Non-affine Motion Research Articles

Nonaffine displacements and the nonlinear response of a strained amorphous solid

We demonstrate that irreversible structural reorganization is not necessary for the observation of yield behavior in an amorphous solid. While the majority of solids strained to their yield point do indeed undergo an irreversible reorganization, we find that a significant fraction of solids exhibits yield via a reversible strain. We also demonstrate that large instantaneous strains in excess of the yield stress can result in complete stress relaxation, a result of the large nonaffine motions driven by the applied strain. The empirical similarity of the dependence of the ratio of stress over strain on the nonaffine mean-square displacement to that for the shear modulus obtained from quiescent liquid at nonzero temperature supports the proposition that rigidity depends on the size of the sampled configurational space only and is insensitive to how this space is sampled.

Nonaffine deformation under compression and decompression of a flow-stabilized solid

Understanding the particle-scale transition from elastic deformation to plastic flow is central to making predictions about the bulk material properties and response of disordered materials. To address this issue, we perform experiments on flow-stabilized solids composed of micron-scale spheres within a microfluidic channel, in a regime where particle inertia is negligible. Each solid heap exists within a stress field imposed by the flow, and we track the positions of particles in response to single impulses of fluid-driven compression or decompression. We find that the resulting deformation field is well-decomposed into an affine field, with a constant strain profile throughout the solid, and a non-affine field. The magnitude of this non-affine response decays with the distance from the free surface in the long-time limit, suggesting that the distance from jamming plays a significant role in controlling the length scale of plastic flow. Finally, we observe that compressive pulses create more rearrangements than decompressive pulses, an effect that we quantify using the statistic for non-affine motion. Unexpectedly, the time scale for the compression response is shorter than for decompression at the same strain (but unequal pressure), providing insight into the coupling between deformation and cage-breaking.

Effect of non-affine motion on the centrifugal instability of circular Couette flow

In the present work, the effect of non-affine motion of the constituents in a complex fluid system (say, a polymeric liquid) is theoretically investigated on the centrifugal instability of circular Couette flow. To achieve this goal, use was made of the linearized Phan-Thien/Tanner (LPTT) model thanks to its allowing non-affine deformation for the polymer strands in a temporary network of junctions through invoking a slip parameter. Knowing the basic-flow velocity and stress fields from the literature, they were subjected to infinitesimally-small, normal-mode perturbations and their time-evolution was monitored using a linear stability analysis for both the axisymmetric and non-axisymmetric modes. An eigenvalue problem was obtained this way which was solved numerically using the pseudo-spectral, Chebyshev-based, collocation method. Based on the results obtained in this work, it is concluded that the non-affine motion can have a stabilizing or destabilizing effect on circular Couette flow depending on the Weissenberg number and the sign/magnitude of the angular velocity ratio.

Microstructural view of burrowing with a bioinspired digging robot.

RoboClam is a burrowing technology inspired by Ensis directus, the Atlantic razor clam. Atlantic razor clams should only be strong enough to dig a few centimeters into the soil, yet they burrow to over 70 cm. The animal uses a clever trick to achieve this: by contracting its body, it agitates and locally fluidizes the soil, reducing the drag and energetic cost of burrowing. RoboClam technology, which is based on the digging mechanics of razor clams, may be valuable for subsea applications that could benefit from efficient burrowing, such as anchoring, mine detonation, and cable laying. We directly visualize the movement of soil grains during the contraction of RoboClam, using a novel index-matching technique along with particle tracking. We show that the size of the failure zone around contracting RoboClam can be theoretically predicted from the substrate and pore fluid properties, provided that the timescale of contraction is sufficiently large. We also show that the nonaffine motions of the grains are a small fraction of the motion within the fluidized zone, affirming the relevance of a continuum model for this system, even though the grain size is comparable to the size of RoboClam.

The role of the Gordon–Schowalter derivative term in the constitutive models—improved flexibility of the modified XPP model

Constitutive models that complete the set of equations describing the flow of polymer melts should respect objective thermodynamics and stability conditions ensuring their validity in the whole range of possible deformation flow. However, in practice, a very good description of flow situations can be achieved with the models not complying with the physical assumptions in all respects. Analogously to the term characterizing yield stress in empirical viscoplastic models, the term represented by the Gordon–Schowalter (GS) derivative in the differential constitutive models contributes to better fitting the experimental data, especially shear thinning. Efficiency of the recently presented modified eXtended Pom-Pom model (just one non-linear parameter per mode) implementing the GS derivative term (one additional non-affine motion parameter per mode) is improved (documented on LDPE, HDPE, and polyvinyl butyral (PVB) materials), and a comparison with the exponential Phan-Tien–Tanner (PTT) and PTT-XPP models (a priori containing the GS derivative term) are presented.

