On stability and posteritical behavior of interactive tension cracks in brittle solids

Abstract

The growth of interactive tension cracks in brittle solids is investigated in general. Cases are considered where the growth regime of several cracks, which are extending simultaneously in a stable manner, may become unstable. The fracture process is assumed to be governed by the Mode I conditions, and only straight-crack extension is examined. First, a variational formulation is presented, and a parametric description of the corresponding potential energy is given. Conditions under which unique crack growth rates correspond to given rates of loading are established from the stationary character of the potential energy, and critical states at which uniqueness, fails, are identified. Four types of critical states are identified, three of which stem from the constraint imposed by the physical conditions associated with crack growth phenomena, whereas the fourth emerges from the vanishing of the Hessian of the second variation of potential energy. The postcritical behavior is determined by minimization of the parametric form of the potential energy. Then a perturbation method is used, and the effect of material inhomogeneity and imperfection sensitivity is examined. Explicit results are obtained for two interacting cracks where imperfection arises from small differences in the fracture toughness of the cracks. It is shown in particular, that if the toughness of the cracks differs by a quantity of the order e, then the strain level at which instability takes place, drops from its value for the case with no imperfection by a quantity of the order e1/2.