(54e) A Constitutive Model for Intermediate, Quasi-Static and Inertial Regimes of Dense Granular Flow

Experiments [Powder Technol. (2003), vol. 131, pp. 23-39; Phys. Rev. E 78 (041306)] and discrete element simulations (DEM) [J. Fluid Mech. (2002), vol. 465, pp. 261-291; Powder Technol. (2006), vol. 169, pp. 10-21] have established that dense granular flow exhibit three different kinds of rheological behavior. In the quasi-static regime the granular shear stress is independent of strain rate, in the intermediate regime the shear stress has a power-law dependence on the strain rate with exponent between 0 and 2, and in the inertial regime the shear stress has a quadratic dependence on the strain rate. A recent assessment of several constitutive models with DEM data for homogeneous sheared dense granular flow [Chem. Eng. Sci. (2012), vol. 72, pp. 20-34] reveals that none of the models is able to capture the granular rheology in the intermediate regime. It has also been recently demonstrated through DEM calculations of silo discharge that the error in granular stress predicted by current constitutive models is directly implicated in accurate prediction of the discharge rate, and the spatial region of this erroneous stress prediction coincides exactly with the spatial extent of the intermediate regime. Analysis of the DEM data in the intermediate regime [Chem. Eng. Sci. (2012), vol. 72, pp. 20-34] also reveals that the dominant contribution to granular stress is from the contact (virial) contribution, while the streaming contribution is negligible. Based on these findings a constitutive model is derived by averaging the DEM expression for the contact stress. This contact stress model (CSM) naturally gives rise to the dependence of average contact stress on mesoscale flow descriptors such as the coordination number and the fabric tensor, as well as the average normal contact force. Appropriate closures for the coordination number and the fabric tensor are provided by solving their modeled evolution equation proposed by Sun and Sundaresan [J. Fluid Mech. (2011), vol.  682, pp. 590-616]. The predictive capability of the proposed contact stress model (CSM) is tested in homogeneous shear flow using DEM data corresponding to the intermediate, quasi-static and inertial regimes. A simple extension of CSM that is obtained by combining it with the refined OP (ROP) concept results in the ROP-CSM model that satisfactorily predict dense granular flow in all regimes: quasi-static, intermediate, and inertial.