Discussion on the Time-Harmonic Elastodynamic Half-Space Green’s Function Obtained by Superposition

Huina Yuan,Ziyang Pan

doi:10.1155/2016/2717810

Abstract

The time-harmonic elastodynamic half-space Green’s function derived by Banerjee and Mamoon by way of superposition is discussed and examined against another semianalytical solution and a numerical solution. It is shown that Banerjee and Mamoon’s solution gives infinitez-displacement response when the depth of the source goes to infinity, which is unreasonable and does not agree with other solutions. A possible problem in the derivation is that it is inappropriate to directly extend the results of Mindlin’s superposition method for the elastostatic half-space problem to the dynamic case. The superposition of the six full-space elastodynamic solutions does not satisfy the required boundary conditions of the half-space elastodynamic problem as in the static case and thus does not solve the dynamic half-space problem.

Highlights

The elastodynamic Green’s function for the half-space is fundamental to the application of boundary element method (BEM) to situations involving semi-infinite media
The time-harmonic elastodynamic half-space Green’s function under discussion, proposed by Banerjee and Mamoon [4], is derived by extending the superposition technique devised by Mindlin [8] for the elastostatic half-space problem to the dynamic case of a periodic point force in a semi-infinite solid
The semi-infinite solid is bounded by the plane z = 0, with the positive z-axis pointing to the interior of the body

Summary

Introduction

The elastodynamic Green’s function for the half-space is fundamental to the application of boundary element method (BEM) to situations involving semi-infinite media. Various derivations of the elastic displacement due to a subsurface transient or time-harmonic point force can be found in the literature [1,2,3,4]. Often expressed in Fourier-Bessel integral forms, numerical evaluation and application of these solutions are usually complex and time-consuming [5,6,7]. The time-harmonic elastodynamic half-space Green’s function under discussion, proposed by Banerjee and Mamoon [4], is derived by extending the superposition technique devised by Mindlin [8] for the elastostatic half-space problem to the dynamic case of a periodic point force in a semi-infinite solid

Banerjee and Mamoon’s Green’s Function

A Possible Problem and Comparison with Other Solutions

Discussion