Constitutive modelling of granular materials with focus to microstructure and its effect on strain-localization

Abstract

The paper presents a micromechanically-based constitutive model for frictional granular materials through a fabric (anisotropy) sensitive stress-dilatancy law. A second order tensor is used to describe fabric whose evolution during plastic deformation is chosen to be purely deviatoric and stress ratio dependent. This produces different fabrics at critical state that are consistent with the corresponding critical void ratio. To illustrate the role of fabric and its evolution, model simulations are presented for mixed drained-undrained stress paths within the context of instability. Shear band localization is also discussed with fabric as a controlling dependent variable. Model results suggest that, in an extreme case, a very loose sand could very well display post peak behaviour and shear banding due to fabric. servation at both microand macro-scales, a stress-dilatancy model describing the collective deformational behaviour of a system of rigid granules undergoing slips and rotations was formulated. It is also worthy to note the work of Nemat-Nasser (2000) which integrates micromechanics in a double sliding model. In this paper, we use the model developed by Wan & Guo (2001b,c), which integrates a micromechanically based dilatancy rule, in order to illustrate the central role of fabric on granular material deformation with focus to both stress and strain paths as well as strain localization. The propensity of the fabric dependent constitutive model to succumb to shear band localization is examined by means of a bifurcation analysis. Hence shear band orientations and shear strains at localization can be computed under drained biaxial stress conditions for sand specimens with different initial fabrics. Model results demonstrate that very loose sand can display post peak behaviour and shear banding due to fabric. 2 MICROSTRUCTURALLY-BASED CONSTITUTIVE MODEL We discuss only the essential features of the model in order to establish the background for the understanding of the constitutive modelling framework. For detailed treatise of mathematical developments, the reader is directed to Wan & Guo (1998, 1999, 2001b,c) and Guo (2000). 2.1 Stress-dilatancy with embedded fabric In view of incorporating microstructural aspects into the formulation of stress-dilatancy, a representative elementary volume (REV) is chosen in which micro-variables are averaged and expressed in terms of macro-variables. For example, as a result of volume averaging, contact forces between particles can be expressed in terms of Cauchy stress, σ , via a so-called fabric tensor, F, that describes the geometrical arrangement of particles. Similarly, global strains e can be linked to fabric and kinematical variables such as particle translations and rotations. Details can be found in Guo (2000) and Chang & Ma (1991). The essence of the dilatancy model is to write energy dissipation considerations at grain contacts that slip and rotate during macroscopic deformations. For energy conservation at both scales, the rate of energy (power) dissipated D& at the microscopic level must be equal to the work rate W& expressed in terms of macro-variables, i.e. ∫ = = sliding v dV D V W ) ( 1 . e e σ & & & (1) over sliding contacts. The power dissipated D& can be further expressed in terms of micro-variables such as average tangential contact forces and relative slip displacements. As a result of such principle, a stress dilatancy equation with embedded micro-variables emerges in the form of a dilatancy rate β defined as the ratio of volumetric strain rate, v e& , over shear strain rate, γ& . These strain rates are wholly plastic. In conventional triaxial (axial symmetry) stress states, we get ) sin sin 1 ( ) sin (sin 3 4 m f f m v φ φ − φ − φ = γ e − = β & & (2)