Comparison of the elongational behavior of various polyolefins in uniaxial and equibiaxial flows

For processing operations with a pronounced elongational component, it was found that the uniformity of extruded items is improved by the presence of strain hardening usually measured in uniaxial elongation. Many processing operations such as foaming, film blowing, and blow molding are dominated by biaxial deformations, however, and therefore, the question arises how strain hardening in uniaxial and biaxial deformation compares. Besides a linear and long-chain branched PP, one classical LDPE, an HDPE pipe extrusion grade with a bimodal MMD, and a LCB-mPE were also characterized. For the measurements in uniaxial elongation the Münstedt tensile rheometer (MTR) and the ARES-EVF were used, while the lubricated flow method was applied for equibiaxial deformation. It was found that the strain hardening in uniaxial elongation is more pronounced. The dependence of strain hardening on strain rate is qualitatively the same in both modes. In the range of strain rates, the chosen long-chain branched LDPE and PP exhibit a strain hardening, which is approximately independent of the elongational rates applied, whereas for the HDPE it becomes smaller with increasing rate.

Structure and rheology of model side-chain liquid crystalline polymers with varying mesogen length

We probed the structural transitions and rheological properties of norbornene-based polymeric liquid crystals with attached side mesogenic groups of varying length. Whereas the short side chain liquid crystals (C5) formed a nematic mesophase, the long ones (C9) formed a smectic-A solid. The degree of polymerization of the main backbone affected the isotropic-mesophase transition temperatures, but did not influence the change of the mesophase dynamics from the isotropic to the anisotropic state. The temperature dependence of the shift factors obtained from the time-temperature superposition were divided into three regions, namely isotropic, mesophase, and transition, all exhibiting Arrhenius behavior except for the nematic fluid, which followed a WLF dependence. A remarkable strain hardening observed in both nematic and smectic samples was attributed to the dynamic coupling of the main chain and the mesogenic side groups. By applying large amplitude oscillatory shear, macroscopic alignment was achieved; the long recovery times upon flow cessation suggested a very slow structural reorganization, which was much longer in the smectics compared to the nematics, possibly due to the involvement of smaller length scales in the former case. Received: 26 December 2000/Accepted: 26 February 2001     

Characterization of polymer dispersions by Fourier transform rheology

Fourier transform rheology is a very sensitive technique to characterize non-linear rheological fluid properties. It has been applied here for the first time to polymer dispersions in water and the results are compared to those from conventional rheology, namely steady and small amplitude oscillatory shear experiments. The investigated systems are mainly based on styrene and n-butylacrylate. A first attempt was made to evaluate how far colloidal parameters like particle volume fraction and ionic strength as well as chemical composition and surface characteristics of the dispersed particles are reflected in FT-rheology spectra. Significantly different non-linearities are observed for highly concentrated dispersions of particles with different T_g. These differences are not detected in linear oscillatory shear and show up in steady shear only at significantly higher shear rates. Particle surface characteristics influence the non-linear response in oscillatory shear significantly and the intensity of the overtones is found to be higher for a dispersion of particles with a “hairy” swollen surface layer as compared to a system of smooth particles, although the solids content was adjusted to match the steady shear viscosity. The intensity of the overtones in FT-rheology strongly decrease upon dilution. At a solid content below 35% no differences are observed in the FT-experiments for the systems investigated here, whereas the differences in steady shear are very pronounced in this concentration range. A significant influence of added salt onto the non-linear response is detected for some systems, which might be correlated to the stability of these systems. The observed phenomena certainly cannot be explained in terms of constitutive equations or microstructural statistical mechanical models at present. Thus, FT-rheology yields information complementary to classical steady or linear oscillatory shear experiments. Received: 11 December 2000 Accepted: 8 April 2001     

Small and large strain rheology of wheat gluten

 The rheological properties of wheat gluten were studied under both small and large deformation and compared with those of the parent flours. The limiting strain of linear viscoelastic behaviour of gluten doughs, 3 × 10⁻², was an order of magnitude larger than that of the flour doughs, 10⁻³. The role of starch in the lower limiting strain of flour doughs was indicated by the exponential decrease in the limiting strain of gluten-starch mixtures with greater quantities of starch. Large strain measurements showed gluten doughs possessed greater shear and elongational viscosities than flour doughs and these differences were greatest at lower shear and elongation rates (0.01 and 0.1 s⁻¹). The larger viscosities of flour and gluten doughs at the low strain rates help to stabilise and prevent the collapse of gas bubbles during bread fermentation and baking. Increasing starch levels in gluten-starch mixtures, at either constant or optimal water levels, lowered the elongational viscosity. Dynamic measurements were, however, more sensitive to the level of water added to the gluten-starch mixtures. The storage modulus decreased with increasing starch levels when constant water levels were used to prepare the mixtures, but when optimal water levels were used the storage modulus increased. Gluten and starch are major contributors to the large and small strain rheological properties of flour doughs; however, gluten-starch mixtures were unable to duplicate exactly the rheological properties of flour doughs, indicating that other flour components such as pentosans, lipids and water soluble proteins also influence dough rheology. Received: 20 March 2001 Accepted: 11 July 2001     

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a ₂, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter describing maximum stretch of the polymer chains.     

