An elastic wave scattering problem relating to the rapid movement of a fracture in a plate

We consider the problem of a semiinfinite through crack in an elastic plate, which starts suddenly under the application of a step load (mode-I). As the crack propagates, microfractures grow rapidly at a small distance ahead of the tip, releasing pressure pulses. This process of microfracture nucleation, pulse emission and subsequent coalescence with the main fracture front continues to occur during crack motion. We crudely simulate this process by employing a single point dilatational source that is situated ahead of the crack tip and moves in unison with it, emitting pulses periodically. The total wavefield is then due to the effect of this point source, as well as the scattering of the pulses by the crack front. We model this elastodynamic problem under the plane stress assumption. A closed form solution is developed for the in-plane displacements of the crack faces due to the scattered field. The surface wave contribution can be pulled out separately and is expected to be significant. In particular, the results are cast into a form that is readily amenable to numerical analysis. We will be presenting the numerical results in Part II of this two part paper.