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.     

A finite element method for computing the flow of multi-mode viscoelastic fluids: comparison with experiments

The numerical computation of viscoelastic fluid flows with differential constitutive equations presents various difficulties. The first one lies in the numerical convergence of the complex numerical scheme solving the non-linear set of equations. Due to the hybrid type of these equations (elliptic and hyperbolic), geometrical singularities such as reentrant corner or die induce stress singularities and hence numerical problems. Another difficulty is the choice of an appropriate constitutive equation and the determination of rheological constants. In this paper, a quasi-Newton method is developed for a fluid obeying a multi-mode Phan-Thien and Tanner constitutive equation. A confined convergent geometry followed by the extrudate swell has been considered. Numerical results obtained for two-dimensional or axisymmetric flows are compared to experimental results (birefringence patterns or extrudate swell) for a linear low density polyethylene (LLDPE) and a low density polyethylene (LDPE).     

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.     

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule.Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted.Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.     

Experimental analysis of rolling contact stresses in a viscoelastic strip

An experimental approach to two-dimensional, viscoelastic, steadily moving rolling contact is described. The photoviscoelastic technique is employed for the analysis of rolling contact stresses between a viscoelastic plate and a rigid rolling cylinder in which the principal axes of stress, strain and birefringence are not coincident with each other. Using an elliptically polarized white light, the distribution of isochromatic fringe order and the principal axes of birefringence at an instant are determined from a single photoviscoelastic image. The time variations of the differences of the principal stresses and strains, as well as their directions, are obtained by use of the optical constitutive equations of photoviscoelasticity. The experimental results involving the time variation of the stresses around the contact surface and their distributions are analyzed.     

45#钢的损伤演化方程和层裂准则研究

通过唯象分析和细观物理统计相结合的方法建立了一种韧性材料的损伤演化方程。在试验结果和内变量理论的基础上得到了45#钢的含损伤热—粘塑性本构关系。用有限差分方法计算了45#钢平板撞击所产生的应力波传播规律、损伤演化规律及层裂进程。通过自由面速度历史的数值模拟,并基于计算结果与试验结果间的最佳一致性,得到了损伤演化方程中的材料参数和极限损伤数值,并以此为依据建立了材料的应力率层裂准则。     

Photoviscoelastic stress analysis of a plate with a central hole

An epoxy resin containing excessive plasticizer was developed and characterized. The material, which deforms viscously at room temperature, has optical properties that depend on stress and strain. A tensile specimen was prepared from the epoxy resin so that the mechanical and optical properties of the epoxy resin could be characterized. The elastic and plastic behavior was determined at 37°C using tensile stresses between 4 and 26 MPa. The birefringence was also recorded as a function of time and stress. From these results, a photoviscoelastic constitutive equation was constructed to describe the dependence of the birefringence on stress and strain. The constitutive equation was then applied to study the deformation of a tensile specimen containing a central circular hole. By using the isochromatic fringes in combination with the isoclinic, the time-dependent variation of the stress field in the specimen was solved.     

Theoretical analysis of non-uniform extensional flows in relation to processing rheology and predictions of some constitutive equations

In this paper, a characteristic equation involving the stream function, already given by one of the authors in a previous work for classifying axisymmetric incompressible flows, is re-considered. Non-uniform nearly extensional flows are derived as particular solutions from this equation. Using experimental data in the literature for polymer solutions and melts, it is proved that particular solutions of the characteristic equation lead to kinematics very close to those encountered in the fiber-spinning process. The kinematic equations satisfactorily correlating the fiber-spinning data are used in order to determine the ability of constitutive equations to predict realistic stresses in the flow domain. The rheological parameters of the fluids, obtained from experiments, are used for computation of differential and integral constitutive equations in the spinning conditions. Comparisons with the stress response of adequate constitutive equations are given and discussed.Also affiliated to: Université Joseph Fourier Grenoble I and Institut National Polytechnique de Grenoble, Associé au CNRS (URA 1510)     

The melt processing of monodisperse and polydisperse polystyrene melts within a slit entry and exit flow

This paper presents experimental results and numerical simulations for mono, blended and polydisperse polstryrenes of different molecular weights flowing within a slit geometry. Flow experiments were carried out on small (less than 10 g) quantities of polymer using a multi-pass rheometer and flow-induced birefringence images were obtained for well-defined flow boundary conditions. Experimental flow birefringence observations illustrate the similarities and differences in the flow behaviour between monodisperse and polydisperse polystyrene. For the case of monodisperse polystyrene a transition from “near-Newtonian” stress patterns for low molecular weight polystyrenes, to a highly unstable flow at high molecular weight was observed. Both blending and polydispersity enabled stable flows to be achieved at high flowrates.Experimental flow birefringence results and some pressure difference predictions were compared with numerical predictions. Two different computational approaches were followed, one using a viscoelastic integral K-BKZ/Wagner model within the finite element method solver Polyflow, and the other using the tube theory-based Pom-Pom constitutive equation and Lagrangian-Eulerian code flowSolve. Both numerical methods were able to capture certain experimental observations reasonably well in the stable flow regime, but were not able to predict the onset of the experimentally observed flow instabilities.     

