Higher order topological derivatives for three-dimensional anisotropic elasticity

Abstract

This article concerns an extension of the topological derivative concept for 3D elasticity problems involving elastic inhomogeneities, whereby an objective function J is expanded in powers of the characteristic size a of a single small inhomogeneity. The O(a 6) approximation of J is derived and justified for an inhomogeneity of given location, shape and elastic properties embedded in a 3D solid of arbitrary shape and elastic properties; the background and the inhomogeneity materials may both be anisotropic. The generalization to multiple small inhomogeneities is concisely described. Computational issues, and examples of objective functions commonly used in solid mechanics, are discussed.

Application of asymptotic analysis to the two-scale modeling of small defects in mechanical structures

“…7, for several choices (given in the figure caption) of inhomogeneity shape, stiffness and orientation θ with respect to the x 1 axis. We observe that R(a) = O(a 2 ) for not-too-small inclusion sizes (Log(a/L) ≥ −3.5); this is the expected theoretical behaviour of R(a) as a → 0, as the O(a) contribution to R(a) is expected to vanish [23,7] for all inhomogeneities with centrally-symmetric shape (such as ellipses). For smaller defects, the theoretical behavior of R(a) = O(a 2 ) (which accounts only for asymptotic approximation errors) is no longer observed due to FE discretization errors becoming comparatively significant (see [24] for an analysis of the interplay between asymptotic and FE errors for the Poisson equation with Dirichlet boundary conditions).…”

“…To ensure computational efficiency, the analysis uses only a FE mesh for the defect-free structure, whose mesh size is hence not influenced by the (small) defect scale. The latter is instead taken into account by means of an asymptotic expansion, as previously done in [1,2,3] for modeling surface-breaking void defects (see also [4] where the concept of topological derivative [5,6,7] is used for predicting the eventual nocivity of surface-breaking small cracks). Here we are addressing the case of a small internal inhomogeneity (or a finite number thereof) embedded in an elastic solid.…”

“…This includes traction-free voids as a special case, thus covering (small) objects variously referred to in the literature (see e.g. [8,9,10]) as inhomogeneities, heterogeneities, cracks, holes, porosities, inclusions... We rely on existing results on small-inhomogeneity asymptotics for elastic solids [11,12,13,14,6,7], which prominently involve elastic moment tensors (EMTs) associated with elastic inhomogeneities [11,12,6], and combine them into a simple computational treatment, whose capabilities (prominently among them the ability to conduct inexpensive parametric studies) are then demonstrated on two examples.…”

Microstructural Topological Sensitivities of the Second-Order Macroscopic Model for Waves in Periodic Media

“…where 0 , µ 0 , 2 , and µ 2 are given by (6) and (8). Note that the contribution of 1 and µ 1 in (16) vanishes as a consequence of Lemma 1, while the o(ε 3 ) remainder (instead of expected O(ε 3 ) residual) stems from the analogous relationship linking 3 and µ 3 to their lower-order companions.…”

“…Topological sensitivity of the effective properties. Let f = f (µ, ρ) stand for any of the effective tensors defined in (6) and (8) for the reference unit cell Y and, similarly, let f a = f (µ a , ρ a ) denote its companion computed for Y a . Our main goal is to determine the TS Df (z) of f due to nucleation of a small inhomogeneity B a at z ∈ Y , defined through the expansion (20) f…”