A micromechanically-based constitutive model for frictional deformation of granular materials
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 and 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, , which is essentially the same as that of the unit branch vectors, , is represented bywhere is the second invariant of the fabric tensor , of components (in two dimensions)where ; defines the degree of anisotropy of distribution (Eq. (1a)), and θ0 gives the orientation of the greatest density of the contact normals; 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 . From Eq. (2b), the hydrostatic tension becomeswhere is some suitable intermediate value of for 0<θ<π. The pressure p is therefore given bySince it is difficult to obtain an explicit expression for the contact force , consider the following alternative approach. Divide , 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, , 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 β≈S. The sliding directions in loading make angles of 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 . 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, :In terms of the components of the stress-difference tensor, S, the deviatoric components of the deformation rate tensor can be rewritten as
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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Multiscale modelling of granular materials in boundary value problems accounting for mesoscale mechanisms
2021, Computers and GeotechnicsCitation Excerpt :The complexity of the constitutive behavior of granular assemblies then naturally emerges from the multiplicity of such local interactions and the evolution of the interaction network. A thorough review of such multiscale approaches to the mechanics of granular media can be found in the seminal works of the past few decades; see for instance Nemat-Nasser and Mehrabadi, 1984; Jenkins and Strack, 1993; Mehrabadi et al., 1993; Balendran and Nemat-Nasser, 1993a, 1993b; Nemat-Nasser, 2000; Nemat-Nasser and Zhang, 2002. In the continuity of microplane models (Zienkiewicz and Pande, 1977), the microdirectional model (Nicot and Darve, 2005) can be interpreted as a micromechanical application of the multislip theory to granular materials.
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2021, Computers and GeotechnicsCitation Excerpt :These behaviours of granular materials are closely associated with the stability and buckling of force chains at a mesoscopic scale and sliding and rolling at contacts, thus governed by the grain-scale structural characteristics and processes. To capture the fabric features, numerous fabric tensors describing the spatial distribution of different microstructural quantities, statistical representation of the microstructural fabric, have been developed in the literature (e.g. reviewed by Li et al. (Li et al., 2009), and many of them have been incorporated into constitutive models for granular materials as essential internal variables (Hu et al., 2020; Li and Dafalias, 2012; Gao et al., 2014; Tobita, 1989; Wan and Guo, 2004; Dafalias et al., 2004; Yang et al., 2018; Petalas et al., 2019; Nemat-Nasser, 2000; Nicot et al., 2005; Zhu et al., 2006; Wang et al., 2020). Under shearing, the fabric of granular materials may be regarded as unchanged only at a very low level of strain, typically at the order of 10−5 (Tatsuoka, 1999).
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2020, International Journal of Solids and StructuresCitation Excerpt :When fabric is incorporated into constitutive models, a corresponding evolution law becomes necessary. Such evolution laws for fabric anisotropy have generally been developed for continuum-mechanical models (Li and Dafalias, 2011; Gao et al., 2014; Gao and Zhao, 2017; Nemat-Nasser, 2000; Sun and Sundaresan, 2011; Oboudi et al., 2016; Oda, 1993; Wan and Guo, 2004; Lashkari and Latifi, 2008; Woo and Salgado, 2015; Yuan et al., 2019; Fang et al., 2019). One type of these evolution laws involves the loading index associated with the macro-scale yield surface (see for instance (Li and Dafalias, 2011; Yang et al., 2018; Gao et al., 2014; Gao and Zhao, 2017; Nemat-Nasser, 2000; Woo and Salgado, 2015; Petalas et al., 2019b)).
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