Continuous and discontinuous contact problems of a functionally graded layer loaded by a rigid stamp and resting on an elastic half plane

In this study, continuous and discontinuous contact problems of a functionally graded layer resting on an elastic half-plane and loaded by a rigid stamp are solved using the theory of elasticity. The layer considered in the problem is functionally graded (FG) and its shear modulus and density vary according to an exponential function. The elastic half-plane is considered as homogeneous; in the solution of the problem, the body force of the elastic half-plane is neglected, while the body force of the FG layer is taken into consideration. It is assumed that Poisson's ratios of both the FG layer and the elastic half-plane do not change. Furthermore, all surfaces are frictionless and only compressive tractions are allowed at the interfaces in the discussed problem. Using the elasticity theory and the integral transform techniques, the continuous contact problem is reduced to an integral equation in which the contact stress under the rigid stamp is unknown; in the case of discontinuous contact, the problem was reduced to two integral equations in which the contact stress under the rigid stamp and the slope of the separation at the interface between the FG layer and the elastic half-plane are unknown. Using Gauss–Chebyshev integration formula, these integral equations are numerically solved separately for cases wherein the rigid stamp profile is either flat or circular. As a result of the study, the contact stress distribution under the rigid stamp, the normal stresses at the axis of symmetry, the shear stresses near the axis of symmetry, the initial separation load, and the initial separation distance between the FG layer and the elastic half-plane are obtained in the case of continuous contact for various dimensionless parameters. Further, a parametric study is conducted for the stress distributions between the FG layer and the elastic half-plane, the starting and ending points of the separation zone, and the separation amount in the case of discontinuous contact. In the event that the rigidity or density decreases from the upper surface to the lower surface of the FG layer, the separation between the FG layer and the elastic half-plane can be occurred easier.