Non-affine Motion Research Articles

Generalized continuum theory for nematic elastomers: Non-affine motion and characteristic behavior

We develop a physically-motivated mechanical theory for predicting the behavior of nematic elastomers — a subset of liquid crystal elastomers (LCEs). We begin with a statistical description of network geometry that naturally incorporates independent descriptors for the mesogens, which create the nematic phase, and the polymer chains, which are assumed to not deform affinely with global deformations. From here, we develop thermodynamically consistent constitutive laws based on classical continuum mechanics principles and ultimately provide simple governing equations with transparent physical interpretation. We found that our framework converges identically to two previously developed mechanical theories, including the well-known neo-classical theory, when considering the extreme ends of our parametric space. We then explore the new predictive capabilities of our model inside these two extremes and illustrate its unique predictions at finite strains, which are distinct in form from other theories. We validate our model using published experimental data from four monodomain nematic liquid crystal elastomers.

Mean-field elastic moduli of a three-dimensional, cell-based vertex model

The mechanics of a foam depends on bubble shape, bubble network topology, and the material at hand, be it metallic or polymeric, for example. While the shapes of bubbles are the consequence of minimizing surface area for a given bubble volume in a space-filling packing, if one were to consider biological tissue as a foam-like material, the zoology of observed shapes of cells perhaps motivates different energetic contributions. Building on earlier two-dimensional results, here, we focus on a mean field approach to obtain the elastic moduli for an ordered, three-dimensional vertex model. We use the space-filling shape of a truncated octahedron and an energy functional containing a restoring surface area spring and a restoring volume spring. The tuning of the three-dimensional shape index exhibits a rigidity transition via a compatible–incompatible transition. Specifically, for smaller shape indices, both the target surface area and volume cannot be achieved, while beyond some critical value of the three-dimensional shape index, they can be, resulting in a zero-energy state. In addition to analytically determining the location of the transition in mean field, we find that the rigidity transition and the elastic moduli depend on the parameterization of the cell shape. This parameterization effect is more pronounced in three dimensions than in two dimensions given the zoology of shapes that a polyhedron can take on (as compared to a polygon). We also uncover nontrivial dependence of the elastic moduli on the deformation protocol in which some deformations result in affine motion of the vertices, while others result in nonaffine motion. Such dependencies on the shape parameterization and deformation protocol give rise to a nontrivial shape landscape and, therefore, nontrivial mechanical response even in the absence of topology changes.

Effects of polydispersity on the plastic behaviors of dense two-dimensional granular systems under shear.

We study particle-scale motion in sheared highly polydisperse amorphous materials, in which the largest particles are as much as ten times the size of the smallest. We find strikingly different behavior from the more commonly studied amorphous systems with low polydispersity. In particular, an analysis of the nonaffine motion of particles reveals qualitative differences between large and small particles: The smaller particles have dramatically more nonaffine motion, which is induced by the presence of the large particles. We characterize how the nonaffine motion changes from the low- to high-polydispersity regimes. We further demonstrate a quantitative way to distinguish between "large" and "small" particles in systems with broad distributions of particle sizes. A macroscopic consequence of the nonaffine motion is a decrease in the energy dissipation rate for highly polydisperse samples, which is due both to a geometric consequence of the changing jamming conditions for higher polydispersity and to the changing character of nonaffine motion.

General theory of the viscosity of liquids and solids from nonaffine particle motions.

A microscopic formula for the viscosity of liquids and solids is derived rigorously from a first-principles (microscopically reversible) Hamiltonian for particle-bath atomistic motion. The derivation is done within the framework of nonaffine linear response theory. This formula may lead to a valid alternative to the Green-Kubo approach to describe the viscosity of condensed matter systems from molecular simulations without having to fit long-time tails. Furthermore, it provides a direct link between the viscosity, the vibrational density of states of the system, and the zero-frequency limit of the memory kernel. Finally, it provides a microscopic solution to Maxwell's interpolation problem of viscoelasticity by naturally recovering Newton's law of viscous flow and Hooke's law of elastic solids in two opposite limits.

Isomorphs in sheared binary Lennard-Jones glass: Transient response.

