Shear-thickening flow of suspensions of carbon nanofibers in aqueous PVA solutions

The suspensions of carbon nanofibers in aqueous poly(vinyl alcohol) solutions were prepared in the presence of spherical carbon black particles, and the steady-shear viscosity and dynamic viscoelasticity were measured for complex suspensions. Although the single suspensions of carbon black are highly stable, the flocculation of carbon nanofibers is promoted by the addition of carbon black particles. The complex suspensions show remarkable shear thickening in the steady-flow and strain hardening in oscillatory shear with large amplitude. The nonlinear responses strongly depend on the carbon black concentration, whereas the dynamic viscoelasticity at low strains in the linear ranges is not significantly influenced. As the highly elastic effects arise from the long-range motion of particles, the possible mechanism may be the orientation of nanofibers in strong shear fields. The suspensions show the time-dependent behavior of viscosity when the time-scale of measurements is shorter than that of structural recovery to the isotropic states.     

Rheology of concentrated microgel solutions

Viscosity, modulus, and yield stress for 0–6 wt% aqueous solutions of Carbopol 941 were investigated using constant shear rate, constant shear stress, and dynamic oscillatory experiments. The microgel character of the polymer was evident from the solid-like behavior of the solutions above 1 wt%. Yield stress increased with concentration, but yield occurred at a critical shear strain of 40%, independent of concentration. The static stress-strain relationship became non-linear at ~ 25% strain, in fair agreement with the onset of non-linear response in the storage modulus at ~ 10% strain. Small strain moduli from static and low frequency measurements agreed rather well; modulus values obtained from the recoverable strain after yielding were 30–40% smaller. Solutions flowed at near-constant stress in the low shear rate regime; at higher rates the stress increases with shear rate more rapidly. The viscosity did not obey the Cox-Merz rule. Steady-state viscosity scaled with polymer concentration to the 3/4 power. Results were interpreted using a cellular, deformable sphere model for the polymer, in analogy to emulsions and foams.     

Volume changes in glassy polymers

Summary Constitutive equations are derived for compressible glassy polymers at non-isothermal loading with finite strains. The model is based on the theory of temporary networks in its version of adaptive links concept. The stress–strain relations are applied to the analysis of uniaxial extension of a viscoelastic bar. Explicit formulas are developed for time-dependent Young's modulus and Poisson's ratio of the bar at small strains. Results of numerical simulation are compared with experimental data for polycarbonate, polyethylene, and poly(methyl methacrylate). It is demonstrated that (i) longitudinal stresses do not affect the specific free volume in the region of linear viscoelasticity at strains up to about 0.2%, and cause substantial changes in the free volume in the region of nonlinear viscoelasticity at strains about 1.0%; (ii) in the latter case, the increment of the free-volume fraction is proportional to the increase in the specific volume. Received 3 April 1998; accepted for publication 22 June 1998     

A critical state sand model with elastic–plastic coupling

Soil elastic moduli are highly pressure-dependent. Experimental findings have indicated that the elastic shear modulus of sands depends on p^χ, where p is mean principal effective stress and χ is a non-dimensional parameter. χ practically remains unchanged for shear strains less than 10⁻⁵ where the mechanical behavior is purely elastic. However, experiments have revealed that the emergence of plasticity for shear strains larger than 10⁻⁵ provokes a gradual increase in χ. Technically, this observation is an elastic–plastic coupling effect in which plasticity causes to change the elastic characteristics. Here, this issue is considered in hyper-elasticity framework in conjunction with a critical state compatible bounding surface plasticity platform for granular soils. To this aim, constitutive equations linking χ to a proper kinematic hardening parameter are presented. Then, using the proposed approach, a hyper-elastic theory is modified to consider the mentioned elastic–plastic coupling effect in the whole domain of the elastoplastic behavior. Adopting the improved hyper-elasticity necessitates the modification of a number of basic plasticity platform elements. In this regard, dilatancy and plastic hardening modulus of the bounding surface platform are modified. Successful performance of the modified constitutive model is presented against experimental data of loading/unloading triaxial tests.     

