The BKZ as an alternative to the Wagner model for fitting shear and elongational flow data of an LDPE melt

For the low density polyethylene Melt I, which is the melt for which the most complete set of shear and elongational data exists, the semi-empirical single integral Wagner model gives an excellent data-fit, but suffers the drawback of having no entropic constitutive equation, that is a relationship between strain history and elastic free energy from which viscous heating and cohesive failure can be predicted. We show here that the BKZ model, which does possess an entropic constitutive equation, gives as good a fit as does the Wagner model to both the shear and elongational data.     

Elongational flow behavior of polymeric fluids according to the Leonov model

The viscoelastic behavior of polymeric systems based upon the Leonov model has been examined for (i) the stress growth at constant strain rate, (ii) the stress growth at constant speed and (iii) the elastic recovery in elongational flow. The model parameters have been determined from the available rheological data obtained either in steady shear flow (shear viscosity and first normal-stress difference as a function of shear rate) or oscillatory flow (storage and loss moduli as a function of frequency in the linear region) or from extensional flow at very small strain rates (time-dependent elongation viscosity in the linear viscoelastic limit). In addition, the effect of the parameter characterizing the strain-hardening of the material during elongation has also been studied. The estimation of this parameter has been based upon the structural characteristics of the polymer chain which include the critical molecular weight and molecular weight of an independent segment. Five different polymer melts have been considered with varying number of modes (maximum four modes). Resulting predictions are in fair agreement with corresponding experimental data in the literature.     

A modification of the convective constraint release mechanism in the molecular stress function model giving enhanced vortex growth

The molecular stress function model with convective constraint release (MSF with CCR) constitutive model [M.H. Wagner, P. Rubio, H. Bastian, The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release, J. Rheol. 45 (2001) 1387] is capable of fitting all viscometric data for IUPAC LDPE, with only two adjustable parameters (with difference found only on reported “steady-state” elongational viscosities). The full MSF with CCR model is implemented in a backwards particle-tracking implementation, using an adaptive method for the computation of relative stretch that reduces simulation time many-fold, with insignificant loss of accuracy. The model is shown to give improved results over earlier versions of the MSF (without CCR) when compared to well-known experimental data from White and Kondo [J.L. White, A. Kondo, Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation, J. Non-Newtonian Fluid Mech. 3 (1977) 41]; but still to under-predict contraction flow opening angles. The discrepancy is traced to the interaction between the rotational dissipative function and the large stretch levels caused by the contraction flow. A modified combination of dissipative functions in the constraint release mechanism is proposed, which aims to reduce this interaction to allow greater strain hardening in a mixed flow. The modified constraint release mechanism is shown to fit viscometric rheological data equally well, but to give opening angles in the complex contraction flow that are much closer to the experimental data from White and Kondo. It is shown (we believe for the first time) that a constitutive model demonstrates an accurate fit to all planar elongational, uniaxial elongational and shear viscometric data, with a simultaneous agreement with this well-known experimental opening angle data. The sensitivity of results to inaccuracies caused by representing the components of the deformation gradient tensor to finite precision is examined; results are found to be insensitive to even large reductions in the precision used for the representation of components. It is shown that two models that give identical response in elongational flow, and a very similar fit to available shear data, give significantly different results in flows containing a mix of deformation modes. The implication for constitutive models is that evaluation against mixed deformation mode flow data is desirable in addition to evaluation against viscometric measurements.     

The rational mechanics and thermodynamics of polymeric fluids based upon the concept of a variable relaxed state

The conceptual framework of polymer continuum mechanics based upon Eckart's idea of a variable relaxed state is developed. No constitutive models are explicitly used. The theory admits four constitutive functions only, the scalar specific internal energy, the vectorial heat flux, and two tensorial fluxes representing non-elastic stress and flow (slippage). The non-linearity of the constitutive relations includes self-induced anisotropy (Leonov) with Reiner-Rivlin's equation representing a special example for this. — The effectiveness of this non-linear theory is demonstrated by treating elongational flows of polymer melts.     

