Frictional systems subjected to oscillating loads

If a discrete elastic system with frictional interfaces is subjected to periodic loading, the eventual steady-state response may depend on the initial condition or an initial transient phase of the loading history. In cases where shakedown is possible, it is known that it will occur for all initial conditions if there is no coupling between slip displacements and normal contact tractions, but that when coupling is present, counter-examples can be developed where the steady-state depends on the initial conditions. In this paper, we explore the conjecture that this is a special case of a more general theorem that the time-varying terms in the steady-state solution for an uncoupled system are always independent of initial conditions. In such problems, the ‘memory’ of previous events can only be stored in the slip displacements at nodes that are presently strictly within the friction cone. If all the nodes slip at some point in the cycle, this memory must be continually exchanged between nodes, with a consequent degradation, or loss of memory, resulting in an asymptotic approach to a unique steady state. This behaviour is illustrated using a simple two-node example. When there exists a set of ‘permanently stuck nodes’, these constitute a repository for the system memory, but in uncoupled problems the displacements at these nodes have no effect on the normal tractions at the slipping nodes and hence on the time-varying terms in the solution These arguments are illustrated in the context of two examples: a random distribution of frictional microcracks in a block loaded in plane strain and a generalized Hertzian contact problem with friction.