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Key Engineering Materials
ISSN: 1662-9795, Vol. 754, pp 226-229
381-381
doi:10.4028/www.scientific.net/KEM.754.381
doi:10.4028/www.scientific.net/KEM.754.226
© 2017 Trans Tech Publications, Switzerland
Online: 2017-09-05
2017-09-11
On the Scale-Transition in Multiscale Modeling of Ductile Damage
Tomislav Lesičar, Jurica Sorić, Zdenko Tonković
Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb
Ivana Lučića 5, 10000 Zagreb, Croatia
tomislav.lesicar@fsb.hr, jurica.soric@fsb.hr, zdenko.tonkovic@fsb.hr
Keywords: heterogeneous materials, nonlocal ductile damage, computational homogenization,
periodic boundary conditions
Abstract. The standard numerical methods cannot adequately capture damage response of ductile
heterogeneous materials, since they are unable to encompass the governing microstructural
mechanisms. This issue is resolved by the multiscale approach, where the coarse scale problem
incorporates behavior at the lower scales. In this paper, a multiscale scheme is employed for scaling
of ductile damage occurring at the microstructural level to the macrolevel. A nonlocal implicit
damage model is applied on the RVE, while a scale transition is performed by means of the firstorder homogenization scheme. The influence of the boundary conditions used at the microscale on
the macroscale response, as well as physical consistency of the RVE mechanical behavior are
tested. It has been shown that displacement boundary conditions inhibit development of the
softening band inside of the RVE and induce artificial material stiffness. The derived algorithms are
embedded into the finite element software ABAQUS.
Introduction
In recent years, in order to fulfil requirements on the load carrying capacity, development of new
materials is needed, where ductile materials are very attractive due to their mechanical properties.
Thus, development of new numerical techniques which can describe behavior of ductile materials
more realistically, especially in the softening regime has become very popular. Material
irregularities at the microstructural level lead to anisotropic behavior, which governs different kinds
of mechanisms, such as microcracking, dislocation plasticity, etc. Unfortunately, the usual
numerical methods cannot adequately predict mechanical behavior of such materials, because they
are unable to capture the relevant microstructural mechanisms. Partial differential equations
associated with a material damage become ill-posed, resulting in their loss of ellipticity. On the
other hand, the solutions obtained reveal dependence on the mesh refinement and alignment [1].
Hence, a special attention has been directed to the investigation of the relations between the
macroscopic material behavior and its microstructure. In the multiscale approach, the response of a
coarse scale problem incorporates physical understanding of material behavior at lower scales. In
recent formulations the computational homogenization approach has become the most popular [2,
3]. In this computational scheme, the results obtained by the simulation of a microscopic
representative sample of material, named Representative Volume Element (RVE), are used at the
macrolevel analysis. Therein, the first-order computational homogenization approach has shown
high versatility [4, 5]. In this computational strategy comprises the local continuum theory is
adopted at both scales.
In this paper, a multiscale procedure employing computational homogenization scheme is
developed, in order to consistently scale the microlevel strain localization towards a macroscopic
fracture. The particular attention is directed towards the application of appropriate boundary
conditions (b. c.) on the RVE, and their influence on the physical material response. To keep the
aforementioned problem as simple as possible, the first-order computational homogenization
scheme assuming small strain and material nonlinearity will be employed. Discretization of both
scales is performed by means of the quadrilateral finite elements. At the microscale, the implicit
nonlocal ductile damage model is employed [6]. The procedure is implemented into the FE software
ABAQUS via user subroutine UMAT.
