example consider two alignments of a fibrous a? PSfrag replacements a ψ ψ? Figure 9: A fibrous material with orientations a and a? is subjected to the same global deformation state. The corresponding stored energy are ψ and ψ ? . material, a and a? and assume that these orientations are linked with an orthogonal transformation, i.e. a? = QT a. In both cases the material is subjected to the same global deformation state, see figure 9 for a visualization. The stored free energies are ψ = ψ(F ) and ψ ? = ψ(F QT ) . (2.66) Fundamentals of Continuum Mechanics 21 In general, both function values will differ, ψ 6= ψ ? . The set of all transformations Q ∈ O(3) yielding the same value for ψ and ψ ? is denoted as material symmetry group of the tensor function ψ, i.e. {Q ∈ O(3) | ψ(F ) = ψ(F QT )}. For a priori objective functions ψ = ψ(C) the material symmetry group is determined by Gψ := {Q ∈ O(3) | ψ(C) = ψ(QCQT )} . (2.67) Section 3 reviews the symmetry groups for anisotropic solids in detail and section 4 discusses the construction of tensor functions for given material symmetry groups. 23 Material Symmetries — Classification of Solids 3. Material Symmetries — Classification of Solids The goal of this section is to present the standard concept of physics to classify solids due to the symmetry of their microscopic structure. There is a wide variety of textbooks in physics treating this topic, e.g. Jagodzinski [57], Voigt [142], Borchardt-Ott [24] or Kleber, Bautsch & Bohm [61]. For the approach presented here we refer especially to the books of Whittaker [152] and Kennon [60]. A comprehensive work is the “International Tables for Crystallography” edited by Hahn [45]. For an introduction to the group theoretical treatment of symmetry we refer to Hamermesh [47] among many others. Condensed matter is distinguished between solids having amorphous and those having crystalline microstructure. We focus here on the latter ones. Any ideal crystal is a threedimensional pattern of atoms. The basic concept to describe the atomic structure is to distinguish between a motif and the scheme whereby the motif is periodically repeated in space to generate the pattern. The motif of a crystal depends upon the chemical identity of the material of the crystal and can be a single atom, a single molecule or a group of molecules. The periodic spatial arrangement of the motif is mathematically described by a space lattice. Any space lattice can be constructed by composing infinitely many parallelepipeds denoted as elementary bricks face to face without cleavage. In a first step, we concentrate on the external geometry of the elementary bricks and ignore the motif. We examine restrictions on the shape so that a brick is infinitely repeatable, thus forming a space lattice. We end up with 14 possible bricks, all having different geometry and therefore representing 14 different lattices denoted as Bravais lattices. Each of these 14 space grids is completely determined by specifying a so-called unit-cell of the Bravais lattice. This could be the elementary brick itself, but for some of the grids, its structure becomes clearer choosing not the brick itself as a construction unit but a larger part of the grid. According to the inherent symmetry elements of the unit-cells, the Bravais lattices are classified into seven crystal systems, denoted as triclinic, monoclinic, orthorhombic, tetragonal, cubic, trigonal and hexagonal. Taking into account the motif, a further classification of the seven crystal systems into 32 crystal classes is possible. 3.1. Construction of a Space Lattice The layout of the bricks building up a crystal follows the principles of a three-dimensional mathematical point grid. The construction of such a grid is done in three steps. (i) Grid line. Starting from an arbitrary point P0 located at x0 in space, we obtain new points Pi by translating P0 by a so-called grid vector a Pi : xi = x0 + ia , i∈Z. (3.1) The distance between two consecutive points is the grid constant a = kak. (ii) Grid plane. Taking a second grid vector b not co-linear to a and translating the whole grid line from (i), we obtain the points Pij : xij = x0 + ia + jb , i, j ∈ Z (3.2) 24 Material Symmetries — Classification of Solids building a grid plane. The parallelogram spanned from a and b is called unit-mesh. The knowledge of the shape and dimensions of the unit-mesh allow the construction of the whole grid plane. (iii) The space lattice is obtained by translating the grid plane in a direction outside the plane, described by a grid vector c. So we have Pijk : xijk = x0 + ia + jb + kc , i, j, k ∈ Z . (3.3) Grids for the two and three dimensional case are depicted in figure 10. One obvious choice PSfrag replacements c Sfrag replacements b a a b Figure 10: Construction of a mathematical 2d and 3d grid. The grids are spanned by the grid vectors a, b and c. For an existing grid the choice of the unit-cell is not unique. In the 2d grid several possible unit meshes are highlighted. Three primitive cells containing one grid point and at the bottom right a centered cell containing two grid points. For the space grid only one primitive unit-cell is signified. for a unit-cell is the parallelepiped spanned by a, b and c. The whole grid is uniquely determined by the unit-cell, which is described either by the grid vectors a, b and c or the three grid constants a, b, c and their corresponding angles α = 6 (bc), β = 6 (ac) and γ = 6 (ab). Angles and lengths of the unit-cell are the metric of the grid. Attached to each grid point of a 3d lattice reside eight unit-cells. The unit-cell indicated in figure 10 contains one (eight times an eighth) grid point and is therefore called primitive. Non-primitive unit-cells are characterized by having more than one grid point inside. As depicted in the 2d grid, an infinite number of primitive and non-primitive unit-cells can be chosen, all defining the same grid. Though it is always possible to describe a grid with primitive unit-cells, the symmetry of some grids become clearer when one chooses non-primitive cells. All in all, there exist only 14 different 3d space lattices. They are named after the physicist Bravais. 3.2. Symmetry Transformations A symmetry transformation of a space lattice is an operation which maps the lattice onto itself. We only consider such symmetry transformations where at least one point remains fixed. One can derive the 14 possible space grids either by introducing mirror planes as done in Cowin & Mehrabadi [33] or by introducing rotation axes and rotation-inversion axes as done in the following. The differences are only formal, because rotation-inversions and mirror planes cause each other. 3.2.1. Rotations A symmetry operation, consisting in a rotation around an axis with angle α = 2π/n, n ∈ N is called a rotational symmetry and the corresponding n-fold rotation axis is denoted by 25 Material Symmetries — Classification of Solids A(n) or simply n. Due to the postulated periodicity for the lattice, 5-fold and n-fold axis with n ≥ 7 are not possible, see figure 11 for a 5-fold symmetry. In figure 12 the five possible rotations are illustrated by stereographic projections. a1 PSfrag replacements a2 Figure 11: Parallel grid vectors have different lengths ka1 k 6= ka2 k which contradicts the translation periodicity (3.3). Consequently 5-fold axes are not possible for a space lattice. PSfrag replacements A(1) A(2) A(3) A(4) A(6) Figure 12: Stereographic projections of points on a sphere visualizing 1-, 2-, 3-, 4- and 6-fold symmetry. The outlined symbols at the centers denote the different rotation axis. 3.2.2. Rotation-Inversions Let M ∈ A(n) denote a point which is left unchanged when applying a symmetry operation. A rotation-inversion or rotoinversion is obtained by the composition of a rotation and a central inversion on M . Corresponding to the five possible rotation axes there are five rotation-inversion axes which are denoted by J (n) or n̄ for short. They are visualized in figure 13. Observe that 2̄ is equivalent to a plane of mirror symmetry perpendicular to the two-fold inversion axis J (2) and containing M . That plane is denoted by m . PSfrag replacements J (1) J (2) J (3) J (4) J (6) Figure 13: Stereographic projections of points on a sphere visualizing 1̄-, 2̄-, 3̄-, 4̄- and 6̄fold rotoinversions. The filled symbols at the centers denote the different inversion-rotation axes. Points on the lower half of the sphere are marked by crosses and points on the upper half of the sphere by shaded circles. The center of inversion M is located at the centers of the spheres 3.2.3. Tensor Representations Rotations and rotoinversions can be described with orthogonal tensors. An arbitrary rotation with angle α around an axis a with kak = 1 is performed by the rotation tensor Qαa := cos α 1 + (1 − cos α) a ⊗ a + sin α ²a . (3.4) 26 Material Symmetries — Classification of Solids The latter formula is known as Euler-Rodrigues formula. Rotation-inversions can be described by the composition of the central inversion with a rotation tensor which is abbreviated by −Qαa := (−1)Qαa . 3.2.4. Symmetry Groups The set of all symmetry transformations of a certain lattice has group structure and is therefore denoted as symmetry group G. It is a subgroup of the orthogonal group O(3) with the following properties (i) (ii) (iii) (iv) if Q1 ∈ G and Q2 ∈ G, then also Q1 Q2 ∈ G multiplication is associative (Q1 Q2 )Q3 = Q1 (Q2 Q3 ) the group G contains identity 1 defined for Q ∈ G by 1Q = Q each element Q ∈ G has an inverse Q−1 ∈ G defined by QQ−1 = 1 . (3.5) The number of elements in G is termed order of the group. Thereupon one speaks of a finite group, if its order is finite, otherwise of an infinite group. Both types play an important role in continuum mechanics. 3.3. The 14 Bravais Lattices and 7 Crystal Systems Different lattices can be distinguished due to their inherent symmetry. We will see that there exist seven different crystal systems, each having its own symmetry group. Within these systems, several grid metrics are possible. We differentiate 14 Bravais lattices belonging to the seven crystal systems according to the shape of their unit-cells. Starting with the full anisotropic case where all grid constants and all angles are different, we claim symmetry operations transforming the grid onto itself. This yields restrictions on the metric of the grid. 3.3.1. Triclinic Symmetry The most general form of a unit-cell is that of triclinic symmetry. It is depicted in figure 14. As it is a parallelepiped, the only possible symmetry operations are identity and PSfrag replacements β a c α γ b Figure 14: Primitive triclinic unit-cell. The grid metric is not restricted. Identity and inversion are the only symmetry operations transforming the unit-cell onto itself. inversion. The cell is spanned by the three grid vectors a, b and c, satisfying α 6= γ 6= β and a 6= b 6= c . (3.6) 27 Material Symmetries — Classification of Solids 3.3.2. Monoclinic Symmetry A monoclinic system is characterized by one inherent two-fold axis A(2) which induces a plane of mirror symmetry denoted by m, see Section 3.2.2. There are two possibilities for the location of that axis or mirror plane, respectively. In figure 15a the axis is introduced parallel to a. The restrictions arising constitute the standard monoclinic shape α = γ = 90◦ 6= β and a 6= b 6= c . (3.7) The name “monoclinic” indicates that only one angle is inclined. If A(2) is introduced parallel to b − a as depicted in figure 15b, the metric of the primitive unit-cell is restricted to α = β 6= 90◦ , a = b 6= c. The symmetries of the space grid become much clearer when choosing the non-primitive C-centered unit-cell shown in figure 15c instead, which again has the standard monoclinic shape (3.7). Because the choice of the unit-cell is not unique, PSfrag replacements c PSfrag replacements c b PSfrag replacements β b c a a b0 PSfrag replacements b c a0 a b0 a0 a. b. c. d. Figure 15: Bravais unit-cells for monoclinic symmetry. Main characteristic is inherent twofold axis, yielding two different primitive unit-cells (a) and (b). Instead of (b) the C-centered cell (c) is used because of its standard monoclinic metric. I-centered cell (d) is alternative to C-centered Bravais cell (c). one could alternatively choose the I-centered cell of figure 15d. 3.3.3. Orthorhombic Symmetry An orthorhombic system is characterized by three perpendicular two-fold axes which come ahead with three mutually perpendicular planes of mirror symmetry. Introducing the additional two-fold axes into figure 15a, we obtain the primitive orthorhombic cell of figure 16a. Its metric constitutes standard orthotropic shape α = β = γ = 90◦ and a 6= b 6= c . (3.8) From the monoclinic unit-cell 15b we get the unit-cell depicted in figure 16b by introducing the axes parallel to the bisecting of a and b and parallel to c. Its metric is a = b 6= c, α = β = 90◦ 6= γ. Instead of this unit-cell one uses its corresponding C-centered Bravais cell of orthotropic shape (3.8) as depicted in figure 16c. Another way is to put the one axis along the body diagonal and one parallel to c as visualized in figure 16d. Here the cell itself with metric a = b, c = 2a cos α, α = β 6= 90◦ 6= γ is not symmetric any more, but the lattice is. Instead of the primitive unit-cell 16d one chooses the I-centered Bravais cell having orthotropic shape (3.8), see figure 16e. The third possibility is to introduce the axis into the primitive monoclinic cell 15b as proposed in figure 16f. This again yields symmetry in the lattice but not the cell itself as shown in 16g. It has the metric a = b, c = −2a cos α, α = β 6= 90◦ 6= γ. Again there exists a non-primitive Bravais cell of shape (3.8). The grid points are located in the corners and in the midpoints of the faces. The cell is therefore called face-centered or for short F-centered. It contains four grid points. 28 Material Symmetries — Classification of Solids c PSfrag replacements c PSfrag replacements PSfrag replacements b a. β a b. PSfrag replacements a a b c γ b γ c. PSfrag replacements PSfrag replacements a b c a b c d. PSfrag replacements a b c e. f. a b g.c Figure 16: Bravais cells for orthotropic symmetry. (a) Primitive unit-cell with standard orthotropic metric. Unit-cells (b), (d) and (f) are not used to describe the structure of space grids. The non-primitive (c) C-, (e) I- and (g) F-centered cells are used instead. 3.3.4. Tetragonal Symmetry Up to this point, the introduced symmetry operations only affected the angles of the unit-cells, not the ratio of the edges. This changes, when converting one two-fold axis of an orthorhombic cell into a four-fold one which is the main characteristic of a tetragonal system. We obtain a primitive tetragonal unit-cell having two edges of equal lengths, see figure 17a, with restrictions α = β = γ = 90◦ and a = b 6= c (3.9) for its shape. The cell thus obtained is also symmetric with respect to reflections on the two diagonal planes containing the four-fold axis. Furthermore it has two diagonal two-fold rotation axes as depicted in figure 17a. Converting the two-fold axis of an I- a. b. Figure 17: Bravais cells for tetragonal symmetry. In addition to the primitive unit-cell (a) an I-centered Bravais cell (b) exists. The restrictions for the corresponding unit-cell not shown here are given in table 1. centered orthorhombic cell into a four-fold one, the tetragonal I-centered cell of figure 17b is obtained. Doing the same thing with a C-centered orthorhombic cell, the resulting Ccentered lattice can be described by means of a smaller primitive cell of tetragonal shape and therefore yields no new space lattice. 29 Material Symmetries — Classification of Solids 3.3.5. Cubic Symmetry To derive the Bravais cells for cubic symmetry, we introduce four three-fold axes along the body diagonals into an orthorhombic system. As a consequence, all edges have the same length α = β = γ = 90◦ and a = b = c (3.10) and the faces are squares. Accompanying, all two-fold axes of the orthorhombic system convert to four-fold ones, we have six additional diagonal mirror planes and six twofold-diagonal-rotation axes in the system. The primitive unit-cell together with these symmetries is shown in figure 18. In the same way, the I- and F-centered Bravais cells PSfrag replacements a b c β Figure 18: The unit-cell for cubic symmetry is obtained by introducing a threefold diagonal axis to the primitive orthorhombic unit cell. In the same way the I- and F-centered Bravais lattices are obtained by their orthotropic counterparts. follow from their orthotropic counterparts by adding a threefold axis along the diagonal. As all the faces have to be equal, a C-centered cell does not exist in cubic systems. It is impossible to find unit-cells having higher symmetry than that of a cubic system. 3.3.6. Trigonal and Hexagonal Symmetry Starting again with a triclinic system, we introduce one three-fold axis along one body diagonal. This leads to the following restrictions α = β = γ 6= 90◦ and a = b = c (3.11) for the primitive unit-cell as depicted in figure 19a. Automatically we obtain three twoα α a α PSfrag replacements a. a PSfrag replacements a γ b. Figure 19: Unit-cells for trigonal (a) and hexagonal (b) symmetry. The trigonal symmetry can be seen in the unit-cell itself where as the hexagonal symmetry can only be found in stacks of unit-cells as depicted in (b). fold axes and three mirror planes as indicated. The shape is denoted as rhombohedral. 30 Material Symmetries — Classification of Solids Six-fold symmetry cannot be inserted into a parallelepiped, but the lattice itself can have that symmetry. In figure 19b a unit-cell for hexagonal symmetry is visualized. The unit-cell is obtained through figure 15b, setting α = β = 90◦ and γ = 120◦ . 3.3.7. Summary The results obtained in this section are summarized in the tables 1 and 2. Lattices can be distinguished due to their inherent symmetry properties. This yields 14 different grids having 14 different primitive unit-cells. Conventionally, the lattices are represented by the 14 Bravais cells. Seven of them are primitive, the remaining seven are larger parts of the grid containing more than one grid point and are denoted as centered cells. Each shape of these centered cells is identical to one of the seven primitive cells. So we can describe all grids with seven crystallographic coordinate systems {a, b, c}, referred to as crystal systems. In table 1 the metric of the 14 primitive unit-cells are listed together with the crystal system and the Bravais cell to which they belong. Table 2 shows the 14 Bravais cells and their affiliation to the seven crystal systems. Besides the metrics of the crystal systems are given. Table 1: Unit-Cells for the 14 Bravais Lattices. Crystal System triclinic monoclinic orthorhombic tetragonal cubic trigonal hexagonal Restriction on Lattice Vectors α 6= β 6= γ a 6= b 6= c α = γ = 90◦ 6= β a 6= b 6= c α = β 6= 90◦ 6= γ a = b 6= c α = β = γ = 90◦ a 6= b 6= c α = β = 90◦ 6= γ a = b 6= c α = β 6= 90◦ 6= γ a = b, c = 2a cos α α = β 6= 90◦ 6= γ cos2 γ/2 a = b, c = −a cos α α = β = γ = 90◦ a = b 6= c √ α = β = sin−1 ( 2 sin γ/2) a = b, c = −2a cos α α = β = γ = 90◦ a=b=c √ α = β = cos−1√ (−1/ 3) γ = 2 sin−1 (1/√ 3) a = b, c = 2a/ 3 α = β = 120◦ , γ = 90◦ a=b=c α = β = γ 6= 90◦ a=b=c α = β = 90◦ , γ = 120◦ a = b 6= c Symmetry of Bravais cell Symmetry of crystal system Centering 1̄ identity P 2 m one 2–axis P 2 m one 2–axis C 2 2 2 mmm three 2–axes P 2 2 2 mmm three 2–axes C 2 2 2 mmm three 2–axes I 2 2 2 mmm three 2–axes F 4 2 2 mmm one 4–axis P 4 2 2 mmm one 4–axis I 2 4 m 3̄ m four 3–axes P 2 4 m 3̄ m four 3–axes I 4 2 m 3̄ m four 3–axes F 2 3̄ m one 3–axis P 6 2 2 mmm one 6–axis P 31 Material Symmetries — Classification of Solids Table 2: Bravais-Cells of the 14 Bravais-Lattices Primitive C-centered I-centered Symmetry F-centered a triclinic: PSfrag replacements a, b, c arbitrary, β α c γ α, β, γ arbitrary b a a monoclinic: replacements PSfrag replacements a, b,PSfrag c arbitrary, π c β c β α = γ = 2 6= β b b orthorhombic: a a a a a, b, c arbitrary, π c c replacements c replacementsc replacements PSfrag replacements PSfrag PSfrag α =PSfrag β=γ= 2 b b tetragonal: a a = b 6= c, π c α =PSfrag β=γ= replacements 2 b b a PSfrag replacementsc a a α trigonal: a a = b = c, π a α =PSfrag β = γ 6= 2 replacements α α a hexagonal: γ a =PSfrag b 6= c, replacementsc a α = β = π2 , γ = 3π 2 cubic: a = b = c, α=β=γ= a a π 2 a a PSfrag replacements a a a a PSfrag replacements PSfrag replacements a a 32 Material Symmetries — Classification of Solids 3.4. The 32 Crystal Classes 3.4.1. The Motif — Inner Symmetries So far, the restrictions on the unit-cell to describe a space lattice were discussed. To describe ideal crystals, it remains to specify the motif which is located at the grid points, inside the cells. Obviously, the motif can reduce the so called outer symmetry of the grid but not extend it, see Figure 20 for a two-dimensional visualization. The combination PSfrag replacements a. b. c. Figure 20: The symmetry of the motif located at the grid points may reduce the overall symmetry. Two-dimensional quadratic unit-cell with (a) quadratic motif has 4 mirror planes and one four-fold axis. (b) Rectangular motif reduces symmetries to two mirror planes and one two-fold axis. (c) Triangle motif allows only one mirror symmetry operation. of the so called inner symmetry of the motif with the symmetry of the 14 Bravais cells leads to 32 possible combinations, referred to as crystal classes. They are summarized in table 3. Introducing a motif is equivalent to a subdivision of the symmetry groups of the crystal system into proper subgroups yielding the 32 crystal classes. The generators of the symmetry groups of the crystal classes are given in table 3. 3.4.2. Notation of Symmetry 3.4.2.1. Herrmann-Mauguin Symbols. According to Hermann-Mauguin n = 1, 2, 3, 4, 6 denote n-fold rotations and n̄ = 1̄, 2̄, 3̄, 4̄, 6̄ n-fold inversion-rotations. The transformation 1̄ is equivalent to the inversion itself and is also denoted as i. The transformation 2̄ is equivalent to a reflection with respect to a plane perpendicular to J (2) including the symmetry center M and denoted by m. Each crystal class has a symbol indicating the present symmetries. Symbols related to the same axis are separated with a slash and symbols related to different axes are listed one after another. A two-fold axis with a perpendicular mirror plane is indicated by 2/m or m2 . For a compact notation, the symmetries are coupled with the directions as specified in table 5. 3.4.2.2. Schoenflies Symbols. Another notation for the crystals and quasi-crystal classes goes back to Schoenflies. This notion is commonly used in chemistry. A summary is given in table 6. 3.5. Icosahedral, Cylindrical and Spherical Symmetry Classes If we consider quasi crystals or engineering materials like composites or biological materials like soft tissues, further symmetry operations than those so far considered will have to be taken into account. Here discrete rotations around an axis A(n) with α = 2π/n, n ∈ N are possible as well as their corresponding rotoinversions. For example quasi crystals with a 33 Material Symmetries — Classification of Solids Table 3: The 32 Crystal Classes Crystal Classes Symmetry triclinic monoclinic orthorhombic tetragonal trigonal hexagonal cubic 1) No. Aniso. Notation Order Generators Type Hermann-Mauguin Schoenflies 1 2 3 4 5 6 7 8 1 1 2 2 2 3 3 3 1 1̄ 2 m 2/m 222 mm2 mmm C1 Ci C2 C1h C2h D2 C2v D2h 1 2 2 2 4 4 4 8 1 −1 Qπc −Qπc Qπc , −1 Qπc , Qπa Qπc , −Qπa Qπc , Qπa , −1 9 4 4 4 Qc2 10 4 4̄ C4 2) 4 11 4 4/m S4 8 12 5 422 C4h −Qc2 8 Qc2 , Qπa 13 5 4mm D4 8 14 5 4̄2m C4v 8 15 5 4/mmm1) D2d Qc2 , −Qπa D4h 16 16 6 3 3 17 6 3̄ C3 18 7 32 19 7 3m 20 7 21 π π π Qc2 , −1 π π π −Qc2 , −Qπa π Qc2 , Qπa , −1 2π Qc3 2π S6 2) 6 D3 6 Qc3 , Qπa 6 3̄m1) C3v D3d 12 Qc3 , −Qπa Qc3 , Qπa , −1 8 6 6 Qc3 22 8 6̄1) C6 6 23 8 6/m 12 24 9 622 C6h −Qc3 25 9 6mm 26 9 6̄2m1) 27 9 28 C3h Qc3 , −1 2π 2π 2π π π π Qc3 , −1 π D6 12 Qc3 , Qπa C6v 12 12 6/mmm D3h Qc3 , −Qπa D6h 24 10 23 12 Qk3 , Qπa , Qπb 29 10 m3̄1) T Th 24 Qk3 , Qπa , Qπb , −1 30 11 432 24 Qk3 , Qa2 , Qπb 31 11 4̄32 O 24 32 11 m3̄m1) Oh 48 Qk3 , −Qa2 , Qπb alternative Hermann-Mauguin symbols: 2 4 2 2 3̄m = 3̄ m 4/mmm = m mm 2 6 2 2 m3̄ = m 3̄ 6/mmm = m mm 3 3 6̄2m = m 2m 6̄ = m 4 2 m3̄m = m 3̄ m 2 Td π π −Qc3 , −Qπa π Qc3 , Qπa , −1 2π 2π π 2π π 2π 2π ) alternative Schoenflies symbols: S4 = C2i S6 = C3i π Qk3 , Qa2 , Qπb , −1 34 Material Symmetries — Classification of Solids Table 4: Non-Crystal Classes Non-Crystal Classes Symmetry cylindrical spherical icosahedral 1) alternative ∞/m = ∞/mm = 2∞ = m∞ ¯ = m3̄5̄ = No. Aniso. Notation Order Generators Type Hermann-Mauguin Schoenflies 33 34 35 36 37 38 39 - ∞ ∞/m1) ∞2 ∞m ∞/mm1) 2∞1) m∞ ¯ 1) C∞ 2),3) C∞h 3) D∞ 3) C∞v 3) D∞h K 3) Kh ∞ ∞ ∞ ∞ ∞ ∞ ∞ Qαc Qαc , −1 Qαc , Qπa Qαc , −Qπa Qαc , Qπa , −1 Qαc , Qβa Qαc , Qβa , −1 40 - 235 60 Qc5 , Qā3 , Qπb 41 - m3̄5̄1) I 120 Qc5 , Qā3 , Qπb , −1 Hermann-Mauguin symbols: ∞ ¯ 2 2 ∞m ¯ =∞ ¯m m m =∞ ∞∞ ∞ ¯∞ ¯ 2 3̄ m 5̄ 2) 3) Ih 2π 2π alternative Schoenflies symbols: C∞h = S∞ = C∞i 3) alternative symbols: C∞ = T1 C∞v = C∞h = T3 D∞h = D∞ = T 5 K = SO(3) Kh = 2π 2π T2 T4 O(3) Table 5: Orientation of Symmetry Elements for Herrmann-Mauguin Symbols Crystal System Orientation triclinic: no convention needed monoclinic: A(2) -axis parallel to [010] orthorhombic: twofold axis and / or mirror planes are denoted in the order of [100], [010], [001] tetragonal: (i) A4 -axis parallel to [001], (ii) direction [100] and (iii) direction [110] cubic: (i) direction [100], (ii) [111] direction and (iii) the direction [110] tri- and hexagonal: (i) A(6) -axis parallel to [001], (ii) direction [100], (iii) only in hexagonal systems direction [120] Schoenflies Cn Cni Cs Sn Cnh Cnv Dn Dnd Dnh Table H.-M. n n̄ m n̄ = 4̄, 6̄ n/m nm n2 n̄m n/mm 6: Systematics of Schoenflies Symbols Symmetry Element n-fold rotation axis, odd order rotation axis and inversion center mirror plane n-fold rotation-reflection axis (only S4 and S6 are used) n-fold axis normal to mirror plane n-fold axis parallel to n mirror planes n-fold axis normal to n 2-fold axis like Dn plus mirror planes bisecting the 2-fold axis, like Dn plus mirror plane normal to the n-fold axis 35 Material Symmetries — Classification of Solids 5-fold axis belong to the icosahedral classes with the Hermann-Mauguin symbols 235 and m3̄5̄. The five-fold symmetry operation maps an icosahedron and a dodecahedron onto itself. The five cylindrical and two spherical symmetry groups are infinite point groups. They are characterized by one or two rotation axes A(∞) with rotation angles α, β ∈ [0, 2π]. In the case of one rotation axis we speak of transversal isotropy. Combination with a two-fold axis perpendicular to the infinite rotation axis and with the central inversion yields the five cylindrical symmetry groups visualized in figures 21a-e. A material with one family of aligned fibers is a typical example for this type of symmetry. Two perpendicular infinite rotation axes constitute spherical symmetry. In combination with the central inversion one distinguishes between hemitropic and isotropic symmetry. The discussed symmetry groups are summarized in table 4. A(∞) A(∞) A(∞) A(∞) A(∞) PSfrag replacements a. b. c. d. A(∞) A(∞) f. e. J (∞) J (∞) g. Figure 21: Visualization of the infinite point-groups. Cyclic groups: (a) C∞ , (b) C∞h , (c) D∞ , (d) C∞v , (e) D∞h . Spherical groups: (f) K and (g) Kh . 3.6. Classification into 14 Types of Anisotropy An often used classification of solid materials bases on the structure of the generators of their symmetry groups. The symmetry of a grid lattice is described by one of the eleven symmetry groups of the crystal classes 2, 5, 8, 11, 15, 17, 20, 23, 27, 29 and 32. All these classes have in common that they contain the central inversion explicitly. For all these classes the motif does not reduce the symmetry of the lattice. In other words the motif features at least the symmetries of the lattice. Considering also the cylindrical and spherical systems, the inversion is contained in the classes 34, 37 and 39. These 14 symmetry groups are referred to as the mechanics symmetry groups or anisotropy types. They are of special importance when considering quadratic tensor functions as for instance for the free energy density or a plastic potential. Anticipating the results of section 4.6, the classes 23, 27, 34 and 37 as well as the classes 29 and 32 yield quadratic tensor functions with identical symmetries so that the 14 mechanics symmetry classes mentioned above reduce to ten. 37 Representations of Anisotropic Tensor Functions 4. Representations of Anisotropic Tensor Functions Constitutive behaviour of materials can be described mathematically with tensor functions. When setting up these functions, several restrictions have to be considered, cf. section 2.4. Especially, the constitutive functions have to account for the inherent material symmetries. The main goal of this section is to discuss how to construct tensor functions that obey symmetry properties with respect to a given material symmetry group. 4.1. Definitions and Notions Consider a map f of elements from a domain D into the domain W as a general representative of a constitutive function f :D→W, x 7→ f (x) . (4.1) The domain D is assumed to consist of a set of vectors {v i }i=1...a and symmetric and skew-symmetric second-order tensors {Ai }i=1...b , {W i }i=1...c D := Va × Ab × Wc (4.2) where V denotes the Euclidean vector space, A the symmetric tensor space and W the skew-symmetric one. The range W of f can be either R, V, A or W. For the sake of brevity, we denote an element of the domain D with x := {v 1 , . . . , v a , A1 , . . . , Ab , W 1 , . . . , W c } ∈ D . (4.3) In order to keep the notation short, we define the action of the orthogonal group O(3) on the r-th order tensor space Tr by the Rayleigh products r=0 (Q, c) 7→ Q ? c := c (Q, v) 7→ Q ? v := vi Qei r = 1 (4.4) ? : O(3) × Tr → Tr : (Q, T ) 7→ Q ? T := Ti1 ...ir Qei1 ⊗ · · · ⊗ Qeir r > 1 where Q ∈ O(3). This notation can be extended to the set of tensors (4.3) by Q ? x := {Q ? v 1 , . . . , Q ? v a , Q ? A1 , . . . , Q ? Ab , Q ? W 1 , . . . , Q ? W c } . (4.5) The function f is classified according to its transformation properties under the action of the orthogonal group O(3) or subgroups of the orthogonal group G ⊂ O(3). A scalarvalued function f with the property f (Q ? x) = Q ? f (x) ∀Q ∈ G (4.6) is denoted as isotropic invariant if G = O(3), as hemitropic invariant if G = SO(3), otherwise as anisotropic invariant. For tensor-valued functions f the corresponding terms are isotropic form-invariant, hemitropic form-invariant and anisotropic form-invariant. The set of transformations G has the properties of a group and is denoted as symmetry group of the function f . The function f in (4.6) is said to be G-invariant. The main task of representation theory is to determine the general structure of f . Let I := {I1 (x), I2 (x), . . . , In (x)} (4.7) 38 Representations of Anisotropic Tensor Functions be a set of n scalar-valued functions Ii=1...n that are all invariant with respect to a given symmetry group G. The set I is denoted as functional basis if for any G-invariant function α(x) there exists a mapping f : Rn → R so that α(x) = f (I1 (x)), I2 (x), . . . , In (x)) ∀x ∈ D . (4.8) For tensor-valued functions f , tensor functions G1 (x), G2 (x), . . . , Gm (x) form a generating set if for all f satisfying (4.6) there exist invariants α1...m such that f (x) = α1 (x)G1 (x) + · · · + αm (x)Gm (x) . (4.9) A functional basis or a generating set are termed complete representations. They are called irreducible, if no proper subset is again a complete representation. It is important to note that for a given function f the choice of a complete and irreducible basis is not unique. Even the number of elements in two different complete and irreducible bases may differ. This fact leads to the notion of a minimal integrity basis. 4.2. Isotropic Extension of Anisotropic Tensor Functions A substantial task in material modeling is the construction of tensor functions which are invariant with respect to a given material symmetry group G. Results frequently used for representations of anisotropic functions go back to the publications of Lokhin & Sedov [67], Boehler [21] and Liu [66]. The key idea of these works is to reduce the problem of constructing an anisotropic function of a set of tensors to the problem of constructing an isotropic function. This is possible when extending the set of argument tensors by socalled constant first- and second-order structural tensors that are G-invariant. Based on results of the representation theory for isotropic functions, this approach leads to complete representations for some of the crystal classes, namely functions with triclinic, monoclinic and orthorhombic symmetry as well as for transversal isotropic functions. Recently, Xiao [156, 157] extended this approach to obtain complete representations for all 32 crystal classes and all non-crystal classes by introducing vector-valued and secondorder tensor-valued G-invariant tensor functions — denoted as structural functions or isotropic extension functions — instead of constant structural tensors, v ei : D → V, Aei : D → A, W ei : D → W. These functions are combined in an ordered set Ξ(x) := {v e1 (x), . . . , v er (x), Ae1 (x), . . . , Aes (x), W e1 (x), . . . , W et (x)} . (4.10) Assume that these structural functions are invariant with respect to symmetry transformations of a given symmetry group G, i.e. Ξ(Q ? x) = Q ? (Ξ(x)) ∀Q ∈ G . (4.11) The key observation is as follows. Any G-invariant tensor function of an argument x can be represented in terms of an isotropic tensor function with an extended set of arguments. For isotropy, the invariance condition (4.6) poses the following restriction on the function f (x, Ξ(x)) = f (Q ? x, Q ? (Ξ(x)) ∀Q ∈ O(3) . (4.12) Inserting (4.11), one observes the desired anisotropic behaviour at frozen structural functions, i.e. f (x, Ξ(x)) = f (Q ? x, Ξ(Q ? x)) ∀Q ∈ G . (4.13) Representations of Anisotropic Tensor Functions 39 Using this isotropic extension method, the problem of finding representations for anisotropic functions is shifted to the problem of finding representations for isotropic functions. The latter is well known for functions of a set of first- and second-order tensors and discussed in section 4.3. Note that the type of anisotropy is solely determined by the material symmetry group G of the structural functions (4.10). Remark 4.1: In literature this extension method with constant structural tensors is referred to as the isotropicization theorem. It can easily be extended to higher-order structural tensors. Zheng [162] specifies a single structural tensor characterizing the symmetry group G for each crystal class. But the essential drawback of constant structural tensors is that only some symmetry groups, namely those of transverse isotropy, triclinic, monoclinic and rhombic anisotropy can be characterized by sets of constant first- and second-order structural tensors. For the remaining symmetry groups, the structural tensors are of higher order than two. Unfortunately, representations of tensor functions with higher-order tensor arguments are only known in some particular cases, see for example Zheng & Betten [160] and Betten & Helisch [14, 15]. 4.3. Isotropic Functions of First- and Second-Order Tensors 4.3.1. Wang’s Approach A tool for finding representations of scalar- and tensor-valued functions is the framework of group theory. In Wang [143] a general representation theorem is presented which is applied in [144, 145] to scalar-, first- and second-order tensor-valued functions of first- and second-order tensors. As an answer to the criticism of Smith [129], Wang republished a revised extensive version [146, 147] with corrigendum [148]. 4.3.1.1. Underlying Idea. The group theoretical approach used by Wang [143] to obtain a general representation theorem can be divided into two parts. The first part is associated with restrictions on functions resulting from the requirements of objectivity and material symmetry separately. In a second step the restrictions resulting from a simultaneous fulfillment of both, objectivity and material symmetry, lead to the formulation of the general representation theorem. From the discussion of restrictions on constitutive equations in section 2.4 we saw that functions with Lagrangian agencies as for instance the right Cauchy-Green tensor C are a priori objective. For such functions it is sufficient to consider restrictions resulting from symmetry requirements. Having in mind that anisotropic tensor functions can be extended to isotropic functions by adding structural functions, the restrictions due to isotropy have to be investigated. In the following, the ideas underlying [143] and [144, 145] are discussed. Wang considers a scalar- or tensor-valued function as defined in (4.1), i.e. f : D → W, x 7→ f (x) which is invariant with respect to orthogonal transformations. Such a function is denoted as isotropic invariant for W = R or isotropic form-invariant in all other cases. The generalized isotropy condition according to (4.6) has the form f (Q ? x) = Q ? f (x) ∀Q ∈ O(3) . (4.14) Restriction (4.14) can be split up into two separate equations. This is done in the following. 