Indentation of solids with gradients in elastic properties: Part I. Point force

Analytical and computational results are presented on the evolution of stresses and displacements due to indentation from a normal point force on an elastic substrate whose Young’s modulus E varies as a function of depth, z, beneath the indented surface. With the exception of variation of properties with depth, the material is assumed to be linearly elastic and locally isotropic. Closed-form solutions are derived for several fixed values of the Poisson ratio, v, and variations in E which follow two prescribed functions of z: (1) a simple power law, E = E,#, where 0 < k < 1 is a non-dimensional exponent ; (2) an exponential law, E = E,e”‘, where E0 is Young’s modulus at the surface, c( < 0 denotes a hardened surface (e.g., graded ceramic coatings on metallic substrates) and CI > 0 denotes a soft surface (e.g., modulus variations measured as a function of depth beneath the earth’s surface for solids and rocks). The analytical solutions are checked with detailed finite element simulations. It is shown that, for the power law case, there exists a critical Poisson ratio, v,,, above which drastic changes occur in the stress distribution under the point load. Finite element results reveal, however, that the response is relatively less sensitive to the variation of Y (on either side of v~,) than to the variation of E with depth. Applications of the present results are discussed, wherever appropriate, to surface treatments of engineering structures, thick coatings, thin-film multilayers for microelectronic devices, and settling of foundations in the context of soil mechanics and rock mechanics. 0 1997 Elsevier Science Ltd