On the predictive/fitting capabilities of the advanced differential constitutive equations for linear polyethylene melts

This contribution examines the capabilities of three differential constitutive models (XPP, PTT-XPP, and modified Leonov) in predicting rheological properties of two virtually linear polyethylene materials (HDPE Tipelin FS 450-26, mLLDPE Exact 0201) with specific attention to both steady as well as transient shear and uniaxial elongational flow situations. For each situation the (dis)advantages of the individual models are discussed and both, qualitative and quantitative model efficiency evaluation has been carried out.     

On the predictive/fitting capabilities of the advanced differential constitutive equations for branched LDPE melts

Predictive/fitting capabilities of the XPP, PTT–XPP and modified Leonov constitutive equations are compared in both steady as well as transient shear and uniaxial elongational flows using two, highly branched LDPE materials (Escorene LD165BW1 and Bralen RB0323). It has been found that even if all three tested models exhibit very good fitting capability for steady uniaxial extensional viscosity curve, their predicting capabilities may differ significantly for shear viscosity as well as first and second normal stress coefficients.     

Remarks on orthogonal superposition of small amplitude oscillations on steady shear flow

The paper demonstrates that experimental data (Simmons, 1968) for orthogonal superposition of small amplitude oscillations on steady shear flow, coincide well enough with the theoretical predictions (Leonov et al., 1976) of simple multi-modal version of Leonov model (Leonov, 1976, 1987; Leonov et al., 1976). It was also shown that the recent theoretical calculations (Wong and Isayev, 1989) of the problem, which used the same Leonov model, are wrong.     

Simulation of planar converging flow of a Leonov viscoelastic fluid

An efficient finite element algorithm is presented to simulate the planar converging flow for the viscoelastic fluid of the Leonov model. The governing equation set, composed of the continuity, momentum and constitutive equations for the Leonov fluid flow, is conveniently decoupled and a two-stage cyclic iteration technique is employed to solve the velocity and elastic strain fields separately. Artificial viscosity terms are imposed on the momentum equations to relax the elastic force and data smoothing is performed on the iterative calculations for velocities to further stabilize the numerical computations. The calculated stresses agree qualitatively with the experimental measurements and other numerically simulated results available in the literature. Computations were successful to moderately high values of Deborah number of about 27·5.     

Quantitative evaluation of extrudate swell from viscoelastic properties of polystyrene

A theory of extrudate swell for short, intermediate or long dies is presented. In our experiment, we consider that the swelling phenomenon is mainly due to the recoverable elongational strain induced by the converging flow at the die entrance, as well as by recoverable shear strain originating within the die. From these concepts, an equation has been derived for the quantitative prediction of extrudate swell from the elastic material properties such as the entrance pressure drop, the relaxation modulus and the recoverable shear strain. Excellent agreement is found between predicted and measured values of extrudate swell obtained on commercial polystyrene melt, using capillaries of length-to-diameter ratios ranging from 1 to 20 and in a wide range of shear rates.     

Elastic properties of polyethylene melts at high shear rates with respect to extrusion

At high shear rates a steady state of shear flow with constant shear rate, constant shear stress, and constant recoverable shear strain is observed in the short-time sandwich rheometer after some few shear units already. The melt exhibits rather high elastic shear deformations and the recovery occurs at much higher speed than it is observed in the newtonian range. The ratio of first normal stress difference and twice the shear stress, being equal to the recoverable strain in the second-order fluid limit, significantly underestimates the true elastic shear strains at high shear rates. The observed shear rate dependence of shear stress and first normal stress difference as well as of the (constrained) elastic shear strain is correctly described on the basis of a discrete relaxation time spectrum. In simple shear a stick-slip transition at the metal walls is found. Necessary for the onset of slip is a critical value of shear stress and a certain amount of elastic shear deformation or orientation of the melt.     

Orthogonal superposition of small and large amplitude oscillations upon steady shear flow of polymer fluids

The orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered. Theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation. Steady-state components, amplitudes and phase angles of the oscillatory components of the shear stress, the first and second normal stress differences as functions of shear rate, deformation amplitude and frequency have been calculated. These oscillatory components include the first and third harmonic of the shear stresses and the second harmonic of the normal stresses. In the case of small amplitude superposition, the effect of the steady shear flow upon the frequency-dependent storage modulus and dynamic viscosity has been determined and compared with experimental data available in literature for polymeric solutions. The predicted results have been found to be in fair agreement with the experimental data at low shear rates and only in qualitative agreement at high shear rates and low frequencies. A comparison of the present theoretical results has also been made with the predictions of other theories.In the case of large amplitude superposition, the effect of oscillations upon the steady shear flow characteristics has been determined, indicating that the orthogonal superposition has less influence on the steady state shear stresses and the first difference of normal stresses than the parallel superposition. However, in the orthogonal superposition a more pronounced influence has been observed for the second difference of normal stresses.     

