Well-posed continuum equations for granular flow with compressibility and $\mu (I)$-rheology

Abstract

Continuum modelling of granular flow has been plagued with the issue of ill-posed equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent $\mu (I)$-rheology is ill-posed when the non-dimensional strain-rate $I$ is too high or too low. Here, incorporating ideas from Critical-State Soil Mechanics, we derive conditions for well-posedness of PDEs that combine compressibility with $I$-dependent rheology. When the $I$-dependence comes from a specific friction coefficient $\mu (I)$, our results show that, with compressibility, the equations are well-posed for all deformation rates provided that $\mu (I)$ satisfies certain minimal, physically natural, inequalities.

References in corpus (1)

• From Elasticity to Hypoplasticity: Dynamics of Granular Solids

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• A note on Stokes' problem in dense granular media using the $\mu (I)$--rheology
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