Non-affine Motion Research Articles

Acoustic Probing of the Jamming Transition in an Unconsolidated Granular Medium

Experimentally determined dispersion relations for acoustic waves guided along the mechanically free surface of an unconsolidated granular packed structure provide information on the elasticity of granular media at very low pressures that are naturally controlled by the gravitational acceleration and the depth beneath the surface. The experiments confirm recent theoretical predictions that relaxation of the disordered granular packing through nonaffine motion leads to a peculiar scaling of shear rigidity with pressure near the jamming transition corresponding to zero pressure.

Modeling the dynamic response of visco-elastic open-cell foams

This article introduces a mesoscopic formulation for modeling the dynamic response of visco-elastic, open-cell solid foams. The effective material response is obtained by enforcing on a representative 3D unit cell the principle of minimum action for dissipative systems. The resulting model accounts explicitly for the foam topology, the elastic and viscous properties of the cell wall, and the inertial effects arising from non-affine motion within the cells. The microinertial effects become significant in retarding the foam collapse during exceedingly high strain-rate loading. As an application example, a heterogenous case of compressive deformation at high strain rate is simulated utilizing the present model as a constitutive update in a non-linear finite element analysis code. This FEM simulation shows the ability of the model to capture the progressive foam collapse during the dynamic compression as observed in experimental studies. Using the microscopic model, the inertial and viscous strain-rate effects are investigated through the foam density, viscosity, and relative density. Based on the physics incorporated into the local cell model, we provide insight into the physical mechanisms responsible for the experimentally observed strain-rate effects on the behavior of dynamically loaded foam materials.

Plastic response of a two-dimensional amorphous solid to quasistatic shear: Transverse particle diffusion and phenomenology of dissipative events

We perform extensive simulations of a two-dimensional Lennard-Jones glass subjected to quasistatic shear deformation at T=0. We analyze the distribution of nonaffine displacements in terms of contributions of plastic, irreversible events, and elastic, reversible motions. From this, we extract information about correlations between plastic events and about the elastic nonaffine noise. Moreover, we find that nonaffine motion is essentially diffusive, with a clearly size-dependent diffusion constant. These results, supplemented by close inspection of the evolving patterns of the nonaffine tangent displacement field, lead us to propose a phenomenology of plasticity in such amorphous media. It can be schematized in terms of elastic loading and irreversible flips of small, randomly located shear transformation zones, elastically coupled via their quadrupolar fields.

Slippage and migration in Taylor–Couette flow of a model for dilute wormlike micellar solutions

We explore the rheological predictions of a constitutive model developed for dilute or semi-dilute worm-like micellar solutions in an axisymmetric Taylor–Couette flow. This study is a natural continuation of earlier work on rectilinear shear flows. The model, based on a bead-spring microstructure with non-affine motion, reproduces the pronounced plateau in the stress–strain-rate flow curve that is observed in laboratory measurements of steady shearing flows. We also carry out a linear stability analysis of the computed steady-state solutions. The results show shear-banding in the form of sharp changes in velocity gradients, spatial variations in number density, and in alignment or stretching of the micelles. The velocity profiles obtained in numerical solutions show good qualitative agreement with those of laboratory experiments.

The link between discrete and continuous modeling of liquid foam at the level of a single bubble

In order to test the adaptability of a homogenization scheme originally developed for dry granular media, a micromechanical constitutive law for three-dimensional monodisperse wet foam is developed. The advantage of the homogenization scheme is that it provides a link between the discrete bubble-scale microstructure and the continuum quantities of stress and deformation. Furthermore, the resulting constitutive law is in terms of physical material properties (e.g., fluid viscosity, surface tension, and bubble radius) rather than the more phenomenological parameters, such as power-law indices, etc. Hence, predictions of the continuum model can be, and are, compared directly to experiments and simulations. Many of the general features of foam behavior are reproduced, including small-strain elasticity, dependence on gas fraction, and Bingham plastic type deformation rate dependency. However, the current micromechanical continuum model substantially overestimates the static shear modulus and yield stress of foams in comparison to simulations and experiments. This discrepancy is due to the adoption of a mean-field assumption for the deformation in the current model. It is then argued that future micromechanical continuum models for foams cannot neglect the nonaffine motions associated with bubble rearrangements. Finally, for both foams and granular materials, our study indicates that the uniform deformation assumption is only valid near the critical packing of monodisperse spherical particles.