Surface tension driven jet break up of strain-hardening polymer solutions

Experimental studies attempting to ascertain the influence of viscoelasticity on the atomization of polymer solution are often hindered by the inability to decouple the effect of shear thinning from the effect of extensional hardening. Here, the influence of viscoelasticity on the jet break up of a series of non-shear-thinning viscoelastic fluids is quantified. Previous characterization using an opposed-nozzle rheometer identified the critical extensional rates for strain hardening of these model fluids. The strain hardening fluids exhibit a beads-on-string structure with reduction or elimination of satellite drops. Capillary instabilities grow on the filaments connecting the spheres and eventually break the filaments up into a string of very small drops about one order of magnitude smaller than the satellite drops formed by a Newtonian fluid with the same shear viscosity, surface tension, and density. These results confirm that strain hardening is the key rheological property in jet break up and that the critical extensional rate of a fluid is pertinent in determining the final characteristics of break up. Results suggest that the opposed-nozzle rheometer does probe extensional behavior in the range of extensional rates that are relevant to jet break up, providing a tool to roughly predict jet break up.     

Application of a discrete polycrystal model to the analysis of cyclic straining in copper

A discrete polycrystal model, designed to simulate a metal aggregate macro-element, is applied to the study of cyclic straining in copper. The numerical method of solution (an adaptation of the “finite element method”) incorporates a convergent discrete Green's function within the constrained minimum principle which governs the (crystallographic) plastic shear increments at each load step. Isothermal elastic moduli of copper crystals and Taylor's hardening rule with constant hardening modulus are used in the calculations. Numerical results are obtained for macroscopic elastic properties, cyclic stress-strain curves (which indicate the contribution of aggregate heterogeneity to macroscopic hardening), macroscopic plastic work, and residual (latent) strain energy through four loading cycles between fixed macrostrain limits. Other estimates for elastic properties also are included, and all results are compared, both qualitatively and quantitatively, with published experiments. The predictions of the model are in general satisfactory.     

A Study of Large Plastic Deformations in Dual Phase Steel Using Digital Image Correlation and FE Analysis

Large plastic deformation in sheets made of dual phase steel DP800 is studied experimentally and numerically. Shear testing is applied to obtain large plastic strains in sheet metals without strain localisation. In the experiments, full-field displacement measurements are carried out by means of digital image correlation, and based on these measurements the strain field of the deformed specimen is calculated. In the numerical analyses, an elastoplastic constitutive model with isotropic hardening and the Cockcroft–Latham fracture criterion is adopted to predict the observed behaviour. The strain hardening parameters are obtained from a standard uniaxial tensile test for small and moderate strains, while the shear test is used to determine the strain hardening for large strains and to calibrate the fracture criterion. Finite Element (FE) calculations with shell and brick elements are performed using the non-linear FE code LS–DYNA. The local strains in the shear zone and the nominal shear stress-elongation characteristics obtained by experiments and FE simulations are compared, and, in general, good agreement is obtained. It is demonstrated how the strain hardening at large strains and the Cockcroft–Latham fracture criterion can be calibrated from the in-plane shear test with the aid of non-linear FE analyses. An erratum to this article can be found at     

Suspensions of monodisperse spheres in polymer melts: particle size effects in extensional flow

Suspensions in polymeric, viscoelastic liquids have been studied in uniaxial extensional flow. The fibre wind-up technique has been used for this purpose. The effects of particle size and particle volume fraction have been investigated, using monodisperse, spherical particles. The results have been compared with shear flow data on the same materials. The values of the relative extensional viscosities at low stretching rates are in agreement with the relative shear viscosities and relative moduli. This indicates that hydrodynamic forces are stronger than the particle interaction forces. At larger strain rates strain hardening occurs; it is suppressed when particles are added. Small aggregating particles reduce the strain hardening more strongly than larger particles; strain hardening can even be totally eliminated. When further increasing the stretching rate, hydrodynamic effects dominate again and the effect of particle size effect on strain hardening disappears.     

