Nonlinear contact force law for spherical indentation of FGM coated elastic substrate: An extension of Hertz's solution

This paper develops a nonlinear contact force law for the indentation of an elastic substrate with thin functionally graded material (FGM) coating by a rigid smooth sphere. The nonlinear contact force law is explicitly expressed in form of extended Hertz's solution with a correction factor, which is related to the coating thickness, radius of the indenter, contact interference and the modulus ratio. The explicit expression of the correction factor for arbitrary coating modulus gradation can be determined statistically with extensive numerical results. For computational efficiency and accuracy, an analytical contact model based on multilayered half space is developed and the associated mixed boundary value problem is converted to a Fredholm integral equation of the second kind. This model discretizes the thin FGM coating into n dissimilar and fully bonded sub-layers. Each sub-layer is a homogeneous elastic seam of finite thickness and constant elastic modulus and Poisson's ratio. Variation of the coating elastic properties along the thickness direction is accurately approximated by the multilayered system. The non-linear contact force laws in closed-form are specifically given for both linear and exponential modulus gradations. The load-displacement relations predicted by these force laws are shown to be in exact agreement with the numerical results from the Fredholm integral equation. Additionally, this nonlinear contact force law for FGM coating is further incorporated into the Greenwood and Williamson model, which provides a feasible and effective way for modeling the contact of FGM coated rough surfaces.