Session H53: Focus Session: Continuum Descriptions of Discrete Materials

Failure of a loose packing of grains

One-sided repulsive interactions and history-dependent friction forces can cause the mechanics of real granular systems to deviate significantly from that of cohesive solids. Marked deviations from elastic behavior can be seen in the mechanical response and structure of sedimented loose packings of frictional spheres even for very delicate perturbations. In our experiments, particles' displacements are observed with 3D fluorescent imaging as a shear plane is displaced through the packing. As anticipated, we find the shear force is approximately linear with the displacement between discrete yielding events. However, even in this apparently linear region, structural aging continues to occur for the smallest displacements that we can apply. This suggests the inaccessibility of a reversible shear deformation regime in loose packings of granular material. We present a spatial characterization of the particle motion that distinguishes between these continuous instances of irreversibility and the larger discrete failure events.   

Jamming in Hopper Flows: Analysis of Survival Times

Many granular systems experience a transition from a fluid-like state to a solid-like state characterized by a sudden arrest in dynamics, or ``jamming.'' Recent experiments by the Behringer Group at Duke University suggest a probabilistic model of jamming in hopper flows. We will show the results of numerical simulations of dense, gravity-driven, granular flows in a two-dimensional hopper with a tapered outlet [\emph{PRE} \textbf{79}, 011303 (2009)]. We will present results for the statistics of mass flow at the outlet, and the probability of survival without a jam. We will correlate the survival times with velocity and density distributions near the hopper opening.   

Experimental analysis on how grain properties affect the performance of jammed granular systems for variable stiffness robotic applications

Jamming of granular media has become increasingly utilized as a variable stiffness mechanism for industrial and robotic applications. The goal of our work is to better understand how grain properties affect jamming so that granular systems can be designed to fulfill the requirements of a given application. We have primarily focused on experimental studies to analyze how certain grain properties---such as shape, surface roughness, size distribution, and shape distribution---affect the performance of granular systems. Potential applications that utilize jamming would typically require that a contained granular system transition between effective solid states (e.g., when particular shape or strength needs to be maintained) and effective liquid states (e.g., when the system needs to be compliant such that it can be shaped or actuated by its environment). Therefore, we are interested in quantifying 1) the strength of compacted granular systems in their effective solid states and 2) the ``ease of flow'' and compliance of granular systems in their effective liquid states.   

Shear-Transformation-Zone Theory of Glassy Diffusion, Stretched Exponentials, and the Stokes-Einstein Relation

The success of the shear-transformation-zone (STZ) theory in accounting for broadly peaked, frequency-dependent, glassy viscoelastic response functions is based on the theory's first-principles prediction of a wide range of internal STZ transition rates. Here, I propose that the STZ's are the dynamic heterogeneities frequently invoked to explain Stokes-Einstein violations and stretched-exponential relaxation in glass-forming materials. I find that, to be consistent with observations of Fickian diffusion near $T_g$, an STZ-based diffusion theory must include cascades of correlated events, but that the temperature dependence of the Stokes-Einstein ratio is determined by an STZ-induced enhancement of the viscosity. Stretched-exponential relaxation of density fluctuations emerges from the same distribution of STZ transition rates that predicts the viscoelastic behavior.   

A granular flow law based on nonlocal fluidity

A general, three-dimensional law to predict granular flow in an arbitrary geometry has been an elusive goal for decades. Recently, an elasto-plastic continuum model has shown the ability to approximate steady flow and stress profiles in multiple inhomogeneous flow environments. However, the model does not capture some phenomena observed in the slow, creeping flow regime. As normalized flow-rate decreases, granular stresses are observed to become largely rate-independent and a dominating length-scale emerges in the mechanics. This talk attempts to account for these effects using the notion of nonlocal fluidity, which has proven successful in treating nonlocal effects in emulsions. The idea is to augment the usual granular fluidity law with a diffusive second-order term scaled by the particle size that spreads flowing zones accordingly. Below the yield stress, the local contribution vanishes and the fluidity becomes rate-independent, as we require. We implement the modified law in multiple geometries and validate its predictions for velocity, shear-rate, and stress against discrete particle simulations.   

