Solution to the Frictional Contact Problem via the Method of Memory Diagrams for General 3D Loading Histories

This paper is concerned with the method of memory diagrams developed for solving a problem of frictional elastic contact. Our goal is to establish a link between contact force and displacement for a general plane contact between two arbitrarily-shaped bodies; rolling and torsion are not considered. Description of mechanical interactions between two solids in contact in the presence of friction is a nontrivial task since the desired force-displacement relationships have hysteretic (memory-dependent) character. Arbitrarily changing applied force (or displacement) creates a cumbersome shear stress distribution in the contact zone that has to be adequately parameterized and accounted for. In that regard, it is suggested to consider, instead of complex shear stress distributions, a simpler functional form called memory diagram that contains the same memory information. We have established two integral relationships that link the force and displacement vectors with that internal functional dependency. The integral relationships are supplemented with two other evolution rules for memory diagrams that eventually follow from the Coulomb friction law. The memory diagram is updated with the help of these rules following a given force history. Then the calculated memory diagram is used to update the history of displacement i.e. to produce the desired force-displacement relationship.