Thermodynamics, stability and non-linear oscillations of viscoelastic solids .2. History type solids

We study the thermodynamics and stability of a history type viscoelastic solid with an exponentially decaying relaxation function. The relaxation modulus must be non-negative and the Helmholtz free energy must be at a minimum at equilibrium in order for the material model to be compatible with thermodynamics. We construct a stability theorem for a history type solid which undergoes mechanically isolated motions by showing that the intrinsic motion of the body with respect to the center of mass is stable in the sense of Lyapunov. We then exemplify this stability theorem by studying the free radial motion of cylindrical and spherical shells through the construction of a history dependent Lyapunov functional. Numerical simulations indicate that the equilibrium state is globally stable. For materials with fast relaxation, the motion is similar to that of Newtonian viscous solids. However, for sufficiently slow relaxation where the history effect becomes dominant, we see that the solution trajectory traces a cyclone pattern in the phase plane. (C) 1997 Elsevier Science Ltd.