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arXiv · 1610.05135

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

T. Barker·D. G. Schaeffer·M. Shearer·J. M. N. T Gray
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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 $μ(I)$-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 $μ(I)$, our results show that, with compressibility, the equations are well-posed for all deformation rates provided that $μ(I)$ satisfies certain minimal, physically natural, inequalities.

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T. Barker, D. G. Schaeffer, M. Shearer, J. M. N. T Gray. 2016-10-14. Well-posed continuum equations for granular flow with compressibility and $μ(I)$-rheology. https://doi.org/10.1098/rspa.2016.0846

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