Session J30: Focus Session: Continuum Descriptions of Discrete Materials

Homogenized Mechanical Behavior of Cross-Linked Fiber Networks Embedded in Matrix

Most biological and some biomimetic materials are made from fiber networks embedded in an elastic medium. The mechanical behavior of these composites depends in interesting ways on the elasticity of the matrix. In this work we study this issue using both 2D and 3D models, with the goal of deriving expressions linking microstructural parameters and the composite elastic properties. We show that the strong interaction between network and matrix precludes the use of linear superposition of effects and that the effective moduli are a complex function of the constituent moduli. The internal distribution of stresses is also studied and discussed in relation with failure mechanisms.   

Size Effects in the Mechanical Behavior of Sparsely Cross-Linked Fiber Networks

Random fiber networks are structural elements in many biological and man-made materials and the prediction of their mechanical properties is desirable in many applications. In this work we first address the problem of the scale of homogeneity of these discrete systems, i.e. the size of the model above which the elastic response is model size- independent. Further, using models large enough to eliminate the size effect, we determine a structure-property relation for networks with variable concentration of cross-links.   

Role of Inhomogeneity in Mechanochemically Active Polymers

Mechanically-induced reactivity is a promising means for designing self sensing and autonomous materials. Mechanically sensitive chemical groups termed mechanophores can be covalently linked into polymers in order to trigger specific chemical reactions upon mechanical loading. The mechanophore reaction kinetics, as determined by ab initio steered molecular dynamics, are exponential in force. As such the mechanochemical behavior of a solid-state polymer is highly sensitive to stress carried by that polymer, including local spatial and temporal fluctuations. Previously we developed microstructurally-based continuum models for fluorescence response in spiropyran-linked rubbery (poly methacrylate) and glassy (poly methylmethacrylate) polymers. The homogenization scheme in each relied on assigning mean effective forces acting on the mechanophores. Here we explore the theoretical influence of nanoscale spatial force distributions and fast temporal force fluctuations on the mechanochromic response of these systems. The effect of each is found to be significant and highly dependent on the intrinsic polymer mechanical behavior.   

Marginal Matter

All around us, things are falling apart. The foam on our cappuccinos appears solid, but gentle stirring irreversibly changes its shape. Skin, a biological fiber network, is firm when you pinch it, but soft under light touch. Sand mimics a solid when we walk on the beach but a liquid when we pour it out of our shoes. Crucially, a marginal point separates the rigid or jammed state from the mechanical vacuum (freely flowing) state - at their marginal points, soft materials are neither solid nor liquid. Here I will show how the marginal point gives birth to a third sector of soft matter physics: intrinsically nonlinear mechanics. I will illustrate this with shock waves in weakly compressed granular media, the nonlinear rheology of foams, and the nonlinear mechanics of weakly connected elastic networks.   

Capturing nonlocal effects in 2D granular flows

There is an industrial need, and a scientific desire, to produce a continuum model that can predict the flow of dense granular matter in an arbitrary geometry. A viscoplastic continuum approach, developed over recent years, has shown some ability to approximate steady flow and stress profiles in multiple inhomogeneous flow environments. However, the model incorrectly represents 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, in the simplified case of 2D, using the notion of nonlocal fluidity, which has proven successful in treating nonlocal effects in emulsions. The idea is to augment the local granular fluidity law with a diffusive second-order term scaled by the particle size, which 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 flow and stress predictions in multiple geometries compared against discrete particle simulations. In so doing, we demonstrate that the nonlocal relation proposed is satisfied universally in a seemingly geometry-independent fashion.   

Predicting dense granular flows: continuum modeling with a length-scale

Dense granular materials display a complicated set of flow properties, which differentiate them from ordinary fluids. In particular, slowly-flowing granular media form clear, experimentally-robust features; most notably, shear bands, which can have a variety of possible widths and which decay non-trivially into the surrounding quasi-rigid material. Despite the ubiquity of granular flows, no model has been developed that captures or predicts these complexities, posing an obstacle in industrial and geophysical applications. We present a three-dimensional constitutive model for well-developed, dense granular flows aimed at filling this need. The key ingredient of the theory is a grain-size-dependent nonlocal rheology -- inspired by efforts for emulsions -- in which flow at a point is affected by both the local stress as well as the flow in neighboring material. With a single new material parameter, we show that the model is able to quantitatively describe dense granular flows in an array of different geometries. Of particular importance, it is the first model to pass the stringent test of capturing all aspects of the highly-nontrivial flows observed in split-bottom cells -- a geometry that has resisted modeling efforts for nearly a decade.   

Changes in fluctuation patterns of a granular hopper flow near jamming

Jams in gravity-driven flows in a vertical hopper with rigid walls occur under extremely inhomogeneous conditions, distinct from what is observed in spatially homogeneous flows. In this work, we use event-driven simulations to study velocity fluctuations in a collisional, 2D gravity-driven flow near jamming. We find a heterogeneous spatial distribution of velocity autocorrelation relaxation times, with the spatial structure changing significantly as the flow approaches jamming. At high flow rates, the flow at the center has lower kinetic temperatures and longer autocorrelation times than at the boundary. Unexpectedly, however, this trend reverses itself as the flow rate slows, with fluctuations relaxing more slowly at the boundaries though the kinetic temperatures remain high in that region. We suggest that this behavior is an indication of the flow becoming glassy close to the boundaries as jamming is approached.   

