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.     

A differential constitutive equation for polymer melts

A modification of the Giesekus constitutive equation is derived by incorporating (approximately, via the Peterlin approximation) the finite extensibility of polymer molecules into dumbbell kinetic theory along with the anisotropic hydrodynamic drag suggested by Giesekus. The constitutive equation that is obtained retains much of the simplicity of Giesekus' constitutive equation, but it involves terms that are cubic in the stress as well as those that are quadratic. It is shown that the constitutive equation quantitatively describes the steady elongational viscosity of the IUPAC polymer melt A (including the strain softening of the melt), but it cannot describe the elongational and shear viscosities simultaneously. It is also shown that the constitutive equation satisfies the Lodge-Meissner relation for shear strains less than unity.     

Convected derivatives for differential constitutive equations

To date, no differential constitutive equation has been proposed that agrees with each of four important experimental observations in relaxation after step shear strains: that the stress is often factorable into time and strain-dependent functions, that the strain-dependent function is shear thinning, that the ratio of first normal stress difference to shear stress equals the shear strain—that is, the Lodge-Meissner relationship holds, and that there is a negative second normal stress difference. The Johnson-Segalman model satisfies three of these, but fails to satisfy the Lodge-Meissner relationship, because in step strains the principal stress and strain axes do not rotate together. Using a mathematical technique for forcing co-rotation of stress and strain axes in an arbitrary deformation, we here present an explicit differential constitutive equation that satisfies all four of the above experimental observations.     

Comparison of linear viscoelastic constitutive equations for non-uniform polymer melts

Linear-viscoelastic properties of polydisperse and randomly-branched polymer melts were fit with several proposed relaxation functions by non-linear regression. Three polymer systems were investigated, including 1) crosslinked polyethylenes, 2) polydisperse linear poly(dimethylsiloxane)s, and 3) Marlex polyethylenes, which are polydisperse and probably contain long-chain branching. Four relaxation functions were evaluated, including the Rouse, reptation, stretched-exponential, and stretched-exponential-power-law (SEPL) relaxation functions. The SEPL best described each series of polymers, and therefore may be a general relaxation function for non-uniform polymer melts. The flow activation energy for crosslinked polyethylene may be coupled to a breadth-of-relaxation index, indicating that a coupling between a characteristic short relaxation time and longest relaxation time, as suggested by Ngai and Plazek (J. Polym. Sci. Polym. Phys. Ed. 1985, 23:2159–2180), may hold for some non-uniform polymers.     

On the burst of branched polymer melts during inflation

Two molten low-density polyethylene melts, shaped as plates, have been inflated into a circular cylinder during isothermal conditions. Lowering the inflation rates allow the plates to be inflated into a larger volume of the cylinder before bursting. Numerical simulations of the inflations have been performed, using a time-strain separable constitutive K-BKZ equation based on the potential function from the Doi–Edwards theory. The material parameters in the constitutive model are based on liner viscoelastic and time dependent uniaxial elongational viscosities. The numerical calculations show quantitative agreement with the experiments, including the appearance of the burst, for a wide range of experimental conditions. This strongly suggests that the initiation of the burst in the polymer melts is a hydrodynamic phenomenon.     

On constitutive equations for electrorheological materials

Constitutive equations for electrorheological (ER) fluids have been based on experimental results for steady shearing flows and constant electric fields. The fluids have been modeled as being rigid until a yield stress is reached. Additional stress is then proportional to the shear rate. Recent experimental results indicate that ER materials have a regime of solid-like response when deformed from a rest state. They behave in a viscoelastic-like manner under sinusoidal shearing and exhibit time-dependent response under sudden changes in shear rate or electric field. In this work, a constitutive theory for ER materials is presented which accounts for these recent experimental observations. The stress is given by a functional of the deformation gradient history and the electric field vector. Using the methods of continuum mechanics, a general three-dimensional constitutive equation is obtained. A sample constitutive equation is introduced which is then used to determine the response of an ER material for different shear histories. The calculated shear response is shown to be qualitatively similar to that observed experimentally.     

