Unilateral interparticle interactions have an effect on the elastic response of granular materials due to the opening and closing of contacts during quasi-static shear deformations. A simplified model is presented, for which constitutive relations can be derived. For biaxial deformations the elastic behavior in this model involves three independent elastic moduli: bulk, shear, and anisotropy modulus. The bulk and the shear modulus, when scaled by the contact density, are independent of the deformation. However, the magnitude of the anisotropy modulus is proportional to the ratio between shear and volumetric strain. Sufficiently far from the jamming transition, when corrections due to non-affine motion become weak, the theoretical predictions are qualitatively in agreement with simulation results.

The evolution of the force distributions during the isotropic compression of two-dimensional packings of soft frictional particles is investigated numerically. Regardless of the applied deformation, the normal contact force distribution P(f(n)) can be fitted by the product of a power law, and a stretched exponential, while the tangential force distribution P(f(t)) is fitted well by a Gaussian. With increasing strain, the asymptotic behavior at large forces does not change, but both P(f(n)) and P(f(t)) exhibit a broadening, even though, when scaled with the average forces, their widths decrease. Furthermore, the distribution of friction mobilization P(η) is a decreasing function of η=|f(t)|/(μf(n)), except for an increased probability of fully mobilized contacts (η=1). The excess coordination number of the packings increases with the applied strain, indicating that the more a packing is compressed the more stable it becomes.

In this numerical study, measurements of the contact forces inside a periodic two-dimensional sheared system of soft frictional particles are reported. The distribution P(f n ) of normalized normal forces f n =F n / F n exhibits a gradual broadening with increasing the pure shear deformation γ, leading to a slower decay for large forces. The process however slows down and P(f n ) approaches an invariant shape at high γ. By introducing the joint probability distribution P(f n , α) in sheared configurations, it is shown that for a fixed direction α, the force distribution decays faster than exponentially even in a sheared system. The overall broadening can be attributed to the averaging over different directions in the presence of shear-induced stress anisotropy. The distribution of normalized tangential forces almost preserves its shape for arbitrary applied strain. Introduction -The contact forces in disordered materials, such as colloidal suspensions, foams, emulsions, and granular media are remarkably organized into highly heterogeneous force networks [1]. A statistical mechanical description of stress transmission in disordered media should provide a way to understand and predict the contact force distributions. The tail behaviour of the normalized normal force distribution P (f n ≡F n / F n ) has received much attention, and several theoretical models with different assumptions and approaches [2,3] predict an exponential as well as a Gaussian tail. While early experiments and numerical simulations [4][5][6] favoured the exponential decay, further studies revealed that the decay can also be faster than exponential [7][8][9][10][11]. A recent numerical study [12] of frictional soft particle systems under pure compression showed that, independent of the distance from jamming, the tail behaviour can be described by a stretched exponential with an exponent around 1.8, which slightly depends on the choice of the contact force law, the friction coefficient, and the relative particle stiffness in tangential and normal directions.

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Abstract. The modeling of the elastic properties of disordered or granular solids requires a theory of elasticity that takes non-affine deformations into account. Using a linearized force law, the non-affine elastic deformations are calculated. Based on the microscopically exact expressions for the local strain and stress fields, a way to calculate maps of the local linear elastic constants for fictional granular packings is presented. The elastic constants are found to be scale and system size independent withing an appropriate parameter range. Deviations from classical elasticity due to the micropolar nature of granular materials is studied.