Two-dimensional contact of two different power-law graded elastic bodies

TL;DR

This paper develops a mathematical model for the contact mechanics of two elastic bodies with power-law graded properties, introducing a novel solution method and analyzing effects of material mismatch and surface energy on contact behavior.

Contribution

It introduces a new integral equation approach using Gegenbauer polynomials for two inhomogeneous elastic bodies with different power-law exponents, providing explicit formulas and analysis.

Findings

Exact solutions for contact zone length and pressure distribution when exponents match.

Effects of Young's modulus exponent mismatch on contact characteristics.

Comparison of Hertzian and adhesive contact models on pressure and surface profiles.

Abstract

Previous study of contact of power-law graded materials concerned the contact of a rigid body (punch) with an elastic inhomogeneous foundation whose inhomogeneity is characterized by the Young modulus varying with depth as a power function. This paper models Hertzian and adhesive contact of two elastic inhomogeneous power-law graded bodies with different exponents. The problem is governed by an integral equation with two different power kernels. A nonstandard method of Gegenbauer orthogonal polynomials for its solution is proposed. It leads to infinite system of linear algebraic equations of a special structure. The integral representations of the system coefficients are evaluated, and the properties of the system are studied. It is shown that if the exponents coincide, the infinite system admits a simple exact solution that corresponds to the case when the Young moduli are different but the exponents are the same. Formulas for the length of the contact zone, the pressure distribution, and the surface normal displacements of the contacting bodies are obtain in the form convenient for computations. Effects of the mismatch in the Young moduli exponents are studied. A comparative analysis of the Hertzian and adhesive contact models clarifies the effects of the surface energy density on the contact pressure, the contact zone size, and the profile of the contacting bodies outside the contact area.