Elastic bars with cohesive energy

Most quasi-static variational models of fracture are based on the splitting of the energy functional into the sum of two terms: bulk, depending on the displacement gradient, and surface, depending on the displacement discontinuities. In this paper we consider the simplest one-dimensional problem of this type, a bar stretched by a given axial displacement, and systematically compare two alternative interpretations of the surface energy term. In the first interpretation (elastic model), the surface energy is viewed as a cohesive energy which is stored and can be recovered. In the second (inelastic model), it is irreversibly lost. We show that by assuming an evolution scheme based on local minimization and by varying the convexity-concavity properties of the surface energy the elastic model can reproduce a broad class of macroscopic material responses which have been traditionally treated as unrelated. These responses are associated with monotone loading and range from brittle fracture to rate independent plasticity. However, a realistic description for both loading and unloading is achieved only within the inelastic model.