Crack initiation in brittle materials

In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t -> tg(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time t(i) > 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functional epsilon(upsilon, Gamma) := integral(Omega)f(x, del upsilon)dx + kH(1)(Gamma), where Omega subset of R-2, k > 0 and f is a suitable Caratheodory function. We prove that if the uncracked configuration u of Omega relative to a boundary displacement psi has at most uniformly weak singularities, then configurations (u(Gamma), Gamma) with H-1(Gamma) small enough are such that epsilon(u, empty set) < epsilon(u(Gamma), Gamma).