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.