SHAPE AND TOPOLOGY SENSITIVITY ANALYSIS FOR CRACKS IN ELASTIC BODIES ON BOUNDARIES OF RIGID INCLUSIONS A.M. KHLUDNEV, A.A. NOVOTNY, J. SOKO LOWSKI, AND A. ˙ ZOCHOWSKI Abstract. We consider an elastic body with a rigid inclusion and a crack located at the boundary of the inclusion. It is assumed that non-penetration 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. 1. Introduction The problem associated to 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 modelling as well as the shape and topology sensitivity analysis associated to the limit case of rigid inclusions in elastic bodies with a crack at the interface. The mechanical modelling is based on the assumption of non-penetration 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 to 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 point of views. The paper is organized as follows. The problem formulation associated to 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 to the energy shape functional are calculated in Section 4. We provide some closed formulas for the case of nucleation of spherical holes in three spatial dimensions and circular elastic inclusions in two spatial dimensions. In 2D case, two limit passages with respect to the so-called contrast factor are performed in such ways that the elastic inclusion become a void or a rigid inclusion. 2. Problem formulation Let Ω ⊂ R 3 be a bounded domain with smooth boundary Γ, and ω ⊂ Ω be a subdomain with smooth boundary Ξ such that ω ∩ Γ= ∅. We assume that Ξ consists of two parts γ and Ξ \ γ , meas(Ξ \ γ ) > 0, where γ is a smooth 2D surface described as x i = x i (y 1 ,y 2 ), (y 1 ,y 2 ) ∈ D, i =1, 2, 3, with bounded domain D ⊂ R 2 having a smooth boundary ∂D, and the rank of the matrix ∂x ∂y is equal to 2. Denote by ν =(ν 1 ,ν 2 ,ν 3 ) a unit outward normal vector to Ξ, see Fig. 1. The subdomain ω is assumed to correspond to a rigid inclusion, and the surface γ describes a crack located on Ξ. Key words and phrases. Rigid inclusion, crack growth, unilateral condition, frictionless contact, shape deriva- tive, topological derivative, asymptotic analysis. 1