Fundamental solutions in 3D elastodynamics for the BEM: A review

Abstract

In this work, we attempt to collect all available fundamental solutions in 3D elastodynamics that would be suitable for use in Boundary Element Method (BEM) formulations. Given that research on transient and steady state cases involving the elastic, homogeneous and isotropic continuum commenced in the 1950′s, it is practically impossible to reference all work done since then, so we apologize a priori for any omissions on our part. When we use the term fundamental solution, what comes to mind is the elastic full space under a point load in space and time and with the radiation condition arising because the medium has no boundaries, i.e. it extends to infinity. However, this is far from being the end: If the full space becomes a half-space, then boundary conditions enter the picture and now we are talking about Green's functions. These are more desirable for BEM formulations because it now becomes unnecessary to discretize the free surface. In general, the more particular features of the problem at hand are included in the Green's function, the less discretization is necessary in the BEM formulation. What we aim for in this review is to present and briefly discuss the basic fundamental solutions and more specialized Green's functions in a 3D elastic continuum, in either the frequency or the time domain, for the following type of materials: (1) Isotropic and homogeneous; (2) isotropic and inhomogeneous; (3) anisotropic and homogeneous; (4) anisotropic and inhomogeneous. We also look at poroelastic materials with all above possible combinations. We note at this point that the continuously inhomogeneous (e.g., functionally graded) material is understood as one having its material parameters as functions of position. There is, of course, the category of layered media, considered here as discretely inhomogeneous materials. One final note of caution has to do with the numerical implementation of these solutions: Some are very difficult to program, despite the fact that they come in closed form, because they may involve integrals, or there may be turning points in the solution where the form available depends on the frequency, or because of round-off errors in computing special functions. In closing, there is a trade-off between easy to implement fundamental solutions that require substantial spatial discretization effort and advanced ones based on Green's functions that are difficult to implement but require minimal discretization effort. This points out the relevance of speeding up BEM computations, a subject that is treated in the Appendix.