Comparison of various models for strain-softening

This paper presents a comparison of various models for strain-softening due to damage such as cracking or void growth, as proposed recently in the literature. Con-tinuum-based models expressed in terms of softening stress-strain relations, and fracture-type models expressed in terms of softening stress-displacement relations are distinguished. From one-dimensional wave propagation calculations, it is shown that strain-localization into regions of finite size cannot be achieved. The previously well-documented spurious convergence is obtained with continuum models, while stress-displacement relations cannot model well smeared-crack situations. Continuum models may, however, be used in general if a localization limiter is implemented. Gradient-type localization limiters appear to be rather complicated; they require solving higher-order differential equations of equilibrium with additional bourdary conditions. Non-local localization limiters, especially the non-local continuum with local strain, in which only the energy dissipating variables are non-local, is found to be very effective, and also seems to be physically realistic. This formulation can correctly model the transition between homogeneous damage states and situations in which damage localizes into small regions that can be viewed as cracks. The size effect observed in the experimental and numerical response of specimens in tension or compression is shown to be a consequence of this progressive transition from continuum-type to fracture-type formulations.