40 Representations of Anisotropic Tensor Functions Two points x and y of the domain D are said to be equivalent, if they are related via an orthogonal transformation y = Q ? x with Q ∈ O(3). The coset α associated with a given point x consists of all the points conjugate to x, i.e. α := {y ∈ D | y = Q ? x ∀Q ∈ O(3)} . (4.15) An important fact is that the orbits of two points x1 and x2 either coincide or have no points in common, see Wineman & Pipkin [154]. As a consequence, the symmetry condition (4.14) for an arbitrary point x of coset α takes the form Q ? f (xα ) = f (x) where x = Q ? xα ∈ α . (4.16) Here xα is a fixed representative point of the coset α. Let Oxα := {Q ∈ O(3) | xα = Q ? xα } denote the set of transformations which leave xα unchanged. If, additionally to (4.16), for all cosets α the function is invariant with respect to transformations of O xα , f (xα ) = Q ? f (xα ) ∀Q ∈ Oxα , (4.17) then f satisfies the generalized symmetry condition. Equation (4.16) states that in each fixed orbit where a point x is equivalent to a reference point xα , the value of f follows from the value at the reference point applying the same equivalence relation. (4.17) requires invariance of the value at the reference point for all transformations leaving the reference point invariant. The procedure of determining representations for f consists of two steps. (i) The cosets in D have to be characterized, and (ii) the function f has to be constructed in such a way, that the isotropy condition is satisfied on each fixed coset. For a scalar-valued function ψ(x) equation (4.16) states, that ψ(x) = ψ(xα ) = constant on each coset. In this case, (4.17) is trivially fulfilled. Characterization of Cosets. Any coset can be characterized by invariants. Consider two sets of tensors belonging to the same domain D x = {v i=1...a , Ai=1...b , W i=1...c } and x̄ = {v̄ i=1...a , Āi=1...b , W̄ i=1...c } (4.18) and assume that they belong to the same coset. Then they are related by x = Q ? x̄ and denoted as equivalent. The problem of finding a set of invariants that characterize the orbit can be treated geometrically. The characteristic space of a nonzero vector v is spanned by the unitvector v/kvk and is denoted as oriented line. The characterization of a symmetric second-order tensor A depends on the number of different eigen-values. For three different eigen-values, the eigen-directions are well defined from the corresponding eigen-value-problem and each eigen-vector spans a characteristic line. In the case of two equal eigen-values, two eigen-vectors lie in the plane perpendicular to the third one but the orientation inside that plane is arbitrary. Here the orientation of the unique eigen-vector and the plane are characterized by one line. A skew-symmetric second-order tensor is characterized by its axial vector spanning its characteristic line, the so-called axial line. 41 Representations of Anisotropic Tensor Functions On the basis of these geometric definitions of oriented lines, lines and axial lines the relative orientation between elements of the sets x or x̄ can be measured. Two sets x and x̄ are equivalent, if and only if their sets of characteristic spaces are congruent and the characteristic values of corresponding characteristic spaces are the same. Thereby the characteristic values are determined by the fundamental invariants vi · vi ; tr[Ai ] , tr[A2i ] , tr[A3i ] ; tr[W 2i ] (4.19) of each of the elements of x or x̄, respectively. The relative orientation of the elements is measured by relative or simultaneous invariants taking into account at least two different elements. An important result of Wang [146] is the Equivalence Theorem. It states that two lists x and x̄ are equivalent if and only if all their corresponding subsets up to order four are equivalent. Construction of the Function. As already mentioned, isotropic tensor-valued functions are restricted besides (4.16) also by (4.17). The admissible tensor space containing the function values f satisfying (4.17) is spanned by a set of so-called tensor generators {f i }i=1,...,g which are elements of V, A or W, depending on the range W. An isotropic tensor-valued function then is a linear combination of these tensor generators with coefficients fi which are scalar-valued isotropic tensor-functions, i.e. f (x) = g X fi (x)f i (x) . (4.20) i=1 Of course, the set of tensor generators depends on the set x of tensor arguments of the function under consideration. For details on the derivation of the generating sets we refer to Wang [147]. 4.3.1.2. Example. We illustrate the characterization of cosets with invariants with an example. Consider two equivalent sets, each consisting of a vector and a symmetric second-order tensor, x = {v, A} and x̄ = Q ? x = {v̄, Ā}, respectively. Firstly, each set is partially characterized by its fundamental invariants v · v = v̄ · v̄ and {tr[A], tr[A2 ], tr[A3 ]} ≡ {tr[Ā], tr[Ā2 ], tr[Ā3 ]} . (4.21) Furthermore, all possible relative orientations of v̄ and Ā that result from the given set {v, A} by the transformation Q ∈ O(3) have to be characterized. Therefore it is sufficient to consider only those transformations that leave A unchanged and investigate the transformation of v. Let λi denote the eigen-values, {ni }i=1,2,3 and {n̄i }i=1,2,3 denote the eigen-vectors of A and Ā, respectively, and assume that they are connected by eight orthogonal transformations ni = Qn̄i so that n1 = ±n̄1 , n2 = ±n̄2 and n3 = ±n̄3 (4.22) where Q ∈ O(3). The eight transformations defined in this way leave A invariant, i.e. A= 3 X i=1 λi n i ⊗ n i = 3 X i=1 λi n̄i ⊗ n̄i = Ā . (4.23) 42 Representations of Anisotropic Tensor Functions The vector v = vi ni = v̄i n̄i transforms according to Qv = v̄i ni = ±v̄i n̄i . (4.24) In contrast to the transformation of the second-order tensor, here, any of the eight transformations (4.22) results in a different coordinate representation. As a measure of the possible eight orientations between the tensors v and A, two additional relative invariants are introduced v · Av = v̄ · Āv̄ and v · A2 v = v̄ · Ā2 v̄ . (4.25) To prove (4.25), assume the representation v = vm where v = v · v is the length of the vector and m = v/kvk its direction. The orientation of the vector v relative to the characteristic space of A, represented by three non-oriented lines along ni , is characterized by two of the three angles cos θi := m · ni i = 1, 2, 3 (4.26) due to the linear dependence cos2 θ1 + cos2 θ2 + cos2 θ3 = 1. With these definitions at hand, the invariants on the left hand side in (4.25) appear in the form ¾ v · Av = v 2 (λ1 − λ3 ) cos2 θ1 + v 2 (λ2 − λ3 ) cos2 θ2 . (4.27) v · A2 v = v 2 (λ21 − λ23 ) cos2 θ1 + v 2 (λ22 − λ23 ) cos2 θ2 The representation for the corresponding invariants of x̄ is analogously. Insertion into (4.25) yields the result cos2 θ1 = cos2 θ̄1 and cos2 θ2 = cos2 θ̄2 . (4.28) This determines the possible orientations of the tensor v. Summarizing, the set consisting of six fundamental and relative invariants I := {v · v, tr[A], tr[A2 ], tr[A3 ], v · Av, v · A2 v} (4.29) constitutes a functional basis for scalar-valued isotropic functions of a single vector and an symmetric second-order tensor, because these invariants characterize the coset of x = {v, A}. Remark 4.2: The assumption of the existence of three distinct eigen-values of A also covers the cases of two and three equal eigen-values. In the first case, the characteristic space of A is a single line. The orientation of v with respect to this line is then covered by one angle which is represented by (4.25)1 . The second invariant (4.25)2 is superfluous. In the second case, the relative orientation of v and A is unspecified. Both invariants (4.25) can be dropped and the coset is completely described by the fundamental invariants only. 4.3.2. Smith’s Approach A systematic treatment of deriving representations for isotropic scalar-, first-order and second-order tensor-valued functions of first-order and symmetric and skew-symmetric second-order tensors is presented in Smith [130]. The basic idea goes back to the work of Rivlin & Ericksen [108] who derived a representation for two symmetric second-order tensors. Representations of Anisotropic Tensor Functions 43 4.3.2.1. Underlying Idea. Consider a scalar-valued function ψ : D → R of a set x ∈ D of first-order, symmetric and skew-symmetric second-order tensors as defined in (4.3). The isotropy condition states ψ(Q ? x) = ψ(x) ∀Q ∈ O(3) . (4.30) A function basis for isotropic invariants of the tensor arguments is a set of isotropic invariants I := {I1 (x), I2 (x), . . . , In (x)} so that any scalar-valued function restricted by (4.30) can be expressed as a scalar-valued function of the invariants I, ψ(x) = ψ(I1 , I2 , . . . , In ) . (4.31) For a given coordinate representation of the argument tensors, the function basis determines the values of the invariants uniquely. Or the other way round, a set of invariants governing the coordinates of the argument tensors uniquely in a given coordinate system constitutes a function basis. The latter was done by Rivlin & Ericksen [108] for two symmetric second-order tensors and extended by Smith [130] to sets of first-order and symmetric as well as skew-symmetric second-order tensors. So the main task is to determine a set of isotropic invariants I := {I1 , I2 , . . . , In } so that the coordinates of the tensor agencies are uniquely determined by the values of the invariants through the equations [v s ]i = f (I1 , I2 , . . . , In ) s = 1, . . . a [As ]ij = f (I1 , I2 , . . . , In ) s = 1, . . . b (4.32) [W s ]ij = f (I1 , I2 , . . . , In ) s = 1, . . . c within suitable coordinate systems. The choice of special coordinate systems means no loss of generality because once the coordinates are known, they can be transformed into any arbitrary coordinate system. The choice of suitable coordinate systems depends on the structure of the set of argument tensors. According to Smith [130], the eight cases listed in table 7 have to be distinguished. The invariants obtained from these eight cases are summarized in table 8. Table 7: Classification of Argument Sets due to Smith No. Argument Set to be Considered 1. 1i. three linear independent vectors two linear independent vectors 2. no vectors, two skew-symmetric tensors with linear independent axial vectors 2i. no vectors, all axial vectors collinear 2ii. no vectors, no skew-symmetric tensors 3. all ial 3i. all 3ii. all vectors collinear, two skew-symmetric tensors with collinear axvectors vectors collinear, all axial vectors collinear vectors collinear, no skew-symmetric tensors 44 Representations of Anisotropic Tensor Functions Table 8: Invariants for Scalar-Valued Functions, cf. Boehler [20] Arguments Invariants v A W A1 , A2 A, W A, v W 1, W 2 W,v v1, v2 A1 , A2 , A3 , A1 , A2 , W , A, W 1 , W 2 W 1, W 2, W 3 A1 , A2 , v A, v 1 , v 2 W 1, W 2, v W , v1, v2 A1 , A2 , v 1 , v 2 A, W , v 1 , v 2 W 1, W 2, v1, v2 v·v tr[A], tr[A2 ], tr[A3 ] tr[W 2 ] tr[A1 A2 ], tr[A21 A2 ], tr[A1 A22 ], tr[A21 A22 ] tr[AW 2 ], tr[A2 W 2 ], tr[A2 W 2 AW ] v · Av, v · A2 v tr[W 1 W 2 ] v · W 2v v1 · v2 tr[A1 A2 A3 ] tr[A1 A2 W ], tr[A21 A2 W ], tr[A1 A22 W ], tr[A1 W 2 A2 W ] tr[AW 1 W 2 ], tr[AW 21 W 2 ], tr[AW 1 W 22 ] tr[W 1 W 2 W 3 ] v · A 1 A2 v v 1 · Av 2 , v 1 · A2 v 2 v · W 1 W 2 v, v · W 21 W 2 v, v · W 1 W 22 v v1 · W v2, v1 · W 2v2 v 1 · A 1 A2 v 2 − v 2 · A 1 A2 v 1 v 1 · AW v 2 − v 2 · AW v 1 v1 · W 1W 2v2 − v2 · W 1W 2v1 4.3.2.2. Examples. We illustrate the afore discussed ideas by two short examples. Consider case one, where there are three linear independent vectors a, b, c ∈ x among the elements of x. Choose an orthonormal coordinate system so that these vectors have the following representation a = a 1 e1 , b = b 1 e1 + b 2 e2 , c = c 1 e 1 + c 2 e 2 + c 3 e3 (4.33) where all coordinates are positive. In this coordinate system, the coordinates of a, b and c can be obtained from the following set of invariants a · a, a · b, a · c, b · b, b · c, c · c . (4.34) Then the coordinates of any other vector v ∈ x can be obtained from the three invariants a · v, b · v, c · v . (4.35) The coordinates of any symmetric second-order tensor A ∈ x are determined by the following set of six invariants a · Aa, a · Ab, a · Ac, b · Ab, b · Ac, c · Ac . (4.36) Finally, for any skew-symmetric second-order tensor W ∈ x the coordinates can be computed from the three invariants a · W b, a · W c, b · W c . (4.37) 45 Representations of Anisotropic Tensor Functions Table 9: First Example: Relations between Invariants and Coordinates No. 1 2 3 4 Invariants Coordinates a · a, a · b, a · c, b · b, b · c, c · c a · v, b · v, c · v a · Aa, a · Ab, a · Ac, b · Ab, b · Ac, c · Ac a · W b, a · W c, b · W c a 1 , b1 , b2 , c 1 , c 2 , c 3 v 1 , v2 , v3 A11 , A22 , A33 , A12 , A23 , A13 W12 , W23 , W13 These dependencies can be summarized as done in table 9. It has to be read as follows: once the invariants up to line No. i are given, the coordinates of all of the tensors entering these invariants can be computed. Thus, the invariants listed in table 9 constitute a functional basis for the set x, supposed that there exist three linear independent vectors among the v i in x. In a second example, we discuss case 3ii for the case of one symmetric tensor A with three different eigen-values and a vector v. Introducing an auxiliary tensor X = v ⊗ v, this case can be cast back to the situation of two symmetric second-order tensors. Therefore, the relation between invariants and coordinates is given in table 10. It can be proved as follows. In the orthonormal eigen-basis {ni }i=1,2,3 of A, the two tensors v and A have Table 10: Second Example: Relations between Invariants and Coordinates No. 1 2 Invariants Coordinates tr[A], tr[A2 ], tr[A3 ] tr[X], tr[AX], tr[A2 X] A11 , A22 , A33 X11 , X22 , X33 the representations v = v i ni and A = 3 X i=1 λi n i ⊗ n i (4.38) where we assume the order of the eigen-values so that λ1 > λ2 > λ3 and vi=1,2,3 ≥ 0. The eigen-values follow from the characteristic equation λ3i − I1 λ2i + I2 λi − I3 = 0 (4.39) in terms of the invariants I1 = tr[A], I2 = 12 {tr[A]2 − tr[A2 ]} and I3 = 13 {tr[A3 ] − 3 tr[A2 ] tr[A] + 21 tr[A]3 }. The invariants of the second line of table 10 constitute the 2 system of equations tr[X] = v · v = v12 + v22 + v32 tr[AX] = v · Av = λ1 v12 + λ2 v22 + λ3 v32 (4.40) tr[A2 X] = v · A2 v = λ21 v12 + λ22 v22 + λ23 v32 which can be solved for the coordinates of v. 46 Representations of Anisotropic Tensor Functions 4.3.3. Comparison of the Results Obtained by Wang and Smith The invariants obtained by Wang and Smith that serve as a function basis differ slightly. Boehler [20] compared the results obtained by the two approaches. The function bases were corrected and modified and a unified representation is derived. The resulting function basis is given in table 8. From that table the invariants that constitute a complete, irreducible scalar-valued function basis for a function f (x) as specified in (4.1) and (4.3) is obtained by taking all possible disordered combinations of one, two, three and four tensors of the set x. The irreducibility of the function basis was proven by Pennisi & Trovato [101]. For a discussion of this topic see section 4.5. Note that although the basis specified in table 8 is irreducible for arbitrary tensor arguments x, a specific choice of these tensors may result in redundant invariants. For irreducible functional bases in two-dimensions we refer to Korsgaard [63]. 4.4. Isotropic Polynomials of First- and Second-Order Tensors In this section, we discuss the construction of isotropic tensor polynomials. Depending on the structure of the tensorial argument set x, the restriction from the before discussed tensor functions to tensor polynomials may result in larger sets of invariants which serve as a polynomial basis. In spite of this drawback, the representation theory for polynomials is often applied in literature. In the context of continuum mechanics it is mostly used for those symmetries and sets of argument tensors where the resulting basis is identical to those obtained for tensor functions. In any case, a basis for tensor polynomials always serves as a function basis but the converse is not true, see Pipkin & Wineman [103]. The theory of isotropic tensor polynomials is well known and often discussed in literature. We here refer to the classical textbooks of Schur [119], Weyl [151], Grace & Young [42] and Gurewich [44] among others. Summaries of the theory in view to the formulation of constitutive polynomials in continuum mechanics and applications can be found in the publications by Boehler [22], Spencer [132, 133] and Betten [13]. 4.4.1. Definitions In contrast to the framework of isotropic tensor functions, we here consider tensor polynomials p:D→R, x 7→ p(x) (4.41) where x ∈ D with D := Va × Tb2 is a set of a first-order and b second-order tensors x := {v 1 , . . . , v a , A1 , . . . Ab } . (4.42) Here T2 denotes the space of second-order tensors. To determine the general structure of p, consider a set on n scalar-valued polynomials Ii=1...n which are invariant with respect to a given symmetry group G, I := {I1 (x), I2 (x), . . . , In (x)} with Ii (Q ? x) = Ii (x) ∀Q ∈ G . (4.43) The set I is denoted as integrity basis if any scalar-valued G-invariant tensor polynomial α(x) can be expressed in the form α(x) = p(I1 (x), I2 (x), . . . , In (x)) ∀x ∈ D . (4.44) Representations of Anisotropic Tensor Functions 47 In the following, the forms of the polynomial invariants of I will be specified. This is done in two steps, first for vectors only and then for first-order tensors and second-order ones. 4.4.2. Integrity Basis for Sets of First-Order Tensors In a first step, assume that the set of argument tensors consists solely of vectors, x = {v 1 , v 2 , . . . , v a } . (4.45) A classical theorem of representation theory is Peano’s Theorem. Applied to the above specified set x, it states that with the possible exception of the determinant det[v i , v j , v k ], i, j, k ∈ [1, a], every polynomial invariant of these vectors can be expressed as a polynomial in the invariants of two of the vectors and invariants obtained from these two by the polarization operator Dij [·] := v i · ∂vj [·] . (4.46) Observe that the determinant is a hemitropic invariant due to its transformation property det[Qv 1 , Qv 2 , Qv 3 ] = det[Q] det[v 1 , v 2 , v 3 ] and has not to be considered in the context of isotropic polynomials. Based on Peano’s Theorem, an integrity basis for scalar-valued functions can be constructed as follows. First note that polarization of the scalar-product of two vectors yields no new invariants, i.e. Dab [v i · v j ] = (v a · v j )δib + (v a · v i )δjb . (4.47) An invariant of a single vector v has to be a polynomial of its coordinates. Consider three orthonormal coordinate systems {ei }, {ēi } and {¯ēi } with i = 1, 2, 3, so that v has the following representations v = v1 e1 + v2 e2 + v3 e3 = v̄ 1 ē1 + v̄ 2 ē2 = v̄¯1¯ē1 (4.48) The invariant I is a polynomial in the coordinates, i.e. ¯ v̄¯ ) . ¯ 1 , v̄2 ) = I( I = I(v1 , v2 , v3 ) = I(v̄ 1 (4.49) It cannot be a hemitropic invariant because reflections on planes containing ¯ē1 would yield the zero polynomial ¯ v̄¯ , 0, 0) = 0 . ¯ v̄¯ , 0, 0) = −I( I( 1 1 (4.50) ¯ v̄ ¯ v̄ ¯ ) we ¯ ) = Q ? I( Thus I is an isotropic invariant. From the invariance property I( ¯ conclude that I has to be a quadratic function of v̄¯1 in the special case where Q = −1. Transformation into the unbarred coordinate system yields I = I(v̄¯12 ) = I(v̄12 + v̄22 ) = I(v12 + v22 + v32 ) . (4.51) Consequently the scalar product constitutes an integrity basis for isotropic functions of a single vector argument. 48 Representations of Anisotropic Tensor Functions Now extend the set x by a second vector w. The coordinate systems are chosen in such a way that ē3 is collinear with v × w and the coordinate representations are given by v = v̄1 ē1 + v̄2 ē2 and w = w̄1 ē1 + w̄2 ē2 . (4.52) From a reflection operation on the plane spanned by ē1 and ē2 on can conclude that any invariant of two vectors has to be isotropic. Any orthogonal transformation preserves the area A of the parallelogram spanned by v and w, for which the relation ¯ ¯ ¯ v̄12 + v̄22 ¯ v̄ w̄ + v̄ w̄ 1 1 2 2 2 2 ¯ A = (v̄1 w̄2 − v̄2 w̄1 ) = ¯¯ (4.53) 2 2 v̄1 w̄1 + v̄2 w̄2 w̄1 + w̄2 ¯ holds. From (4.53) we conclude that the invariant I is a polynomial in the scalar products v·v , w·w , v·w . (4.54) Application of Peano’s theorem to (4.54) simply reproduces the scalar products and therefore any invariant of a set of vectors is a polynomial in the elements of the integrity basis I := {v i · v i , v i · v j } i, j ∈ [1, a] . (4.55) 4.4.3. Isotropic Tensors Before proceeding with the construction of isotropic polynomials of a set of first- and second-order tensors, the notion of isotropic tensors is introduced. The key property of an isotropic tensor is that its coordinates are invariant under a change of reference frame. As a result of that a rotation yields the same tensor. Consider the coordinate representation of an m-order tensor A Ai1 i2 ...im ei ⊗ e2 ⊗ · · · ⊗ em = Āi1 i2 ...im ēi ⊗ ē2 ⊗ · · · ⊗ ēm (4.56) where the basis vectors are related according to ei = Qēi with Q ∈ O(3). The tensor A is assumed to be isotropic and the above mentioned properties are Ai1 ,i2 ...im = Āi1 ,i2 ...im and Q ? A = A . (4.57) Any isotropic tensor is of even order because of the transformation properties Q ? A = det[Q] (det[Q] Q) ? A with det[Q]Q ∈ SO(3) . (4.58) Consider the following invariant of a set of an even number m of vectors and an isotropic tensor A of order m I(v 1 , . . . , v m ; A) = (((A · v a1 ) · v a2 ) · . . . ) · v am = (((A · v̄ a1 ) · v̄ a2 ) · . . . ) · v̄ am (4.59) where v̄ ai = Qv ai and the indices a1 , . . . , am are a permutation of 1, . . . , m. As the indices ai are all different, the invariant is multi-linear in the vectors. According to the results of the previous section, any isotropic invariant of a set of vectors can be represented Representations of Anisotropic Tensor Functions 49 as a polynomial in terms of scalar-products of all possible pairs of vectors. Due to the multi-linearity of I only mixed scalar-products have to be taken into account so that I = p(J ) where J = {v ai · v aj } with i 6= j ∈ [1, m] . (4.60) A typical term of that polynomial then has the form c (v a1 · v a2 )(v a3 · v a4 ) · · · (v am−1 · v am ) where c ∈ R is a scalar value. The coordinates of A follow by differentiation Aj1 ...jm = c va1 ·i1 va2 ·i2 · · · vam−1 ·im−1 vam ·im ∂mI = = c δi1 j1 δi2 j2 · · · δim jm (4.61) ∂v a1 · · · ∂v am va1 ·j1 va2 ·j2 · · · vam−1 ·jm−1 vam ·jm and can be represented exclusively in terms of Kronecker deltas. From (4.61) one can conclude that any isotropic tensor can be represented as a polynomial in products of Kronecker deltas. 4.4.4. Integrity Bases for Sets of First-Order and Second-Order Tensors 4.4.4.1. Structure of the Invariants. Isotropic polynomials of a set of m first-order and n second-order tensors x = {v 1 , . . . v m , A1 , . . . An } can be constructed with isotropic tensors. Consider therefore the invariant I := (((((A · v a1 ) · · · · ) · v am ) : Ab1 ) : · · · ) : Abn (4.62) where ai ∈ [1, m], bi ∈ [1, n] and A is an isotropic tensor. When expressing the coordinates of A in terms of Kronecker deltas as specified in (4.61), one observes that any isotropic polynomial I(x) coincides with a polynomial in expressions of either one of the types tr[Ab1 Ab2 · · · Abm ] or v ai · Ab1 Ab2 · · · Abm v aj . (4.63) Type (4.63)1 belongs to invariants of argument sets solely consisting of second-order tensors, whereas (4.63)2 belongs to sets of first-order tensors and second-order ones. Probably the most important result in the theory of invariants is Hilbert’s Theorem, see Weyl [151], which states that for any finite set of tensor agencies x an integrity basis consisting of a finite number of invariants exists. This theorem motivates all further investigations in deriving complete and irreducible integrity bases. 4.4.4.2. Traces of Tensor Products. We restrict our attention to second-order tensor agencies only x = {A1 , A2 , . . . , Aλ }. The results of the previous section is that every invariant of a set of second-order tensors can be expressed in terms of traces of tensor products of that set. While the number of possible tensor products Π := As1 As2 . . . Asn ; s1 , s2 , . . . , sn ∈ [1, . . . , λ] (4.64) with tensors Asi ∈ x is infinite, due to the arbitrary number of factors n, the number of traces of these products is not. To derive restrictions for the traces, we consider tensor polynomials with coefficients being polynomials in traces of tensor products, i.e. X p(x) = p(tr[Π(x)])Π(x) . (4.65) The basic idea in order to limit the number of relevant tensor products is to show that concrete Π’s can be expressed in terms of traces of tensor products of lower order tr[Π(x)] = tr[p(x)] with ord(Π) > ord(p) (4.66) 50 Representations of Anisotropic Tensor Functions which is abbreviated by tr[Π(x)] ≡ 0. In what follows, we illustrate the procedure as outlined for instance in Spencer [132, 22]. Point of departure is the multiplication of two permutation tensors of the n+1 dimensional space, which is zero due to a restriction of the indices to i, j ∈ [1, n], ²i1 i2 ...in+1 ²j1 j2 ...jn+1 = 0 ∀ ² ∈ Rn n ×n ; ik , jk ∈ [1; n] (4.67) The product of permutation tensors can be expressed as determinant of Kronecker deltas, see for example Betten [13]. Multiplication with n tensors A, B, . . . , N ∈ Rn×n yields ¯ ¯ ¯ δi j δi1 j2 · · · δi1 jn+1 ¯¯ 1 1 ¯ ¯ δi j δ i 2 j2 δi2 jn+1 ¯¯ 2 1 ¯ (4.68) ¯ ¯ A i 1 j1 B i 2 j2 . . . N i n jn = 0 . .. .. .. ... ¯ ¯ . . . ¯ ¯ ¯ δin+1 j1 δin+1 j2 . . . δin+1 jn+1 ¯ For n = 3 and A, B, C ∈ x follows Rivlin’s identity, see Rivlin [109] ABC + ACB + BCA + BAC + CAB + CBA −A(tr[BC] − tr[B] tr[C]) − B(tr[CA] − tr[C] tr[A]) −C(tr[AB] − tr[A] tr[B]) − (BC + CB) tr[A] −(CA + AC) tr[B] − (AB + BA) tr[C] −I(tr[A] tr[B] tr[C] − tr[A] tr[BC] − tr[B] tr[CA] − tr[C] tr[AB] + tr[ABC] + tr[CBA]) = 0 , (4.69) which can be seen as a generalization of the Cayley-Hamilton theorem. The latter is obtained by setting A = B = C and with det[A] = 13 {tr[A3 ] − 23 tr[A2 ] tr[A] + 21 tr[A]3 } A3 − A2 tr[A] + 1 2 A(tr[A]2 − tr[A2 ]) − 1 det[A] = 0 . (4.70) Multiplication of Rivlin’s identity with D ∈ x and applying the trace operator shows that the first line in (4.69) can be expressed in terms of traces of tensor polynomials of lower order than 4. This is indicated by tr[(ABC + ACB + BCA+ BAC + CAB + CBA)D] ≡ 0 . (4.71) Setting A = B = C we obtain the first result tr[A3 D] ≡ 0 (4.72) which states that traces of tensor products where one factor has the exponent 3 can be expressed with traces of tensor polynomials of lower degree. This result is conform to the well-known fact that an isotropic scalar-valued function of a single argument tensor ψ(C) can be expressed in terms of the three fundamental invariants tr[C], tr[C 2 ] and tr[C 3 ] and no higher order terms have to be taken into account. Setting A = B in (4.71) we get tr[(A2 C + ACA + CA2 )D] ≡ 0 (4.73) 51 Representations of Anisotropic Tensor Functions which motivates the convention that tensor products with two identical factors are to be expressed according to tr[ACAD] = tr[(A2 C +CA2 )D]+p(A, C, D), where p(A, C, D) stands for the tensor polynomial obtained from (4.69) with ord(p) < 4. Setting D = AE in (4.73) shows that the last term is reducible and it remains tr[(A2 CA + ACA2 )E] ≡ 0 (4.74) with the consequence that only such factors have to be taken into account, where A precedes A2 in any product with both A and A2 as factors. To obtain a complete set of rules, traces of tensor products up to an order of 7 have to be considered. We do not want to go through the whole theory but summarize the results discussed in detail in Spencer [132] in table 11. Based on these rules, finite integrity Table 11: Summary. Rules for Relevant Traces of Tensor Products A representation of an isotropic tensor polynomial ψ(x) of a set of λ second-order tensors x = {A1 , A2 . . . Aλ } in terms of traces of tensor products has the form ψ(x) = p(tr[Πi ]) . The tensor products Πi follow from the set of all possible tensor products of the argument tensors A ∈ x by application of the following rules: 1. The factors are of no higher degree than 3. 2. If one factor is of degree 3, there are no other factors. 3. The first and the last factor are no powers of the same tensor. 4. No two factors are the same. 5. Ar precedes A2r in any product with both Ar and A2r as factors. 6. If A2r and A2s , r 6= s, are both factors, they are consecutive factors. 7. No product has a degree greater than six. bases can be deduced. Irreducibility of the so obtained bases is not guaranteed and has to be treated separately. We do not list the integrity bases for different set of argument tensors here but refer e.g. to Spencer [132]. 4.5. Irreducibility of Integrity Bases and Functional Bases An important notion in the context of functional bases is the irreducibility. A functional basis is irreducible if none of its invariants be expressed as a function of the remaining ones. For the invariants of table 8 irreducibility was proved by Pennisi & Trovato [101]. When dealing with integrity bases, irreducibility of a basis states that none of the invariants can be expressed as a polynomial of the remaining ones. The key idea is as follows. To show that one invariant Ik (x) of a function basis I = {I1 (x), . . . , In (x)} is irreducible one has to find two distinct sets of tensors x1 and x2 so 52 Representations of Anisotropic Tensor Functions that all invariants — except the one under consideration — take the same values, i.e. Ik (x1 ) 6= Ik (x2 ) and Ii (x1 ) = Ii (x2 ) ∀i 6= k ∈ [1, n] . (4.75) Obviously, the invariant Ik cannot be uniquely expressed as a function of the remaining ones. Finding such sets for all invariants of a function basis proves their irreducibility. As an example consider the function basis for a single symmetric second-order tensor I := {tr[A], tr[A2 ], tr[A3 ], } . (4.76) The special choices for A1 and A2 k A1 A2 √ √ 3 1 diag[−1, −1, 2] diag[1, 1, − 3 2] Ā 2Ā 2 3 diag[−1, −1, 2] diag[1, 1, −2] 0 1 0 where Ā := 1 0 0 0 0 0 (4.77) prove the irreducibility of I. In general, it is preferable to have functional bases consisting of a small number of invariants because it leads to more concise functions. Nevertheless, the investigations on anisotropic quadratic functions of a symmetric second-order tensor and their second derivative that will be carried out in the next chapter are not based on irreducible functional bases as for example specified in Xiao [155, 158] or Bruhns, Xiao & Meyers [26]. Their use requires the knowledge of how to combine the invariants to obtain a function that completely covers the symmetries under investigation. This question leads to the notion of syzygies that appear in the theory of polynomial invariants. The determination of syzygies is a non-trivial task, cf. Bao & Smith [6]. 4.6. Quadratic Functions of a Symmetric Second-Order Tensor For a broad range of applications in continuum mechanics the constitutive behaviour of materials can be described with scalar-valued tensor functions of a single symmetric second-order tensor argument. Typical examples appearing in this work are energy storage functions, yield criteria or level-set functions and dissipation functions. In this chapter we apply the above discussed framework of anisotropic tensor functions to the important case of quadratic scalar-valued potential functions for the 14 mechanics symmetry groups specified in section 3.6. Without loss of generality, the results of this chapter are derived by means of a free energy function. We consider a scalar-valued function ψ(C) of a symmetric second-order tensor C. Beside the specification of functional bases we derive coordinate-free representations for the second derivative ψ,CC . The corresponding coordinate forms are well known and listed in many publications, e.g. Schmid & Boas [113], Voigt [142] and Hosford [54] in the context of crystal plasticity, Cowin & Mehrabadi [33] and Forte & Vianello [40] in the context of elasticity among many others. These representations are typically derived by direct exploitation of the restrictions of symmetry transformations on the coordinates of the second derivative as carried out in detail for example in Hund [55]. Here the coordinate forms are obtained from the coordinate-free representations when choosing a 53 Representations of Anisotropic Tensor Functions specific orientation of the global coordinate system with respect to the principal axes of anisotropy, cf. also Baumberger [8]. The structure of the subsequent derivations is as follows. In a first step the isotropic extension functions Ξ(C) given in Xiao [156] are specified for the symmetry group G under consideration. For the extended set {C, Ξ(C)} of second-order valued tensor functions a possible function basis for an isotropic scalar-valued function is obtained from table 8. Depending on the structure of the extension functions the thus obtained basis may obtain linearly dependent invariants which have to be dropped. In a second step, the consideration is restricted to quadratic functions in C and their second derivative ψ,CC . Because ψ(C, Ξ(C)) is invariant among the central inversion at fixed structural functions Ξ(C), only the thirteen classes no. 2, 5, 8, 11, 15, 17, 20, 23, 27, 29, 32, 34 and 37 of the tables 3 and 4 have to be investigated that include this symmetry transformation. It is convenient and sufficient to consider quadratic polynomials. Therefore the invariants are combined in all possible combinations that are quadratic in C. Together with the linear invariants they constitute the complete set of monomials for a quadratic polynomial. Dropping out the linear dependent elements which result from the combination of the linear invariants, the final set I := {I1 . . . In } is obtained. It is irreducible with respect to linear dependencies among the invariants. Thus the anisotropic G-invariant polynomial is of the form P (4.78) ψ = ψ(C, Ξ(C)) = ni=1 αi Ii . The set of monomials I may consist of many invariants. Nevertheless the constant second derivative ψ,CC can be reduced to an expression in terms of four fourth-order tensor functions that will be referred to as prototypes. Let F and G denote linear functions in C, L a quadratic function in C and M and N constant tensors. The four types of derivatives are defined by the functions 2 I1 := ∂CC tr[L, M ] = Lijabcd Mji 2 I2 := ∂CC tr[F , M ]2 = 2Fijab Mji Fklcd Mlk 2 I3 := ∂CC tr[F , G, M ] = Fijab Gjkcd Mki + Fijcd Gjkab Mki (4.79) 2 I4 := ∂CC tr[F , M ] tr[G, N ] = Fijab Mji Gklcd Mkl + Fijcd Mji Gklab Mkl . Thereby the fourth-order tensors denote the first and second derivatives of the arguments Fijab = Fij,Cab , and Isym ijab = 1 2 Gijab = Gij,Cab , Lijabcd = Lij,Cab Ccd (4.80) (δia δjb + δib δja ) denotes the symmetric fourth-order identity tensor. 4.6.1. Triclinic Functions — Symmetry Group Ci 4.6.1.1. Isotropic Extension Functions. A set of second-order tensor functions that allows to construct a triclinic Ci -invariant tensor function is specified in Xiao [156] as Ξ(C) : {N , V } with N := ²a, V := ²b . (4.81) 54 Representations of Anisotropic Tensor Functions The orientation of the material is determined by two orthonormal vectors a and b. Any Ci -invariant function can be represented in terms of the elements of a functional basis for the symmetric second-order tensor C and the two skew-symmetric second-order tensors N and V . Table 8 lists invariants that constitute such a functional basis. For the set of argument tensors under consideration the basis consists of the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], (4.82) the relative invariants of every two tensor arguments tr[CN 2 ], tr[C 2 N 2 ], tr[C 2 N 2 CN ], tr[CV 2 ], tr[C 2 V 2 ], tr[C 2 V 2 CV ], tr[N V ] (4.83) and the relative invariants of all three arguments tr[CN V ], tr[CN 2 V ], tr[CN V 2 ] . (4.84) 4.6.1.2. Application to Quadratic Functions. As far as quadratic functions in C are concerned only the linear and quadratic invariants of (4.82), (4.83) and (4.84) have to be taken into account. Having in view the specification of a concrete Ci -invariant quadratic polynomial function it is of advantage to assemble every possible combination of invariants — up to order two in C. With the above given invariants this set contains 27 combined invariants, i.e. I := { tr[C]2 , tr[CN 2 ]2 , tr[CV 2 ]2 , tr[CN V ]2 , tr[CN 2 V ]2 , tr[CN V 2 ]2 , tr[C 2 ], tr[C 2 N 2 ], tr[C 2 V 2 ], tr[C] tr[CN V ], tr[C] tr[CN 2 V ], tr[C] tr[CN V 2 ], tr[CN 2 ] tr[CN V ], tr[CN 2 ] tr[CN 2 V ], tr[CN 2 ] tr[CN V 2 ], tr[CV 2 ] tr[CN V ], tr[CV 2 ] tr[CN 2 V ], 2 2 2 (4.85) 2 tr[CV ] tr[CN V ], tr[CN V ] tr[CN V ], tr[CN V ] tr[CN V ], tr[CN 2 V ] tr[CN V 2 ], tr[C], tr[CN 2 ], tr[CV 2 ], tr[CN V ], tr[CN 2 V ], tr[CN V 2 ] } . A straightforward combination of the invariants in (4.82), (4.83) and (4.84) would yield three additional terms to those given in (4.85). But these are linear functions of the other invariants. Enumerating the invariants of the set I with I1 , . . . , I27 these dependencies read tr[C] tr[CN 2 ] = − 21 (I1 + I2 + I4 + I5 − I7 − I8 ) tr[C] tr[CV 2 ] = − 12 (I1 + I3 + I4 + I6 − I7 − I9 ) 1 tr[CN 2 ] tr[CV 2 ] = I + I4 − 32 I7 − I8 − I9 . 2 1 (4.86) Note that these dependencies stated above permit several possibilities for the choice of which invariants to drop. With this basis at hand, quadratic polynomials ψ(C) that are invariant with respect to symmetry transformations of the group Ci can be formulated in terms of the elements of the monomials in I P ψ = ψ(C, N , V ) = ψ(I1 , ..., I27 ) = 27 (4.87) i=1 αi Ii . 55 Representations of Anisotropic Tensor Functions The second derivative with respect to C gives the constant fourth-order tensor P ψ,CC = 21 where Ci := Ii,CC . i=1 αi Ci (4.88) In the context of constitutive modeling, the constant coefficients α1 , . . . , α27 are denoted as material parameters. 21 of these 27 parameters are related to quadratic terms in C. The second derivatives of the invariants in I can be expressed in terms of the four prototypes (4.79). We get C1 = I2 (C, 1) C2 = I2 (C, N 2 ) C3 = I2 (C, V 2 ) C4 = I2 (C, N V ) C5 = I2 (C, N 2 V ) C6 = I2 (C, N V 2 ) C7 = I3 (C, C, 1) C8 = I3 (C, C, N 2 ) C9 = I3 (C, C, V 2 ) C10 = I4 (C, 1, C, N V ) C11 = I4 (C, 1, C, N 2 V ) C12 = I4 (C, 1, C, N V 2 ) 2 C13 = I4 (C, N , C, N V ) 2 C16 = I4 (C, V , C, N V ) 2 C15 = I4 (C, N , C, N V ) 2 2 C18 = I4 (C, V 2 , C, N V 2 ) C14 = I4 (C, N , C, N V ) C17 = I4 (C, V , C, N V ) 2 (4.89) 2 2 C19 = I4 (C, N V , C, N 2 V ) C20 = I4 (C, N V , C, N V 2 ) C21 = I4 (C, N 2 V , C, N V 2 ) . A coordinate representation for the second derivative requires the specification of the orientations of the anisotropy axes with respect to the global Cartesian frame. For the choice a = e1 and b = e2 the form ψ,CC c11 c12 c13 c22 c23 c33 = sym. c14 c24 c34 c44 c15 c25 c35 c45 c55 c16 c26 c36 c46 c56 c66 for G = Ci (4.90) is obtained. The coordinates are functions of the material parameters α1 -α21 , i.e. c11 = 2(α1 + α3 + α7 ) c22 = 2(α1 + α2 + α7 − α8 ) c33 = 2(α1 + α2 + α3 + α7 − α8 ) c44 = c55 = c66 = 1 2 1 2 1 2 α4 + α 7 − 1 2 α6 + α 7 − α 8 α8 (α5 − α8 ) + α7 c12 = 2 α1 c13 = 2(α1 + α3 ) c14 = c15 = c16 = 1 2 1 2 1 2 c25 = 1 2 1 2 1 2 1 2 1 2 1 2 (α14 − α11 ) (α10 − α16 ) c34 = (α17 − α11 ) c36 = (α10 − α13 ) c46 = − (α9 + α19 ) (α18 − α12 ) c23 = 2(α1 + α2 ) c24 = c26 = (α15 − α12 ) c35 = c45 = − c56 = 1 4 (α10 − α13 − α16 ) (α15 + α18 − α12 ) (α14 + α17 − α11 ) 1 4 1 4 (4.91) α20 α21 . 4.6.2. Monoclinic Functions — Symmetry Group C2h 4.6.2.1. Isotropic Extension Functions. The constitutive behaviour of materials of the prismatic crystal class with symmetry group C2h can be described by an isotropic function of C and the extension functions Ξ(C) : {N , D} with N := ²n, D := a ⊗ a − b ⊗ b . (4.92) The unit vector n is aligned to the two-fold axis, a and b is a pair of arbitrary orthonormal vectors perpendicular to n. Any C2h -invariant scalar-valued tensor function can be 56 Representations of Anisotropic Tensor Functions described in terms of the elements of the function basis for two symmetric second-order tensors C and D and a skew-symmetric second-order tensor N . From table 8 the functional basis consisting of the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], (4.93) the relative invariants of each two arguments tr[C 2 N 2 CN ], tr[CD], tr[C 2 D], tr[CD 2 ], tr[C 2 D 2 ], tr[DN 2 ], tr[D 2 N 2 ], tr[D 2 N 2 DN ] (4.94) and the relative invariants of all three argument tensors tr[CDN ], tr[C 2 DN ], tr[CN 2 DN ] (4.95) is obtained. In the above given functional basis the following dependencies are already taken into account − tr[CD 2 ] = tr[CN 2 ], − tr[C 2 D 2 ] = tr[C 2 N 2 ], − tr[CDN ] = tr[CN 2 DN ], tr[CD 2 N ] = 0 . (4.96) The choice which of the dependent invariants to eliminate is arbitrary. 