Optimal design of fractured media with prescribed macroscopic strain

Abstract

In this work we consider an optimal design problem for two-component fractured media for which a macroscopic strain is prescribed. Within the framework of structured deformations, we derive an integral representation for the relaxed energy functional. We start from an energy functional accounting for bulk and surface contributions coming from both constituents of the material; the relaxed energy densities, obtained via a blow-up method, are determined by a delicate interplay between the optimization of sharp interfaces and the diffusion of microcracks. This model has the far-reaching perspective to incorporate elements of plasticity in optimal design of composite media.

Upscaling and spatial localization of non-local energies with applications to crystal plasticity

“…Therefore, for belonging to (E) with ∞ Lipschitz in the second variable uniformly with respect to the first, it is immediate to see that ∞ is a surface energy density that satisfies hypotheses (ψ1), (ψ2), (ψ3) and (ψ4) (see [15,Remark 3.3] and [31,Remark 3.1]). Thus, the relaxation/upscaling process (5.11) is the same as that of Theorem 5.1 for a local energy of the type (5.1) whose densities are W (x, A) := W (x, A) + (x, 0) and ψ(x, λ, ν) := ψ(x, λ, ν) + ∞ (x, λ ⊗ ν).…”

Relaxation for an optimal design problem inBD(Ω)

“…This generalizes to the full space BD(Ω), and to the case of densities f 0 depending explicitly on (x, u), the results obtained in [12]. We will make use of their work in § 4.2 to prove both lower and upper bounds for the density of the Cantor part of the measure F(χ, u; •), by means of an argument based on Chacon's biting lemma which allows us to fix χ at an appropriately chosen point x 0 , as in [41]. The contents of this paper are organized as follows.…”

Relaxation for an optimal design problem in $BD(Ω)$

“…This generalises to the full space BD(Ω), and to the case of densities f 0 depending explicitly on (x, u), the results obtained in [12]. We will make use of their work in Subsection 4.2 to prove both lower and upper bounds for the density of the Cantor part of the measure F (χ, u; •), by means of an argument based on Chacon's Biting Lemma which allows us to fix χ at an appropriately chosen point x 0 , as in [41]. The contents of this paper are organised as follows.…”