Frictionless contact in a layered piezoelectric medium characterized by complex eigenvalues

A local/global stiffness matrix formulation is presented to study the response of an arbitrarily multilayered piezoelectric half-plane indented by a rigid frictionless parabolic punch. The methodology is extended to accommodate the presence of piezoelectric layers that are characterized by complex eigenvalues. Different arrangements of elastic and transversely orthotropic piezoelectric materials within the multilayered medium are considered. A generalized plane deformation is used to obtain the governing equilibrium equations for each individual layer. These equations are solved using the infinite Fourier transform technique. The problem is then reformulated using the local/global stiffness method, in which a local stiffness matrix relating the stresses and electric displacement to the mechanical displacements and electric potential in the transformed domain is formulated for each layer. Then it is assembled into a global stiffness matrix for the entire half-plane by enforcing continuity conditions along the interface. Application of the mixed boundary conditions reduces the problem to an integral equation for the unknown pressure in the contact area. This integral has a divergent kernel that is decomposed into a Cauchy-type and regular parts using the asymptotic properties of the local stiffness matrix. The resulting equation is numerically solved for the unknown contact pressure using a technique based on the Chebyshev polynomials.