Linear and Nonlinear Elasticity of Granular Media: Stress-Induced Anisotropy of a Random Sphere Pack

[+] Author and Article Information
D. L. Johnson, L. M. Schwartz

Schlumberger-Doll Research, Old Quarry Road, Ridgefield, CT 06877-4108

D. Elata, J. G. Berryman

Earth Sciences Division, Lawrence Livermore National Laboratory, Mail Stop L-202, P.O. Box 808, Livermore, CA 94551

B. Hornby

Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge, CB3 0EL, UK

A. N. Norris

Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08855-0909

J. Appl. Mech 65(2), 380-388 (Jun 01, 1998) (9 pages) doi:10.1115/1.2789066 History: Received March 19, 1997; Revised September 21, 1997; Online October 25, 2007


We develop an effective medium theory of the nonlinear elasticity of a random sphere pack based upon the underlying Hertz-Mindlin theory of grain-grain contacts. We compare our predictions for the stress-dependent sound speeds against new experimental data taken on samples with stress-induced uniaxial anistropy. We show that the second-order elastic moduli, Cijkl, and therefore the sound speeds, can be calculated as unique path-independent functions of an arbitrary strain environment, kl}, thus generalizing earlier results due to Walton. However, the elements of the stress tensor, σij, are not unique functions of kl} and their values depend on the strain path. Consequently, the sound speeds, considered as functions of the applied stresses, are path dependent. Illustrative calculations for three cases of combined hydrostatic and uniaxial strain are presented. We show further, that, even when the additional applied uniaxial strain is small, these equations are not consistent with the usual equations of third-order hyperelasticity. Nor should they be, for the simple reason that there does not exist an underlying energy function which is simply a function of the current state of the strain. Our theory provides a good understanding of our new data on sound speeds as a function of uniaxial stress.

Copyright © 1998 by The American Society of Mechanical Engineers
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