Partial Slip Contact Problem in a graded half plane

We consider in this paper the two-dimensional nonlinear partial slip contact problem between a non-homogeneous isotropic graded half plane and a rigid punch of an arbitrary profile subjected to a monotonically increasing normal load. The graded medium is modeled as a non-homogeneous isotropic material with an exponentially varying shear modulus and a constant Poisson's ratio. Using standard Fourier Transform, the problem is formulated under plane strain conditions and is reduced to a set of singular integral equations. An asymptotic analysis is performed to extract the proper singularities from the kernels, resulting in two integral equations which are solved numerically using Gauss-Chebechev integration formulas. Based on the Goodman approximation, the contact problem is simplified and an iterative method is developed to determine the stick-slip zone, as well as the normal and tangential tractions in the entire contact zone. The objective of this paper is to study the effect of the graded medium non-homogeneity parameter and the friction coefficient on the size of the stick zone and the contact stresses for the cases of flat and circular stamp profiles. The proposed solution method can be potentially used to study piezoelectric indentation problems in nanocomposites involving flexoelectricity and polarization gradient effects.