Granular packings: Nonlinear elasticity, sound propagation, and collective relaxation dynamics

Experiments on isotropic compression of a granular assembly of spheres show that the shear and bulk moduli vary with the confining pressure faster than the 1/3 power law predicted by Hertz-Mindlin effective medium theories of contact elasticity. Moreover, the ratio between the moduli is found to be larger than the prediction of the elastic theory by a constant value. The understanding of these discrepancies has been a long-standing question in the field of granular matter. Here we perform a test of the applicability of elasticity theory to granular materials. We perform sound propagation experiments, numerical simulations, and theoretical studies to understand the elastic response of a deforming granular assembly of soft spheres under isotropic loading. Our results for the behavior of the elastic moduli of the system agree very well with experiments. We show that the elasticity partially describes the experimental and numerical results for a system under compressional loads. However, it drastically fails for systems under shear perturbations, particularly for packings without tangential forces and friction. Our work indicates that a correct treatment should include not only the purely elastic response but also collective relaxation mechanisms related to structural disorder and nonaffine motion of grains.

A full-chain, temporary network model with sliplinks, chain-length fluctuations, chain connectivity and chain stretching

A full-chain, temporary network model is proposed for nonlinear flows of linear, entangled polymeric liquids. The model is inspired by the success of a recent reptation model, but contains no beads or tubes. Instead, each chain uses a different (and smaller) set of dynamic variables: the location of each entanglement and the number of Kuhn steps in chain strands between entanglements. As before, the model requires only a single phenomenological parameter that is fit by linear viscoelasticity. The number of Kuhn steps varies stochastically from imbalances in chemical potential and Brownian forces. In the language of reptation, the model exhibits chain connectivity, chain-length fluctuations, chain stretching, and tube dilation. The current implementation in this framework does not include constraint release, although its addition is possible. The entanglements are assumed to move affinely. Because of the affinity assumption and lack of constraint release, the model should be expected to approximate well a linear chain in a matrix of fixed obstacles, and somewhat less accurately a polymer melt. Straightforward modifications to these assumptions allow us to consider chains of any architecture in concentrated solutions or melts. A simulation algorithm for the model is described in detail. Stress results are encouraging, since the model performs at least as well as a complete tube model without constraint release at much lower computational cost. Finally, we consider possible generalizations of the proposed model, to include additional physics, such as constraint release, branching, and nonaffine motion.

A non-affine transient network model

A new formulation of non-affine motion for the network model is proposed and its applications are presented in this manuscript. The major improvement is that a finite elongational viscosity is predicted for finite elongational rates contrary to the infinite elongational viscosities existing at some elongational rates predicted by previous network models. Comparisons with experimental data of the shear viscosity, normal stress coefficient and elongational viscosity are given, in terms of the same set of model parameters. Model predictions for the stress growth are also shown.

Modelling of non-isothermal viscoelastic flows

The modelling of non-isothermal flow of viscoelastic materials using differential constitutive equations is investigated. The approach is based on the concept of a slip tensor describing the nonaffine motion of the stress-carrying structure of the fluid. By specifying a slip tensor and the elastic behaviour of the structure, the constitutive model is determined. This slip tensor also appears in the energy equation and thus, for a given constitutive model, the form of the energy equation is known. Besides the partitioning between dissipated and elastically stored energy, also the difference between entropy and energy elasticity is discussed. Numerical simulations are based on a stabilized Discontinuous Galerkin method to solve the mass, momentum and constitutive equations and a Streamline Upwind Petrov-Galerkin formulation to solve the energy equation. Coupling is achieved by a fixed point iteration. The flow around a confined cylinder is investigated, showing differences between viscous and viscoelastic modelling, and between limiting cases of viscoelastic modelling.