Rheology of discontinuously shear thickening suspensions beyond simple shear

The behavior of discontinuously shear thickening suspensions in flows other than simple shear is not well understood, in part due to unresolved experimental challenges. For example, such suspensions thicken most easily close to rigid boundaries due to the no-slip condition. This makes experiments highly dependent on the shape and size of the container used. We show that by placing a lubricating layer of oil between the suspension and the container we can generate flows where thickening is nearly independent of rigid boundaries. This method is particularly useful in creating quasi one- and two-dimensional flows, which can be easily visualized. We use this method to conduct capillary breakup experiments with thickening suspensions of silica and cornstarch particles, in which we observe the formation of beads-on-a-string morphologies with multiple satellite and sub-satellite bead generations, similar to the morphologies observed in breakup of viscoelastic fluids. Using a one-dimensional continuum model, we show how nonlinear rheology of thickening suspensions results in the creation of these complex morphologies.   

Activated processes in a stress landscape: a rheological model for dense driven granular materials

We model (Phil. Trans. R. Soc. A 2009 {\textbf {367}}) the rheological behavior of granular materials based on a stress-based statistical ensemble and the Soft Glassy Rheology framework (SGR). It takes into account the disordered nature of granular packings and the metastability of jammed states, as well as spatial heterogeneity and intermittency. In this model, mesoscopic subregions of a driven granular material undergo activated processes in a {\it stress} landscape with a broad distribution of barrier heights. Due to the athermal nature of granular materials, the activated processes are induced not by the thermodynamic temperature, but by a temperature-like quantity which is a measure of the fluctuation of stresses. Results and predictions of the model have been successfully applied to analyze experiments in a Couette geometry. We will discuss applications of the stress-activated framework in, for example, recent experiments that study non-local rheology in dense flows (PRL \textbf{106} 108301(2011)).   

Shear-transformation-zone theory of plasticity in hard-sphere materials

The dynamics of sheared, dense granular materials exhibits features, such as a dynamic yield stress and a glass transition, similar to those of other amorphous solids. However, strictly granular, hard-sphere systems fundamentally differ from traditional glassy and colloidal systems because at microscopic scales their dynamics and interaction energies are insensitive to thermal temperature. In this talk we present a theory of plasticity for sheared, granular materials that combines Shear Transformation Zone (STZ) theory with Edwards' statistical theory of granular materials. We find that the dynamics of a strictly granular system, like other amorphous solids, can be captured statistically in terms of entropic mechanisms. In our analysis of sheared hard spheres, the volume $V$ replaces the energy $E$ as a function of entropy $S$ in conventional statistical mechanics. In place of an effective disorder temperature, a central feature of the STZ theory for traditional glassy systems, the compactivity $X = \partial V / \partial S$ characterizes the internal state. We derive the STZ equations of motion for a granular material accordingly, and predict its macroscopic properties such as shear viscosity and macroscopic frictional coefficient under different shearing conditions.   

Width of Shear Zones in Gravity-Driven Granular Flows

Gravity-driven granular flow in a vertical hopper exhibits a flow profile that consists of a plug near the center and a shear zone near the boundary walls. It has been observed that the width of the shear zone is a few particle diameters and is independent of the channel width, however, the mechanism by which the width is selected remains unclear. Using event-driven simulations of granular flow in a two-dimensional hopper, we investigate the width of the shear zone as a function of the channel width and the boundary conditions at the wall. We focus on the role played by fluctuations in the stress as the source of activated slips near the wall as a candidate mechanism for the shear zone.   

