Session X13: Focus Session: Continuum Description of Particulate Media

A nonlocal enhancement to granular elasto-plasticity

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 of the characteristic 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 with a nonlocal correction term that modifies the continuum law when the inertial number drops below a critical value. The correction depends on stress and strain-rate gradients and brings in a natural dependence on the particle diameter. We implement the modified law in multiple geometries and validate its predictions against discrete particle simulations.   

Effective temperature in elastoplasticity of amorphous solids

An effective temperature $T_{\rm eff}$ which differs from the bath temperature is believed to play an essential role in the theory of elasto-plasticity of amorphous solids. The definition of a measurable $T_{\rm eff}$ in the literature on sheared solids suffers however from being connected to a fluctuation-dissipation theorem which is correct only in equilibrium. Here we introduce a natural definition of $T_{\rm eff}$ based on measurable structural features without recourse to any questionable assumption. The value of $T_{\rm eff}$ is connected, using theory and scaling concepts, to the flow stress and the mean energy that characterize the elasto-plastic flow.   

Shock driven jamming and periodic fracture of particulate rafts

A tenuous monolayer of hydrophobic particles at the air-water interface often forms a scum or raft. When such a monolayer is disturbed by localized surfactant introduction, a radially divergent shock emanates and packs the particles into a jammed, compact, annular band that grows with time. The resulting two-dimensional, disordered, elastic solid locally has a packing fraction that saturates at random close packed density ($\phi_{RCP}$) and fractures as it is driven radially outwards, to form periodic triangular cracks with robust geometrical features. We find that the number of cracks $N$ varies monotonically with the initial particulate packing fraction $\phi_{init}$, as does the compaction band radius $R^*$ at fracture onset. However, its width $W^*$ is constant across all $\phi_{init}$. A simple geometric theory that treats the compaction band as an elastic annulus, and accounts for mass conservation allows us to deduce that $N \simeq 2\pi R^*/W^* \simeq 4\pi \phi_{RCP}/\phi_{init}$, a result that we experimentally verify over the range ($0.1 \le \phi_{init} \le 0.64$).   

Application of Classical Nucleation Theory to Cavitation in Metallic Glass

In order to predict the fracture toughness of amorphous solids such as metallic glasses it is necessary to understand the physics of the process zone. Theories of plastic deformation provide information about response to shear, but on their own these theories provide limited insight into the microscopic mechanisms that mediate the free surface generation critical to crack propagation. Previous molecular dynamics simulations indicate that cavitation likely plays this role. We have undertaken a series of molecular dynamics simulations of cavitation under hydrostatic tension in a binary metallic glass analog using pair-wise potentials. We compare the rate of cavity nucleation directly to homogeneous nucleation theory to examine the role of surface energy and irreversible deformation in the cavitation process. We find that both the reduction of the surface energy at small cavity size and the plastic deformation required for the cavity to grow play important roles in setting the strain-dependent free energy barrier to cavitation.\\[4pt] Work done in collaboration with Michael Spector, Materials Science and Engineering, Johns Hopkins University, Baltimore; Shuo Lu, Materials Science and Engineering, Beihang University; and Pavan K. Valavala, Materials Science and Engineering, Johns Hopkins University.   

Macroscopic transport and topological transitions in ordered suspensions in parallel-wall channels

Our recent investigations of ordered suspensions in parallel-wall channels revealed complex nonlinear dynamics, including formation of defects in a particle lattice, dynamic order-disorder transitions, buckling of particle lattice, and fingering instabilities. We will describe hydrodynamic mechanisms that govern this collective particle behavior.   

Continuum Mean-Field Theories for Molecular Fluids, and Their Validity at the Nanoscale

We present a calculation of the physical properties of solid triglyceride particles dispersed in an oil phase, using atomic- scale molecular dynamics. Significant equilibrium density oscillations in the oil appear when the interparticle distance, $d$, becomes sufficiently small, with a global minimum in the free energy found at $d \approx$ 1.4 nm. We compare the simulation values of the Hamaker coefficient with those of models which assume that the oil is a homogeneous continuum: (i) Lifshitz theory, (ii) the Fractal Model, and (iii) a Lennard-Jones 6-12 potential model. The last-named yields a minimum in the free energy at $d \approx$ 0.26 nm. We conclude that, at the nanoscale, continuum Lifshitz theory and other continuum mean-field theories based on the assumption of homogeneous fluid density can lead to erroneous conclusions.   