Role of micro-convection due to non-affine motion of particles in a mono-disperse suspension

Non-affine translation of suspended particles in a flowing suspension causes micro-convection of the fluid in the vicinity of the particle which affects the local transport mechanisms in the suspension. To investigate the importance of micro-convection on heat transport, a unit cell approach is adopted. A parametric study investigates the influence of particle fall velocity in a quiescent fluid and particle volume fraction on micro-convection in a sedimenting suspension. The thermal flux across the unit cell is shown to be a function of modified Péclet number that includes the influence of volume fraction for the range of volume fractions (0 < ε < 0.3) and for Pe ≈ O(1). This is used to investigate the influence of hindered settling on transport processes such as heat transfer occurring in a sedimenting suspension.

A generalized Giesekus constitutive model with retardation time and its association to the spurt effect

This paper presents a simple generalization of the single-mode Giesekus constitutive model with retardation time. It is based on the utilization of the Gordon Schowalter derivatives for the stress and rate-of-deformation tensors, which account for the non-affine motion of polymer strands, and make the model more versatile in terms of its ability to predict the complex flow behaviour. The model is consistent with molecular kinetic theory, and has been tested in several well-characterized shear flows, whereas its ability to describe shear-free flows has been proved as well. Finally, a comparison against steady shear data of nearly monodisperse polybutadienes demonstrates its superiority with respect to other continuum or molecular models exhibiting non-monotonic constitutive behaviour.

A non-affine network model for polymer melts

In this paper, a network model of polymer melts is proposed in which network junction points move non-affinely. In this non-affine motion, junction points follow particle paths as seen by an observer rotating at the fluid element's net rigid-rotation rate. The speed at which junction points move is reduced as the network segments near their maximum extensions. In order to maintain a frame invariant model, it is necessary that the vorticity be decomposed into two portions, such that, ω = ωR + ωD. The deformational vorticity, ωD, arises from shear deformation and is frame invariant while the rigid vorticity, ωR, is frame dependent. A constitutive equation based on this finitely extensible network strand (FENS) motion is developed. The model illustrates how rotations that cause changes in the relative orientation of a fluid element with its surroundings can be incorporated into a constitutive equation using the deformational vorticity. The FENS model predicts a shear-thinning viscosity, and the Trouton viscosity predicted by the model is finite for all elongation rates. Finally, stochastic simulation results are presented to justify a mathematical approximation used in deriving the constitutive equation.

Problems originating from the use of the gordon-schowalter derivative in the johnson-segalman and related models in various shear flow situations

The predictions of the Johnson-Segalman model have been compared to experimental results in some simple flows, namely steady state shearing flow and stress growth in shear, for various molten polyethylenes having different molecular characteristics as far as their molecular weight level, polydispersity index and branching degree are concerned. Attention has been focused on this particular model because it has been shown to be an approximation of the Phan-Thien- Tanner equation in simple shear flows, the later model being frequently used in numerical calculations for complex flows. The Johnson-Segalman model is a codeformational rheological equation of state, which allows non-affine motion of the network junctions by introduction of a single parameter ( a) named the slip factor. This assumption provides a great improvement in the predictions of non-linear response to flows involving large deformations, either in shear or in elongation. However, the determination of the slip parameter from the various experiments has shown that a single value of the slip factor can not be used to describe both tangential stresses (τ 12) and normal stresses (τ 11—τ 22) in polyethylene melts. This has been related to a discrepancy of the model concerning the violation of the Lodge-Meissner rule as a consequence of the use of the Gordon-Schowalter derivative. Similar discrepancy can be expected in the case of the Phan-Thien-Tanner equation. Nevertheless, the slip factor, that can be determined either by fitting tangential stresses or normal stresses, is found to be nearly constant for a particular material, whatever the flow regime is (transient or steady). Moreover, for the various linear polymers (high density or linear low density polyethylenes), these parameters appear to be nearly independent of the molecular weight distribution. In this case, the differences in the non-linear behaviour in shear can only be attributed to differences in the linear relaxation modulus. On the other hand, a highly branched material (low density polyethylene) shows very different values of the slip parameters.

