Integral representation for functionals defined on SBDpSBD^p in dimension two

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

Phase field approximation of cohesive fracture models

We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=min\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings

Existence of minimizers for the 22d stationary Griffith fracture model

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove existence of strong minimizers, that is deformation fields which are continuously differentiable outside a closed jump set and which minimize the relevant energy. To this aim, we show that minimizers of the weak formulation of the problem, set in the function space $SBD^2$ and for which existence is well-known, are actually strong minimizers following the approach developed by De Giorgi, Carriero, and Leaci in the corresponding scalar setting of the Mumford-Shah problem

Optimal existence results for the 2d elastic contact problem with Coulomb friction

In this article, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini-Coulomb problem) is unraveled and optimal existence results are proved for the most general bidimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray-Lions operators, so that a result of Br\'ezis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray-Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the bidimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini-Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient