Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T10:23:40.081Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ,
Eduardo Godoy  and
Mario Durán
Valeria Boccardo
Affiliation:
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*
Affiliation:
Ingenieros Matemáticos Consultores Asociados S.A. (INGMAT), José Miguel de la Barra 412, 4to piso, Santiago, Chile
Mario Durán*
Affiliation:
Facultad de Ingeniería, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Macul, Santiago, Chile
*
*Corresponding author. Email: vhboccar@puc.cl (V. Boccardo), eduardo.godoy@ingmat.com (E. Godoy), mduran@ing.puc.cl (M. Durán)
*Corresponding author. Email: vhboccar@puc.cl (V. Boccardo), eduardo.godoy@ingmat.com (E. Godoy), mduran@ing.puc.cl (M. Durán)
Get access

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.

Keywords

35Q7435C1074G15Elastic half-spacesemi-infinite domainaxisymmetric domainhemispherical pitsemi-analytical solution
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 3 , June 2015 , pp. 295 - 322
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Amenzade, Yu. A., Theory of Elasticity, Mir Publishers, Moscow, 1979.Google Scholar
[2]Andrews, L.C., Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Co., New York, 1985.Google Scholar
[3]Arfken, G.B and Weber, H.J, Mathematical Methods for Physicists, Harcourt/Academic Press, London, 2001.Google Scholar
[4]Bell, W.W., Special Functions for Scientists and Engineers, Van Nostrand, London, 1968.Google Scholar
[5]Björck, Ǻ., Numerical Methods for Least Square Problems, SIAM, Philadelphia, 1996.CrossRefGoogle Scholar
[6]Eubanks, R.A., Stress concentration due to a hemispherical pit at a free surface, J. Appl. Mech., 21 (1954), pp. 57–62.CrossRefGoogle Scholar
[7]Fujita, T., Sadayasu, T., Tsuchida, E. and Nakahara, I., Stress concentration due to a hemispherical pit at a free surface of a thick plate under all-around tension, Bull. Japan Soc. Mech. Eng., 21 (1978), pp. 561–565.CrossRefGoogle Scholar
[8]Fujita, T., Tsuchida, E. and Nakahara, I., Asymmetric problem of a semi-infinite body having a hemispherical pit under uniaxial tension, J. Elasticity, 12 (1982), pp. 177–192.Google Scholar
[9]Givoli, D. and Keller, J., A finite element method for large domains, Comput. Methods Appl. Mech. Eng., 76 (1989), pp. 41–66.Google Scholar
[10]Givoli, D. AND Vigdergauz, S., Artificial boundary conditions for 2D problems in geophysics, Comput. Methods Appl. Mech. Eng., 110 (1993) pp. 87–101.Google Scholar
[11]Golub, G.H. and Van Loan, C.F., Matrix Computations, 3rd Edition, Johns Hopkins university Press, Baltimore, 1996.Google Scholar
[12]Gradshteyn, I.S. and Ryzhik, I.M., Table of integrals, Series and Products, 7th Edition, Elsevier/Academic Press, Amsterdam, 2007.Google Scholar
[13]Han, H. and Wu, X.Approximations of infinite boundary conditions and its application to finite elements methods, J. Comput. Math., 3 (1985), pp. 179–192.Google Scholar
[14]Han, H. and Wu, X., The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its application, Math. Comput., 59 (1992), pp. 21–37.Google Scholar
[15]Sadd, M.H., Elasticity: Theory, Applications, and Numerics, Elsevier Butterworth-Heinemann, Boston, 2005.Google Scholar
[16]Stacey, T.R. and Page, C.H., Practical Handbook for Underground Rock Mechanics, Trans Tech Publications, 1986.Google Scholar
[17]Turcotte, D.L. AND Schubert, G., Geodynamics, Cambridge University Press, New York, 2002.Google Scholar
[18]Verruijt, A., A complex variable solution for a deforming circular tunnel in an elastic half-plane, Int. J. Numer. Anal. Met., 21 (1997), pp. 77–89.3.0.CO;2-M>CrossRefGoogle Scholar
[19]Verruijt, A., Deformations of an elastic half-plane with a circular cavity, Int. J. Solids Struct., 21 (1998), pp. 2795–2804.Google Scholar