Volume 7, Issue 3
An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit

Valeria Boccardo, Eduardo Godoy & Mario Durán

Adv. Appl. Math. Mech., 7 (2015), pp. 295-322.

Published online: 2018-05

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  • Abstract

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.

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@Article{AAMM-7-295, author = {Valeria Boccardo , and Eduardo Godoy , and Mario Durán , }, title = {An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {7}, number = {3}, pages = {295--322}, abstract = {

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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2014.m544}, url = {http://global-sci.org/intro/article_detail/aamm/12049.html} }
TY - JOUR T1 - An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit AU - Valeria Boccardo , AU - Eduardo Godoy , AU - Mario Durán , JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 295 EP - 322 PY - 2018 DA - 2018/05 SN - 7 DO - http://doi.org/10.4208/aamm.2014.m544 UR - https://global-sci.org/intro/article_detail/aamm/12049.html KW - AB -

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.

Valeria Boccardo, Eduardo Godoy & Mario Durán. (1970). An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit. Advances in Applied Mathematics and Mechanics. 7 (3). 295-322. doi:10.4208/aamm.2014.m544
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