In this study, the response of coupled frictional contact problems to cyclic loading is investigated. We first consider the receding contact problem for the important case where the loading contains a mean and a periodic component. Dundurs (1975) has shown that if the contact area in a frictionless elastic system under load is equal to or smaller than that before loading, the extent of the contact area is load-independent, and the stress field varies linearly with load. Similar results apply to problems with Coulomb friction as long as the loading is monotonic, but otherwise the system approaches a steady periodic state relatively slowly, and in this final state there is continuous variation of the contact area, with the minimum occurring at the minimum applied load.;Second, we explore whether Melan's theorem can be applied to the coupled system. The evolution of the system history is conveniently represented graphically by tracking the instantaneous condition in the slip-displacement space. The frictional inequalities define directional straight line constraints in this space that tend to 'sweep' the operating point towards the safe shakedown condition if one exists. However, if the safe shakedown region is defined by a triangle in which two adjacent sides correspond to the extremal positions of the two frictional constraints for the same node, initial conditions leading to cyclic slip can be found.;Finally, we study an analytical method to predict friction-induced instability for the case where Coulomb friction conditions may fail to define a unique solution for a quasi-static evolution algorithm. In an attempt to resolve this problem, we develop an elasto-dynamic formulation by introducing the effect of viscous damping. As a result, we find that the final state is determined by the particular values at the discontinuity point. Therefore, it is possible for us to define the unique final state of the system without involving the transient dynamic analysis.
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