Predicting the deflection and sub-surface stress field within two-dimensional inhomogeneously elastic bonded layered solids under pressure

This paper describes a Fourier series based solution method for the displacements and sub-surface stresses within a graded elastic layered solid under pressure. The solid is assumed to be in a state of plane strain and thus the derived solution is valid for two-dimensional problems. Whilst this method provides a fully analytic solution when the contact pressure is known exactly, it may also be used when the contact pressure is only known numerically (see Section 4). The solution given in this paper is generic and easily utilised to solve real problems as it requires only known physical characteristics of the solid under study and an applied surface pressure. The solid consists of two distinct regions which are considered to be perfectly bonded. These comprise a graded elastic coating whose shear modulus varies exponentially with the depth coordinate and a homogeneously elastic substrate. As the stresses and displacements induced by the applied pressure decay very quickly outside of the contact region, the contact problem need only be solved in a small piece of the solid as the remainder is unaffected. It is found that accurate results are obtained when the contact problem is solved over a region of the solid 10 times larger than the contact region. This method as a result is computationally cheap to use as the number of Fourier modes needed to accurately capture the solution is small.