We have studied shear deformation of binary Lennard-Jones glasses to investigate the extent to which the transient part of the stress strain curves is invariant when the thermodynamic state point is varied along an isomorph. Shear deformations were carried out on glass samples of varying stability, determined by cooling rate, and at varying strain rates, at state points deep in the glass. Density changes up to and exceeding a factor of two were made. We investigated several different methods for generating isomorphs but none of the previously developed methods could generate sufficiently precise isomorphs given the large density changes and nonequilibrium situation. Instead, the temperatures for these higher densities were chosen to give state points isomorphic to the starting state point by requiring the steady-state flow stress for isomorphic state points to be invariant in reduced units. In contrast to the steady-state flow stress, we find that the peak stress on the stress strain curve is not invariant. The peak stress decreases by a few percent for each ten percent increase in density, although the differences decrease with increasing density. Analysis of strain profiles and nonaffine motion during the transient phase suggests that the root of the changes in peak stress is a varying tendency to form shear bands, with the largest tendency occurring at the lowest densities. We suggest that this reflects the effective steepness of the potential; a higher effective steepness gives a greater tendency to form shear bands.

Acoustic Localization of an Intruder in a Strongly Scattering Medium

Localizing an intruder submerged in a strongly scattering medium, such as a dense granular suspension, is a practical challenge. Here we extract the coherent ultrasonic echo from a steel ball submerged in a dense glass-bead packing saturated by water, by using a standard single-element ultrasonic transducer thanks to a configuration averaging process. Different configurations of the granular packing are created by the nonaffine motion of the beads with a mixing blade, akin to the Brownian motion, in the vicinity of the intruder. We investigate the efficiency of this process to reduce the so-called material noise from multiply scattered ultrasound, as a function of the configuration number, the shear rate and the blade-intruder distance. Nonaffine motions of the beads in the shear zone induced by the blade are then analyzed on the basis of a split-bottom rotating shear cell. This method helps to develop not only ultrasonic imaging tools of buried objects in turbid marine sediments, but also the local rheology based on a ball falling monitored by ultrasonic tracking.

Non-affine motion and selection of slip coefficient in constitutive modeling of polymeric solutions using a mixed derivative

Constitutive models for the dynamics of polymer solutions traditionally rely on closure relations for the extra stress or related microstructural variables (e.g., conformation tensor) linking them to flow history. In this work, we study the eigendynamics of the conformation tensor within the GENERIC framework in mesoscopic computer simulations of polymer solutions to separate the effects of nonaffine motion from other sources of non-Newtonian behavior. We observe that nonaffine motion or slip increases with both the polymer concentration and the polymer chain length. Our analysis allows to uniquely calibrate a mixed derivative of the Gordon–Schowalter type in macroscopic models based on a micro-macromapping of the dynamics of the polymeric system. The presented approach paves the way for better polymer constitutive modeling in multiscale simulations of polymer solutions, where different sources of non-Newtonian behavior are modelled independently.

Label-free cell tracking enables collective motion phenotyping in epithelial monolayers

SummaryCollective cell migration is an umbrella term for a rich variety of cell behaviors, whose distinct character is important for biological function, notably for cancer metastasis. One essential feature of collective behavior is the motion of cells relative to their immediate neighbors. We introduce an AI-based pipeline to segment and track cell nuclei from phase-contrast images. Nuclei segmentation is based on a U-Net convolutional neural network trained on images with nucleus staining. Tracking, based on the Crocker-Grier algorithm, quantifies nuclei movement and allows for robust downstream analysis of collective motion. Because the AI algorithm required no new training data, our approach promises to be applicable to and yield new insights for vast libraries of existing collective motion images. In a systematic analysis of a cell line panel with oncogenic mutations, we find that the collective rearrangement metric, D2min, which reflects non-affine motion, shows promise as an indicator of metastatic potential.

Theory of sound attenuation in amorphous solids from nonaffine motions

We present a theoretical derivation of acoustic phonon damping in amorphous solids based on the nonaffine response formalism for the viscoelasticity of amorphous solids. The analytical theory takes into account the nonaffine displacements in transverse waves and is able to predict both the ubiquitous low-energy diffusive damping ∼k 2, as well as a novel contribution to the Rayleigh damping ∼k 4 at higher wavevectors and the crossover between the two regimes observed experimentally. The coefficient of the diffusive term is proportional to the microscopic viscous (Langevin-type) damping in particle motion (which arises from anharmonicity), and to the nonaffine correction to the static shear modulus, whereas the Rayleigh damping emerges in the limit of low anharmonicity, consistent with previous observations and macroscopic models. Importantly, the k 4 Rayleigh contribution derived here does not arise from harmonic disorder or elastic heterogeneity effects and it is the dominant mechanism for sound attenuation in amorphous solids as recently suggested by molecular simulations.

