Physica B 279 (2000) 134}138 Nonlinear elasticity of granular media David Linton Johnson*, HernaH n A. Makse, Nicolas Gland, Lawrence Schwartz Schlumberger-Doll Research, Old Quarry Road, Ridgexeld, CT 06877-4108, USA Ecole Normale Superieure, 45 rue d'ULM, F-75005, Paris, France Abstract The linear and nonlinear elastic properties of granular media are analyzed within the context of e!ective medium theories, as well as with numerical molecular dynamic simulations, assuming the validity of the Hertz}Mindlin theory at the single contact level. There is a crucial distinction between force laws which are path independent, leading to a hyper-elastic e!ective medium theory, and those which are path dependent, for which the deformation history must be followed explicitly. The e!ective medium theories provide a reasonable description of existing experimental data, considered as a function of applied stress, but there are signi"cant discrepancies. Numerical simulations resolve the question as to whether the problem lies with the treatment of the individual grain}grain contact or with the e!ective medium approximation (ema). We "nd that the problem lies principally with the latter: The bulk modulus is well described by the ema but the shear modulus is not, principally because the ema does not correctly allow for the grains to relax from the a$ne motion assumed by the ema.  2000 Elsevier Science B.V. All rights reserved. PACS: 81.06.Rm; 81.40.Ji Keywords: Granular media; Acoustics; Nonlinear elasticity 1. Introduction The elastic properties of granular aggregates, such as sedimentary rocks, can be enormously nonlinear as com- pared with the properties of nonporous materials [1]. The end member of such systems may be taken to be a loose/unconsolidated aggregate of glass beads which acquire a sti!ness solely as a result of applied stress. This is because if two grains are just touching, the force, considered as a function of displacement, does not ini- tially grow linearly, as with most systems, but it has a power law behavior (Eqs. (1) and (2) below). Aside from posing an interesting problem in the physics of dis- ordered systems, these systems are unusually nonlinear in their response and they can exhibit path dependence. By this we mean that the work done in deforming the system can depend upon whether one "rst compresses the * Corresponding author. Fax: #1-203-438-3819. E-mail address: djohnson@SDR.SLB.com (D.L. Johnson) system, then shears it, or "rst shears then compresses, or compresses and shears simultaneously, etc. The result depends upon the path taken in    space and not just on the "nal state of strain   ("nal). Here, we review some recent theoretical research we have undertaken in an attempt to understand these systems. We "rst discuss e!ective medium theories of the elastic properties, then we present our molecular dynamic simulations, and we end with a brief summary. 2. E4ective medium theories The starting point is the behavior of a single grain} grain contact, which we assume to be describable by the Hertz}Mindlin theory and variations thereof. Two touching grains are displaced in compression along a line joining their centers by an amount 2w. They may also su!er a transverse relative displacement of their centers 0921-4526/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 7 0 0 - 0