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.
This paper is concerned with an investigation into the two-dimensional frictionless contact problem of an inhomogeneously elastic material under a rigid punch and in particular the induced subsurface stress fields. The inhomogeneous solid is deemed to comprise three distinct regions which represent a homogeneously elastic coating and substrate joined together by a functionally graded transition layer whose shear modulus depends exponentially on the vertical coordinate. Using the assumption that the effects of the contact pressure die quickly away from the contact region, we formulate closed form solutions for the horizontal and vertical displacements of the solid which are analytic if the contact pressure is known exactly. These solutions are further used to derive a fast and efficient algorithm from which the contact footprint may be computed.A selection of numerical results are presented using this method and it is found that our model compares well with those of authors in two particular limiting cases. We then investigate the effects of material inhomogeneity and coating thickness on the circular stamp problem and it is found that hard coatings experience much larger contact pressures than soft coatings but act over a smaller area. These effects are exacerbated by decreased interlayer thickness where hard coatings achieve their maximum pressure and soft coatings achieve their minimum. This has a knock-on effect on the sub-surface stress fields as the maximum principal stress attained within hard coatings when the interlayer is thin is much larger than when the interlayer is thick. This indicates that the maximum principal stress is highly dependent on not only material inhomogeneity but interlayer thickness as well.
This paper attempts to solve analytically for the stresses present in a graded elastic solid resulting from pressure applied to its surface by computing the Airy stress function. The horizontal dimensions of the solid are assumed finite and hence we form the solution of the stress function as a Fourier series rather than an inverse Fourier transform. Finally, a selection of contour plots is presented to exhibit the behavior of this new model.
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