Semiempirical models for steady‐shear viscosity and prediction of first normal stress function

AbstractIn order to describe viscosity function behavior dependent on shear rate in steady‐shear flow and to predict the first normal stress function without using discrete relaxation time specturm, we combined the microscopic entanglement concept based on Graessley's entanglement network theory with the macroscopic constitutive model. The viscosity function derived from the entanglement model has three adjustable parameters and it was assumed to have the form of the inverse contangent function of the shear rate. Then the first normal stress function is obtained using the Wagner's relationship. The model prediction for the first normal stress coefficient is compared with experimental data of polymer melts. Even though viscosity function and the first normal stress function vary over a wide range of shear rate, they were in good agreement for polymer melts at high shear rate. First normal stress function do not show any numerical artifacts. However, discrepancy in the first normal stress function occurs at low shear rate. This is due to the irreversibility of the entangled molecule's motion that was not taken into consideration in the entanglement model. Irreversible nonaffine motion is introduced by adding a term similar to the White‐Metzner model's irreversible factor. Addition of irreversibility in the calculation of the first normal stress function considerably improves the agreement. Other comparisons are also presented and related discussions are given. © 1994 John Wiley &amp; Sons, Inc.

Network Recovery from Uniaxial Extension: II. The Origin of the Mullins Effect

Abstract The Mullins effect refers to the dissipation in crosslinked rubber of mechanical energy beyond that due to linear relaxation processes or irreversible structural changes. Physically, Mullins softening probably arises from adjustment of local imbalances in segment density and from contraction of the primitive path of network chain ends. The resulting more random chain configuration contributes to the relaxation of the stress, but at a sufficiently rapid rate that it is omitted from the experimentally measured linear relaxation spectrum. This omission is made apparent upon reversal of the applied strain. The mechanism underlying the hysteresis can be modeled empirically by the introduction of irreversibility into the damping function. Although the Mullins effect is a reflection of nonaffine network motion during deformation of rubber, it is independent of the extent to which the junctions of the network are affinely displaced at elastic equilibrium. Randomly crosslinked networks of widely varying structure exhibited comparable degrees of Mullins softening. Corroboration that the network chain ends are a major source of Mullins softening could be obtained by a similar study using end-linked networks. Such experiments on end-linked polytetrahydrofuran rubber are currently in progress.

The Mullins Effect in Crosslinked Rubber

The Mullins effect refers to the dissipation in cross‐linked rubber of mechanical energy in apparent excess of that due to linear relaxation processes or irreversible structural changes. Although it reflects nonaffine network motion in response to an applied deformation, contrary to previous assertions Mullins softening is found to be independent of the extent to which the junctions of the network are affinely displaced at elastic equilibrium. Rubbers of widely varying network structure were observed to exhibit comparable degrees of Mullins softening. Physically this softening arises from equilibration of local imbalances in the force along the network chains and from contraction of the primitive path of network chain ends. These mechanisms, which transpire too rapidly to be included in the linear relaxation spectrum, can be modeled empirically by the introduction of irreversibility into the damping function in the constitutive equation.

