Modeling of microdamage accumulation in anisotropic metals and alloys M. N. Krivosheina, M. A. Kozlova, S. V. Kobenko, E. V. Tuch, and A. I. Lotkov Citation: AIP Conference Proceedings 1683, 020105 (2015); View online: https://doi.org/10.1063/1.4932795 View Table of Contents: http://aip.scitation.org/toc/apc/1683/1 Published by the American Institute of Physics Modeling of Microdamage Accumulation in Anisotropic Metals and Alloys M. N. Krivosheina1, a), M. A. Kozlova1, b), S. V. Kobenko2, c), E. V. Tuch1, d), and A. I. Lotkov3, e) 1 Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055 Russia 2 Nizhnevartovsk State University, Nizhnevartovsk, Russia 3 National Research Tomsk State University, Tomsk, 634050 Russia a) Corresponding author: marina_nkr@mail.ru b) kozlova_ma@mail.ru c) sergeyvk@inbox.ru d) tychka2012@mail.ru e) lotkov@ispms.tsc.ru. Abstract. This paper studies the processes of fracture of barriers using the fracture criterion, allowing for consideration of large anisotropy for ultimate values of total plastic strains in various symmetry axes of an anisotropic material in conditions of sign-variable strain. All calculations were performed using the finite-element method under conditions of dynamic loading in three-dimensional graphics with the use of original software. It was shown that when modeling processes of fracture in barriers made of anisotropic and isotropic materials the differences in size and localization of fracture zones increase with the increase in the initial speed of shock loading of barriers. INTRODUCTION Strain anisotropy of mechanical characteristics occurs in metals and alloys as a result of formation, each of them occurring to different extents. As a rule, strength characteristics have the maximum anisotropy. When breaking-off, the values of the ultimate stress of samples which are cut out in three mutually perpendicular directions may differ by 10–15%, and the values of ultimate strains corresponding to them by 10 times. Calculating the cumulative plastic strain in sign-variable processes allows for consideration of accumulation of microdamage in dynamic processes of material loading. For modeling the fracture of metals and alloys within the limits of isotropic deformation models under dynamic conditions of loading, especially under conditions of compression, the fracture criterion containing the value of cumulative plastic strain—the Odkvist parameter is often used. The Odkvist parameter for anisotropic materials reflects only the anisotropy of elastic and plastic properties, but does not reflect anisotropy of strength characteristics. Therefore, the fracture criterion using values of ultimate cumulative plastic strains in symmetry axes of a material was used. This criterion allows for modeling of the microdamage accumulation in these axes in anisotropic materials. The relative residual strain under tension and shear in constructional materials is used as values of ultimate cumulative plastic strains. The strain in samples was calculated in three symmetry axes of materials. It is especially important for materials with a high degree of anisotropy of plastic properties because when fracturing in some directions they can be considered low-plastic, and in others, under fracture, they have developed a plastic strain [1–4]. This research aims to study the fracture of barriers using a fracture criterion which allows for the consideration of large anisotropy of ultimate values of total plastic strains in various symmetry axes of an anisotropic material in conditions of sign-variable strain occurring under wave pattern of deformation. Advanced Materials with Hierarchical Structure for New Technologies and Reliable Structures AIP Conf. Proc. 1683, 020105-1–020105-4; doi: 10.1063/1.4932795 © 2015 AIP Publishing LLC 978-0-7354-1330-6/$30.00 020105-1 FIGURE 1. Initial volumetric configuration of the projectile and the barrier PROBLEM STATEMENT All calculations were performed with the finite-element method [5] in three-dimensional graphics with the use of original software. Discretization of computational regions of the projectile (D1) and the barrier (D2) was carried out by