Indentation of a Periodically Layered, Planar, Elastic Half-Space

We investigate indentation by a smooth, rigid indenter of a two-dimensional half-space comprised of periodically arranged linear-elastic layers with different constitutive responses. Identifying the half-space’s material parameters as periodic functions in space, we utilize the theory of periodic homogenization to approximate the layered heterogeneous material by a linear-elastic, homogeneous , but anisotropic medium. This approximation becomes exact as the layer thickness becomes infinitesimal. In this way, we reduce the original problem to the indentation of an anisotropic, homogeneous, linear-elastic half-space by a smooth, rigid indenter. The latter is solved analytically by formulating and resolving the corresponding matrix Riemann–Hilbert boundary-value problem in complex analysis. Thus, we obtain an approximate, but analytical solution for the indentation of a layered heterogeneous medium. We then compare this solution with finite element computations of the indentation on the original layered, heterogenous half-space. We conclude that (a) the contact pressure on the layered, heterogenous half-space is well approximated by that obtained through homogenization, and the approximation improves as the layer thickness is decreased, or if the indentation force is increased; (b) the upper bound of the difference between the two contact pressures depends only upon the ratio of the Young’s moduli of the two materials constituting the heterogenous medium and their Poisson’s ratio; and (c) the average variation of the discontinuous von Mises stress in the layered half-space is well approximated by the one found in the homogenized half-space. The approach presented here can be utilized for a diverse array of indentation and contact problems of finely mixed heterogeneous media, and is also amenable to systematic improvements.