A two-dimensional direct boundary integral method for elastodynamics

This paper presents the mathematical and numerical development of a two-dimensional boundary element model for elastodynamics problems in rock mechanics. The model is based on a time-domain formulation of the direct boundary integral method, and is capable of computing the transient response of an underground excavation to dynamic loading conditions. All boundary contours are approximated by straight line segments, joined end to end, and the displacements and tractions within a time step are assumed to vary quadratically over each segment. This type of discretization facilitates analytical evaluation of the integrals involved in the numerical model and allows one to deal directly with the important case of discontinuous loading caused by the passage of elastic waves across an element. Two example problems for a suddenly pressurized circular hole in an infinite plate are used to illustrate the numerical procedures described in the paper. In one case the hole is isolated, while in the other it is close to another circular hole with a traction-free boundary. When the pressurized hole is isolated, the elastic waves radiate outward and never return; when it is near another hole, wave reflections at both boundaries produce a much more complicated time-dependent response. The boundary element model appears to handle the wave reflections properly, provided a sufficiently small time step is chosen.