Micromechanics of shear bands

Abstract

In the past, we have developed a micromechanically-based constitutive model of a 2D, monodisperse granular assembly consisting of circular particles, in which the tangential displacements at particle–particle contacts were limited to microslip only i.e. particles do not slide relative to each other. This constitutive law was later extended, using slightly more advanced contact laws, to include sliding contacts, along with the potential for loss of contacts. Furthermore, through these contact laws, evolution of the distribution of contact modes (non-sliding or microslip contacts, sliding contacts and loss of contacts), contact forces and the density of contact directions, can be determined as the deformation proceeds i.e. deformation-dependent anisotropies. In this paper we apply this latter constitutive model to shear band formation in a bi-axial test. Using an initially isotropic sample, we demonstrate that the constitutive model can reproduce the various anisotropies that have been observed in experiments and simulations. Moreover, the predicted shear band properties (e.g. thickness, inclination, prolonged localisation, void ratio) show even better agreement with experimental observations than previously found using our past models. These results take on particular significance when one considers that, in contrast to the constitutive equations traditionally used for granular materials, the micromechanically-based constitutive model presented here contains a direct link to the physical and measurable properties of particles (e.g. particle–particle friction coefficient, particle stiffness coefficients) and so arguably contains no fitting parameters.

Introduction

One of the difficulties encountered when handling granular matter is the lack of a well-accepted constitutive model. By now it should be widely acknowledged that the behaviour of granular materials is governed by the interaction between constituent particles. That is, the finite size of particles cannot be ignored in a continuum model. For example, the ability of particles to rotate significantly reduces the strength of granular materials (Oda et al., 1982). With this in mind, if we are to remain in a continuum setting, the classical models of plasticity need to be extended for granular materials to include, at the very least, the effects of a length-scale and particle rotation. This extension can be made within the framework of micropolar and/or higher order strain gradient theories. A less phenomenological approach would be to develop constitutive models based on averaging the discrete interactions between particles over a region, to obtain continuum laws. This is the approach taken here.

In this paper, a micromechanically based micropolar constitutive law is developed for two-dimensional, dry assemblies of uniformly-sized, circular particles. The homogenisation procedure used to obtain the constitutive law is the small strain scheme of Tordesillas and Walsh (2002) and is outlined here in Section 2. In particular, this scheme is considered high resolution, as it is based around the contacts of a single particle and its nearest neighbours to enable fine-scale structures to be captured. Section 3 introduces specific strain dependent contact laws into the homogenisation procedure, which incorporate sliding and non-sliding contacts, a rolling resistance and loss of contacts. These strain dependent contact laws are obtained from a mean-field approximation to the motion around a contact. Hence, these strain dependent contact laws naturally result in an evolving contact anisotropy and contact force anisotropy. To demonstrate this anisotropy development, the constitutive laws are presented in a form that assumes that the particle contacts are initially isotropically distributed in both direction and force. Finally, in Section 4, the problem of shear band formation and evolution in bi-axial test is studied to test the constitutive model. The shear band analysis adopted here is based on the method presented in Tordesillas et al. (in press). The method is a combination of the one-dimensional analysis first introduced for micropolar continua by Mühlhaus and Vardoulakis (1987), and an incremental step-by-step procedure for the post localisation analysis. Additionally, from the onset of deformation, the quantities defining the characteristics of the material (e.g. contact modes, contact anisotropy and contact force anisotropy) are updated in a stepwise manner after each small increment of strain. Although this method of shear band analysis is restrictive, in comparison for example to a finite element method, the semi-analytic solutions provided by the one-dimensional simplification help to identify the limitations of the shear band analysis versus the constitutive model (Tordesillas et al., 2003).