A General Theory for the Viscoplasticity of Dry and Fluid-Saturated Granular Media

J. D. Goddard

Department of Aerospace and Mechanical Engineering

University of California, San Diego

La Jolla, CA, 92093-0411, USA

jgoddard@ucsd.edu

This paper revisits the "purely dissipative" model proposed several years ago [1] as a general continuum model for the history-dependent viscoplasticity of non-colloidal particle dispersions. Essential to the model is a positive-definite fourth-rank viscosity tensor h depending on the history of deformation. In the reduced form considered here, h is an isotropic function of a history-dependent 2nd-rank "texture" or "fabric" tensor A, which gives stress as a tensor-valued function of fabric and strain-rate tensors. This paper considers several special cases appropriate to systems ranging from Stokesian suspensions to dry granular media. For Stokesian suspensions, a formula for h(A) given by the analogous theory of linear elasticity, together with a corotational memory integral for A, provides a compelling model of transient viscosity and normal stresss evolution in simple shear [5,3]. However, one extremely rapid mode of relaxation is required to mimic the incomplete reversal of stress on abrupt reversal of shearing. This suggests that non-hydrodynamic effects are implicated, and it establishes a kinship to liquid-saturated granular media with sustained particle contact. In the case of granular media, the isotropic version of the above model reduces to the Reiner-Rivlin form proposed previously [2], which encompasses a quasi-linear model proposed recently for dense rapid granular flow [4]. Since isotropic models cannot represent the effect of flow-induced anisotropy on yield surfaces and viscometric normal stresses, attention is given here to a more general forms, involving nonlinear dependence on both fabric and strain rate. Two important time scales are highlighted, a grain-inertia time scale for dry granular media [2, 4], and a viscous-frictional time scale that appears to be implicated in recent experiments on completely saturated granular media.

References

[1] J. D. Goddard. Dissipative materials as models of thixotropy and plasticity. Journal of Non-Newtonian Fluid Mechanics, 14:141-60, 1984.

[2] J. D. Goddard. Dissipative materials as constitutive models for granular materials. Acta Mechanica, 63:3-13, 1996.

[3]J.D. Goddard. A dissipative anisotropic fluid model for non-colloidal particle dispersions. Journal of Fluid Mechanics, to appear, 2006.

[4] P. Jop, Y. Forterre, and O. Pouliquen. Crucial role of sidewalls in granular surface flows: consequences for the rheology. Journal of Fluid Mechanics, 541:167-92, 2005.

[5] V. G. Kolli, E. J. Pollauf, and F. Gadala-Maria. Transient normal stress response in a concentrated suspension of spherical particles. Journal of Rheology, 46:321-34, 2002.