Analytical and numerical treatment of a dynamic crack model

We discuss the propagation of a running crack in a bounded linear elastic body under shear waves for a simplified 2D-model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked, bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u(y, t) and the one-dimensional crack tip trajectory h = h(t). We assume that the crack grows straight. Based on a paper of Nicaise-Sändig, we derive an improved formula for the ordinary differential equation of motion for the crack tip, where the dynamical stress intensity factor occurs. The numerical simulation is an iterative procedure starting from the wave field at time t = ti . The dynamic stress intensity factor will be extracted at t = ti . Its knowledge allows us to compute the crack-tip motion h(ti+1) with corresponding nonuniform crack speed assuming (ti+1 − ti ) is small. Now, we start from the cracked configuration at time t = ti+1 and repeat the steps. The wave displacements arecomputed with the FEM-packagePDE2D.Somenumerical examples demonstrate the proposed method. The influence of finite length of the crack and finite size of the sample on the dynamic stress intensity factor will be discussed in detail