Affine and nonaffine motions in sheared polydisperse emulsions.

We study dense and highly polydisperse emulsions at droplet volume fractions ϕ≥0.65. We apply oscillatory shear and observe droplet motion using confocal microscopy. The presence of droplets with sizes several times the mean size dramatically changes the motion of smaller droplets. Both affine and nonaffine droplet motions are observed, with the more nonaffine motion exhibited by the smaller droplets which are pushed around by the larger droplets. Droplet motions are correlated over length scales from one to four times the mean droplet diameter, with larger length scales corresponding to higher strain amplitudes (up to strains of about 6%).

Fluctuations of particle motion in granular avalanches - from the microscopic to the macroscopic scales.

In this study, we have investigated the fluctuations of particle motion, i.e. the non-affine motion, during the avalanche process, discovering a rich dynamic from the microscopic to the macroscopic scales. We find that there is a strong correlation between the magnitude of the velocity fluctuation and the velocity magnitude in the spatial and temporal domains. The possible connection between this finding and the theory of the shear transformation zones is discussed based on the direct measurement of the T1 events. In addition, the velocity magnitude of the system and the stress fluctuations of the system are strongly temporally correlated. Our finding will pose challenges to the development of more rigorous theories to describe avalanche dynamics based on the microscopic approach. Moreover, our finding presents a plausible mechanism of particle entrainment in a simple system.

Image-Based Lagrangian Analysis of Granular Kinematics

AbstractIn this paper, algorithms to describe mesoscale and microscale kinematics of granular flow using image analysis are described. At the mesoscale, digital image correlation (DIC) is employed to derive displacement fields, from which rigid body rotation and strains are calculated using continuum mechanics descriptions of kinematics. Moreover, Lagrangian and Eulerian trajectories are obtained from DIC analysis. At the microscale, individual particle kinematics is resolved using particle identification and tracking algorithms. Microscale analysis of kinematics is then performed using definitions of affine and nonaffine local motion. In order to demonstrate the application of the algorithms an experimental setup of pile penetration in a plane strain calibration chamber is presented. Mesoscale and microscale analysis is performed on high-resolution images acquired from incremental jacking of the pile. It is shown through analysis of captured images that the developed tools can be effectively employed in ...

Patterned Nonaffine Motion in Granular Media

Vortex-like flow patterns are often observed in experiments on granular media for which uniform strain is expected based on the loading boundary conditions. These deformations become apparent when the motion associated with uniform strain is subtracted from the total particle motion. Besides presenting an interesting phenomenon that begs explanation, these vortex patterns suggest multiscale structure for nonaffine motion as suggested by modern continuum theories. Further, the authors note that the rotational velocity field added to a uniform strain field produces a planar slip field. Thus, these structures are associated with the slip-band fields that eventually form, which are generally associated with bifurcations in the solution path of the governing partial differential equations. The authors present a procedure to extract these motion fields from discrete-element simulations along with conjugate forces associated with these motions. A key finding from the simulations is that the motions that eventually lead to shear band formation develop throughout the loading history rather than arising as a distinct bifurcation. Further, the pattern of rotational fields, and hence the shear banding pattern, are controlled by the boundary conditions. A question, only partly resolved here, is the origin of forces driving the rotational fields.

Disorder-Assisted Melting and the Glass Transition in Amorphous Solids

The mechanical response of solids depends on temperature, because the way atoms and molecules respond collectively to deformation is affected at various levels by thermal motion. This is a fundamental problem of solid state science and plays a crucial role in materials science. In glasses, the vanishing of shear rigidity upon increasing temperature is the reverse process of the glass transition. It remains poorly understood due to the disorder leading to nontrivial (nonaffine) components in the atomic displacements. Our theory explains the basic mechanism of the melting transition of amorphous (disordered) solids in terms of the lattice energy lost to this nonaffine motion, compared to which thermal vibrations turn out to play only a negligible role. The theory is in good agreement with classic data on melting of amorphous polymers (for which no alternative theory can be found in the literature) and offers new opportunities in materials science.

Stress partition analysis of granular materials

Writing the stress as a sum over contact forces allows one to partition stress increments according to their microscopic, grain-level causes. We present two ways of doing this: the first way differentiates between normal and tangential contact forces, while the second ascribes different parts of a change in stress to different physical processes such as contact network anisotropy, non-affine motions and sliding contacts. These two partitioning methods can be combined. We then use them to analyze simulations of failure in biaxial tests, leading to several results. We show that the granular material becomes effectively frictionless well before failure. In addition, by partitioning the second order work, one can attribute a part of the destabilization to each of the various physical processes mentioned above.