4.6.2.2. Application to Quadratic Functions. In order to describe a quadratic function, the linear and quadratic invariants of the irreducible functional basis are taken into account. The linear invariants are combined multiplicatively in all possible combinations that are quadratic in C. Thus a quadratic polynomial function can be formulated with the combined invariants of the set I := { tr[C]2 , tr[CD]2 , tr[CD 2 ]2 , tr[CDN ]2 , tr[C] tr[CD], tr[C] tr[CD 2 ], tr[C] tr[CDN ], tr[CD] tr[CD 2 ], tr[CD] tr[CDN ], tr[CD 2 ] tr[CDN ], tr[C 2 ], tr[C 2 D], tr[C 2 DN ], (4.97) tr[C], tr[CD], tr[CD 2 ], tr[CDN ] } . In this set I we have dropped one linear dependent quadratic invariant. Labeling the invariants of I with I1 - I17 , the relation tr[C 2 D 2 ] = − 21 I1 + 1 4 1 4 I2 − I3 + 1 4 I4 + I 6 + 1 2 I11 holds. This set serves as a basis for C2h -invariant tensor functions of the form P ψ = ψ(C, N , D) = ψ(I1 , ..., I17 ) = 17 i=1 αi Ii . The first 13 parameters enter the constant second derivative. It runs P ψ,CC = 13 where Ci := Ii,CC . i=1 αi Ci (4.98) (4.99) (4.100) The second derivatives of the combined invariants are C1 = I2 (C, 1) C5 = I4 (C, 1, C, D) C9 = I4 (C, D, C, DN ) C2 = I2 (C, D) C6 = I4 (C, 1, C, D ) C10 = I4 (C, D 2 , C, DN ) C3 = I2 (C, D 2 ) C7 = I4 (C, 1, C, DN ) C11 = I3 (C, C, 1) C4 = I2 (C, DN ) 2 2 C8 = I4 (C, D, C, D ) C12 = I3 (C, C, D) C13 = I3 (C, C, DN ) . (4.101) 57 Representations of Anisotropic Tensor Functions From the coordinate free representation (4.100) a coordinate representation is obtained when specifying the orientation of the anisotropy axes with respect to a global coordinate system. Setting a = e1 , b = e2 and n = e3 the second derivative takes the form ψ,CC c11 c12 c13 c22 c23 c33 := sym. c14 c24 c34 c44 0 0 0 0 0 0 0 0 c55 c56 c66 for G = C2h . (4.102) The coordinates cij are in a one-to-one relation with the 13 coefficients, i.e. c11 = 2(α1 + α2 + α5 + α6 + α8 + α11 + α12 ) + α3 c22 = 2(α1 + α2 − α5 + α6 − α8 + α11 − α12 ) + α3 c12 = 2(α1 − α2 + α6 ) + α3 c13 = 2 α1 + α5 + α6 c33 = 2 α1 + 2 α11 c14 = α7 + α9 + α10 + α13 c44 = 2 α4 + α11 c23 = 2 α1 − α5 + α6 c55 = α11 − c66 = α11 + 1 2 1 2 α12 (4.103) c24 = α7 − α9 + α10 + α13 α12 c34 = α7 c56 = 1 2 α13 . 4.6.3. Orthorhombic Functions — Symmetry Group D2h 4.6.3.1. Isotropic Extension Functions. To construct functions that are invariant under symmetry transformations of the group D2h of the rhombic-dipyramidal crystal class the extension functions consisting of a single constant second-order tensor Ξ(C) : {D} with D := a ⊗ a − b ⊗ b (4.104) can be used as additional arguments besides C. The orthonormal vectors a and b are aligned along two of the three two-fold axis that characterize D2h . Any isotropic function with respect to transformation of all arguments can be expressed in terms of the invariants obtained from table 8 for two symmetric second-order tensors C and D. Thus a functional basis consists of the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], (4.105) and the relative invariants of the two arguments tr[CD], tr[C 2 D], tr[CD 2 ], tr[C 2 D 2 ] . (4.106) 4.6.3.2. Application to Quadratic Functions. Quadratic combinations of the linear invariants in C gives together with the a priori quadratic invariants the set of combined invariants I := { tr[C]2 , tr[CD]2 , tr[CD 2 ]2 , tr[C] tr[CD], tr[C] tr[CD 2 ], tr[CD] tr[CD 2 ], tr[C 2 ], tr[C 2 D], tr[C 2 D 2 ], tr[C], tr[CD], tr[CD 2 ] } . (4.107) 58 Representations of Anisotropic Tensor Functions This set serves as a basis for quadratic D2h -invariant polynomial functions. The twelve invariants in I are linear independent. A scalar-valued polynomial can be expressed in terms of these invariants, i.e. P (4.108) ψ = ψ(C, D) = ψ(I1 , ..., I12 ) = 12 i=1 αi Ii . Therefrom, we get the second derivative P ψ,CC = 9i=1 αi Ci with Ci := Ii,CC . (4.109) The second derivatives of the invariants are functions of the nine material parameters C1 = I2 (C, 1) C4 = I4 (C, 1, C, D) C7 = I3 (C, C, 1) 2 (4.110) C2 = I2 (C, D) C5 = I4 (C, 1, C, D ) C8 = I3 (C, C, D) C3 = I2 (C, D 2 ) C6 = I4 (C, D, C, D 2 ) C9 = I3 (C, C, D 2 ) . Aligning the anisotropy axis with the global coordinate axes, a = e1 and b = e2 , the well known coordinate representation for the second derivative is obtained ψ,CC c11 c12 c13 0 0 c c 0 0 22 23 c 0 0 33 := c44 0 sym. c55 0 0 0 0 0 c66 (4.111) for G = D2h . The coordinates of the tensor are obtained from the material parameters, i.e. 1 2 1 2 c11 = 2(α1 + α2 + α3 + α4 + α5 + α6 + α7 + α8 + α9 ) c66 = α7 + c22 = 2(α1 + α2 + α3 − α4 + α5 − α6 + α7 − α8 + α9 ) c12 = 2(α1 − α2 + α3 + α5 ) c44 = α7 + α9 c23 = 2 α1 − α4 + α5 . c33 = 2 α1 + 2 α7 c55 = α7 − 1 2 α8 + 1 2 α9 α8 + α9 c13 = 2 α1 + α4 + α5 (4.112) 4.6.4. Tetragonal Functions — Symmetry Group C4h 4.6.4.1. Isotropic Extension Function. The set of second-order tensor functions that allow for modeling of a tetragonal-dipyramidal C4h -invariant tensor function with representation theorems for isotropic functions is given by Ξ(C) : {A, B, N } with D4h := a ⊗ a ⊗ a ⊗ a + b ⊗ b ⊗ b ⊗ b A := D4h : C, B := D4h : C 2 N := ²n . (4.113) The vector n is of unit length and coincides with the four-fold axis that is characteristic for C4h symmetry. a and b are two orthonormal vectors perpendicular to n. The functional basis has to be set up for three symmetric second-order tensors C, A, and B combined 59 Representations of Anisotropic Tensor Functions with one skew-symmetric second-order tensor N . From table 8 we can read off the elements of a function basis. It consists of the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], tr[A], tr[A2 ], tr[A3 ], tr[B], tr[B 2 ], tr[B 3 ] , (4.114) the relative invariants of each two arguments tr[C 2 A], tr[CA2 ], tr[C 2 A2 ], tr[CB], tr[C 2 B], tr[CB 2 ], tr[C 2 B 2 ], tr[AB], tr[A2 B], tr[AB 2 ], tr[A2 B 2 ], tr[C 2 N 2 CN ], tr[A2 N 2 AN ], tr[B 2 N 2 ], tr[B 2 N 2 BN ] (4.115) and the relative invariants of each three argument tensors tr[CAB], tr[CAN ], tr[C 2 AN ], tr[CA2 N ], tr[CBN ], tr[C 2 BN ], (4.116) tr[CB 2 N ], tr[CN 2 BN ], tr[ABN ], tr[A2 BN ], tr[AB 2 N ], tr[AN 2 BN ] . Linearly dependencies among the invariants may occur as a consequence of the structure of the extension functions. In particular the identities tr[A] = − tr[CN 2 ] = − tr[AN 2 ] tr[B] = − tr[C 2 N 2 ] = − tr[BN 2 ] tr[A2 ] = tr[CA] = − tr[A2 N 2 ] tr[CAN ] = − tr[CN 2 AN ] (4.117) among the linear and quadratic invariants have been taken into account. 4.6.4.2. Functional Basis for Quadratic Functions. The linear invariants can be combined to quadratic terms which together with the a priori quadratic invariants enter the irreducible set I := { tr[C]2 , tr[A]2 , tr[C] tr[A], tr[C 2 ], tr[A2 ], tr[B], tr[CAN ], tr[C], tr[A] } (4.118) that constitutes a basis for a function of the type ψ = ψ(C, A, B, N ) = ψ(I1 , ..., I9 ) = P9 i=1 α i Ii . The constant second derivative of ψ is obtained as P ψ,CC = 7i=1 αi Ci with Ci := Ii,CC . (4.119) (4.120) The seven derivatives of the quadratic invariants in I can be expressed in terms of the four prototypes (4.79). We get C1 = I2 (C, 1) C3 = I4 (C, 1, A, 1) C5 = I3 (A, A, 1) C2 = I2 (A, 1) C4 = I3 (C, C, 1) C6 = I1 (B, 1) C7 = I3 (C, A, N ) . (4.121) In contrast to the up to now treated symmetries, the extension functions for the considered tetragonal symmetry are no more constant but depend on C. Therefore the derivatives of the arguments of the prototypes (4.80) do not vanish any more, i.e. Aijab = Aij,Cab = D4h ijab 4h 4h 4h Bijabcd = Bij,Cab Ccd = 21 (D4h ijac δbd + Dijad δbc + Dijbc δad + Dijbd δac ) . (4.122) 60 Representations of Anisotropic Tensor Functions For the choice a = e1 , b = e2 and n = e3 of the anisotropy axes, the second derivative can be given in the matrix form ψ,CC c11 c12 c13 c14 0 c11 c13 −c14 0 c33 0 0 := c 0 44 sym. c55 0 0 0 0 0 c55 for G = C4h . (4.123) Finally we specify the dependence of the matrix coordinates on the polynomial coefficients c11 = 2(α1 + α2 + α3 + α4 + α5 + α6 ) c33 = 2 α1 + 2 α4 c44 = α4 + α6 c55 = α4 + 1 2 c12 = 2(α1 + α2 + α3 ) α6 c13 = 2 α1 + α3 c14 = 1 2 (4.124) α7 . 4.6.5. Tetragonal Functions — Symmetry Group D4h 4.6.5.1. Isotropic Extension Functions. For the second tetragonal anisotropy type the tetragonal-dipyramidal crystal class D4h has to be investigated. The appropriate extension functions are obtained from (4.113) by replacing the skew-symmetric tensor N = ²n with M := n ⊗ n, i.e. Ξ(C) : {A, B, M } with D4h := a ⊗ a ⊗ a ⊗ a + b ⊗ b ⊗ b ⊗ b A := D4h : C, B := D4h : C 2 M := n ⊗ n . (4.125) Here n is aligned to the principal four-fold axis, a and b coincide with the two-fold axes. All three vectors are of unit length. The functional basis has to be set up for four symmetric second-order tensors C, A, B and M . Exploitation of table 8 gives a basis consisting of the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], tr[A], tr[A2 ], tr[A3 ], tr[B], tr[B 2 ], tr[B 3 ], (4.126) the relative invariants of each two tensors tr[C 2 A], tr[CA2 ], tr[C 2 A2 ], tr[CB], tr[C 2 B], tr[CB 2 ], tr[C 2 B 2 ], tr[AB], tr[A2 B], tr[AB 2 ], tr[A2 B 2 ], tr[B 2 M ], (4.127) and the invariants of each three of the tensors tr[CAB], tr[CBM ], tr[ABM ] . (4.128) In the above specified functional basis some linearly dependent invariants have been dropped. Besides the identity M = M 2 the following relations between linear and quadratic invariants have been used tr[AM ] = tr[A2 M ] = tr[BM ] = tr[ACM ] = 0 tr[A ] = tr[CA], tr[CM ] = tr[C] − tr[A], tr[C 1 M ] = tr[C 2 ] − tr[B] 2 to get a small number of elements. (4.129) 61 Representations of Anisotropic Tensor Functions 4.6.5.2. Application to Quadratic Functions. Construction of all possible combinations of quadratic invariants then leads to a set of monomials suitable to set up quadratic polynomial functions. The result is the set of eight invariants I := {tr[C]2 , tr[A]2 , tr[C] tr[A], tr[C 2 ], tr[A2 ], tr[B], tr[C], tr[A]} that can be used to construct a D4h -invariant function according to P ψ = ψ(C, A, B, M ) = ψ(I1 , ..., I8 ) = ψ = 8i=1 αi Ii . Two-fold derivation gives the constant fourth-order tensor P ψ,CC = 6i=1 αi Ci with Ci := ∂CC Ii . (4.130) (4.131) (4.132) Here the second derivatives of the monomials are determined by the relations C1 = I2 (C, 1) C3 = I4 (C, 1, A, 1) C5 = I3 (A, A, 1) C2 = I2 (A, 1) C4 = I3 (C, C, 1) C6 = I1 (B, 1) . (4.133) The derivatives of the structural functions that enter the expressions (4.79) via (4.80) are Aijab = Aij,Cab = D4h ijab , 4h 4h 4h Bijabcd = Bij,Cab Ccd = 12 (D4h ijac δbd + Dijad δbc + Dijbc δad + Dijbd δac ) . (4.134) The classical matrix representation for the second derivative is obtained by setting the anisotropy axes according to a = e1 , b = e2 and n = e3 , so that ψCC c11 c12 c13 0 0 c11 c13 0 0 c 0 0 33 = c44 0 sym. c55 0 0 0 0 0 c55 for G = D4h . (4.135) The coordinates are obtained from the coefficients by the relations 1 2 c11 = 2(α1 + α2 + α3 + α4 + α5 + α6 ) c55 = α4 + c33 = 2(α1 + α4 ) c12 = 2(α1 + α2 + α3 ) c44 = α4 + α6 c13 = 2(α1 + α3 ) . α6 (4.136) 4.6.6. Trigonal Functions — Symmetry Group S6 4.6.6.1. Isotropic Extension Functions. Rhombohedral functions possessing the symmetry group S6 can be described with isotropic tensor functions by using Ξ(C) : {W , V , N } with P3 D3d := i=1 ²ai ⊗ ai ⊗ ai W := D3d : C, V := D3d : C 2 N := ²n (4.137) 62 Representations of Anisotropic Tensor Functions as isotropic extension functions. n is a unit vector indicating the principal axis of S 6 . The three unit vectors ai are all orthogonal to n and inclined at 120◦ to each other. With the above specified isotropic extension functions a scalar-valued tensor function that is S6 -invariant can be formulated in terms of a functional basis for one symmetric secondorder tensor C and three skew-symmetric second-order tensors W , V and N . The basis obtained from table 8 contains the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], tr[W 2 ], tr[V 2 ] , (4.138) the relative invariants of each two tensors tr[CW 2 ], tr[C 2 W 2 ], tr[C 2 W 2 CW ], tr[CV 2 ], tr[C 2 V 2 ], tr[C 2 V 2 CV ], tr[CN 2 ], tr[C 2 N 2 ], tr[C 2 N 2 CN ], tr[W V ] (4.139) and the relative invariants of each three tensors tr[CW V ], tr[CW 2 V ], tr[CW V 2 ], tr[CW N ], tr[CW 2 N ], tr[CW N 2 ], tr[CV N ], tr[CV 2 N ], tr[CV N 2 ], tr[W V N ] . (4.140) Here the relations tr[W N ] = tr[V N ] = 0 are already taken into consideration. 4.6.6.2. Application to Quadratic Functions. Combining the two linear invariants (4.138)1 and (4.139)7 gives together with the quadratic invariants (4.138)2,4 , (4.139)8,12 a redundant set of seven quadratic invariants. One of these invariants can be eliminated using the relation tr[C 2 N 2 ] = 1 2 (tr[C]2 + tr[C 2 ]) + 1 4 tr[CN 2 ]2 + tr[C] tr[CN 2 ] + 92 tr[W 2 ] . Thus a functional basis for quadratic scalar-valued tensor functions consisting of six quadratic and two linear invariants that are assembled in the set I := {tr[C]2 , tr[CN 2 ]2 , tr[C] tr[CN 2 ], tr[W 2 ], tr[C 2 ], tr[CW N ], tr[CW N 2 ], tr[C], tr[CN 2 ]} . (4.141) Thus a possible complete representation for a quadratic polynomial function reads P ψ = ψ(C, W , V , N ) = ψ(I1 , ..., I9 ) = 10 i=1 αi Ii . Deriving this function two times with respect to C yields the constant fourth-order tensor P ψ,CC = 7i=1 αi Ci with Ci := ∂CC Ii . (4.142) The derivatives of the invariants can be specified in terms of the prototypes (4.79), i.e. C1 = I2 (C, 1) C3 = I4 (C, 1, A, N 2 ) C5 = I3 (W , W , 1) C2 = I2 (C, N 2 ) C4 = I3 (C, C, 1) C6 = I3 (C, W , N ) C7 = I3 (C, W , N 2 ) . (4.143) Their evaluation requires the derivative Wijab = Wij,Cab = 1 2 D3d ijab + 1 2 D3d ijba . (4.144) 63 Representations of Anisotropic Tensor Functions For an orientation of the anisotropy axes according to a1 = e1 , a2 = − 12 e1 + the classical matrix form of c11 c12 c11 ψ,CC = sym. √ 3 e, 2 2 a3 = − 12 e1 , − √ 3 e 2 2 and n = e3 the second derivative is obtained as c13 0 c15 c16 c13 0 −c15 −c16 c33 0 0 0 for G = S6 . 1 (c11 − c12 ) −c16 c15 2 c55 0 c55 (4.145) (4.146) Thereby the coordinates are determined uniquely by functions of the parameters α1 , . . . α7 c11 = 2(α1 + α2 − α3 + α4 ) − 9 4 α5 c55 = α4 c33 = 2 α1 + 2 α4 c12 = 2(α1 + α2 − α3 ) + 9 4 α5 c13 = 2 α1 − α3 c15 = c16 = 3 8 3 8 α7 (4.147) α6 . 4.6.7. Trigonal Functions — Symmetry Group D3d 4.6.7.1. Isotropic Extension Functions. From the second subset of the trigonal crystal system th hexagonal-scalenohedral class with symmetry group D3d will be investigated. For this class the set P3 D3d := i=1 ²ai ⊗ ai ⊗ ai Ξ(C) : {W , V , M } with (4.148) W := D3d : C, V := D3d : C 2 M := n ⊗ n acts as a isotropic extension. It differs only in the constant structural tensor from the extension (4.137). The unit vector n characterizes the direction of the principal axis of D3h and the three vectors ai coincide with the three two-fold axes. With the above specified extension, a functional basis for D3h -invariant scalar-valued functions consists of the invariants of two symmetric tensors C and M as well as the two skew-symmetric tensors W and V specified in (4.148). A possible functional basis is obtained with the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], tr[W 2 ], tr[V 2 ], (4.149) the relative invariants of each two tensor arguments tr[CM ], tr[C 2 M ], tr[CW 2 ], tr[C 2 W 2 ], tr[C 2 W 2 CW ], tr[CV 2 ], tr[C 2 V 2 ], tr[C 2 V 2 CV ], tr[M W 2 ], tr[M 2 W 2 M W ], tr[M V 2 ], tr[M 2 V 2 M V ], tr[W V ] and the relative invariants of each three tensors tr[CM W ], tr[C 2 M W ], tr[CW 2 M W ], tr[CM V ], tr[C 2 M V ], tr[CV 2 M V ], tr[CW V ], tr[CW 2 V ], tr[CW V 2 ], tr[M W V ], tr[M W 2 V ], tr[M W V 2 ] . To obtain a small number of invariants, the functional dependencies tr[M W 2 ] = 1 2 tr[CM ], have already been taken into account. and M = M 2 (4.150) (4.151) 64 Representations of Anisotropic Tensor Functions 4.6.7.2. Application to Quadratic Functions. Combining the linear terms of the above given invariants to quadratic terms in C gives together with the a priori quadratic invariants a set of nine monomials that serve to construct quadratic polynomial functions. The set obtained in this way can be reduced because of the linear dependence tr[C 2 M ] = 1 4 (tr[CM ]2 − tr[C]2 ) + 1 2 (tr[C] tr[CM ] + tr[C 2 ]) + 92 tr[W 2 ] . (4.152) Thus a quadratic D3d -invariant scalar-valued tensor function is obtained with the set I := {tr[C]2 , tr[CM ]2 , tr[C] tr[CM ], tr[C 2 ], tr[W 2 ], tr[CM W ], tr[C], tr[CM ]} . (4.153) and has the representation ψ = ψ(C, W , V , M ) = ψ(I1 , ..., I8 ) = P8 i=1 α i Ii . (4.154) The second derivative of ψ with respect to C then gives a constant fourth-order tensor. Its coordinate free representation reads ψ,CC = P6 i=1 αi C i with Ci := ∂CC Ii (4.155) where the second derivatives of the invariants are C1 = I2 (C, 1) C3 = I4 (C, 1, C, M ) C5 = I3 (W , W , 1) C2 = I2 (C, M ) C4 = I3 (C, C, 1) C6 = I3 (C, W , M ) . (4.156) This representation requires the derivative of the extension function, i.e. Wijab = 1 2 D3d ijab + 1 2 D3d ijba . (4.157) √ For an orientation of the anisotropy axes according to a1 = e1 , a2 = − 21 e1 + 23 e2 , √ a3 = − 12 e1 , − 23 e2 and n = e3 the classical matrix form of the second derivative is obtained as ψ,CC c11 c12 c13 c11 c13 c33 := sym. 0 c15 0 0 −c15 0 0 0 0 1 (c11 − c12 ) 0 c15 2 c55 0 c55 for G = D3d . (4.158) The six distinct coordinates are functions of the six parameters c11 = 2(α1 + α4 ) − 49 α5 c33 = 2(α1 + α2 + α3 + α4 ) c12 = 2 α1 + 94 α5 c55 = α4 c13 = 2 α1 + α3 c15 = 38 α6 . (4.159) 65 Representations of Anisotropic Tensor Functions 4.6.8. Hexagonal Functions — Symmetry Group C6h 4.6.8.1. Isotropic Extension Functions. A hexagonal-dipyramidal function with symmetry group C6h can be constructed with the isotropic extension functions Ξ(C) : {A, B, N } with D 6h A B N P3 := i=1 ai ⊗ ai ⊗ ai ⊗ ai ⊗ ai ⊗ ai := (D6h : C) : C := (D6h : C 2 ) : C 2 := ²n . (4.160) Here n denotes a vector of unit length aligned with the principal six-fold direction of C 6h . The three vectors ai all lie in one plane characterized by the normal n. They are inclined at 120◦ . A functional basis for a C6h -invariant scalar-valued tensor function is obtained from table 8 for the three symmetric tensors C, A, B and one skew-symmetric tensor N defined above. The basis contains the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], tr[A], tr[A2 ], tr[A3 ], tr[B], tr[B 2 ], tr[B 3 ], (4.161) the relative invariants of each two tensors tr[CA], tr[C 2 A], tr[CA2 ], tr[C 2 A2 ], tr[CB], tr[C 2 B], tr[CB 2 ], tr[C 2 B 2 ], tr[AB], tr[A2 B], tr[AB 2 ], tr[A2 B 2 ], tr[CN 2 ], tr[C 2 N 2 ], tr[C 2 N 2 CN ], tr[A2 N 2 ], tr[A2 N 2 AN ], tr[BN 2 ], tr[B 2 N 2 ], tr[B 2 N 2 BN ] (4.162) and the relative invariants of each three tensor arguments tr[CAB], tr[CAN ], tr[C 2 AN ], tr[CA2 N ], tr[CN 2 AN ], tr[CBN ], tr[C 2 BN ], tr[CB 2 N ], tr[CN 2 BN ], tr[ABN ], tr[A2 BN ], tr[AB 2 N ], tr[AN 2 BN ] . (4.163) This representation accounts for the dependence tr[A] = − tr[AN 2 ] among the quadratic invariants. Possible linear dependencies of invariants of higher order in C have not been checked. 4.6.8.2. Application to Quadratic Functions. A combination of the invariants of the above introduced basis to a set that serves for the formulation of quadratic tensor functions gives the set I := {tr[C]2 , tr[CN 2 ]2 , tr[C] tr[CN 2 ], tr[C 2 ], tr[A], tr[C], tr[CN 2 ]} . (4.164) Thereby one linear dependency was considered when setting up (4.164), i.e. tr[C 2 N 2 ], = 1 2 tr[C]2 + 43 tr[CN 2 ]2 + tr[C] tr[CN 2 ] − 1 2 tr[C 2 ] − 2 3 tr[A] (4.165) and the function ψ may be expressed in the form ψ = ψ(C, A, B, N ) = ψ(I1 , ..., I7 ) = P7 i=1 α i Ii . Deriving this function two times with respect to C the constant tensor P ψ,CC = 5i=1 αi Ci with Ci := ∂CC Ii (4.166) (4.167) 66 Representations of Anisotropic Tensor Functions is obtained in terms of the derivatives of the invariants of I C3 = I4 (C, 1, C, N 2 ) C1 = I2 (C, 1) 2 C2 = I2 (C, N ) C5 = I1 (A, 1) C4 = I3 (C, C, 1) . (4.168) This representation requires the derivative Aijabcd = Aij,Cab Ccd = 2D6h ijabcd . Setting a1 = e1 , a2 = − 12 e1 + coordinate form ψ,CC C11 C12 C13 C11 C13 C33 := sym. √ 3 e, 2 2 a3 = − 12 e1 , − 0 0 0 1 (C 11 − C12 ) 2 0 0 0 0 C55 0 0 0 0 0 C55 (4.169) √ 3 e 2 2 and n = e3 the classical for G = C6h (4.170) in matrix form is obtained. The coordinates are C11 = 2 α1 + 2 α2 − 2 α3 + 2 α4 + 49 α5 C33 = 2 α1 + 2 α4 C55 = α4 C12 = 2 α1 + 2 α2 − 2 α3 + 34 α5 C13 = 2 α1 − α3 . (4.171) in terms of the material parameters α1...5 . 4.6.9. Hexagonal Functions — Symmetry Group D6h 4.6.9.1. Isotropic Extension Functions. From the second anisotropy type with hexagonal symmetry we investigate the class with D6h symmetry. The set of structural secondorder tensor functions differs from that for the first hexagonal type only in the constant tensor, i.e. Ξ(C) : {A, B, M } with D 6h A B N P3 := i=1 ai ⊗ ai ⊗ ai ⊗ ai ⊗ ai ⊗ ai := (D6h : C) : C := (D6h : C 2 ) : C 2 := n ⊗ n . (4.172) Here n denotes a vector of unit length aligned with the principal six-fold direction of D6h . The three vectors ai all lie in one plane characterized by the normal n. They are inclined at 120◦ and coincide with the two-fold axes. A functional basis for a D6h -invariant scalar-valued tensor function is obtained from table 9 for the four symmetric tensors C, A, B and N that enter the function ψ as additional arguments defined above. The basis contains the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], tr[A], tr[A2 ], tr[A3 ], tr[B], tr[B 2 ], tr[B 3 ], (4.173) 67 Representations of Anisotropic Tensor Functions the relative invariants of each two of the tensors tr[CA], tr[C 2 A], tr[CA2 ], tr[C 2 A2 ], tr[CB], tr[C 2 B], tr[CB 2 ], tr[C 2 B 2 ], tr[CM ], tr[C 2 M ] tr[AB], tr[A2 B], tr[AB 2 ], tr[A2 B 2 ], tr[AM ], tr[A2 M ], tr[BM ], tr[B 2 M ], (4.174) and the relative invariants of each three of the arguments tr[CAB], tr[CBM ], tr[ACM ], tr[ABM ] . (4.175) In the above given listing the identities tr[AM ] = 0 and M = M 2 have already been considered. 4.6.9.2. Application to Quadratic Functions. Combination of the invariants to set up a quadratic function in C and accounting that tr[C 2 M ] = 1 4 tr[C]2 + 43 tr[CM ]2 − 1 2 tr[C] tr[CM ] + 1 2 tr[C 2 ] − 2 3 tr[A] leads to a set of invariants that permit the construction of a complete D6h -invariant scalar-valued tensor function, i.e. I := {tr[C]2 , tr[CM ]2 , tr[C] tr[CM ], tr[C 2 ], tr[A], tr[C], tr[CM ]} . (4.176) A polynomial is then of the form ψ = ψ(C, A, BM ) = ψ(I1 , ..., I7 ) = P7 i=1 α i Ii . The second derivative gives the constant fourth-order tensor P ψ,CC = 5i=1 αi Ci with Ci := ∂CC Ii (4.177) (4.178) based on the second derivatives of the invariants C1 = I2 (C, 1) C3 = I4 (C, 1, C, M ) C2 = I2 (C, M ) C4 = I3 (C, C, 1) C5 = I1 (A, 1) . (4.179) The sixth-order tensor in (4.80)3 runs Aijabcd = Aij,Cab Ccd = 2D6h ijabcd . Setting a1 = e1 , a2 = − 12 e1 + coordinate form c11 c12 c13 c11 c13 c33 ψ,CC := sym. √ 3 e, 2 2 a3 = − 12 e1 , − 0 0 0 1 (c11 − c12 ) 2 0 0 0 0 c55 0 0 0 0 0 c55 (4.180) √ 3 e 2 2 and n = e3 the classical for G = D6h (4.181) in matrix notation. The coordinates are determined by the functions c11 = 2 α1 + 2 α4 + 94 α5 c12 = 2 α1 + 34 α5 c33 = 2 α1 + 2 α2 + 2 α3 + 2 α4 c13 = 2 α1 + α3 c55 = α4 in terms of the five parameters α1...5 . (4.182) 68 Representations of Anisotropic Tensor Functions 4.6.10. Cubic Functions — Symmetry Group Oh 4.6.10.1. Isotropic Extension Functions. A hexoctahedral functions with symmetry group Oh can be obtained by application of representation theorems for isotropic function with the help of the set of two structural functions Ξ(C) : {A, B} with ½ P Oh := 3i=1 ai ⊗ ai ⊗ ai ⊗ ai A := Oh : C, B := Oh : C 2 . (4.183) The three unit vectors ai are orthogonal to each other. Each vector coincides with a fourfold symmetry axis. For the extension functions (4.183) a functional basis is obtained from table 8 for three symmetric second-order tensors. We get the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], tr[A3 ], tr[B 2 ], tr[B 3 ], (4.184) the relative invariants of each two arguments tr[CA], tr[C 2 A], tr[CA2 ], tr[C 2 A2 ], tr[CB], tr[C 2 B], tr[CB 2 ], tr[C 2 B 2 ], tr[AB], tr[A2 B], tr[AB 2 ], tr[A2 B 2 ] (4.185) and one relative invariant of all three arguments tr[CAB] . (4.186) When setting up this list, the following identities have been taken into account tr[C] = tr[A] , tr[C 2 ] = tr[B] , tr[A2 ] = tr[CA] . 4.6.10.2. Application to Quadratic Functions. Taking the linear and quadratic invariants, the set of combined invariants I := {tr[C]2 , tr[C 2 ], tr[A2 ], tr[C]} (4.187) is obtained. The four elements are linearly independent and constitute a functional basis that allows the construction of a complete quadratic scalar-valued Oh -invariant tensor polynomial. In order to derive a representation of the constant second derivative of the function ψ we consider the polynomial P ψ = ψ(C, A, B) = ψ(I1 , ..., I4 ) = ψ = 4i=1 αi Ii . (4.188) This form leads to the derivative P ψ,CC = 3i=1 αi Ci with Ci := ∂CC Ii . (4.189) The second derivatives of the quadratic invariants are C1 = I2 (C, 1) C2 = I3 (C, C, 1) C3 = I3 (A, A, 1) (4.190) based on the derivatives of the structural functions Aijab = Aij,Cab = Ohijab , Bijabcd = Bij,Cab Ccd = 21 (Ohijac δbd + Ohijad δbc + Ohijbc δad + Ohijbd δac ) . (4.191) 69 Representations of Anisotropic Tensor Functions The well known coordinate representation is obtained for specific orientations of the anisotropy axes. Setting ai = ei one gets the well-known representation ψ,CC c11 c12 c12 0 0 c c 0 0 11 12 c11 0 0 := c44 0 sym. c44 0 0 0 0 0 c44 for G = Oh (4.192) in matrix notation. The coordinates of the tensor are c11 = 2 α1 + 2 α2 + 2 α3 , c12 = 2 α1 and c44 = α2 (4.193) in terms of three constant parameters α1...3 . 4.6.11. Cubic Functions — Symmetry Group Th 4.6.11.1. Isotropic Extension Functions. The second type of cubic anisotropy consists of the two classes T and Th . We focus here on the latter one because it contains the central inversion as a symmetry operation. The set Ξ(C) = {Th : C} with Th := Tah + Tsh Ts := P h (i,j)∈Q ai ⊗ ai ⊗ aj ⊗ aj − aj ⊗ aj ⊗ ai ⊗ ai P a Th := (i,j,k)∈P ²ai ⊗ (aj ⊗ ak + ak ⊗ aj ) (4.194) constitutes extension functions for the considered symmetry class. The set P in the definition of the tensor Tah consists of the even permutations, the indices of the set Q appearing in the definition of Tsh take the values given below, i.e. Q := {(12), (23)(31)} and P := {(1, 2, 3), (2, 3, 1), (3, 1, 2)} . The three unit vectors ai are co-linear to the two-fold axes. A functional basis for a Th -invariant scalar-valued tensor function is a linear combination of the functional basis for a symmetric tensor C and a skew symmetric tensor W := Tah : C and the functional basis for two symmetric tensors C and A := Tsh : C. Thus we obtain from table 8 the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ], tr[A2 ], tr[A3 ], tr[W 2 ], (4.195) and the relative invariants of each two tensors tr[C 2 A], tr[CA2 ], tr[C 2 A2 ], tr[CW 2 ], tr[C 2 W 2 ], tr[C 2 W 2 CW ] . Here the identities tr[A] = 0 and tr[CA] = 0 have been considered. (4.196) 70 Representations of Anisotropic Tensor Functions 4.6.11.2. Application to Quadratic Functions. Based on the above specified invariants that constitute a function basis, the following set of combined invariants is obtained I := {tr[C]2 , tr[C 2 ], tr[A2 ], tr[C]} . (4.197) Thereby the invariant tr[W 2 ] = 34 (tr[C]2 + tr[A2 ]) − 4 tr[C 2 ] was dropped out. The set I permits the construction of an Th -invariant function of the form P ψ = ψ(C, A, W ) = ψ(I1 , ..., I4 ) = 4i=1 αi Ii . (4.198) From that polynomial form we obtain the second derivative P ψ,CC = 3i=1 αi Ci with Ci := ∂CC Ii (4.199) based on the second derivative of the invariants C1 = I2 (C, 1) C2 = I3 (C, C, 1) C3 = I3 (A, A, 1) (4.200) and the derivative of the symmetric part of the extension function Aijab = Aij,Cab = Ths ijab . (4.201) In the case where the two-fold axes coincide with the global coordinate axes, i.e. a i = ei , the coordinate form ψ,C C c11 c12 c12 0 0 c11 c12 0 0 c 0 0 11 := c44 0 sym. c44 0 0 0 0 0 c44 for G = Th (4.202) is obtained. The coordinates are given as functions of three material constants, i.e. c11 = 2α1 + 2α2 + 4α3 c12 = 2 α1 − 2 α3 and c44 = α2 . (4.203) 4.6.12. Transversely Isotropic Functions — Symmetry Group C∞h 4.6.12.1. Isotropic Extension Functions. For the transversely isotropic symmetry class with symmetry group C∞h the set Ξ : {N } with N := ²n (4.204) can be used to construct a C∞h -invariant scalar-valued tensor function. The unit vector n is aligned to the principal anisotropy axes. A complete functional basis consists of the invariants of the symmetric second-order tensor C and the skew symmetric tensor N . Thus from table 8 we get the fundamental invariants of C tr[C], tr[C 2 ], tr[C 3 ] (4.205) and the relative invariants tr[CN 2 ], tr[C 2 N 2 ], tr[C 2 N 2 CN ] . (4.206) 71 Representations of Anisotropic Tensor Functions 4.6.12.2. Application to Quadratic Functions. A combination of the invariants listed above to quadratic terms yields together with the linear and a priori quadratic invariants the functional basis I := {tr[C]2 , tr[CN 2 ]2 , tr[C] tr[CN 2 ], tr[C 2 ], tr[C 2 N 2 ], tr[C], tr[CN ]} . (4.207) The second derivative of the polynomial gives the constant fourth-order tensor P ψ,CC = 5i=1 αi Ci P7 i=1 αi Ii (4.208) with Ci := ∂CC Ii (4.209) ψ = ψ(C, N ) = ψ(I1 , ..., I7 ) = in terms of the second derivatives of the quadratic invariants of the basis I C1 = I2 (C, 1) C3 = I4 (C, 1, C, N 2 ) C2 = I2 (C, N 2 ) C4 = I3 (C, C, 1) C5 = I3 (C, C, N 2 ) . (4.210) For n = e3 the well-known coordinate form in matrix notation is obtained, i.e. ψ,CC c11 c12 c13 c11 c13 c33 := sym. 0 0 0 1 (c 11 − c12 ) 2 0 0 0 0 c55 0 0 0 0 0 c55 for G = C∞h . (4.211) The coordinates are c11 = 2(α1 + α2 − α3 + α4 − α5 ) c33 = 2 α1 + 2 α4 c55 = α4 − 1 2 α5 c12 = 2(α1 + α2 − α3 ) c13 = 2 α1 − α3 (4.212) in terms of the coefficients of the polynomial. 4.6.13. Transversely Isotropic Functions — Symmetry Group D∞h 4.6.13.1. Isotropic Extension Functions. From the second type of transversely isotropic symmetry classes the one with material symmetry group D∞h is investigated. The set Ξ : {M } with M := n ⊗ n (4.213) is an isotropic extension. n is a unit vector that coincides with the principal direction of D∞h . Once the isotropic extension functions are known, any transversely isotropic scalar-valued tensor function with symmetry group D∞h can be represented in terms of the elements of a functional basis for two symmetric second-order tensors C and M . 72 Representations of Anisotropic Tensor Functions Taking into account the property M = M 2 of the structural tensor, a possible functional basis consists of the fundamental invariants tr[C], tr[C 2 ], tr[C 3 ] (4.214) and the two relative invariants of both argument tensors tr[CM ] and tr[C 2 M ] . (4.215) 4.6.13.2. Application to Quadratic Functions. The above given elements are combined such that the order in C of the combined terms is at most two. The thus obtained set of monomials reads I := {tr[C]2 , tr[CM ]2 , tr[C] tr[CM ], tr[C 2 ], tr[C 2 M ], tr[C], tr[CM ] } . (4.216) The basis I consists of seven invariants that may enter a transversely isotropic polynomial. A typical form is ψ = ψ(C, M ) = ψ(I1 , ..., I7 ) = P7 i=1 α i Ii . (4.217) From that representation, the second derivative is obtained in terms of the first five coefficients ψ,CC = P5 i=1 αi C i with Ci := ∂CC Ii . (4.218) The derivatives are specified in terms of the prototypes (4.79) by C1 = I2 (C, 1) C3 = I4 (C, 1, C, M ) C2 = I2 (C, M ) C4 = I3 (C, C, 1) C5 = I3 (C, C, M ) . (4.219) Setting n = e3 , a coordinate representation is obtained. In matrix notation it is ψ,CC c11 c12 c13 c11 c13 c33 := sym. 0 0 0 1 (c 11 − c12 ) 2 0 0 0 0 c55 0 0 0 0 0 c55 for G = D∞h . (4.220) For the coordinates the following relations hold c11 = 2 α1 + 2 α4 c12 = 2 α1 c33 = 2(α1 + α2 + α3 + α4 + α5 ) c13 = 2 α1 + α3 . c55 = α4 + 1 2 α5 (4.221) Representations of Anisotropic Tensor Functions 73 4.6.14. Conclusion The investigations of the representations for quadratic functions were carried out for thirteen of the fourteen mechanics symmetry groups. Comparing the structure of the coordinate-forms of the second derivatives, those for C6h , D6h , C∞h and D∞h as well as those for Th and Oh symmetry are identical. Consequently, the fourth-order tensors of these groups obey the same symmetries, respectively. All in all only ten different types appear. This is a consequence of the restriction to quadratic functions which results in the loss of information. Zheng & Boehler [163] discuss this observation. They introduce the notion of physical symmetry for the symmetries that are covered by certain classes of constitutive functions. Anisotropic Elasticity 75 5. Anisotropic Elasticity In continuum mechanics isotropic tensor functions have a huge spectrum of applications. A representative example is the description of the elastic material behaviour of crystals. The elastic strains are typically small so that a quadratic potential function as discussed in chapter 4.6 can be used to describe the stored free energy in a material point. In this case the second derivatives are the elastic moduli. Clearly, the elastic anisotropies of crystals comprise the 32 crystal classes. Measured material constants for some crystals can be found in Sutcliffe [135]. Further examples are multi-field problems as for instance thermo-elasticity or electro-elasticity. The symmetries belonging to the non-crystal classes come typically up when considering quasi crystals and non-crystalline materials. Engineering materials like composites or bio-materials like soft tissues which have a fibrous structure are typical representatives. Homogenized material behaviour or overall behaviour observed on a macro-scale is also often characterized by the symmetries of the non-crystal classes. A poly crystal for example is made up of grains of mono-crystals. For randomly orientated grains the elastic overall behaviour is isotropic. Special distributions of the orientations of the grains can result from forming processes and generally lead to other symmetries of the elastic material response. The well-known textbooks of Spencer [133] and Boehler [22] as well as the article of Spencer [132] provide an introduction into the constitutive theory for anisotropic materials. Some articles with a strong focus on elasticity are listed below. The constitutive description of fiber-reinforced composites in the small strain context for plane problems is part of Zheng, Betten & Spencer [161] and Zheng & Betten [159]. BischoffBeiermann & Bruhns [19] derive an alternative representation for transverse isotropy at small strains. Thereby the introduced invariants are physically interpretable. Applications to elasticity and visco-elasticity at small and finite strains are given in Kaliske [58] and Holzapfel & Gasser [52]. A micro-mechanically based approach to hyperelasticity is provided by Bischoff, Arruda & Grosh [17]. They derive constitutive equations for polymers and polymer-like materials showing a macroscopic orthotropic behaviour. A transversely isotropic model for biological soft tissues and its finite element implementation is discussed by Weiss, Maker & Govindjee [150]. Menzel & Steinmann [76] compare different approaches to the description of orthotropic elasticity at finite strains. To guarantee the existence of minimizing solutions in the context of a variational formulation of elasticity, the stress potential function must be quasiconvex. The condition of quasiconvexity is rather hard to handle. Alternatively, the stronger condition of polyconvexity can be used. This is done by Schröder & Neff [118]. They derive polyconvex functions for transversal isotropic materials and propose an extension to account for orthotropic symmetries. A slightly weaker restriction than quasiconvexity is that of rank-1-convexity. Both restrictions are very close to each other and from the viewpoint of engineering applications, the latter is considered to be sufficient. Bruhns, Xiao and Meyers [27] investigate strain energy functions in terms of logarithmic strains and derive ranges where the functions are infinitesimally rank-1-convex. 5.1. General Framework The main characteristic of elastic materials is that the applied work in a loading process of the material is stored without any loss and is gained back when unloading. The stress 76 Anisotropic Elasticity states are in a one-to-one relation to the strain states. Loading and unloading takes place on the same path in the stress-strain space, hysteresis effects are not present, i.e. D = 0 in (2.63). In the ensuing parts of this chapter we concentrate on a Lagrangian description of elasticity. As discussed in section 2.4.1 functions formulated in terms of the right Cauchy-Green tensor C := F T gF are a priori objective. Let G denote a given material symmetry group representing the symmetries of the material under consideration. The set of G-invariant second-order tensor functions that allows the modeling of the free energy function ψ as an isotropic function is denoted by Ξ(C). Then a G-invariant function of C is obtained from an isotropic function of C and Ξ(C) at fixed structural functions. Thus ψ = ψ(C, Ξ(C)) = ψ(I1 , . . . , In ) (5.1) where I1,...,n are the invariants of the set {C, Ξ(C)} that constitute a functional basis. Within a Lagrangian framework, the second Piola-Kirchhoff stresses are obtained by the derivative of the free energy function with respect to C. Application of the chain rule yields the representation S := 2∂C ψ = 2 Pn i=1 ψ,Ii Ii,C . (5.2) The numerical treatment within a computer code requires the second derivative of the free energy function. Application of the chain rule gives o Pn n Pn 2 C := 4∂CC ψ = 4 i=1 . (5.3) j=1 ψ,Ii Ij Ii,C ⊗ Ij,C + ψ,Ii Ii,CC The split of the stresses and moduli in (5.2) and (5.3) into scalar derivatives with respect to the invariants and the derivatives of the invariants with respect to C leads to convenient implementations of constitutive models. It offers a high flexibility and allows a fast implementation of different constitutive functions, as long as the basis of invariants remains unchanged. This is because the derivatives Ii,C remain unchanged in such a scenario. 5.2. Model Problem 5.2.1. Fiber-Reinforced Technical Rubber A typical example for an anisotropic elastic material is a rubber that is reinforced with one family of fibers. The orientation of the fibers is described by the vector a. The isotropic extension function for a D∞h symmetry group is the constant structural tensor Ξ := {a ⊗ a} . (5.4) In order to account for the incompressibility of rubber-like materials, the free energy function is split into a volumetric part ψvol and an isochoric part ψiso . The volume map (2.30) is a measure for the volumetric deformation of a material point. This motivates the split of the deformation gradient into a spherical part J 1/3 1 and a unimodular part F̄ := J −(1/3) F ∈ SL(3). An a priori objective free energy function is then formulated in T terms of C̄ := F̄ g F̄ . For the isochoric elastic strain measure of the Seth-Hill family Ē = 1 2 (C̄ − G) (5.5) 77 Anisotropic Elasticity the free energy function is assumed to have the decoupled form ψ = ψvol (J) + ψiso (Ē) . (5.6) The volumetric part is a scalar-valued scalar function and can therefore not serve to describe anisotropic behaviour. The anisotropy is completely modeled by the isochoric part. A function basis can be obtained from table 9. It consists of the five invariants 2 3 2 I := {tr[Ē], tr[Ē ], tr[Ē ], tr[Ē(ā ⊗ a)], tr[Ē (a ⊗ a)]} . (5.7) We consider a model problem, where the isochoric part of the free energy function depends on two invariants only, i.e. P (5.8) ψiso = 3i=1 ci I1i + α1 I42 + α2 I1 I4 . The isotropic part is a Yeoh model. It is described by three material parameters c i . The anisotropic part is associated with the stiffness of the fibers. For the volumetric part we set ψvol = κ(J − ln[J] − 1). The stresses and moduli are obtained by differentiation with respect to C̄. Based on the first and second-order derivatives with respect to Ē T := ∂Ē ψ = ψP ,I1 I1 , Ē + Pψ,I4 I4 , Ē 2 E := ∂Ē Ē ψ = i=1,4 { j=1,4 ψ,Ii Ij Ii,Ē ⊗ Ij,Ē + ψ,Ii Ii,Ē Ē } (5.9) they are obtained by purely geometric mappings, i.e. 2 T S̄ = 2∂C̄ ψ = T : P and C̄ = 4∂C̄ C̄ ψ = P : E : P + T : L . (5.10) The mapping tensors P := 2Ē ,C̄ and L := 4Ē ,C̄ C̄ were introduced by Miehe & Lambrecht [88]. They specify efficient spectral-decomposition-based algorithms to perform mappings of the type (5.10) for all strain tensors of the Seth-Hill family. The transposition operator in (5.10) is associated with the first and last pair of indices. 