Analytical solution of stress and displacement in axisymmetric inhomogeneous half space under normal concentrated surface loading

In this paper, an analytical solution for stress and displacement in an inhomogeneous half space under the action of concentrated normal surface loading is investigated. The Young modulus is considered to vary with the spherical radius R in a power law form of order n, while the Poisson’s ratio is taken to be constant. The problem is solved analytically using an elasticity approach and considering a semi-inverse method in which, based on equilibrium equations on the surface of an arbitrary hemisphere in the half-space and centered at the point of application of load, some stress components are assumed to be proportional to 1/R2. It is then shown that this assumption is valid and all stress components in this axisymmetric problem are proportional to 1/R2, while displacements are proportional to 1/R(n+1). and their variation with azimuthal coordinate φ is in the form of a special function called hyper-geometric function. Illustrative examples are presented, which show variations of stresses and displacements both in R and φ directions. It is seen that the inhomogeneity parameter has a significant effect on both of these field variables.