Integral representation for functionals defined on $SB{D}^p$ in dimension two

Abstract

We prove an integral representation result for functionals with growth conditions which give coercivity on the space $SB{D}^p(\Omega )$, for $\Omega \subset {\mathbb{R}}^2$. The space $SB{D}^p$ of functions whose distributional strain is the sum of an $L^p$ part and a bounded measure supported on a set of finite ${\mathcal{H}}^1$-dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by $W^{1,p}$ functions. We also obtain a generalization of Korn's inequality in the $SB{D}^p$ setting.

Cited by in corpus (8)

• Approximation of a Brittle Fracture Energy with a Constraint of Non-Interpenetration
• A density result in $GSB{D}^p$ with applications to the approximation of brittle fracture energies
• Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity
• Approximation of fracture energies with $p$-growth via piecewise affine finite elements
• Functionals defined on piecewise rigid functions: Integral representation and $\Gamma$-convergence
• On the approximation of $SBD$ functions and some applications
• From atomistic systems to linearized continuum models for elastic materials with voids
• Phase-field approximation of functionals defined on piecewise-rigid maps