On the stability of quasi-static paths of a linearly elastic system with friction

Abstract

In this paper we discuss the stability of quasi-static paths of a single degree of freedom linearly elastic system with Coulomb friction and known normal force. A common and useful approximation for the equations that govern the slow evolution of many mechanical systems is to neglect inertia effects in the dynamic balance equations, and replace them by static equilibrium equations. Slow evolutions calculated with this approximation are called quasi-static evolutions. The relationship of this issue with the theory of singular perturbations has been established in [1], where the existence of fast (dynamic) and slow (quasi-static) time scales was recognized: a change of variables is performed that replaces the (fast) physical time t by a (slow) loading parameter λ, whose rate of change with respect to time, e = dλ/dt, is decreased to zero. This change of variables leads to a system of dynamic differential equations or inclusions that defines a singular perturbation problem: the small parameter e multiplies some of the highest order derivatives in the system. The concept of stability of quasi-static paths used here is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces, e (as in singular perturbation problems). This study applies for the first time to a nonsmooth context, the definition of stability of quasi-static paths, recently proposed by Martins et al. ([2], [3]).