Stress Distributions Analyzed in Bispherical Co-Ordinates

Abstract

Boussinesq-Papkovich potentials are used in conjunction with the bispherical co-ordinate system to analyze three problems in the classical theory of linear elasticity: (a) The extension of the Boussinesq point-load problem to that in which the half-space contains a spherical cavity; (b) the determination of the stress distribution in an eccentric spherical shell under uniform internal pressure; (c) the determination of the stress distribution in a half-space containing a uniformly pressurized spherical cavity. Numerical results are presented for representative configurations and load distributions in each case.

Elastic stress and deformation near a finite spherical magma body: Resolution of the point source paradox

“…choice of illustrative values of the hoop stress is an unfortunate one because the departure from the leadingorder solution, aooC(ø)(e, 0)--1/2, is small at these points.Nonetheless, the present analysis shows reasonable agreement with Tsuchida and Nakahara[-1972] and reinforces their contention that the solution ofMitchell and Weese [1960], although correct, was not evaluated properly in the original paper.…”

“…Exact analytical solutions for a pressurized, spherical cavity in a homogeneous, isotropic, elastic half-space have been presented in the engineering mechanics literature. Mitchell and Weese [1960] solved the problem in bispherical coordinates, in which the spherical cavity boundary and the plane free surface are defined by constant values of one of the orthog-onal coordinates. Tsuchida and Nakahara [1972] solved the problem with a combination of both spherical and cylindrical coordinate systems.…”

“…Second, both solutions take the form of infinite series with very complex coefficients that are difficult to evaluate. Indeed, Tsuchida and Nakahara [1972] suggest that errors in the evaluation of the solution of Mitchell and Weese [1960] may explain the discrepancies from their own results. Finally, the approximate solutions presented here isolate the leading-order effects of the interaction of the pressurized cavity with the free surface, and these effects are the principal interest in the present context.…”

Analytical models of magma chamber stability: An abridged critical review of key concepts

“…In the context of the volcanological literature, three works are most commonly quoted. Two of these works offer exact analytical solutions using alternative coordinate systems (Mitchell and Weese, 1960;Tsuchida and Nakahara, 1972) and a third offers an approximate solution based on a spherical coordinate system (McTigue, 1987). Also, similar to the case of a hole, there is some disagreement between the results reported on those works.…”

Reaction mechanism and mechanics in shale-induced concrete popouts

“…The problem of the stresses induced into an elastic media by a pressurized spherical cavity in a half-space has been solved by Mitchell and Weese (19). They found that the tangential tensile stress at the top of the cavity-matrix interface was about 1.14 times the compressive stress in the cavity.…”