Mechanical Failure in Microstructural Heterogeneous Materials

Various heterogeneous materials with multiple scales and multiple phases in the microstructure have been produced in the recent years. We consider a mechanical failure due to the initiation and propagation of cracks in places of high pore density in the microstructures. A multi–scale method based on the asymptotic homogenization theory together with the mesh superposition method (s-version of FEM) is presented for modeling of cracks. The homogenization approach is used on the global domain excluding the vicinity of the crack where the periodicity of the microstructures is lost and this approach fails. The multiple scale method relies on efficient combination of both macroscopic and microscopic models. The mesh superposition method uses two independent (global and local) finite element meshes and the concept of superposing the local mesh onto the global continuous mesh in such a way that both meshes not necessarily coincide. The homogenized material model is considered on the global mesh while the crack is analyzed in the local domain (patch) which allows to have an arbitrary geometry with respect to the underlying global finite elements. Numerical experiments for biomorphic cellular ceramics with porous microstructures produced from natural wood are presented. 1 Global–Local Approach for Heterogeneous Materials Consider a domain Ω ⊂ Rd, d = 2, 3, occupied by a heterogeneous material with microstructures of periodically distributed constituents. Suppose that the boundary of Ω, denoted by Γ , consists of a prescribed displacement boundary ΓD (meas ΓD > 0) and a prescribed traction boundary ΓT , such that Γ = ΓD ∪ ΓT , ΓD ∩ ΓT = ∅, as shown in Figure 1. Assume that the periodic cells in the macrostructure are infinitely many but infinitely small and repeated periodically through the medium. The unit microstructure consists of different material constituents and a pore. Both the This work has been partially supported by the German National Science Foundation (DFG) under Grant No.HO877/5-3. The third author has also been supported in part by the Bulgarian Ministry for Education and Science under Grant I1402/2004. T. Boyanov et al. (Eds.): NMA 2006, LNCS 4310, pp. 533–541, 2007. c © Springer-Verlag Berlin Heidelberg 2007 534 S. Bordas, R.H.W. Hoppe, and S.I. Petrova