Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity

Abstract

We extend the results about existence of minimizers, relaxation, and approximation proven by Chambolle et al. in 2002 and for an energy related to epitaxially strained crystalline films, and by Braides, Chambolle, and Solci in 2007 for a class of energies defined on pairs of function-set. We study these models in the framework of three-dimensional linear elasticity, where a major obstacle to overcome is the lack of any a priori assumption on the integrability properties of displacements. As a key tool for the proofs, we introduce a new notion of convergence for (d−1)-rectifiable sets that are jumps of GSBD p functions, called σ p sym -convergence.2010 Mathematics Subject Classification. 49Q20, 26A45, 49J45, 74G65.

A derivation of Griffith functionals from discrete finite-difference models

“…The aforementioned compactness result, which is the content of Proposition 1, determines the functional space domain of the limit: we benefit from the recent results [21,25] and prove that sequences (u ε , v ε ) with equibounded energies (1.4) converge (up to subsequences) to a limit pair (u, v) ∈ G S B D 2 ∞ (Ω) × {1}. We refer the reader to Sect.…”

“…Inspired by the previous compactness result, in [25] a space of G S B D 2 functions which may also attain a limit value ∞ has been introduced, as we recall. The spaceR d := R d ∪{∞} (with its sum given by a + ∞ = ∞ for any a ∈R d ) is in a natural bijection with S d = {ξ ∈ R d+1 : |ξ | = 1} through the stereographic projection of…”

Topics in the mathematical design of materials

“…In recent years, there has been extensive study of existence, regularity and stability of equilibrium configurations obtained by energy minimization. In [41], the authors extended to the three-dimensional setting what was previously obtained in two dimensions (see [42–48]), although in [49,50] the three-dimensional case was considered in the geometrically nonlinear setting and with linear elasticity under antiplane-shear assumption.…”

Two phase models for elastic membranes with soft inclusions

“…A main motivation for such results has derived from investigations on epitaxially strained films [BC02,CS07]. We refer to [CF20] for latest results and a detailed account of the literature in that direction.…”