Stability of the three‐dimensional Coulomb friction law

Abstract

It is known that problems arise over existence and uniqueness of solution in quasi–static contact problems involving large coefficients of Coulomb friction. This paper considers the behaviour of an elastically supported mass with three translational degrees of freedom that can make contact with a rigid Coulomb friction support. A critical coefficient of friction is identified, above which the quasi–static solution can be non–unique. A numerical solution of the corresponding dynamic problem shows that the state realized then depends on the initial conditions. Even below the critical coefficient of friction, the dynamic solution can deviate from the quasi–static even for arbitrarily small loading rates. Typical dynamic responses include limit–cycle oscillations in velocity, oscillations involving a brief period of zero velocity (stick) and ‘stick–slip’ motion in which the mass spends significant periods in a state of stick, interspersed with short rapid–slip periods. All these non–steady motions involve non–rectilinear motion, even in cases where the time derivative of the applied load is constant in direction. A perturbation analysis is performed on the quasi–static frictional slip solution and predicts instability for certain slip directions, depending on functions of the off–diagonal stiffness components and the coefficient of friction. These results are presented in dimensionless form in stability diagrams. It is also shown that quasi–static slip that is stable when there is no damping can be destabilized if the damping matrix has sufficiently large off–diagonal components.