Abstract
We present a micro-mechanical model that is able to predict localization in a sheared planar aggregate of spheres. We assume a non-linear contact law between interacting particles that deform differently from an affine deformation. Equilibrium determines this deviation. The aggregate is isotropically compressed and then sheared so anisotropy develops because of contacts deletion and a non-linear contact law. Because of anisotropy and fluctuations in particles deformation, the resulting macroscopic stiffness tensor, which relates increments in the average stress with increments in the average strain, is characterized by a lack of major symmetry, Aijkl ≠ Aklij . At given shear strain and coordination number it is possible to detemine a plane over which discontinuity in strain occurs; this is identified as localization.
Highlights
The authors have recently proposed a micro-mechanical model that highlights a possible mechanism for localization in granular materials [1]
This is equivalent to a solution for the zero eigenvalue of the acoustic tensor associated with the macroscopic stiffness tensor of the aggregate [5]
While it is well-known that localization in granular materials is associated with the solution of the vanishing of the acoustic tensor related to the stiffness tensor of the aggregate, it is less obvious why the stiffness tensor should not be characterized by a major symmetry
Summary
The authors have recently proposed a micro-mechanical model that highlights a possible mechanism for localization in granular materials [1]. Following previous work [2,3,4], we identify localization with the possibility of a discontinuity in a component of the incremental strain at a predicted value of the shear strain and a predicted orientation relative to the axis of greatest compression This is equivalent to a solution for the zero eigenvalue of the acoustic tensor associated with the macroscopic stiffness tensor of the aggregate [5]. In general this condition is possible when the stiffness tensor, that relates increments in the average stress with increments in the average strain, exhibits a lack of major symmetry, Ai jkl Akli j. This occurs when the incremental response of the aggregate, at relatively high shear stress, is essentially frictionless and the incremental contribution associated with the tangential force is almost negligible [8]
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