Nonaffine measures of particle displacements in sheared colloidal glasses

The nonaffine motion of particles is central to the relaxation and flow of glasses. It is usually assumed in plasticity theories that nonaffine rearrangements are localized and uncorrelated. Here we present evidence that this assumption may not hold. We investigate and compare systematically different measures of nonaffinity in a sheared colloidal glass by tracking the motion of the individual particles directly with confocal microscopy. We show that besides differences in the appearance and degree of localization of nonaffine displacements, the nature of their fluctuations is very similar. At intermediate times, all spatial correlation functions display robust power-law behavior, clearly demonstrating long-range correlations and critical behavior of the driven glass, in contrast to the assumptions of plasticity theories. We show that on long-time scales, correlations become finite and plasticity theories may apply.

Unilateral interactions in granular packings: a model for the anisotropy modulus

Unilateral interparticle interactions have an effect on the elastic response of granular materials due to the opening and closing of contacts during quasi-static shear deformations. A simplified model is presented, for which constitutive relations can be derived. For biaxial deformations the elastic behavior in this model involves three independent elastic moduli: bulk, shear, and anisotropy modulus. The bulk and the shear modulus, when scaled by the contact density, are independent of the deformation. However, the magnitude of the anisotropy modulus is proportional to the ratio between shear and volumetric strain. Sufficiently far from the jamming transition, when corrections due to non-affine motion become weak, the theoretical predictions are qualitatively in agreement with simulation results.

Compression and shear-wave velocities in discrete particle simulations of quartz granular packings: Improved Hertz-Mindlin contact model

The Hertz-Mindlin (HM) contact model has been a cornerstone for the development of several effective medium theories (EMTs) aimed at describing the mesoscopic and macroscopic mechanical behavior of granular materials like unconsolidated sands. In addition, this model is at the core of most of the discrete particle method designs used to numerically solve for the responses of these heterogeneous materials to external perturbations, like acoustic and stress-strain experiments. However, this model has shown shortcomings in the description of the shear response characterization of granular materials, partly due to the non-affine motions experienced by the grains. We have developed a correction of the model based on a detailed calibration of our acoustic numerical results with previous empirical data. Using a microscopic approach to the grain-grain contact surfaces, the nature of the corrections found appear to be related to the shear resistant asperities and the smaller scale of the grain-grain contact areas compared to the total area assumed by the HM model. An improved HM model characterized by a tangential stiffness weakening is based on these surface corrections. Using this observation an enhanced EMT theory emerges based not only on the tangential stiffness modification but also on the velocity-pressure dependence obtained during the calibration of our numerical model.

The molecular kink paradigm for rubber elasticity: Numerical simulations of explicit polyisoprene networks at low to moderate tensile strains

Based on recent molecular dynamics and ab initio simulations of small isoprene molecules, we propose a new ansatz for rubber elasticity. We envision a network chain as a series of independent molecular kinks, each comprised of a small number of backbone units, and the strain as being imposed along the contour of the chain. We treat chain extension in three distinct force regimes: (Ia) near zero strain, where we assume that the chain is extended within a well defined tube, with all of the kinks participating simultaneously as entropic elastic springs, (II) when the chain becomes sensibly straight, giving rise to a purely enthalpic stretching force (until bond rupture occurs) and, (Ib) a linear entropic regime, between regimes Ia and II, in which a force limit is imposed by tube deformation. In this intermediate regime, the molecular kinks are assumed to be gradually straightened until the chain becomes a series of straight segments between entanglements. We assume that there exists a tube deformation tension limit that is inversely proportional to the chain path tortuosity. Here we report the results of numerical simulations of explicit three-dimensional, periodic, polyisoprene networks, using these extension-only force models. At low strain, crosslink nodes are moved affinely, up to an arbitrary node force limit. Above this limit, non-affine motion of the nodes is allowed to relax unbalanced chain forces. Our simulation results are in good agreement with tensile stress vs. strain experiments.

The Origins of Strain Stiffening in Stiff Biopolymer Networks

The extracellular matrix proteins fibrin and collagen form biopolymer networks which are major constituents of tissues, tendons and blood clots. Their mechanical properties are important for proper function in the body. Previous work using atomic force microscopy (AFM) has characterized the mechanical properties of individual collagen and fibrin fibers, but there is currently no model which can predict bulk material properties from its constituent properties. Here, we study fibrin and collagen networks undergoing shear on a confocal microscope and compare this to bulk rheological measurements. We track individual fiber branchpoints as function of strain. We characterize the non-affinity of the motion and show that the low strain, linear regime corresponds to highly non-affine motion while the high strain, linear regime corresponds to affine motion. We also characterize individual fiber strain as a function of overall system strain and show that linear elastic beams are sufficient to describe the overall strain stiffening response measured in rheology.