applying tetrahedrons (Fig. 1). MODELING OF ELASTO-PLASTIC DEFORMATION OF AN ORTHOTROPIC MATERIAL The system of equations describing non-stationary adiabatic motions of the compressible anisotropic medium includes [6] equation of continuity: wU div U v 0, (1) wt equation of continuous medium motion: U dv k dt wV ki Fk, wxi (2) energy equation: dE 1 Vij eij . (3) dt U Here U—medium density, v—velocity vector, V ij —contravariant components of symmetric stress tensor, E— specific internal energy; 1 ( i v j j vi ), 2 eij —components of symmetric strain velocity tensor, vi —components of velocity vector, i, j = 1, 2, 3. eij (4) Let us suppose that full strain is representable in the form of the sum of elastic and plastic strains, the plastic flow of an anisotropic material does not depend on hydrostatic pressure, and the elastic properties of a material do not change under plastic deformation. Elastic deformation of a material is described by the generalized Hooke’s law dVij Cijkl ekl , (5) dt Cijkl —components of the elastic constant tensor. In the field of plastic strains the full tension tensor is decomposed into spherical and deviatoric parts Vki PGki S ki , ki where Ɋ—hydrostatic pressure, S —components of stress deviator and G ki —Kronecker delta. 020105-2 (6) The dependence of hydrostatic stress on current density and specific internal energy has the form (GrüneisenMie EOS): n §V · ª §V · º P ¦ K n ¨ 1¸ «1 K 0 ¨ 0 1¸ 2 » K 0UE , n 1 V V © ¹ ¼ ¬ © 0 ¹ 3 (7) where K 0 , K1 , K 2 , K3 —constants of a material, V0 , V—initial and current volumes. Take the associated flow law in the form of dHijp dO wF , wVij (8) parameter dO = 0 under elastic strain, under plastic strain it is always positive, it is determined by the yield criterion, H ijp —components of plastic strain, F—plasticity function. The von Mises–Hill ’48 criterion is used as a plasticity function. The fracture criterion is used p T kl d E kl , k , l 1, ..., 3, where Ekl —relative residual strain under fracture in conditions of tension and shear, and increment strain of every component of plastic strain tensor separately Tpkl (9) T pkl —cumulative plastic p ³ d Hkl , k , l 1, ..., 3. Tensions determined in the element which is rigidly turned in space are recalculated by the Jaumann derivative. Elasto-plastic deformation of a material of the projectile was determined within the limits of the presented model for a special case of isotropic material. RESULTS AND DISCUSSION By means of the suggested criterion [7] the fracture of barriers made of transtropic aluminum alloy D16T is modeled as well as ones made of a material having isotropic characteristics received by averaging the characteristics of transtropic material. The results of numerical modeling of the fracture of barriers are given at initial loading velocities of 200 and 600 m/s with a steel compact projectile of cylindrical form and 20 g in mass. The elastic, plastic and strength characteristics of a barrier material [8] are the following: U = 2700 kg/m3, Ex = 86.7GPa, E y Ez = 92.1 GPa, Q xy = 0.32, Q zx = 0.34, Q yz = 0.33, Gxy Gxz = 33 GPa, Gyz = 31 GPa, V xs = 290 MPa, V ys V zs = 350 MPa, W xys fracture were E x = 0.14, E y W xzs = 150 MPa, W yzs = 180 MPa. The values of ultimate cumulative strains under E z = 0.2, E xy E xz = 0.07, E yz = 0.1. Here Ei —Young’s moduli, Gij —shear moduli, Qij —Poisson’s ratios of a transtropic material, Vis —tensile yield limits, Wijs —shear yield points, Eij —values of ultimate strains under fracture in conditions of tension and shear. The characteristics of isotropic material of the barrier received by averaging the characteristics of a transtropic material are the following: E = 87883MPa, G = 32934 MPa, Vs = 330 MPa, E = 0.18. The direction of shock loading is oriented along the axis OZ (Fig. 1), and a special axis of a transtropic material of the barrier is the axis 0X so the non axisymmetric stress-strain state occurs in the barrier. To illustrate the differences arising in various sections of the barrier, Figure 2 presents two halves of barrier sections completed in planes ZOY (left parts) and ZOX (the right parts). Figures 2a and 2b present distributions