Stress,dilatancy and fabric in granular materials

On the basis of a simple micromechanical model, expressions are developed for the overall stress and deformation rate measures in a granular mass which carries the applied loads through contact friction. Various measures of the granular fabric are examined, and explicit connections between a fabric measure and the stress and the deformation rate tensors are obtained. Then, from the balance of energy, a general stress-dilatancy equation is deduced, which includes the effect of fabric. Several other published dilatancy equations are discussed and compared with the new equation.     

A continuum model for granular materials: Considering dilatancy and the Mohr-Coulomb criterion

Summary In this paper we will explore the consequences of the Mohr-Coulomb criterion on the constitutive equation proposed by Rajagopal and Massoudi [1]. This contunuum model which is based on the earlier works of Cowin [2] has also the ability to predict the dilatancy effect which is related to the normal stress effects. At the same time, if a proper representation is given to some of the material parameters, this model would also comply with the Mohr-Coulomb criterion. We also present, as a special case, an exact solution for the case of simple shear flows.     

Evaluation of energy contributions using inter-particle forces in granular materials under impact loading

A drop-tower based experimental setup was developed for the impact testing of 2D assembly of cylinders with impactor velocity of around 6 m/s. This drop tower setup was used to load 2D granular assemblies of polyurethane and polycarbonate cylinders of 1\(^{\prime \prime }\)–1.25\(^{\prime \prime }\) length with three different diameters of 1/4\(^{\prime \prime }\), 3/8\(^{\prime \prime }\) and 1/2\(^{\prime \prime }\). A high speed camera was used for recording the images at speeds between 10,000 and 55,000 fps to monitor the deformation of the cylinders. Kinematic and strain fields in individual grains during each experiment were measured using digital image correlation. These experimentally measured strain and kinematic fields were used as inputs for the granular element method (GEM) based force inference technique and the inter-particle forces in normal and tangential direction were determined at every contact in each experiment. The inter-particle forces at each contact can facilitate the calculation of frictional work done at each contact. The GEM based inter-particle forces for a simple 2 particle granular assembly were found to be in good agreement with predictions from ABAQUS explicit based FEM simulation. The influence of different model parameters was also characterized such as grain stiffness, frictional co-efficient was investigated qualitatively. The impact response of the various ordered granular assemblies was also investigated using the GEM approach and the effect of local defects such as voids or layering of granular materials on the wave propagation phenomena is also studied. The presence of the point or line defects have significant effects on the wave propagation in the granular assemblies due to wave scattering and attenuation.     

Experimental and numerical study of granular medium-rough wall interface friction

Wall roughness plays a crucial role in granular medium - rough wall interface friction. In this study, an experimental device has been designed to study the influence of boundary conditions, more specifically wall roughness, on the behavior of sheared granular medium. The study is based on use of an analog model, and consists of simulating roughness by means of notches and grains in the medium by monodisperse beads and on use of a numerical model based on the discrete element method. The test protocol entails displacing at fixed speed notched rods under confined granular medium. Movement of the beads layer near the rods as well as friction of the beads against the rods are both studied herein. Results indicate that the parameter controlling friction at the granular medium - rough wall interface is primarily the depth of beads embedment in surface asperities. The objective of the associated numerical modeling is to supplement the experimental results.     

Waves of dilatancy in a granular material with incompressible granules

As a densely packed granular material begins to flow through a constriction, dilatant waves are often observed. These waves initiate at the constriction and propagate outward into the granular material with an increase in the porosity of the material behind the wave front. As a model for dilatant waves, we consider curved waves carrying a jump in the gradient of porosity in the context of the continuum theory for the flow of granular materials presented by Goodman and Cowin. Their speed of propagation is determined and it is shown that such waves carry a jump in the stress acting on planes transverse to their direction of propagation. For the case when a wave of dilatancy propagates into a uniform region at rest, an explicit formula for the amplitude of the wave is derived for waves of general shape and then specialized to give results for plane, cylindrical, and spherical waves. In this case, the wave is purely longitudinal.     

