Initial planar deformation of dilatant granular materials

A theory for the initial planar deformation of dilatant granular materials based on a kinematic proposal of R. Butterfield and R.M. Harkness (1972) is presented. The theory introduces an additional parameter called the angle of dilatancy into the traditional structure of plasticity theories for granular materials and soils. When the angle of dilatancy is zero, the present theory reduces to the theory introduced by A.J.M. Spencer in 1964. When the angle of dilatancy is equal to the angle of internal friction, the present theory reduces to the planar form of the theory introduced by D. C. Drucker and W. Prager in 1952. The properties of the theory presented here include coincidence of the stress and velocity characteristics, realistic energy dissipation predictions, and, in general, non-coincidence of the principal axes of stress and strain-rate. However, the angle of dilatancy is assumed to be a constant in this analysis and it does not decrease to zero with increased monotonic shearing deformation as experiment requires that it should, the theory therefore being limited to the initial deformation of dilatant granular materials.     

Three-dimensional constitutive relations for granular materials based on the dilatant double shearing mechanism and the concept of fabric

An extension of a three-dimensional model proposed by Anand and Gu (2000) for amorphous granular materials to include the effects of initial and induced anisotropy is presented in this paper. The proposed model can also be considered as a three-dimensional generalization of a model recently developed by Zhu et al. (2005) for the planar deformation of granular materials. The main ingredients of the model include the dilatant double shearing mechanism (Spencer, 1964, Mehrabadi and Cowin, 1978), the concept of fabric (Oda, 1972), and an extension of the Mohr–Coulomb yield criterion (Shield, 1955, Spencer, 1982) to three dimensions.The constitutive equations are implemented in the finite element program ABAQUS/Explicit (ABAQUS, 2001) by developing a user-material subroutine to conduct numerical triaxial compression tests for samples of granular materials with different initial anisotropy. The numerical results agree with the observed behavior and show that the extended constitutive model is capable of capturing the strength anisotropy of granular materials. Employing the anisotropic model developed here, we have also repeated the numerical simulation of the stress state in a static conical sand pile conducted earlier by Anand and Gu (2000). We find that fabric has little or no influence on the vertical stress distribution except at the base of the sand pile where the peak value of this stress is slightly higher than that predicted by the model of Anand and Gu (2000) which does not include the effects of fabric. We also find that the direction of the principal compressive stress changes from vertical at points away from the center of the pile to almost horizontal at points close to the center of the pile. This result provides a possible explanation for the observed dip in the vertical stress distribution in sand piles.     

Dynamics and wave propagation in dilatant granular materials

The equations of motion for dilatant granular material are obtained from a Hamiltonian variational principle of local type in the conservative case. The propagation of nonlinear waves in a region with uniform state is studied by means of an asymptotic approach that has already appeared useful in an investigation on wave propagation in bubbly liquids and in fluid mixtures. When the grains are assumed to be incompressible, it is shown that the material behaves as a continuum with latent microstructure.

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Sommario Si ricavano le equazioni di moto per i materiali granulari dilatanti da un principio variazionale Hamiltoniano di tipo locale nel caso conservativo. Si studia la propagazione delle onde non lineari in una regione di stato costante per mezzo di un approccio asintotico già rivelatosi utile nello studio della propagazione di onde nei liquidi con bolle e nelle miscele di fluidi. Quando si supponga che i granuli siano incomprimibili, si dimostra che il materiale si comporta come un continuo con microstruttura latente.

     

Multi-mechanism anisotropic model for granular materials

By representing the assembly by a simplified column model, a constitutive theory, referred to as sliding–rolling theory, was recently developed for a two-dimensional assembly of rods subjected to biaxial loading, and then extended to a three-dimensional assembly of spheres subjected to triaxial (equibiaxial) loading. The sliding–rolling theory provides a framework for developing a phenomenological constitutive law for granular materials, which is the objective of the present work. The sliding–rolling theory provides information concerning yield and flow directions during radial and non-radial loading. In addition, the theory provides information on the role of fabric anisotropy on the stress–strain behavior and critical state shear strength. In the present paper, a multi-axial phenomenological model is developed within the sliding–rolling framework by utilizing the concepts of critical state, classical elasto-plasticity and bounding surface. The resulting theory involves two yield surfaces and falls within the definition of the multi-mechanism models. Computational issues concerning the solution uniqueness for stress states at the corner of yield surfaces are addressed. The effect of initial and induced fabric anisotropy on the constitutive behavior is incorporated. It is shown that the model is capable of simulating the effect of anisotropy, and the behavior of loose and dense sands under drained and undrained loading.     