A constitutive equation derived from Lodge's network theory

A constitutive equation is presented, derived from Lodge's molecular network theory. The equation is of a form similar to the one presented previously by Carreau. The rate of creation of junctions and the probability of loss of junctions depend here also on the second invariant of the rate-of-strain tensor. The dependence, however, is through simple exponential functions, resulting in easy-to-use equations.Material functions are presented for the viscosity, the primary normal stress coefficient, the complex viscosity, the stress relaxation after cessation of steady simple shear, the stress growth after onset of steady simple shear and for elongational flow. The usefulness of the simplified (series truncated) equations is discussed and the model is evaluated with typical viscoelastic data.     

Practical comparison of differential viscoelastic constitutive equations in finite element analysis of planar 4:1 contraction flow

Finite element modeling of planar 4:1 contraction flow (isothermal incompressible and creeping) around a sharp entrance corner is performed for favored differential constitutive equations such as the Maxwell, Leonov, Giesekus, FENE-P, Larson, White-Metzner models and the Phan Thien-Tanner model of exponential and linear types. We have implemented the discrete elastic viscous stress splitting and streamline upwinding algorithms in the basic computational scheme in order to augment stability at high flow rate. For each constitutive model, we have obtained the upper limit of the Deborah number under which numerical convergence is guaranteed. All the computational results are analyzed according to consequences of mathematical analyses for constitutive equations from the viewpoint of stability. It is verified that in general the constitutive equations proven globally stable yield convergent numerical solutions for higher Deborah number flows. Therefore one can get solutions for relatively high Deborah number flows when the Leonov, the Phan Thien-Tanner, or the Giesekus constitutive equation is employed as the viscoelastic field equation. The close relationship of numerical convergence with mathematical stability of the model equations is also clearly demonstrated.     

Transient phenomena in thixotropic systems

A nonlinear rheological model which accounts for the time-dependent elastic, viscous and yielding phenomena is developed in order to describe the flow behavior of thixotropic materials which exhibit yield stress. A key feature of the formulation is a smooth transition from an ‘elastically’ dominated response to a ‘viscous’ response without a discontinuity in the stress–strain curve. The model is phenomenological and is based on the kinetic processes responsible for structural changes within the thixotropic material. As such, it can predict thixotropic effects, such as stress overshoot during start-up of a steady shear flow and stress relaxation after cessation of flow. Thus this model extends a previously proposed viscoplastic model [J. Rheol. 34 (1991) 647] to include thixotropy.An analysis and comparison to experimental data involving oscillatory shear flow are provided to evaluate the accuracy of the model and to estimate the model parameters in a prototype concentrated suspension. The experiments were conducted using a series of concentrated suspensions of silicon particles and silicon carbide whiskers in polyethylene. The data obtained with this experimental system indicated much better agreement between the theory and experiments that obtained in earlier work.     

VISCOELASTIC MODELLING OF ENTRANCE FLOW USING MULTIMODE LEONOV MODEL

A simulation of planar 2D flow of a viscoelastic fluid employing the Leonov constitutive equation has been presented. Triangular finite elements with lower-order interpolations have been employed for velocity and pressure as well as the extra stress tensor arising from the constitutive equation. A generalized Lesaint–Raviart method has been used for an upwind discretization of the material derivative of the extra stress tensor in the constitutive equation. The upwind scheme has been further strengthened in our code by also introducing a non-consistent streamline upwind Petrov–Galerkin method to modify the weighting function of the material derivative term in the variational form of the constitutive equation. A variational equation for configurational incompressibility of the Leonov model has also been satisfied explicitly. The corresponding software has been used to simulate planar 2D entrance flow for a 4:1 abrupt contraction up to a Deborah number of 670 (Weissenberg number of 6·71) for a rubber compound using a three-mode Leonov model. The predicted entrance loss is found to be in good agreement with experimental results from the literature. Corresponding comparisons for a commercial-grade polystyrene, however, indicate that the predicted entrance loss is low by a factor of about four, indicating a need for further investigation. © 1997 by John Wiley & Sons, Ltd.     

