Shakedown of coupled two-dimensional discrete frictional systems

Recent results have established that Melan's theorem can be applied to discrete elastic systems governed by the Coulomb friction law only when the normal contact reactions are uncoupled from the tangential (slip) displacements. For coupled systems, periodic loading scenarios can be devised which lead to either shakedown or cyclic slip depending on the initial condition. Here we explore this issue in the simplest coupled system involving two contact nodes. The evolution of the system ‘memory’ is conveniently represented graphically by tracking the instantaneous condition in 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 can be found leading to cyclic slip. The critical value of a loading parameter at which this occurs can be determined by requiring that three of the four constraint lines intersect in a point. Below this value, shakedown occurs for all initial conditions. Similar concepts can be extended to multi-node systems.