Contact force measurements and local anisotropy in ellipses and disks

Experimental measurements of contact forces are limited to spheres and disks in three and two dimensions, making the evaluation of shape effect and universality of force distributions and comparison between experiments and theories extremely difficult. Here we present precise measurements of vector contact forces in photoelastic ellipses and disks subject to isotropic compression and pure shear. We find the local, instead of the global, stress ratio controls the width of force distributions for forces larger than the mean, regardless of particle shape and preparation protocols. By taking advantage of anisotropic particle shape, we can determine the anisotropic growth of contacts in ellipses subject to isotropic compression, revealing the role of nonaffine particle motions in homogenizing force distributions. Our results uncover the role of local anisotropy in the statistical framework of granular materials and open a regime of exploring the role of particle shape on the mechanical and dynamical properties of granular materials in depth.

Local micromorphic non-affine anisotropy for materials incorporating elastically bonded fibres

There has been increasing experimental evidence of non-affine elastic deformation mechanisms in biological soft tissues. These observations call for novel constitutive models which are able to describe the dominant underlying micro-structural kinematic aspects, in particular relative motion characteristics of different phases. This paper proposes a flexible and modular framework based on a micromorphic continuum encompassing matrix and fibre phases. In addition to the displacement field, it features so-called director fields which can independently deform and intrinsically carry orientational information. Accordingly, the fibrous constituents can be naturally associated with the micromorphic directors and their non-affine motion within the bulk material can be efficiently captured. Furthermore, constitutive relations can be formulated based on kinematic quantities specifically linked to the material response of the matrix, the fibres and their mutual interactions. Associated stress quantities are naturally derived from a micromorphic variational principle featuring dedicated governing equations for displacement and director fields. This aspect of the framework is crucial for the truly non-affine elastic deformation description. In contrast to conventional micromorphic approaches, any non-local higher-order material behaviour is excluded, thus significantly reducing the number of material parameters to a range typically found in related classical approaches. In the context of biological soft tissue modelling, the potential and applicability of the formulation is studied for a number of academic examples featuring anisotropic fibre-reinforced composite material composition to elucidate the micromorphic material response as compared with the one obtained using a classical continuum mechanics approach.

Shear response of granular packings compressed above jamming onset.

We investigate the mechanical response of jammed packings of repulsive, frictionless spherical particles undergoing isotropic compression. Prior simulations of the soft-particle model, where the repulsive interactions scale as a power law in the interparticle overlap with exponent α, have found that the ensemble-averaged shear modulus 〈G(P)〉 increases with pressure P as ∼P^{(α-3/2)/(α-1)} at large pressures. 〈G〉 has two key contributions: (1) continuous variations as a function of pressure along geometrical families, for which the interparticle contact network does not change, and (2) discontinuous jumps during compression that arise from changes in the contact network. Using numerical simulations, we show that the form of the shear modulus G^{f} for jammed packings within near-isostatic geometrical families is largely determined by the affine response G^{f}∼G_{a}^{f}, where G_{a}^{f}/G_{a0}=(P/P_{0})^{(α-2)/(α-1)}-P/P_{0},P_{0}∼N^{-2(α-1)} is the characteristic pressure at which G_{a}^{f}=0,G_{a0} is a constant that sets the scale of the shear modulus, and N is the number of particles. For near-isostatic geometrical families that persist to large pressures, deviations from this form are caused by significant nonaffine particle motion. We further show that the ensemble-averaged shear modulus 〈G(P)〉 is not simply a sum of two power laws, but 〈G(P)〉∼(P/P_{c})^{a}, where a≈(α-2)/(α-1) in the P→0 limit and 〈G(P)〉∼(P/P_{c})^{b}, where b≳(α-3/2)/(α-1), above a characteristic pressure that scales as P_{c}∼N^{-2(α-1)}.

Using delaunay triangularization to characterize non-affine displacement fields during athermal, quasistatic deformation of amorphous solids.

