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An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit

Published online by Cambridge University Press:  28 May 2015

Valeria Boccardo
Facultad de Ingeniería, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Macul, Santiago, Chile Facultad de Ingeniería, Universidad Mayor, Av. Manuel Montt 367, Providencia, Santiago, Chile
Eduardo Godoy*
Ingenieros Matemáticos Consultores Asociados S.A. (INGMAT), José Miguel de la Barra 412, 4to piso, Santiago, Chile
Mario Durán*
Facultad de Ingeniería, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Macul, Santiago, Chile
*Corresponding author. Email: (V. Boccardo), (E. Godoy), (M. Durán)
*Corresponding author. Email: (V. Boccardo), (E. Godoy), (M. Durán)
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This paper presents an efficient method to calculate the displacement and stress fields in an isotropic elastic half-space having a hemispherical pit and being subject to gravity. The method is semi-analytical and takes advantage of the axisymmetry of the problem. The Boussinesq potentials are used to obtain an analytical solution in series form, which satisfies the equilibrium equations of elastostatics, traction-free boundary conditions on the infinite plane surface and decaying conditions at infinity. The boundary conditions on the free surface of the pit are then imposed numerically, by minimising a quadratic functional of surface elastic energy. The minimisation yields a symmetric and positive definite linear system of equations for the coefficients of the series, whose particular block structure allows its solution in an efficient and robust way. The convergence of the series is verified and the obtained semi-analytical solution is then evaluated, providing numerical results. The method is validated by comparing the semi-analytical solution with the numerical results obtained using a commercial finite element software.

Research Article
Copyright © Global-Science Press 2015 

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