The Hemispherical Inhomogeneity Subjected to a Concentrated Force

Abstract

The presence of two or more different constituents in an elastic material has a substantial effect on its mechanical behavior under thermal or mechanical loading. The overall, as well as the local, properties of such a material may bear little relation to those of the components, even though the components retain their integrity within the composite. Stress fields in composite materials under applied stresses can be simulated by the inclusion problem, when fibers in the composite are replaced by inhomogeneities. A considerable number of problems can be found in the literature involving the determination of the local stress and displacement fields in the vicinity of a single inhomogeneity or impurity, which is embedded in an infinitely extended surrounding material, i.e., the matrix. In the present paper, an analytical solution for the three-dimensional mixed boundary value problem of a hemispherical inhomogeneity is presented. The inhomogeneity is embedded at the surface of an elastic half-space and is subjected to a concentrated force. First, the problem is solved by assuming perfect bonding along the interface between the matrix and the inhomogeneity. Next, the shear-traction-free (sliding) boundary is considered and the two results are compared. For both cases, the elastic field is deduced in a series form, using Boussinesq’s displacement potentials. Several calculations are performed for various combinations of elastic constants. Finally, stresses and displacements along the free surface and the matrixinhomogeneity interface are evaluated for typical situations, in order to assess the significance of inhomogeneities in composite material applications.