Evaluation of rheological constitutive equations for branched polymers in step shear strain flows

The pom-pom rheological constitutive equation for branched polymers proposed by McLeish and Larson is evaluated in step shear strain flows. Semianalytic expressions for the shear-stress relaxation modulus are derived for both the integral and approximate differential versions of the pom-pom model. Predictions from the thermodynamically motivated differential pompon model of ?ttinger are also examined. Single-mode integral and differential pom-pom models are found to give qualitatively different predictions, the former displays time–strain factorability after the backbone stretch is relaxed, while the latter does not. We also find that the differential pompon model gives quantitatively similar predictions to the integral pom-pom model in step strain flows. Predictions from multimode integral and differential pom-pom models are compared with experimental data on a widely characterized, low-density polyethylene known as 1810H. The experiments strongly support time–strain factorability, while the multimode pom-pom model predictions show deviations from this behavior over the entire range of time that is experimentally accessible.     

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.Financial support for this research was given by the National Science Foundation, the Office of Naval Research, and the Defense Research Projects Agency. Computational resources were supplied by a grant from the Pittsburgh National Supercomputer Center and by the MIT Supercomputer Facility.     

Alternative derivation of differential constitutive equations of the Oldroyd-B type

Now that almost 60 years have passed since the pioneering works of J.G. Oldroyd it seems appropriate as an homage to consider here constitutive equations that can be viewed as generalisations of the by now classical Oldroyd-B model. In this short communication we shall address heuristically the theme of differential constitutive models and will provide an alternative way of deriving a “modified FENE” equation (FENE-M) and inter-relating the PTT and FENE-P-like models.     

On constitutive equations for fiber-reinforced nonlinearly viscoelastic solids

In this paper, we are interested in developing constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation. These invariants are analyzed, and we obtain restrictions such as positivity of some of them.     

On the stochastic interpretation of gradient-dependent constitutive equations

The paper elaborates on the statistical interpretation of a class of gradient models by resorting to both microscopic and macroscopic considerations. The microscopic stochastic representation of stress and strain fields reflects the heterogeneity inherently present in engineering materials at small scales. A physical argument is advanced to conjecture that stress shows small fluctuations and strong spatial correlations when compared to those of strain; then, a series expansion in the respective constitutive equations renders unimportant stress gradient terms, in contrast to strain gradient terms, which should be retained. Each higher-order strain gradient term is given a physically clear interpretation. The formulation also allows for the underlying microstrain field to be statistically non-stationary, e.g., of fractal character. The paper concludes with a comparison between surface effects predicted by gradient and stochastic formulations.     

On constitutive equations for branching of response with selectivity

Materials, such as elastic-plastic, which exhibit distinct regimes of response are usually modeled by different constitutive equations in each regime. The present paper explores a method for the construction of a unified constitutive equation from these separate relations. The main idea is to write this unified equation in an implicit form which contains these separate solutions as non-unique solutions. The form is chosen in order to utilize the notions of branch points and branches. Different solutions, corresponding to constitutive equations for different regimes of response, are then regarded as bifurcations at branch points from the fundamental response. The choice of the appropriate branch at a branch point is governed by a selectivity condition which depends on the nature of the response under consideration. A detailed example is provided for elastic-plastic response, with and without the effect of strain rate dependence.     

On the constitutive equations for cyclic plasticity under nonproportional loading

Experimental results on 316 stainless steel, at room temperature, under strain controled axial-torsion loadings, point out a large difference in strengthening between proportional and nonproportional loadings. The maximum difference is produced by 90° out of phase straining. In this paper we discuss the possibility of describing this additional hardening, without any new internal variable, by only modifying the expression for the yield criterion.     