A unified constitutive theory for polymeric liquids II. Applications to basic problems

A unified constitutive theory for polymeric liquids has been recently proposed. Its derivation is based on a combination of continuum mechanical approach, transient-network concept and thermodynamics of irreversible processes. In the resulting model, many modes may be present for each of which there are two time scales, associated with the loss rate and the nonaffine motion of transient network junctions, respectively. A single effective relaxation time, constructed from the two time scales, governs the behavior in the linear regime of deformation. Two new parameters for each mode, in comparison with other models, are introduced: (i) the ratio “r” of the two time scales, and (ii) the index “a” distinguishing the rates of loss and creation of junctions. Both are important only for the nonlinear regime of deformation. In this paper, the theory is applied to predict the following cases: (i) stress growth at constant shear strain rate, (ii) steady shear-rate-dependent viscosity and first normal-stress difference and (iii) transient elongational viscosity at constant elongational strain rate. Determination of the model parameters based on usual characterization experiments is described. Comparison of calculated and observed behavior of low-density polyethylene at 150 °C available in the literature are presented. In general, the agreement of the predictions with experiment appear gratifying even with the simplest version of the new model.

On configuration-dependent generalized oldroyd derivatives

After a short derivation of the upper and lower Oldroyd derivatives, linear combinations of them are examined with regard to their suitability for describing the non-affine motion of polymer structures occuring because of hampered flow. This leads to the introduction of a new type of convected derivative called a “configuration-dependent” derivative. While the combined Oldroyd derivatives can be considered to be upper convected derivatives supplemented by the addition of terms containing the rate of deformation tensor, in the new derivatives the irreversible rate of deformation tensor is substituted instead. This tensor, first introduced by Leonov, is strongly related to the configuration tensor, which is a measure of recoverable strain, at least in so-called one-mode models. The force-free motion of particles in steady simple shear flow as predicted by the respective generalized derivatives is analysed and it is found that the configuration-dependent derivative supplies a more realistic result for hampered flow than the combined Oldroyd derivative, except in the case of permanently isotropic structures for which both predictions are identical. For the special case of application of the configuration-dependent derivative to the configuration tensor, the expression is equivalent to that obtained by the application of the simple upper Oldroyd derivative itself. Thus it is concluded that the effects predicted by simple constitutive equations, when combined Oldroyd derivatives are used but not when only the upper Oldroyd derivative is used, may be more appropriately described by taking into account the non-isotropic character of the relative motion of structural elements occuring because of mutual interactions.

Description of the non-affine motions of dilute polymer solutions by the porous molecule model

The equations of porous molecule theory are solved to find the flow field near a polymer molecule. This is used to determine a scalar parameter which measures the non-affineness of the solvent flow. It is found that non-affineness is a function of a dimensionless permeability and is independent of the distribution of permeability in the molecule and of the molecular topology. Furthermore, non-affineness is an important contributing factor to many of the non-linear properties of polymer fluids. Results are also given for intrinsic viscosity and molecular friction factor for molecules with a uniform and Gaussian distribution of inverse permeability.

Cooperative diffusion in a temporary network

Local rearrangements following connectivity changes in a temporary network are taken into explicit account by introducing a friction coefficient for non-affine adaptive motions of the general junction. In this way a simple model of internal diffusion is obtained, in which connectivity changes act as driving forces. A transport equation is then derived from the microscopic equation of motion. This transport equation contains a regeneration term, a convective term and an additional diffusive contribution, depending on past deformation history. Retaining the basic assumptions of the simple network theory on regeneration kinetics, a calculation of the diffusive term is performed for tetrafunctional networks; preliminary results obtained for semi-permanent networks undergoing regeneration pulses are extended to temporary systems on the basis of “mean field” arguments. Two characteristic time constants appear in the constitutive equation: the chain lifetimeτ0 and the diffusive constantτ1. A preliminary application to elongational flow shows that the elastic modulusG and the lifetimeτ0 practically determine the limit response of the material at low flow rates, whereas the magnitude of the diffusive time constantτ1 relative toτ0 is responsible for the ultimate properties of the fluid. Ifτ0/τ1 >τc ≃ 0.933 the elongational viscosity curve shows a vertical asymptote, whereas a turbulent transition occurs at finite velocity gradients ifτ0/τ1 <τc. The latter range may correspond to the physical situation actually found in very concentrated systems, such as gels and polymer melts.