A plastic flow rule at a yield vertex

A yield vertex having three distinct facets at the uniaxial stress point is considered. An associated plastic flow rule is constructed using a new 3 × 3 coupled hardening matrix. This has the property that the incipient shear modulus is less than the elastic value. The approach is novel and distinct from previous work having that conclusion. In particular, all the relevant incipient moduli governing fully active in-plane loading can be fitted, if desired, to those values which J₂-deformation theory would require. To that extent the proposed incremental theory therefore legitimizes the use of the latter moduli, for example in certain ‘paradoxical’ buckling problems. When the moduli of the incremental theory are so chosen, the domain of stress-rate vectors enforcing loading is a calculable pyramid which contracts from the Mises half-space as the stress increases beyond yield. This domain becomes wider, at a given stress, as the initial smooth curvature of the stress-strain curve is imagined to become sharper.     

In search of physical meaning: defining transient parameters for nonlinear viscoelasticity

A complete set of model-independent viscoelastic functions for understanding responses to transient nonlinear rheological tests is presented, using large-amplitude oscillatory shear strain as a model nonlinear protocol. The derivation makes no assumptions about symmetries, and is therefore applicable to the responses to any input, allowing researchers to unambiguously define time-dependent moduli, viscosities, compliances, fluidities, and normal stress coefficients. A legend for interpreting the dynamic trajectories in modulus space is provided, along with explicit definitions of the rates at which the moduli change. These provide a quantitative mechanism to identify when, and by how much, a material response stiffens, softens, thickens, or thins while being deformed. In addition to providing analytical expressions for the moduli, the derivation requires the definition of a conceptually new term. This means there exist three, not two, time-dependent nonlinear viscoelastic functions by which any response can be fully described. The third function accounts for nonlinear properties such as yield stresses and the shifting of the strain equilibrium. This complete analysis scheme is unique in making a distinction between the strains in the lab and material frames. The quantitative sequence of physical process analysis, which is fully developed in this work, allows for comprehensive physical interpretations of responses to transient deformations of any kind to be made, including the steady alternance responses to large-amplitude oscillatory shear (LAOS), time-dependent oscillatory shear startup responses, and thixotropic and anti-thixotropic responses.     

Prediction of the initial thickness of shear band localization based on a reduced strain gradient theory

Plastic flow localization in ductile materials subjected to pure shear loading and uniaxial tension is investigated respectively in this paper using a reduced strain gradient theory, which consists of the couple-stress (CS) strain gradient theory proposed by Fleck and Hutchinson (1993) and the strain gradient hardening (softening) law (C–W) proposed by Chen and Wang (2000). Unlike the classical plasticity framework, the initial thickness of the shear band and the strain rate distribution in both cases are predicted analytically using a bifurcation analysis. It shows that the strain rate is obviously non-uniform inside the shear band and reaches a maximum at the center of the shear band. The initial thickness of the shear band depends on not only the material intrinsic length l_cs but also the material constants, such as the yield strength, ultimate tension strength, the linear hardening and softening shear moduli. Specially, in the uniaxial tension case, the most possible tilt angle of shear band localization is consistent qualitatively with the existing experimental observations. The results in this paper should be useful for engineers to predict the details of material failures due to plastic flow localization.     

