Frictionless Contact of Layered Half-Planes, Part I: Analysis

A method is presented for the solution of frictionless contact problems on multilayered half-planes consisting of an arbitrary number of isotropic, orthotropic, or monoclinic layers arranged in any sequence. A displacement formulation is employed and the resulting Navier equations that govern the distribution of displacements in the individual layers are solved using Fourier transforms. A local stiffness matrix in the transform domain is formulated for each layer which is then assembled into a global stiffness matrix for the entire multilayered half-plane by enforcing continuity conditions along the interfaces. Application of the mixed boundary condition on the top surface of the medium subjected to the force of the indenter results in an integral equation for the unknown pressure in the contact region. The integral possesses a divergent kernel which is decomposed into Cauchy type and regular parts using the asymptotic properties of the local stiffness matrix and the ensuing relation between Fourier and finite Hilbert transform of the contact pressure. For homogeneous half-planes, the kernel consists only of the Cauchy-type singularity which results in a closed-form solution for the contact stress. For multilayered half-planes, the solution of the resulting singular integral equation is obtained using a collocation technique based on the properties of orthogonal polynomials. Part I of this paper outlines the analytical development of the technique. In Part II a number of numerical examples is presented addressing the effect of off-axis plies on contact stress distribution and load versus contact length in layered composite half-planes.