A General Approximate Solution for the Slightly Non-Axisymmetric Normal Contact Problem of Layered and Graded Elastic Materials

Abstract

We present a general approximate analytical solution for the normal contact of layered and functionally graded elastic materials for almost axisymmetric contact profiles. The solution only requires knowledge of the corresponding contact solution for indentation using a rigid cylindrical flat punch. It is based on the generalizations of Barber’s maximum normal force principle and Fabrikant’s approximation for the pressure distribution under an arbitrary flat punch in an inhomogeneous case. Executing an asymptotic procedure suggested recently for almost axisymmetric contacts of homogeneous elastic media results in a simple approximate solution to the inhomogeneous problem. The contact of elliptical paraboloids and indentation using a rigid pyramid with a square planform are considered in detail. For these problems, we compare our results to rigorous numerical solutions for a general (bonded or unbonded) single elastic layer based on the boundary element method. All comparisons show the quality and applicability of the suggested approximate solution. Based on our results, any compact axisymmetric or almost axisymmetric contact problem of layered or functionally graded elastic materials can be reduced asymptotically to the problem of indenting the material using a rigid cylindrical flat punch. The procedure can be used for different problems in tribology, e.g., within the framework of indentation testing or as a tool for the analysis of local features on a rough surface.

Modeling of the Combined Effect of the Surface Roughness and Coatings in Contact Interaction

“…As follows from (8), the nominal contact pressure is determined by the following parameters: the coefficients A 1 and A 2 , which depend on the ratio of the mechanical properties of the layer and the half-space and boundary condition between them, the punch curvature radius R, the layer thickness h, and the parameters B and β depending on the roughness parameters. The coefficient A 1 and the layer thickness h are included in (9) through the parameter C.…”

“…Finally, from the solution of the problem at the microscale with roughness parameters NR 2 a = 4, it was found (see Section 4.1) that B = 0.1R a /E ′β 1 and β = 0.5. Thus, we have all the necessary dimensionless parameters (9) to solve the integral Equation ( 10) numerically.…”

“…In [8], a general algorithm, which is based on dividing the contact region into annular areas, for solving such problem for the axisymmetric punch is presented. In [9], the boundary element method is used to validate an approximate solution of the contact problem of indenting an almost axisymmetric punch into layered elastic materials. The finite element method is also very popular, since it makes it possible to investigate a wide range of effects that occur during the contact of bodies with coatings [10].…”