Causality constraint for fractures with linear slip weakening

We consider the problem of dynamic fracture growth characterized by linear slip-weakening properties. If the problem is attacked by standard linear spectral methods, we show that the elastic wave radiation from the slip in the weakening zone has noncausal terms. The nonphysical behavior is specifically attributable to the application of a weak causality constraint in standard linear spectral analysis to a problem with an active boundary condition. We give the correct analytic solution, derived from linear spectral analysis applied to a boundary integral procedure.