Finite element technique to solve the elastic strain for Leonov fluid flow

The finite element method is used to find the elastic strain (and thus the stress) for given velocity fields of the Leonov model fluid. With a simple linearization technique and the Galerkin formulation, the quasi-linear coupled first-order hyperbolic differential equations together with a non-linear equality constraint are solved over the entire domain based on a weighted residual scheme. The proposed numerical scheme has yielded efficient and accurate convective integrations for both the planar channel and the diverging radial flows for the Leonov model fluid. Only the strain in the inflow plane is required to be prescribed as the boundary conditions. In application, it can be conveniently incorporated in an existing finite element algorithm to simulate the Leonov viscoelastic fluid flow with more complex geometry in which the velocity field is not known a priori and an iterative procedure is needed.     

Cyclic stress-softening model for the Mullins effect in compression

This paper models the cyclic stress softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic stress softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.     

Dissipative stresses in dilute polymer solutions

Planar extensional flows of a dilute polymer solution are investigated using a free-draining bead-rod model. For steady flows, an analytic expression for the probability density of the polymer configuration is available. It is found that part of the associated steady polymer stress is unambiguously viscous at all time scales, in the sense that on cessation of flow it disappears instantaneously, but, except at very high flow rates, the elastic component is larger.A Brownian dynamics simulation of the chain is constructed for start-up flows for which no analytic expression is known. A stress that is apparently viscous is found to develop alongside the elastic stress, having comparable magnitude at moderate flow rates. An interpretation of this result for a system having a wide spectrum of relaxation times is given. This feature is not captured by conventional FENE constitutive equations, and a novel model is developed. The consequences for calculations of complex flows are briefly discussed.     

Investigation of the rheological properties of short glass fiber-filled polypropylene in extensional flow

The behavior of short glass fiber–polypropylene suspensions in extensional flow was investigated using three different commercial instruments: the SER wind-up drums geometry (Extensional Rheology System) with a strain-controlled rotational rheometer, a Meissner-type rheometer (RME), and the Rheotens. Results from uniaxial tensile testing have been compared with data previously obtained using a planar slit die with a hyperbolic entrance. The effect of three initial fiber orientations was investigated: planar random, fully aligned in the stretching flow direction and perpendicular to it. The elongational viscosity increased with fiber content and was larger for fibers initially oriented in the stretching direction. The behavior at low elongational rates showed differences among the various experimental setups, which are partly explained by preshearing history and nonhomogenous strain rates. However, at moderate and high rates, the results are comparable, and the behavior is strain thinning. Finally, a new constitutive equation for fibers suspended into a fluid obeying the Carreau model is used to predict the elongational viscosity, and the predictions are in good agreement with the experimental data.     

On stretching maxwell models

The transient response of selected Maxwell models to various modes of spatially homogeneous uniaxial elongation is studied qualitatively. New results include the effect of non-linearity on the creep curve (constant stress elongation), which leads to a delay in attaining steady flow, and the difference in recoverable strain estimates obtained from extrapolating the linear part of the creep curve back to zero time and from reducing the applied stress to zero. Constant force elongation is found to lead to mathematical difficulties (non-existence of solutions) for some models including the lower convected Maxwell model.The notion of a “limiting stretch rate” is re-examined critically and is seen to have meaning only once a mode of elongation is specified. For example, the maximum rate of strain attainable in constant force elongation of an upper convected Maxwell model is twice that attainable in constant stress elongation, and in fact any desired constant rate of strain may be obtained provided only that the stress is increased rapidly enough.The correspondence of a class of rate equations (Gordon-Schowalter-Everage) to integral equations based on non-linear strain measures due to Seth is established for elongational flow and is exploited to obtain initial values of strain and rate of strain when a load is applied suddenly. Extension of the results to more complicated models is discussed and the relative merits of some of the simple (three-parameter) models are examined.