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.     

Two-dimensional viscoelastic flows: experimentation and modeling

Extensive experimental data on the birefringence in converging and diverging flows of a polymeric melt have been obtained. The birefringence and pressure drop measurements were carried out in working cells of planar geometry having different contraction angles and contraction ratios. For investigation of diverging or abrupt expansion flow, the direction of flow in the cells was reversed. The theoretical predictions are based upon the Leonov constitutive equation and a finite element scheme with streamwise integration.In contrast to Newtonian and second-order fluids, viscoelastic fluids at high shear rates show significant differences in pressure drop and birefringence (i.e. stresses) in converging and diverging flows. For a constant flow rate, the pressure drop is higher and the birefringence smaller in diverging flows than in converging flows. This difference increases with increasing flow rate. Further, for the same contraction ratio but different contraction angles, the birefringence maximum increases considerably with contraction angle. In addition, an increase in contraction ratio has the same effect.The viscoelastic constitutive equation of Leonov has been shown to describe all the above viscoelastic effects observed in the experiments. In general, a reasonable agreement between theory and experiment has been obtained, which shows the usefulness of the Leonov model in describing actual flows.     

Inelastic constitutive equations for complex flows

A new class of inelastic constitutive equations is presented and discussed. In addition to the rate-of-strain tensor, the stress is assumed to depend also on the relative-rate-of-rotation tensor, a frame-indifferent quantity that brings information about the nature of the flow. The material functions predicted by these constitutive equations are given for simple shear and uniaxial extension. A special case of these equations takes the Newtonian form, except that the viscosity is a function of the invariants of both kinematic tensors on which the stress depends. This simple constitutive equation has potential applications in liquid flow process simulations, since it combines simplicity with the capability of responding independently to shear and extension, as real liquids seem to do. Finally, possible forms for the new viscosity function are discussed.     

Onset and dynamic shallow flow of a viscoplastic fluid on a plane slope

The shallow flow of a viscoplastic fluid on a plane slope is investigated. The material constitutive law may include two plasticity (flow/no-flow) criteria: Von-Mises (Bingham fluid) and Drucker–Prager (Mohr–Coulomb). Coulomb frictional conditions on the bottom are included, which implies that the shear stresses are small and the extensional and in-plane shear stress becomes important. A stress analysis is used to deduce a Saint-Venant type asymptotic model for small thickness aspect ratio. The 2D (asymptotic) constitutive law, which relates the average plane stresses to the horizontal rate of deformation, is obtained from the initial (3D) viscoplastic model.The “safety factor” (limit load) is introduced to model the link between the yield limit (material resistance) and the external forces distribution which could generate or not the shallow flow of the viscoplastic fluid. The DVDS method, developed in [I.R. Ionescu, E. Oudet, Discontinuous velocity domain splitting method in limit load analysis, Int. J. Solids Struct., doi:10.1016/j.ijsolstr.2010.02.012], is used to evaluate the safety factor and to find the onset of an avalanche flow.A mixed finite element and finite volume strategy is developed. Specifically, the variational inequality for the velocity field is discretized using the finite element method while a finite volume method is adopted for the hyperbolic equation related to the thickness variable. To solve the velocity problem, a decomposition–coordination formulation coupled with the augmented lagrangian method, is adapted here for the asymptotic model. The finite volume method makes use of an upwind strategy in the choice of the flux.Several boundary value problems, modeling shallow dense avalanches, for different visoplastic laws are selected to illustrate the predictive capabilities of the model.     

Simulation of two-dimensional planar flow of viscoelastic fluid

A numerical scheme based on the Finite Element Method has been developed which uses a relaxation factor in the momentum equation with the stresses being evaluated via a streamwise integration procedure. A constitutive equation introduced by Leonov has been used to represent the rheological behavior of the fluid. The convergence of the scheme has been tested on a 2 : 1 abrupt contraction problem by successive mesh refinement for non-dimensional characteristic shear rates, of 5 and 50 for polyisobutylene Vistanex at 27 °C. The recirculation region is shown to increase in size with non-dimensional characteristic shear rate.Theoretical predictions have been compared with the experimental data which include birefringence and pressure loss measurements. In general, the comparisons have been reasonably good and demonstrates the usefulness of the present numerical scheme and the Leonov constitutive equation to describe real polymer flows.     

Sensitivity of the macroscopic elasticity tensor to topological microstructural changes

A remarkably simple analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological microstructural changes of the underlying material is proposed. The derivation of the proposed formula relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. The derived sensitivity—a symmetric fourth order tensor field over the RVE domain—measures how the estimated two-dimensional macroscopic elasticity tensor changes when a small circular hole is introduced at the microscale level. This information has potential use in the design and optimisation of microstructures.     