5.3. Numerical Example: Tension Test of a Fiber-Reinforced Bar In a first example we demonstrate the effects of the orientation of the transversal structural tensor a ⊗ a. Consider a rectangular bar made of two layers of a fiber-reinforced incompressible rubber material. The fiber’s orientations in the upper layer are defined by the structural vector au , those in the lower layer by al . Figure 22 shows the specimen. u/2 PSfrag replacements au ϕu ϕl al u/2 Figure 22: Tension test of a rectangular bar. The bar of dimensions 10 × 5 × 1mm is deformation driven pulled up to a total elongation of u = 10mm. The bar consists of two layers of fiber-reinforced rubber material whereby the orientations of the fibers differ. 78 Anisotropic Elasticity Its dimensions are 10 × 5 × 1mm. The bar is stretched to its double size in a deformation controlled process. The material is assumed to be quasi-incompressible. The isochoric part of the free energy is assumed to be of the form (5.8). The three material parameters ci describe the isotropic response. Kaliske [58] has identified the following values for a filled rubber: c1 = 0.5894, c2 = −0.1203, c3 = 0.04484. Their units are N/mm2 . The anisotropic part is determined by the constants α1 = 1.0N/mm2 and α2 = 0.2N/mm2 . The bulk modulus is set to κ = 1000N/mm2 . The deformed shape of the bar strongly depends on the orientation of the structural tensors. We investigate two settings. In the first one the vectors are oriented at ϕl = +45◦ and ϕu = −45◦ . This leads to a torsion of the bar. The rotation is such that the orientations of the fibers align with the loading axis because the stiffness of the material rises to its highest value in direction of the fibers. See figure 23a for the deformed shapes. The plot in figure 24a quantifies the rotation of the cross section at the end of the bar. It shows the angle of rotations of the fictitious straight edges through the corner nodes of the cross section. If we set ϕl = 90◦ and replacements PSfrag replacements u u a1 a2 a1 a2 200 200 replacements PSfrag replacements u u a1 a2 a1 a2 200 200 replacements PSfrag replacements u u a1 a2 a1 a2 200 200 replacements PSfrag replacements u u a1 a2 a1 a2 200 a. 200 b. Figure 23: Tension test of a rectangular bar. Deformed configurations for (a) ϕl = +45◦ , ϕu = −45◦ and (b) ϕl = 90◦ , ϕu = 0◦ . The two lower corner points of the cross section of the back face are fixed in vertical direction. 79 Anisotropic Elasticity 0 140 1600 3200 4800 6400 angle [◦ ] −40 −60 −80 −100 Sfrag replacements −120 −140 2 a. 4 6 8 0 2 4 6 8 10 displacement u [mm] 300 −0.3 −0.4 −0.5 PSfrag replacements 0 2 4 6 8 displacement u [mm] 1600 3200 4800 6400 250 load [N/mm2 ] deflection [mm] 40 b. 1600 3200 4800 6400 −0.2 c. 60 0 displacement u [mm] −0.1 −0.6 80 10 0 Sfrag replacements 100 20 PSfrag replacements 0 1600 3200 4800 6400 120 load [N/mm2 ] −20 200 150 100 50 0 10 d. 0 2 4 6 8 10 displacement u [mm] Figure 24: Tension of a rectangular bar. Twisting bar: (a) Relative torsion of the fictitious edges obtained by straight interpolation of the corner nodes of the cross-sections at the ends: average rotations. The upper curves document the rotation of the long edges at the top and bottom surface, the lower curves belong to the short edges on the left and right side. (b) Load displacement curve. Bending bar: (c) Deflection of the mid-points of the cross-section. (d) Load displacement figure. ϕu = 0◦ the deformation mode is completely different. In the lower layer the contraction is blocked by the stiff fibers. Consequently the bar deforms to a groove as shown in figure 23b. The deflection in the mid-point of the cross sections is plotted in figure 24c. Figures 24b,d show the load displacement figures for both settings. The numerical computations were carried out with Q1P0 elements as described for example in Miehe [77]. The bar was discretized with 1600, 3200, 4800 and 6400 elements. The results of all simulations are close up to a total deformation of u ≈ 3.5mm. Then the deflections obtained with the different discretizations diverge. Small deviations are also observed for the angle of rotation of the bar. The load displacement figures are independent of the discretization. 5.4. Numerical Example: Inflation of a Fiber-Reinforced Sheet In a second example we simulate the inflation of an elastic fiber-reinforced membrane. The specimen is depicted in figure 25. The radius is 400mm and the thickness is 30mm. It is loaded on its lower surface with a pressure p = 10N/mm2 . The boundary of the upper surface is fixed in all three directions. The constitutive model is identical to the one used for the rectangular bar, the material parameters are chosen as follows: κ = 1000, c1 = 0.5894, c2 = −0.1203, c3 = 0.04484, α1 = 1.0 and α2 = 0.25, all in N/mm2 . To 80 Anisotropic Elasticity PSfrag replacements p Figure 25: Inflation of a fiber reinforced sheet. The circular sheet has a radius of 400mm. Its thickness is 30mm. The parallel lines symbolize the fibers. The boundary of the top face is completely fixed, the bottom face is loaded with a pressure p = 10N/mm 2 . perform the simulation the material model has been implemented into the non-linear finite element program Abaqus Standard via the “umat” user interface. The specimen is discretized with 474 C3D8I eight-node continuum elements arranged in a single layer. The element design bases on an extension of the incompatible mode method to the nonlinear regime as suggested by Simó & Armero [123]. The results of the simulation are documented in figure 26. While for isotropic material behaviour the deformed shape is a sphere, the fibers prevent the uniform extension of the specimen. As the anisotropic terms of the free energy function can be associated with contributions coming from fibers that are added to the isotropic matrix material, the extension along the fiber direction is expected to be less than in other directions. This effect can clearly be observed in the plots in figure 26. a. b. c. Figure 26: Inflation of a sheet. Different stages of inflation: (a) side view, (b) view from below and (c) perspective view. 81 Approaches to Anisotropic Plasticity 6. Approaches to Anisotropic Plasticity PSfrag replacements 6.1. Kinematic Approach in Terms of a Plastic Map 6.1.1. Geometry of Multiplicative Plasticity, Stress Tensors The key assumption of plastic-map plasticity is the multiplicative decomposition of the deformation gradient into an elastic map F e and a plastic map F p , i.e. F = F e F p . This split was first introduced by Lee [65]. Many models of crystal plasticity base on this ϕ F e C, S Fp g, τ F = ∇ϕ x X C̄, S̄ B Fe Fp S X B̄ Figure 27: Kinematic setting for multiplicative plasticity. A point X of the Lagrangian configuration B is mapped by the non-linear point map ϕ onto the point x of the Eulerian configuration S. Motivated by crystal plasticity, the tangent map F = ∇ϕ is multiplicatively decomposed into a plastic part F p and a part F e := F F p−1 accounting for the elastic deformation and the rigid body rotation of the slip systems. This split introduces a local incompatible intermediate configuration B̄ with convected metric C̄ and symmetric stresses S̄, the counterparts of C, S and g, τ of the reference and current configurations. kinematic ansatz. Therein F p describes the part of the deformation that results from plastic flow on crystallographic planes whereas F e describes the reversible distortion of the lattice plus the rigid body rotation of the lattice. The elastic map is defined by the composition F e := F F p−1 (6.1) of the deformation gradient and the inverse plastic map. F p is considered as an internal variable that develops according to a constitutive ansatz with initial condition F p (t = t0 ) = 1. The plastic map defines locally a stress-free state associated with a plastic intermediate configuration B̄, see figure 27 for an illustration. In this configuration the current metric C̄ := F eT gF e (6.2) is a function solely of the elastic part of the deformation. Hence it is a suitable measure for the stored free energy function. The symmetric stress tensor in the intermediate configuration is defined as the pull-back of the Kirchhoff stress tensor, i.e. S̄ := F e−1 τ F e−T . (6.3) Alternatively the quantities in (6.2) and (6.3) are obtained by a push-forward of the Lagrangian tensors C and S, respectively. By construction, the quantities C̄ and S̄ are 82 Approaches to Anisotropic Plasticity work conjugate. The transformation g P̄ := F e S̄ defines the first Piola-Kirchhoff stress tensor with respect to the intermediate configuration. It is work conjugate to the elastic map F e . Within the subsequent development, the spatial velocity gradient l := grad[ ẋ] = Ḟ F −1 PSfrag replacements that appears in the Clausius-Duhem inequality plays an important role. The pulled back quantity L̄ := F e−1 lF e to the intermediate configuration decomposes additively into an elastic and a plastic part, i.e. ½ e L̄ := F e−1 Ḟ e e p (6.4) L̄ = L̄ + L̄ with L̄p := Ḟ p F p−1 . These two parts represent the temporal change of the elastic and plastic maps separately with respect to the intermediate configuration. A visualization of the introduced tensors is given in figure 28. X x L̄ F TX B S Fe p TX B̄ S̄ P̄ p P P̄ F e−T F p−T TX? B TX? B̄ τ, σ Tx S Σ̄ Tx? S Figure 28: Multiplicative decomposition of the tangent map F into an elastic part F e and a plastic part F p in plastic-map plasticity. 6.2. Constitutive Model for Plastic-Map Plasticity 6.2.1. Energy Storage and Elastic Stress Response As pointed out in section 2.4.1, an a priori objective free energy function is obtained by assuming a functional dependence on the elastic deformation measure C̄ defined in (6.2) with respect to the intermediate configuration, i.e. ψ = ψ(C̄, q̄) . (6.5) Here ψ = ρ0 Ψ is the locally stored energy per unit reference volume and q̄ a generalized ˙ vector of internal variables. The latter is governed by a constitutive assumption for q̄, evolving from the initial condition q̄(X, t = t0 ) = 0. Any thermodynamically consistent constitutive model has to satisfy the second law of thermodynamics. The temporal evolution of the free energy is obtained straight forward with (6.1) as ψ̇ = 2[gF e ψ,C̄ F p−T ] : Ḟ − 2[C̄ψ,C̄ F p−T ] : Ḟ p + ψ,q̄ · q̄˙ . (6.6) From the dissipation inequality (2.63)2 one obtains the first Piola-Kirchhoff stresses by standard argumentation gP := ψ,F e F p−T = 2gF e ψ,C̄ F p−T . (6.7) 83 Approaches to Anisotropic Plasticity Furthermore the stress-like internal variables conjugate to the plastic deformation F p and the internal variable vector q̄ are ḠP p := F eT ψ,F e F p−T = 2C̄ψ,C̄ F p−T and Q̄ := −ψ,q̄ . (6.8) The actual dissipated energy is then given in terms of the Mandel tensor Σ̄ by ρ0 D = Σ̄ : L̄p + Q̄ · q̄˙ ≥ 0 with Σ̄ := F eT ψ,F e = 2C̄ψ,C̄ . (6.9) A visualization of these tensors as mappings in between tangent and co-tangent spaces is p given in figure 28. Both, the plastic evolution operator L̄ and the Mandel tensor Σ̄ are mixed variant objects. 6.2.2. Dissipation and Plastic Flow Response Consider a non-smooth convex elastic domain E that bounds the internal forces E := {(Σ̄, Q̄)|f α (Σ̄, Q̄) − cα ≤ 0 ; α = 1, . . . , m} . (6.10) in terms of m level-set functions f α describing a hyper-surface in the plastic stress space. The functions φα := f α − cα are denoted as yield criteria functions. The principle of maximum dissipation governs the evolution of the internal variables. This constrained optimization problem can be solved with a Lagrange multiplier method, i.e. P ˙ + α λα (f α (Σ̄, Q̄) − cα ) → stat. L(Σ̄, Q̄, λα ) = −(Σ̄ : L̄p + Q̄ · q̄) (6.11) The solution is given by the evolution equations P P p L̄ = α λα f,αΣ̄ and q̄˙ = α λα f,αQ̄ (6.12) that are governed by the loading conditions λα > 0, (f α − cα ) < 0 and λα (f α − cα ) = 0. Alternatively, the constrained optimization problem can be solved approximately by a penalty method. Then one has to solve the minimization problem p ˙ + P(Σ̄, Q̄) := −(Σ̄ : L̄ + Q̄ · q̄) 1 α (f − cα )+2 → min. 2η (6.13) With this approach stress states outside of the elastic domain are admissible but penalized. The superscript “+” has the meaning (•)+ := 12 {|(•)|+(•)} and η is a material parameter. The evolution equations resulting from (6.13) are identical to those from the Lagrangian multiplier method but the plastic parameters λα are identified as 1 λα := (f α − cα )+ η (6.14) and replace the loading conditions. In this case the yield criteria are replaced by viscous pseudo yield functions φαv := φα − ηλα that give the rate-independent functions in the limit for η → 0. (6.15) 84 Approaches to Anisotropic Plasticity The plastic flow rule (6.12)1 can be decomposed into a symmetric part and a skewsymmetric part. Multiplication of the flow operator from the left side with the convected metric gives a covariant tensor that can be decomposed according to p p C̄ L̄ = D̄ + W̄ p (6.16) in terms of the definitions p p p p D̄ := sym[C̄ L̄ ] and W̄ := skew[C̄ L̄ ] . (6.17) Making use of the definition of the Mandel tensor in (6.9), the derivative of the level-set function appearing in the flow rule can be expressed as f,αΣ̄ = C̄ −1 α f,S̄ (6.18) where S̄ := 2ψ,C̄ denotes the second Piola-Kirchhoff stress tensor in the intermediate configuration. Thus the symmetric part of the flow rule takes the form P p (6.19) D̄ = α λα f,αS̄ p and the plastic spin vanishes, W̄ = 0. 6.2.3. Decoupling of the Constitutive Functions For the subsequent developments, we consider a decoupling of the constitutive functions into macroscopic and microscopic parts. This is typically done when considering crystalline materials. The elastic response function is assumed to be of the form ψ = ψ e (C̄) + ψ i (A) (6.20) where the internal variable vector is degenerated to a single scalar value, q̄ = {A}, the socalled equivalent plastic strain. The macroscopic part ψ e accounts for the stored energy resulting from macroscopic lattice deformations. The microscopic or internal part ψ i describes contributions resulting from micro-stress fields as a consequence of dislocations or point defects in the crystal lattice. The plastic stresses are Σ̄ := 2C̄ψ,eC̄ i . and B := −ψ,A (6.21) These stresses are restricted to the elastic domain, which is defined by yield criterion functions. Here we assume the decoupled form of the level-set functions f α = f eα (Σ̄) + f iα (B) . (6.22) In the following we denote the gradients of the level-set functions as normals defined by α N̄ := f,eα Σ̄ iα and N α := f,B . (6.23) The evolutions of the plastic strains (6.12) then run P P α L̄p = α λα N̄ and Ȧ = α λα N α (6.24) F p ∈ SL(3) (6.25) together with the corresponding loading conditions. It is observed in experiments that the plastic deformation of metals is volume preserving. This property can be integrated into the constitutive model by restricting the plastic map to the special linear group, i.e. ⇔ det[F p ] = +1 . 85 Approaches to Anisotropic Plasticity 6.2.4. Continuous Tangent Moduli The considered model problem is formulated with respect to the intermediate configuration. Consequently, the tangent moduli have to relate the relative rates of the stress tensor and the deformation tensor in that configuration. These rates are given by the objective Lie-derivatives ) ˙ + L̄pT C̄ + C̄ L̄p = 2D̄ £vp [C̄] = C̄ . (6.26) £vp [S̄] = S̄˙ − L̄p S̄ − S̄ L̄pT Here D̄ is the rate of deformation tensor. The continuous tangent moduli C̄ep are then defined by £vp [S̄] = C̄ep : 1 2 £vp [C̄] = C̄ep : D̄ . (6.27) For a more compact representation we define the deformation dependent tensors α α α P̄ := sym[C̄ N̄ ] and Q̄ := sym[S̄ N̄ αT ] (6.28) in terms of the normal onto the α-th yield surface as defined in (6.23). Together with the flow operator (6.24) the first equation in (6.26) then takes the form ˙ = 2D̄ − 2P λ P̄ α C̄ α α (6.29) and with the elastic moduli C̄ = 2S̄ ,C̄ , equation (6.26)2 can be recast to P α α £vp [S̄] = C̄ : D̄ − α λα [C̄ : P̄ + 2Q̄ ] . (6.30) The plastic multipliers λα are obtained from the consistency conditions, stating that in the case of plastic loading where λα 6= 0 the stress state remains on the yield surface, i.e. P P β α β f˙α = 2f,eα : ( D̄ − λ ) − (6.31) P̄ β β β λβ N KN = 0 . C̄ i Herein the hardening module is abbreviated by K := ψ,AA . Inserting the identity α α f,eα C̄ = N̄ : Σ̄,C̄ = Q̄ + 1 2 α P̄ : C̄ (6.32) and solving for λα yields λα = with the matrix P β [gαβ ] α −1 β β (2Q̄ + P̄ : C̄) : D̄ α (6.33) β gαβ := (2Q̄ + P̄ : C̄) : P̄ + N α KN β . (6.34) Finally, insertion of this equation into (6.30) allows the identification of the in general unsymmetric continuous elastic-plastic tangent moduli as C̄ep := C̄ − P P α β [gαβ ] −1 α α β β (C̄ : P̄ + 2Q̄ ) ⊗ (2Q̄ + P̄ : C̄) relative to the intermediate configuration. (6.35) 86 Approaches to Anisotropic Plasticity 6.3. Algorithmic Formulation 6.3.1. Outline of the Standard Stress Update Algorithms We consider deformation controlled simulations where F n+1 is prescribed. Standard stress update algorithms integrate the system of evolution equations of the internal variables I in a discrete time interval [tn ; tn+1 ] while accounting for the restriction of the plastic forces F to the elastic domain. This constraint is typically formulated by yield criterion functions in terms of level-set functions of the time-continuous setting, f α (F , I) − cα − η γα = 0 ∀α ∈ A ∆t (6.36) that covers both, rate-dependent and rate-independent behaviour. γα := λα (tn+1 − tn ) are the algorithmic parameters. The set A consists of those numbers of the flow systems on which plastic loading takes place. In general this set is not known at the beginning of the time step and has therefore to be determined by an iterative procedure as discussed for example in Miehe & Schröder [91]. See also the literature therein. An approach circumventing the difficulties arising from the non-uniqueness of the activity of the slip systems is proposed by Schmidt-Baldassari [114]. For the moment this set is assumed to be known. The vectors of the internal variables and the internal forces are related by I := {F p , q̄} and F := {ḠP p , Q̄} to the quantities introduced at the beginning of this chapter. They are related by the constitutive equation F + ∂I ψ(C, I) = 0 . (6.37) The evolution of internal variables is specified by an integration algorithm which is assumed to be a function of the internal variables, the internal forces and the active algorithmic parameters and to have the form A(I, F , γα ) = 0 . (6.38) Assembling the variables that have to be updated in a vector p := [I, F , γα ]T (6.39) leads to the following compact format of the non-linear system of evolution equations F + ∂I ψ(C, I) =0. A(I, F , γα ) r(C, p) := (6.40) η α −f (F , I) + cα + ∆t γα For strain-driven deformation processes in the above system the deformation gradient F is fixed within the time step. The iterative solution based on a Newton scheme requires the linearization Lin[r] = r + k · ∆p in terms of the matrix ψ,II I 0 (6.41) k := r ,p = A,I A,F A,γβ . η α α −f,I −f,F ∆t δαβ The vector p introduced in (6.39) is updated according to p ⇐ p − k−1 r (6.42) 87 Approaches to Anisotropic Plasticity until convergence is obtained in the sense that krk ≤ tol. Symmetry of the tangent matrix k is obtained if the conditions A,I = I , α A,γα = −f,F α and f,I =0 (6.43) are satisfied. In single surface plasticity this is for example the case for standard implicit update algorithms of the form A := I − In − γ∂F f (F ) with level set functions which depend on the internal force only. Once the inelastic variables p := [I, F , γα ]T are determined by the above-outlined local Newton iteration,the stresses are obtained by the constitutive expression S = 2∂C ψ(C, I) (6.44) from the free energy function. The sensitivity of the stresses with respect to the total deformation ∆S = Calgo : 1 2 ∆C (6.45) is governed by the fourth-order consistent tangent moduli tensor Calgo , which appears in the form 2 2 Calgo = 4∂CC ψ + 4∂Cp ψ · p,C . (6.46) Here the sensitivity p,C of the inelastic variables with respect to the total deformation is obtained implicitly from the condition (6.40) yielding r ,C + k · p,C = 0 and therefore p,C = −k−1 · r ,C (6.47) in terms of the tangent matrix k evaluated at the solution point p of the local iteration. Insertion into (6.46) gives the representation 2 2 Calgo = 4∂CC ψ − 4∂Cp ψ · k−1 · r ,C (6.48) 2 of the tangent moduli. Insertion of the explicit representations for ∂CC ψ and r ,C gives the final form −1 ψ,II I 0 ψ,IC ψ,CI (6.49) Calgo = 4ψ,CC − 4 0 · A,I A,F A,γβ · 0 η α α δ f,I f,F 0 0 ∆t αβ of the tangent moduli, consisting of the elastic contribution and a softening term due to the accumulation of inelastic deformation in the time interval under consideration. Note that the tangent moduli are symmetric if the conditions (6.43) are satisfied. Observe furthermore that Calgo can be represented in the form Calgo = 4ψ,CC − 4ψ,CI · k̃ · ψ,IC where k̃ denotes the quadratic sub-matrix of dimension (len[I] × len[I]) of k −1 −1 ψ,II I 0 k̃ := A,I A,F A,γβ . η α α δ f,I f,F ∆t αβ (6.50) (6.51) 88 Approaches to Anisotropic Plasticity As for fast numerical computations within finite element codes, the general structure outlined here can be modified so that the numerical effort is reduced. The main drawback of the above-outlined algorithm is that the iteration is performed directly with the plastic internal variables. This leads to a high number of local iteration steps. Taking into account the structure of concrete algorithms allows the iteration with elastic quantities, which need less iteration steps to converge. This is done in the next sections. 6.3.2. Implicit Stress Update Algorithm (U1) The evolution of the plastic map F p is integrated by using an exponential map that automatically preserves the plastic incompressibility property (6.25). In the context of computational plasticity it was first introduced by Weber & Anand [149]. A backward Euler scheme is applied to the integration of the equivalent plastic strain. Thus the implicit integration scheme of the evolution equations in a time interval [t n , tn+1 ] that accounts for the bounded plastic stress space is given by £ P α¤ F p = e−1 F pn with e := exp − α∈A γα N̄ P α A = An + α∈A γα N (6.52) γ ≥ 0 ; f α − c − η γ ≤ 0 ; γ (f α − c − η γ ) = 0 . α α ∆t α α α ∆t α α Here γα := λα (tn+1 − tn ) are denoted as incremental plastic parameters, the normals N̄ PSfrag αreplacements and N are the gradients of the level-set functions as defined in (6.23). Note that only the F X F pn TX B x F e? Tx S Fe Fp e−1 X TX B̄ Figure 29: Mappings in the stress update algorithm in plastic-map plasticity. active constraints at the end of the time step enter the equations in (6.52). The numbers of the active flow systems are collocated in the active set A := {α ∈ [1, m] | γα > 0} . (6.53) A procedure for its determination is specified in box 1 that summarizes the algorithm. For the subsequent developments, we postulate that A is known. The update (6.52)1 is equivalent to F e = F e? e with F e? := F F np−1 (6.54) where F e? is the elastic trial value. For a geometrical interpretation of these mappings see figure 29. On the basis of the following trial values ? C̄ := F e?T F e? , ? ? S̄ := 2ψ,eC̄ (C̄ ) , ? ? Σ̄ := C̄ S̄ ? i and B ? := −ψ,A (An ) (6.55) 89 Approaches to Anisotropic Plasticity Box 1: Implicit Standard Stress Update Algorithm (U1) History data: internal variables In = {F pn , An } 1. Compute trial values ? F e? = F F np−1 , C̄ = F e?T F e? , ? ? ? ? ? i S̄ = 2ψ,eC̄ (C̄ ), Σ̄ = S̄ C̄ and B = −ψ,A Set initial values ? ? C̄ = C̄ , A = A? = An , γα=1...m = 0, A = An , S̄ = S̄ , B? = B ? 2. Check for plastic loading based on trial state f α? = f eα (Σ̄ ) + f iα (B ? ) if f α? ≤ cα ∀ α ∈ [1, m] exit 3. Compute derivatives of potential functions S̄ = 2ψ,eC̄ , Σ̄ = C̄ S̄ , i , B = −ψ,A α N̄ = f,eα , Σ̄ iα N α = f,B 4. Compute residuals and check convergence P P α ? α RC̄ = −C̄ + expT [− α∈A γα N̄ ]C̄ exp[− α∈A γα N̄ ] P RA = −A + An + α∈A γα N α η γα Rf α = f eα (Σ̄) + f iα (B) − cα − ∆t q P if kRC̄ k2 + kRB k2 + | α∈A Rf α |2 < tol goto 7 5. Compute increments P −1 −1 : RC̄ − Rf β ,A · RA,A · RA ) ∆γα = β∈A [gαβ ]−1 (Rf β − Rf β ,C̄ : RC̄, C̄ −1 : RC̄,γβ + Rf α ,A · RA,A with gαβ = Rf α ,C̄ : R−1 · RA,γβ + C̄,C̄ η ∆t δαβ 6. Update internal variables γα ⇐ γα + ∆γα P C̄ ⇐ C̄ − R−1 : (R + C̄ α∈A RC̄,γα ∆γα ) C̄,C̄ P −1 A ⇐ A − RA,A (RA + α∈A RA,γα ∆γα ) goto 3 7. Check for negative plastic multipliers if γα < 0 for some α ∈ A then A ⇐ A\α? where α? := argα∈A {min{γα | α ∈ A}} γ α? = 0 goto 3 endif 8. Check yield criteria η if f α > cα + ∆t γα for some α ∈ [1, m] then ? A ⇐ A\α where α? := argα∈[1,m] {max{f α |f α > cα + goto 3 endif η ∆t γα }} 90 Approaches to Anisotropic Plasticity one can decide whether or not plastic loading occurs. In order to solve the constrained ? problem (6.52) in the non-trivial case of plastic loading, indicated by f α (Σ̄ , B ? ) − cα − η γ > 0 for at least one α, the following residuals are defined ∆t α ? RC̄ := −C̄ + eT C̄ e P α (6.56) RA := −A + An + α γα N Rf α := f eα (Σ̄) + f iα (B) which have to vanish in the solution point. This non-linear coupled system of equations is solved iteratively by a Newton scheme. Therefore (6.56) is linearized with respect to C̄, A and γα , i.e. P Lin RC̄ = RC̄ + RC̄,C̄ : ∆C̄ + α RC̄,γα ∆γα = 0 P (6.57) Lin RA = RA + RA,A ∆A + α RA,γα ∆γα = 0 Lin Rf α = Rf α + Rf α ,C̄ : ∆C̄ + Rf α ,A ∆A + Rf α ,γα ∆γα = 0 . Solving (6.57)1,2 for the strain increments ∆C̄ and ∆A then allows the computation of the algorithmic parameters from (6.57)3 . Resubstitution finally yields the updates for the strain variables. The algorithm is summarized in box 1. 6.3.3. Explicit Stress Update Algorithm (U2) Using an explicit integration scheme for the evolution equations of the internal variables, the stress update algorithm takes a much simpler structure than its implicit counterpart discussed in the last section. The integration scheme for the evolution equations in a time interval [tn , tn+1 ] reads £ P α¤ F p = e−1 F pn with e := exp − α∈A γα N̄ n P α A = An + α∈A γα Nn (6.58) η η γ ≥ 0 ; f α − c − γ ≤ 0 ; γ (f α − c − γ ) = 0 . α α ∆t α α α ∆t α The above equations differ from the implicit ones in (6.52) only in the normals, here they are evaluated at the beginning of the time step. As before, γα := λα (tn+1 − tn ) are the incremental plastic parameters. The set A := {α ∈ [1, m] | γα > 0} determines the active constraints at the end of the time step and is assumed to be known for the moment. Based on the trial values ? C̄ := F e?T F e? ; ? ? S̄ := 2ψ,eC̄ (C̄ ) ; ? ? Σ̄ := C̄ S̄ ? i and B ? := −ψ,A (An ) (6.59) that are obtained with the definitions F e = F e? e and F e? := F F np−1 one can determine the loading state of the material in the considered time step. In the following, plastic η loading indicated by f α − cα − ∆t γα > 0 for at least one α ∈ [1, m] is considered. The only unknowns in (6.58) are the algorithmic parameters γα . They are determined by η γα = 0 that have to hold for all α ∈ A at the end the consistency conditions f α − cα − ∆t of the time step. These equations are solved iteratively by a Newton scheme. Therefore consider the linearization P P η iα ∆γα = 0 . (6.60) f α − cα + f,eα : C̄ ∆γ + f · B · ,γ β ,A ,B β β∈A β∈A A,γβ ∆γβ − C̄ ∆t 91 Approaches to Anisotropic Plasticity Box 2: Explicit Standard Stress Update Algorithm (U2) α History data: internal variables In = {F pn , An } and normals {N̄ n , Nnα } 1. Compute trial values ? F e? = F F np−1 , C̄ = F e?T F e? ? ? ? ? ? i (A ) S̄ = 2ψ,eC̄ (C̄ ) , Σ̄ = S̄ C̄ , B ? = −ψ,A n Set initial values ? C̄ = C̄ , A = An , γα=1...m = 0, A = An , e = 1, S̄ = S̄ ? ? 2. Check for plastic loading based on trial state f α? = f eα (Σ̄ ) + f iα (B ? ) if f α? ≤ cα ∀ α ∈ [1, m] exit 3. Compute derivatives of potential functions S̄ = 2ψ,eC̄ , i , B = −ψ,A Σ̄ = C̄ S̄ , α , N̄ = f,eα Σ̄ iα N α = f,B i C̄ = 4ψ,eC̄ C̄ , K = ψ,AA 4. Check convergence qP α if α∈A (f − cα − η 2 ∆t γα ) ≤ tol goto 8 5. Compute second-order tensors α α P̄ = sym[C̄ N̄ ] α? N̄ = sym[−e−1 e,γα ] α αT Q̄ = sym[S̄ N̄ ] α? α? P̄ = sym[C̄ N̄ ] 6. Compute increments P ∆γα = β∈A [gαβ ]−1 (f eβ + f iβ − cβ − α α with gαβ := (2Q̄ + P̄ : C̄) : P̄ β? η ∆t γβ ) + N α KN β + η ∆t δαβ 7. Update internal variables γα e C̄ A ⇐ γα + ∆γα P α = exp[− α∈A γα N̄ n ] ? ⇐ sym[eT C̄ e] P ⇐ A + α∈A ∆γα Nnα goto 3 8. Check for negative plastic multipliers if γα < 0 for some α ∈ A then A ⇐ A\α? where α? := argα∈A {min{γα | α ∈ A}} γ α? = 0 goto 3 endif 9. Check yield criteria η if f α > cα + ∆t γα for some α ∈ [1, m] then ? A ⇐ A\α where α? := argα∈[1,m] {max{f α |f α > cα + goto 3 endif η ∆t γα }} 92 Approaches to Anisotropic Plasticity Introducing the symmetric tensors P̄ (6.32) gives the linear equation f α − cα + P β∈A © α α α and Q̄ defined in (6.28) and using the relation α 1 2 (2Q̄ + P̄ : C̄) : C̄ ,γβ − N α KN β − ª η δαβ ∆γβ = 0 . ∆t (6.61) The sensitivities of the elastic right Cauchy-Green tensor with respect to the algorithmic parameters follow directly from the definition (6.2). Introducing the algorithmic tensors N̄ α? := −e−1 e,γα (6.62) then gives a representation for the algorithmic sensitivities of C̄ with respect to γα that α are analogous to the definitions (6.28) of the tensors P̄ in the continuous setting, i.e. P̄ α? ? α? := − 12 C̄ ,γα = − sym[eT C̄ e,γα ] = sym[C̄ N̄ ] . (6.63) With these definitions at hand, the algorithmic parameters are obtained from (6.61) as P ∆γα = β∈A [gαβ ]−1 (f β − cβ ) (6.64) in terms of the generally unsymmetric matrix α α gαβ := (2Q̄ + P̄ : C̄) : P̄ β? + N α KN β + η δαβ . ∆t (6.65) Finally, it remains to update the internal variables according to the integration scheme (6.58). A summary of the algorithm is given in box 2. 6.3.4. Algorithmic Tangent Moduli The tangent moduli in the intermediate configuration are defined by £vp? [S̄] = C̄algo : 1 2 £vp? [C̄] (6.66) in terms of the incremental Oldroyd rates p? p?T £vp? [S̄] = ∆S̄ − L̄ S̄ − S̄ L̄ p?T p? ? £vp? [C̄] = ∆C̄ + L̄ C̄ + C̄ L̄ = 2D̄ ) (6.67) The quantities that are superscribed with a star denote the algorithmic counterparts of the variables introduced in the continuous setting. The incremental plastic flow operator denotes the increment of the plastic map with respect to the intermediate configuration. p? With (6.52) it is defined by L̄ := ∆F p F p−1 = ∆e−1 e = −e−1 ∆e. Making use of the fact that e depends on C̄ via Σ̄ and directly on γα , we get P α? p? (6.68) L̄ = α∈A N̄ ∆γα + N̄? : ∆C̄ with the definition N̄? := −e−1 e,C̄ . Insertion into (6.67)2 and solving for ∆C̄ yields P α? ? (6.69) ∆C̄ = 2B̄ : (D̄ − α∈A P̄ ∆γα ) where B̄−1 := I + 2 sym12 [C̄ N̄? ] and sym12 [·] means the symmetric part with respect to the first two indices. The increments of the algorithmic plastic parameters are obtained 93 Approaches to Anisotropic Plasticity from the consistency conditions. In the case of plastic loading the stress state at the end of the time step has to lie on the yield surface, i.e. P η ∆γα ∀α ∈ A . (6.70) ∆f α = f,eα : ∆C̄ + β∈A f,γiαβ ∆γβ = ∆t C̄ i . Insertion The derivative f,eα is defined in (6.32) and f,γiαβ = −N α KN β where K := ψ,AA C̄ of the strain increment (6.69) into (6.70) then allows solving for the plastic parameters. The result is P α ? α β β ∆γα = R̄ : D̄ where R̄ := β∈A [gαβ ]−1 (2Q̄ + P̄ : C̄) : B̄ . (6.71) The matrix gαβ is defined by α α gαβ := (2Q̄ + P̄ : C̄) : B̄ : P̄ β? + N α KN β + η δαβ . ∆t (6.72) Substitution of the increments of the plastic parameters in (6.69) yields P ? α? α ∆C̄ = R̄ : D̄ where R̄ := B̄ : (I − α∈A P̄ ⊗ R̄ ) (6.73) and then in turn the incremental flow operator in (6.68) as a function of the algorithmic ? rate of deformation tensor D̄ , i.e. P α? α p? ? (6.74) L̄ = L̄ : D̄ where L̄ := (N̄? : R̄ + α∈A N̄ ⊗ R̄ ) . This expression leads, when inserted into (6.67), together with the constitutive equation ∆S̄ = C̄ : 12 ∆C̄ to the form 12 C̄algo := C̄ − C̄ : sym 12 [C̄ L̄] − 2 sym 12 [S̄ L̄ T ] . (6.75) Making use of the structure of L in (6.74), the final form of the algorithmic moduli C̄algo := C̄ − (C̄ : P̄ + 2Q̄) : R̄ − P α∈A (C̄ : P̄ α? α? + 2Q̄ ) ⊗ R̄ α (6.76) is obtained. Here we have defined the algorithmic counterparts of (6.28) by P̄ α? α? := sym[C̄ N̄ ] and Q̄ α? := sym[S̄ N̄ α?T ] (6.77) and have introduced the fourth-order tensors 12 P̄ := sym 12 [C̄ N̄? ] and Q̄ := sym 12 [S̄ N̄? T ] . (6.78) Specification of the Derivatives. In the following, the derivatives appearing in the derivations of the stress update algorithm and the consistent tangent moduli are specified in index notation. The key ingredient is the exponential map. Let m denote its argument, then the derivative with respect to γα is P α eij,γα = eij,mab N̄ab with mij = − α∈A γα N̄ijα . (6.79) 94 Approaches to Anisotropic Plasticity α The normal N̄ to the yield surface is a function of the Mandel stress tensor Σ̄. Its derivative with respect to C̄ is then given by the formulas Σ̄ij,C̄kl = δik S̄lj + 1 2 C̄ia C̄ajkl α α and N̄ij, C̄kl = N̄ij,Σ̄ab Σ̄ab,C̄kl . (6.80) The derivatives of the residuals (6.56) are RC̄ij ,C̄kl RC̄ij ,γα RA,A RA,γα Rf α ,C̄ij Rf α ,γβ Rf α ,A = = = = = = = ? −δik δjl + 2 sym ij [eai C̄ab ebj,mcd ? 2 symij [eai C̄ab ebj,γα ] P iα i −1 − α∈A γα f,BB ψ,AA Nα α Σ̄ab,C̄ij N̄ab i −N α ψ,AA Nβ −N α ψ,AA . P α γ N̄ ] α α∈A cd,C̄kl (6.81) There are some papers treating the numerical computation of the exponential map of an unsymmetric argument and its derivatives. Ortiz, Radovitzky & Repetto [97] compare two different methods for the computation of the exponential map and its first and second derivatives. One of them bases on a Taylor series expansion of the exponential map, the other one on a spectral representation of the argument tensor. In Itskov [56] analytical expressions for the exponential map and its first derivative are obtained by means of the Dunford-Taylor integral. Simplifications for Explicit Integration. In cases where the normals onto the yield α α surfaces are constant, i.e. N̄ ,C̄ = 0 and N,A = 0 for all α ∈ [1, m], the above derived equations can be simplified. This is for example the case in Schmid-type crystal plasticity, where the structural tensors are constant in the intermediate configuration and for the explicit integration scheme proposed in section 6.3.3. Then we have N̄? = 0 and B̄ = I. The moduli turn out to have a structure similar to the continuous setting, i.e. C̄algo := C̄ − P α∈A P β∈A [gαβ ] −1 (C̄ : P̄ α? α? β β + 2Q̄ ) ⊗ (2Q̄ + P̄ : C̄) . (6.82) Consider the yield surface of single slip crystal plasticity specified by a level-set function of the type f e = |Σ̄ : s̄ ⊗ m̄| where s̄ is a unit vector characterizing the slip direction and m̄ ? is a unit vector associated with the normal of the slip plane. Then the identities P̄ = P̄ ? and Q̄ = Q̄ hold and the moduli turn out to be symmetric. This follows immediately from a series expansion of the exponential map in (6.68). This completes the classical algorithmic treatment of crystal plasticity. 6.4. Variational Formulation The constitutive model of elasto-plasticity outlined in section 6.2 belongs to the class of so-called standard materials This generic type of material model can be related to the publications by Biot [16], Ziegler [164], Germain [41], Halphen & Nguyen [46], see also the recent treatments in the textbooks by Maugin [73] and Nguyen [96]. It is described by two fundamental scalar-valued constitutive functions, an energy storage function and a dissipation function. 95 Approaches to Anisotropic Plasticity 6.4.1. Standard Formulation of Rate-Independent Plasticity Assume a functional dependence of the stored energy ψ in a material point on the objective right Cauchy-Green tensor C and an internal variable vector I. It governs the constitutive equation for the second Piola-Kirchhoff stresses S = 2∂C ψ(C, I) (6.83) as well as the reduced dissipation inequality through the internal forces, ρ0 D = F · İ ≥ 0 with F := −∂I ψ(C, I) . (6.84) The dissipation function φ is assumed to depend upon the flux of the internal variables and on the internal variables themselves. To cover the properties of a normal dissipative mechanism, the following properties must hold: φ is convex with respect to the flux İ, φ(0, I) = 0, φ(İ, I) ≥ 0 and φ is homogeneous of degree one with respect to the flux İ. The dissipation function determines the evolution of I in time by the constitutive differential equation 0 ∈ ∂I ψ(C, I) + ∂İ φ(İ, I) with I(0) = I0 (6.85) often referred to as Biot’s equation of standard dissipative systems, cf. Biot [16] and Nguyen [96]. The two constitutive equations (6.83) and (6.85) determine the stress response of a normal-dissipative material in a deformation-driven process where C is prescribed. To account for a non-smooth dissipation function, the derivatives appearing in (6.85) are understood to be sub-differentials. Plasticity and dry friction are timeindependent and non-viscous irreversible processes governed by non-smooth dissipation functions. These functions are positively homogeneous of degree one in the fluxes, φ(²İ, I) = ²φ(İ, I) , (6.86) have cone-like structures and are not differentiable at İ = 0 as visualized in figure 30. An introduction to sub-differential calculus in the context of plasticity theory can be found e.g. in Han & Reddy [48] or Maugin [73]. PSfrag replacements φ? φ(İ, I) İ F ∈ ∂İ φ(İ, I) F İ ∈ ∂F φ? E Figure 30: Dissipation function φ(İ, I) for rate-independent plasticity. The dual function φ? (F , I) is the indicator function of the elastic domain E := ∂İ φ(0, I). Based on the definition (6.84)2 of the internal forces F , one introduces a dual dissipation function φ? depending on the forces F by the Legendre-Fenchel transformation φ? (F , I) = sup{ F · İ − φ(İ, I) } , İ (6.87) 96 Approaches to Anisotropic Plasticity which is convex and positively homogeneous of degree one in the internal forces. The definitions (6.84)2 and (6.87) induce the two alternative representations F ∈ ∂İ φ(İ, I) and İ ∈ ∂F φ? (F , I) (6.88) of Biot’s equation (6.85)1 . The internal force defined in (6.88)1 is an element of the cone E := ∂İ φ(0, I) := { F | F · İ ≤ φ(İ, I) ∀İ } (6.89) that is identified with the elastic domain of (6.10). See figure 30 for an illustration. In the case of plastic loading where İ 6= 0 the reduced dissipation inequality (6.84)1 takes the form ρ0 D = ∂İ φ(İ, I) · İ = φ(İ, I) ≥ 0 (6.90) where the property (6.86) has been made use of, cf. appendix C. This inequality is satisfied by the above-assumed properties of φ. 6.4.2. Incremental Variational Formulation Now we proceed with the construction of an integrated version of constitutive equations giving a consistent approximation of the continuous differential equation (6.85) in a finite increment [tn , tn+1 ] ∈ R+ of time. Following conceptually the recent publications from Miehe [85] and Miehe, Schotte & Lambrecht [90], we define an incremental stress potential function W depending on the deformation measure C n+1 := C(tn+1 ) at time tn+1 that determines the stresses S n+1 at time tn+1 by the quasi-hyperelastic function evaluation S n+1 = 2∂C W (C n+1 ) . (6.91) Clearly, this function W must cover characteristics of the storage function ψ and the dissipation function φ introduced above. To this end, consider the variational problem W (C n+1 ) := inf I∈Gp Z tn+1 [ ψ̇ + φ ] dt with I(tn ) = In . (6.92) tn This problem defines R t the incremental stress potential function W as a minimum of the generalized work tnn+1 [ψ̇ + φ] dt done on the material in the time increment under consideration. Starting with the given initial condition I(tn ) = In , the minimum problem defines an optimal path of the internal variables I(t) for t ∈ [tn , tn+1 ] including the right boundary value In+1 := I(tn+1 ). The internal variables can be restricted to be elements of a certain group, as for example F p ∈ SL(3) to guarantee plastic incompressible flow, see (6.25). This is indicated by I ∈ Gp . The two equations (6.91) and (6.92) provide an approximately variational counterpart to the continuous setting (6.83) and (6.85) of the constitutive equations in the discrete time step [tn , tn+1 ] under consideration. In order to show the consistency, we at first recast (6.92) into the form ½ ¾ Z tn+1 tn+1 W (C n+1 ) = inf [ψ(C, I)]tn + φ(İ, I) dt . (6.93) I∈Gp tn 97 Approaches to Anisotropic Plasticity The necessary condition for the minimum problem is that the variation with respect to the internal variables of the term in brackets vanishes, i.e. [∂I ψ · δI]ttn+1 n + Z tn+1 [ ∂İ φ · δ İ + ∂I φ · δI ] dt = 0 . tn (6.94) For smooth functions integration by parts yields the expression [( ∂I ψ + ∂İ φ ) · δI]ttn+1 n + Z tn+1 tn [− d (∂ φ) + ∂I φ ] · δI dt = 0 . dt İ (6.95) Thus the variational problem (6.92) yields Biot’s equation (6.85) ∂I ψ + ∂İ φ = 0 for t = tn+1 (6.96) at the discrete right boundary of the interval [tn , tn+1 ]. The minimizing path of the internal variables inside the interval is determined by the Euler equation − d (∂ φ) = 0 for t ∈ [tn , tn+1 ] . dt İ (6.97) In the limit tn+1 → tn , the form of the minimization path becomes irrelevant because the time increment degenerates to a discrete time tn+1 . Equation (6.96) still holds in this case and therefore it is shown that the variational formulation (6.92) represents a consistent point-wise approximation of Biot’s normal-dissipative evolution equation (6.85). Furthermore, taking the derivative of the incremental potential function with respect to the strains C n+1 , we have ∂C W (C n+1 ) = ∂C ψ(C n+1 , In+1 ) , (6.98) where In+1 is considered to be given by the minimization problem (6.92). Comparison with (6.83) then shows the consistency of the potential equation (6.91) with the continuous setting. 