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, 2018)
This paper presents an efficient method to calculate the displacement andstress fields in an isotropic elastic half-space having a hemispherical pit and being subjectto gravity. The method is semi-analytical and takes advantage of the axisymmetryof the problem. The Boussinesq potentials are used to obtain an analytical solutionin series form, which satisfies the equilibrium equations of elastostatics, traction-freeboundary 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 minimisationyields a symmetric and positive definite linear system of equations for the coefficientsof the series, whose particular block structure allows its solution in an efficient androbust way. The convergence of the series is verified and the obtained semi-analyticalsolution is then evaluated, providing numerical results. The method is validated bycomparing the semi-analytical solution with the numerical results obtained using acommercial finite element software.