Comparison of experimental data with the numerical simulation of planar entry flow: Role of the constitutive equation

The goal of this research was to determine whether there is any interaction between the type of constitutive equation used and the degree of mesh refinement, as well as how the type of constitutive equation might affect the convergence and quality of the solution, for a planar 4:1 contraction in the finite eiement method. Five constitutive equations were used in this work: the Phan-Thien–Tanner (PTT), Johnson–Segalman (JS), White–Metzner (WM), Leonov-like and upper convected Maxwell (UCM) models. A penalty Galerkin finite element technique was used to solve the system of non-linear differential equations. The constitutive equations were fitted to the steady shear viscosity and normal stress data for a polystyrene melt. In general it was found that the convergence limit based on the Deborah number De and the Weissenberg number We varied from model to model and from mesh to mesh. From a practical point of view it was observed that the wall shear stress in the downstream region should also be indicated at the point where convergence is lost, since this parameter reflects the throughput conditions. Because of the dependence of convergence on the combination of mesh size and constitutive equation, predictions of the computations were compared with birefringence data obtained for the same polystyrene melt flowing through a 4:1 planar contraction. Refinement in the mesh led to better agreement between the predictions using the PTT model and flow birefringence, but the oscillations became worse in the corner region as the mesh was further refined, eventually leading to the loss of convergence of the numerical algorithm. In comparing results using different models at the same wall shear stress conditions and on the same mesh, it was found that the PTT model gave less overshoot of the stresses at the re-entrant corner. Away from the corner there were very small differences between the quality of the solutions obtained using different models. All the models predicted solutions with oscillations. However, the values of the solutions oscillated around the experimental birefringence data, even when the numerical algorithm would not converge. Whereas the stresses are predicted to oscillate, the streamlines and velocity field remained smooth. Predictions for the existence of vortices as well as for the entrance pressure loss (ΔP_ent) varied from model to model. The UCM and WM models predicted negative values for ΔP_ent.     

Mathematical modelling of rate-independent pseudoelastic SMA material

The paper presents a procedure for the formulation of constitutive equations for rate-independent pseudoelastic SMA material models. The procedure applies a rheological scheme representing mechanical properties of the material. An additive decomposition of strains into two parts is proposed. The first part describes strains of a perfectly elastic body while the second part may be represented by a combination of a rigid perfectly elastic body and a rigid perfectly plastic body. It is demonstrated that the key problem of formulation of constitutive relationships is to derive the 1st order differential equation with respect to the tensor describing the second part of the strain field. This equation may be obtained in explicit form starting from the variational inequalities defining non-elastic parts of rheological model. The uniqueness of the obtained differential equation has been proved. A numerical implementation of the constitutive relationships of SMA material was done through the user subroutine module VUMAT within the FE commercial code ABAQUS/Explicit. As an example we analyzed the problem of vibration of a simple 3D structure made of SMA.     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

Molecular model and flow calculation: I. The numerical solutions to multibead-rod models in homogeneous flows

The Galerkin method is used to solve the diffusion equation of the distribution function in configurational space for a multibead-rod model, and the dimensionless components of the extra stress tensor are then calculated by means of the expression of ensemble average. The material functions for steady-state shear flow and uniaxial flow and the mechanical properties of rigid-rodlike molecule suspensions in superposed flows are obtained numerically. The results indicate that it is promising to employ the multibead-rod models without the constitutive equation in numerical simulations of flows of suspensions. The project supported by the National Natural Science Fundation of China.     

A stable unstructured finite volume method for parallel large-scale viscoelastic fluid flow calculations

A new stable unstructured finite volume method is presented for parallel large-scale simulation of viscoelastic fluid flows. The numerical method is based on the side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face, while the pressure term and the extra stress tensor are defined at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance the pressure–velocity–stress coupling. The log-conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix–logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] has been implemented in order improve the limiting Weissenberg numbers in the proposed finite volume method. The time stepping algorithm used decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. The resulting algebraic linear systems are solved using the FGMRES(m) Krylov iterative method with the restricted additive Schwarz preconditioner for the extra stress tensor and the geometric non-nested multilevel preconditioner for the Stokes system. The implementation of the preconditioned iterative solvers is based on the PETSc library for improving the efficiency of the parallel code. The present numerical algorithm is validated for the Kovasznay flow, the flow of an Oldroyd-B fluid past a confined circular cylinder in a channel and the three-dimensional flow of an Oldroyd-B fluid around a rigid sphere falling in a cylindrical tube. Parallel large-scale calculations are presented up to 523,094 quadrilateral elements in two-dimension and 1,190,376 hexahedral elements in three-dimension.     

