Session E34: Continuum Descriptions of Discrete Materials I

Fabric variables in dense sheared suspensions

The rheology of granular flows and dense suspensions can be described in terms of their effective shear and bulk viscosities as a function of packing fraction. Using stress partition and equivalence between frictional and viscous descriptions in the dense state, we show that the effective viscosities can be expressed in terms of the force-network anisotropy. This is supported by our extensive DEM-LBM simulations for a broad range of inertial and viscous parameters.   

Jammed granular cones affect frictional resistive forces at the onset of intrusion

Characterizing the functional form of granular resistive forces has allowed for analysis of the locomotion of animals and robots on and within dry granular media. Resistive force theory (RFT) has been an effective tool in predicting these forces for various locomotive gaits within the ``frictional fluid'' regime, where intrusions are sufficiently slow such that granular inertial effects are negligible. These forces have been typically described by a linear dependence to submersion depth. However, recent experiments on robotic jumping [Aguilar {\&} Goldman, Nature Physics, 2015] have revealed the importance of considering the nonlinear effects at the onset of intrusion to accurately predict robot kinematics. Particle image velocimetry (PIV) analysis of sidewall grain flow during foot intrusion reveals a jammed granular cone that develops beneath the foot at the onset of intrusion. A geometric model of cone development combined with empirical RFT forces on angled conical surfaces was able to predict the non-linear force trajectory vs. depth for experimental intrusions of various foot sizes, suggesting that intruders experience non-linear frictional forces according to the shape of the granular jamming fronts that form at the onset of movement.   

Jammed Clusters and Non-locality in Dense Granular Flows

We investigate the micro-mechanisms underpinning dense granular flow behaviour from a series of DEM simulations of pure shear flows of dry grains. We observe the development of transient clusters of jammed particles within the flow. Typical size of such clusters is found to scale with the inertial number with a power law that is similar to the scaling of shear-rate profile relaxation lengths observed previously\footnote{K. Kamrin and G. Koval, Physical Review Letters 108, (2012); M. Bouzid, et al., Physical Review Letters 111, (2013); P. G. Rognon, et al., Journal of Fluid Mechanics 764, 171 (2015)}. Based on the simple argument that transient clusters of size $\ell$ exist in the dense flow regime, the formulation of steady state condition for non-homogeneous shear flow results in a general non-local relation, which is similar in form to the non-local relation conjectured for soft glassy flows\footnote{Goyon, et al., Nature 454, 84 (2008)}. These findings suggest the formation of jammed clusters to be the key micro-mechanism underpinning non-local behaviour in dense granular flows.   

Velocity profiles and rheology of a granular bed sheared by a fluid flow

We discuss an experimental investigation of motion of a granular bed driven by a laminar fluid flow as a function of applied shear rate. This is a model system to investigate a variety of examples where such a situation arises including wind blowing over sand, sediment transport in rivers, slurries, and turbidity currents. We have developed an experimental apparatus which allows examination of the fluid as well as the grain dynamics both at the surface as well as deep into the bed under steady state conditions with refractive index matching technique. This allows us to obtain both the applied local shear stress by the fluid as well as the local strain rate inside the bed. We find that that the granular flux as a function of depth decays exponentially into the bed. Further, the velocity profile is observed to exhibit a crossover from a regime where particles are fully suspended to where there is bed load transport. We will discuss the observed velocity and density profiles in light of various models of granular suspensions.   

Modeling Shear Banding in Amorphous Solids, from Atomistic to Continuum

Molecular dynamics simulations of strain localization are carried out using different materials systems and interatomic potentials including CuZr modeled via the embedded-atom method (EAM), amorphous Si modeled using Stillinger-Weber (SW) and a binary Lennard-Jones (LJ) system. Quench schedules and strain rates are varied. Different systems exhibit marked similarities in plastic behavior. Systematic differences between systems are analyzed in the context of Shear Transformation Zone (STZ) theory in the effort to develop a generalized constitutive framework for plasticity in glasses. Effective temperature inferred from the potential energy is explored as a local coarse-grained measure of the degree of disorder.   

Identifying shear transformation zones in amorphous solids via a virtual strain method

One outstanding problem in the mechanical response of amorphous solids is the identification of flow defect sites, so called shear transformation zones (STZs), a priori in the structure. Many methods have been utilized in order to predict local STZ sites including short-range order, soft-mode analysis and machine learning. Here we directly probe local regions of the material via shear in order to detect nearby saddle points that can result in transformations. This non-perturbative method gives excellent correlation with global shear of the system. It also provides a means to cross-correlate the existence of such local transition pathways with other proposed diagnostics such as the soft-spot method of Manning and Liu. We use the information gained by this method to consider the coarse-graining necessary to connect atomistic methods to continuum theories.   

Stick-slip instabilities in sheared granular flow: The role of friction and acoustic vibrations

We propose a theory of shear flow in dense granular materials. A key ingredient of the theory is an effective temperature that determines how the material responds to external driving forces such as shear stresses and vibrations. We show that, within our model, friction between grains produces stick-slip behavior at intermediate shear rates, even if the material is rate strengthening at larger rates. In addition, externally generated acoustic vibrations alter the stick-slip amplitude, or suppress stick-slip altogether, depending on the pressure and shear rate. We construct a phase diagram that indicates the parameter regimes for which stick-slip occurs in the presence and absence of acoustic vibrations of a fixed amplitude and frequency. These results connect the microscopic physics to macroscopic dynamics and thus produce useful information about a variety of granular phenomena, including rupture and slip along earthquake faults, the remote triggering of instabilities, and the control of friction in material processing.   

A microstructural description of shear thickening in dense suspensions.

