L. Rothenburg

Micromechanical definition of the strain tensor for granular materials

In order to develop constitutive relations for granular materials from the micromechanical viewpoint, general expressions relating macroscopic stress and strain to contact forces and particle displacements are required. Such an expression for the stress tensor under quasi-static conditions is well established in the literature, but a corresponding expression for the strain tensor has been lacking so far. This paper presents such an expression for two-dimensional assemblies. This expression is verified by computer simulations of biaxial and shear tests. As a demonstration of the use of the developed expression, a study is made of the elastic moduli of two-dimensional, isotropic assemblies of bonded, nonrotating disks. Theoretical expressions are given for the elastic moduli in terms of micromechanical parameters, such as coordination number and contact stiffnesses. Comparison with the results from computer simulations show that the agreement is fairly good over a wide range of coordination numbers and contact stiffness ratios.

Observations on stress-force-fabric relationships in idealized granular materials

Abstract The paper presents a series of theoretical developments and numerical experiments directed at quantifying important features of the micromechanical behavior of granular media by introducing average characteristics of fabric anisotropy and certain statistical averages of contact forces. The introduced characteristics are explicitly linked to stress through a stress-force-fabric relationship. Numerical simulations are used to verify this relationship for plane assemblies and to trace the evolution of induced anisotropy in contact orientations and contact forces. It is shown that at large strain the degree of anisotropy in contact orientations achieves some limiting value and the macroscopic angle of friction at large strains can be expressed directly in terms of this value. Effects of the angle of interparticle friction and magnitude of interparticle stiffness on the shear capacity of numerical assemblies at large strain are investigated.

NUMERICAL SIMULATION OF IDEALIZED GRANULAR ASSEMBLIES WITH PLANE ELLIPTICAL PARTICLES

Abstract The paper presents the development of algorithms that have been implemented in a computer program used to simulate the performance of idealized granular systems composed of elliptical-shaped particles. The work is an extension of numerical simulation methods which have been successfully used in micromechanics research on disk-shaped or polygon-shaped particles by the authors and others. The simulation of elliptical-shaped particles offers the possibility to explore the influence of particle shape on the micromechanical behaviour of plane assemblies of particles and the stress-strain behaviour of these systems at the macro scale. Typical results of simulation runs are presented to illustrate the importance of particle shape on the macroscopic stress-strain response of plane systems.

`Stress–force–fabric' relationship for assemblies of ellipsoids

Abstract Stress–force–fabric relationship (S–F–F) stands for the explicitly formulated relationship between the components of the stress tensor for an assembly of particles in static equilibrium and anisotropy tensors representing either existing or load-induced biases in the spatial distributions of some relevant microscopic parameters. In this paper, the S–F–F relationship is formulated for ellipsoids and verified using results of three-dimensional numerical simulations with assemblies of prolate spheroids. The implications of the relationship regarding the effect of particle shape and contributions of different forms of anisotropy to the material shear strength of such assemblies are discussed.

NOTE ON A RANDOM ISOTROPIC GRANULAR MATERIAL WITH NEGATIVE POISSON'S RATIO

Abstract Poissons ratio in generalized Hookes law for an isotropic continuum is subject to the restriction − 1 ≤ v≤ 0.5 . With the exception of recently developed low-density polymer foams, solids including granular materials have not been known to exhibit negative Poissons ratio. This paper is concerned with the verification of constitutive stress-strain relationships proposed by Rothenburg (Micromechanics of idealized granular systems. Ph.D. thesis, Carleton University, 1980) which describe macroscopic behaviour of idealized bonded granular materials by elastic parameters ( K and v ) that are formulated explicitly in terms of microstructural parameters such as interparticle stiffness, contact density and average interparticle distance. The theory includes an expression for Poissons Ratio which is a function only of the ratio of tangential (shear) to normal contact stiffness λ A negative Poissons ratio is predicted for both planar and three-dimensional random isotropic systems when the tangential stiffness is greater than the normal stiffness (i.e. λ > 1 ). The results of numerical simulation of bonded disc assemblies used to verify constitutive relationships show that systems with λ > 1 do exhibit negative Poissqns ratio. Similar theoretical developments are summarized for three-dimensional random isotropic assemblies of bonded spheres and an analogous expression for Poissons ratio is presented for these systems. It is noted that while a negative Poissons ratio is theoretically possible, the micromechanieal condition for λ > 1 is physically unlikely for particles of natural materials.

An algorithm for detecting inter-ellipsoid contacts

Abstract An inter-ellipsoid contact detection algorithm was developed and used for simulating the behaviour of assemblies of ellipsoid-shaped particles using the well-known discrete element method (DEM). The contact algorithm was implemented in the modified version of the DEM program TRUBAL originally written to simulate the behaviour of assemblies of spheres. The modified program was used to perform deviatoric and axisymmetric compression tests on a 1000 prolate spheroid assembly in periodic space. The obtained stress–strain–dilation curves conform with the experimental evidence both qualitatively and quantitatively.

Statistical theories for the elastic moduli of two-dimensional assemblies of granular materials

In this micro-mechanical study of the behaviour of granular materials, a minimum potential energy and a minimum complementary energy principle are derived for the elastic moduli of two-dimensional, bonded assemblies of particles. These extremum principles are used to obtain bounds on the elastic moduli. Statistical versions of these energy principles are developed. The theoretical results are verified by computer simulations using the discrete element method. Good agreement between theory and simulations is observed.

Probability density functions of contact forces for cohesionless frictional granular materials

A theory is developed for the probability density functions of contact forces for cohesionless, frictional granular materials in quasi-static equilibrium. This theory is based on a maximum information entropy principle, with an expression for information entropy that is appropriate for granular materials. Entropy is maximized under the constraints of a prescribed stress and that the normal component of the contact force is compressive and that the tangential component of the contact force is limited by Coulomb friction. The theory results in a dependence of the probability density function for the tangential contact forces on the friction coefficient. The theoretical predictions are compared with results from discrete element simulations on isotropic, two-dimensional assemblies under hydrostatic stress. Good qualitative agreement is found for means and standard deviations of contact forces and the shape of the probability density functions, while the quantitative agreement is fairly good. Discrepancies between theory and simulations, such as the difference in shape of the probability density function for the normal force and the observed dependence on elastic properties of the exponential decay rate of tangential forces, are attributed to the fact that the method does not take into account any kinematics, which are essential in relation to elastic effects.

Statistics of the elastic behaviour of granular materials

The elastic behaviour of isotropic assemblies of granular materials consisting of two-dimensional, bonded and non-rotating particles is studied from the micromechanical viewpoint. Discrete element simulations have been performed of assemblies of 50,000 particles with various coordination numbers (average number of contacts per particle) and various ratios of tangential to normal stiffness. Statistics, such as mean, standard deviation and probability distribution function, have been computed for geometrical quantities and contact displacements under compressive and shearing loading. A linear relation is observed between the average number of contacts of a particle and its circumference. Average displacements do not conform to (generalised) uniform strain as is often assumed. The probability density functions for normal and tangential displacements are nearly Gaussian. None of the considered energies satisfy an equipartition of energy between normal and tangential modes.

Influence of particle eccentricity on micromechanical behavior of granular materials

Abstract The paper describes the results of numerical simulation of 2D assemblies of elliptical-shaped particles. The study is focussed on the influence of the magnitude of particle eccentricity on assembly strength, density and coordination number.