Rheology of weakly wetted granular materials: a comparison of experimental and numerical data

Shear cell simulations and experiments of weakly wetted particles (a few volume percent liquid binders) are compared, with the goal to understand their flow rheology. Application examples are cores for metal casting by core shooting made of sand and liquid binding materials. The experiments are carried out with a Couette-like rotating viscometer. The weakly wetted granular materials are made of quartz sand and small amounts of Newtonian liquids. For comparison, experiments on dry sand are also performed with a modified configuration of the viscometer. The numerical model involves spherical, monodisperse particles with contact forces and a simple liquid bridge model for individual capillary bridges between two particles. Different liquid content and properties lead to different flow rheology when measuring the shear stress-strain relations. In the experiments of the weakly wetted granular material, the apparent shear viscosity $\eta _g$ η g scales inversely proportional to the inertial number $I$ I , for all shear rates. On the contrary, in the dry case, an intermediate scaling regime inversely quadratic in $I$ I is observed for moderate shear rates. In the simulations, both scaling regimes are found for dry and wet granular material as well.     

The influence of inter-particle friction and the intermediate stress ratio on soil response under generalised stress conditions

Previous research studies have used either physical experiments or discrete element method (DEM) simulations to explore, independently, the influence of the coefficient of inter-particle friction (μ) and the intermediate stress ratio (b) on the behaviour of granular materials. DEM simulations and experiments using photoelasticity have shown that when an anisotropic stress condition is applied to a granular material, strong force chains or columns of contacting particles transmitting relatively large forces, form parallel to the major principal stress orientation. The combined effects of friction and the intermediate stress ratio upon the resistance of these strong force chains to collapse (buckling failure) are considered here using data from an extensive set of DEM simulations including triaxial and true triaxial compression tests. For all tests both?μ and b affected the macro- and micro-scale response, however the mechanisms whereby the force chain stability was improved differ. While friction clearly enhances the inherent stability of the strong force chains, the intermediate stress ratio affects the contact density and distribution of orthogonal contacts that provide lateral support to the force chains.     

The role of contact friction in the dynamic breakage behavior of granular materials

Based on the discrete element method, a multi-scale model is employed to investigate the role of contact friction in the dynamic compression responses of brittle granular materials. Four numerical granular samples with different particle friction coefficients ranging from 0.0 to 2.0 are tested and the particle breakage extent is quantified with the Einav breakage index. It is observed that the relationship of the breakage extent with the axial stress is apparently non-monotonic concerning the particle friction coefficient. At the same stress level, the breakage extent exhibits a minimum when the particle friction coefficient is around 0.1 but increases significantly with the particle friction coefficient to both sides. The micro physical origin of this non-monotonic behavior is a distinct transition in dominant particle-breakage modes from tension to shear. Moreover, energy analyses also show non-monotonic evolution of the frictional and damping dissipation with the particle friction coefficient. The joint effect of these two dissipation terms contributes to the non-monotonic behavior of particle breakage. In addition, the accuracy and competence of two frequently-used micro quantities, fraction of sliding contacts and average coordination number, are discussed.

     

Failure of cemented granular materials under simple compression: experiments and numerical simulations

We investigate the strength and failure properties of a model cemented granular material under simple compressive deformation. The particles are lightweight expanded clay aggregate beads coated by a controlled volume fraction of silicone. The beads are mixed with a joint seal paste (the matrix) and molded to obtain dense cemented granular samples of cylindrical shape. Several samples are prepared for different volume fractions of the matrix, controlling the porosity, and silicone coating upon which depends the effective particle–matrix adhesion. Interestingly, the compressive strength is found to be an affine function of the product of the matrix volume fraction and effective particle–matrix adhesion. On the other hand, it is shown that particle damage occurs beyond a critical value of the contact debonding energy. The experiments suggest three regimes of crack propagation corresponding to no particle damage, particle abrasion and particle fragmentation, respectively, depending on the matrix volume fraction and effective particle–matrix adhesion. We also use a sub-particle lattice discretization method to simulate cemented granular materials in two dimensions. The numerical results for crack regimes and the compressive strength are in excellent agreement with the experiments.     

