On viscoplastic regularisation of strain-softening rocks and soils

Constitutive models for rocks and soils typically incorporate some form of strain softening. Moreover, many plasticity models for frictional materials use a nonassociated flow rule. Strain softening and nonassociated flow rules can cause loss of well-posedness of the initial-value problem, which can lead to a severe mesh dependence in simulations and poor convergence of the iterative solution procedure. The inclusion of viscosity, which is a common property of materials, seems a natural way to restore well-posedness, but the mathematical properties of a rate-dependent model, and therefore the effectiveness with respect to the removal of mesh dependence, can depend strongly on how the viscous element is incorporated. Herein, we show that rate-dependent models, which are commonly applied to problems in the Earth's lithosphere, such as plate tectonics, are very different from the approach typically adopted for more shallow geotechnical engineering problems. We analyse the properties of these models under dynamic loadings, using dispersion analyses and one-dimensional finite difference analyses, and complement them with two-dimensional simulations of a typical strain localisation problem under quasi-static loading conditions. Finally, we point out that a combined model, which features two viscous elements, may be the best way forward for modelling time-dependent failure processes in the deeper layers of the Earth, since it not only enables modelling of the creep characteristics typical of long-term behaviour but also regularises the initial/boundary-value problem.