Session X18: Continuum Descriptions of Discrete Materials

Retrogressive failure of a layer of granular material on an inclined plane

The flow of granular materials down an inclined plane is closely related to many natural hazards, such as landslides and avalanches, which can cause serious damage to life and property. Avalanches can be triggered by many different factors, such as human activities, new material falling, wind or earthquakes. When an avalanche is triggered by a local disturbance, it is not only material downstream of this disturbance that is dislodged. Material upslope of the disturbance may also collapse, through an upwards propagating erosion wave through the granular layer, or `retrogressive failure', which separates the regions of flowing and static material. This retrogressive failure is critically dependent on physics beyond the $\mu(I)$-rheology and, despite being one of the basic waves in granular flow, has not been modelled in detail before. We use small scale lab experiments, novel theory and numerical simulations to model retrogressive failure, and apply our results to both geophysical and industrial contexts.   

Formation of levees, troughs and elevated channels by avalanches on erodible slopes

Snow avalanches are typically initiated on marginally stable slopes with a layer of fresh snow that may easily be incorporated into the avalanche. The net balance of erosion and deposition of snow determines whether an avalanche grows, starves away or propagates steadily. We present the results of small scale experiments in which particles are released on a rough inclined plane coated with a static erodible layer of the same grains. For thick static layers on steep slopes the initial avalanche grows rapidly in size by entraining grains. On shallower slopes an elevated channel forms and material is eventually brought to rest due to a greater rate of deposition than erosion. On steep slopes with thinner erodible layers it is possible to generate avalanches that have a perfect balance between erosion and deposition, leaving a constant width trough with levees. We then show, by combining Pouliquen \& Forterre (2002)'s friction law with Gray \& Edwards (2014)'s depth-averaged $\mu(I)$-rheology, that it is possible to develop a simple 2D shallow water-like avalanche model that qualitatively captures all of the experimental behaviours. Hence this model may have important practical implications for modeling the initiation, growth and decay of snow avalanches for hazard risk assessment.   

Manifestation of Particle Morphology on the Mechanical Response of a Granular Ensemble

We present the effect of particle morphology (grain shape) on the mechanical response of granular materials at an ensemble level. We chose two model systems with extreme differences in morphology, i.e. spherical glass ballotini and angular sand in our experimental programme. We conducted a series of continuum elemental tests under these model materials reconstituted to the same packing. We arrive at the failure locus on the octahedral plane experimentally for these two systems. The ballotini shows increased dilation at the outset of the test, however, at large strains, the particle rearrangement in the angular sand and the increased interlocking leads to higher strength. The effect of individual particle morphology is manifested in both the increased friction angle and a larger sized failure locus in stress space with increase in angularity. The stresses developed in these two model materials are also accompanied by intriguing volume change behaviour. The glass ballotini despite a lower strength presents a predominantly dilative response while the angular sand shows showing a contractive response. Such an ensemble manifestation of individual particle morphology is useful in interpreting the extensive DEM simulations that are available in literature.   

Brittle to ductile transition in a model of sheared granular materials

Understanding the fundamental mechanisms of deformation and failure in sheared fault gouge is critical for the development of physics-based earthquake rupture simulations that are becoming an essential ingredient in next generation hazard and risk models. To that end, we use the shear transformation zone (STZ) theory, a non-equilibrium statistical thermodynamics framework to describe viscoplastic deformation and localization in gouge materials as a first step towards developing multiscale models for earthquake source processes that are informed by high-resolution fault zone physics. We will describe an implementation of this theory in a 2D/3D finite element framework, accounting for finite deformation, under both axial and shear loading and for dry and saturated conditions. We examine conditions under which a localized shear band may form and show that the initial value of disorder plays an important role. In particular, our simulations suggest that if the material is more compact initially, the behavior is more brittle and the plastic deformation localizes with large strength drop. On the other hand, an initially loose material will show a more ductile response and the plastic deformations will be distributed more broadly. We will further show that incorporation of pore fluids alters the localization pattern and changes the stress slip response due to coupling between gouge volume changes (compaction and dilation) and pore pressure build up. Finally, we discuss the implications of our model for gouge friction and dynamic weakening.   

