*Archive for Rational Mechanics and Analysis*

**Inserted:** 1 oct 2015

**Last Updated:** 31 oct 2016

**Year:** 2016

**Abstract:**

We prove an integral representation result for functionals with growth conditions which give coercivity on the space $SBD^p(\Omega)$, for $\Omega\subset \mathbb{R}^2$ a bounded open Lipschitz set, $p\in(1,\infty)$. 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 Kornâ€™s inequality in the $SBD^p$ setting.

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