Experimental strain analysis of the losipescu shear test specimen

An experimental strain analysis of the losipescu shear test specimen was performed, utilizing a 20-ply AS4/3501-6 carbon/epoxy unidirectional composite. Using three-element strain-gage rosettes, it was shown that the presence of loading-point-induced transverse normal strains in the gage section do not affect the measured shear strain. Thus, the shear modulus determined using the standard notch specimen is not affected. Likewise, modulus determination is not influenced by cracking at the notch tips, since this occurs at strains beyond the range over which modulus is determined. To further evaluate the effect of notch-tip cracking, material was removed adjacent to the standard V-notches where these cracks initiate. The measured shear strength was unaffected by removing this material, although the shear modulus was reduced slightly (by as much as eight percent for the more grossly exaggerated geometries). E.Q. Lewis, former graduate student, is now Engineer, Lockheed Corporation.     

A Kolsky Torsion Bar Technique for Characterization of Dynamic Shear Response of Soft Materials

A novel Kolsky torsion bar technique is developed and successfully utilized to characterize the high strain rate shear response of a rate-independent end-linked polydimethylsiloxane (PDMS) gel rubber with a shear modulus of about10 KPa. The results show that the specimen deforms uniformly under constant strain rate and the measured dynamic shear modulus follows reasonably well the trend determined by dynamic mechanical analysis (DMA) at lower strain rates. For comparison, Kolsky compression bar experiments are also performed on the same gel material with annular disk specimens. The dynamic moduli obtained from compression experiments, however, are an order of magnitude higher than those obtained by the torsional technique, due to the pressure caused by the radial inertia and end constraints.     

Stability and finite strain of homogenized structures soft in shear: Sandwich or fiber composites,and layered bodies

The stability theories energetically associated with different finite strain measures are equivalent if the tangential moduli are transformed as a function of the stress. However, for homogenized soft-in-shear composites, they can differ greatly if the material is in small-strain and constant elastic moduli measured in small-strain tests are used. Only one theory can then be correct. The preceding variational energy analysis showed that, for sandwich columns and elastomeric bearings, respectively, the correct theories are Engesser’s and Haringx’s, associated with Green’s and Almansi’s Lagrangian strain tensors, respectively. This analysis is reviewed, along with supporting experimental and numerical results, and is then extended to arbitrary multiaxially loaded homogenized soft-in-shear orthotropic composites. It is found that, to allow the use of constant shear modulus when the material is in small strain, the correct stability theory is associated with a general Doyle–Ericksen finite strain tensor of exponent m depending on the principal stress ratio. Further it is shown that the standard updated Lagrangian algorithm for finite element analysis, which is associated with Green’s Lagrangian finite strain, can give grossly incorrect results for homogenized soft-in-shear structures and needs to be generalized for arbitrary finite strain measure to allow using constant shear modulus for critical loads at small strain.     

Effects of particle softness on the rheology and yielding of colloidal glasses

We studied the linear and nonlinear rheology of colloidal glasses consisting of hard spheres and soft core-shell particles at several volume fractions to explore the effects of particle softness on the mechanical properties and yielding. Creep and recovery and oscillatory shear measurements were used to determine the shear elastic modulus and the yield strain. Both hard and soft sphere glasses exhibited ‘entropic cage elasticity’ below random close packing, whereas for compressed soft spheres at higher effective volume fractions, the yield strain was determined by shell elasticity. The shear modulus followed a strong increase with volume fraction for hard spheres and a much weaker one for soft particles reflecting their interparticle potential. Nonlinear effects, revealed as strong distortions of the stress signal during yielding, were analyzed via Fourier transform rheology and Lissajous plots. The significant contribution of the nonlinearities was analyzed in terms of strain softening and hardening mechanisms within a cycle of oscillation and discussed in relation to particle softness.     

Moving crack for functionally grated material in an infinite length strip under antiplane shear

Considered is a Yoffe crack in an infinite strip of functionally grated material (FGM) subjected to antiplane shear. The shear moduli in two directions of FGM are assumed to be of exponential form. The dynamic stress intensity factor and strain energy density factor at the crack tip are obtained by using integral transforms and dual-integral equations. The numerical results show that the decrease of the strain energy density factor varies with the shear moduli gradient, and the increase of the strain energy density factor varies with the increase of the moving crack speed. The ratio of shear moduli in material vertical orientation has a great influence on the strain energy density factor.     

