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 2007 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 $GSB{D}^p$ functions, called ${\sigma }_{\mathrm{sym}}^p$-convergence.

References in corpus (8)

• A density result in $GSB{D}^p$ with applications to the approximation of brittle fracture energies
• Integral representation for functionals defined on $SB{D}^p$ in dimension two
• Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization
• Study of island formation in epitaxially strained films on unbounded domains
• Approximation of fracture energies with $p$-growth via piecewise affine finite elements
• Stability of equilibrium configurations of elastic films in two and three dimensions
• On the approximation of $SBD$ functions and some applications
• The surface diffusion flow with elasticity in the plane

Cited by in corpus (2)

• From atomistic systems to linearized continuum models for elastic materials with voids
• Area quasi-minimizing partitions with a graphical constraint: relaxation and two-dimensional partial regularity