Detection and quantification of industrial polyethylene branching topologies via Fourier-transform rheology,NMR and simulation using the Pom-pom model

The significance of sparse long-chain branching in polyolefines towards mechanical properties is well-known. Topology is a very important structural property of polyethylene, as is molecular weight distribution. The method of Fourier-transform rheology (FTR) and melt state nuclear magnetic resonance (NMR) is applied for the detection and quantification of branching topology (number of branches per molecule), for industrial polyethylenes of various molecular weight and molecular weight distributions. FT rheology consists of studying the development of higher harmonics contribution of the stress response to a large amplitude oscillatory shear deformation. In particular, when applying large-amplitude oscillatory shear (LAOS), one observes the development of mechanical higher harmonic contributions at 3ω ₁, 5ω ₁,..., in the shear stress response. We correlate the relative intensity, I _3/1, and phase Φ ₃ of these harmonics with structural properties of industrial polyethylene, i.e. polymer topology and molecular weight distribution. Experiments are complemented by numerical simulations, using a multimode differential Pom-pom constitutive model (DCPP formulation), by fitting to the experimental linear and nonlinear viscoelastic behaviours. Simulation results in the nonlinear regime are correlated with molecular properties of the “pom-pom” macromolecular architecture. Qualitative agreement is found between predicted and experimental FT rheology results.     

Melt rheology and structure of silicone resins

Rheological investigation and X-ray analysis of silicone resins have been performed to get a fundamental understanding of the relationship between melt rheology and structure. Rheological properties of the melts of silicone resins were characterized by dynamic shear measurements. Samples were a series of silicone resins having different side groups, such as methyl, phenyl, methylphenyl, and propylphenyl. A time-temperature superposition based on the WLF rule was successfully applicable. The free volume fraction at the glass transition temperature and the free volume expansion coefficient were estimated from the C₁ and C₂ values of the WLF-equation. The loss modulus G′′ was found to be proportional to the angular frequency in a double-logarithmic plot over a wide frequency range. However, the storage modulus G′ exhibited a bending in a intermediate frequency region for all silicone resins. The shape of the G′ curve is unexpected from the results of gel permeation chromatography (GPC) and differential scanning calorimeter (DSC), which demonstrate that the silicone resins are amorphous polymers of relatively low molecular weight. To get more insight into the rheological properties of the silicone resins which indicate a heterogeneous structure, X-ray analysis was carried out. The X-ray measurement displayed two peaks, one broad peak around 4.5?Å of d-spacing which corresponds to a random amorphous structure, and the other higher intensity peak from 8.5?Å to 12?Å. This peak strongly depends on the size of organic side groups. From the rheological characterization and the X-ray analysis, it can be concluded that the silicone resins consist of two components; one of them shows a random amorphous and the other a regular structure even in the molten stage.     

Melt shear rheology of carbon nanofiber/polystyrene composites

The rheological behavior and morphology of carbon nanofiber/polystyrene (CNF/PS) composites in their melt phase have been characterized both through experimental measurements and modeling. Composites prepared in the two different processes of solvent casting and melt blending are contrasted; melt-blended and solvent-cast composites were each prepared with CNF loadings of 2, 5, and 10 wt%. A morphological study revealed that the melt blending process results in composites with shorter CNFs than in the solvent-cast composites, due to damage caused by the higher stresses the CNFs encounter in melt blending, and that both processes retain the diameter of the as-received CNFs. The addition of carbon nanofiber to the polystyrene through either melt blending or solvent casting increases the linear viscoelastic moduli, G′ and G″, and steady-state viscosity, η, in the melt phase monotonically with CNF concentration, more so in solvent cast composites with their longer CNFs. The melt phase of solvent-cast composites with higher CNF concentrations exhibit a plateau of the elastic modulus, G′, at low frequencies, an apparent yield stress, and large first normal stress difference, N ₁, at low strain rates, which can be attributed to contact-based network nanostructure formed by the long CNFs. A nanostructurally-based model for CNF/PS composites in their melt phase is presented which considers the composite system as rigid rods in a viscoelastic fluid matrix. Except for two coupling parameters, all material constants in the model for the composite systems are deduced from morphological and shear flow measurements of its separate nanofiber and polymer melt constituents of the composite. These two coupling parameters are polymer–fiber interaction parameter, σ, and interfiber interaction parameter, C _I. Through comparison with our experimental measurements of the composite systems, we deduce that σ is effectively 1 (corresponding to no polymer–fiber interaction) for all CNF/PS nanocomposites studied. The dependence of CNF orientation on strain rate which we observe in our experiments is captured in the model by considering the interfiber interaction parameter, C _I, as a function of strain rate. Applied to shear flows, the model predicts the melt-phase, steady-state viscosities, and normal stress differences of the CNF/PS composites as functions of shear rate, polymer matrix properties, fiber length, and mass concentration consistent with our experimental measurements.     