Numerical solution to the shearing flow of granular materials between two plates

We study the shearing flow of granular materials between two horizontal flat plates where the top plate is moving with a constant speed. The constitutive relation used for the stress is based on the continuum model proposed by Rajagopal and Massoudi (DOE Report, DOE/PETC/TR-90/3, 1990). The material coefficients such as viscosity and normal stress coefficients are based on the model of Boyle and Massoudi (Int. J. Eng. Sci 28 (1990) 1261). The governing equations are non-dimensionalized and the resulting system of non-linear differential equations is solved numerically using finite difference technique.     

Quantification of large and localized deformation in granular materials

Quantifying large deformation in granular assemblies using concepts originating from continuum mechanics is a challenging task because of (1) the discontinuous nature of granular displacement, which does not allow the definition of a continuum measure of deformation, and (2) the almost inevitable shear band localization. These problems exist in both real-world granular materials and their numerical idealizations using particle-based simulations. In this work a new method is developed in order to address these issues. Instead of creating a meshed equivalent continuum for quantifying small engineering strains, the new method performs independent random queries on the velocity gradient characteristics of arbitrary sub-domains in the assembly through the novel concept of overlapping reference triangles, thus, enabling rigorous handling of large deformations which are usually associated with localization. The proposed method is illustrated and validated by discrete element method (DEM) simulation of a biaxial compressive test, in which apparent shear banding takes place. The homogenized deformation quantifications based on the new method match the estimations from the imposed boundary conditions. The numerical examples are also applied to (1) quantifying the heterogeneous distribution of deformation over the specimen, (2) visualizing the nucleation process of shear bands, and (3) characterizing shear flow patterns in shear bands. An investigation on the effects of the reference triangle sizes yields some inspiring and practically significant results.     

Influence of porosity on the non-linear deformation of granular materials

An algorithm is suggested to determine non-linear deformative properties of granular composite materials with non-linearly deformed porous components. Stochastic differential equations of physically non-linear elasticity theory with the following use of conditional averaging are assumed as a basis. Solution of non-linear equations with respect to conditional moment is constructed by the method of successive approximation. The volume content of the components and pores in the components have been studied for their effect on the character of the strain curves of porous granular composite.     

A simplified second gradient model for dilatant materials: Theory and numerical implementation

This paper deals with the development of a new second gradient model, its numerical implementation and its validation. In order to remedy to the spurious mesh dependency of the post localized computation enhanced models incorporating some internal length are necessary. These models are very time consuming. In this paper we present a simplified theory within the framework of constrained micromorphic models involving only the micro volumetric strain. Provided the use of an additional penalty term in the numerical treatment, this model is quite efficient to regularize problems modelling behaviors exhibiting plastic volumetric strain such as the ones of geomaterials. More over this model is notably less time consuming than the more general ones.     

Micromechanical definition of an entropy for quasi-static deformation of granular materials

A micromechanical theory is formulated for quasi-static deformation of granular materials, which is based on information theory. A reasoning is presented that leads to the definition of an information entropy that is appropriate for quasi-static deformation of granular materials. This definition is based on the hypothesis that relative displacements at contacts with similar orientations are independent realisations of a random variable. This hypothesis is made plausible based on the results of Discrete Element simulations. The developed theory is then used to predict the elastic behaviour of granular materials in terms of micromechanical quantities. The case considered is that of two-dimensional assemblies consisting of non-rotating particles with an elastic contact constitutive relation. Applications of this case are the initial elastic (small-strain) deformation of granular materials. Theoretical results for the elastic moduli, relative displacements, energy distribution and probability density functions are compared with results obtained from the Discrete Element simulations for isotropic assemblies with various average numbers of contacts per particle and various ratios of tangential to normal contact stiffness. This comparison shows that the developed information theory is valid for loose systems, while a theory based on the uniform-strain assumption is appropriate for dense systems.     

Micro-mechanical analysis of deformation characteristics of three-dimensional granular materials

