An artificial viscosity approach to quasistatic crack growth

In this paper we consider the quasistatic crack growth in brittle materials in the particular case of a preassigned crack path Γ, and propose a new notion of irreversible quasistatic evolution which is based on a local stability criterion for the energy functional, rather than on a global one. To better focus on this aspect we present our approach in the simplest model case of a homogeneous isotropic material subject to antiplane shears. We assume that the reference configuration Ω is a bounded Lipschitz domain in R, and that the crack path Γ is a regular arc with one endpoint on the boundary of Ω. Moreover, we assume that there exists an initial connected crack starting from the boundary point and that the crack remains connected during the evolution. Hence, such a crack will be completely determined by its length σ. The evolution is supposed to be irreversible, so that the length of the crack will be increasing in time, and quasistatic, i.e. at each time the configuration describing the body is in equilibrium. By configuration we mean a pair (u, σ) where u represents the displacement orthogonal to the plane of Ω, and σ is the length of the crack. The choice of the total energy of a configuration (u, σ) is inspired by Griffith’s idea [10] that the evolution of cracks in brittle materials is the result of the competition between the elastic energy of the body and the energy needed to extend the crack. In our case the bulk part of the energy is given by the square of the L-norm of the gradient of u, while the surface energy will be simply given by the length σ of the crack (i.e. the toughness of the material will be assumed to be equal to one). The evolution is driven by time-dependent imposed boundary displacements ψ(t) on a part ∂DΩ of the boundary, and applied boundary forces g(t) on the remaining