13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- irreversibility of the previous sliding into account. This procedure in combination with approximation of the slip band by two closely located layers is capable to account for irreversible fraction of sliding. Hardening is assumed to result from a rising dislocation density. Therefore in the model the dislocation density is raised according to the plastic slip deformation in the slip band. After the slip band mechanisms are well defined in the simulation model in the following the numerical method is presented. 4. Numerical Method The boundary element method used in this study combines the traditional displacement BEM as well as the displacement discontinuity BEM [9]. By doing so, sliding displacements can be directly evaluated in slip bands and element discretization is confined to outer boundaries such as grain boundaries and on slip bands. The BEM used in this study is described by means of a problem statement, which consists of a two-dimensional homogeneous, anisotropic and linear elastic solid containing a finite slip line as shown in Fig. 3. Figure 3. An anisotropic solid including a slip line On the external boundary Γb displacements and tractions with components ui and pi are prescribed, while relative displacements Δui and stresses σiα are considered on one face Γs of the slip line. Here, relative displacements consist only of tangential relative displacements, because slip lines in contrast to cracks cannot perform opening - only sliding. Throughout the analysis the conventional summation rule over double indices is applied, Roman and Greek indices can only have the values 1 and 2. The procedure is based on two boundary integral equations: the displacement boundary integral equation, which is applied on the external boundary, and the stress boundary integral equation, which is used on the slip line face. The displacement boundary integral equation for a solid containing a slip line can be written as [9,10]: * * * ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) , b s ij i ij i ij i y ij i y b c u u p p u d p u d Γ Γ ⎡ ⎤ ⋅ = ⋅ − ⋅ Γ + ⋅Δ Γ ∈Γ ⎣ ⎦ ∫ ∫ x x y y x y y x y y x , (4) where cij equals 0.5 when Γb is smooth and * ( , ) ij u x y and *( , ) ij p x y are the displacement and the traction fundamental solutions. Vector x denotes the positions, where displacements are determined, and y denotes the integration points on the boundaries Γb and Γs. The stress boundary integral equation is obtained by substituting equation (4) into Hooke's law: * * * ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) , b s j ij i ij i y ij i y s d p s u d s u d Γ Γ ⎡ ⎤ = ⋅ − ⋅ Γ + ⋅Δ Γ ∈Γ ⎣ ⎦ ∫ ∫ x x y y x y y x y y x γ γ γ γ σ , (5) where * ( , ) ij d γ x y and * ( , ) ij s γ x y are the stress and the higher-order stress fundamental solutions.
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