Manifestation of phase separation processes in oscillatory shear: droplet-matrix systems versus co-continuous morphologies

Phase separation processes in mixtures of poly-α-methylstyrene-co-acrylonitrile (PαMSAN) and poly-methylmethacrylate (PMMA) with lower critical solution temperature (LCST) behavior have been studied, focusing on the manifestation of the interface in oscillatory shear measurements. By using blends of different composition, systems with a droplet-matrix morphology or a co-continuous structure are generated during the phase separation process. The feasibility of probing this morphology development by rheological measurements has been investigated. The development of a disperse droplet phase leads to an increase in the elasticity of the blend at low frequency, showing up as a shoulder in the plot of storage modulus versus frequency. Here, the droplet growth is unaffected by the shear amplitude up to strains of 0.2; therefore the resulting dynamic data are suitable for quantitative analysis. In contrast, for blends in which phase separation leads to a co-continuous structure, the storage modulus shows a power law behavior at low frequency and its value decreases as time proceeds. For the latter systems, effects of the dynamic measurement on the morphology development have been observed, even for strain amplitudes as low as 0.01. To probe the kinetics of morphology evolution in droplet-matrix systems, measurements of the time dependence of the dynamic moduli at fixed frequency should be performed (for a whole series of frequencies). Only from such measurements, curves of the frequency dependence of the moduli at a well defined residence time can be constructed. From fitting these curves to the emulsion model of Palierne, the droplet diameter distribution at that particular stage in the phase separation and growth process can be obtained. It is not appropriate to use a simplified version of the Palierne model containing only the average droplet size, because a morphology with too broad a size distribution is generated. Received: 15 February 1999 Accepted: 20 May 1999     

Large amplitude oscillatory shear rheology of three different shear-thickening particle dispersions

We present a large amplitude oscillatory shear rheology (LAOS) investigation of three different shear-thickening particle dispersions - fumed silica in polyethylene oxide (FLOC), fumed silica in polypropylene glycol (HydroC), and cornstarch in water (JAM). These systems shear-thicken by three different mechanisms - shear-induced formation of particle clusters flocculated by polymer bridging, hydrocluster formation, and jamming. The viscoelastic non-linearities of the three fluids were studied as a function of strain and strain-rate space through the use of Lissajous-Bowditch curves and local nonlinear viscoelastic moduli of an oscillatory shear cycle. The nonlinear behaviors of the three fluids were compared and contrasted to understand the nonlinear shear-thickening mechanism of each. Both HydroC and JAM dispersions were found to exhibit strong strain stiffening of the elastic moduli and strain thickening of the loss moduli behavior associated with possible hydrocluster formation and particle jamming. However, the FLOC dispersion, in contrast, showed strong strain softening and strain thinning behavior at large strain amplitudes associated with yielding of the microstructure. The expected thickening of the loss modulus of FLOC in LAOS with increasing strain was not observed even though viscosity of FLOC was found to shear-thicken in steady-shear measurements. This disagreement is likely due to very large strain amplitudes required for shear-thickening to occur by shear-induced polymer bridging mechanism. The hypothesis was confirmed through stress growth experiments. Conversely, the HydroC and JAM dispersions required relatively small applied strains for shear-thickening to occur by hydrocluster and jamming mechanism. The comparison of local intra-cycle nonlinearity through Lissajous-Bowditch plots and nonlinear viscoelastic parameters indicated that the elastic nonlinearities of all three systems are primarily driven by a strong dependence on the magnitude of the applied strain-rates within an oscillatory cycle rather than the amplitude of the applied strain. A close inspection of the LAOS data reveals strong differences in the viscoelastic nonlinearities of these three different shear-thickening dispersions which can be used to create a nonlinear rheological fingerprint for each and offers valuable new insights into the nonlinear dynamics associated with each of the shear-thickening mechanisms.