13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 3. Finite Elements (FE) calculations 3.1. FE model and loading A 2D matrix is considered, embedding a surface grain, main grain (MG), which contains a slip band. Constitutive laws in each material are: - Both isotropic and cubic elasticity laws in SB. As shown in Fig. 2a, it is characterized by a quasi-perfect-plastic flow, indeed, a low hardening coefficient (H0 ~ 1MPa) is assumed for avoiding numerical convergence problems. Only, one slip system is activated in the SB, which slip plane is (111) and slip direction is [101]. A yield stress 0τ , equals to 60 MPa, is initial SB critical shear stress. - Both isotropic and cubic elasticity laws in the MG. - The matrix obeys isotropic elasticity, defined by a Young’s modulus and Poisson ration values. Crystalline FE elements are used in the mesh shown in Fig. 2b to allow FE calculations using Cast3M code. A tensile loading is imposed alongx ρ direction and plane strain is assumed. In addition, many FE computations proved that this model is insensitive to both the mesh size and the time stepping. Figure 2.a) Perfect plasticity behavior of the slip band b) Zoom on the mesh: main grain (MG) and slip band (SB) c) Intersection of the SB and the GB. The applied tensile stress 0Σ is high enough to lead to slip band plastic flow and (Table 1) shows both isotropic and cubic elasticity parameters used in the calculations [30]. C11, C12 and C44 are the crystalline elasticity parameters. Table 1. Isotropic and cubic elasticity parameters E (in matrix, MG and SB) 180 GPa ν (in matrix, MG and SB) 0.33 Isotropic elasticity Cubic elasticity (MG and SB) (MG and SB) C11 267 GPa 267 GPa C12 131 GPa 131 GPa C44 68 GPa 224.4 GPa a 1 3.3 The anisotropy coefficient is defined by 12 11 44 / 2 a C C C− = and if it equals to one, crystalline elasticity is isotropic. In case of copper or austenitic stainless steel, the Young’s modulus along the <111> directions is more than 3 times the one along the <100> directions. That is why it would be worth to numerically account for such anisotropy in our contribution. Matrix Localized band GB r grain 0 20 40 60 80 100 120 140 160 180 0 0,001 0,002 0,003 0,004 0,005 0,006 0,007 0,008 0,009 0,01 strain stress (MPa) 10-3s-1 10-5s-1 a) b) c)
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