6.4.3. Specification to Multi-Surface Plasticity In the subsequent development we consider a non-smooth convex elastic domain that is described by m functions f α (F , I) depending on the internal forces and the internal variables. The level-surfaces f α (F , I) = cα with the thresholds cα > 0 are assumed to describe the boundary ∂E of the domain E := {F | f α (F , I) ≤ cα ; α = 1, . . . , m} , (6.99) which is identical to (6.10). Here the functions f α are assumed to be (i) convex with respect to the plastic forces, (ii) positively homogeneous of degree one with respect to F and (iii) zero at the origin, f α (0, I) = 0. Application of the principle of maximum plastic dissipation determines the evolution of the internal variables φ(İ, I) = sup {F · İ} . F ∈E (6.100) 98 Approaches to Anisotropic Plasticity That maximization problem with inequality constraints can be solved by a Lagrangian multiplier method. Therefore the dissipation function is extended by the weighted constraints and the solution is then obtained by P (6.101) φ(İ, I) = sup inf {F · İ + α λα (f α (F , I) − cα )} F ∈E λα where the Lagrange parameters λα are determined by the loading conditions λα ≥ 0 , f α − cα ≤ 0 , λα (f α − cα ) = 0 . (6.102) For known Lagrange multipliers (6.101) can be interpreted as the Legendre-Fenchel transformation of the dual dissipation potential P φ? (F , I) = α λα (f α (F , I) − cα ) . (6.103) Insertion into (6.88)2 yields the evolution equation P İ = α λα ∂F f α (F , I) . (6.104) Alternatively, problem (6.100) can be solved approximately with a penalty method. Introducing the inverse penalty parameter η, the approximative solution X 1 (f α (F , I) − cα )+2 } (6.105) φ(İ, I) = sup {F · İ − 2η F ∈E α is obtained. Physically, the inverse penalty parameter can be interpreted as the viscosity of the material. This approach regularizes the rate-independent ansatz (6.101). Equation (6.105) may be interpreted as the Legendre-Fenchel transformation of the dual dissipation function P 1 α (f (F , I) − cα )+2 . (6.106) φ? (F , I) = α 2η The evolution equation for the internal variables follows from (6.88)2 in the same form as in the rate-independent case that is given in (6.104) but with the parameters 1 λα := (f α − cα )+ η (6.107) that replace the loading conditions (6.102). The evolution equation (6.104) can be viewed as a split of the evolution into normal directions ∂F f α and the amounts λα of plastic flow. Evaluation of the dissipation (6.100) at the solution point with the rate (6.104) and exploiting the homogeneity of the functions f α gives the representation P (6.108) φ(λ1 , . . . , λm ) = α λα f α ≥ 0 of the dissipation. Thus the image of the level-set function f α can be considered as the force driving the amount of flow λα on the flow system α. From (6.102)3 we get f α = cα in the case of rate-independent plastic loading and from (6.107) f α = ηλα + cα for ratedependent plastic loading. Both cases are covered by the scalar dissipation function of the form P (6.109) φ(λ1 , . . . , λm ) = α φα (λα ) with φα := λα cα + η2 λ2α which is a function of the plastic multipliers only. The rate-independent case is obtained for η = 0. 99 Approaches to Anisotropic Plasticity 6.4.4. Implicit Discrete Variational Formulation (V1) The integration of the rate ψ̇ and the dissipation function φ requires function evaluations of the free energy and the dissipation function at time tn+1 and therefore bases on the internal variables at the end of the time step. The internal variables are obtained by some integration algorithm that accounts for the constrained plastic forces to the elastic domain. In contrast to the standard stress update algorithm outlined in section 6.3.1 we now formulate this constraint by algorithmic yield criteria functions which lead to a symmetric formulation of the update algorithm. 6.4.4.1. Integration of the Free Energy. A key kinematic quantity in the timediscrete setting that allows the construction of a symmetric formulation is the relative plastic deformation gradient defined by p f̄ := F p F np−1 . (6.110) The current metric in the intermediate configuration is a function of the relative deformation gradient C̄ := F eT gF e = F p−T CF p−1 = f̄ p−T ? C̄ f̄ p−1 (6.111) ? where C̄ := F e?T F e? is the trial metric in the increment evaluated with the elastic trial deformation gradient F e? := F F np−1 . Thus in the time increment [tn , tn+1 ] the relative deformation gradient replaces the internal variable F p and the free energy function ψ(C̄, q̄) can alternatively be parameterized by the total deformation and the incremental internal variables, i.e. ψ = ψ(C, I ? ) (6.112) p where I ? := {f̄ , q̄} is the vector of algorithmic internal variables. The algorithmic plastic forces dual to the algorithmic internal variables are given by the derivatives F ? := −∂I ? ψ(C, I ? ) (6.113) The integration of the algorithmic internal variables is performed by some algorithm in the form A(I ? , F ? , γα ) = 0 (6.114) where the algorithmic parameters are defined by γα := λα (tn+1 − tn ). 6.4.4.2. Integration of the Dissipation Function. Before integrating the dissipation function over the time interval, the dissipation is reformulated in terms of the relative deformation gradient associated with the considered time interval that is introduced above. In the dissipation inequality (6.9), we express the macroscopic part of the dissipation by the scalar product of two algorithmic quantities associated with the time-interval under consideration, p p Σ̄ : L̄ = Σ̄ : (Ḟ F np−1 F pn F p−1 ) = σ̄ : f̄˙ p . (6.115) 100 Approaches to Anisotropic Plasticity p Herein f̄ denotes the algorithmic relative deformation gradient defined in (6.110) and σ̄ the associated work conjugate algorithmic stress tensor. The latter is related to the Mandel stress tensor via p σ̄ = −∂f̄ p ψ(C, f̄ ) = Σ̄f̄ p−T (6.116) and is part of the vector of the algorithmic plastic forces F ? := {σ̄, Q̄}. The dissipation can be expressed by these algorithmic tensors as φ = ρ0 D = F ? · İ ? , (6.117) cf. (6.115). Approximating the incremental flux by İ ? ≈ [I ? − In? ]/∆t gives together with the assumption that F ? is constant within the time interval the expression Z tn+1 φ dt ≈ F ? : (I ? − In? ) . (6.118) tn 6.4.4.3. Discrete Setting of the Principle of Maximum Dissipation. A decisive step towards a symmetric formulation is the restriction of the algorithmic plastic forces F ? instead of the forces F to an elastic domain, i.e. E := {F ? |f α (F ? ) − cα ≤ 0 α = 1, . . . , m} . (6.119) Obviously, the evaluation of the level-set functions with F ? and F will give different results as can be seen from (6.116). The deviation depends on the size of the time step but is proven to be negligible for typical time increments used in numerical simulations. In the limit tn+1 → tn we get the identity of both stress tensors because f p → 1. Applying the principle of maximum plastic dissipation to the incremental setting defines the following Lagrange function P (6.120) L(F ? , γα ) := −F ? : (I − In ) + α γα (f α (F ? ) − cα ) → stat. with Lagrange multipliers γα . The corresponding Karush-Kuhn-Tucker equation is P α (6.121) A = ∇F ? L = −I + In + α γα f,F ? = 0 which is nothing but the evolution equation for the internal variables. Furthermore the loading conditions γα ≥ 0, f α − cα ≤ 0 and γα (f α − cα ) = 0 must hold at the solution point. 6.4.4.4. Discrete Setting of the Variational Problem. For the restriction to a single surface yield criterion or to a system of orthogonal flow systems as obtained by a Kelvin-mode decomposition, cf. section 6.6.5 and appendix A.2, it is possible to specify an incremental potential function that yields the identical symmetric algorithm as it is obtained from the principle of maximum dissipation discussed above. Inserting the approximation of the dissipation (6.118) and the algorithm (6.114) into the variational problem (6.92) gives the representation W (C n+1 ) = inf [W h (C n+1 , I ? )] ? I ∈Gp (6.122) 101 Approaches to Anisotropic Plasticity which is a finite dimensional approximation of the continuous formulation (6.92) of the incremental variational problem. (6.122) minimizes the function W h = ψ(C n+1 , I ? ) − ψn + cγ (6.123) p with respect to the algorithmic internal variables I ? = {f̄ , q̄}. The solution of (6.122) has to guarantee that the dissipation in the time increment is positive and is obtained as W (C n+1 ) = inf sup[W h (C n+1 , I ? ) − νγ] . ? I ∈Gp (6.124) ν Here we consider the plastic multiplier to be a function of the internal variable vector, i.e. γ = γ(I ? ). The necessary conditions read W,I ? = −F ? + cγ,I ? − νγ,I ? = 0 , γ≥0, ν≥0, νγ = 0 . (6.125) The sensitivity of the plastic multiplier with respect to the algorithmic internal variable vector is obtained from the integration algorithm A = −I ? + In? + γf,F ? (F ? ) = 0 for the internal variables when considering F ? · A,I ? = 0 ⇒ γ,I ? = F ? /f (F ? ) (6.126) where we have exploited the homogeneity properties of the level-set function discussed in appendix C. Plastic loading with γ > 0 enforces ν = 0 because of (6.125)4 . Then (6.125)1 gives the negative yield condition −f + c = 0. Thus the discrete variational formulation results in the same set of equation as the principle of maximum plastic dissipation. 6.4.5. Algorithmic Solution of the Discrete Variational Formulation (V1) Both, application of the principle of maximum dissipation to the incremental quantities and the discrete variational problem (6.122) lead to the same set of equations, i.e. the yield criteria functions, the integration algorithm for the internal variables and the constitutive equation for the algorithmic forces. In order to solve these equations the iteration variables are assembled in a vector p := [I ? , F ? , γα ]T and the governing equations are combined in a residual vector e ? F ? + ∂P I ? ψ (C, I ) r(C, p) := I ? − In? − α γα ∂F ? f α (F ? ) = 0 η −f α (F ? ) + cα + ∆t γα (6.127) (6.128) which has to vanish in a strain driven process where F and because of that also C are prescribed. The solution is obtained by applying a Newton scheme. The variable vector p is updated according to p ⇐ p − k−1 r (6.129) 102 Approaches to Anisotropic Plasticity until convergence is reached, i.e. krk < tol. The iteration matrix k appears within the linearization of (6.128) and has the structure k := r ,p = ψ,I ? I ? sym. I 0 P β α . −f,F − α γα f,F ?F ? ? η δ ∆t αβ (6.130) In contrast to the standard stress update algorithm outlined in section 6.3.1 this variational formulation is always symmetric. 6.4.6. Stresses and Algorithmic Tangent Moduli (V1) Once the internal variables are determined by the solution of the constrained minimization problem the stresses and algorithmic tangent moduli have to be computed. The second Piola-Kirchhoff stresses follow from the definition (6.91) by a function evaluation, i.e. S = 2dC W h (C, p) = 2∂C W h + 2∂p W h · p,C . (6.131) The last part vanishes due to the necessary condition (6.130)1 so that the stresses are determined by the constitutive equation S = 2∂C ψ(C, I ? ) . (6.132) The sensitivity of the stresses with respect to the deformation is given by the second derivative which turns out to have the form 2 Calgo = 4d2CC W h (C, p) = 4∂CC W h + 4∂Cp W h · p,C . (6.133) The sensitivity of the variable vector with respect to the deformation is obtained from the consistency condition dC r = ∂C r + ∂p r · p,C = 0. With the derivatives 2 ∂Cp W h = [ψ,CI ? , 0, 0]T and ∂C r = [ψ,I ? C , 0, 0]T (6.134) the moduli are obtained in the form 2 2 2 C = ∂CC ψ − ∂CI ? ψ · k̃ · ∂I ? C ψ (6.135) where k̃ denotes the quadratic upper left sub-matrix of the inverse iteration matrix k −1 with dimension (len[I ? ] × len[I ? ]), compare (6.50) and (6.51). 6.4.7. Application of the Algorithm (V1) to the Model Problem We now concretize the discussed framework of an implicit variational formulation of elastoplasticity to the model problem of section 6.2.3. Recall the assumed decoupled structure of the stored energy function (6.20) and the level-set functions (6.22) which here take the forms p ψ = ψ e (C, f̄ ) + ψ i (A) and f α = f αe (σ̄) + f αi (B) . (6.136) 103 Approaches to Anisotropic Plasticity For the integration we apply an algorithm based on an operator split. In a first step the evolution equations are integrated by a backward Euler algorithm, i.e. ¾ P p α = 0 f̄ − 1 − Pα∈A γα N̄ . (6.137) A − An − α∈A γα N α = 0 p The algorithmic variable vector p := [f̄ , A, σ̄, B, γα ]T is updated iteratively in terms of the residual p σ̄ + ∂f̄ p ψ e (C̄, f̄ ) B +P ∂A ψ i (A) p eα =0 f̄ − 1 − Pα γα ∂σ̄ f (σ̄) r(C, p) := (6.138) iα A − An − α γα ∂B f (B) η −f eα (σ̄) − f iα (B) + cα + ∆t γα and the local symmetric matrix ψ,f̄ p f̄ p 0 I 0 T ψ,AA 0 1 P k := r ,p = − α γα f,ασ̄σ̄ 0 P α sym. − α γα f,BB 0 0 −f,βσ̄ β −f,B η δ ∆t αβ (6.139) as specified in (6.129). It remains to specify the derivatives appearing in (6.138) and (6.139) for concrete forms of the constitutive functions (6.136). The derivatives of the free energy function are p ψ,f¯ijp = ψ,C̄ab C̄ab,f¯,ij p p p p p p ψ,f¯ij f¯kl = ψ,C̄ab C̄cd C̄ab,f¯,ij C̄cd,f¯,k,l + ψ,C̄ab C̄ab,f¯,ij f¯,kl (6.140) ψ,Cij = ψ,C̄ab C̄ab,Cij ψ,f¯ijp Ckl = ψ,C̄ab C̄cd C̄ab,f¯ijp C̄cd,Ckl + ψ,C̄ab C̄ab,f¯ijp Ckl p ¯ p C̄ p . ψ,Cij f¯klp = ψ,C̄ab f¯cd f + ψ C̄ ¯ ab,Cij cd,Ckl ,C̄ab ab,Cij fkl They base on the following derivatives of the right Cauchy-Green tensor C̄ij,f¯klp = −2 symij [f¯lip−1 C̄kj ] p−1 ¯p−1 p−1 ¯ ¯ p p ] C̄ij,f¯klp f¯mn = 2 symij [flm fni C̄kj − fli C̄kj,f¯mn p−1 p−1 ¯ ¯ C̄ij,Ckl = fki flj p−1 ¯p−1 ¯p−1 p−1 ¯p−1 ¯p−1 p flm fnj fni flj − f¯ki = −f¯km C̄ij,Ckl f¯mn p−1 ¯p−1 ¯p−1 ¯ C̄ ¯p = −2 sym [f f f ] . ij,fkl Cmn ij li mk (6.141) nj If the local iteration has converged, the relative plastic deformation gradient has to be corrected. This is because of the additive update that violates the incompressibility constraint f p ∈ SL(3). We suggest a correction of the form p p p f̄ ⇐ det[f̄ ]−1/3 f̄ . (6.142) 104 Approaches to Anisotropic Plasticity 6.4.8. Explicit Discrete Variational Formulation (V2) We now discuss the solution process of the incremental variational problem (6.92) for explicit integration algorithms, leading to a tremendous reduction of the numerical effort. The integration of the rate ψ̇ requires a function evaluation of the free energy at time tn+1 . Therefore the internal variables at the end of the time step have to be known. Typical one-step algorithms iterate solely the scalar algorithmic multipliers γ α := λα (tn+1 − tn ), A(γα ) = 0 . (6.143) The internal variable vector is a function of the plastic parameters, I = I(γ α ). A wellP known representative is the forward Euler scheme I = In + α γα ∂Σ̄ f α (Fn , In ). For the dissipation function one obtains with (6.108) the expression Z tn+1 P P 1 η 2 (6.144) α φα (λα ) dt = α cα γα + 2 ∆t γα . tn The plastic multipliers λα are assumed to be constant in the time increment. When inserting (6.144) and the algorithm (6.143) into the variational problem (6.92) the following representation is obtained W (C n+1 ) = inf W h (C n+1 , γ1 , . . . , γm ) . γα ≥0 (6.145) The continuous formulation (6.92) of the incremental variational problem is approximated by the finite-dimensional problem (6.145) that minimizes the function P η 2 γα (6.146) W h := ψ(C, I) − ψn + α cα γα + 21 ∆t with respect to the algorithmic incremental parameters γα . 6.4.9. Algorithmic Solution of the Discrete Variational Formulation (V2) The minimization problem (6.145) with inequality constraints can be solved by a Lagrange multiplier method. The solution is the saddle-point of the associated Lagrange function P (6.147) W (C n+1 ) = inf sup[ W h (C n+1 , γα ) − α γα να ] γα να which is characterized by the loading conditions and the constraints W,γhα − να = 0 , να ≥ 0 , γα ≥ 0 , ν α γα = 0 . (6.148) The set (6.148) boils down to the restrictions W,γhα ≥ 0 , γα ≥ 0 and W,γhα γα = 0 (6.149) that determine the algorithmic parameters γα . In the case of plastic loading a non-empty set of active constraints exists A := {α | γα 6= 0} (6.150) 105 Approaches to Anisotropic Plasticity and an improved solution of the incremental parameters is obtained by a Newton step X γα ⇐ γα + ∆γα where ∆γα = − [W,γhα γβ ]−1 [W,γhβ ] ∀α ∈ A . (6.151) β∈A The iteration is terminated if the residual is below a given tolerance, i.e. £P ¤ h 2 1/2 (W ) ≤ tol . ,γα α∈A (6.152) During the iteration the set of active constraints may change. We apply the update procedure of the active set proposed by Miehe, Schotte & Lambrecht [90] in the context of crystal plasticity. The solution of the minimization problem and the integration of the internal variables is coupled and has to be treated simultaneously. The minimization problem can be viewed as an energetic counterpart of the consistency condition of the standard formulation. In contrast to the standard formulation the update of the plastic increments in the variational framework is solely determined by this so-called energetic consistency condition. The update of the internal variables in any iteration step is performed according to the integration algorithm (6.143). 6.4.10. Stresses and Algorithmic Tangent Moduli (V2) Once the constrained minimization problem (6.145) is solved, the stresses and elasticplastic moduli are obtained by function evaluation of the derivatives of the incremental stress potential function W . According to (6.91), the derivative with respect to the strains C n+1 yield the stresses S n+1 . Application of the chain rule gives the expression P (6.153) dC W = ∂C W h + α∈A W,γhα γα,C . In the solution point the last term drops out because (6.149)3 gives W,γhα = 0 for plastic loading where γα > 0 and γα,C = 0 in the case of elastic behaviour, where γα = 0 = constant. Thus the stresses are S n+1 = 2∂C W h . (6.154) The sensitivity of the stresses with respect to the strains is governed by the algorithmic tangent moduli. Like the moduli in elasticity theory they are obtained by the second derivative of the stress potential function in the solution point P h h 2 (6.155) Cep n+1 := 4dCC W (C n+1 ) = 4W,CC + 4 α∈A W,Cγα ⊗ γα,C . The sensitivity of the incremental plastic parameter with respect to the strains is obtained by linearization of the necessary condition (6.149)3 with respect to the deformation. Inserting the result P γα,C = − β∈A [W,γhα γβ ]−1 W,γhβ C (6.156) into (6.155) gives the algorithmic elastic-plastic moduli h Cep n+1 = 4W,CC − 4 P α∈A P −1 h h β∈A [W,γα γβ ] W,Cγα ⊗ W,γhβ C . (6.157) They consist of an elastic contribution and a softening part. The latter is a consequence of a change in the internal variables of the material within the time step under consideration. The algorithm is summarized in box 3. 106 Approaches to Anisotropic Plasticity Box 3: Explicit Variational Stress Update Algorithm (V2) History data: internal variables In = {F pn , An } and normals {N n , Nn } 1. Set initial values γα=1...m = 0, A = An , 2. Determine current state of internal variables Perform integration step of algorithm A(γα ) = 0 3. Evaluate local minimization function P W h = ψ(C, I) − ψn + α∈A cα γα + 1 η 2 2 ∆t γα and its derivatives h W,C = ψ,C h W,CC = ψ,CC η W,γhα = ψ,γα + ∆t γα η h δαβ W,γα γβ = ψ,γα γβ + ∆t h = ψ,Cγα W,Cγ α 4. Check convergence qP if ( α∈A W,γhα )2 ≤ tol goto 7 5. Perform Newton update of algorithmic parameters P γα ⇐ γα − β∈A [W,γhα γβ ]−1 W,γhβ 6. Check for negative parameters α? = argα∈A [min{γα |α ∈ A}] if γα? ≤ 0 then determine scaling parameter ξ := 1 − γα? /∆γα? perform scaling P γα ⇐hγα − (1 − ξ)∆γα h if ∆W := α∈A W,γα ≤ 0 remove flow system, A = A\α? goto 2 endif 7. Check necessary condition for minimum α? = argα=1...m [min{W,γhα ] if Wγhα? ≤ 0 then A ⇐ {A ∪ α? } goto 2 endif 8. Set stresses S n+1 = 2∂C W h Approaches to Anisotropic Plasticity 107 6.4.11. Application to Model Problem We now apply the above discussed framework of incremental variational plasticity to the model problem of section 6.2.3. Recall the assumed decoupled structure of the stored energy function (6.20) and the level-set functions (6.22), i.e. ψ = ψ e (C̄) + ψ i (A) and f α = f αe (Σ̄) + f αi (B) . (6.158) The integration of the internal variables is performed with the explicit scheme (6.58). The solution of the minimization problem (6.145) with a Newton scheme bases on the derivatives of the function W h with respect to the algorithmic parameters γα . The stresses and moduli require the derivatives with respect to the total strains. So the following expressions have to be specified ) η h W,γhα = ψ,γα + cα + ∆t γα W,C = ψ,C̄ : C̄ ,C W,γhα C = ψ,γα C̄ : C̄ ,C (6.159) η h h = C̄ ,C : ψ,C̄γα δαβ WCC = C̄ ,C : ψ,C̄ C̄ : C̄ ,C W,Cγ W,γhα γβ = ψ,γα γβ + ∆t α for the functions given in (6.158). The following compact expressions for the derivatives of the free energy with respect to the elastic deformation measure C̄ are obtained ψ,γα = 2ψ,C̄ : 12 C̄ ,γα + ψ,A A,γα 1 1 1 ψ,γα γβ = 2 C̄ ,γα : 4ψ,C̄ C̄ : 2 C̄ ,γβ + 2ψ,C̄ : 2 C̄ ,γα γβ + A,γα ψ,AA A,γβ (6.160) ψ,C̄γα = ψ,C̄ C̄ : C̄ ,γα + 2 sym[e−1 e,γα ψ,C̄ ] ψ,γα C̄ = ψ,C̄γα . Within the iterative solution procedure of the minimization problem the internal variables are updated according to ¾ ? C̄ ⇐ eT (γα )P C̄ e(γα ) (6.161) A ⇐ A + α∈A A,γα ∆γα . It remains to specify the derivatives appearing in (6.160). They are obtained in terms of derivatives of the exponential map, i.e. ¾ ? 1 C̄ ,γα = sym[eT C̄ e,γα ] = sym[C̄e−1 e,γα ] 2 (6.162) ? ? 1 C̄ ,γα γβ = sym[eT,γβ C̄ eγα ] + sym[eT C̄ e,γα γβ ] . 2 6.5. Model Problem: Double Slip Plasticity Metallic solids belong to the class of crystalline materials and show anisotropic elastic behaviour. The anisotropic response is caused by the periodically distributed matter, i.e. the crystalline atomic structure. A typical example is copper whose lattice can be constructed with a cubic face-centered Bravais-cell. A cubic free energy function with symmetry group Oh is specified with respect to the isoclinic intermediate configuration. It is assumed to be of the form ψ(C̄) = ψ e (I1 , I2 , I3 , I4 ) + ψ i (A) (6.163) 108 Approaches to Anisotropic Plasticity consisting of a macroscopic part ψ e and a microscopic part ψ i . The four invariants I1 , . . . , I4 constitute the basis I := {tr[C̄], 1 2 (tr[C̄]2 − tr[C̄ 2 ]), det[C̄], tr[(Ō : C̄)2 ]} (6.164) which is of course far from being complete. The first three invariants are the principal invariants of the current metric in the intermediate configuration C̄ := F eT gF e , the fourth invariant accounts for cubic symmetry. The fourth-order tensor Ō is defined in terms of the three orthonormal anisotropy directions that characterize cubic symmetry, P O := 3i=1 ai ⊗ ai ⊗ ai ⊗ ai . (6.165) For the considered model problem we assume the elastic constitutive function −β/2 ψ = µ2 (I1 − 3) + βµ (I3 − 1) + 1 4 α(I4 − 2I1 + 3) + ψ i (A) (6.166) where ψ i = 12 hA2 +(y∞ −y0 )(A+ ω1 exp[−ωA]) accounts for non-linear isotropic hardening. The first two terms in the free energy function represent the isotropic part in form of a compressible Neo-Hooke model, whereas the latter two parts model the anisotropic response. The shear modulus is denoted by µ, the parameter β is related to the Poisson ration according to β= 2ν . 1 − 2ν (6.167) α is a parameter for the cubic part. The plastic contribution is governed by the hardening modulus h, the initial yield stress y0 and the saturation yield stress y∞ . In metallic materials plastic deformation is associated with movements of dislocations in so-called slip planes. The deformations take place when the Schmid-stress in a slip plane reaches a critical value. The corresponding yield criterion functions read α φα = f eα (Σ̄, M̄ ) + f iα (B) − y0α ≤ 0 . (6.168) α The second-order tensors M̄ := s̄α ⊗ m̄α characterize the slip systems by two orthogonal vectors. Here m̄α denotes the normal of the slip plane and s̄α specifies the direction in which plastic flow takes place. Thus the following relations hold s̄α · m̄α = 0 and ks̄α k = km̄α k = 1 . (6.169) For the model problem we assume level-set functions of the form α f eα := |Σ̄ : M̄ | and f iα := B . (6.170) 6.5.1. Numerical Example: Rotation of the Slip Systems An interesting effect in plastic-map plasticity is the rotation of the slip systems. This rotation is contained in the elastic map and can be determined from the polar decomposition F e = v e Re . Small elastic deformations are characterized by a small left stretch 109 Approaches to Anisotropic Plasticity tensor in the sense that v e ≈ 1 and consequently Re ≈ F e . To visualize this effect, a simple shear test in plane strain state is considered. Plastic flow can take place on two slip lines that are inclined at 60◦ . Both slip systems have identical plastic material parameters. Setting y0 = y∞ = 80N/mm2 and h = 0N/mm2 specifies rate-independent ideal plastic response. The elastic material response has cubic symmetry, the shear modulus is µ = 75000N/mm2 , the exponent β = 0.8067 and the cubic parameter α = −51500N/mm2 . The deformation is prescribed through the deformation gradient 1 γ 0 (6.171) F := 0 1 0 ei ⊗ ej , 0 0 1 parameterized with the scalar γ. The evolution of the slip systems’ orientations during the deformation process is depicted in figure 31 by the two lines inside of the elements. At the beginning of the deformation process both slip lines rotate rapidly clockwise. The Figure 31: Rotation of the slip planes in a simple shear test. The lines indicate the slip orientations. The sequence shows deformation states at γ = 0.0, 0.5, 1.0 and γ = 2.5. rotation becomes slower as the orientation tends towards a preferred alignment, where one of the slip systems is in line with the shearing direction. The rotation of the slip systems is not proportional to the global deformation of the specimen. Some more insight to what is going on during the deformation process is gained by the plot in figure 32. There the angles of the rotations contained in the elastic and plastic maps obtained from a polar decomposition are plotted. The sum of the rotation angles from the elastic and plastic maps coincide very well with the overall rotation contained in F . At the beginning of the process, the elastic rotation increases faster than the overall rotation and converges towards a final value of −60◦ . The plastic rotation first compensates the fast elastic rotation by turning to the opposite direction. When the latter slows down, the plastic 40 F Fe Fp Fe + Fp angle [◦ ] 20 PSfrag replacements 0 −20 −40 −60 −80 0 1 2 3 4 5 6 7 8 9 10 shear parameter γ Figure 32: Rotation of the slip planes in a simple shear test. The plot documents the rotations contained in the maps F , F e and F p . The structural tensors map with the elastic map and rotate counter clockwise up to an angle of −60◦ . 110 Approaches to Anisotropic Plasticity rotation more and more develops like the global rotation. For the application of a similar double slip model to the simulation of the necking of a metallic strip we refer to Miehe [81] for rate-independent crystal plasticity and to Steinmann & Stein [134] in the context of rate-dependent response as well as preceding publications by other authors cited therein. 6.5.2. Numerical Example: Drawing of a Flange To show the performance of the double slip formulation, we investigate a drawing process of a flange under plane strain conditions. The geometry and the boundary conditions are depicted in figure 33. The inside of the flange is drawn uniformly in radial direction 200 PSfrag replacements a2 200 u a1 200 200 Figure 33: Drawing of a circular flange in plane strain. The inner boundary is pulled towards the center up to a final deformation of u = 75mm. The crosses indicate the tangent vectors of the slip planes aligned at 60◦ to each other. All length in mm. towards the center until a final deformation of u = 75mm is reached. In the computation increments of ∆u = 0.1mm are used. The slip systems are inclined at 60◦ to each other. Their orientation is specified by the crosses in the figure. The elastic behaviour is described by three elastic constants. For the purely isotropic parameters we set µ = 75000N/mm 2 and β = 0.8067. The cubic coupling parameter is set to α = −51500N/mm2 . This choice reflects the elastic behaviour of copper. The non-linear plastic response is governed by a hardening modulus of h = 100N/mm2 , an initial yield stress of y0 = 80N/mm2 and the saturation stress of y∞ = 110N/mm2 . The saturation parameter is set to ω = 16, the viscosity is η = 0.1Ns/mm2 . The specimen is discretized by 644 six-node triangular elements. The computation is performed with the implicit algorithms U1 and V1. Figure 34 shows the results of the simulation. One clearly sees the softer behaviour along the a. b. c. Figure 34: Drawing of a circular flange in plane strain. Distribution of the equivalent plastic strain for displacements of (a) 24mm, (b) 48mm and (c) 75mm of the inner boundary. 111 Approaches to Anisotropic Plasticity vertical axis a2 that has smaller angles with the slip systems than the horizontal axis a1 . The distribution of the equivalent plastic strain obtained with both algorithms are indistinguishable. 6.6. Model Problem: Plasticity based on Quadratic Functions 6.6.1. Elastic Response We now specify the free energy (6.20) for anisotropic elasticity. The function is assumed to depend on the elastic Hencky strain tensor defined in the intermediate configuration. Restricting the considerations to quadratic functions, the canonical form reads ψ(C̄, A) = 1 2 e e kĒ kĒ + ψi (A) where Ē = 1 2 ln[C̄] (6.172) where ψ i = 12 hA2 +(y∞ −y0 )(A+ ω1 exp[−ωA]) accounts for non-linear isotropic hardening. √ e e e Here kĒ kĒ := Ē : Ē : Ē is the norm of the Hencky strain tensor with respect to a constant fourth-order tensor Ē that characterizes the macroscopic elasticity moduli associated with a fictitious lattice of the intermediate configuration. Due to the fact that Ē can be derived from a potential function as discussed in section 4.6 and the symmetry e of Ē and the logarithmic stresses ψ,Ē e , the elasticity tensor has the major and minor symmetries, i.e. ĒĀB̄ C̄ D̄ = ĒC̄ D̄ĀB̄ = ĒB̄ ĀC̄ D̄ = ĒĀB̄ D̄C̄ . (6.173) The scalar variable A accounts for energy storage due to micro-stress fields. 6.6.2. Plastic Response For the inelastic response, the level-set function (6.22) is specified for multi-surface plasticity by f α (Σ̄, B) = kΣ̄kH̄α + q 2 B 3 . (6.174) The unsymmetric tensor Σ̄ = C̄ S̄ is referred to as Mandel tensor, cf. (6.9)2 , and B is the stress variable dual to A. The constant fourth-order tensors H̄α are assumed to possess the major and minor symmetries, i.e. B̄ C̄ D̄ D̄ ĀB̄ ĀC̄ D̄ B̄ D̄ C̄ H̄Ā = H̄C̄ = H̄B̄ = H̄Ā . α α α α (6.175) In order to model incompressible plastic flow, the plasticity tensors are restricted to the deviatoric subspace of the fourth-order tensors possessing major and minor symmetries (6.175) by the constraints H̄α : 1 = 0 . (6.176) Then obviously H̄α = P : H̄α : P where P is the deviatoric projection tensor defined by P := I − 13 1 ⊗ 1. In the context of single surface plasticity, (6.174) with restrictions (6.176) leads to a yield criterion which is similar to the widely-used Hill-criterion derived in the classical textbook by Hill [50]. The construction of a coordinate-free deviatoric representation of H̄ for some types of anisotropy is discussed in the next section. 112 Approaches to Anisotropic Plasticity Box 4: Model Problem for Hill-Type Plastic-Map Plasticity F e := F F p−1 , C̄ := F eT gF e p p Ē := 21 ln[C̄] , L̄ = Ḟ F p−1 e free energy ψ = 21 ||Ē ||Ē + 12 hA2 + (y∞ − y0 )(A + ω1 exp[−ωA]) e e stresses S̄ = 2(Ē : Ē ) : Ē ,C̄ plastic force Σ̄ = C̄ S̄ hardening stress B = −ψ,A = −hA − (y∞ − y0 )(1 − exp[−ωA]) eα level set functions f = ||Σ̄||H̄α p iα f = 2/3 B P α α p λα N̄ ; N̄ = H̄α : Σ̄/||Σ̄||H̄α flow rules L̄ = p Pα α Ȧ = N α = 2/3 α λα N ; loading conditions λα ≥ 0, f α ≤ cα , λα (f α − cα ) = 0 kinematics 1. 2. 3. 4. 5. 6. 6.6.3. D2h -, Oh -, D∞h and O(3)-invariant Fourth-Order Tensors We now discuss a unified representation for the constant fourth-order tensors Ē and H̄ that covers the symmetries described by the symmetry group D2h , cubic symmetry with symmetry group Oh , transversal symmetry with symmetry group D∞h and isotropic response by appropriate choices of parameters. Tensors possessing theses symmetries were already derived in section 4, equations (4.111), (4.192) and (4.219). In the following, an alternative function basis will be given that leads to a convenient and simple structure of the orthotropic fourth-order tensor. Furthermore we comment on how to satisfy the incompressibility condition (6.176). Recall the function basis already specified for D2h -invariant functions in (4.105) and (4.106). For a symmetric second-order argument tensor η it reads I1 = {tr[η], tr[η 2 ], tr[η 3 ], tr[M η], tr[M 2 η], tr[M η 2 ], tr[M 2 η 2 ]} . (6.177) This basis can be transformed to the well-known alternative basis I2 = {tr[η], tr[η 2 ], tr[η 3 ], tr[m1 η], tr[m2 η], tr[m1 η 2 ], tr[m2 η 2 ]} (6.178) for orthotropic response in terms of two of the structural tensors mi := ai ⊗ ai for i = 1, 2, 3. Based on these invariants, we develop another basis I3 yielding a convenient and transparent representation for the fourth-order tensor. To this end, at first we reformulate the isotropic invariants of η by ¾ tr[η] = tr[m1 η] + tr[m2 η] + tr[m3 η] (6.179) tr[η 2 ] = tr[m1 η 2 ] + tr[m2 η 2 ] + tr[m3 η 2 ] and replace tr[η 3 ] by det[η] by using Cayley-Hamilton’s theorem. With the identity tr[mi η 2 ] − tr[mi η]2 = tr[mij η]2 + tr[mik η]2 (6.180) that is valid for every even and odd permutation of (i, j, k), by insertion of (6.179) and (6.180) into (6.178) we obtain the new function basis I3 = {tr[m1 η], tr[m2 η], tr[m3 η], tr[m12 η]2 , tr[m23 η]2 , tr[m13 η]2 , det[η]} (6.181) 113 Approaches to Anisotropic Plasticity with the definitions mij := 21 (ai ⊗aj +aj ⊗ai ). A coordinate form of this coordinate-free representation has been proposed by Smith & Rivlin [131]. Having this basis at hand, we define the fourth-order tensor as the second derivative of an isotropic function χ, i.e. 2 M := ∂ηη χ(I1 , . . . , I7 ) , (6.182) where Ii=1...7 denote the elements of the function basis I3 . For a quadratic function of the form χ = 21 α1 I12 + 21 α2 I22 + 12 α3 I32 + α 4 I1 I2 + α 5 I2 I3 + α 6 I1 I3 + α 7 I4 + α 8 I5 + α 9 I6 (6.183) we obtain the constant fourth-order structural tensor M = α 1 m1 ⊗ m 1 + α 2 m2 ⊗ m 2 + α 3 m3 ⊗ m 3 + α4 Sym[m1 ⊗ m2 ] + α5 Sym[m2 ⊗ m3 ] + α6 Sym[m1 ⊗ m3 ] + 2α7 m12 ⊗ m21 + 2α8 m23 ⊗ m32 + 2α9 m13 ⊗ m31 . (6.184) Here we use the abbreviation Sym[mi ⊗mj ] := 12 (mi ⊗mj +mj ⊗mi ). Clearly, the tensor M has the major and minor symmetries. A comparable approach to the construction of integrity bases and structural tensors functions for orthotropic anisotropy is outlined in Boehler [21]. In a Cartesian coordinate system aligned to the axes of orthotropy {ai }i=1,2,3 , the tensor appears in the simple coordinate representation α1 α4 α6 0 0 0 α2 α5 0 0 0 α 0 0 0 3 M= (6.185) 1 α 0 0 2 7 1 α 0 sym. 2 8 1 α 2 9 in terms of nine material parameters. The deviatoric property (6.176) is satisfied for the three dependencies α4 = 1 2 (α3 − α1 − α2 ) ; α5 = 1 2 (α1 − α2 − α3 ) ; α6 = 1 2 (α2 − α1 − α3 ) (6.186) of the parameters. The representation (6.185) of the orthotropic fourth-order tensor includes other symmetry types as special cases: (i) Transverse isotropy with the only preferred direction a1 yields the dependencies α1 6= α2 = α3 , α4 = α6 6= α5 , α8 = α2 − α5 ) 6= α7 = α9 . (6.187) (ii) Cubic symmetry has equal material response with respect to all three axes, i.e. α1 = α 2 = α 3 , α4 = α5 = α6 , α7 = α8 = α9 . (6.188) (iii) Isotropy has no privileged direction and is obtained by setting α1 = α 2 = α 3 = α 4 + α 7 , α4 = α5 = α6 , α7 = α8 = α9 . (6.189) The elasticity tensor Ē in the free energy function is specified to anisotropic response by setting Ē = M. The function is then described in terms of nine elastic constants 114 Approaches to Anisotropic Plasticity α1,...,9 . For isotropic elastic material behaviour, these are related to the well-known Lamé constants λ and µ by α1,2,3 = λ + 2µ, α4,5,6 = λ and α7,8,9 = 2µ. Setting H̄ = M specifies the plasticity tensor for the considered symmetries. Orthotropic plastic yielding for incompressible plastic flow is governed by the six parameters α 1,2,3 and α7,8,9 while α4,5,6 are determined by condition (6.186). These parameters are related to the initial yield stresses yij with respect to the principal axes of orthotropy. For the simple tension p modes one obtains, by evaluating the level-set function (6.174) with threshold c = 2/3y0 in box 4 line (4), the relations α1 = 2 y02 ; 2 3 y11 α2 = 2 y02 ; 2 3 y22 α3 = 2 y02 . 2 3 y33 (6.190) 1 y02 ; 2 3 y23 α9 = 1 y02 . 2 3 y13 (6.191) For the three simple shear modes we get α7 = 1 y02 ; 2 3 y12 α8 = √ The isotropic case is characterized by y11 = y22 = y33 = y0 and y12 = y23 = y13 = y0 / 3. 