Geometrically non-linear modelling of contact problems involving thin elastic layers

The mechanical behaviour of thin elastic films undergoing large displacements and small strains is considered. Using the asymptotic expansion method from a three-dimensional analysis of the layer, a two-dimensional model is derived, under the assumptions of large displacements and small strains; it is further assumed that the elastic layer is more flexible than the other solids. The leading term of the solution of the asymptotic development is such that the displacement field varies linearly through the layer thickness and the stress tensor is constant. Convergence of the asymptotic expansion is studied and estimates of the error produced by approximating the original three-dimensional solution by the limit solution within the adhesive are obtained. As an application, the problem of the contact between a rigid hemisphere and a thin elastic layer strongly bonded on a rigid plane support is studied, which can be thought of as an adhesive displaying a geometrically non-linear behaviour due to the change of contact area. A quasi-linear relation is obtained between the area of contact and the penetration of the hemisphere within the layer, and the variation with penetration of the compressive load exerted by the hemisphere is seen to give satisfactory agreement with experiment. Copyright (C) 1996 Elsevier Science Ltd