Tensor Random Fields in Continuum Physics

We discuss the basic properties of tensor random fields (TRFs) of statistically homogeneous kind, with focus on isotropic correlation functions with generally anisotropic realizations. We give explicit representations of TRFs of 2nd, 3rd, and 4th ranks [1-3]. We also find the corresponding spectral expansions. Next, we examine the consequences dictated by field equations on TRFs of displacement, stress and strain in classical continua and, similarly, for temperature and heat flux in conductivity. Then, we report analogous consequences for TRFs of rotation, curvature-torsion, and couple-stress in stochastic micropolar theories [4]. 1. A. Malyarenko and M. Ostoja-Starzewski, Statistically isotropic tensor random fields: Correlation structures, MEMOCS 2(2), 209-231, 2014. 2. A. Malyarenko and M. Ostoja-Starzewski, Spectral expansions of homogeneous and isotropic tensor-valued random fields, ZAMP 67(3), paper 59, 2016. 3. A. Malyarenko and M. Ostoja-Starzewski, A random field formulation of Hooke's law in all elasticity classes, arXiv:1602.09066 4. M. Ostoja-Starzewski, L. Shen and A. Malyarenko, Tensor random fields in conductivity and classical or microcontinuum theories, Math. Mech. Solids 20(4), 418-432, 2015.   

Improved particle simulation with rigid elasticity?

There are two primary methods for simulating the dynamic contact interactions between discrete bodies: the discrete element method (DEM) which treats the particles as elastic, and contact dynamics (CD), which assumes rigid particles. CD offers stability at much larger timesteps than DEM. However, it gives rise to indeterminate forces in highly coordinated packings, as the forces are not derived from physical interaction laws. We propose a method to resolve the indeterminacy by imposing an elastic compatibility condition to the contact forces while retaining the stability of CD. We verify the uniqueness of the force solutions of our method by demonstrating cases where traditional CD cannot recover the DEM force distribution.   

Size-dependence of the flow threshold in dense granular materials

The flow threshold in dense granular materials is typically modeled by a local criterion, involving only a quantity given through the stress - typically the ratio of the shear stress to the pressure. However, nonlocal effects lead to phenomena that cannot be captured with such local criteria. In a widely studied example, flows of thin layers of grains down an inclined surface exhibit a size effect whereby thinner layers require more tilt to flow, and hence, sufficiently thin layers will not flow, even when the stress in the layer exceeds the flow threshold. In this talk, we consider whether the size-dependence of the flow threshold observed in inclined plane flow is configurationally general. Specifically, we consider two additional examples of inhomogeneous flow - planar shear with gravity and vertical chute flow - using two-dimensional discrete element method (DEM) calculations and show that the flow threshold is indeed size-dependent in these flow configurations, displaying additional strengthening as the system size is reduced. We then show that the nonlocal granular fluidity model - a recently-proposed, nonlocal continuum model for dense granular flow - is capable of quantitatively capturing the observed size-dependent strengthening of thin granular bodies in all flow configurations.   

Well-posed continuum equations for granular flow with compressibility and $\mu(I)$-rheology

Continuum modelling of granular flow has been plagued with the issue of ill-posed equations for a long time. Equations for incompressible, two-dimensional flow based on the Coulomb friction law are ill-posed regardless of the deformation, whereas the rate-dependent $\mu(I)$-rheology is ill-posed when the non-dimensional strain-rate $I$ is too high or too low. Here, incorporating ideas from Critical-State Soil Mechanics, we derive conditions for well-posedness of PDEs that combine compressibility with $I$-dependent rheology. When the $I$-dependence comes from a specific friction coefficient $\mu(I)$, our results show that, with compressibility, the equations are well-posed for all deformation rates provided that $\mu(I)$ satisfies certain minimal, physically natural, inequalities.   

Experimental Tests of Nonlocal Rheology in Dense Granular Flows

Several nonlocal granular rheology models have been proposed to address shortcomings in local rheology models. One such model, developed by Kamrin $\&$ Koval, is based on extending a local Bagnold-type granular flow law by including a Laplacian term that accounts for the grain size and cooperative effects. We perform experiments to test this model in a quasi-2D annular shear geometry with a fixed outer wall and a rotating inner wall. We obtain the speed profile by particle tracking. We measure the inner wall torque, and calculate the pressure and shear stress on the outer wall using deformable laser-cut leaf springs. This allows us to calculate the relationship between the stress ratio $\mu$ and the inertial number $I$ at different inner wall rotation speeds and packing fractions. The results are compared with nonlocal models.   