The deformation characteristics of idealized granular materials have been studied from the micro-mechanical viewpoint, using Bagi’s three-dimensional micro-mechanical formulation for the strain tensor [Bagi, K., 1996. Mechanics of Materials 22, 165–177]. This formulation is based on the Delaunay tessellation of space into tetrahedra. The set of edges of the tetrahedra can be divided into physical contacts and virtual contacts between particles. Bagi’s formulation expresses the continuum, macro-scale strain as an average over all edges, of their relative displacements (between two successive states) and the complementary-area vectors. This latter vector is a geometrical quantity determined from the set of edges, i.e. from the structure of the particle packing.Results from Discrete Element Method simulations of isotropic and triaxial loading of a three-dimensional polydisperse packing of spheres have been used to investigate statistics of the branch vectors and complementary-area vectors of edges (subdivided into physical and virtual contacts) and of the relative displacements of edges. The investigated statistics are probability density functions and averages over groups of edges with the same orientation. It is shown that these averages can be represented by second-order Fourier series in edge orientation.Edge orientations are distributed isotropically, contrary to contact orientations. The average lengths of the branch vectors and the normal component of the complementary-area vectors are distributed isotropically (with respect to the edge orientation) and their average values are related to each other and to the volume fraction of the assembly. The other two components of the complementary-area vector are zero on average.The total deformation of the assembly, as given by the average of the relative displacements of the edges of the Delaunay tessellation follows the uniform-strain prediction. However, neither the deformation of the physical contact network nor of the virtual contact network has this property. The average relative displacement of physical edges in the normal direction (determined by the branch vector) is smaller than that according to the uniform-strain assumption, while that of virtual contacts is larger. This is caused by the high interparticle stiffness that hinders compression. The reverse observation holds for the tangential component of the relative displacement vector. The contribution of the deformation of the empty space between physical contacts to the continuum, macro-scale strain tensor is therefore very important for the understanding and the prediction of the macro-scale deformation of granular materials.     

A microstructural elastoplastic model for unsaturated granular materials

The homogenization technique is used to obtain an elastoplastic stress–strain relationship for dry, saturated and unsaturated granular materials. Deformation of a representative volume of material is generated by mobilizing particle contacts in all orientations. In this way, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. For the non-saturated state, capillary forces at the grain contacts are added to the contact forces created by an external load. They are calculated as a function of the degree of saturation, depending on the grain size distribution and on the void ratio of the granular assembly. Numerical simulations show that the model is capable of reproducing the major trends of a partially saturated granular assembly under various stress and water content conditions. The model predictions are compared to experimental results on saturated and unsaturated samples of silty sands under undrained triaxial loading condition. This comparison shows that the model is able to account for the influence of capillary forces on the stress–strain response of the granular materials and therefore, to reproduce the overall mechanical behavior of unsaturated granular materials.     

A kinetic model for rapid flows of granular materials

A kinetic model for rapid flows of granular material is considered. An evolution equation for the first order distribution function is developed which includes the effects of intergranular friction force in an average sense. The collision operator is approximated by the BGK relaxation model. The fundamental equations of motion including an equation for dynamics of the fluctuation energy are derived and discussed. The gravity and the plane Couette flows of granular materials are treated as examples of the applications of the present theory.     

An elasto-plastic model for granular materials with microstructural consideration

In this paper, we have extended the granular mechanics approach to derive an elasto-plastic stress–strain relationship. The deformation of a representative volume of the material is generated by mobilizing particle contacts in all orientations. Thus, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. Essential features such as continuous displacement field, inter-particle stiffness, and fabric tensor are discussed. The predictions of the derived stress–strain model are compared to experimental results for sand under both drained and undrained triaxial loading conditions. The comparisons demonstrate the ability of this model to reproduce accurately the overall mechanical behavior of granular media and to account for the influence of key parameters such as void ratio and mean stress. A part of this paper is devoted to the study of anisotropic specimens loaded in different directions, which shows the model capability of considering the influence of inherent anisotropy on the stress–strain response under a drained triaxial loading condition.     

基于细观方向平均模型的颗粒材料宏观Cosserat连续体本构关系

提出了基于细观微-方向模型（Micro—Directional Model）的宏观Cosserat连续体本构关系。在细观尺度上考虑颗粒旋转自由度及接触力矩,微结构的影响通过接触分布函数体现。给出均质各向同性Cosserat连续体模型弹性常数的细观参数表达式,并建议了二维情况下内尺度参数的细观力学表达式。对颗粒材料宏观行...     

Generalization of the Prandtl solution to the case of axisymmetric deformation of materials obeying the double shear model

A semianalytic solution of the problem on the compression of an annular layer of a plastic material obeying the double shear model on a cylindrical mandrel is obtained. The approximate statement of boundary conditions, which cannot be satisfied exactly in the framework of the constructed solution, is based on the same assumptions as the statement of the classical plasticity problem of compression of a material layer between rough plates (Prandtl’s problem). It is assumed that the maximum friction law is satisfied on the inner surface of the layer. The solution is singular near this surface. The strain rate intensity factor is calculated, and its dependence on the process and material parameters is shown.     

A model for compaction and shear hysteresis in saturated granular materials

Fluid-saturated sands exhibit irreversible compaction and shear hysteresis under cyclic shear loads in both free draining and undrained conditions. Constitutive relations of differential-type are constructed heuristically from typical qualitative response. An influence of pore pressure on compaction is incorporated, and the generation of pore pressure under cyclic shearing is investigated. Parameter variations in the shear relations allow a variety of hysteresis loop behaviours to be described.