The topological derivative of stress-based cost functionals in anisotropic elasticity

The topological derivative of cost functionals J that depend on the stress (through the displacement gradient, assuming a linearly elastic material behavior) is considered in a quite general 3D setting where both the background and the inhomogeneity may have arbitrary anisotropic elastic properties. The topological derivative DJ(z) of J quantifies the asymptotic behavior of J under the nucleation in the background elastic medium of a small anisotropic inhomogeneity of characteristic radius a at a specified location z. The fact that the strain perturbation inside an elastic inhomogeneity remains finite for arbitrarily small a makes the small-inhomogeneity asymptotics of stress-based cost functionals quite different than that of the more usual displacement-based functionals. The asymptotic perturbation of J is shown to be of order O(a3) for a wide class of stress-based cost functionals having smooth densities. The topological derivative of J , i.e. the coefficient of the O(a3) perturbation, is established, and computational procedures then discussed. The resulting small-inhomogeneity expansion of J is mathematically justified (i.e. its remainder is proved to be of order o(a3)). Several 2D and 3D numerical examples are presented, in particular demonstrating the proposed formulation of DJ on cases involving anisotropic elasticity and non-quadratic cost functionals.