On the stability of rate-dependent solids with application to the uniaxial plane strain test

摘要

The linear stability criterion, proposed for structural models in an earlier paper, is now extended for a general class of elastic--viscoplastic continua. The time-dependent trajectories, whose stability is under investigation, are functions of two characteristic times: the relaxation time of the viscous solid and the rate of the applied loading, with their ratio denoted by T. It is assumed that the loading conditions of the trajectory are not modified by the perturbation. The criterion predicts that a solid is initially unstable if there exists a unit norm perturbation in the velocity field whose time derivative is positive. This condition is equivalent to finding a positive eigenvalue in the self-adjoint part of the operator relating the initial first and second rate of the displacement perturbations. If the dominant eigenvalue is obtained from the non self-adjoint operator, the change in sign of its real part is a sufficient condition for instability. For solids with an associated flow rule, it is shown that the exclusion of instability in a trajectory, in the limit of vanishing T, coincides with stability of the corresponding rate-independent solid in the sense of Hill. The theory is applied to the analysis of a finitely strained rectangular block under uniaxial tension and compression, for different elastic--viscoplastic solids of the von Mises and Drucker--Prager type.