6.6.4. Comparison of the Stress Update Algorithms We now compare the proposed four stress update algorithms for multiplicative anisotropic plasticity U1: standard implicit, U2: standard explicit, V1: variational implicit and V2: variational explicit. Therefore a simple shear test and a simple tension test are investigated. The considered deformation gradients that drive the tests are 1 γ 0 λ 0 0 0 ei ⊗ ej , F := 0 1 0 ei ⊗ ej and F := 0 f (λ) (6.192) 0 0 1 0 0 f (λ) respectively. The simple shear test is parameterized by the shear parameter γ. To obtain the corresponding stresses a single pass through the return scheme is required. The simple tension test is parameterized by the stretch λ along the e1 axis. The contraction f (λ) of the material in the direction perpendicular to the stretch direction has to be determined iteratively so that the stresses along e2 and e3 vanish. For that test, the return algorithm is passed several times in a single time step. The computations are carried out using a single surface Hill-type plasticity model as discussed in section 6.6. The material parameters are κ = 164200N/mm2 and µ = 80190N/mm2 for isotropic elasticity. Two different symmetry properties of the level2 set function are investigated. The reference yield stress is set to y0 = 450N/mm √ . An O(3)-invariant function is then completely determined, i.e. y11 = y0 and y12 = y0 / 3. For Oh -invariant cubic symmetry we set y11 = y0 , y12 = 129.904N/mm2 . Thus the resistance to shear stresses is weakened compared with the isotropic case. The principal axes of anisotropy ai coincide with the global coordinate axes. All computations are carried out for rate-independent ideal elasto-plasticity. The load-displacement curves for the tests specified in (6.192) are plotted in figure 35. The quality of the stress response depends on the size of the increments that is used to reach the final shear deformation of γ = 4 and the final stretch of λ = 4. The size of the 115 Approaches to Anisotropic Plasticity stretch λ angle [◦ ] 290 280 stress τ12 [N/mm2 ] shear parameter γ 270 260 stress τ11 [N/mm2 ] stress τ12 [N/mm2 ] Sfrag replacements tress τ11 [N/mm2 ] Isotropic Material Response PSfrag replacements 500 250 240 U1 U2 V2 U1 V1 V1 230 220 210 200 a. angle [◦ ] 1.E-02 1.E-03 1.E-03 1.E-01 1.E-01 1.E-02 480 460 440 420 400 380 360 0 0.5 1 1.5 2 2.5 3 3.5 4 0 b. shear parameter γ 0.5 1 1.5 2 U1 U2 V1 V1 V2 V2 1.E-02 1.E-01 1.E-02 1.E-01 1.E-02 1.E-01 2.5 3 3.5 4 3.5 4 3.5 4 stretch λ Cubic Material Response 300 stretch λ angle [◦ ] 300 250 PSfrag replacements stress τ11 [N/mm2 ] 200 stress τ12 [N/mm2 ] stress τ12 [N/mm2 ] Sfrag replacements tress τ11 [N/mm2 ] 150 100 U1 U1 U1 V1 V1 50 0 c. 0 0.5 1 1.5 2 stretch λ angle [◦ ] 5.E-01 1.E-01 1.E-02 1.E-01 1.E-02 2.5 3 3.5 250 200 150 100 50 0 4 d. shear parameter γ Sfrag replacements angle [◦ ] 0.5 1 1.5 2 1.E-02 1.E-03 1.E-03 1.E-02 2.5 3 shear parameter γ 60 480 PSfrag replacements stress τ11 [N/mm2 ] stress τ12 [N/mm2 ] 460 angle [◦ ] stress τ11 [N/mm2 ] tress τ12 [N/mm2 ] shear parameter γ 0 U1 U2 V2 V2 440 420 400 380 e. 0 0.5 1 1.5 2 U1 V1 V1 V2 V2 U2 1.E-02 1.E-02 1.E-01 1.E-01 1.E-02 1.E-02 2.5 3 stretch λ stretch λ 40 20 0 −20 −40 −60 F Fe Fp Fe + Fp −80 −100 3.5 4 −120 f. 0 0.5 1 1.5 2 2.5 3 shear parameter γ Figure 35: Comparison of the stress update algorithms U1: standard implicit, U2: standard explicit, V1: variational implicit and V2: variational explicit. Isotropic material response: (a) simple shear test and (b) simple tension test. Cubic material response: (c) and (d) simple shear test, (e) simple tension test and (f) angle of rotation from polar decomposition of the deformation maps. The numbers denote the size of the time increment used. Note the ranges of the τ11 -axes in (b) and (e)! 116 Approaches to Anisotropic Plasticity increments was increased as long as the results were reasonable and the thus obtained curves are plotted in figure 35. For the shear test with the isotropic material the results obtained and depicted in figure 35a can be summarized as follows. Deviations from the solutions obtained with the algorithms U1, U2 and V2 that use the yield criterion function of the continuous setting are observed for the algorithm V1. This is mainly because the latter uses algorithmic stresses that enter the yield function. The increment of 1.E-01 is too large for V1, the results obtained with 1.E-02 does not improve for smaller steps. Figure 35b shows the tensile stresses for the tension test. The algorithms U1, U2 and V2 all tend to an identical result while V1 shows slight deviations. Note the range of the stress axis! Figures 35c-f document the results for cubic material behaviour. The cubic symmetry is reflected in the periodical stress response. This is because of the linearly increasing rotation of the flow system documented in figure 35f. Recall that the rigid body rotation of the flow system is part of the elastic map F e . In figure 35c the implicit algorithms U1 and V1 are compared. For a step size of 1.E-02 both yield identical results. For small steps, the explicit algorithms U2 and V2 give the same stress response. It is identical to the one obtained with the implicit algorithm U1 but the step size is only a tenth of that for the latter. If one increases the step size, the algorithms become unstable as documented in figure 35d. Finally, figure 35e shows the results for the simple tension test. As for the isotropic material, the stresses from algorithm V1 slightly deviate from the results of the other algorithms, even when the step size is refined. Note again the range of the stress axis. 6.6.5. Kelvin-Mode Decomposition of Fourth-Order Tensors A class of phenomenological multi-surface elasto-plasticity models bases on the spectral decomposition of the plasticity tensor that governs the anisotropy of the level-set function. The concept of this so-called Kelvin-mode decomposition dates back to Kelvin [59] and is topic of several publications. Rychlewski [110] uses this idea in the context of thermo-elasticity. Qi & Bertram [104] apply this concept to the description of creep damage of single crystal superalloys and Mahnken [69] treats the modeling of creep phenomena. Applications in context of the modeling of elastic plastic material behaviour is the contents of the works of Schreyer & Zuo [117] and Arramon, Mehrabadi, Martin & Cowin [3]. A constitutive elasto-plasticity model in the logarithmic strain space is recently discussed by Himpel [51]. The spectral decomposition of the elasticity tensor is topic of Mehrabadi & Cowin [74] and Sutcliffe [135]. In the latter, a collection of elastic constants for several crystalline materials is given. The graphical representation of fourth-order elasticity tensors is contents of Böhlke & Brüggemann [23]. The Kelvin-mode decomposition of a fourth-order tensor with major and minor symmetries is outlined in appendix A.2. In general, for given orientation, the eigen-bases depend on the coordinates of the tensor that is to be decomposed. The only two exceptions are isotropic and cubic tensors. For an isotropic tensor the well-known volumetric-isochoric decomposition of the unit tensor holds I sym = Pvol + Piso with ½ Pvol := 13 1 ⊗ 1 Piso := Isym − Pvol . (6.193) 117 Approaches to Anisotropic Plasticity For a cubic tensor, where the axes of anisotropy coincide with the global coordinate axes, the decomposition reads ½ P3 Pcub1 = Ni ⊗ Ni sym Pi=2 (6.194) I = Pvol + Pcub1 + Pcub2 with 6 Pcub2 = i=4 N i ⊗ N i where N i are the eigen-tensors of the cubic tensor. For the first cubic projection tensor, they are defined by 0 0 0 −2 0 0 1 1 0 , N3 = 0 1 0 . N2 = 0 1 (6.195) 2 6 0 0 −1 0 0 1 The associated deformation mode is a volume-preserving distortion of a unit cube where all faces remain rectangles. The definition of the second cubic projection tensor bases on the three eigen-tensors 0 1 0 0 0 0 0 0 1 1 1 1 (6.196) N4 = 1 0 0 , N5 = 0 0 1 , N6 = 0 0 0 2 2 2 0 0 0 0 1 0 1 0 0 that are associated to shear deformations of a unit cube along the anisotropy axes. A cubic Hill-type plasticity model with plastic incompressible flow is then defined by two level-set functions in terms of the cubic projection tensors, i.e. 1 f (Σ) := kΣkPcub1 + q 2 B 3 2 and f (Σ) := kΣkPcub2 + q 2 B 3 . (6.197) 6.7. Constitutive Model for Plastic-Metric Plasticity In the remainder of this chapter we discuss a class of plasticity models that base on the introduction of a plastic metric instead of a plastic map. A special class of models is obtained for an additive combination of the total deformation measure C and the plastic metric. In conjunction with logarithmic strains these models turn out to have a structure identical to the geometric linear theory and are proven to be very powerful. Because of that this topic is treated separately in the next chapter. 6.7.1. Energy Storage and Elastic Stress Response A special class of plasticity models beside the plastic-map approach can be derived from the latter by proposing invariance of the intermediate configuration with respect to superimposed rigid body rotations, cf. Casey & Naghdi [30]. Recall the general form of an objective free energy function, i.e. ψ = ψ(C, F p ) . (6.198) Superimposed rotations onto the intermediate configuration restrict the free energy to ψ(C, QF p ) = ψ(C, F p ) ∀Q ∈ SO(3) . (6.199) 118 Approaches to Anisotropic Plasticity In complete analogy to the introduction of the right Cauchy-Green tensor as an objective measure into the free energy, this invariance condition is a priori satisfied by the reduced form ψ(C, Gp ) where Gp := F pT ḠF p . (6.200) The second-order tensor Gp is denoted as plastic metric because of its symmetry and positive definiteness. In contrast to the unsymmetric plastic map F p ∈ SL(3) the plastic metric Gp ∈ Sym+ (3) cannot describe rotations. Observe that the function ψ in (6.200) solely depends on Lagrangian quantities. Within a plastic-metric framework the introduction of an intermediate configuration is superfluous. The plastic metric describes the plastic deformation of the material and is governed by a constitutive evolution law with initial condition Gp (t = t0 ) = Gp0 . In the subsequent development we consider additional Lagrangian internal variables that are assembled in the generalized vector q. For the functional dependency ψ = ψ(C, Gp , q) exploitation of the dissipation inequality ρ0 D = S : definitions of stress variables S := 2ψ,C , S p := −ψ,Gp (6.201) 1 2 Ċ − ψ̇ ≥ 0 motivates the following and Q := −ψ,q . (6.202) With these definitions, the reduced form of the dissipation inequality reads ρ0 D = S p : Ġp + Q · q̇ ≥ 0 . (6.203) 6.7.2. Dissipation and Plastic Flow Response Consider a non-smooth convex elastic domain E that bounds the internal forces E := {(S p , Q)|f α (S p , Q) − cα ≤ 0 ; α = 1, . . . , m} (6.204) in terms of m level-set functions. The evolution equations for the internal variables are obtained postulating maximal dissipation for the deformation process. This ansatz gives a constrained optimization problem that can be solved with a Lagrangian multiplier method. It has the solution P P iα eα and q̇ = α λα f,Q (6.205) Ġp = α λα f,S p together with the loading conditions λα ≥ 0, (f α − cα ) ≤ 0 and λα (f α − cα ) = 0. 6.7.3. Decoupling of the Constitutive Functions For the subsequent developments, we consider a decoupling of the constitutive functions into two parts, ψ(C, Gp , Q) = ψ e (C, Gp ) + ψ i (q) . (6.206) Yield criteria formulated in the reference configuration restrict the stress according to φα = f eα (S p ) + f iα (Q) − cα ≤ 0 . (6.207) 119 Approaches to Anisotropic Plasticity 6.7.4. Continuous Tangent Moduli As the proposed elasto-plasticity model is formulated with respect to the reference configuration the derivation of the Lagrangian continuous tangent moduli is straight forward. They link the rate of the right Cauchy-Green tensor to the rate of stresses, i.e. Ṡ = Cep : 1 2 Ċ . (6.208) The stress tensor is defined in (6.202)1 . Together with the decoupled ansatz of the free energy function (6.206) and the evolution equation (6.205)1 its rate takes the form P eα p . Ṡ = 2S ,C : 12 Ċ + α λα S ,Gp : f,S (6.209) The amounts λα of plastic flow are determined by the consistency conditions f˙α = 0 in the case of plastic loading of the flow systems α. Thus we obtain P p eβ (6.210) λα = − β [gαβ ]−1 {f,S p : S ,C : Ċ} iβ eβ p iα eα with the matrix gαβ := f,S p : S p : f p + f,Q · Q,q · f ,Q . Insertion into (6.209) then gives ,S G by comparison with (6.208) the elasto-plastic tangent moduli Cep := C + 4 P P α β [gαβ ] −1 p eβ eα S pT ,C : f,S p ⊗ f,S p : S ,C . (6.211) Here we have exploited the interchangeability of the second derivatives S ,Gp = −2S pT ,C . 6.8. Algorithmic Formulation 6.8.1. Implicit Stress Update Algorithm The evolution equations (6.205) and the loading conditions are discretized using an implicit scheme. In a typical time increment [tn ; tn+1 ] we get the following set of equations P Gp = Gpn + P α∈A γα N α q = q n + α∈A γα M α (6.212) α α γα ≥ 0 , f − cα ≤ 0 , γα (f − cα ) = 0 . α iα eα := f,Q . The set A contains the numbers of the flow systems that Here, N α := f,S p and N are active. The constrained set of equations (6.212) is solved iteratively with a Newton scheme. Therefore, the residuals P RG := −Gp + Gpn +P α∈A γα N α = 0 Rq := −q + q n + α∈A γα M α = 0 (6.213) α Rfα := f − cα = 0 are defined, which have to vanish in the solution point at the end of the time step. For its determination see the exposition in section 6.3.1. The linearization at fixed C with respect to the variables Gp , q and γα reads P Lin RG = RG − [I − N : S p,Gp ] : ∆Gp + α∈A ∆γα N α = 0 P = 0 Lin Rq = Rq − [I − M · Q,q ] · ∆q + α∈A ∆γα M α (6.214) p p α α α α = 0. Lin Rf = Rf + N : S ,Gp : ∆G + M · Q,q · ∆q 120 Approaches to Anisotropic Plasticity Here we have abbreviated the weighted sum over the second derivatives of the level-set function with P P iα eα p p . (6.215) and M := α∈A γα f,QQ N := α∈A γα f,S S To obtain a short and compact representation, the following definitions are introduced Ξ := [ (S p,Gp )−1 + N ]−1 and Υ := [ (Q,q )−1 + M ]−1 . (6.216) Within the considered algorithm, the stresses are determined for prescribed deformations, i.e. ∆C = 0. This system of linear equations in the increments is solved for the incremental plastic multiplier. The result is P P (6.217) ∆γα = − α∈A β∈A [gαβ ]−1 (Rf β + N β : Ξ : RG + M β · Υ · Rq ) in terms of the matrix gαβ := N α : Ξ : N β + M α · Υ · M β . Now the internal variables can be updated according to P Gp ⇐ Gp + [I − N : S p,Gp ]−1 : (RG + α∈A ∆γα N α ) P q ⇐ q + [I − M · Q,q ]−1 · (Rq + α∈A ∆γα M α ) γα ⇐ γα + ∆γα . (6.218) (6.219) p P The iteration has terminated when kRG k2 + kRq k2 + α∈A (Rf α )2 < tol, where tol denotes the machine-dependent numerical zero. 6.8.2. Algorithmic Tangent Moduli The algorithmic tangent moduli are the discrete counterpart of the continuous moduli in equation (6.211). They are defined by ∆S = Calgo : 1 2 ∆C . (6.220) In contrast to the situation in the stress update algorithm, now the global deformation prescribed by C is no more constant. In the solution point determined by the stress update algorithm, the residuals in (6.213) are zero. Then from (6.212) the increments of the internal variables ¾ P ∆Gp = (S p,Gp )−1 : Ξ : ( α∈A N α ∆γα + N : S p,C ) : ∆C P (6.221) ∆q = (Q,q )−1 · Υ · α∈A M α ∆γα are obtained. The linearization of an active consistency condition (6.212)3 in the solution point reads f˙α = N α : S p,Gp : ∆Gp + M α · Q,q · ∆q + N α : S p,C : ∆C = 0 . (6.222) Together with the increments in (6.221) the algorithmic parameter is obtained as P ∆γα = − β∈A [gαβ ]−1 {N β : Ξ : N : S p,C + N α : S p,C } : ∆C (6.223) 121 Approaches to Anisotropic Plasticity with the matrix gαβ as defined in (6.218). The linearization of the second Piola-Kirchhoff stresses ∆S = C : 1 2 ∆C + S Gp : ∆Gp (6.224) then yields together with (6.221)1 the algorithmic tangent moduli as p −1 : Ξ : N : S p,C Calgo := C − 4 S pT ,C : (S ,Gp ) P P p −1 : Ξ : N α ⊗ (N β : Ξ : N + N β ) : S p,C . + 4 α∈A β∈A S pT ,C : (S ,Gp ) (6.225) 123 Additive Plasticity in the Logarithmic Strain Space 7. Additive Plasticity in the Logarithmic Strain Space This section discusses the essential steps of a Lagrangian geometric approach to anisotropic finite plasticity in terms of a plastic metric that was motivated in chapter 6. The restriction of the framework to the logarithmic strain space yields a modular structure. It consists of a pre- and post-processing module that surrounds a constitutive model. The structure of that model is identical with the geometric linear theory. The subsequent discussion follows the ideas of the recent paper by Miehe, Apel & Lambrecht [87]. 7.1. Kinematic Approach in Terms of a Plastic Metric PSfrag replacements 7.1.1. Current Metric, Plastic Metric and Stresses ϕ C, Gp , S p g, cp , τ F = ∇ϕ x X S B Figure 36: Kinematic setting for additive plasticity. A point X of the Lagrangian configuration B is mapped by the non-linear point map ϕ onto the point x of the Eulerian configuration S. The total deformation is measured by the convected metrics C and g in the reference and current settings, respectively. The plastic deformation is governed by the reference plastic metric Gp and its current counterpart cp . Within a Lagrangian setting, the total deformation of a body B can be measured with the a priori objective right Cauchy-Green-tensor C := F T gF (7.1) which represents the current metric g in the Lagrangian manifold. According to the works of Coleman & Gurtin [32] and Lubliner [68], the history of the inelastic deformation process can be described by internal variables. Following the works of Miehe [82, 83] the plastic deformation is to be described in terms of a so-called plastic metric Gp ∈ Sym+ (3). It is a co-variant second-order tensor which develops within the elastic-plastic deformation process starting from the initial condition Gp (t0 ) = G , (7.2) where G is the Lagrangian metric tensor defined in (2.26) and t0 denotes the time at the beginning of the deformation process which is assumed to be stress free. 7.1.1.1. Geometric Preprocessing into the Logarithmic Strain Space. A key point in the setting up of a framework of finite plasticity is the definition of an elastic strain measure E e which enters the stored free energy function. We assume this strain measure to be a function of the above introduced current and plastic-metric tensors, i.e. E e = E e (C, Gp ) . (7.3) 124 Additive Plasticity in the Logarithmic Strain Space In Miehe [82, 83] several possible definitions of the elastic strain variable are given. Here we consider the elementary additive form E e := E − E p (7.4) in terms of the logarithmic Lagrangian total and plastic strains E := 1 2 ln[C] and E p := 1 2 ln[Gp ] , respectively. A key property of the logarithmic tensor function in the elasto-plasticity is the mapping of the large strain multiplicative plastic constraint onto an additive restriction. The structure of that additive same as the one in the small strain theory. Reformulation of (7.5)2 Jacobian p J p := det[Gp ] = exp[tr[E p ]] (7.5) context of metal incompressibility restriction is the gives the plastic (7.6) that is a measure for the change of volume due to plastic deformation. The constraint J p = 1 takes the additive form tr[E p ] = 0 (7.7) similar to the geometric linear theory. The one-to-one relation between G p in the Euclidean space and E p in the logarithmic space suggests the usage of E p as internal variable. Thus a constitutive model can be formulated exclusively in terms of quantities of the logarithmic strain space. 7.1.1.2. Constitutive Model in the Logarithmic Strain Space. As already stated, an important fact is that a constitutive model can exclusively be formulated in the logarithmic strain space. Assume therefore a functional dependence of the free energy on the logarithmic strain tensor E defined in (7.5)1 and a set of internal variables I := {E p , . . . } consisting of the logarithmic plastic strain tensor defined in (7.5)2 and some additional hardening variables. The rate of the total logarithmic strains is related to the rate of the right Cauchy-Green-tensor according to Ė = P : 1 2 Ċ with P := 2E ,C , (7.8) where P denotes a geometric fourth-order transformation tensor. Insertion of (7.8)1 into the dissipation inequality (2.63) then gives a representation exclusively in quantities of the logarithmic strain space, i.e. ρ0 D := T : Ė − ψ̇(E, I) ≥ 0 . (7.9) The constitutive model can be viewed as a “material box”. Its input is the logarithmic strain tensor and a set of internal variables. The output is the current stress tensor T dual to the logarithmic strain tensor and the associated elastic-plastic tangent moduli Eep , i.e. {E, I} ⇒ Model ⇒ {T , Eep } . (7.10) Additive Plasticity in the Logarithmic Strain Space 125 The elasto-plastic tangent moduli relate the rate of total logarithmic strains to the rate of logarithmic stresses Ṫ = Eep : Ė . (7.11) The attractive feature of the constitutive model in the logarithmic space is that it can preserve the structure of models of the geometrically linear theory. If so, also the structures of the algorithms are identical to those of small strain theory. 7.1.1.3. Geometric Postprocessing from the Logarithmic Strain Space. Once the stresses and moduli are obtained in the logarithmic space, they have to be mapped to the Lagrangian stresses and moduli. This step is a purely geometric transformation denoted as geometric postprocessing. The stresses and moduli are obtained by S := T : P and Cep := PT : Eep : P + T : L (7.12) where the transformation tensor P is defined in (7.8)2 and L := 2P,C = 4E ,CC is a sixthorder geometric transformation tensor. The elasto-plastic moduli relate the Lagrangian rate of strains to the stresses, i.e. Ṡ = Cep : 1 2 Ċ . (7.13) Closed-form spectral-decomposition-based algorithmic approaches to these types of tensors have been discussed in the context of Seth-Hill strain measures by Miehe & Lambrecht [88]. 7.2. Constitutive Model in the Logarithmic Strain Space This section specifies the constitutive model of equation (7.10) for metal plasticity. We take into account the modeling of elastic and plastic anisotropies as well as induced anisotropy effects as in the case of kinematic hardening. As already mentioned before, due to the modular structure and the logarithmic strains, the structure of geometrically linear constitutive models is preserved. 7.2.1. Energy Storage and Elastic Stress Response Consider a free energy function describing the energy with respect to unit reference volume that is locally stored in a material point of the form ψ = ψ(E e , q) . (7.14) The logarithmic elastic strains are defined in (7.4) and (7.5). q denotes a set of internal variables accounting for hardening effects. To obtain a thermodynamically consistent model, we insert (7.14) into the Clausius-Planck-inequality (7.9). According to the standard argumentation of rational thermodynamics or Coleman’s method, the inequality must hold for arbitrary processes. This motivates the definition of the stress-like variables T := +∂E e ψ(E e , q) T p := −∂E e ψ(E e , q) (7.15) e Q := −∂q ψ(E , q) . 126 Additive Plasticity in the Logarithmic Strain Space The first equation is the constitutive equation for the stresses in the logarithmic strain space. The remaining two define so-called thermodynamical forces, dual to the internal variables. Observe that the additive combination of E and E p results in T p = T . The forces and internal variables are combined in the sets F := {T , Q} and I := {E p , q}, respectively. With these definitions, the dissipation inequality (7.9) reduces to ρ0 D := F · İ = T : Ė p + Q · q̇ ≥ 0 (7.16) where Q · q̇ denotes the generalized scalar product of the dual quantities assembled in the sets Q and q, respectively. 7.2.2. Dissipation and Plastic Flow Response Consider the plastic flow to be constrained by a non-smooth convex elastic domain in the space of the plastic forces E := {(T , Q) | f α (T , Q) ≤ cα ; α = 1, . . . , m} (7.17) which is defined in terms of m scalar-valued functions denoted as level-set functions f α = f α (T , Q) (7.18) and constant threshold parameters cα ≥ 0 associated with the initial yield stresses acp cording to cα := 2/3y0 . The level-set functions are gauges with respect to the thermodynamical forces T and Q. A canonical form of the evolution equations for the internal variables in the framework of an associative plasticity theory is determined by the constrained thermodynamical extremum principle of maximum plastic dissipation min {−ρ0 D} with f α (T , Q) − cα ≤ 0 ∀α ∈ [1, m] . (T ,Q)∈E (7.19) In general (7.19) is a non-linear coupled problem which cannot be solved directly. The solution bases on the definition of the Lagrange function with λα ≥ 0 P L(T , Q, λα ) := −T : Ė p − Q · q̇ + α λα (f α − cα ) → stat. (7.20) which converts the problem (7.19) into a saddle point problem. The solution is given by ∇L = 0 yielding the Karush-Kuhn-Tucker equations P ∂T L = 0 = −Ė p + α∈A λα ∂T f α P α . (7.21) ∂Q L = 0 = −q̇ + α∈A λα ∂Q f α ∂λ α L = 0 = f (T , Q) − cα where in the last equation (7.21)3 only the active constraints appear. They are combined in the active set defined by A := {α | f α (T , Q) − cα = 0} . (7.22) The internal variables are assumed to be zero at the beginning of the deformation process E p (t = t0 ) = 0 and q(t = t0 ) := 0 . (7.23) Additive Plasticity in the Logarithmic Strain Space 127 7.2.3. Considered Model Problem In metal plasticity one often assumes a decoupling ψ = ψ e + ψ p of the free energy function defined in (7.14) into two parts. ψ e describes the energy storage due to macroscopic lattice deformations and ψ p an energy storage due to micro-stress fields associated with dislocations and point defects, cf. Rice [107] for a micro-mechanical motivation. Here we consider a fully decoupled representation ψ = ψ e (E e ) + ψ k (A) + ψ i (A) ψe = ψk = ψi = 1 2 1 2 1 2 kE e k2E kkAk2 hA2 + (y∞ − y0 )(A + exp[−ωA]/ω) (7.24) in terms of the set q := {A, A} of internal variables consisting of 7 scalar fields. A is a symmetric second-order tensor for the description of the kinematic hardening and A a scalar variable that models isotropic hardening. The quadratic function kEk 2E = E : E : E in (7.24)2 is a typical example and can be replaced by any anisotropic function. For the structure of the fourth-order elasticity tensor E we refer to section 4.6. The material parameters k ∈ R+ and h ∈ R+ are associated with the kinematic and isotropic hardening, respectively. The dual forces are T = E : Ee , B = −kA and B = −hA − (y∞ − y0 )(1 − exp[−ωA]) . (7.25) A classical form of the yield function suitable for the description of the Bauschinger effect of kinematic hardening identifies the internal force B with a so-called back-stress. The level-set functions are assumed to have the particular functional dependencies f eα f α = f eα (T + B) + f iα (B) p = kT + BkHα , f iα = 2/3B (7.26) on a stress T + B relative to the negative back-stress B. Instead of the quadratic levelset function in (7.26) any other anisotropic function can be used. In the case of single surface plasticity, possible structures of the plasticity tensor H are given in section 4.6. Applications of the proposed form in multi-surface plasticity on the basis of Kelvin-modes can be found for example in Himpel [51]. To model plastic incompressible flow as is observed in metals, the plasticity tensors Hα have to be restricted according to Hα : 1 = 0 . (7.27) Insertion of (7.26)1 into (7.21) yields the evolution equations and loading conditions ) P P P Ė p = α∈A λα ∂T f eα , Ȧ = α∈A λα ∂B f eα , Ȧ = α∈A λα ∂B f iα (7.28) λα ≥ 0 , f α − c α ≤ 0 , λ α f α = 0 . As a consequence of the normality rule and the initial conditions (7.23) the kinematic hardening strains turn out to equal the plastic strains and with (7.25)2 a simple one-toone relation to the kinematic hardening stress is obtained, i.e. A = Ep and B = −kE p . (7.29) 128 Additive Plasticity in the Logarithmic Strain Space Box 5: Anisotropic Additive Plasticity in the Logarithmic Strain Space Geometric Preprocessor E := 12 ln[F T gF ] 1. 2. 3. 4. 5. 6. 7. 8. 9. Constitutive Model internal variables I := {E p , A, A} free energy ψ = 21 ||E − E p kE + k2 ||A||2 + ψ i (A) with ψ i = + h2 A2 + (y∞ − y0 )(A + ω1 exp[−ωA]) stresses T = E : (E − E p ) back stresses B = −kA internal stress B = −hA q level set functions f α = ||T + B||Hα + 23 B P flow rules Ė p = α λα Hα : (T + B)/||T + B||Hα evolution Ȧ = Ė p q P evolution Ȧ = 23 α∈A λα Geometric Postprocessor C = T : P with P := 2∂C E This response is often denoted as Prager-type kinematic hardening. A summary of the model is given in box 5. 7.2.4. Continuous Elastic-Plastic Tangent Moduli The elasto-plastic tangent moduli Eep relate the logarithmic total strain rate to the logarithmic stress rate Ṫ = Eep : Ė. The rates of the stress-like variables are Ṫ = E : Ė e , Ḃ = −K : Ȧ and Ḃ = −K Ȧ (7.30) where we have introduced the tensors E := ∂E e E e ψ, K := ∂AA ψ and K := ∂AA ψ to abbreviate the second derivatives of the free energy function. It follows from the consistency condition (7.28)6 that the evolutions of the level-set functions f˙α = ∂T f eα : Ṫ + ∂B f eα : Ḃ + ∂B f iα Ḃ = 0 (7.31) have to vanish for plastic loading, where λα 6= 0. With Ė e = Ė − Ė p and the evolution of the plastic strains (7.28)1 , the elastic stress rate takes the form X Ṫ := E : Ė − E : λα ∂T f eα . (7.32) α∈A Insertion of (7.32), (7.30)2,3 and (7.28)2,3 into (7.31) yields X f˙α = ∂T f eα : E : Ė − λβ gαβ = 0 (7.33) β∈A in terms of the symmetric matrix gαβ := ∂T f eα : E : ∂T f eβ + ∂B f eα : K : ∂B f eβ + ∂B f iα K∂B f iβ . (7.34) 129 Additive Plasticity in the Logarithmic Strain Space The plastic multipliers now can be obtained by solving (7.33) for λα , i.e. X λα = [gαβ ]−1 ∂T f eβ : E : Ė for α ∈ A . (7.35) β∈A Insertion into (7.32) provides the elasto-plastic tangent modulus for plastic loading Eep = E − XX α∈A β∈A [gαβ ]−1 E : ∂T f eα ⊗ ∂T f eβ : E . (7.36) 7.3. Algorithmic Formulation In this section we propose an algorithmic formulation of the constitutive elasto-plasticity model in the logarithmic strain space which was discussed in the previous sections. The key point is an implicit integration algorithm for the evolution equations of the internal variables (7.28)1,2,3 that accounts for the loading conditions (7.28)4,5,6 . 7.3.1. Stress Update Algorithm Consider a time interval [tn , tn+1 ] and let E pn = An and An be the initial data at time tn . Application of an implicit integration scheme to the evolution equations and loading conditions (7.28) gives the discrete equations P E p = E pn + α∈A γα ∂T f eα P iα , (7.37) A = An + α∈A γα ∂B f α α γα ≥ 0 , f − cα ≤ 0 , γ α f = 0 with the incremental plastic parameters γα := λα ∆t on the flow-systems. Note that only the active constraints enter (7.37). They are combined in a so-called active set A := {α | f α − cα = 0} (7.38) where f α := f α (T n+1 , B n+1 , An+1 ). In order to solve the problem (7.37) the first step is to check, whether plastic loading occurs or not. Therefore trial states are defined which are obtained by freezing the internal variables, i.e. E e? := E − E pn , E p? = E pn T ? := ∂E e ψ ? A? := An B ? := −∂A ψ ? (7.39) ? ? ? A := An B := −∂A ψ with ψ ? := ψ(E e? , A? , A? ). Insertion of these trial values into (7.37)4 gives the trial levelset functions f α? := f α (T ? , B ? , B ? ) and plastic loading occurs if at least on one flow system the loading conditions are not fulfilled, i.e. f α? − cα ≥ 0 for some α ∈ [1, m] ⇔ plastic loading . (7.40) In the case of plastic loading, the non-linear coupled system (7.37) has to be solved iteratively with a general return algorithm for γα , E p , A and A. Since the active set A at the end of the time step is not known from the beginning and may change during the iterative solution procedure, an active set search strategy is required. 130 Additive Plasticity in the Logarithmic Strain Space We apply a formalism identical to that for the algorithms U1 in plastic-map plasticity discussed in section 6.3.2. For the moment this set is assumed to be known. The iterative Newton-type solution algorithm bases on the definition of the residuals P RE := −E p + E pn + α∈A γα ∂T f eα = 0 RB := −A + An + γα ∂B f iα = 0 α Rf α := f (T , B, B) − cα = 0 ∀α∈A. A linearization of the above defined residuals yields the equations (7.41) P Lin RE = RE − ∆E p + α∈A [∆γα ∂T f eα − γα ∂T T f eα : A : ∆E p ] = 0 P Lin RA = RA − ∆A + α∈A [∆γα ∂B f iα − γα ∂BB f iα K∆A] = 0 Lin Rf α = Rf α − ∂T f eα : A : ∆E p + ∂B f iα K∆A = 0 (7.42) which have to vanish in the solution point. Within the time step the total logarithmic strains E are constant so that the kinematic relation ∆E p = −∆E e holds. Furthermore, the second derivatives of the free energy function are abbreviated by E := ψ,E e E e , K := ψ,AA and K := ψ,AA . (7.43) The fourth-order tensor appearing in (7.42) is defined by A := E+K. The strain residuals (7.42)1,2 can be solved for the strain increments. With the definitions Ē := (A−1 + P α∈A ∂T T f eα −1 ) and Ē := (K−1 + these strain increments are obtained as P α∈A ∂BB f iα −1 ¾ P ∆E p = A−1 : Ē : (REP+ α∈A ∆γα ∂T f eα ) ∆A = K−1 Ē (RA + α∈A ∂B f iα ) . ) (7.44) (7.45) Insertion into the linearized discrete consistency condition (7.42)3 gives the increments of the plastic multipliers ∆γα = X β∈A [gαβ ]−1 (f β − ∂T f eβ : A : Ē : RE − ∂B f iβ KĒRA ) (7.46) with the definition of the matrix gαβ = ∂T f eα : Ē : ∂T f eβ + ∂B f iα : Ē : ∂B f iβ . (7.47) Now the plastic multiplier can be updated according to γα ⇐ γα + ∆γα ∀α∈A. (7.48) Equations (7.45) determine the updates for the strain increments ∆E p = ∆A and ∆A. If any of the γα ∈ A at the end of the iteration is negative, we have to update the active set and restart the iteration. A summary of the algorithm is given in box 6. Additive Plasticity in the Logarithmic Strain Space Box 6: General Return Algorithm for Multi-Surface Elasto-Plasticity 1. Set initial values E e = E e,? = E − E pn , E p = E pn , A = A? = An , A = A? = An , γα = 0 and compute trial stresses T ? = T = ∂E e ψ, B ? = −∂A ψ, B ? = −∂A ψ 2. Compute trial states f α? = f (T ? , B ? , Bα? ) α = 1, . . . , m and setup active set† A = {α | f α? − cα > 0}. Check for plastic loading if (A = { }) −→ elastic else −→ plastic 3. Compute stress variables and the derivatives of the level set functions T = ∂E e ψ, N α = ∂T f eα , B = −∂A ψ, N α = ∂B f eα , B = −∂A ψ N α = ∂B f iα 4. Compute residuals and check tolerance P P RA = −A + An + α∈A γα N α RE = −E p + E pn + α∈A γα N α , P Rf α = f α ∀ α ∈ A RA = −A + An + α∈A γα , N α p P if kRE k2 + kRA k2 + |RA |2 + α∈A |Rf α |2 < tol → exit 5. Compute second derivatives and matrix gαβ E = ∂E e E e ψ, K = ∂AA ψ, K = ∂AA ψ A = E + K, P N = α∈A γα ∂T T f eα , P Ē = (A−1 + α∈A γα ∂BB f eα )−1 P Ē = (K−1 + α∈A γα ∂BB f iα )−1 , gαβ = N α : Ē : N β + N α ĒN β 6. Compute plastic increment for α ∈ A P ∆γα = β∈A [gαβ ]−1 (f β − N β : A : Ē : RE − N β KĒRA ) 7. Update internal variables Ep A A γα ⇐ ⇐ ⇐ ⇐ goto 2. endif † P E p + A−1 : Ē : (RE + α∈A N α ∆γα ) Ep P A + K−1 Ē (RA + α∈A N α ∆γα ) ∀α ∈ A γα + ∆γα ∀ α ∈ A Active set search strategy as in the algorithm U1 131 132 Additive Plasticity in the Logarithmic Strain Space 7.3.2. Algorithmic Tangent Moduli The algorithmic elastic-plastic tangent moduli relate the increment of the total strains to the increment of the stresses, i.e. ∆T = Eep : ∆E . (7.49) Point of departure for the derivation of Eep is the additive decomposition of the strains (7.4) together with the incremental elasticity law ∆T = E : (∆E − ∆E p ) (7.50) where E are the elastic moduli defined in (7.30)1 . The increments of the plastic strains are obtained from (7.37)1,2 as ¾ P ∆E p = Pα∈A A−1 : Ē : ∂T f eα ∆γα + A−1 : Ē : N : E : ∆E (7.51) iα −1 ∆A = α∈A K Ē ∂B f ∆γα . Here we have abbreviated the sum of the active second derivatives of the level set funcP tions by N := α∈A γα ∂T T f eα . The increments of the plastic multipliers remain to be determined. Therefore the incremental consistency conditions ∆f β = ∂T f eβ : E : ∆E − ∂T f eβ : A : ∆E p − ∂B f iβ K∆A = 0 have to be evaluated. Inserting the strain increments (7.51) gives P ∆γα = { β∈A [gαβ ]−1 (∂T f eβ : (I − Ē : N) : E} : ∆E , (7.52) (7.53) in terms of the matrix gαβ defined in (7.47). Evaluating the strain increments (7.51) and insertion into (7.50) then allows the identification of the algorithmic tangent moduli by comparison with (7.49). Defining the fourth-order tensors Ξ := E : A−1 : Ē = {(I − Ē : N) : E}T and B := (N−1 + A)−1 (7.54) leads to the compact representation of the symmetric consistent algorithmic elastic-plastic moduli Eep = E − E : B : E − XX α∈A β∈A [gαβ ]−1 (Ξ : ∂T f eα ⊗ ∂T f eβ : ΞT ) . (7.55) The transposition of the fourth-order tensors in (7.54) and (7.55) is associated with the first and second pairs of indices, i.e. [(•)ijkl ]T = (•)klij . 7.4. Variational Formulation in the Logarithmic Strain Space The constitutive model of elasto-plasticity outlined in section 7.2 belongs to the class of standard materials. The theoretical framework for this type of materials has already been dealt with in section 6.4. In the sequel we discuss the characteristics due to the logarithmic additive strain measure. The results obtained in the last chapter can be transferred to the logarithmic strain space when the deformation measure C is replaced by E and a functional dependence of the dissipation function on the flux of the internal variables only is considered. The latter is a consequence of the geometrical linear structure of the constitutive model inside the logarithmic strain space, cf. Miehe [85]. A detailed discussion can be found in Miehe, Apel & Lambrecht [87]. Additive Plasticity in the Logarithmic Strain Space 133 7.4.1. Standard Formulation of Inelasticity A constitutive model belonging to the class of generalized standard media is described by two fundamental constitutive functions, an energy storage function and a dissipation function. The energy storage function ψ is assumed to depend on the logarithmic strain E and an internal variable vector I. It governs the constitutive equation for the stresses T = ∂E ψ(E, I) (7.56) and the reduced dissipation inequality ρ0 D = F · İ ≥ 0 with F := −∂I ψ(E, I), see (7.15) and (7.16), respectively. The dissipation function φ is assumed to depend on the flux İ of the internal variables only. It determines the evolution of I by Biot’s equation ∂I ψ(E, I) + ∂İ φ(İ) = 0 with I(0) = I0 . (7.57) The two constitutive equations (7.56) and (7.57) determine the stress response of a smooth normal-dissipative material in a deformation-driven process where the strains E are prescribed. Based on the definition of the internal forces F , one introduces a dual dissipation function φ∗ depending on the forces F by the Legendre-Fenchel transformation φ∗ (F ) = supİ { F · İ − φ(İ) }. This induces the two alternative representations F = ∂İ φ(İ) and İ = ∂F φ∗ (F ) (7.58) of (7.57)1 . The reduced dissipation inequality ρ0 D = ∂İ φ(İ) · İ ≥ 0 serves as a fundamental physically-based constraint on the dissipation function φ. It is a priori satisfied by assuming φ convex and prescribing the properties φ(0) = 0 and φ(İ) ≥ 0. For rate-independent response and a first-order positively homogeneous φ, evaluation of the dissipation functions yields the dissipation, i.e. ρ0 D = φ(İ) ≥ 0 . (7.59) 7.4.2. Incremental Variational Formulation Next we discuss the construction of an integrated version of the constitutive equations giving a consistent approximation of the continuous differential equation (7.57) in a finite increment [tn , tn+1 ] ∈ R+ of time. Following the recent works Miehe [85] and Miehe, Apel & Lambrecht [87], we define an incremental stress potential function W depending on the logarithmic strains E n+1 := E(tn+1 ) at time tn+1 that determines the stresses at tn+1 by the quasi-hyperelastic function evaluation T n+1 = ∂E W (E n+1 ) . (7.60) Clearly, this function must cover characteristics of the storage function ψ and the dissipation function φ introduced above. To this end, we consider the variational problem W (E n+1 ) = inf I Z tn+1 [ ψ̇ + φ ] dt with I(tn ) = In . (7.61) tn For prescribed strains, this problem defines R t the incremental stress potential function W as a minimum of the generalized work tnn+1 [ψ̇ + φ]dt done on the material in the time 134 Additive Plasticity in the Logarithmic Strain Space increment under consideration. Starting with the given initial condition I(t n ) = In , the minimum problem defines an optimal path of the internal variables I(t) for t ∈ [t n , tn+1 ] including the right boundary value In+1 := I(tn+1 ). The two equations (7.60) and (7.61) provide an approximative variational counterpart of the continuous setting (7.56) and (7.57) of the constitutive equations in the discrete time step [tn , tn+1 ] under consideration. The proof that (7.61) represents a consistent point-wise approximation of Biot’s normaldissipative evolution equation (7.57) is analogous to the one given in section 6.4.2. Taking the derivative of W with respect to E n+1 we get ∂E W (E n+1 ) = ∂E ψ(E n+1 , In+1 ) (7.62) where In+1 is given by (7.57). Comparison with (7.56) shows the consistency of the potential equation (7.60) with the continuous setting. 