Rheological properties of branched polystyrenes: nonlinear shear and extensional behavior

Nonlinear shear and uniaxial extensional measurements on a series of graft-polystyrenes with varying average numbers and molar masses of grafted side chains are presented. Step-strain measurements are performed to evaluate the damping functions of the melts in shear. The damping functions show a decreasing dependence on strain with an increase in mass fraction of grafted side chains. Extensional viscosities of the melts of graft-polystyrenes exhibit a growing strain hardening with increasing average number of grafted side chains as long as the side branches have a sufficient molar mass to be entangled. Graft-polystyrenes with side arms below the critical molar mass M _c for entanglements of linear polystyrene but above the entanglement molar mass M _e of linear polystyrene (M _e < M _w,br < M _c) still show a distinct strain hardening. With decreasing molar mass of the grafted side chains (M _w,br < M _e) the nonlinear-viscoelastic properties of the graft-polystyrene melts approach the behavior for a linear polystyrene with comparable polydispersity.Electronic Supplementary Material Supplementary material is available for this article at     

Material aspects of dynamic neck retardation

Neck retardation in stretching of ductile materials is promoted by strain hardening, strain-rate hardening and inertia. Retardation is usually beneficial because necking is often the precursor to ductile failure. The interaction of material behavior and inertia in necking retardation is complicated, in part, because necking is highly nonlinear but also because the mathematical character of the response changes in a fundamental way from rate-independent necking to rate-dependent necking, whether due to material constitutive behavior or to inertia. For rate-dependent behavior, neck development requires the introduction of an imperfection, and the rate of neck growth in the early stages is closely tied to the imperfection amplitude. When inertia is important, multiple necks form. In contrast, for rate-independent materials deformed quasi-statically, single necks are preferred and they can emerge in an imperfection-free specimen as a bifurcation at a critical strain. In this paper, the interaction of material properties and inertia in determining neck retardation is unraveled using a variety of analysis methods for thin sheets and plates undergoing plane strain extension. Dimensionless parameters are identified, as are the regimes in which they play an important role.     

Effect of the kinematic hardening on the plastic anisotropy parameters for metallic sheets

The initial plastic anisotropy parameters are conventionally determined from the Lankford strain ratios defined by $r^{\psi }=\frac{{\epsilon }_{22}^{\mathrm{p}\psi }}{{\epsilon }_{33}^{\mathrm{p}\psi }}$ (ψ being the direction of the loading path). They are usually considered as constant parameters that are determined at a given value of the plastic strain far from the early stage of the plastic flow (i.e. equivalent plastic strain of ${\epsilon }_{\mathrm{eq}}^{\mathrm{p}}=0.2\text{\%}$) and typically at an equivalent plastic strain in between 20% and 50% of plastic strain failure (or material ductility). What prompts to question about the relevance of this determination, considering that this ratio does not remain constant, but changes with plastic strain. Accordingly, when the nonlinear evolution of the kinematic hardening is accounted for, the Lankford strain ratios are expected to evolve significantly during the plastic flow.In this work, a parametric study is performed to investigate the effect of the nonlinear kinematic hardening evolution of the Lankford strain ratios for different values of the kinematic hardening parameters. For the sake of clarity, this nonlinear kinematic hardening is formulated together with nonlinear isotropic hardening in the framework of anisotropic Hill-type (1948) yield criterion. Extension to other quadratic or non-quadratic yield criteria can be made without any difficulty. This parametric study is completed by studying the effect of these parameters on simulations of sheet metal forming by large plastic strains.     