From streamline jumping to strange eigenmodes: Learning from simple continuum models of granular mixing

Simple continuum models of granular flow can provide fundamental insight into how and why granular materials mix. Though a similar kinematic framework can be used to study both fluid and granular mixing, there are striking differences that we explore through a computational--experimental study of granular flow in a slowly rotating quasi-two-dimensional polygonal container. In the Lagrangian frame, for small numbers of revolutions, we show that the mixing pattern is captured by a model termed ``streamline jumping.'' This minimal model, arising at the limit of a vanishingly-thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories that resemble chaos. In the Eulerian frame, meanwhile, we show the presence of naturally-persistent granular mixing patterns (``strange'' eigenmodes) for intermediate numbers of revolutions. Unlike fluid mixing, however, strong diffusive effects (due to particle collisions in granular flows) result in fast decay of these transient patterns in monodisperse mixtures. Meanwhile, segregation leads to permanent excitation of eigenmodes in bidisperse mixtures.   

From liquid to stone: a polydispersive colloidal model for cement setting

The main binding phase of cement is the nano-porous C-S-H gel, (calcium-silicate-hydrate). Here we investigate the temporal evolution of the mechanical properties of cement across the rigidity transition from liquid paste to solid stone, due to the precipitation of C-S-H. This transition is named ``setting'' and occurs several hours after mixing cement powder with water. We present a numerical model of random insertion of colloidal C-S-H nano-particles in the Monte Carlo framework. The particles interact with each other according to a generalized Lennard-Jones potential. Depending on the particle size polidispersity, the packing fraction of the final assemblies ranges between 0.64 and 0.73. The mechanical properties of the assemblies indicate that the packing fraction is the key parameter for continuum mechanics models at larger scales. In fact, both the stiffness and the strength of the assemblies increase with the packing fraction, while the ductility decreases. On the other hand, the evaluation of the specific surface requires an additional parameter fixing the length scale, for example, the characteristic size of the nano-pores. We finally show the relevance of our results for cement setting with a simple semi-analytical model of micro-pore filling.   

Effect of randomness on wave propagation in granular systems

Granular systems have been shown to possess energy absorbing and potential wave mitigation characteristics due to the flexibility in tuning the properties of the particles. The present study focuses on the impact of randomness on wave propagation in 1D and 2D lattices of spherical particles where the randomness is associated with either the mass, Young's modulus or radius of the spheres. The 1D study (motivated by M. Manciu et. al. (2001)) reveals the presence of two distinct regimes of decay in peak compressive force with distance for any level of randomness. The transition between the regimes of exponential and power-law decay is shown to occur when the amplitude of the leading pulse reduces below that of the scatter. Investigation into the ensemble kinetic and potential energies of the system as a function of time shows the gradual transfer of energy from potential to kinetic with increase in the level of randomness. In 2D square packed systems simulated with a modified version of the molecular dynamics package LAMMPS, we note that the decay in peak compressive force is present due to dimensionality as well as randomness. Normalization is then used to quantify the decay due to randomness alone and we investigate the anisotropy of the randomness induced decay.   

Continuum Representation of the Mechanics of Random Fiber Networks

Solving boundary value problems over large domains of random fiber networks is important in the design of fiber-based engineering materials and in the understanding of the biophysics of biological materials. In most of these applications, systems contain a very large number of fibers, which renders nearly impossible solving boundary value problems while resolving every fiber in the problem domain. Therefore, developing a continuum model for the discrete system is desirable. This presentation focuses on conditions under which this mapping can be performed and the minimum size of the problem beyond which the continuum representation is valid. Random fiber networks are highly heterogeneous and exhibit non-affine deformation with correlated fields at different observation length scales. The scale of transition from the discrete to the continuum model must be large enough to capture all the statistically independent subdomains of the network. This scale cannot be determined exclusively based on geometric considerations (e.g. based on fiber density). These considerations, along with a constitutive model for the small deformation of continuum models of fiber networks are discussed in this talk.