7.4.3. Specification to Multi-Surface Models of Elasto-Plasticity For a known elastic domain E specified as in (7.17) by functions f α , the dissipation function φ may in the rate-independent case be defined by a generalization of the classical principle of maximum dissipation of plasticity proposed by Hill. It defines the dissipation function by the constrained maximum problem with inequality constraint φ(İ) = supF ∈E [F · İ] which can approximately be solved by a multiplier method P (7.63) φ(İ) = sup[ F · İ − α λα (f α (F ) − cα ) ] . F The Lagrange parameters λα are determined by the loading conditions λα ≥ 0, f α ≤ cα and λα (f α − cα ) = 0. Equation (7.63) may be interpreted as the Legendre-Fenchel P transformation of the dual dissipation potential φ∗ (F ) = α λα (f α (F ) − cα ). Insertion of (7.58)2 and exploiting the homogeneity of φ yields the one-dimensional representation of the dissipation function which is equal to the dissipation (7.59) P (7.64) ρ0 D = φ(λ1 , . . . , λm ) = α cα λα . Discretization of the variational problem (7.61) bases on an implicit integration algorithm A for the internal variables of the form A(E n+1 , γα ) = 0 . (7.65) Here, the internal variables are viewed as functions of the algorithmic incremental parameters γα := λα ∆t that are elements of the cone K := {γP α |γα ≥ 0}, i.e. I = I(γα ). A typical example is the backward Euler scheme I = In + α γα ∂F f α (F ). In the logarithmic strain space the level set functions depend on the plastic force only in contrast to the setting in multiplicative plasticity where the level set functions depend p on the Mandel stress tensor Σ̄ and therefore on the plastic force F = ḠP̄ and the dual internal variable F p , see (6.8) and (6.9). The integral of the dissipation function is discretized by a fully implicit integration scheme Z tn+1 φ dt = ∆t φ(γα /∆t) . (7.66) tn 135 Additive Plasticity in the Logarithmic Strain Space Box 7: Incremental Variational Formulation of Modular Finite Inelasticity 1. Geometric preprocessor. Let F n+1 ∈ GL(3) be the current deformation gradient and g the Eulerian standard metric. Get Lagrangian logarithmic strains E n+1 := 1 2 ln[F Tn+1 gF n+1 ] 2. Variational update of constitutive model in the logarithmic strain space. Solve incremental variational formulation for given data base {E n+1 , In } in [tn , tn+1 ] W (E n+1 ) = inf I Z tn+1 [ ψ̇ + φ ] dt tn for internal variables In+1 ∈ Rn at time tn+1 and compute stresses and moduli T n+1 = ∂E W (E n+1 ) and 2 En+1 = ∂EE W (E n+1 ) in the logarithmic strain space. Discretization by the algorithm in box 8. 3. Geometric postprocessor. The transformation tensors Pn+1 := 2∂C E n+1 and 2 Ln+1 := 4∂CC E n+1 map stresses and moduli from the logarithmic strain space to the Lagrangian space S n+1 = T n+1 : Pn+1 and ep T Cep n+1 = Pn+1 : En+1 : Pn+1 + T n+1 : Ln+1 Insertion of the integration algorithm for the internal variables and (7.66) into (7.61) defines the function W h (E n+1 , γ1 , . . . , γm ) = ψ(E n+1 , I h (E n+1 , γα )) − ψn + ∆tφ(γα /∆t)) (7.67) and the discretization of the variational problem (7.61) then reads W (E n+1 ) = inf W h (E n+1 , γ1 , . . . , γm ) . γα ∈K (7.68) A summary of the constitutive setting is given in box 7. 7.4.4. Algorithmic Solution The minimization problem (7.68) with inequality constraints can be solved by a Lagrange multiplier method. The solution is the saddle-point of the associated Lagrange function P (7.69) W (E n+1 ) = inf sup[W h (E n+1 , γα ) − α γα να ] γα να which is characterized by the loading conditions and the constraints W,γhα − να = 0 , να ≥ 0 , γα ≥ 0 , ν α γα = 0 . (7.70) 136 Additive Plasticity in the Logarithmic Strain Space The set (7.70) is equivalent to the restrictions W,γhα ≥ 0 , γα ≥ 0 and W,γhα γα = 0 (7.71) that determine the algorithmic parameters γα . In the case of plastic loading a nonempty set of active constraints exists A := {α | γα 6= 0} and an improved solution of the incremental parameters is obtained by a Newton step P γα ⇐ γα + ∆γα where ∆γα = − β∈A [W,γhα γβ ]−1 [W,γhβ ] ∀α ∈ A . (7.72) The iteration is terminated if the residual is below a given tolerance, i.e. P [ α∈A (W,γhα )2 ]1/2 ≤ tol . (7.73) During the iteration the set of active constraints may change. Here we apply the active set search strategy suggested by Miehe, Schotte & Lambrecht [90]. The solution of the minimization problem and the integration of the internal variables are coupled and have to be treated simultaneously. In any iteration step the update of the internal variables is performed according to P (7.74) In+1 ⇐ In+1 + α∈A In+1,γα ∆γα where the increments of the algorithmic multipliers are determined by (7.72). 7.4.5. Stresses and Moduli Once the constrained minimization problem (7.68) is solved, the stresses and elastic-plastic moduli are obtained by function evaluation of the derivatives of the incremental stress potential function W . According to (7.60), the derivative with respect to the strains E n+1 yield the stresses T n+1 . Application of the chain rule gives the expression P h (7.75) ∂E W = W,E + α∈A W,γhα γα,E . In the solution point the last term drops out due to (7.71)3 and so that the stresses are h T n+1 = W,E . (7.76) The sensitivity of the stresses with respect to the strains is governed by the algorithmic tangent moduli. Like the moduli in elasticity theory they are obtained by the second derivative of the stress potential function in the solution point P h h Eep (7.77) n+1 := ∂EE W (E n+1 ) = W,EE + α∈A W,Eγα ⊗ γα,E . The sensitivity of the incremental plastic parameter with respect to the strains is obtained by linearization of the necessary condition (7.71)3 . Inserting the result X (7.78) γα,E = − [W,γhα γβ ]−1 W,γhβ E β into (7.77) gives the algorithmic elastic-plastic moduli h Eep n+1 = W,EE − XX α∈A β∈A h ⊗ W,γhβ E . [W,γhα γβ ]−1 W,Eγ α (7.79) The softening part is a consequence of the change of the internal variables within the time step. The algorithm is summarized in box 8. Additive Plasticity in the Logarithmic Strain Space Box 8: Variational Update Algorithm in Logarithmic Strain Space History data: internal variables In = {E pn , An , An } 1. Set initial values γα=1...m = 0, A = An , 2. Determine current state of internal variables Perform integration step of algorithm A(E n+1 , γα ) = 0 3. Evaluate local minimization function W h = ψ(E n+1 , γα ) − ψn + ∆tφ(γα ) and its derivatives h W,E = ψ,E h W,EE = ψ,EE h W,Eγ = ψ,Eγα α W,γhα = ψ,γα + ∆tφ,γα W,γhα γβ = ψ,γα γβ + ∆tφ,γα γβ 4. Check convergence qP if ( α∈A W,γhα )2 ≤ tol goto 7 5. Perform Newton update of algorithmic parameters P γα ⇐ γα − β∈A [W,γhα γβ ]−1 W,γhβ 6. Check for negative parameters α? = argα∈A [min{γα |α ∈ A}] if γα? ≤ 0 then determine scaling parameter ξ := 1 − γα? /∆γα? perform scaling P γα ⇐hγα − (1 − ξ)∆γα h if ∆W := α∈A W,γα ≤ 0 remove flow system, A = A\α? goto 2 endif 7. Check necessary condition for minimum α? = argα=1...r [min{W,γhα ] if Wγhα? ≤ 0 then A ⇐ {A ∪ α? } goto 2 endif 8. Set stresses T n+1 = ∂E W h 137 138 Additive Plasticity in the Logarithmic Strain Space 7.4.6. Application to Model Problem We now apply the above discussed framework of incremental variational plasticity to the model problem of section 7.2.3. Recall the assumed structure of the stored energy function (7.24) and the level-set functions (7.26), i.e. ψ = ψ e (E e ) + ψ k (A) + ψ i (A) and f α = f eα (T + B) + f iα (B) . (7.80) The solution of the minimization problem (7.68) with a Newton scheme bases on the derivatives of the function W h with respect to the algorithmic parameters γα . The stresses and moduli require the derivatives with respect to the total strains. So the following expressions have to be specified ) h = ψ,γα + cα W,γhα W,E = ψ,E W,γhα E = ψ,γα E (7.81) h h W,γhα γβ = ψ,γα γβ WEE = ψ,EE W,Eγ = ψ ,Eγ α α for the functions given in (7.80). This somewhat lengthy and tricky procedure is discussed in detail in appendix B. Here we solely summarize the results. For the derivatives of the free energy the following compact expressions are obtained ψ,γα = −f α βi βe αi αe ψ,γα γβ = f,T + f,B · Ē · f,B : Ē : f,T (7.82) ψ,E = T ψ,EE = E − E : B : E αi ψ,Eγα = E : A−1 : Ē : f,B . The fourth-order tensors and the scalar are defined as P ∂T T f αe )−1 + A]−1 E := ψ,E e E e B := [( α γαP K := ψ,AA Ē := (A−1 + Pα γα ∂T T f αe )−1 αi −1 ) . A := E + K Ē := (K−1 + α γα f,BB (7.83) Within the iterative solution procedure of the minimization problem, the internal variables are updated according to P E p ⇐ E p + Pα∈A E p,γα ∆γα A ⇐ A + Pα∈A A,γα ∆γα (7.84) A ⇐ A + α∈A A,γα ∆γα based on their sensitivities E p,γα = A,γα = A−1 : Ē : ∂T f α iα and A,γα = K−1 · Ē · f,B (7.85) with respect to the algorithmic parameters. The stresses (7.76) and moduli (7.79) that are obtained with the derivatives (7.82) specified in this section are identical to the those obtained with the general return algorithm in section 7.3. 139 Finite Shell Element Implementation 8. Finite Shell Element Implementation This section outlines an eight-node brick-type finite shell element design based on an additively enhanced current metric relative to the parameter space of the shell-like structure. It is a further development of the gradient-enhanced formulation of Miehe [84] and investigated in detail in Miehe & Apel [86]. This formulation is applied to the simulation of deep drawing processes in Miehe & Schotte [89]. Similar approaches can be found in Betsch [10], Seifert [120] and Klinkel, Gruttmann & Wagner [62]. Defining an interface to strain-driven constitutive algorithms allows the use of existing isotropic and anisotropic constitutive material models. 8.1. Parameterization of the Shell-Like Continuum We consider a shell as a standard continuum where one dimension is small in comparison with its span. Correlated to that dimension is the thickness direction of the shell. The shell is parameterized with convected curvilinear coordinates. Therefore consider the socalled parameter space A ∈ R3 of the shell that has a particular Cartesian structure A = M × H. Here M ∈ R2 is the reference surface and H ∈ R the parameter space of the shell fiber. The points of the parameter space are described by the curvilinear PSfrag replacements coordinates θ1 , θ2 of the shell surface and θ3 along the shell fiber. The parameter space is equipped with a Cartesian orthonormal basis {Ē i }i=1,2,3 . Figure 37 visualizes the notation introduced in this paragraph. ϕ {Gi } {g i } F x X j J B X S θ x A Figure 37: Geometry of a shell. Points X = X(θ i ) of the reference configuration B and their Eulerian counterparts x = x(θ i , t) in the actual configuration are parameterized with the coordinates θ i of the parameter space A of the shell. The associated linear tangent maps are J := ∇θ X and j := ∇θ x. The deformation gradient is the composition F := jJ −1 . The local parameterization of the reference configuration B is determined by the map X = X(θ). The corresponding tangent map is denoted by J : Tθ A → TX B , J = ∇θ X . (8.1) The dual tangent mapping that connects the co-tangent spaces is J −T : Tθ? A → TX? B. In a similar manner the actual configuration S is parameterized. Points θ of the parameter space are mapped onto points of the Eulerian configuration by x := x(θ, t). The tangent map is defined by j : Tθ A → TX S , j = ∇θ x (8.2) 140 Finite Shell Element Implementation and the co-tangent map by j −T : Tθ? A → Tx? S. The covariant current metric g and the reference metric G at θ ∈ A in the representations with respect to the parameter space are denoted by C̄ := j T gj and Ḡ := J T GJ . (8.3) The local deformation of the shell-like continuum can be defined in terms of the above introduced parameterizations of the Eulerian and Lagrangian configurations by ϕ = x ◦ X −1 and F = jJ −1 . (8.4) These compositions are depicted in figure 37. The local stress state in a shell-like continuum is primarily a function of an objective strain tensor which constitutes a relationship between the current and the reference metrics. With regard to the interpretation and identification of the strain tensor it is convenient to consider the geometric setting relative to the parameter space. It is of special advantage when dealing with enhanced and assumed strain modifications. A classical objective strain measure is the Green-Lagrange tensor Ē at θ ∈ A defined by 1 2 Ē := [C̄ − Ḡ] . (8.5) 8.2. Finite Element Approximation PSfrag replacements 8.2.1. Compatible Displacement Approach For the spatial discretization the shell-like continuum B is divided into non-overlapping elements of finite size B e ⊂ B so that B = ∪ne=1 B e . The element nodes are located at the θ3 Ae = M e × H e 4 7 Me θ2 6 1 3 8 5 θ1 2 Figure 38: Parameter space of the shell element. It has Cartesian structure Ae = Me × He associated with the reference surface of the shell and the thickness direction. The four assumed strain points for the thickness strain are marked with a square, the four assumed strain points for the transverse shear strain interpolation are marked with black circles. bottom and top surface of the shell-like continuum. Figure 38 shows the parameter space Ae of a single eight-node brick-type shell element. Using the iso-parametric concept, the interpolations of the geometry of the elements in the reference configuration and their deformed actual counterparts are given by x= 8 X i=1 I 1 2 3 N (θ , θ , θ )xI and X = 8 X i=1 N I (θ1 , θ2 , θ3 )X I . (8.6) 141 Finite Shell Element Implementation The quantities xI and X I denote the discrete nodal coordinates at the bottom and top surface. For the interpolation the standard trilinear shape functions N I (θ1 , θ2 , θ3 ) = 1 8 (1 − θ1 θI1 )(1 − θ 2 θI2 )(1 − θ 3 θI3 ) (8.7) are used. In contrast to standard continuum settings we here assume a particular orientation of the brick by defining the parameter θ 3 to be associated with the thickness direction of the shell. This motivates the decomposition of the element parameter space Ae = Me × He into a reference surface Me ∈ R2 and the part He ∈ R associated with the shell fiber, cf. figure 38. The parameterization of the deformation of the shell is then simply provided by the standard nodal displacement vector of the brick element xI = X I + d I (8.8) for I = 1, . . . , 8. Based on the interpolations (8.6) the compatible Jacobians (8.1) and (8.2) are obtained at any point of the element parameter space, i.e. j= 8 X i=1 x I ⊗ ∇θ N I and J = 8 X i=1 X I ⊗ ∇θ N I . (8.9) This representation allows the computation of the compatible current and reference metrics according to (8.3) C̄ C := j T gj and ḠC := J T GJ . (8.10) The discrete forms of the first and second variations of the current metric tensor are 1 2 δ C̄ C = 8 X B IC δdI and ∆( 12 δ C̄ C ) = i=1 8 X 8 X T GIJ CC (δdI ∆dJ ) (8.11) I=1 J=1 in terms of the nodal B-matrices of dimension 6×3 and the nodal G-matrices of dimension 6 × 1 that are defined by BCI (ij)a := 1 2 [N,iI jaj + jai N,jI ] and GIJ CC (ij) := 1 2 (NiI NjJ + NiJ NjI ) . (8.12) 8.2.2. Assumed Strain Modifications Assumed strain interpolations were first introduced by Dvorkin & Bathe [37], Bathe & Dvorkin [7] to avoid locking effects due to parasitic shear strains. Betsch & Stein [11] extended this method to avoid locking caused by parasitic thickness strains. The strains mentioned above are associated with the coordinates C̄13 , C̄23 and C̄33 of the current metric and Ḡ13 , Ḡ23 and Ḡ33 of its Lagrangian counterpart relative to the parameter space. The assumed strain interpolations take the forms P4 ass 1 2 (θ , θ ) = C̄33 PA=1 6 ass 2 (θ ) = C̄13 A=5 P 8 ass 1 C̄23 (θ ) = A=7 P4 ass 1 2 Ḡ33 (θ , θ ) = P6A=1 2 Ḡass (θ ) = 13 PA=5 8 1 = Ḡass 23 (θ ) A=7 1 4 1 2 1 2 1 4 1 2 1 2 1 2 A (1 + θ1 θA )(1 + θ 2 θA )C̄33 2 A (1 + θ2 θA )C̄13 1 A (1 + θ1 θA )C̄12 1 2 (1 + θ1 θA )(1 + θ 2 θA )ḠA 33 2 (1 + θ2 θA )ḠA 13 1 (1 + θ1 θA )ḠA 12 (8.13) 142 Finite Shell Element Implementation based on the discrete values of the metrics at the collocation points for the assumed strains that are enumerated in figure 38. In the parameter space Ae of the element the assumed strain points 1, . . . , 4 of the Betsch-Stein approach are identical with the points in the corners of the reference surface of the shell. They have the coordinates (−1, −1, 0) , (+1, −1, 0) , (+1, +1, 0) and (−1, +1, 0) . The assumed strain points 5, . . . , 8 of the Dvorkin-Bathe approach are located at the centers of the edges of the reference surface, i.e. (0, −1, 0) , (0, +1, 0) , (−1, 0, 0) and (1, 0, 0) . Note that the assumed strains (8.13) replace the associated values of the compatible setting (8.10). This replacement procedure can be expressed by ˜ = Ḡ + [Ḡass − Ḡ ] ˜ = C̄ + [C̄ ass − C̄ ] and Ḡ C̄ C C C C C C C C (8.14) ˜ denote the modified metric tensors. The bracket terms in (8.14) are ˜ and Ḡ where C̄ C C understood to be present only for the values associated with the transverse shear and the thickness components as depicted in (8.13). These modifications have an effect on the Band G-matrices. The modified terms are P4 1 1 1 2 2 IA BCI (33)a := A=1 4 (1 + θ θA )(1 + θ θA )BC (33)a P 6 1 2 2 IA BCI (13)a := (8.15) A=5 2 (1 + θ θA )BC (13)a P 8 1 I 1 1 IA BC (23)a := A=7 2 (1 + θ θA )BC (23)a for the B-matrix (8.12)1 . The the corresponding entries in the G-matrix (8.12)2 have to be changed accordingly, i.e. P4 1 1 1 2 2 IJ A ḠIJ CC 33 := PA=1 4 (1 + θ θA )(1 + θ θA )GCC 33 6 1 A 2 2 IJ (8.16) ḠIJ CC 13 := PA=5 4 (1 + θ θA )GCC 13 8 1 IJ A 1 1 IJ ḠCC 23 := A=7 4 (1 + θ θA )GCC 23 . 8.2.3. Enhanced Strain Modifications The additively enhanced strain formulation is identical to a straightforward enhancing of the Green-Lagrangian strains defined in (8.5), similar to the approach proposed by Simó & Rifai [127] for continuum elements for small strains. In the context of a brick-type shell element design, the current metric in the representation relative to the parameter space is additively enhanced ˜˜ ˜ . ˜ + C̄ C̄ = C̄ E C (8.17) ˜˜ ˜ C̄ is the assumed enhanced current metric which consists of the compatible part C̄ C ˜ defined in (8.14) and the additional element-wise incompatible contribution C̄ E . The latter is assumed to have the form 1 2 ˜ =B a, C̄ E E (8.18) 143 Finite Shell Element Implementation where B E is a matrix of shape functions that governs the incompatible contributions and a the vector of internal element degrees. In order to set up the matrix B E , we start with the particular interpolation θ 1 α1 θ 1 α4 + θ 2 α5 0 ˜ ? = θ1 α + θ2 α θ 2 α2 0 (8.19) C̄ 4 5 E 3 0 0 θ α3 in terms of five internal parameters a = [α1 α2 α3 α4 α5 ]T . (8.20) These five internal element parameters α1,...,5 govern incompatible modes which enhance membrane and thickness strains. The incompatible membrane shapes follow the classical works of Taylor, Beresford & Wilson [139] and Simó & Rifai [127]. It was applied to four-node shell elements by Betsch, Gruttmann & Stein [12]. The incompatible ˜? thickness shape is adopted from Büchter & Ramm [28]. Observe that the shape C̄ E satisfies the condition Z ˜ ? dV = 0 . C̄ (8.21) E Ae In order to satisfy the patch test, Simó & Rifai [127] suggested a transformation for ˜ ? −1 the ansatz (8.19) of the form C̃ E = (J0 /J)J −T 0 C̄ E J 0 from the parameter space A to the Lagrangian manifold B, yielding the modification C̃ E of the current metric relative to the Lagrangian configuration. Hereby J := det[J ] and the subscript “0” indicates the evaluation at the center of the element, i.e. at θ = (0, 0, 0). In order to achieve the identical result within a computation relative to the parameter space A, the above given ansatz is pulled back. The result is the enhanced part of the metric in (8.17) 1 2 ˜ = C̄ E 1 2 J0 T −T ˜ ? −1 (J J 0 )C̄ E (J 0 J ) =: B E a J (8.22) that defines the enhanced B-matrix B E , cf. (8.18). 8.3. Variational Formulation The variational formulation of the enhanced strain method used here is similar to the Hu-Washizu-type three-field variational formulation presented by Sim ó & Rifai [127] for the small-strain case. Point of departure is the variational problem Z ˜ ]J dV − Π → stat. ˜ ) − S̄ : 1 C̄ ˜ (8.23) Π(u, C̄ E , S̄) = [ψ(C̄ C + C̄ E ext E 2 A with respect to the parameter space of the shell. The above functional depends on the ˜ and actual displacement u = x − X, the enhanced part of the current metric tensor C̄ E the stresses S̄. The external loads are assumed to be dead loads. They are prescribed for the body in the reference configuration, their contributions are Z Z Πext (u) = ρ0 γ̄ · u dV + T̄ · u dA . (8.24) B ∂B 144 Finite Shell Element Implementation Variation of (8.23) yields, after plugging in the discretization, the set of coupled weak forms associated with the element domains in the parameter space R GeC := Ae 21 δ C̄ C : 2ψ,C̄ J dV − Geext = 0 R 1 ˜ e (8.25) = 0 GS := Ae δ S̄ : 2 C̄ E J dV R ˜ : (2ψ − S̄)J dV = 0 GeE := Ae 21 δ C̄ E ,C̄ R R where Geext := Be ρ0 γ̄ · δu dV + ∂Be T̄ · δu dA. Observe that the corresponding Euler equation to (8.25)2 demands that the enhanced part of the right Cauchy-Green-tensor ˜ = 0. Within a finite element approximation vanishes in the continuous setting, i.e. 12 C̄ E this requirement must be weakened, otherwise no improvement of the solution will be obtained. In order to ensure correct convergence towards the exact solution, an element formulation must pass the patch test. This restriction is a priori satisfied by the chosen ansatz (8.19). It guarantees that the element-wise incompatible part of the enhanced right Cauchy-Green tensor are L2 -orthogonal to at least constant stress fields. Based on this fact, the three-field formulation reduces to the two-field formulation ) R GeC := Ae 21 δ C̄ C : 2ψ,C̄ J dV − Geext = 0 (8.26) R 1 ˜ Ge := = 0. δ C̄ : 2ψ J dV E E Ae 2 ,C̄ Linearization and insertion of the finite element interpolations (8.11), (8.15), (8.16) and (8.22) then gives the coupled element equations ¾ GeC + ∆GeC = δdT {Rd + K dd ∆d + K da ∆a} (8.27) GeE + ∆GeE = δaT {Ra + K ad ∆d + K aa ∆a} in terms of the partial element residuals and element stiffness matrices R R ¾ K dd = RAe (B TC CB C + GCC )J dV T Rd = RAe B C S̄J dV (8.28) and K da = RAe B TC CB E J dV Ra = Ae B TE S̄J dV T K aa = Ae B E CB E J dV and K ad = K Tda . The numerical integration must be performed with at least nine Gauss points to avoid under integration. We refer here to Simó, Armero & Taylor [124] as well as the numerical examples in section 9.2 for details. In (8.28) we have introduced the constitutive functions for the stresses and consistent tangent moduli ˜ ) ˜ ) and C := 4ψ (C̄ + C̄ (8.29) S̄ := 2ψ (C̄ + C̄ ,C̄ C E ,C̄ C̄ C E at time tn+1 computed in terms of the assumed enhanced metric (8.17). The parameters a are defined on the element level. Their increments ∆a in (8.27) can be eliminated by static condensation, i.e. ∆a = −K −1 aa (Ra + K ad ∆d) . (8.30) The condensed element residual vector and element tangent matrix are R := Rd − K da K −1 aa Ra and K := K dd − K da K −1 aa K ad . (8.31) A typical global Newton iteration step consists of an assembling procedure of the condensed element residuals R and element tangents K to the global residual vector and tangent matrix, the solution of the associated linear algebraic system and an update of the incremental displacement. The increments for the parameters a on the element level are obtained from (8.30). 145 Finite Shell Element Implementation 8.4. Gradient-type Interface to Constitutive Models The assumed enhanced strain procedure yields modifications of the current and the reference metrics, see (8.14) and (8.17). A general interface to strain-driven constitutive models is provided by the deformation gradient F . It is defined in (8.4) for the compati˜ based on this ble setting. A computation of an assumed enhanced deformation gradient F̃ ˜ of the Eulerian definition needs a recovery of the assumed enhanced Jacobians ˜j̃ and J̃ and Lagrangian parameter maps. In the context of assumed enhanced strain methods the basic idea bases on a polar decomposition of the Jacobians j and J of the Eulerian and Lagrangian parameter maps, respectively, j = ru and J = RU . (8.32) The symmetric and positive definite stretch tensors are obtained from the current and reference metrics defined in (8.10) as 1/2 u = C̄ C 1/2 and U = ḠC . (8.33) In a second step the rotations are obtained by −1/2 r = j C̄ C −1/2 and R = J ḠC . (8.34) The assumed enhanced strain modifications (8.14) and (8.17) affect the metric tensors directly. This yields the modified stretch tensors ˜˜ 1/2 ˜ := Ḡ ˜ 1/2 . ˜ := C̄ ũ (8.35) and Ũ C A basic assumption conceptually outlined in Dvorkin, Pantuso & Repetto [38] considers the rotation unaffected by the assumed strain modification. This assumption defines the enhanced assumed Jacobians of the parameter maps ˜ = RŨ ˜ . ˜j̃ = r ũ ˜ and J̃ (8.36) Insertion of the compatible rotations (8.34) into (8.36) gives the representations ˜˜ 1/2 ˜j̃ = j C̄ −1/2 C̄ C ˜ = J Ḡ−1/2 Ḡ ˜ 1/2 . and J̃ C C (8.37) Finally the enhanced deformation gradient is defined in analogy to (8.4) by ˜ := ˜j̃ J̃ ˜ −1 . F̃ (8.38) This enhanced deformation gradient enters a strain-driven algorithm of an anisotropic constitutive model. The constitutive model then determines the stresses and moduli ˜ ) and C = ∂ 2 ψ(F̃ ˜). P = ∂F̃˜ ψ(F̃ (8.39) ˜˜ F̃ F̃ The stresses and consistent moduli of the parameter space needed for the setting up of the residual and tangent matrix of the mixed shell element are then obtained by a transformation by means of the assumed enhanced parameter maps ˜ −1 )B̄ S̄ ĀB̄ = P aB (˜j̃ −1 )Ā (J̃ (8.40) a B and ˜ −1 )B̄ (˜j̃ −1 )C̄ (J̃ ˜ −1 )D̄ − S̄ B̄ D̄ C̄˜˜ ĀC̄ . C̄ĀB̄ C̄ D̄ = CaBcD (˜j̃ −1 )Ā a (J̃ B c D (8.41) 147 Numerical Examples 9. Numerical Examples 9.1. Necking of an Isotropic Rod In this first example we consider the classical necking problem of a rod for isotropic elasticplastic material response. It is a standard benchmark problem of finite plasticity and has been analyzed by many authors, see for example Simó [121] and Papadopoulos & Lu [99]. The aim of the investigation is to compare the results obtained with the different stress update algorithms, U1: standard implicit, U2: standard explicit, V1: variational implicit and V2: variational explicit. Furthermore the results obtained from the additive logarithmic strain space formulation are compared to those from the multiplicative plasticmap plasticity framework. The length of the rod in its reference configuration is l = 53.34mm, the radius r0 = 6.4135mm. Due to the apparent symmetry of the problem, we discretize one eighth of the specimen by 120 Q1P0 elements as described in Simó, Taylor & Pister [128] and Miehe [77]. Half of the elements are concentrated on a length of 8.98mm in the middle of the rod close to the necked zone. The necking is triggered by an imperfection of the rod in form of a continuous decrease of the radius in the fine discretized region from r0 to r = 0.982r0 at the center cross section. We use the multiplicative Hill-type constitutive model of section 6.6 and the additive Hill-type model of section 7.2.3 with saturation-type non-linear isotropic hardening response. The set of material parameters for rate-independent behaviour is summarized in table 12. a. c. b. 0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.65 0.73 0.81 0.89 0.97 1.05 1.13 Figure 39: Necking of an isotropic rod. Deformed specimen and distribution of equivalent plastic strain: (a) Reference configuration, (b) intermediate state after onset of necking at u = 8.4mm and (c) final state at u = 14mm. The rod is pulled in a deformation-driven simulation in constant increments up to a total elongation of u = 14mm. Figure 39 depicts the initial and two deformed discretized structures with the distribution of the equivalent plastic strain of the additive model. The final displacement was applied in 200 increments. To compare the different stress update algorithms the global load-deflection curves and the radial contractions at the center of the specimens are considered. They are plotted in figure 40. Up to a deformation of around 6mm, the load increases and the whole rod contracts uniformly. When the load is increased further, necking starts at the center of the rod. This is documented by the kinks in the ∆r/u-curves. Figure 40a compares the implicit algorithms at different step sizes. The results from the standard formulation U1 are very close to each other. Deviations come solely from the discretization of the evolution equation. The results from 148 Numerical Examples 4 70 3.5 ∆r [mm] 80 F [kN] 60 replacements PSfrag replacements 50 40 30 V1 U1 U1 V1 V1 20 10 0 a. 0 2 4 6 8 12 b. 4 70 3.5 ∆r [mm] F [kN] 0 14 80 PSfrag replacements 50 40 U1 V1 U2 V2 V2 20 10 0 c. 0 2 4 6 8 10 12 0 14 d. ∆r [mm] 50 40 e. 0 2 4 6 8 10 12 14 10 12 14 10 12 14 200 200 400 200 400 0.5 3.5 0 6 2 70 10 U1 V1 U2 V2 V2 2.5 4 PSfrag replacements plastic metric plastic map 4 u [mm] 3 80 20 2 1 u [mm] 30 0 1.5 200 200 400 200 400 60 F [kN] 2 0.5 u [mm] 30 replacements 2.5 1 60 replacements 3 200 200 50 50 14 1.5 200 200 50 50 14 10 V1 U1 U1 V1 V1 0 2 4 6 8 u [mm] plastic metric plastic map 3 2.5 2 1.5 1 0.5 8 u [mm] 10 12 14 0 f. 0 2 4 6 8 u [mm] Figure 40: Necking of an isotropic rod. Comparison of the update algorithms. U1: standard implicit, U2: standard explicit, V1: variational implicit and V2: variational explicit. (a) and (c) show the load displacement curves, (b) and (d) the radial contraction ∆r of the mid cross section of the rod. The numbers of iteration steps used are specified in the legends. Plots (e) and (f) compare the results from the plastic-map model (U1) with the results of the plastic-metric model obtained in 200 steps. 149 Numerical Examples Table 12: Isotropic Material Parameters. bulk modulus: shear modulus: initial yield stress: infinite yield stress: hardening modulus: saturation parameter: κ µ y0 y∞ h ω = 164.206 kN/mm2 = 80.1938 kN/mm2 = 0.45 kN/mm2 = 0.715 kN/mm2 = 0.12924 kN/mm2 = 16.93 the variational algorithm V1 coincide with those from U1 but differ in the post critical regime for large steps. The variational algorithm allows larger load step increments than the standard formulation. The performance and robustness of the explicit algorithms U2 and V2 is not so good as that of the implicit ones. The algorithms U2 and V2 require 400 steps. For larger step sizes the computations either fail or the curves diverge and oscillate in the post critical range as shown for V2 in the load-displacement figure 40c and plot 40c that documents the contraction of the strip. The characteristics observed in the load-displacement curves can also be found in the graphs 40b,d, where the contraction of the cross section in the middle of the rod is plotted against the displacement. Minor differences between the two frameworks of plasticity models – the plastic-map approach and the plastic-metric approach – become apparent in figure 40e. The simulations were performed in 200 steps. The plastic-map model with the implicit standard integration algorithm U1 is identical to the formulation of multiplicative isotropic plasticity outlined in Simó & Miehe [125], Simó [121] and Miehe [83]. The results are almost identical except a minor deviation in the softening regime where the non-coaxiality of the current and the plastic metric C and Gp seems to separate the results. In the same range we also observe small differences in the radial contraction as depicted in figure 40f. Nevertheless, in the considered case of isotropic finite plasticity, results of the proposed formulation based on the additive kinematic ansatz (7.4) are surprisingly close to the multiplicative kinematic ansatz (6.172). 9.2. Necking of an Isotropic Rectangular Strip In this example we investigate the performance of the additively enhanced finite shell element formulation proposed in chapter 8. For comparison, the multiplicatively enhanced shell element of Miehe [84] has also been implemented. By means of the benchmark problem of necking of a rectangular strip both element designs are compared. Figure 41 depicts the geometry of the specimen and its loading. The strip is deformation driven pulled up to a total length of 29.78mm. On its loaded ends the boundary conditions allow free contraction of the specimen. In the middle of the specimen the width is reduced to 13.5mm in order to trigger necking. The same material as in the first example is used. It is isotropic with non-linear isotropic hardening and is described by the material constants given in table 12. The analysis is performed with the additive plastic-metric formulation of section 7. 150 Numerical Examples u u h = 13.545 PSfrag replacements t = 1.0 l = 17.78 Figure 41: Necking of a Strip. Geometry and boundary conditions. All length in mm. Due to symmetry, one fourth of the strip is discretized using 5 × 10 shell elements. The specimen is stretched in 120 uniform steps up to a final edge displacement of u = 6mm. We investigate five numerical integration schemes having 8, 9, 15, 27 and 64 Gauss points. The coordinates and weights for the 8-, 27- and 64-point integration formulas follow from a one-dimensional scheme by tensor product operations, i.e. (θ1 , θ2 , θ3 )ijk = [a]i ⊗ [a]j ⊗ [a]k ; For the 8-point integration these vectors are q q a = [− 13 , 13 ]T ; wijk = [w]i [w]j [w]k . w = [1, 1]T . (9.1) The 27-point integration scheme is defined by q q a = [− 35 , 0, 35 ]T ; w = [ 59 , 89 , 59 ]T (9.2) and the 64-point scheme by q q q q a = [− 37 + s, − 37 − s, 37 − s, 37 + s, ]T ; w = [ 12 − 1t , 12 + 1t , 12 + 1t , 21 − 1t ]T (9.3) √ 24 with constants s = 245 and t = 6 1.2. In Simó, Armero & Taylor [124], a 9-point quadrature formula was proposed consisting of the eight points resulting from (9.1) but with weights w = [ 95 , 95 ]T and one additional point at the center of the element with weight w = 32 . For a two-dimensional element, they suggested a 5-point formula. The 9 latter can be used in shell elements for integration over the reference surface M e , once the integration over the thickness direction in He has been done. In this case we end up with 15 Gauss points. Three-point thickness integration over H e is done at q q (9.4) θ3 = − 35 , 0, 35 with w = 59 , 98 , 95 . q In the shell surface, the 5-point quadrature formula consists of four points given by q q a = [− 35 , 35 ]T ; w = [ 59 , 95 ]T (9.5) and one additional point at the center of the surface with weight w = 16 . 9 Figure 42 shows the final states of the simulations. With the 8-point integration the strip deforms in the post critical regime completely differently from the ones with the 9-point 151 Numerical Examples a. b. c. d. Figure 42: Necking of a strip. Deformed configurations with distribution of equivalent plastic strain. (a) Additively enhanced and (b) multiplicatively enhanced element design with 8-point quadrature. (c) and (d) show the corresponding results for 9-point integration. 9000 9000 8 9 15 27 64 8000 7000 6000 7000 6000 load [N] PSfrag replacements 5000 4000 3000 5000 4000 3000 2000 2000 1000 1000 0 a. 0 1 2 3 4 5 0 6 b. displacement u [mm] 0 1 2 3 4 5 displacement u [mm] 9000 mul. enh. add. enh. 8000 7000 6000 load [N] load [N] Sfrag replacements 8 9 15 27 64 8000 PSfrag replacements 5000 4000 3000 2000 1000 0 c. 0 1 2 3 4 5 6 displacement u [mm] Figure 43: Necking of a strip. Load-displacement curves for (a) multiplicatively enhanced and (b) additively enhanced element design for 8-, 9-, 15-, 27- and 64-point quadrature formulas. Plot (c) compares the curves for 9-point integration. 6 152 Numerical Examples 0 0 8 9 15 27 64 −1 −2 −3 −0.2 PSfrag replacements ∆t [mm] ∆w [mm] replacements −4 −5 −6 −7 −0.3 −0.4 −0.5 −0.6 −0.7 −8 −0.8 −9 −0.9 −10 a. 0 1 2 3 4 5 −1 6 b. displacement u [mm] 0 −2 PSfrag replacements −4 −5 −6 −7 0 1 2 3 4 5 6 8 9 15 27 64 0 1 2 3 4 5 6 displacement u [mm] 0 mul. enh. add. enh. mul. enh. add. enh. −0.1 −0.2 ∆t [mm] ∆w [mm] 5 −0.8 d. −2 −3 −4 PSfrag replacements −5 −6 −7 4 −0.6 −1.2 6 displacement u [mm] −1 e. 3 −0.4 0 replacements 2 displacement u [mm] −1 −8 −9 1 −0.2 ∆t [mm] ∆w [mm] −3 c. 0 0 8 9 15 27 64 −1 replacements 8 9 15 27 64 −0.1 −0.3 −0.4 −0.5 −0.6 0 1 2 3 4 displacement u [mm] 5 6 −0.7 f. 0 1 2 3 4 5 displacement u [mm] Figure 44: Necking of a strip. Change of width ∆w in the necking cross section and change of thickness ∆t at the center of the strip for (a,b) multiplicatively enhanced and (c,d) additively enhanced shell element design for integration schemes with 8-, 9-, 15-, 27- and 64-Gauss points. Plots (e) and (f) compare the curves for 9-point integration, respectively. 6 153 Numerical Examples or higher-order integration schemes. The necking cross section degenerates to a line. The plastic deformation is essentially restricted to one row of elements and one observes a hourglass-like pattern which renders the results useless. For details on the drawback of this quadrature formula we refer to Simo, Armero & Taylor [124]. The results obtained with higher quadrature formulas are close to each other. In figure 43 the load-displacement curves are plotted. (a) belongs to the additively enhanced and (b) to the multiplicatively enhanced element design. The curves for 8-point integration document the inadequacy of this integration scheme. All other curves almost coincide. In (c) the results obtained with both elements using the 9-point quadrature formula are compared. There is a slight difference in the post-critical regime. In order to document the effect of the integration order, the changes of width ∆w and thickness ∆t in the necking cross section are documented in figure 44. One clearly observes the effect of under-integration of the 8-point quadrature formula in the post-critical regime leading to physically meaningless results. The curves of a higher integration order slightly differ towards the end of the deformation process. 