Shakedown in frictional contact problems for the continuum

Elastic systems with frictional interfaces subjected to periodic loading are often found to 'shake down' in the sense that frictional slip ceases after the first few loading cycles. The similarities in behaviour between such systems and monolithic bodies with elastic-plastic constitutive behaviour have prompted various authors to speculate that Melan's theorem might apply to them - i.e. that the existence of a state of residual stress sufficient to prevent further slip is a sufficient condition for the system to shake down. In this article, we prove this result for 'complete' contact problems in the continuum formulation for systems with no coupling between relative tangential displacements at the interface and the corresponding normal contact tractions. This condition is satisfied for the contact of two half spaces, or of a rigid body with a half space if Dundurs' constant beta = 0. It is also satisfied for the contact of two symmetric bodies of similar materials at the plane of symmetry.