Bubble dynamics and rheology in sheared two-dimensional foams

The flow of foams and emulsions is characterized by a nonlinear relation between stress and strain rate. In simulations of linearly sheared, two-dimensional foam flows we observe two distinct regimes with qualitatively different bubble dynamics: at low shear rates there is strongly non-affine bubble motion while at higher shear rates the bubbles are confined to lanes. The transition between these two regimes is smooth and stretches over one decade in strain rate. We measure the fluctuations in the non-affine bubble motion over a wide range of strain rates in order to elucidate the nature of this transition and connect the bubble dynamics to the bulk rheology. While the first regime is well described by a deformation-relaxation model, the second regime is consistent with a flow model where the shear stress follows directly from the viscous drag between the bubbles.

Nanoplastic Interactions of Surface-Grafted Single-Walled Carbon Nanotubes with Glassy Polymer Chains in Nanocomposites

The nanoscopic interactions between glassy macromolecules and the dispersed single-walled carbon nanotubes (SWCNTs) in a nanocomposite during very large local elongations were revealed. The results also unveiled a unique mode of molecular motions followed by glassy chains during brittle nanoplastic flows. Based on detailed nanomechanical calculations on results from atomic force and electron microscopy, the molecular motions of the “strain-softened” glassy polymer chains were found highly dependent on chain entanglement density (νe), in sharp contrast to the νe-independent elastic reinforcement. Particularly, in the loosely entangled chains the SWCNTs behaved like “phantom tubes”, manifesting no effects on strain hardening of the chain network during plastic flow. This indicates that the “glassy” chains had undergone entanglement clustering, a new and unique mode of nonaffine molecular motions. The results bear important significance in revealing the fundamental behavior of glassy polymer chains and the r...

Beads-on-string formation during filament pinch-off: Dynamics with the PTT model for non-affine motion

Fundamental understanding of the formation and pinch-off of viscoelastic filaments is important in applications involving production of drops (e.g., ink-jet printing, micro-arraying, and atomization). In addition to delaying pinch-off, in some cases, viscoelasticity is known to cause the so-called beads-on-string structure, i.e., a number of small droplets interconnected by thin filaments. In a recent publication [H. Matallah, M.J. Banaai, K.S. Sujatha, M.F. Webster, J. Non-Newtonian Fluid Mech. 134 (2006) 77–104], it was shown that the simulation of an elongating filament modeled by the Phan-Thien/Tanner (PTT) equation with the Gordon–Schowalter (GS) convected derivative, which allows non-affine motion of polymer molecules in the continuum, results in the formation of the beads-on-string structure. On the other hand, such bead formation is not reported in calculations with other viscoelastic models that are also strain-hardening like the PTT model but do not have the GS convected derivative (see, e.g., [M. Yao, S.H. Spiegelberg, G.H. McKinley, J. Non-Newtonian Fluid Mech. 89 (2000) 1–43]). This starkly different behavior of the PTT equation with the GS convected derivative is investigated here. During the elongation of the filament, regions of shear form inside the filament due to its initially curved surface. Because of the presence of the GS convected derivative in the PTT equation – which is known to cause unphysical oscillations in stress in simple step shear flow – the shear stress within the PTT filament exhibits temporal oscillations. The onset of these oscillations coincides with the symmetrical migration of the location of the single maximum in the axial component of the rate-of-strain tensor from the center of the filament to two other locations, one in each half of the filament. This is followed by a similar movement of the location of the maximum in the axial elastic stress inside the filament. These two events eventually lead to the formation of a bead-like structure. The occurrence of the bead is also shown to depend on the extent of the polymer contribution to the total viscosity compared to that of the solvent.

Fiber Dynamics during Strain Stiffening in Stiff Biopolymer Networks

Many biopolymers have the unusual mechanical property of strain stiffening. Previous work and theoretical models have focused largely on semi-flexible biopolymers such as actin. Recently, there has been increased interest in understanding stiff, athermal biopolymers. We compare the dynamics of 2 stiff biopolymer networks: fibrin, which is primarily responsible for the mechanical properties of a blood clot, and collagen, which is the main component of connective tissue. We use confocal microscopy to image these in-vitro¬ networks as they undergo a steady shear. Using image processing techniques we record information about the fiber structure as a function of strain. In particular, we quantify angle distributions and the degree of non-affine motion. This is compared to bulk rheological measurements. We also get qualitative information from reconstructed images of the network when viewed as snapshots at various strain positions.