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans
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(#103310979, University of Auckland, Auckland, New Zealand-11/11/17,22:38:58)
382
Advances
Key Engineering
in Fracture
Materials
and Damage
Vol. 754Mechanics XVI
227
Nonlocal implicit ductile damage formulation
In the following, a brief overview of the finite element implementation of the nonlocal ductile
damage model is presented. Here, besides the usual equilibrium equation
∇ ⋅ σ + f = 0,
(1)
an additional Helmholtz-type equation
(2)
εp − l 2∇ 2 εp = ε p
should be solved. In Eq. (2), ε p is the standard equivalent plastic strain measure, while εp
represents the nonlocal equivalent plastic strain. As can be seen, the Laplacian of εp is multiplied
by the microstructural parameter l . In this way, the solution of εp is continuous over the entire
computational domain Ω , which enables the straightforward satisfaction of the additional boundary
condition [7]
∇ εp ⋅ n = 0
(3)
only on the external boundary ∂Ω , which is an advantage over the explicit approach, where
condition (3) should be prescribed on the evolving elastoplastic boundary. The standard linear
isotropic hardening rule is employed, with modified yield function
(4)
F ( σ, ε p , εp ) = σ e ( σ ) − 1 − D ( εp ) σ y ( ε p ) ,
(
)
where σ e is the equivalent von Mises stress, and σ y denotes the yield stress. Modification of the
(
)
yield function is arising by the multiplier 1 − D ( εp ) , where D is the damage variable depending
on the nonlocal plastic measure. In the finite element formulation, besides the displacements, the
nonlocal equivalent plastic strain is discretized using the corresponding shape functions, along with
the following constitutive relations
(5)
δσ = Cσε δε − Cσε p δεp ; δε p = Cε pε δε − Cε p εp δεp ,
where Cσε , Cσε p , C ε pε ,Cε p εp represent the constitutive matrices. Using the standard procedures, the
mixed-type of the finite element equation is obtained
 K uu K uε p   ∆u   Feu − Fiu 
=


,
 K ε u K ε ε   ∆ε p   Fε p 
p p 
i


 p
(6)
where contributions of the displacement field ( u ) and the nonlocal plastic strain measure ( εp ) are
accounted for. The presented methodology is embedded into the finite element framework by means
of the four-node quadrilateral finite element. The element has three nodal DOF (2 displacements
and nonlocal plastic strain), and it is derived under the plane strain assumption. The presented
algorithm is implemented into the FE software ABAQUS via UEL subroutine.
First-order computational homogenization scheme
In the following, basics of the micro-macro scale transition procedure are explained. All variables
appearing at the microscale are denoted by the subscript “m”, while the macrostructural quantities
are referenced by the index “M”. In the first-order homogenization the RVE displacement field is
dependent on the macrolevel strain ε M as
um = εM ⋅ x + r .
(7)
Here x and r represent the microlevel spatial coordinate and the microfluctuation field,
respectively. Based on the displacement field (7), the microlevel strain tensor may be expressed as
εm = εM + ∇ ⊗ r .
(8)
228
Advances
Key Engineering
in Fracture
Materials
and Damage
Vol. 754Mechanics XVI
383
In order to satisfy the averaging over the RVE, it is obvious that the microfluctuation must vanish
1
1
(9)
( ∇ ⊗ r ) dV = ∫ ( n ⊗ r ) dΓ = 0 ,
∫
VV
V Γ
where Γ stands for an outer RVE boundary. In the case of the displacement b. c. all boundary nodes
have prescribed values based on Eq. (7), where the microfluctuations are suppressed. Furthermore,
introducing the coordinate matrix D , Eq. (7) can be recast into matrix presentation
u b = DT ε M ,
(10)
where u b are the displacements of the RVE boundary nodes. In the case of the periodic b. c., only
corner nodal displacements are prescribed. The kinematics of the remaining boundary nodes is
related based on the periodicity assumption among the opposite edges (Left-Right, Top-Bottom),
leading to the periodicity equations
u R − u L = DTR − DTL ε M ; uT − uB = DTT − DTB ε M .
(11)
(
)
(
)
In the computational homogenization scheme the homogenized macroscale stress tensor can be
computed based on the Hill-Mandel energy condition, leading to
1
(12)
σ M = ∫ σ m dV .