On instabilities of single-integral constitutive equations for viscoelastic liquids

Various types of instabilities are exposed in this paper for time-strain separable single-integral viscoelastic constitutive equations (CE's). They were distinguished into two groups and defined as Hadamard and dissipative type of instabilities. As for the Hadamard-type, previously obtained criteria are found to be necessary only. They are necessary and sufficient only for thermodynamic stability. Improved, stricter Hadamard stability criteria are described briefly in this paper, and then applied to study of stability of several CE's. It is shown that the Currie potential with the K-BKZ equation and the model proposed by Papanastasiou et al. are Hadamard unstable. In the case of dissipative stability, the necessary and sufficient condition for stress boundedness in any regular flow with a given history, is proved. Then, this criterion was applied to the neoHookean, Mooney, and Yen and McIntire specifications of the general K-BKZ model, to exhibit unbounded solutions. In addition, Larson-Monroe potential which is later proved to be Hadamard unstable but satisfies the above criterion of boundedness, is shown to have unstable decreasing branch in steady simple shear flow. At present, to the authors' knowledge, there is no viscoelastic single-integral CE of factorable type proposed in the literature which can satisfy all the Hadamard and dissipative stability criteria.     

On a class of constitutive equations for viscoelastic liquids

Simple rheological equations that describe non-linear viscoelastic phenomena in polymeric liquids have long attracted the attention of many rheologists. Although there are many ways of deriving such equations, only one concept is considered here. This concept is based on the introduction of an internal parameter, the recoverable strain tenson, and arises from a special kinematic study together with the formalism of irreversible thermodynamics. The main part of the paper sets out the theory for a single mode but a multimode extension is demonstrated towards the end and is compared with experimental data. Finally some of the problems that remain unsolved in the theory are discussed.The aim of this paper is to acquaint rheologists with the author's views as the model rheological equations he developed have recently been discussed in the literature without his participation.     

Viscoelastic flow analysis using the software OpenFOAM and differential constitutive equations

Viscoelastic fluids are of great importance in many industrial sectors, such as in food and synthetic polymers industries. The rheological response of viscoelastic fluids is quite complex, including combination of viscous and elastic effects and non-linear phenomena. This work presents a numerical methodology based on the split-stress tensor approach and the concept of equilibrium stress tensor to treat high Weissenberg number problems using any differential constitutive equations. The proposed methodology was implemented in a new computational fluid dynamics (CFD) tool and consists of a viscoelastic fluid module included in the OpenFOAM, a flexible open source CFD package. Oldroyd-B/UCM, Giesekus, Phan-Thien–Tanner (PTT), Finitely Extensible Nonlinear Elastic (FENE-P and FENE-CR), and Pom–Pom based constitutive equations were implemented, in single and multimode forms. The proposed methodology was evaluated by comparing its predictions with experimental and numerical data from the literature for the analysis of a planar 4:1 contraction flow, showing to be stable and efficient.     

On minimal representations for constitutive equations of anisotropic elastic materials

The problem of determining minimal representations for anisotropic elastic constitutive equations is proposed and investigated. For elastic constitutive equations in any given case of anisotropy, it is shown that there exist generating sets consisting of six generators and such generating sets are minimal in all possible generating sets. This fact implies that most of the established results for representations of elastic constitutive equations are not minimal and remain to be sharpened. For elastic constitutive equations in some cases of anisotropy, including orthotropy, transverse isotropy, the trigonal crystal class S ₆, and the classes C _2mh , m=1, 2, 3,..., etc., representations in terms of minimal generating sets are presented for the first time.     

On the evaluation of some differential formulations for the pom-pom constitutive model

In recent years, progress has been made in modelling long chain branched polymers by the introduction of the so-called pom-pom model. Initially developed by McLeish and Larson (1998), the model has undergone several improvements or alterations, leading to the development of new formulations. Some of these formulations however suffer from certain mathematical defects. The purpose of the present paper is to review some of the formulations of the pom-pom constitutive model, and to investigate their possible mathematical defects. Next, an alternative formulation is proposed, which does not appear to exhibit mathematical defects, and we explore its modelling performance by comparing the predictions with experiments in non-trivial rheometric flows of an LDPE melt. The selected rheometric flows are the double step strain, as well as the large amplitude oscillatory shear experiments. For LAOS experiments, the comparison involves the use of Fourier-transform analysis.