Applicability of Resistive Force Theory in design optimization for locomotion in granular beds

Recent developments in the field of Resistive Force hypotheses for granular media has inspired for the discovery of a general dimensionless form for granular locomotion, which instructs how to scale various dimensions and parameters for predicting associated dynamic outputs related to motion of various locomotors in given granular media beds. Scalings are experimentally confirmed with wheel pairs of various shapes with varied dimensions and driving conditions, by measuring the corresponding outputs in various non-cohesive granular media beds. A newly developed class of mutable wheels called the Franken wheel has also been experimented to optimize the performance of flapped wheels locomotors. The experimental results are matched with numerical computations developed using RFT hypothesis which has recently been also revealed to be a special case of local frictional yielding in Columbic plasticity.   

How sand grains stop a high speed intruder

When a speeding intruder impacts on a granular material, it comes rapidly to rest after penetrating only a modest distance. Empirical dynamical models, dating to the 19th century (if not earlier), describe the drag on the intruder in terms of two types of depth-dependent forces: one a static force, which also includes gravity, and the other a collisional force proportional to the square of the instantaneous speed of the intruder. What processes occur in the material to so quickly decelerate the intruder? We address this question through experiments and simulations (work of Lou Kondic and collaborators). We first probe the granular response using quasi-two-dimensional granular materials consisting of photoelastic discs. When such a particle experiences a force, it appears bright under cross-polarized illumination. High speed video reveals dynamic force transmission into the material along force chains that form in response to the intruder motion. These chains are nearly normal to the intruder surface, implying that collisional rather than frictional forces dominate the momentum transfer from intruder to grains. These observations allow the formation of a collision-based model that correctly captures the collisional drag force for both 2D and 3D intruders of a variety of shapes. This talk will develop a collisional picture of impact, and also explore the change in the system response as the impact speed increases. Experimental collaborators include Abe Clark, Cacey Stevens Bester, and Alec Petersen.   

Fast Wheels in Loose Granular Media

We experimentally investigate the mechanics of single wheeled locomotion in loosely packed granular media. A recent study [Slonaker et. al., arXiv:1604.02490] has demonstrated that granular Resistive Force Theory (RFT) [Li et al, Science 2013; Askari & Kamrin, Nature Materials, 2016] successfully models the movement of slowly rotating wheels in the quasistatic limit, predicting that slip (defined as $S = \frac{\omega r - v}{v}$, where $v$ is translational velocity, $r$ is the wheel’s radius, and $\omega$ is its angular velocity ) is $\approx 1.2$ and insensitive to rotational speed. To test if RFT applies at increased $\omega$, we study a 16 cm diameter wheel in a granular bed of loosely packed poppy seeds (~1 mm diam.) driven at constant $\omega$ from 15 to 360 degrees per second. As in previous work, for low $\omega$, $S$ is constant. However, as $\omega$ increases beyond 90 degrees per second, $S$ increases as the wheel loses traction with the granular media. At the highest $\omega$, the wheel fails to move due to a rapid excavation of material underneath. Our results suggest that granular RFT may have to be modified for new effects at higher $\omega$.   

Acoustic probing of a ball sinking in weakly vibrated dense granular suspensions

A convenient method to determine the viscosity of a fluid is to drop a high density ball in it. The ball will first accelerate before it reaches a terminal velocity related to the fluid viscosity. Instead, in a yield stress fluid like a dense granular suspension, the ball will stop sinking at a certain depth due to friction between solid particles. The jamming phase diagram provides a general framework to explain such a transition from a liquid-like state to a solid-like state as a function of packing density and applied shear. However, understanding an intruder sinking in quicksands remains a conceptual and practical challenge. Here, we develop an ultrasound probing to investigate the dynamics of a steel ball sinking in 3D opaque dense granular suspensions under weak vibration. We show that the ball motion in a vibrated granular suspension is consistent with the frictional rheology mu(J) with J the viscous number. Our main finding is that the extracted static friction and viscous coefficients decrease with increasing the vibration intensity, due to vibration-induced contact sliding between particles without significant packing density change. Additionally, we find that these rheological parameters depend on the size of the probing ball, suggesting thus a non-local rheology.