of differences in mass fractions of the fractured material Rd in sections of the barrier which were received in the calculations with transtropic mechanical characteristics of the barrier deducting distributions of mass fractions of the fractured material in similar barrier sections with isotropic mechanical characteristics. In the ZOY sections (the left parts of Fig. 2) the values of mechanical characteristics in isotropic material are less than in transtropic material. Therefore, negative values for Rd arise, i.e. in this section there are more fracture zones in isotropic barrier material. In the ZOX sections (the right parts) the values of mechanical characteristics in isotropic material are larger than in transtropic material. Therefore, the distribution of positive values for Rd presents distribution of additional fracture zones in transtropic barrier material as compared to isotropic material. 020105-3 (a) (b) FIGURE 2. Differences in mass fractions of the fractured material in sections of the barrier made of transtropic and isotropic materials at the time moment 24 μs: (a) 200 m/s, (b) 600 m/s Sections of deformable projectiles in Fig. 2 are removed. With the initial velocity of the projectile at 600 m/s the fracture of the barrier occurs like “knocking-out a stopper”. It can be seen at the back surface of the barrier (Fig. 2b). Additional zones of the fractured material in the barrier (Rd) made of isotropic material are observed in the ZOY section (Fig. 2b, left part). In the ZOX section additional zones of the fractured material arise in the barrier made of transtropic material, especially in the back area of the barrier and near the projectile (Fig. 2b, right part). Figure 2 illustrates that with the initial velocity of the projectile at 600 m/s the differences in distribution Rd in fracture zones in the barriers made of isotropic and transtropic materials increase. The increase in speed of shock loading of the barrier leads to an increase in the cumulative plastic strain in the direction of each of the axes of transtropic material of the barrier. This explains the increase in differences in the volume of zones of the fractured material of the barrier for cases of isotropic and transtropic materials. The distribution of differences of the fractured material in the volume of barrier material is determined by the wave pattern of stress state of the projectile and the barrier. CONCLUSION The use of the fracture criterion allows for the consideration of the process of microdamage accumulation and fracture, taking into account anisotropy of ultimate strains under fracture. This criterion also allows for those fracture zones in the barrier detecting zones that are localized, which are not considered when modeling fracture processes in barriers made of isotropic materials. It has been shown in the case when the minimum elastic, plastic and strength properties of a transtropic material of the barrier are oriented perpendicular to the direction of rolling. The study was conducted as part of fundamental scientific research of the State Academies of Sciences for 2013– 2020. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. F. Barlat, J. C. Brema, J. W. Yoon, K. Chung, R. E. Dicke, D. J. Legea, F. Pourboghratg, S.-H. Choif, and E. Chu, Int. J. Plasticity 19, 1297–1319 (2003). J.W. Yoon, F. Barlat, J.J. Gracio, and E. Rauch, Int. J. Plasticity 21, 2426–2447 (2005). O. Cazacu, B. Plunkett, and F. Barlat, Int. J. Plasticity 22, 1171–1194 (2006). G. Rousselier, F. Barlat, and J. W. Yoon, Int. J. Plasticity 26, 1029–1049 (2010). G. R. Johnson, J. Appl. Mech. 95–100 (1977). L. I. Sedov, Continuum Mechanics (Nauka, Moscow, 1976). Ɇ. N. Krivosheina, ȿ. V. Tuch, S. V. Kobenko, Ɇ. Ⱥ. Kozlova, and I. Yu. Konysheva, Modelling of the Deformation of Highly Porous Metals and Alloys under Dynamic Loading, in Physical Mesomechanics of Multilevel Systems-2014, AIP Conference Proceedings 1623, edited by V. E. Panin, et al. (American Institute of Physics, Melville, NY, 2014), pp. 315–318, doi 10.1063/1.4901487. V. V. Kosarchuk, B. I. Kovalchuk, and A. A. Lebedev, Strength Mater. 4, 50–57 (1986). 020105-4

1/--страниц