Interplay between packing and flow in the shear zone at the wall of a granular hopper flow

A granular medium flowing through a vertical channel has a flat velocity profile in the bulk with a shear zone at the wall. The size of the shear zone and the dependence on flow parameters is poorly understood. To address this issue we image the flow of spherical steel spheres under gravity in a vertical, straight-walled 2-dimensional hopper, where the flow velocity is controlled by a taper at the outlet. ~Our measurements focus on the role of microstructure in controlling the shear zone. ~We have found that the size of this zone is larger in bidisperse, disordered flow that in monodisperse, nearly-crystalline flows. We report the effect of packing as quantified by local dilation, as a function of flow rate for systems of both bidisperse and monodisperse grains.   

Boundary layer model for intruder drag

We propose a boundary layer model for drag on vertical intruders with uniform cross-sections in granular beds. The drag is the surface integral of the stress over a monolayer of particles, where the stress has a simple dependence on depth beneath the surface and angle of the surface normal relative to the direction of flow. This model is in good experimental agreement, accounts for the scale effect and the associated force focusing observed on edges of intruders.   

Rheology and migration in colloidal and noncolloidal suspensions

Suspensions of solid particles in liquids provide a useful setting for development of continuum description of particle-laden fluids. These mixtures can be made density matched, so that the volume fraction is freely variable, and the rheology can be measured in standard rheometric apparatus. This work will describe the rheology of concentrated suspensions and its implications in continuum description of the bulk flow of the mixture; the development will focus on colloidal suspensions where Brownian motion is relevant, with the limit of strong shear taken to describe noncolloidal suspensions. The normal stress response of these suspensions will be shown to be critical to description of the migration of the particles, leading to strong concentration gradients. The normal stress differences as well as the isotropic normal stress of the particle phase, or nonequilibrium osmotic pressure, will be described and related to these migration phenomena. The implications of the normal stress differences in secondary flow generation will also be described.   

How to predict polydisperse hard-sphere mixture behavior using maximally equivalent tridisperse systems

Polydisperse hard sphere mixtures have equilibrium properties which essentially depend on the number density and a reduced number $K$ of moments of the size distribution function. Such systems are equivalent to other systems with different size distributions if the $K$ moments are matched. In particular, a small number $s$ of components, such that $2s-1=K$ is sufficient to mimic systems with continuous size distributions. For most of the fluid phase $K=3$ moments ($s=2$ components) are enough to define an equivalent system, while in the glassy states one needs $K=5$ moments ($s=3$ components) to achieve good agreement between the polydisperse and its maximally-equivalent tridisperse system. With $K=5$ matched moments they are also close in number- and volume-fractions of rattlers. Finally, also the jamming density of maximally-equivalent jammed packings is very close, where the tiny differences can be explained by the distribution of rattlers.   

Finite element modeling of the dynamic effective mass of granular media

Finite sized granular media have a frequency dependent, complex valued effective mass, characterized by several resonant features. In the vicinity of the corresponding frequencies the associated mass can be several times the static mass. This complicated behavior is due to mechanical interactions between neighboring grains. In contrast we investigate the viability of using a continuum approximation for the mechanical response to model the effective mass. We find that the granular medium is suitably represented by a linear elastic stress-strain relationship with viscous damping. The free parameters in the linear elastic model, the elastic modulus and poisson's ratio, are measured using conventional mechanical testing equipment, and a novel sensor which permits the measurement of lateral stress. Moreover, we characterize the frequency dependent displacement profile on the surface of the granular medium. In this talk we demonstrate that our continuum model is suitable for reproducing the frequency dependent effective mass, and the displacement profile at the resonant frequencies.   

A terradynamics of legged locomotion on granular media

The theories of aero and hydrodynamics form the bases for prediction of animal movement and device design in air and water, and allow computation of lift, drag, and thrust forces on wings and fins. While models of terrestrial legged locomotion have focused on interactions with solid ground, many legged animals (and increasingly robots) move on substrates such as sand, gravel, soil, mud, snow, grass, and leaf litter that flow in response to intrusion. However, locomotor-ground interaction models on such flowable ground are often unavailable. Here we develop a resistive force model that predicts forces on arbitrary-shaped legs and bodies moving freely in granular media in the vertical plane. Our resistive force measurements reveal a complex but generic dependence of stresses on an intruder on its depth, orientation, and movement direction in granular media of different particle size, density, friction, and compaction. Our resistive force model and a multi-body simulation predict a small legged robot's locomotion on granular media using various leg shapes and stride frequencies, and give insight into the effects of leg morphology and kinematics on movement on granular media. Our study is an initial but important step in creation of ``terradynamics'' of locomotion on flowable ground.