Abstract
In this article, a general method for shape-topology sensitivity analysis of contact problems is proposed. The method uses domain decomposition combined with specific properties of minimizers for the energy functional. The method is applied to the static problem of an elastic body in frictionless contact with a rigid foundation. The contact model allows a small interpenetration of the bodies in the contact region. This interpenetration is modeled by means of a scalar function that depends on the normal component of the displacement field on the potential contact zone. We present the asymptotic behavior of the energy shape functional when a spheroidal void is introduced at an arbitrary point of the elastic body. For the asymptotic analysis, we use a nonoverlapping domain decomposition technique and the associated Steklov---Poincare pseudodifferential operator. The differentiability of the energy with respect to the nonsmooth perturbation is established, and the topological derivative is presented in the closed form.
Highlights
Topological asymptotic analysis [1,2,3] allows us to obtain an asymptotic expansion of a given shape functional of linear elasticity when a geometrical domain is singularly perturbed by the insertion of holes, inclusions, source-terms, or even cracks
The knowledge of analytical form of topological derivatives is important in numerical methods of shape optimization
There are no asymptotic analysis tools which can be applied in the general case of nonlinear structural models in mechanics
Summary
Topological asymptotic analysis [1,2,3] allows us to obtain an asymptotic expansion of a given shape functional of linear elasticity when a geometrical domain is singularly perturbed by the insertion of holes, inclusions, source-terms, or even cracks. A less restrictive boundary condition on the contact region is obtained by considering the normal compliance model In this kind of models, based on the assumption of small displacement, some interpenetration between the contacting bodies is allowed, and the boundary forces are given as a function of the interpenetration. A new class of models has been presented in [16], using less restrictive boundary conditions that allow small interpenetrations of the bodies In such a model, the small interpenetration is governed by a function that depends on the normal component of the displacement field on the boundary of the potential contact region. We. consider the energy shape functional associated to the frictionless contact problem allowing a small interpenetration between the elastic body and a rigid foundation, developed in [16].
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