تجزیه و تحلیل حساسیت شکل توپولوژی برای ترک در اجسام الاستیک در مرزهای اجزاء سخت

We consider an elastic body with a rigid inclusion and a crack located at the boundary of the inclusion. It is assumed that nonpenetration conditions are imposed at the crack faces which do not allow the opposite crack faces to penetrate each other. We analyze the variational formulation of the problem and provide shape and topology sensitivity analysis of the solution in two and three spatial dimensions. The differentiability of the energy with respect to the crack length, for the crack located at the boundary of rigid inclusion, is established.

The problem associated with cracks in elastic bodies on boundaries of rigid inclusions appears in a vast number of applications in civil, mechanical, aerospace, biomedical and nuclear industries. In particular, some classes of materials are composed by a bulk phase with inclusions inside. When the inclusions are much stiffer than the bulk material, we can treat them as rigid inclusions. In addition, it is quite common to have cracks between both phases. Thus, in this paper we deal with the mechanical modeling as well as the shape and topology sensitivity analysis associated with the limit case of rigid inclusions in elastic bodies with a crack at the interface. The mechanical modeling is based on the assumption of nonpenetration conditions at the crack faces between the elastic material and the rigid inclusion, which do not allow the opposite crack faces to penetrate each other, leading to a new class of variational inequalities. For the sensitivity analysis, we attempt to find the shape derivative of the elastic energy with respect to the perturbations of the crack tip. We also obtain the topological derivatives of the energy shape functional associated with the nucleation of a smooth imperfection in the bulk elastic material. These quantities are very important in design procedures and in numerical solution of some inverse problems. Both the analysis and the shape and topology optimization of this class of problems seem to be new and very useful from the mathematical and also the mechanical points of view. The paper is organized as follows. The problem formulation associated with cracks in elastic bodies on boundaries of rigid inclusions is presented in Section 2. Some results concerning shape sensitivity analysis with respect to the perturbations of the crack tip are given with all details in Section 3. The topological derivatives associated with the energy shape functional are calculated in Section 4. We provide some closed formulas for the case of nucleation of spherical holes in 3D and circular elastic inclusions in 2D. In this last case, we present the limit cases in which the elastic inclusion becomes a hole (void) and also a rigid inclusion.