Sliding contact problems involving inhomogeneous materials comprising a coating-transition layer-substrate and a rigid punch

This paper proposes a semi-analytical model for the two-dimensional contact problem involving a multi-layered elastic solid loaded normally and tangentially by a rigid punch. The solid is comprised of a homogeneous coating and substrate joined together by a graded elastic transition layer whose material properties exhibit an exponential dependence on the vertical coordinate. By applying the Fourier transform to the governing boundary value problem, we formulate analytic expressions for the stresses and displacements induced by the application of line forces acting both normally and tangentially at the origin. The superposition principle is then used to generalise these expressions to the case of distributed normal and tangential tractions acting on the solid surface. A pair of coupled integral equations are further derived for the parabolic stamp problem which are easily solved using collocation methods. The primary aim of this paper is to provide insight into the likely behaviour of graded materials under the combined effects of surface pressure and shear stress. In this study, the assumption of Coulomb friction is invoked and the effects of material gradation, coating/interlayer thickness and friction coefficient upon the contact footprint and sub-surface stress field are investigated in great detail. The results we obtain suggest that the thickness of the transition layer as well as the combined thickness of the coating and transition layer have a significant effect on the maximum sub-surface stress attained through contact. This indicates that small changes in the composition of the coating can lead to significant differences in material behaviour. We additionally find that an increase in the amount of friction present in the contact can cause dramatic changes in the pattern of the stress field and can give rise to a much larger maximum stress. This effect can be offset somewhat under certain conditions by changing the thickness of the transition layer. To the best of the authors belief, this work represents the first attempt made to characterise the effects of friction on the sub-surface stress field within a graded elastic material