Sensitivity series and friction surface analysis of non-metallic friction materials

Nine non-metallic friction material formulations contained fibers, fillers and binder without strong abrasives were designed using Golden Section sequence combined with least-square method. Seven ingredients used without strong abrasives were selected based on the combinatorial approach. Wear (w) and coefficient of friction (μ) are expressed as a function of volume fraction of the ingredients selected as and . The formulations were optimized using the sensitivity series obtained from least-square method. An optimized formulation (S-10) was obtained with a total wear loss of 6.69 wt% and an average μ of 0.375. Friction surface of both brake pad and disc was observed using SEM, EDX and XRD. The non-metallic friction materials without abrasives exhibit unique friction performance and phenomena compared with the non-metallic friction materials with abrasives and semi-metallic brake linings. The temperature measured during friction is lower and the oxidation of the cast iron disc is not rigorous.     

On fluidization of polydisperse granular materials

A method has been developed for calculating the fractional composition of boiling bed particles liquefied at a given speed of filtration with account for the composition of the supplied granular materials. The relation between the maximum size of particles and their mean diameter and the degree of polydispersity has been investigated. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 2, pp. 353–357, March–April, 2008.     

Stress partition analysis of granular materials

Writing the stress as a sum over contact forces allows one to partition stress increments according to their microscopic, grain-level causes. We present two ways of doing this: the first way differentiates between normal and tangential contact forces, while the second ascribes different parts of a change in stress to different physical processes such as contact network anisotropy, non-affine motions and sliding contacts. These two partitioning methods can be combined. We then use them to analyze simulations of failure in biaxial tests, leading to several results. We show that the granular material becomes effectively frictionless well before failure. In addition, by partitioning the second order work, one can attribute a part of the destabilization to each of the various physical processes mentioned above.     

DEM analysis of the influence of the intermediate stress ratio on the critical-state behaviour of granular materials

The critical-state response of granular assemblies composed of elastic spheres under generalised three-dimensional loading conditions was investigated using the discrete element method (DEM). Simulations were performed with a simplified Hertz–Mindlin contact model using a modified version of the LAMMPS code. Initially isotropic samples were subjected to three-dimensional stress paths controlled by the intermediate stress ratio, \(b=[(\sigma '_{2}-\sigma '_{3})/\) \((\sigma '_{1}-\sigma '_{3})]\) . Three types of simulation were performed: drained (with \(b\) -value specified), constant volume and constant mean effective stress. In contrast to previous DEM observations, the position of the critical state line is shown to depend on \(b\) . The data also show that, upon shearing, the dilatancy post-peak increases with increasing \(b\) , so that at a given mean effective stress, the void ratio at the critical state increases systematically with \(b\) . Four commonly-used three-dimensional failure criteria are shown to give a better match to the simulation data at the critical state than at the peak state. While the void ratio at critical state is shown to vary with \(b\) , the coordination number showed no dependency on \(b\) . The variation in critical state void ratios at the same \(p'\) value is apparently related to the directional fabric anisotropy which is clearly sensitive to \(b\) .     

Experimental and numerical determination of representative elementary volume for granular plant materials

A series of physical and numerical tests were conducted to determine representative elementary volume of granular plant material. The load response of pea grain assembly poured into a cuboid test chamber and subjected to uniaxial confined compression was studied. The apparatus was equipped with adjustable side walls that allowed measurement of boundary stresses in samples of varying thickness. It was found that load distribution varied considerably in samples of thickness smaller than three times the size of the particle. Less pressure variation was observed in grain assemblies of thickness equaled to three, five and seven times the particle size. Comparison between experimental data and numerical DEM results have shown qualitative agreement. It was found that the specimen of dimension not smaller than five times the particle size can be used as a representative elementary volume in confined uniaxial compression test of granular plant materials.     

