Quasi-static versus dynamic stability associated with local damage models

We investigate the stability of dynamic and quasi-static strain-driven processes for a large class of micromechanics-based models of damage in brittle solid materials. A special attention is paid to the compatible models having the same equilibrium states where the dynamic damage law is constructed from a quasi-static one. We deduce a simple stability criterion for non-associated damage models. The unstable equilibria of quasi-static models are related to the solution multiplicity, shocks and non-smooth stress-strain curves. Also, we deduce the stability of the dynamic models from the linear stability of the associated damage differential system. For compatible models the dynamic and quasi-static stability criteria coincide. We use the stability criteria to analyse mechanically non-interacting cracks with self-similar growth, and find a critical crack density parameter that distinguishes between stable and unstable behaviours. It depends on the chosen configuration and Poisson's ratio, but it is independent of the other micromechanical properties. The configurations with the crack densities smaller than the critical value are mechanically unstable but cracks are activated at high stress levels, generating a sharp stress drop in the stress-strain diagram. For crack densities above the critical value, the model is stable and the stress varies smoothly. For dynamic loadings of unstable configurations a high strain rate is associated with a smooth stress-strain curve, while a low strain rate induces an abrupt stress drop in the stress-strain relation. This kind of instability is qualitatively the same for porous brittle plates under uni-axial compression and for the self-consistent effective elasticity and the wing crack models.