Coarse-Graining of a Physical Granular System

We present results, including particle displacements and rotations, as well as strain and stress fields, obtained by applying a resolution-controlled coarse-graining method to an experiment comprised of bidisperse disks subject to pure shear. We briefly review the experimental methods which involve determing inter-particle contact forces using the photoelastic properties of the disks. We then consider the philosophical and technical approaches of the coarse-graining methods used here. We particularly consider the emergence of shear bands, which are manifest in the displacements, rotations, and some strain fields, but not in the stress. Correlations of the displacement fluctuations decay on a very small scale, of the order of a few particle diameters, even close to jamming, which in this case, is induced by shear. We report an unexpected but simple correlation between particle rotation angles and the rotation field.   

Nonequilibrium Thermodynamics of Driven Disordered Materials

We present a nonequilibrium thermodynamic framework for describing the dynamics of driven disordered solids (noncrystalline solids near and below their glass temperature, soft glassy materials such as colloidal suspensions and heavily dislocated polycrystalline solids). A central idea in our approach is that the set of mechanically stable configurations, i.e. the part of the system that is described by inherent structures, evolves slowly as compared to thermal vibrations and is characterized by an effective disorder temperature. Our thermodynamics-motivated equations of motion for the flow of energy and entropy are supplemented by coarse-grained internal variables that carry information about the relevant microscopic physics. Applications of this framework to amorphous visco-plasticity (Shear-Transformation-Zone theory), glassy memory effects (the Kovacs effect) and dislocation-mediated polycrystalline plasticity will be briefly discussed.   

Breakdown of Granular Constitutive Relations for Flow through a Narrow Vertical Channel

We have used NMR/MRI techniques to study flow profiles and fluctuations in the dense, gravity-driven flow of a granular medium through a relatively narrow vertical channel (channel diameter approximately 20 grain diameters). Although the flow is macroscopically steady, the NMR experiments reveal large velocity fluctuations that can be characterized as random, short-lived jamming of the flow. Constitutive relations have been successfully developed for granular shear flows in constant pressure conditions, such as flow in open chutes or wide vertical channels. For the narrow-channel flow probed in our experiments, the constant-pressure constitutive relations are not appropriate. An alternative equation of state based on constant-volume conditions may be appropriate for the narrow-channel case, or it can be modeled using a piling-jamming model that abandons the constitutive-equation approach altogether.   

Dynamic Structure Factor and Transport Coefficients of a Homogeneously Driven Granular Fluid in Steady State

We study the dynamic structure factor of a granular fluid of hard spheres, driven into a stationary nonequilibrium state by balancing the energy loss due to inelastic collisions with the energy input due to driving. The driving is chosen to conserve momentum, so that fluctuating hydrodynamics predicts the existence of sound modes. We present results of computer simulations which are based on an event driven algorithm. The dynamic structure factor $F(q,\omega)$ is determined for volume fractions 0.05, 0.1 and 0.2 and coefficients of normal restitution 0.8 and 0.9. We observe sound waves, and compare our results for $F(q,\omega)$ with the predictions of generalized fluctuating hydrodynamics which takes into account that temperature fluctuations decay either diffusively or with a finite relaxation rate, depending on wave number and inelasticity. We determine the speed of sound and the transport coefficients and compare them to the results of kinetic theory.   

Discrete-continuum mapping for fiber network mechanics

Semi-flexible random fiber networks are the structural element of many biological and non-biological systems such as the cytoskeleton, artificial tissue and cellulose-based products. We have shown that in these systems the density, as well as mechanical fields (elastic moduli, strain energy etc), are long-range power-law correlated. The correlation length evolves during deformation. A procedure to map the elasticity of the discrete system to continuum representations is developed. The method is used to solve boundary value problems defined over large fiber network domains. However, the mapping can be performed only in some situations, limitations which are discussed in this talk.