On the two-dimensional solution of both adhesive and non-adhesive contact problems involving functionally graded materials

This paper presents a semi-analytical algorithm for the determination of the contact half width and surface pressure which results from both adhesive and non-adhesive contact problems involving functionally graded materials (FGM). The inhomogeneously elastic solid comprises a graded elastic coating whose shear modulus depends exponentially on the vertical coordinate and a homogeneously elastic substrate. The solid is assumed to be in a state of plane strain and thus a two-dimensional analysis is performed within this work. Using the work of Chidlow et al. (2011a) as a starting point, we derive a pair of integral equations which may be used to determine approximations to the contact pressure when either the surface deflection or the deflection gradient is known over the contact region. As these integral equations are non-singular, we use Galerkin's method to approximate the contact pressure and it is found that relatively small trial spaces allow accurate computation of the pressure. Information about the prescribed load is then used to formulate an iterative algorithm to determine the contact half width. A selection of numerical results are presented using this method and it is found that the solutions computed here compare favourably with those of other authors. A further investigation is then conducted into the solution of adhesive contact problems using the assumptions of Maugis (1992) and Johnson and Greenwood (2008) to inform the nature of the adhesive stresses outside of the contact. It is found that both JKR-like and DMT-like behaviour can be observed in contact problems involving FGMs