A micromechanically-based constitutive model for frictional deformation of granular materials

https://doi.org/10.1016/S0022-5096(99)00089-7Get rights and content

Abstract

A micromechanically-based constitutive model is developed for inelastic deformation of frictional granular assemblies. It is assumed that the deformation is produced by relative sliding and rolling of granules, accounting for pressure sensitivity, friction, dilatancy (densification), and, most importantly, the fabric (anisotropy) and its evolution in the course of deformation. Attention is focused on two-dimensional rate-independent cases. The presented theory fully integrates the micromechanics of frictional granular assemblies at the micro- (grains), meso- (large collections of grains associated with sliding planes), and macro- (continuum) scales. The basic hypothesis is that the deformation of frictional granular masses occurs through simple shearing accompanied by dilatation or densification (meso-scale), depending on the microstructure (micro-scale) and the loading conditions (continuum-scale). The microstructure and its evolution are defined in terms of the fabric and its evolution. While the elastic deformation of most frictional granular assemblies is rather small relative to their inelastic deformation, it is included in the theory, since it affects the overall stresses.

Introduction

A physically-based constitutive model is developed by Balendran and Nemat-Nasser, 1993a, Balendran and Nemat-Nasser, 1993b by considering the frictional anisotropic deformation of cylindrical granular masses and Coulomb’s friction criterion. This model predicts rather well the observed dilatancy and densification effects in monotonic and cyclic loading. The model is based on the double-sliding mechanism originally proposed by Mandel (1947), and further developed by de Josselin de Jong, 1959, Spencer, 1964, Spencer, 1982, Mehrabadi and Cowin, 1978.

The present work seeks to integrate the experimentally observed response of two-dimensional frictional granules into a general theory which includes the above-cited theories as special cases; a comprehensive account of these and related results is contained in an upcoming book by the author. Experiments on various assemblies of two-dimensional photoelastic rods of oval cross section, deformed under biaxial loads and in simple shearing, have shown that:1

  • 1.

    the distributions of the unit contact normals n, and the unit branch vectors2 m, are essentially the same and may be used interchangeably;

  • 2.

    the fabric tensors nn and mm are essentially the same, where 〈φ〉 denotes the volume average of φ;

  • 3.

    the diagonal elements of these fabric tensors are almost constant in simple shearing under a constant confining pressure;

  • 4.

    the off-diagonal elements of these fabric tensors behave similarly to the applied shear stress; and

  • 5.

    the second-order distribution density function of the unit contact normals, En, which is essentially the same as that of the unit branch vectors, Em, is represented byEn11+Ecos2θ−2θ0,where E=12EijEij1/2 is the second invariant of the fabric tensor E, of components (in two dimensions)Eij=4Jij12δij,where Jij=〈ninj; E defines the degree of anisotropy of distribution (Eq. (1a)), and θ0 gives the orientation of the greatest density of the contact normals; θ0+π2 then gives the orientation of the least density; see Subhash et al., 1991, Balendran and Nemat-Nasser, 1993a.

Section snippets

Sliding resistance

It may be assumed that the overall deformation of a granular mass consists of a number of dilatant simple shearing deformations, along the active shearing planes. At the micro-scale, this dilatant simple shear flow occurs on active shearing planes through sliding and rolling of grains over each other at active contacts. In a granular sample with a large number of contacting granules, a mesoscopic shearing plane passes through a large number of contacting granules with various orientations of

A two-dimensional model

Consider two-dimensional deformation of a granular mass. Let f̂=n·f̂. From Eq. (2b), the hydrostatic tension becomes12trσ̄=Nl̄ππ01+Ecos2θ−2θ0f̂dθ=12Nl̄f̂where f̂ is some suitable intermediate value of f̂θ for 0<θ<π. The pressure p is therefore given byp=−12Nl̄f̂Since it is difficult to obtain an explicit expression for the contact force f̂, consider the following alternative approach. Divide τ̄r, the resistance to sliding in the s-direction, into two parts, one due to a Coulomb-type

Meso-scale yield condition

Consider a typical sliding plane at the meso-scale. The resistance to shearing of the granules over this plane is due to interparticle friction and fabric anisotropy, as is expressed by (6). The micromechanical formulation of the preceding subsection provides explicit expressions for the parameters which define this resistance. The resulting quantities, φμ,β,β, and ν0, have physical meanings, and are related to the microstructure of the granular mass. Hence, they can be associated with the

Loading and unloading

It is known that unloading from an anisotropic state may produce reverse inelastic deformation, even against an applied shear stress; see Nemat-Nasser, 1980, Okada and Nemat-Nasser, 1994. Furthermore, in a continued monotonic deformation, the principal axes of the stress and the fabric tensors tend to coincide.

Consider the biaxial loading shown in Fig. 5, and assume that the loading has induced a strong anisotropy, with βS. The sliding directions in loading make angles of ±π4+φμ2 with the SI

A rate-independent compressible double-sliding model

The meso-scale double-sliding formulation of the preceding section will now be used to develop a model for planar deformation of frictional granules, similar to that of crystals. To this end, assume that the total plastic deformation at the continuum level, consists of two superimposed shearing deformations along the active sliding planes. This sliding is accompanied by volumetric changes and induced anisotropy. Based on these concepts, a complete set of constitutive relations is produced in

Constitutive relations for double-sliding model

To complete the constitutive relations, two ingredients are necessary. These are: (1) the elasticity relations; and (2) the evolutionary equation for the variation of the fabric tensor β. Once these two ingredients are provided, then the slip rates can be computed using the yield and the consistency conditions.

A continuum model based on double sliding

Start with definition (14a,b), and noting (17a–e), obtain the following relations for the components of the plastic part of the velocity gradient, Lp=Dp+Wp:Dpkk=γ̇1tanδ1+γ̇2tanδ2,Wp12=12γ̇1γ̇2,Dp11=12γ̇1cos2ψ−φμ−δ1cosδ1+12γ̇2cos2ψ+φμ−δ2cosδ2=−Dp22,Dp12=12γ̇1sin2ψ−φμ−δ1cosδ1+12γ̇2sin2ψ+φμ−δ2cosδ2.In terms of the components of the stress-difference tensor, S, the deviatoric components of the deformation rate tensor can be rewritten asDp11=S112Sγ̇1cosφμ−δ1cosδ1+γ̇2cosφμ−δ2cosδ2+S122Sγ̇1sinφμ−δ1

Acknowledgements

The work reported here has been supported in part by the Army Research Office under Contract DAAHO4-96-1-0376, and in part by the National Science Foundation under Grant CMS-9729053, with the University of California, San Diego.

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