Two-dimensional contact mechanics problems involving inhomogeneously elastic solids split into three distinct layers

This paper investigates the frictionless two-dimensional contact problem of an inhomogeneously elastic material under a rigid punch. The inhomogeneous solid is deemed to comprise three distinct regions which represent a homogeneously elastic coating and substrate joined together by a functionally graded transition layer (interlayer) whose shear modulus depends exponentially on the vertical coordinate.We propose closed form solutions for the horizontal and vertical displacements of the solid which are analytic if the contact pressure is known exactly. These solutions are further used to derive a fast and efficient iterative algorithm from which the contact footprint resulting from the rigid stamp problem may be computed. A selection of numerical results are then presented using this method and it is found that our model compares well with those of other authors in the two particular limiting cases considered here. We then investigate the effects of material inhomogeneity and coating thickness on the cylindrical stamp problem and it is found that the maximum principal stress is highly dependent on the thickness and mechanical properties of the layer. In particular, it is found that the maximum principal stress that occurs in a material with a hard coating may be reduced by increasing the thickness of the transition layer whilst lower stresses are achieved in materials with soft coatings by decreasing interlayer thickness.