An instability condition of the deformation process in elasto-(visco)-nonlinear materials (Q1977283)

The reversible behaviour of materials can be described by a free energy depending on the strain and internal variables, where the stress and thermodynamic forces are connected with constitutive laws involving derivatives of the free energy function and internal variables. The irreversible behaviour can be characterized by a convex domain where the potential function of thermodynamic forces parametrized by internal variables vanishes identically. Is also admitted a pseudo-potential with the same kind of dependence, the derivatives of which in the space of thermodynamic forces are multiplied by a pseudo-visco-nonlinear factor. This factor depends on the relaxation time, and the resistance coefficient depends on the mechanical state of material (described by strain, internal variables, thermodynamic forces and potential). Here the authors perform a stability anlaysis based on small linear harmonic perturbation and on a second-order evolution of stress divergence, which allows to take into account the progressive localization in viscous materials. The first-and second-order time derivatives of stress are needed to derive the corresponding equations. The condition of vanishing of stress divergence leads to a time-perturbation parameter. For moderate viscosity this parameter can be taken of the same order as the viscosity coefficient. Hence the constitutive equations describe elastic moduli by means of complex terms. The onset of stationary discontinuity is related to the vanishing perturbation frequency, which can be expressed through the vanishing of determinant of pseudo--acoustic tensor, whose singularity determines the normal band localization. For rate-independent models the acoustic tensor is different from this tensor for rate-dependent models, and both rate-dependent and rate-independent criteria converge if and only if the above-mentioned factor is linear in pseudolinear potential, and the strain hardening modulus is identical with the resistance coefficient. The applications given in the paper concern a rate-independent model and a viscoplastic model with linear damage term, the yield function depending quadratically on stress. The numerical results show a small sensitivity of the localization to relaxation time, if the relaxation time is small compared to the macroscopic time of experiment. The localization hardening moduli are the same.