The strain-hardening behaviour of linear and long-chain-branched polyolefin melts in extensional flows

By generalizing the Doi-Edwards model to the Molecular Stress Function theory of Wagner and Schaeffer, the extensional viscosities of polyolefin melts in uniaxial, equibiaxial and planar constant strain-rate experiments starting from the isotropic state can be described quantitatively. While the strain hardening of four linear polymer melts (two high-density polyethylenes, a polystyrene and a polypropylene) can be accounted for by a tube diameter that decreases affinely with the average stretch, the two long-chain-branched polymer melts considered (a low-density polyethylene and a long-chain branched polypropylene) show enhanced strain hardening in extensional flows due to the presence of long-chain branches. This can be quantified by a molecular stress function, the square of which is quadratic in the average stretch and which follows from the junction fluctuation theory of Flory. The ultimate magnitude of the strain-hardening effect is governed by a maximum value of the molecular stress, which is specific to the polymer melt considered and which is the only free non-linear parameter of the theory. Received: 1 June 1999/Accepted: 24 November 1999     

Modelling of the internal stress in dislocation cell structures

The nonuniform distribution of dislocations in metals gives rise to material anisotropy and internal stresses that determine the mechanical response. This paper proposes a micromechanical model of a dislocation cell structure that accounts for the material inhomogeneity and incorporates the internal stresses in a physically-based manner. A composite model is employed to describe the material with its dislocation cell structure. The internal stress is obtained as a natural result of plastic deformation incompatibility and incorporated in the composite model. Applications of this model enable the prediction of the mechanical behavior of metals under various nonuniform deformations. The implementation of the model is relatively straightforward, allowing easy use in macroscopic engineering computations.     

Prediction of normal stresses under large amplitude oscillatory shear flow

The dynamic response of viscoelastic fluids under large amplitude oscillatory shear (LAOS) has been a subject of long history. In the LAOS flow, the analysis has been mostly focused on shear stress, possibly due to the lack of accurate measurement of normal stress. However, the normal stress may become larger than shear stress at high-strain amplitudes, and thus it is important that we have a good understanding of the normal stress behavior. Furthermore, with the advancement in the instrumentation, it has become possible to get more reliable data. The purpose of this paper is to develop a research platform to analyze and to understand the normal stress behavior of complex fluids under LAOS flow. In this study, we utilized the Giesekus model as a representative constitutive model, and investigated its diverse responses. We defined the dynamic properties corresponding to normal stress, in a similar way to define dynamic moduli from shear stress, and examine their behavior with various analyzing tools. Experimental data were also compared with model predictions. Despite the fact that it is not yet possible to compare all of the predictions because of instrumental limitation, the prediction has been found to fit well with the experimental data. This study is expected to provide a useful framework for further understanding the nonlinear behavior of complex fluids at large deformation.     

The Modified Rockwell Test: A New Probe for Mechanical Properties of Metals

In the present work a novel methodology is proposed, based on the combination of the Rockwell and the Vickers tests, to provide estimates of the mechanical properties of metal substrates. The analysis is based on some novel invariants obtained from the finite element solution of the Vickers indentation (the imprint diagonal relates to the maximum indentation depth and the residual indentation depth with the average pressure and the elastic modulus). Several other useful results are discussed and experiments are performed with a modified Rockwell apparatus on steel and aluminium alloys. The results are important for the interpretation of micro indentation tests. Inverting the indentation data, reasonably accurate results can be obtained for strain hardening properties for “power law” behaviour, whereas more complex strain hardening would require further investigation.     

Strain hardening behavior of polymer blends with fibril morphology

We investigate the rheological behavior of the polymer blends with fibril morphology, with special focus on the effect of fibril morphology on the extensional properties under uniaxial extension. We add a small amount of the dispersed phase to the matrix, and control the blend morphology by changing the viscosity ratio. When the fibril morphology is maintained, the blend shows not only a significant increase of the extensional viscosity but the strain hardening behavior. The extensional viscosity increases depending on the aspect ratio of the fibers, while the strain hardening behavior originates from the restricted stretching of deformable fibers, which has been confirmed theoretically by introducing the concept of rigidity of the fiber. It suggests a way to induce the strain hardening behavior by introducing deformable fibrils into the matrix, that is, by the design of polymer blends with a small amount of dispersed phase such that the fibril structure is maintained.     

Extension of the static shakedown theorems to softening materials

A generalization of the static shakedown theorems for elastic plastic hardening solids with isotropic [Mech. Res. Commun. 29 (2002) 159] and anisotropic [Acta Mechanica, 2004] damage accounting for the possibility of material softening is proposed.     

A review of some plasticity and viscoplasticity constitutive theories

The purpose of the present review article is twofold:

•

recall elementary notions as well as the main ingredients and assumptions of developing macroscopic inelastic constitutive equations, mainly for metals and low strain cyclic conditions. The explicit models considered have been essentially developed by the author and co-workers, along the past 30 years;     

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.