An analysis of stability of equilibrium and of quasi-static transformations on the basis of the dissipation function

The stability of equilibrium for general dissipative systems and of quasi-static processes for time-independent inelastic systems is discussed in the isothermal case. The analysis relies on the inspection of a functional sum of the reversible and dissipated energies. In particular, the dissipated power is viewed as a constitutive function, i.e., a function of the current state and its time variations. The restrictions induced by a stability postulate are analyzed. They result in a sequence of extremum principles for the time derivatives of the functional of increasing order. In particular, a generalized normality law is retrieved as a necessary condition for stability. Under some restrictions, the time derivatives of the state variables corresponding to stable paths are completely determined by a minimization procedure. For a large class of problems it is proved that this minimization procedure is also equivalent to the evolution equations as directly derived from the equilibrium equations and the constitutive laws. In the case of bifurcation, the minimization of the higher order terms generally allows a complete determination of the evolution.