Indentation of a periodically layered, elastic half-space by a rigid sphere

Abstract

We investigate indentation by a smooth, rigid, spherical indenter of a heterogeneous half-space made up of layers of two different linear-elastic materials arranged periodically, parallel to the free surface. We utilize homogenization theory to approximate the heterogeneous material by a linear-elastic, homogeneous, but transversely isotropic material. This replaces the original system by an analytically tractable indentation problem, whose solution is then compared with that of the former obtained through the finite element (FE) method. We find that the contact pressure obtained through homogenization is a good approximation to that on the layered half-space, and the approximation improves if (1) the layer thickness is reduced, or (2) the indented force is augmented, or (3) the two layers exhibit closer elastic response. We demonstrate that the difference between the pressures obtained from FE and homogenization has an upper bound that depends only upon the materials’ Poisson’s ratio, the ratio of their Young’s moduli, and their volume fractions. We then show that the penetration depth of the indenter in the layered half-space converges to that in the homogenized material as the layers become thinner. Finally, we find that the homogenized material captures well the discontinuous variation of von Mises stress in the layered medium.