On direct Liapunov method in continuum theories

Let S_b be a basic rest state for a elastic body B. In the theory of direct Lyapunov method of stability, we reach two objectives: one method is related to the control of the perturbation to S_b in terms of the data (Lagrange-Dirichlet theorem), for conservative systems; the other method is related to an asymptotic decay to zero for the perturbation, for dissipative systems. In the first section, we prove a Lagrange-Dirichlet theorem for hyperelastic continua. In this case, we prove that the basic configuration is stable if the given configuration is a local minimum for the physical energy, indentified with total mechanical energy for mechanical systems. In the second section, we analize the same continua adding a dissipative behavior. In this case, the time derivative of Lyapunov functionals is strictly negative, and a decay of the motion to the rest is expected in the general abstract context. To prove asymptotic decay, we propose a useful technique which employs a general variational formulation, the algorithm consists in the choice of the test function