This paper presents stability results for rate-independent mechanical systems,
associated with general tangent stiffness matrices including symmetric and
nonsymmetric ones. Conservative and nonconservative as well as associate and
nonassociate elastoplastic systems are concerned by such a theoretical study. Hill’s
stability criterion, also called the second-order work criterion, is here revisited in
terms of kinematically constrained systems. For piecewise rate-independent
mechanical systems (which may cover inelastic and elastic evolution processes), such a
criterion is also a divergence Lyapunov stability criterion for any kinematic
autonomous constraints. This result is here extended for systems with nonsymmetric
tangent matrices. By virtue of a new type of variational formulation on the
possible kinematic constraints, and thanks to the concept of kinematical
structural stability (KISS), both criteria, Hill’s stability criterion and the
divergence stability criterion under kinematic constraints, are shown to be
equivalent.