9.3. Drawing of a Circular Flange This example is concerned with a drawing process of a thin circular flange. It serves as a benchmark for the analysis of anisotropic elastic-plastic response and has already been considered by Papadopoulos & Lu [100]. The problem can be viewed as a simplified model of the outer part of a circular sheet that is deep-drawn into a cup. The benchmark substantially idealizes a three-dimensional deep-drawing experiment because no out-of-plane drawing occurs and therefore no contact elements are needed. Geometry Q a1 PSfrag replacements u P a2 a3 10 a2 200 400 200 Figure 45: Drawing of a flange. Geometry and √ boundary conditions. Coordinates of P √ and Q are P (200/0/0) and Q((200/ 2)/(200/ 2)/0). All length in mm. and boundary conditions are depicted in figure 45. The nodes at the inner side are drawn in radial direction up to a total displacement of u = 75mm. In order to prevent buckling of the blank, the lower layer is supported in vertical direction. The structure is discretized using 10 × 40 brick-type shell elements with the nine-point quadrature. The process is deformation-controlled by applying increments of ∆u = 0.1mm of radial displacement. 154 Numerical Examples We assume isotropic elastic and cubic plastic response and investigate two different approaches to the modeling of the anisotropic yield criteria. On the one hand the anisotropic plastic behaviour is governed by a single surface cubic Hill-type level set function in terms of a constant fourth-order tensor H for plastic-metric plasticity and H̄ for plastic-map plasticity. The restriction to cubic symmetry is obtained by the choice y11 = y22 = y33 and y12 = y23 = y13 . The orientations of the principal axes of anisotropy {ai }i=1,2,3 are specified in figure 45. The ratio of normal stresses to shear yield stresses δ := y11 /y12 serves as a measure for the deviation from the isotropic state. We investigate two sets of material parameters which only differ in this ratio. For the boundary-value problems, considered here, these values are δ = 3.4641 and δ = 0.86603. Throughout all simulations we assume isotropic linear hardening using the material parameters listed in table 13. Table 13: Material Parameters bulk modulus: shear modulus: reference yield stress: normal yield stress: hardening modulus: κ µ y0 y11 h = 164000 N/mm2 = 80190 N/mm2 = 450 N/mm2 = 450 N/mm2 = 100 N/mm2 On the other hand the anisotropic response is the result of a Kelvin-mode decomposition of a cubic fourth-order tensor. The attractive feature of such a decomposition is that in the case of cubic symmetry the eigen-tensors do only depend on the orientation of the fourth-order tensor. For co-axial principal axes of anisotropy {ai } and global coordinate axes {ei }, the eigen-tensors are specified in section 6.6.5. 9.3.1. Comparison of Additive and Multiplicative Plasticity The results of the numerical simulations obtained with the single surface additive plasticmetric model and the multiplicative plasticity model using the stress update algorithm U1 are documented in figures 46, 47 and 48. The additively enhanced shell element of section 8 with 9-point quadrature is used. Figure 46 shows the deformed flanges and the distribution of the equivalent plastic strains for δ = 3.4641, whereas figure √ 47 shows those for δ = 0.86603. Compared with the isotropic setting, where δ = 3, the results for δ = 3.4641 are associated with an increase of plastic deformation in regions of high shear stresses T12 . The onset of yielding is observed at π/4 + nπ/2, n = 0, 1, 2, 3, see figure 46. The choice δ = 0.86603 yields an increase of plastic deformations in regions of high normal stresses T11 and T22 . We observe the onset of yielding at nπ/2, n = 0, 1, 2, 3, see figure 47. For both sets of anisotropic material parameters, the outer rim becomes wavelike during the deformation. This phenomenon is well known in sheet metal forming and is denoted as earing. The plots of equivalent plastic strain from the additive plasticity model are almost indistinguishable from those of the multiplicative model. Figure 48 depicts the development of the nodal forces acting at the two nodes P and Q specified in figure 45. For the two considered sets of anisotropic material parameters the curves for P and Q differ from each other as follows. For δ = 3.4641 the maximum yielding occurs at π/4 Numerical Examples a. b. Figure 46: Drawing of a flange. Deformed meshes and distributions of equivalent plastic strain for u = 25mm, u = 50mm and u = 75mm. (a) Multiplicative plasticity using algorithm U1 with maximal accumulated plastic strain Amax = 0.4435 and (b) additive plasticity with Amax = 0.4465 for anisotropy ratio δ = 3.4641. a. b. Figure 47: Drawing of a flange. Deformed meshes and distributions of equivalent plastic strain for u = 25mm, u = 50mm and u = 75mm. (a) Multiplicative plasticity using algorithm U1 with Amax = 0.9244 and (b) additive plasticity with Amax = 0.9478 for anisotropy ratio δ = 0.86603. 155 156 Numerical Examples PSfrag multiplicative plasticity additive plasticity replacements replacements 50 40 P isotropy Q 30 20 0 a. 0 10 20 30 40 50 10 0 80 b. u [mm] isotropy 30 multiplicative plasticity 70 Q 40 20 60 multiplicative plasticity additive plasticity 50 additive plasticity Q 10 60 nodal force [kN] nodal force [kN] 60 P 0 10 20 30 40 50 60 70 80 u [mm] Figure 48: Drawing of a flange. Nodal forces at P and Q for multiplicative and additive plasticity. Anisotropic response for (a) δ = 3.4641 and (b) δ = 0.86603. in the positive quadrant. In this direction the material is softer and the nodal force at Q is smaller than the one at P . For δ = 0.86603 the softer behavior is along the a1 and a2 axes and the nodal force at P is smaller than the one at Q. Clearly, both curves are identical in the isotropic case. Differences between the additive and the multiplicative model are in correlation to the amount of plastic deformation of the material. The higher the accumulated plastic strain, the stronger the deviation of the nodal forces of both models. Nevertheless, the curves are close to each other. 9.3.2. Comparison of the Stress Update Algorithms We now focus on the four proposed stress update algorithms U1: standard implicit, U2: standard explicit, V1: variational implicit and V2: variational explicit. Figure 49 shows the development of the nodal forces at the points P and Q during the simulation with the single surface Hill-type model. The maximum constant step size of ∆u = 0.1mm has been proven to be appropriate for the simulations. It was determined with the implicit standard algorithm U1. Simulations where the size was doubled after some plastic steps aborted. This was caused by the failure of the local Newton iteration in the stress update algorithm. The results of all algorithms are almost identical. Furthermore we compare the implicit algorithms within a simulation using the multi- replacements 60 U1 V1 U2 V2 50 nodal force [kN] nodal force [kN] 60 40 PSfrag replacements 30 20 10 0 a. U1 V1 U2 V2 50 40 30 20 10 0 10 20 30 40 50 60 displacement u [mm] 70 80 0 b. 0 10 20 30 40 50 60 70 displacement u [mm] Figure 49: Drawing of a flange. Nodal forces at P and Q for the proposed stress update algorithms U1: standard implicit, U2: standard explicit, V1: variational implicit, V2: variational explicit. Anisotropic response for (a) δ = 3.4641 and (b) δ = 0.86603. 80 157 Numerical Examples PSfrag replacements nodal force [kN] displacement u [mm] PSfrag replacements nodal force [kN] displacement u [mm] PSfrag replacements nodal force [kN] displacement u [mm] U1 U1 U1 V1 V1 V1 Figure 50: Drawing of a flange. Deformed meshes with equivalent plastic strain obtained with algorithm U1 and a Kelvin-mode decomposition of a cubic fourth-order tensor. surface model based on the Kelvin-mode decomposition of a cubic fourth-order tensor. In the case of ideal plasticity, the flow systems are decoupled. They are weakly coupled via the equivalent plastic strain if hardening phenomena are modeled. On one of the flow systems yielding occurs due to the normal stresses, while on the other system yielding is caused by the shear stresses, cf. (6.197). In order to stabilize the solution procedure, the viscosity is set to η = 500Ns/mm2 . The resulting deformation of the flange differs from those obtained with the single surface model. A sequence of deformed meshes and the distribution of the equivalent plastic strain is visualized in figure 50. Obviously, the effect of yielding due to normal stresses is dominating. The deformation has similarities to the one from the single surface Hill-type model with anisotropy ratio δ = 0.86603, where the resistance to shear strains is increased. Plastic flow dominates in the regions near the coordinate axes. The evolution of the nodal forces at the points P and Q during the deformation process is plotted in figure 51. As already observed in the boundary value problem with the single surface model, the curves do agree very well. Maximum values for the equivalent plastic strain are Amax = 0.81556 for U1 and Amax = 0.81837 for V2. nodal force [kN] 70 PSfrag replacements 60 50 40 30 20 U1 V1 10 0 0 10 20 30 40 50 60 70 80 displacement u [mm] Figure 51: Drawing of a flange. Nodal forces at P and Q for the implicit stress update algorithms U1 and V1. Plastic anisotropy is introduced into the constitutive model by a Kelvin-mode decomposition of a cubic fourth-order tensor. 158 Numerical Examples a. b. Figure 52: Drawing of a flange. Deformed meshes and distributions of equivalent plastic strain for u = 25mm, u = 50mm and u = 75mm with the additive plasticity model. (a) Multiplicatively enhanced element design with Amax = 0.4512 and (b) additively enhanced element design with Amax = 0.4465 for anisotropy ratio δ = 3.4641. a. b. Figure 53: Drawing of a flange. Deformed meshes and distributions of equivalent plastic strain for u = 25mm, u = 50mm and u = 75mm with the additive plasticity model. (a) Multiplicatively enhanced element design with Amax = 0.9976 and (b) additively enhanced element design with Amax = 0.9478 for anisotropic response δ = 0.86603. 159 Numerical Examples PSfrag replacements 60 mul.-enh. add.-enh. 50 nodal force [kN] nodal force [kN] 60 40 30 PSfrag replacements 20 10 0 a. 0 10 20 30 40 60 70 40 30 PSfrag replacements 20 0 10 20 30 40 50 60 20 30 40 50 60 70 80 70 80 mul.-enh. add.-enh. 50 40 30 20 d. displacement u [mm] 40 PSfrag replacements 30 20 add.-enh. mul.-enh. 0 10 20 30 40 50 60 displacement u [mm] 0 10 20 30 40 50 60 displacement u [mm] 60 9 15 27 64 50 0 10 displacement u [mm] 0 80 nodal force [kN] nodal force [kN] 70 10 e. 0 10 60 add.-enh. mul.-enh. 20 60 10 PSfrag replacements 30 b. mul.-enh. add.-enh. 50 0 40 0 80 nodal force [kN] nodal force [kN] 50 displacement u [mm] c. 50 10 60 PSfrag replacements mul.-enh. add.-enh. 70 80 9 15 27 64 50 40 30 20 10 0 f. 0 10 20 30 40 50 60 70 displacement u [mm] Figure 54: Drawing of a flange. Nodal forces at Q (upper curves) and P (lower curves) for (a) 9-, (b) 15-, (c) 27- and (d) 64-point integration. Plot (e) shows the results for multiplicatively enhanced and (f) for additively enhanced element design. 80 160 Numerical Examples 9.3.3. Comparison of Additive and Multiplicative Shell Element Design To evaluate the additively enhanced finite shell element design we compare the results with the multiplicatively enhanced shell element design proposed by Miehe [84]. For the simulations the additive plasticity model is used. The set of cubic material parameters listed in table 13 and the two anisotropy ratios introduced in the previous section, δ = 3.4641 and δ = 0.86603, are considered. In figures 52 and 53 the deformed flanges are plotted. They show slight differences in the distribution of the equivalent plastic strains. The difference in the maximum equivalent plastic strains is small. For δ = 3.4641 the nodal forces at P and Q differ only slightly towards the end of the simulation. In the other case, where δ = 0.86603, there are stronger differences as documented in detail in figure 54. Both element formulations seem to be sensitive with respect to the accuracy of the numerical integration scheme. One observes a change of the curves from 15 to 27 gauss points. The limit load also depends on the integration scheme and is different for both element designs. 9.4. Deep Drawing of Cubic and Orthotropic Sheets Now we compare the proposed stress update algorithms as well as the discussed additive and multiplicative material models within the simulation of a deep-drawing process. When drawing cups out of single- and poly-crystalline circular sheets, the top of the cups do not show a constant height, as one would expect in the case of isotropic material behavior. Instead, a periodic sequence of lower and higher points can be observed, the so-called troughs and ears, which are caused by the plastic anisotropy of the blank. An experimental result is shown in figure 56a. The experimental configuration considered is sketched in figure 55, it is axisymmetric with respect to the punch direction. The circular sheet is 39.5 27.0 PSfrag replacements 20.6 4.6 0.81 5.1 22.0 Figure 55: Earing in deep drawing. Geometry and boundary conditions. Lengths in mm. fixed upon the die by a ring. This so-called blank-holder is not pressed against the blank, but is fixed in a distance equal to the initial thickness of the sheet. When pushing down the punch, the specimen is increasingly deformed until it finally drops out of the machine. For the numerical simulation of this problem, both the material models and the shell elements were implemented into the finite element program ABAQUS Standard. Three different contact pairs occur between top and bottom of the blank and the machine tool, which is modeled as a rigid surface. We assume very low friction with a coefficient of 1%. At the final state of the process that prevents rigid body motion of the specimen, when it is no longer fixed by the blank-holder. The discretization of the blank is done by taking 161 Numerical Examples into account the symmetry conditions of the level set function, leading to sectors of 90 ◦ in case of orthotropic symmetry and 45◦ for cubic symmetry, see figure 56b,c. Table 14 lists the material parameters used in the calculations. Table 14: Material Parameters bulk modulus: shear modulus: normal yield stress: normal yield stress: normal yield stress: shear yield stress: shear yield stress: shear yield stress: initial yield stress: hardening modulus: κ µ y11 y22 y33 y12 y23 y13 y0 h = 128000 N/mm2 = 45000 N/mm2 = 200/250 N/mm2 = 200/150 N/mm2 = 200/200 N/mm2 = 80 N/mm2 = 80 N/mm2 = 80 N/mm2 = 200 N/mm2 = 100 N/mm2 9.4.1. Comparison of Additive and Multiplicative Plasticity In order to document the deep-drawing process, figures 57 and 59 show sequences of deformed meshes for both the additive and the multiplicative plasticity model. The latter uses the standard implicit update U1. The additively-enhanced assumed shell element with 9-point quadrature is used. For the cubic level-set function, figure 57, as well as for the orthotropic level-set function, figure 59, the deformed sheet has four ears. For cubic symmetry, all four ears are of equal shape. This is observed in experiments with single-crystal sheets with [100] orientation, see for example Tucker [141]. From the simulation with the orthotropic parameter set, one observes the development of four ears, where the shape and distribution of equivalent plastic strains of opposing ears are equal. This behavior is experimentally observed in the deep drawing of poly-crystalline sheets with rolling texture, see for example Balasubramanian & Anand [5]. The profiles of the ears obtained at the end of the simulations are documented in figure 58. It shows the relative heights of the cup profiles as suggested by Tucker [141] for both sets of material parameters. The corresponding load deflection PSfrag replacementssheet sheet 45◦ a2 PSfrag replacements a1 a1 a. b. 90◦ a2 c. Figure 56: Earing in deep drawing. (a) Experimental observation of earing in deep drawing of a poly-crystalline sheet, taken from Wilson & Butler [153]. (b) and (c) show the circular sheet. The symmetries of the level set functions are pictured by the curved lines. For cubic symmetry (b) one eighth of the sheet, for orthotropic symmetry (c) one quarter is discretized. 162 Numerical Examples a. b. Figure 57: Earing in deep drawing. Formation of four equal ears using the cubic level set function. (a) Multiplicative plasticity using algorithm U1 with maximum equivalent plastic strain Amax = 0.7670 and (b) additive plasticity with Amax = 0.7600. rel. height 1.80 replacements additive multiplicative 1.60 1.40 1.20 1.00 0 30 60 90 120 150 180 210 240 270 300 330 angle [◦ ] Figure 58: Earing in deep drawing. Profiles from multiplicative plasticity with algorithm U1 and additive plasticity. Cubic symmetry leads to four equal ears, the considered orthotropic symmetry yields two equal pairs of ears. The heights are given relative to the reference height of href = 21mm as suggested by Tucker [141]. 360 163 Numerical Examples a. b. Figure 59: Earing in deep drawing. Formation of two equal pairs of ears using the orthotropic level set function. (a) Multiplicative plasticity using algorithm U1 with maximum equivalent plastic strain Amax = 0.8510 and (b) additive plasticity with Amax = 0.8430. 3500 3500 multiplicative additive 3000 2500 load [N] load [N] 2500 Sfrag replacements 2000 1500 PSfrag replacements 1000 500 0 a. multiplicative additive 3000 2000 1500 1000 500 0 5 10 15 20 25 30 35 punch displacement [mm] 40 45 0 b. 0 5 10 15 20 25 30 35 punch displacement [mm] Figure 60: Earing in deep drawing. Load deflection curves of the punch for (a) cubic and (b) orthotropic plastic anisotropy. The results obtained with the multiplicative approach with algorithm U1 and the additive plasticity formulation coincide. 40 45 164 Numerical Examples curves of the punch forces are plotted in figure 60 for both sets of parameters. The curves from the additive and multiplicative plasticity model are identical. 9.4.2. Comparison of the Stress Update Algorithms The simulations of the deep-drawing process were carried out with all four stress update algorithms, U1: standard implicit, U2: standard explicit, V1: variational implicit and V2: variational explicit. The resulting profiles of the cups obtained from sheets with the cubic and orthotropic level set functions are shown in figure 61. The curves match very well. Figure 62 shows the global load displacement curves for the punch. 1.80 rel. height replacements U2 U1 U1 V1 V2 1.60 1.40 1.20 1.00 0 30 60 90 120 150 180 angle 210 240 270 300 330 360 [◦ ] Figure 61: Earing in deep drawing. Comparison of the profiles obtained with the different stress update algorithms. Four equal ears are obtained with the cubic level set function. The orthotropic level set functions brings about two equal pairs of ears. 3500 3500 U1 U2 V1 V2 3000 replacements 2500 load [N] load [N] 2500 2000 PSfrag replacements 1500 2000 1500 1000 1000 500 500 0 a. 0 5 10 15 20 25 30 35 punch displacement [mm] U1 U2 V1 V2 3000 40 45 0 b. 0 5 10 15 20 25 30 35 40 45 punch displacement [mm] Figure 62: Earing in deep drawing. Load-displacement curves obtained with the four proposed stress update algorithms for (a) cubic and (b) orthotropic plastic symmetry. 9.4.3. Comparison of Multiplicative and Additive Shell Element Design In order to compare the additively and the multiplicatively enhanced finite shell elements, the additive plasticity model of section 7 is used with the material parameters listed in table 14. The deformation process of the simulation with cubic level-set function is documented in figure 63 while figure 65 shows deformed plastic orthotropic sheets. One clearly observes the inherent symmetry of the model in the distribution of the equivalent plastic strains. The range of the equivalent plastic strain differs and the plots show slight differences. Nevertheless the profiles are in very good accordance for both the multiplicativelyand additively-enhanced finite shell elements. They are plotted in figure 64. The loaddisplacement curves in figure 66 for the punch also show negligible differences. 165 Numerical Examples a. b. Figure 63: Earing in deep drawing. Formation of four ears with a sheet with cubic level set function using additive plasticity. (a) Multiplicatively enhanced element design with Amax = 0.8480 and (b) additively enhanced element design with Amax = 0.7600. 3500 mul. enh. add. enh. 3000 load [N] 2500 PSfrag replacements orthotropic cubic 2000 1500 1000 500 0 0 5 10 15 20 25 30 35 40 punch displacement [mm] Figure 64: Earing in deep drawing. Load displacement curves of the punch obtained with multiplicative plasticity with algorithm U1 and additive plasticity for cubic and orthotropic symmetry of the level set function. 166 Numerical Examples a. b. Figure 65: Earing in deep drawing. Formation of four ears with a sheet with orthotropic level set function using additive plasticity. (a) Multiplicatively enhanced element design with maximal equivalent plastic strain Amax = 0.8540 and (b) additively enhanced element design with Amax = 0.8430. add. enh. mul. enh. rel. height 1.80 replacements 1.60 1.40 1.20 1.00 0 30 60 90 120 150 180 210 240 270 300 330 angle [◦ ] Figure 66: Earing in deep drawing. The cubic level set function yields four equal ears whereas in the orthotropic case two equal pairs of ears develop. Results from multiplicatively enhanced shell-element design fall in line with those from additively enhanced design. 360 Summary and Conclusion 167 10. Summary and Conclusion This thesis deals with theoretical and computational approaches to the modeling of anisotropic material behaviour at finite elastic-plastic strains. To this end, several different fundamental topics are considered. In the first part, the classification of solids due to their inherent symmetries are discussed. In the second part we focus on the representation theory of isotropic tensor functions and discuss how they can be used to model anisotropic material behaviour. Here recently developed results by Xiao [156, 157] are picked up and applied to the formulation of the important class of quadratic potential functions. Compact coordinate-free representations of fourth-order moduli tensors are given for all classes of anisotropy. To the knowledge of the author, up to now only coordinate representations exist in literature. One should get further experience in the formulation of anisotropic tensor functions with those symmetries where structural functions are required. To this end the interpretation of invariants of special interest. The construction of interpretable invariants is still an open question. Another aspect is related to the identification of the material parameters from experimental tests. Here one is interested on functional bases that yield combined invariants that lead to stress expressions that are sensitive with respect to the material parameters. An essential part of this thesis is the discussion of approaches to anisotropic plasticity. First we discuss the plastic-map approach that bases on a multiplicative decomposition of the deformation gradient into elastic and plastic parts. On the numerical side implicit and explicit stress update algorithms are presented. Beside the standard unsymmetric formulations two symmetric algorithms are developed. The latter ones are related to variational formulations recently developed by Miehe [85] and Miehe, Schotte & Lambrecht [90], cf. also Miehe, Apel & Lambrecht [87]. Furthermore a specific framework of additive plasticity in terms of logarithmic strains is developed. Its characteristic is an attractive modular structure consisting of (i) a pre-processor that determines the elastic strain measure that enters the free energy function, (ii) a constitutive model with a structure similar to models of the geometrically linear theory and (iii) a post-processor that maps the stresses and moduli obtained from the constitutive model in the logarithmic strain space back to the standard Euclidean space. The key difference to existing formulations in logarithmic strains is its modular structure which is here part of the model and not only of the algorithm. The additive plasticity framework provides a powerful approach to a broad spectrum of constitutive models. Scheday successfully [112] used this framework for the identification of material parameters of aluminium and rubbers. It has already been extended to non-local plasticity by Zimmermann [165]. To the authors opinion, this approach provides a good basis for the development of advanced constitutive models and offers many possibilities for future research. For the simulation of shell-like structures we propose a brick-type mixed finite element design. It is the additively enhanced counterpart of a multiplicatively enhanced shell element proposed by Miehe [84]. The additive enhancing simplifies the element formulation enormously. A gradient-type interface to arbitrary deformation driven anisotropic constitutive models is presented. All of the proposed formulations are tested and evaluated by representative numerical 168 Summary and Conclusion examples. Driver shear tests of an isotropic and a cubic material with different load increments are used to investigate the stress update algorithms for rate-independent plasticity. As all integration algorithms are approximations, their accuracy increases with decreasing load step size. Due to the algorithmic stresses that enter the yield criterion function, deviations of the curves are observed for the implicit variational formulation also for small time steps. The implicit formulations behave stable while the explicit ones fail or start to oscillate if the load size step is too large. This phenomenon is not subject of investigations in this thesis, here further research must be done. An evaluation of the quality of the stress update algorithms can be obtained from numerical simulations of boundary value problems. Within the simulation of the necking of an isotropic rod the influence of the step size is investigated. For moderate load step sizes the resulting load displacement curves and the radial contraction of the necking zone are in good accordance. As in the driver test, oscillations are observed at comparable small load step sizes for the explicit codes. In this isotropic test the implicit codes allow very large steps, the variational formulation seems to be more robust than its standard counterpart. The immense difference in the sensitivities of the algorithms with respect to the load step size reduces when considering anisotropic material behaviour. One criterion that allows an evaluation of the four stress update algorithms is the development of the equivalent plastic strains. To this end we consider a simulation of a drawing process of a circular flange using the newly developed additively enhanced shell element design. Furthermore a comparison of nodal forces in the zones of maximum plastic flow is provided for cubic single-surface and a Kelvin-mode decomposition multi-surface phenomenological model. We do not document the influence of the load increment size explicitly, but the chosen increment size is such that the simulations using the standard implicit update cannot be carried out with larger constant increments. The obtained results are in very good accordance. The same quality of the results is observed in the simulation of the deep drawing of a metallic sheet. Here we consider cubic and orthotropic plastic anisotropic response. The update algorithms are compared by means of the global load displacement curves of the punch and the profiles of the cups as well as the distributions of the equivalent plastic strain. An evaluation of the update algorithms in the context of rate-dependent plasticity remains open. The simulation of the above-mentioned drawing processes has also been carried out with the additive plasticity model. The deviations of the curves from those obtained with the plastic map approach using the implicit standard stress update algorithm are very small. Only in the simulation of the drawing process of the flange the nodal forces differ slightly towards the end of the deformation process. The performance of the proposed finite shell element design is documented in detail by comparing the results with those obtained with the multiplicatively enhanced formulation published by Miehe [84]. As a benchmark problem we consider the necking of an isotropic rectangular strip. Here we compare the load displacement curves and the contraction of the necking cross section. The results obtained with both element designs are close together, both elements show the same characteristics concerning the choice of the quadrature rule for the numerical integration. 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Coordinate Representation For the implementation of anisotropic material models into a computer code, we briefly review a notation which is convenient to use and reduces the numerical effort for symmetric second-order and fourth-order tensors with minor symmetries Aij = Aji and Aijkl = Ajikl = Aijlk = Ajilk . (A.1) The basis of the tensors are assumed to be orthonormal. As a consequence of these symmetries that restrict the tensors to the space of symmetric second-order tensors and fourth-order tensors with minor symmetry, their coordinates can be represented as vectors and matrices of a six-dimensional space. A computer implementation can make use of this fact by storing the coordinates of second-order tensors according to the scheme √ √ √ A = Aα = [A11 A22 A33 2A12 2A23 2A13 ]T . (A.2) In order to distinguish between the tensor notation and the corresponding matrix storage form, Latin indices belong to the tensor representation and take the values 1, 2, 3, while Greek letters belong to their matrix representations and take the values from one to six. The coordinates of fourth-order tensors are stored as a 6 × 6 matrix according to the scheme √ √ √ A1111 A1122 A1133 2A 2A 2A 1112 1123 1113 √ √ √ A 2A 2A A2222 A2233 2212 2223 2211 √ √ √2A2213 2A3312 2A3323 2A3313 A3322 A3333 √A3311 √ √ A = Aαβ = . (A.3) √2A1211 √2A1222 √2A1233 2A1212 2A1223 2A1213 √2A2311 √2A2322 √2A2333 2A2312 2A2323 2A2313 2A1311 2A1322 2A1333 2A1312 2A1323 2A1313 The representations (A.2) and (A.3) date back to Kelvin [59] and can frequently be found in literature. They have some nice properties. Due to the preservation of the Froebenius norm of a second-order tensor kAij k = kAα k (A.4) one also speaks of a norm equivalent notion. Moreover, a double contraction of two tensors corresponds to a scalar-product of two vectors, a matrix-vector or matrix-matrix composition in the six-dimensional space, i.e. A : B = Aij Bij = A α Bα A : A = Aijkl Akl = Aαβ Aβ A : B = Aijst Bstkl = Aασ Bσβ . (A.5) The inverse of a fourth-order tensor with minor symmetry is defined by A : A−1 = I or Aijst A−1 stkl = Iijkl = 1 2 (δik δjl + δil δjk ) (A.6) and has the corresponding matrix representation based on a standard matrix inversion Aαδ A−1 δβ = Iαβ = δαβ . (A.7) 180 On Matrix Representations of Symmetric Tensors The Rotation of a tensor with an orthogonal tensor Q = Qij ei ⊗ ej ∈ SO(3) can be expressed by the Rayleigh product as Q ? A = Qia Qjb Aab = Θαβ Aβ . Q ? A = Qia Qjb Qkc Qld Aabcd = Θαγ Θβδ Aγδ (A.8) The corresponding 6 × 6 rotation matrix Θαβ has the coordinate representation Θαβ := Qia Qjb = Q2 Q2 11 Q221 2 √ Q31 2Q Q31 21 √ √2Q11 Q31 2Q11 Q21 12 Q222 2 √ Q32 2Q Q32 22 √ √2Q12 Q32 2Q12 Q22 Q213 Q223 2 √ Q33 2Q Q33 23 √ 2Q Q33 13 √ 2Q13 Q23 √ √2Q12 Q13 √2Q22 Q23 2Q33 Q32 Q22 Q33 + Q23 Q32 Q12 Q33 + Q13 Q32 Q12 Q23 + Q13 Q22 √ √2Q11 Q13 √2Q21 Q23 2Q33 Q31 Q21 Q33 + Q31 Q23 Q11 Q33 + Q13 Q31 Q11 Q23 + Q21 Q13 √ √2Q11 Q12 √2Q22 Q21 2Q31 Q32 Q21 Q32 + Q31 Q22 Q11 Q32 + Q31 Q12 Q11 Q22 + Q21 Q12 with the properties det[Θ] = +1 and ΘT = Θ−1 . (A.9) A.2. Spectral Decomposition of Symmetric Fourth-Order Tensors Consider a fourth-order tensor A having minor symmetries as in (A.1) and major symmetry, Aijkl = Aklij . The space of fourth-order tensors with minor and major symmetries is denoted by T and A ∈ T . Consider the eigen-value problem A : N i = λi N i ; Ni : Ni = 1 . (A.10) The six eigen-values λi and eigen-directions N i can directly be computed using the normequivalent representations (A.2) and (A.3). Assume a labeling of the different eigen-values λ1 < · · · < λr where 2 ≤ r ≤ 6. For each tensor A ∈ T there exactly exists one orthogonal decomposition of T , T = P1 ⊕ · · · ⊕ P r , (A.11) with the subspaces Pi ⊥ Pj for i 6= j. The tensor A can be uniquely decomposed into these subspaces according to A = λ 1 P1 + · · · + λ r Pr (A.12) where Pi are fourth-order eigen-projection tensors, which map a second-order tensor to the space Pi . They constitute a proper orthogonal decomposition of the fourth-order identity tensor ½ Pi for i = j sym P1 + · · · + P r = I ; Pi : Pj = . (A.13) 0 otherwise . In the case of six distinct eigen-vectors we have six uniquely determined eigen-tensors N i . For r < 6 some eigen-values coincide. In this case only the subspace corresponding to the eigen-value with a multiplicity greater than one is uniquely determined, not its basis. For an eigen-value of multiplicity m the projection tensor related to Pi is given by Pi = m X k=1 (λi ) Nk (λi ) ⊗ Nk = r Y A − λi Isym j=1 j6=i λi − λ j , (A.14) 181 Incremental Variational Formulation — Derivatives (λ ) i where N k=1...m denote an arbitrary orthonormal basis spanning Pi . The second equation in (A.14) allows the computation of the projection tensors, solely based on the knowledge of the distinct eigen-values λi=1...r of the given tensor in a closed form. B. Incremental Variational Formulation — Derivatives In this section we deduce the derivatives of the free energy function that occur in the framework of the incremental variational formulation in section 7.4.6. Recall the decoupled structure of the potential functions of the model problem, i.e. ψ(E, I) = ψ e (E e ) + ψ k (A) + ψ i (A) and f α (F ) = f αe (T + B) + f αi (B) . (B.1) These functions depend on the internal strain-like variables and forces that are combined in the sets I := {E p , A, A} and F := {T , B, B}, respectively. The forces are defined by T := ψ,E e ; B := −ψ,A ; B := −ψ,A . (B.2) The evolution equations of the internal variables are integrated using an implicit scheme yielding the discrete forms X X X αe αe αi E p = E pn + γα f,T , A = An + γα f,B , A = An + γα f,B . (B.3) α α α The following identities are direct consequences of the assumed decoupled structure of the level set functions in (B.1) eα αe f,T = f,B , αe αe αe αe f,T T = f,BT = f,BT = f,BB and E p = A . To get a compact notation we introduce the abbreviations E := ψ,E e E e ; K := ψ,AA ; A := E + K and K := ψ,BB for the second derivatives. Furthermore the fourth-order tensors P P αe −1 αi −1 Ē := [A−1 + α γα f,T ; Ē := [K−1 + α γα f,BB ] T] (B.4) (B.5) and B := [( are defined. P α γα ∂T T f αe )−1 + A]−1 (B.6) The derivatives of the internal variables with respect to the algorithmic parameters and the total strains are P p αe αe e e f,T E p,γα = f,T − T : (ψE E + ψ,AA ) : E ,γα P α γααe −1 αe = [I + α γα f,T T : A] f,T (B.7) αe = A−1 : Ē : f,T P βi αi A,γα = f,B − β∈A γβ f,BB · ψ,AA A,γα P βi αi (B.8) = [1 + β∈A γβ f,BB · ψ,AA ]−1 f,B αi −1 = K Ē f,B P P p αe αe E p,E = α f,T T : E − ,T T : A : E ,E α γα fP α γP αe −1 αe (B.9) = (I + α γα f,T : α γα f,T T : A) T : E = B:E. 182 Positively Homogenous Functions of Degree One The derivatives of free energy function take the forms ψ,γα = −T : E p,γα − B : A,γα − B · A,γα P βe p αe = −(T + B) : (f,T − β∈A f,T T : A : E ,γα ) P βi −B · (∂B f αi − β∈A γβ f,B · K · A,γα ) αe αi α = −f − f = −f (B.10) αe αi ψ,γα γβ = f,T : (ψ,E p E p + ψ,AA ) : E p,γβ + f,B · K · A,γβ βe βi αe αi = f,T : Ē : f,T + f,B · Ē · f,B (B.11) ψ,E = ψ,E e : (EP − E p ),E αe = T − T : α γα f,T T : (T + B),E = T (B.12) ψ,EE = T ,E e : (E P − E p ),E αe −1 = E − E : [( α γα f,T + A]−1 : E T) = E−E:B:E (B.13) p ψ,Eγα = ψ,E e E e : (E ,γα P− E )αe αe = −E : (I + α γα f,T T : A)−1 : f,T αe . = E : A−1 : Ē : f,T (B.14) Here we have exploited the fact that the level set functions are homogeneous functions of degree one, i.e. αe T : f,T = f αe αe and T : f,T T = 0 . (B.15) C. Positively Homogenous Functions of Degree One The framework of standard dissipative materials requires that the dissipation function and the level set functions are positively homogeneous of degree one. For a scalar-valued function f (A) of a single second-order tensor A homogeneity of degree one is defined by the property f (²A) = ²1 f (A) (C.1) where ² ∈ R is a scalar constant. A key property of f then is dA f : A = f (A) and d2AA f : A = 0 . (C.2) The first property directly follows from the definition of the Gateaux-derivative of f dA f : A = d d [f (A + ²A)]|²=0 = [(1 + ²)f (A)]|²=0 = f (A) . d² d² (C.3) The second property follows by deriving (C.3) with respect to A 0= d [dA f : A − f (A)] = d2AA f : A + dA f − dA f = d2AA f : A . dA (C.4) Curriculum Vitae Name: Nikolas Apel Date of birth: October 18th , 1970 Place of birth: Ulm, Germany Nationality: German Marital Status: Married, one child Education: 1977 Elementary School: Obereschach (Ravensburg) 1978 – 1981 Elementary School: “Deutsche Schule Lissabon”, Estoril (Portugal) 1981 – 1982 Secondary School: “Deutsche Schule Lissabon”, Lisbon (Portugal) 1982 – 1990 Secondary School: “Gymnasium Weingarten”, Weingarten Community Service: 1990 – 1991 Community service at “Körperbehinderten Zentrum Oberschwaben”, Weingarten Education: 1991 – 1993 Study of Physics at the University of Tübingen 1993 – 1998 Study of Civil Engineering at the University of Hannover Professional Occupation: 1998 – 2003 Assistant Lecturer at the Institute of Mechanics (Civil Engineering), University of Stuttgart since 2003 Computational Engineer at DaimlerChrysler AG, Stuttgart (Germany) In dieser Schriftenreihe bisher erschienene Berichte: I-1(1996) Theoretische und algorithmische Konzepte zur phänomenologischen Beschreibung anisotropen Materialverhaltens, J. Schröder, Dissertation, 1996. I-2(1996) Zur Theorie und Numerik finiter elastoplastischer Deformationen von Schalenstrukturen, B. Seifert, Dissertation, 1996. I-3(1996) Zur Modellierung des künstlichen Infrarot-Dichroismus in Polymerfolien bei großen Verformungen, J. Buhler, Dissertation, 1996. I-4(1998) Verfahren zur Ermittlung der Erdbebenlasten mit Berücksichtigung des stochastischen Charakters des Bebens, S. Zhang, Dissertation, 1998. I-5(1998) Zur Beschreibung finiter Deformationen von Polymeren: Experimente, Modellbildung, Parameteridentifikation und Finite-Elemente-Formulierung, J. Keck, Dissertation, 1998. I-6(1999) Berechnungsverfahren instationärer Systeme im Frequenzbereich, A. Jaworek, Dissertation, 1999. I-7(2000) Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitätsproblemen, J. Schröder, Habilitation, 2000. I-8(2002) Theorie und Numerik von Materialinstabilitäten elastoplastischer Festkörper auf der Grundlage inkrementeller Variationsformulierungen, M. Lambrecht, Dissertation, 2002. I-9(2002) Mikromechanisch motivierte Modelle zur Beschreibung finiter Deformationen gummiartiger Polymere: physikalische Modellbildung und numerische Simulation, F. Lulei, Dissertation, 2002. I-10(2003) Adaptive Finite-Elemente-Berechnungen der nichtlinearen Festkörpermechanik bei kleinen und großen Verzerrungen, A. Koch, Dissertation, 2003. I-11(2003) Theorie und Numerik der Parameteridentifikation von Materialmodellen der finiten Elastizität und Inelastizität auf der Grundlage optischer Feldmeßmethoden, G. Scheday, Dissertation, 2003. I-12(2004) Approaches to the Description of Anisotropic Material Behaviour at Finite Elastic and Plastic Deformations. Theory and Numerics, N. Apel, Dissertation, 2004.

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