We investigate the non-affine displacement fields that occur in two-dimensional Lennard-Jones models of metallic glasses subjected to athermal, quasistatic simple shear (AQS). During AQS, the shear stress versus strain displays continuous quasi-elastic segments punctuated by rapid drops in shear stress, which correspond to atomic rearrangement events. We capture all information concerning the atomic motion during the quasi-elastic segments and shear stress drops by performing Delaunay triangularizations and tracking the deformation gradient tensor Fα associated with each triangle α. To understand the spatio-temporal evolution of the displacement fields during shear stress drops, we calculate Fα along minimal energy paths from the mechanically stable configuration immediately before to that after the stress drop. We find that quadrupolar displacement fields form and dissipate both during the quasi-elastic segments and shear stress drops. We then perform local perturbations (rotation, dilation, simple and pure shear) to single triangles and measure the resulting displacement fields. We find that local pure shear deformations of single triangles give rise to mostly quadrupolar displacement fields, and thus pure shear strain is the primary type of local strain that is activated by bulk, athermal quasistatic simple shear. Other local perturbations, e.g. rotations, dilations, and simple shear of single triangles, give rise to vortex-like and dipolar displacement fields that are not frequently activated by bulk AQS. These results provide fundamental insights into the non-affine atomic motion that occurs in driven, glassy materials.

The weakly nonlinear response and nonaffine interpretation of the Johnson–Segalman/Gordon–Schowalter model

We derive new analytical solutions for the nonaffine Johnson–Segalman/Gordon–Schowalter (JS/GS) constitutive equation with a general relaxation kernel in medium-amplitude oscillatory shear (MAOS) deformation. The results show time-strain separable (TSS) nonlinearity, therefore providing new physically meaningful interpretation to the heuristic TSS nonlinear parameter in MAOS [Martinetti and Ewoldt, Phys. Fluids 31, 021213 (2019)]. The upper-convected, lower-convected, and corotational Maxwell models are all subsets of the results presented here. The model assumes that the microscale elements causing stress in the material slip compared to the continuum deformation. We introduce a visualization of the nonaffine deformation field that acts on stress-generating elements to reinforce the physical interpretation of the JS/GS class of models. Finally, a case study is presented where previously published results, from fitting TSS models to MAOS data, can be reinterpreted based on the concept of nonaffine motion of the JS/GS framework.

Spatial correlation and temporal evolution of plastic heterogeneity in sheared granular materials

Granular materials will yield and undergo serrated plastic flow when subjected to loading. The serrated plastic flow is related to the spatial concentration and temporal evolution of the microscopic plastic events. Limited by the temporal resolution of experimental techniques available, the initialization and localization of the plastic events within timescale much less than the structural relaxation time is still poorly understood. We use combined finite and discrete element method to model the simple shearing of an assembly of irregularly shaped particles. At small temporal scales, nonaffine motions originate at particles experiencing large stress fluctuations. The cooperative dynamics and long-ranged dynamic heterogeneity is responsible for the annihilation and merging of plastic zones at the timescale close to the structural relaxation time, and the plasticity is manifested as irreversible rearrangement at large enough strain scales. The spatiotemporal variability of plastic activity is confirmed by the transient nature of the high mobile clusters.

Effects of Cyclic Loading and Time-Recovery on Microstructure and Mechanical Properties of Particle-Binder Composites

Abstract We propose a systematic experimental procedure and quantitative analyses to investigate the effect of cyclic loading, and time-recovery, or aging, on the mechanical properties and microstructure of particle-binder composites. Specifically, we study three compositions that differ in aluminum content from the mock sugar formulation of plastic-bonded explosive PBXN-109. Cast cylindrical specimens are subjected to high-amplitude quasi-static cyclic compressive loading, before and after a 4-week time-recovery period, and their microstructures are analyzed using micro-computed tomography (CT). For quantitative analysis, we develop a procedure for identifying the spatial distribution of primary components of the formulation, including pore space, from micro-CT images. The study shows that the stress–strain response is highly nonlinear, without a distinct yield point, and exhibits hysteresis and cyclic stress softening, or Mullins effect, with cyclic stabilization. Specimens without aluminum exhibit considerable gain in stiffness and strength after the time-recovery or aging period, owing to the development of increased sucrose particle–particle interactions during the first cyclic loading. In contrast, specimens with aluminum micro-sized powder exhibit permanent loss of stiffness and strength, owing to large ductile plastic flow and irrecoverable damage. Further insight from micro-CT analysis is gained by observing that, for all compositions, the majority of microstructural changes occur near the specimen core. Specifically, affine radial deformation of the soft and debonded binder, as it is compressed by the non-affine longitudinal motion of stiffer sucrose crystals, is observed in the formulation without aluminum, whereas non-affine rearrangement of the binder toward the specimen core, and affine radial flow of sucrose particles away from the core due to ductile macroscopic deformation of the specimen, is observed in the formulations with aluminum content.