The mechanism of shear thickening in dense suspensions has been recently linked to a transition from a lubricated ``frictionless'' to an unlubricated ``frictional'' rheology. Recent particle simulations have been successful to quantitatively reproduce both the continuous and discontinuous shear thickening as observed experimentally. However, a microstructural description of these suspensions is still lacking, which would aid in understanding and predicting the flow behavior of shear thickening suspensions. To tackle this challenging issue, we explore various microscopic properties, like the inter-particle force distribution, the particle motion correlations, and the anisotropy (in both contact and force network). Further, we also attempt to link the observed rheological behavior observed at the macro scale to mean displacement and fluctuations at the particle scale.   

Deformation in Metallic Glass: Connecting Atoms to Continua

Metallic glasses like other amorphous solids experience strain localization as the primary mode of failure. However, the development of continuum constitutive laws which provide a quantitative description of disorder and mechanical deformation remains an open challenge. Recent progress has shown the necessity of accurately capturing fluctuations in material structure, in particular the statistical changes in potential energy of the atomic constituents during the non-equilibrium process of applied shear. Here we directly cross-compare molecular dynamics shear simulations of a ZrCu glass with continuum shear transformation zone (STZ) theory representations. We present preliminary results for a methodology to coarse-grain detailed molecular dynamics data with the goal of initializing a continuum representation in the STZ theory.   

Size effects and internal length scales in the elasticity of random fiber networks.

Random fiber networks are the structural element of many biological and man-made materials, including connective tissue, various consumer products and packaging materials. In all cases of practical interest the scale at which the material is used and the scale of the fiber diameter or the mean segment length of the network are separated by several orders of magnitude. This precludes solving boundary value problems defined on the scale of the application while resolving every fiber in the system, and mandates the development of continuum equivalent models. To this end, we study the intrinsic geometric and mechanical length scales of the network and the size effect associated with them. We consider both Cauchy and micropolar continuum models and calibrate them based on the discrete network behavior. We develop a method to predict the characteristic length scales of the problem and the minimum size of a representative element of the network based on network structural parameters and on fiber properties.   

Is the microscopic stress computed from molecular simulations in mechanical equilibrium?

The microscopic stress field connects atomistic simulations with the mechanics of materials at the nano-scale through statistical mechanics. However, its definition remains ambiguous. In a recent work [1,2] we showed that this is not only a theoretical problem, but rather that it greatly affects local stress calculations from molecular simulations. We find that popular definitions of the local stress, which are continuously being employed to understand the mechanics of various systems at the nanoscale, violate the continuum statements of mechanical equilibrium. We exemplify these facts in local stress calculations of defective graphene, lipid bilayers, and fibrous proteins. Furthermore, we propose [1,3] a new physical and sound definition of the microscopic stress that satisfies the continuum equations of balance, irrespective of the many-body nature of the inter-atomic potential. Thus, our proposal provides an unambiguous link between discrete-particle models and continuum mechanics at the nanoscale. [1] Torres-Sanchez, A; Vanegas, J. M.; Arroyo, M.; Phys. Rev. Lett. 114, 258102 (2015). [2] Vanegas, J. M.; Torres-Sanchez, A; Arroyo, M.; J. Chem. Theor. Comput., 10 (2), 691–702 (2014). [3] Torres-Sanchez, A; Vanegas, J. M.; Arroyo, M.; Submitted to J. Mech. Phys. Solid   

Extending two-phase theories of soft composite solids to the non-dilute regime

Composite materials are ubiquitous in the natural environment and in engineered materials and hence capture the interest of a wide audience. Eshelby's 1957 theory treats the interaction of macroscopic stress fields with isolated inclusions within an elastic solid, and it has been widely used to treat the mechanics of composite materials. However, due to its neglect of interface stress, which is a particularly key effect in soft materials, the theory breaks down whenever the typical inclusion size $R$ is of order or less than the elastocapillary lengthscale $L$. In this regime, under external stress, the effect of inclusion shape becomes strongly size-dependent. Here, we develop two new non-dilute theories, estimate the elastic moduli of composites comprised of an isotropic, compressible, linear-elastic compliant solid hosting a non-dilute spatially-random distribution of identical liquid droplets. The composite stiffness depends on a single dimensionless parameter $L/R$, and we find significant elastic moduli corrections for inclusions sizes $R$ as large as 100 $L$. By generalizing the exact theory recently developed for the corresponding dilute case, we find that when $R < 3L/2$ ($R=3L/2$) liquid inclusions stiffen (cloak the far-field signature of) of the host solid.   

Continuum modeling of dense granular flow down heaps

Dense, dry granular flows display many manifestations of grain-size dependence, or nonlocality, in which the finite-size of grains has an observable impact on flow phenomenology. Such behaviors make the formulation of an accurate continuum model for dense granular flow particularly difficult, since local continuum models are not equipped to describe size-effects. One example of grain-size dependence is seen when avalanches occur on a granular heap -- a situation which is frequently encountered in industry, as in rotating drums, as well as in nature, such as in landslides. In this case, flow separates into a thin, quickly flowing surface layer and a slowly creeping bulk. While existing local granular flow models are capable of capturing aspects of the flowing surface layer, they fail to even predict the existence of creeping flow beneath, much less being able to quantitatively describe the flow fields. Recently, we have proposed a new, scale-dependent continuum model -- the nonlocal granular fluidity (NGF) model -- that successfully predicted steady, slow granular flow fields, including grain-size-dependent shear-band widths in a variety of flow configurations. In this talk, we extend our model to the rapid flow regime and show that the model is capable of quantitatively predicting all aspects of gravity-driven heap flow. In particular, the model predicts the coexistence of a rapidly flowing, rate-dependent top surface layer and a rate-independent, slowly creeping bulk -- a feature which is beyond local continuum approaches.