(22c) Constitutive Modeling of Slow Flows of Dense Granular Assemblies

Slow flows of dense granular materials are encountered widely in industries and in nature. Reliable constitutive models for the rheological behaviors of dense granular flows that are linked to particle-level properties are not yet available; as a result, quantitative prediction of the flow characteristics in such systems through solution of continuum models remains elusive for the most part. In this presentation, we will describe a dissipative plasticity model [1, 2] constructed to meet such prediction needs for dense quasi-static flows.

To link the constitutive model to particle-level properties, we perform micromechanical analysis of dense granular assemblies under different types of deformations, including isotropic compression, simple shear, and triaxial compression. Towards this end, the discrete element method (DEM) is used to simulate the motion of individual particles in deforming assemblies; the stress and kinematics information at the continuum level are then obtained through statistical averaging. The simulations also afford detailed information about internal variables such as the fabric tensor and the coordination number of particles involved in force chains, which characterize the evolution of the microstructure of the assembly. These results are then analyzed to first identify the internal variables that are natural candidates to describe the microstructure and stress evolution and then seek quantitative relations that form the basis for continuum rheological models.

The dissipative plasticity model is thus composed of a rate-independent stress constitutive equation and the pressure and stress ratio relations, which are expressed as functions of the fabric tensor and coordination number associated with the force chains. The microstructure is evolved by evolution equations for these internal variables and the stress state varies accordingly. The evolution equations are developed by simplifying the general frame-indifferent functional forms of the time derivatives of the microstructure variables, as used for anisotropic fluids [3] and dense suspensions [4]. We will show that the model is properly defined so that the material constants depending on particle-level properties can be calibrated by the steady and unsteady simply shear data; the calibrated model is able to describe the rheological behaviors under those conditions with a good agreement. We will further demonstrate that the model is capable of predicting the rheological behaviors under other complex shearing conditions or different loading paths. Specifically, comparisons between model predictions and DEM data will be given for flows under cyclic shear with small strain amplitude, for Reynolds dilatancy and for flows under triaxial compression and extension.

[1] J. D. Goddard. Dissipative materials as models of thixotropy and plasticity. Journal of Non-Newtonian Fluid Mechanics, 14:141?160, 1984.

[2] J. D. Goddard. A dissipative anisotropic fluid model for non-colloidal particle dispersions. Journal of Fluid Mechanics, 568:1?17, 2006.

[3] G. L. Hand. A theory of anisotropic fluids. Journal of Fluid Mechanics, 13:33?46, 1962.

[4] J. J. Stickel, R. J. Phillips, and R. L. Powell. A constitutive model for microstructure and total stress in particulate suspensions. Journal of Rheology, 50:379?413, 2006.