History-dependent deformation of a rotated granular pile governed by granular friction

We experimentally examined the history dependence of the rotation-induced granular deformation. As an initial state, we prepared a quasi-two-dimensional granular pile whose apex is at the rotational axis and its initial inclination is at the angle of repose. The rotation rate was increased from 0 to 620 (rpm) and then decreased back to 0. During the rotation, deformation of the rotated granular pile was captured by a camera. From the acquired image data, granular friction coefficient $\mu$ was measured as a function of the ratio between centrifugal force and gravity, $\mathrm{\Gamma }$. To systematically evaluate the variation of $\mu$ both in the increasing (spinning up) and decreasing (spinning down) rotation-rate regimes, surface profiles of the deformed granular piles were fitted to a model considering the force balance among gravity, friction, and centrifugal force at the surface. We found that $\mu$ value grows in the increasing $\mathrm{\Gamma }$ regime. However, when $\mathrm{\Gamma }$ was reduced, $\mu$ cannot recover its initial value. A part of the history-dependent behaviors of the rotated granular pile can be understood by the force balance model.     

Moisture influence on the resilient deformation behaviour of unbound granular materials

The influence of moisture on the resilient deformation properties of unbound granular materials was investigated based on repeated load triaxial tests. Results showed that the resilient modulus (M_R) decreased with increasing moisture for a relatively low number of load cycles (N) where the deformation behaviour was mostly resilient with a negligible amount of associated accumulated permanent deformation (PD). Modelling attempts on this behaviour were quite satisfactory. Furthermore, the M_R showed an increasing trend with increasing moisture, up to the optimum, when the N was relatively large with a significant amount of accumulated PD. Above the optimum, the M_R generally decreased. Further investigation suggested that moisture aided the post-compaction (PC) and possible particle rearrangement that resulted in the increased PD and increased M_R. The existing model did not work in this case indicating that the effect of PC on M_R should be considered in modelling.     

Relationship between contact stiffness ratio and structural stiffness coefficient of a rectangular granular sample: a numerical analysis

In this study, the relationship between contact stiffness ratio and stiffness coefficient of a rectangular sample of granular material is investigated using the granular element method (GEM). Randomly packed granular samples are studied with different contact force models and particle size distributions in numerical simulation using the GEM. Initial arrangements of particles in the granular samples are generated using different algorithms such as inwards packing method, iterative growth method and compression method. It is demonstrated that the relationship between the stiffness coefficient of the granular sample and the contact stiffness ratio is basically logarithmic (Eq. 6). For granular samples generated by inwards packing method, the deviation for logarithmic relationship is relatively small. For granular samples generated by iterative growth method and compressive method, the deviation for logarithmic relationship is relatively large.     

The influence of rolling resistance on the stress-dilatancy and fabric anisotropy of granular materials

The effects of rolling resistance on the stress-dilatancy behavior and fabric anisotropy of granular materials were investigated through a three-dimensional discrete element method (DEM). A rolling resistance model was incorporated into the DEM code PFC3D and triaxial DEM simulations under simulated drained and undrained conditions were carried out. The results show that there existed a threshold value of the rolling friction. When the rolling friction was smaller than this value, the mechanical behavior of granular materials under both drained and undrained conditions were substantially influenced by the rolling friction, but the influence diminished when it was larger than the threshold value. A linear relationship has been observed between the dilatancy coefficient and the natural logarithm of the rolling-friction coefficient when it was smaller than the threshold value. An increase in the rolling friction led to an increase in the fabric anisotropy of all strong contacts under both testing conditions until the threshold value was attained. The investigation on the effect of rolling friction on the microstructure of granular materials revealed that the rolling friction enhanced the stability of force chains, which resulted in the difference in the stress-dilatancy behavior. Finally, the relationship between the stress ratio q/p\(^{\prime }\) and the fabric measure at strong contacts \(\hbox {H}_{\mathrm{d}}^{\mathrm{s}} /\hbox {H}_{\mathrm{m}}^{\mathrm{s}}\) was found independent of the inter-particle friction, rolling friction and testing conditions.