The plane contact problem for a prestressed incompressible elastic layer

The problem of the indentation of a rigid punch into the upper face of a layer when a uniform field of initial stresses is present in the layer is considered. A model of an isotropic incompressible non-linearly elastic material, specified by the Mooney elastic potential, is used. The case when the layer rests on the lower face without friction is investigated. It is assumed that the additional stresses, due to the punch indentation, are small compared with the initial stresses. This assumption enables the problem of determining the initial stresses to be linearized. It is later reduced to the solution of an integral equation of the first kind with a difference kernel with respect to the pressure in the contact region. Depending on the dimensionless parameter λ, characterizing the relative thickness of the layer, asymptotic solutions are constructed for large and small values of this parameter. A solution for a whole range of values of the parameter, investigated by the “large” and “small” λ methods, is also obtained using a modified Multhopp–Kalandiya method.