Initiation of cracks with cohesive force models: a variational approach

In the spirit of the variational approach of Fracture Mechanics initiated in [Del Piero, G., 1997. One-dimensional ductile-brittle transition, yielding and structured deformations. In: P. Argoul, M. Frémond (Eds.), Proceedings of IUTAM Symposium “Variations de domaines et frontières libres en mécanique”, Paris, 1997, Kluwer Academic] and [Francfort, G.A., Marigo, J.-J., 1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319–1342], we define the loss of stability of the elastic response of the body as the criterion of initiation of cracks. The result is very sensitive to the choice of the surface energy density. On one hand, if we adopt the Griffith assumption, then the elastic state is generally always stable. On the other hand, in the case of a surface energy of the Barenblatt type, i.e. a surface energy depending non-trivially on the jump of the displacement and inducing cohesive forces, the elastic response remains stable only if the stress field does not reach a critical value. In the full three-dimensional context of an isotropic material, we prove that this yield stress criterion is equivalent to a maximal traction criterion and a maximal shear criterion if the surface energy density is Fréchet differentiable at the origin. When the surface energy density is only Gâteaux differentiable, we obtain a yield stress criterion based on an intrinsic curve in the Mohr diagram. In any case, the domain of the admissible stress tensors is convex, unbounded in the direction of the hydrostatic pressures and depends only on the extreme eigenvalues of the stress tensor.