Slip-Spring and Kink Dynamics Models for Fast Extensional Flow of Entangled Polymeric Fluids.

We combine a slip-spring model with an ‘entangled kink dynamics’ (EKD) model for strong uniaxial extensional flows (with Rouse Weissenberg number ) of long ( for polystyrene) entangled polymers in solutions and melts. The slip-spring model captures the dynamics up to the formation of a ‘kinked’ or folded state, while the kink dynamics simulation tracks the dynamics from that point forward to complete extension. We show that a single-chain slip-spring model using affine motion of the slip-spring anchor points produces unrealistically high tension near the center of the chain once the Hencky strain exceeds around unity or so, exceeding the maximum tension that a chain entangled with a second chain is able to support. This unrealistic tension is alleviated by pairing the slip links on one chain with those on a second chain, and allowing some of the large tension on one of the two to be transferred to the second chain, producing non-affine motion of each. This explicit pairing of entanglements mimics the entanglement pairing also used in the EKD model, and allows the slip spring simulations to be carried out to strains high enough for the EKD model to become valid. We show that results nearly equivalent to those from paired chains are obtained in a single-chain slip-spring simulation by simply specifying that the tension in a slip spring cannot exceed the theoretical maximum value of where , and are the friction per unit length, strain rate and contour length of the chain, respectively. The effects of constraint release (CR) and regeneration of entanglements is also studied and found to have little effect on the chain statistics up to the formation of the kinked state. The resulting hybrid model provides a fast, simple, simulation method to study the response of high molecular weight () polymers in fast flows (), where conventional simulation techniques are less applicable due to computational cost.

Dynamical density functional theory analysis of the laning instability in sheared soft matter.

Using dynamical density functional theory (DDFT) methods we investigate the laning instability of a sheared colloidal suspension. The nonequilibrium ordering at the laning transition is driven by nonaffine particle motion arising from interparticle interactions. Starting from a DDFT which incorporates the nonaffine motion, we perform a linear stability analysis that enables identification of the regions of parameter space where lanes form. We illustrate our general approach by applying it to a simple one-component fluid of soft penetrable particles.

Atomic-scale origin of dynamic viscoelastic response and creep in disordered solids.

Viscoelasticity has been described since the time of Maxwell as an interpolation of purely viscous and purely elastic response, but its microscopic atomic-level mechanism in solids has remained elusive. We studied three model disordered solids: a random lattice, the bond-depleted fcc lattice, and the fcc lattice with vacancies. Within the harmonic approximation for central-force lattices, we applied sum rules for viscoelastic response derived on the basis of nonaffine atomic motions. The latter motions are a direct result of local structural disorder, and in particular, of the lack of inversion symmetry in disordered lattices. By defining a suitable quantitative and general atomic-level measure of nonaffinity and inversion symmetry, we show that the viscoelastic responses of all three systems collapse onto a master curve upon normalizing by the overall strength of inversion-symmetry breaking in each system. Close to the isostatic point for central-force lattices, power-law creep G(t)∼t^{-1/2} emerges as a consequence of the interplay between soft vibrational modes and nonaffine dynamics, and various analytical scalings, supported by numerical calculations, are predicted by the theory.

3D in situ observations of the compressibility and pore transport in Sheet Moulding Compounds during the early stages of compression moulding

The properties of Sheet Moulding Compounds (SMCs) are altered by their porosity induced during the manufacturing or moulding of these prepregs. To characterise the pore scale mechanisms occurring during SMC flow, 3D synchrotron X-ray images were acquired during the compression of two uncured SMCs. For the high fibre content SMC (50wt%), with a high porosity, pores mainly decreased in size and disappeared during SMC consolidation. These mechanisms were related to the elasticity of the fibrous networks and to the permeability of the porous phase, estimated using pore scale CFD simulation. For the standard fibre content SMC (29wt%), with a lower porosity, the compressibility was limited and closed pores were transported towards the external surface of samples with non-affine motion, i.e., faster than the SMC flow and with tortuous trajectories. Besides, pores coalesced and decreased in size during compression. The size decrease was mainly related to the dissolution of pore gases.