The Giesekus fluid in ω–D form for steady two-dimensional flows

Using the fact that for simple fluids the most general constitutive equation in constant stretch history flows for the extra stress tensor τ is known in an explicit form, the Giesekus fluid model is cast into this (ω–D) form for two-dimensional flows. The three material functions needed to characterize τ are listed. The explicit results for simple shear and planar elongation reveal that the parameter α should be restricted to values less than 0.5. It is demonstrated that in this explicit form the constitutive equation is free from thermodynamic objections and can thus be used as a starting point for numerical calculations of general, but steady, two-dimensional flows. Received: 9 November 1998 Accepted: 20 May 1999     

Elongational-flow predictions of the Leonov constitutive equation

The basic thermodynamic ideas from rubber-elasticity theory which Leonov employed to derive his constitutive model are herein summarized. Predictions of the single-mode version are presented for homogeneous elongational flows including stress growth following start-up of steady flow, stress decay following sudden stretching and following cessation of steady flow, elastic recovery following cessation of steady flow, energy storage in steady-state flow, and the velocity profile in constantforce spinning. Using parameters of the multiple-mode version which fit the linearviscoelastic data, the Leonov-model predictions of elongational stress growth during, and elastic recovery following, steady elongation are calculated numerically and compared to the experimental results for Melt I and to the Wagner model. It is found that the Leonov model, as originally formulated, agrees qualitatively with the data, but not quantitatively; the Wagner model gives quantitative agreement, but requires much nonlinear data with which to fit model parameters. Quantitative agreement can be obtained with the Leonov model, if the nonequilibrium potential which relates recoverable strain to strain rate is adjusted empirically. This can most simply be done by making each relaxation time dependent upon the recoverable strain. The Leonov model, unlike the Wagner model, is derived from an entropic constitutive equation, which is advantageous for calculating stored elastic energy or viscous dissipation. The Leonov model also has an appealingly simple differential form, similar to the upper-convected Maxwell model, which, in numerical calculations, may be an important advantage over the integral Wagner model.     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

Flows of constant stretch history for polymeric materials with power-law distributions of relaxation times

It is herein shown that for separable integral constitutive equations with power-law distributions of relaxation times, the streamlines in creeping flow are independent of flow rate.For planar flows of constant stretch history, the stress tensor is the sum of three terms, one proportional to the rate-of-deformation tensor, one to the square of this tensor, and the other to the Jaumann derivative of the rate-of-deformation tensor. The three tensors are the same as occur in the Criminale-Ericksen-Filbey Equation, but the coefficients of these tensors depend not only on the second invariant of the strain rate, but also on another invariant which is a measure of flow strength. With the power-law distribution of relaxation times, each coefficient is equal to the second invariant of the strain rate tensor raised to a power, times a function that depends only on strength of the flow. Axisymmetric flows of constant stretch history are more complicated than the planar flows, because three instead of two nonzero normal components appear in the velocity gradient tensor. For homogeneous axisymmetric flows of constant stretch history, the stress tensor is given by the sum of the same three terms. The coefficients of these terms again depend on the flow strength parameter, but in general the dependences are not the same as in planar flow.     

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.     

Flow of a quasi-Newtonian fluid through a planar contraction

Flows through abrupt contractions are dominated by the rapid extension experienced in passing through the contraction. Thus, it is useful to employ a fluid model which considers the extensional viscosity explicitly in its constitutive equation. In this paper, the quasi-Newtonian fluid model, which admits shear thinning and extension thickening of the viscosity depending on the local type of flow as proposed by Schunk and Scriven [P. Schunk, L. Scriven, J, Rheol 34 (1990) 1085], is applied to the numerical simulation of the flow of a dilute polyacrylamide solution through a planar 4 : 1 contraction. In this theory the extra stress tensor does not only depend on the rate of strain tensor but also on the relative rate of rotation of the fluid. The material function – the viscosity function – is allowed to depend on the invariants of these two kinematic tensors yielding a local distinction between extensional, shear or rotation dominated flow. The governing equations are discretized using a finite volume method. Different model parameters are varied and the simulation results are compared with the generalized Newtonian fluid and experimental data.     

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.     

Three‐dimensional simulation of planar contraction viscoelastic flow by penalty finite element method

The planar contraction flow is a benchmark problem for the numerical investigation of viscoelastic flow. The mathematical model of three‐dimensional viscoelastic fluids flow is established and the numerical simulation of its planar contraction flow is conducted by using the penalty finite element method with a differential Phan‐Thien–Tanner constitutive model. The discrete elastic viscous split stress formulation in cooperating with the inconsistent streamline upwind scheme is employed to improve the computation stability. The distributions of velocity and stress obtained by simulation are compared with that of Quinzani's experimental results detected by laser–doppler velocimetry and flow‐induced birefringence technologies. It shows that the numerical results agree well with the experimental results. The numerical methods proposed in the study can be well used to predict complex flow patterns of viscoelastic fluids. Copyright © 2009 John Wiley & Sons, Ltd.