Strain energy change to the insertion of inclusions associated to a thermo-mechanical semi-coupled system

Abstract

The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations. In addition, the topological asymptotic analysis associated to multi-physics problems has been reported in the literature only on the level of mathematical analysis of singularly perturbed geometrical domains. In this work, we present the topological derivative in its closed form for the total potential mechanical energy associated to a thermo-mechanical semi-coupled system, when a small circular inclusion is introduced at an arbitrary point of the domain. In particular, we consider the linear elasticity system (modeled by the Navier equation) coupled with the steady-state heat conduction problem (modeled by the Laplace equation). The mechanical coupling term comes out from the thermal stress induced by the temperature field. Since this term is non-local, we introduce a non-standard adjoint state, which allows to obtain a closed form for the topological derivative. Finally, we provide a full mathematical justification for the derived formulas and develop precise estimates for the remainders of the topological asymptotic expansion.