Stress -and structure-anisotropy (bulk)

Physical experiments on wave propagation on granular material control two macroscopic parameters that are assumed to characterize the response of a given assembly: the isotropic pressure p0 and the solid volume fraction φ. Here, by means of numerical simulation, we investigate the effect of the coordination number Z (the average number of contacts per particle) and the fluctuation of the number of contacts per particle Z. We adopt a numerical protocol to create several initial packings characterized by the same volume fraction and, for a given isotropic pressure, we are able to obtain different coordination numbers. That is, pressure and coordination number, for a given volume fraction, are independent. The result is that packings with fixed volume fraction and isotropic pressure exhibit a different elastic response. We attribute this behavior only to the coordination number as we find, surprisingly, that Z is linked to Z .

The steady-state shear rheology of granular materials is investigated in slow quasistatic and inertial flows. The effect of gravity (thus the local pressure) and the often-neglected contact stiffness are the focus of this study. A series of particle simulations are performed on a weakly frictional granular assembly in a split-bottom geometry considering various magnitudes of gravity and contact stiffnesses. While traditionally the inertial number, i.e., the ratio of stress to strain-rate time scales, is used to describe the flow rheology, we report that a second dimensionless number, the ratio of softness and stress time scales, must also be included to characterize the bulk flow behavior. For slow, quasistatic flows, the density increases while the effective (macroscopic) friction decreases with increase in either particle softness or local pressure. This trend is added to the μ I ( )rheology and can be traced back to the anisotropy in the contact network, displaying a linear correlation between the effective friction coefficient and deviatoric fabric in the steady state. When the external rotation rate is increased towards the inertial regime, for a given gravity field and contact stiffness, the effective friction increases faster than linearly with the deviatoric fabric.

It is widely recognized in particle technology that adhesive powders show a wide range of different bulk behavior due to the peculiarity of the particle interaction. We use Discrete Element simulations to investigate the effect of contact cohesion on the steady state of dense powders in a slowly sheared split-bottom Couette cell, which imposes a wide stable shear band. The intensity of cohesive forces can be quantified by the granular Bond number (Bo), namely the ratio between maximum attractive force and average force due to external compression. We find that the shear banding phenomenon is almost independent of cohesion for Bond numbers Bo < 1, but for Bo ≥ 1 cohesive forces start to play an important role, as both width and center position of the band increase for Bo > 1. Inside the shear band, the mean normal contact force is always independent of cohesion and depends only on the confining stress. In contrast, when the behavior is analyzed focusing on the eigen-directions of the local strain rate tensor, a dependence on cohesion shows up. Forces carried by contacts along the compressive and tensile directions are symmetric about the mean force (larger and smaller respectively), while the force along the third, neutral direction follows the mean force. This anisotropy of the force network increases with cohesion, just like the heterogeneity in all (compressive, tensile and neutral) directions.

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Physical experiments can characterize the elastic response of granular materials in terms of macroscopic state variables, namely volume (packing) fraction and stress, while the microstructure is not accessible and thus neglected. Here, by means of numerical simulations, we analyze dense, frictionless granular assemblies with the final goal to relate the elastic moduli to the fabric state, i.e., to microstructural averaged contact network features as contact number density and anisotropy. The particle samples are first isotropically compressed and then quasi-statically sheared under constant volume (undrained conditions). From various static, relaxed configurations at different shear strains, infinitesimal strain steps are applied to "measure" the effective elastic response; we quantify the strain needed so that no contact and structure rearrangements, i.e. plasticity, happen. Because of the anisotropy induced by shear, volumetric and deviatoric stresses and strains are cross-coupled via a single anisotropy modulus, which is proportional to the product of deviatoric fabric and bulk modulus (i.e., the isotropic fabric). Interestingly, the shear modulus of the material depends also on the actual deviatoric stress state, along with the contact configuration anisotropy. Finally, a constitutive model based on incremental evolution equations for stress and fabric is introduced. By using the previously measured dependence of the stiffness tensor (elastic moduli) on the microstructure, the theory is able to predict with good agreement the evolution of pressure, shear stress and deviatoric fabric (anisotropy) for an independent undrained cyclic shear test, including the response to reversal of strain.

A local constitutive model for anisotropic granular materials is introduced and applied to isobaric (homogeneous) axial-symmetric deformation. The simplified model (in the coordinate system of the bi-axial box) involves only scalar values for hydrostatic and shear stresses, for the volumetric and shear strains as well as for the new ingredient, the anisotropy modulus.The non-linear constitutive evolution equations that relate stress and anisotropy to strain are inspired by observations from Discrete Element Method (DEM) simulations. For the sake of simplicity, parameters like the bulk and shear modulus are set to constants, while the shear stress ratio and the anisotropy evolve with different rates to their critical state limit values when shear deformations become large.When applied to isobaric deformation in the bi-axial geometry, the model shows ratcheting under cyclic loading. Fast and slow evolution of anisotropy with strain (relative to the evolution of anisotropy in stress) lead to dilatancy and contractancy, respectively. Furthermore, anisotropy acts such that it works "against" the strain/stress, e.g., a compressive strain builds up anisotropy that creates additional stress acting against further compression.

The bulk properties of powders depend on material characteristics and size of the primary particles. During storage and transportation processes in the powder processing industry, the material undergoes various modes of deformation and stress conditions, e.g., due to compression or shear. In many applications, it is important to know when powders are yielding, i.e. when they start to flow under shear; in other cases it is necessary to know how much stress is needed to keep them flowing. The measurement of powder yield and flow properties is still a challenge and will be addressed in this study. In the framework of the collaborative project T-MAPPP, a large set of shear experiments using different shear devices, namely the Jenike shear tester, the ELE direct shear tester, the Schulze ring shear tester and the FT4 powder rheometer, have been carried out on eight chemically-identical limestone powders of different particle sizes in a wide range of confining stresses. These experiments serve two goals: i) to test the reproducibility/ consistency among different shear devices and testing protocols; ii) to relate the bulk behaviour to microscopic particle properties, focusing on the effect of particle size and thus inter-particle cohesion. The experiments show high repeatability for all shear devices, though some of them show more fluctuations than others. All devices provide consistent results, where the FT4 powder rheometer gives lower yield/steady state stress values, due to a different pre-shearing protocol. As expected, the bulk cohesion decreases with increasing particle size (up to 150 μm), due to the decrease of inter-particle cohesion. The bulk friction, characterized in different ways, is following a similar decreasing trend, whereas the bulk density increases with particle size in this range. Interestingly, for samples with particle sizes larger than 150 μm, the bulk cohesion increases slightly, while the bulk friction increases considerably-presumably due to particle interlocking effects-up to magnitudes comparable to those of the finest powders. Furthermore, removing the fines from the coarse powder samples reduces the bulk cohesion and bulk density, but has a negligible effect on the bulk friction. In addition to providing useful insights into the role of microscopically attractive, van der Waals, gravitational and/or compressive forces for the macroscopic bulk powder flow behaviour, the experimental data provide a robust database of cohesive and frictional fine powders for industrially relevant designs such as silos, as well as for calibration and validation of models and computer simulations.