Exact Closed-form Solutions for Lamb's Problem

TL;DR

This paper presents an exact closed-form solution for the displacement caused by a buried point source in an elastic half-space, extending Lamb's problem to interior points and providing explicit formulas for practical use.

Contribution

The authors derive explicit closed-form solutions for Lamb's problem applicable to interior points, expanding previous surface-only solutions and validating with numerical results.

Findings

Closed-form solutions match Johnson's numerical results

Explicit formulas involve elementary algebra and elliptic integrals

Solution valid for interior points using reciprocity theorem

Abstract

In this article, we report on an exact closed-form solution for the displacement at the surface of an elastic half-space elicited by a buried point source that acts at some point underneath that surface. This is commonly referred to as the 3-D Lamb's problem, for which previous solutions were restricted to sources and receivers placed at the free surface. By means of the reciprocity theorem, our solution should also be valid as a means to obtain the displacements at interior points when the source is placed at the free surface. We manage to obtain explicit results by expressing the solution in terms of elementary algebraic expression as well as elliptic integrals. We anchor our developments on Poisson's ratio 0.25 starting from Johnson's (1974) integral solutions which must be computed numerically. In the end, our closed-form results agree perfectly with the numerical results of Johnson (1974), which strongly confirms the correctness of our explicit formulas. It is hoped that in due time, these formulas may constitute a valuable canonical solution that will serve as a yardstick against which other numerical solutions can be compared and measured.