Some sufficient conditions for lower semicontinuity in SBD and applications to minimum problems of Fracture Mechanics

TL;DR

This paper establishes lower semicontinuity conditions for energies in the space of special functions of bounded deformation, with applications to fracture mechanics problems involving minimization of such energies.

Contribution

It provides new lower semicontinuity results in SBD space and demonstrates their application to fracture mechanics minimization problems.

Findings

Derived sufficient conditions for lower semicontinuity in SBD.

Provided examples illustrating the application of these conditions.

Applied results to specific fracture mechanics energy minimization problems.

Abstract

We provide some lower semicontinuity results in the space of special functions of bounded deformation for energies of the type $$ %\int_\O {1/2}({\mathbb C} \E u, \E u)dx + \int_{J_{u}} \Theta(u^+, u^-, \nu_{u})d \H^{N-1} \enspace, \enspace [u]\cdot \nu_u \geq 0 \enspace {\cal H}^{N-1}-\hbox{a. e. on}J_u, $$ and give some examples and applications to minimum problems. \noindent Keywords: Lower semicontinuity, fracture, special functions of bounded deformation, joint convexity, $BV$-ellipticity.