Dynamic modelling of the flat 2-D crack by a semi-analytic BIEM scheme

We present an e cient numerical method for solving indirect boundary integral equations that describe the dynamics of a at two-dimensional (2-D) crack in all modes of fracture. The method is based on a piecewise-constant interpolation, both in space and time, of the slip-rate function, by which the original equation is reduced to a discrete convolution, in space and time, of the slip-rate and a set of analytically obtained coe cients. If the time-step interval is set su ciently small with respect to the spatial grid size, the discrete equations decouple and can be solved explicitly. This semi-analytic scheme can be extended to the calculation of the wave eld o the crack plane. A necessary condition for the numerical stability of this scheme is investigated by way of an exhaustive set of trial runs for a kinematic problem. For the case investigated, our scheme is very stable for a fairly wide range of control parameters in modes III and I, whereas, in mode II, it is unstable except for some limited ranges of the parameters. The use of Peirce and Siebritsʹ -scheme in time collocation is found helpful in stabilizing the numerical calculation. Our scheme also allows for variable time steps.