13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- dense arrangement of mono-sized spheres, the produced grains are convex and the result is exactly a Voronoï diagram (see Fig. 3a). The use of multi-sized arbitrary shaped polyhedra leads to disordered microstructures (see Fig. 3b). A constant thickness B joint between the grains can be obtained by considering the function < : 9, : − 9,@ : B. 4. LS-XFEM discretisation for material heterogeneities The complexity of generating finite element meshes for the generated RVEs motivates the use of an alternative discretisation method. The eXtended Finite Element Method [5], that does not require meshes conforming with the material boundaries, is therefore used. In addition to be defined as an extension of the standard finite element scheme, this method uses the level set formalism to describe the RVE geometry, which allows its seamless integration with the RVE generator. (a) (b) Figure 3. (a) convex Voronoï-like cells produced by a circle packing, (b) disordered cells produced by an arbitrary shaped multi-sized inclusion packing The principle of XFEM is to use a non conforming regular mesh with additional degrees of freedom related to additional shape functions (denoted the enrichment) introducing the strain jumps induced by material heterogeneities. This treatment, concentrated on finite elements intersected by a material interface (e.g. inclusion/matrix boundary), uses signed distance functions to construct the enrichment and to subdivide elements by material at the stiffness integration stage. The interpolation of each displacement field components therefore reads CDE' F2 2 2 F6 Ψ H6 6 (17) where the first term represents the usual finite element polynomial interpolation containing the standard shape functions as a partition of unity. The second term introduces the XFEM enrichment with H6 the additional unknowns and Ψ the enrichment functions. For heterogeneous materials, the 9, level set (distance) function was shown to introduce the required strain jump at the material boundary. This principle is illustrated in Fig. 4 for the 1D case. Likewise, the pressure field within the heterogeneous microstructure can be described using a similar principle. #CDE' F2 #2 2 F6 Ψ 6 6 (18) The XFEM methodology was implemented in a 3D setting, and was coupled with periodic homogenization. The result of the mechanical loading of a RVE can subsequently be used in the
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