Discrete Coulomb Frictional Systems Subjected to Periodic Loading

Abstract

If elastic systems with frictional interfaces are subjected to periodic loading, the system may shake down, meaning that frictional slip is restricted to the first few cycles, or it may settle into a steady periodic state involving cyclic slip. Furthermore, if the system posesses a rigid-body mode, the slip may also cause an increment of rigid-body motion to occur during each cycle — a phenomenon known as ratcheting.

Here we investigate this behaviour for discrete systems such as finite element models, for which the contact state can be described in terms of a finite set of nodal displacements and forces. If the system is ‘uncoupled — i.e. if the stiffness matrix is such that the tangential nodal displacements are uninfluenced by the normal nodal forces, a frictional Melan’s theorem can be proved showing that shakedown will occur for all initial conditions if there exists a safe shakedown state for the periodic loading in question. For coupled systems, we develop an algorithm for determining the range of periodic load amplitudes within which the long-time state might be cyclic slip or shakedown, depending on the initial condition. The problem is investigated using a geometric representation of the motion of the frictional inequality constraints in slip displacement space. Similar techniques are used to explore ratcheting behaviour in a low-order system.