Integral representation for functionals defined on $SBD^p$ in dimension two

TL;DR

This paper establishes an integral representation for energy functionals on the space $SBD^p$ in two dimensions, crucial for fracture modeling, by constructing local $W^{1,p}$ approximations and generalizing Korn's inequality.

Contribution

It provides the first integral representation result for $SBD^p$ functionals in 2D, including a generalized Korn's inequality and a method for local approximation by $W^{1,p}$ functions.

Findings

Integral representation for $SBD^p$ functionals in 2D.

Generalized Korn's inequality in the $SBD^p$ setting.

Method for local approximation by $W^{1,p}$ functions.

Abstract

We prove an integral representation result for functionals with growth conditions which give coercivity on the space $SBD^p(\Omega)$, for $\Omega\subset\mathbb{R}^2$. The space $SBD^p$ 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 $SBD^p$ setting.