VV
To account the contribution of the heterogeneous microstructure, the macrolevel constitutive
, obtained by the static condensation
behavior is dependent on the condensed RVE stiffness K
bb
procedure, which leads to
1 T
C = DK
(13)
bb D .
V
Numerical example
In order to test the influence of boundary conditions on the homogenized behavior of the damaged
heterogeneous material, a tension test has been performed. An academic example of a steel
consisting of 13% of porosities has been used. The matrix material has Young’s modulus
E = 210 GPa and Poisson’s ratio ν = 0.3 , exhibiting linear isotropic hardening with yield stress
σ y = 200 MPa and elastoplastic tangent modulus of 20000 MPa . The material microstructure is
presented by the RVE of side length 0.2 mm, discretized by 508 finite elements, as presented in
Fig. 1. The linear damage model, D = ε p ε pc is adopted, where the critical value of the nonlocal
plastic strain is ε pc = 5 ⋅ 10 −4 . The results presented by the reaction force vs. displacement diagram at
the macrolevel are shown in Fig. 2. As evident, the periodic b. c. provide more compliant behavior,
especially in the post peak regime. It is known that the displacement b. c. have stiffer response
compared to the periodic b. c. in a general case when damage is not involved. Thus, this effect is
also apparent in the softening regime, as expected. However, from Figs. 3 and 4 can be seen that the
displacement b. c. are not suitable in the softening regime at all. In those figures the distribution of
the nonlocal plastic strain is displayed. The grey areas represent the damaged zones, where the
critical value ε pc is reached. It is clear that the displacement b. c. inhibit development of localization
band inside the RVE. Eventually, the localization band will be developed, but at much higher
strains. Due to the prescribed displacements on the boundary, the artificial stiffness is induced,
leading to unphysical stiff behavior of the microstructure, which makes this type of b. c.
unfavourable for the softening RVE.
Advances
Key Engineering
in Fracture
Materials
and Damage
Vol. 754Mechanics XVI
Reaction force, N
384
350
300
250
200
150
100
50
0
229
Displacement b. c.
Periodic b. c.
0
0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08
Displacement, mm
Fig. 1 RVE.
Fig. 3 Distribution of ε p for displacement b. c.
Fig. 2 Force vs. displacement diagram.
Fig. 4 Distribution of ε p for periodic b. c.
Conclusions
The paper deals with the modeling of ductile damage in a multiscale scheme employing the firstorder homogenization. Details on the finite element implementation of a nonlocal implicit damage
model defined at the microscale are explained. The micro-macro scale transition methodology is
discussed. The displacement and the periodic b. c. are tested on an academic RVE. It has been
concluded that in the softening regime the displacement b. c. increase material stiffness resulting
with the non-physical response and inhibit development of the shear bands.
Acknowledgements
The work has been supported in part by Croatian Science Foundation under the project “Multiscale
Numerical Modeling of Material Deformation Responses from Macro- to Nanolevel” (2516). The
investigations are also part of the project supported by Alexander von Humboldt Foundation,
Germany.
References
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[2]
[3]
[4]
[5]
[6]
[7]
G. Pijaudier-Cabot and Z. P. Bazant, Journal of engineering mechanics 113(10) (1987) 15121533.
T. Wu, İ. Temizer, et al., Cement and Concrete Composites 35(1) (2013) 59-70.
T. Lesičar, J. Sorić, et al., Computer Methods in Applied Mechanics and Engineering 298
(2016) 303-324.
V. Kouznetsova, W. A. M. Brekelmans, et al., Computational Mechanics 27(1) (2001) 37-48.
C. Miehe and A. Koch, Archive of Applied Mechanics 72(4-5) (2002) 300-317.
R. A. B. Engelen, M. G. D. Geers, et al., International Journal of Plasticity 19(4) (2003) 403433.
R. De Borst and J. Pamin, International Journal for Numerical Methods in Engineering
39(14) (1996) 2477-2505.
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