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

Area of Science:

• Geophysics
• Continuum Mechanics
• Soil Mechanics

Background:

• Continuum modelling of granular flow often suffers from ill-posed dynamic equations.
• Incompressible flow models based on Coulomb friction are inherently ill-posed.
• Rate-dependent rheology, specifically μ(I)-rheology, becomes ill-posed at extreme inertial numbers (I).

Purpose of the Study:

• To derive conditions for well-posedness in partial differential equations for granular flow.
• To combine compressibility effects with rate-dependent rheology.
• To address the long-standing issue of ill-posed dynamic equations in granular flow modelling.

Main Methods:

• Incorporating principles from critical-state soil mechanics.
• Developing partial differential equations that integrate compressibility and I-dependent rheology.
• Analyzing the conditions for mathematical well-posedness.

Main Results:

• Established conditions for well-posedness by combining compressibility with I-dependent rheology.
• Demonstrated that compressibility resolves ill-posedness issues for rate-dependent rheology.
• Showed that well-posedness is achieved for all deformation rates if the friction coefficient μ(I) meets specific inequalities.

Conclusions:

• The proposed framework ensures well-posedness for continuum modelling of granular flow.
• Compressibility is a key factor in stabilizing dynamic equations for granular materials.
• The findings provide a robust foundation for simulating granular flows across various deformation rates.