Dynamics and control of viscoelastic solids with contact and friction effects

Abstract

THE DYNAMICSand conlrol of distributed systems subjected to frictional resistive forces has been an open problem in partial differential equations for many years, largely owing to the absence of a meaningful existence theory for dynamical systems with friction. The mathcmatical difficulties associated with Coulomb friction in problems of elastodynamics were pointed out in 1972 by Duvaut and Lions (see (I)) and has been the subject of much study in the intervening years. The finite-dimensional case has proved to be somewhat more tractable, and conditions for the existence of solutions of discrete dynamical problems with friction have been recently reported by Lbtstedt (2) and Jean and Pratt (3). The complexity of the problem can be appreciated by reviewing the work on dynamics of systems with frictionless contacts by, for example. Schatzman (4), Carrero and Pascali (5), Lotstedt (6), Amerio and Prouse (7). Amerio (8), and others. In recent papers, Oden and Martins (9, 10) pointed out that one of the principal sources of mathematical difficulty was the definition of frictional stresses on the contact surfaces charac- terized by Coulomb's law. However, an overwhelming volume of experimental data accumu- lated over a half century suggests that this law is inadequate for modeling actual contacts and resisIive forces on deformable bodies. By characterizing the actual normal compliance of elastic interfaces. a constitutive equation for an interface can be developed which yields results in agreement with a sizable collection of experimental results on static and dynamic friction (9). Moreover. the use of such interface constitutive laws in mathematical models of elastostatics, elastodynamics, and viscoelastodynamics problems with friction produces a tractable theory: results on the existence and local uniqueness of solutions to static problems in elaslicity with the Oden-Martins (9) nonlinear friction laws were recently established by Rabier el of. (II) and of dynamic problems on elasticity and viscoelasticity by Martins and Oden (10). These new theories and results set the stage for the study of the optimal control of such systems, taken up in the present paper. In the present study, we establish the existence of a class of optimal controls for a broad class of problems in the dynamics of linear viscoelastic bodies with contact and friction laws of the type introduced by Oden and Martins (9). We are able to show, under very mild restrictions