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.     

沥青混合料一维非线性粘弹本构方程

本文根据Green-Rivlin提出的Frechet级数,提出了一个兼具长时粘流效应及短时粘弹效应的一维非线性本构方程,并对一种沥青混合料采用蠕变试验方式拟合了本构方程的参数向量。拟合结果表明:这一本构方程形式简洁,物理意义清晰,易于拟合,并对不同应力历史下的应变响应具有广泛的适用性。     

汽车玻璃高温蠕变特性的实验与理论研究

通过对汽车用挡风玻璃进行不同温度、不同应力水平下的单轴拉伸蠕变实验,揭示了其非线性蠕变变形特性.实验发现,不同温度和载荷下的蠕变曲线形状类似,高温(550℃～590℃)下该材料的减速蠕变变形不很明显,主要表现为稳态和加速蠕变变形,且其断裂数据符合Monkman-Grant关系,应力和温度对其影响不显著.基于实验结果,本文建立了非线性高温蠕变本构方程.理论预测与实验数据进行了比较,结果表明所提出的本构方程能较好地描述汽车玻璃高温蠕变变形的全过程.     

Birefringence measurements on polymer melts in an axisymmetric flow cell

 The stress-optical rule relates birefringence to stress. Consequently, measurement of flow birefringence provides a non-intrusive technique of measuring stresses in complex flows. In this investigation we explore the use of an axisymmetric geometry to create a uniaxial elongational flow in polymer melts. In axisymmetric flows both birefringence and orientation angle change continuously along the path of the propagating light. The cumulative influence of the material's optical properties along the light's integrated path makes determination of local birefringence in the melt impossible. One can nevertheless use birefringence measurements to compare with predictions from computer simulations as a means of evaluating the constitutive equations for the stress. More specifically, in this investigation we compare the light intensity transmitted through the experimental set-up vs entry position, with the theoretically calculated transmitted intensity distribution as a means of comparing experiment and simulation. The main complication in our experiments is the use of a flow cell that necessarily consists of materials of different refractive indices. This introduces refraction and reflection effects that must be modeled before experimental results can be correctly interpreted. We describe how these effects are taken into account and test the accuracy of predictions against experiments. In addition, the high temperatures required to investigate polymer melts mean that a further complication is introduced by thermal stresses present in the flow cell glass. We describe how these thermal-stresses are also incorporated in the simulations. Finally, we present some preliminary results and evaluate the success of the overall method. Received: 2 April 2001 Accepted: 27 August 2001     

A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments

A one-dimensional constitutive equation for the Mullins effect in rubberlike materials, which is motivated by the two phase microstructural material model proposed by Mullins and Tobin [I], is developed in [2]. The constitutive equation is used in [2] to predict the effect of stress softening on the small transverse vibration of a rubber string loaded in uniaxial extension. The two functions which comprise the constitutive equation were assumed to be monotone, but no further analysis of the actual nature of these functions was necessary.In this paper, we examine more closely how the results of a stress controlled uniaxial extension experiment can be used to gain insight into the specific nature of the microstructural strain and the strain amplification functions which comprise the constitutive equation. We examine experimental representations of the two functions which are independent of any special microstructural interpretations. Stress controlled uniaxial extension experiments with buna-n, neoprene, and silicone rubber cords are examined. We demonstrate how the experimental data can be applied to yield representations of the functions of interest to within a multiplicative constant; but no attempt is made to find specific analytical representations of these functions. For buna-n and neoprene samples, we observe behavior consistent with our monotone assumptions, while anomalous behavior is observed with silicone rubber.     

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.     

Constitutive equations of elastoplastic isotropic materials that allow for the stress mode

The constitutive equations describing the elastoplastic deformation of an isotropic material and taking into account the stress mode are validated against available experimental data. We propose a method for the approximate determination of the base functions appearing in the constitutive equations and relating the first and second invariants of the stress tensors and the linear components of finite strains. The strain components obtained by this method are compared to experimental data     

A stress-optic law for photoelastic analysis of orthotropic composites

The feasibility for utilizing transparent filament-resin composites for photoelastic stress analysis was investigated. Satisfactory photoelastic stress patterns were demonstrated in simple models with undirectional and bidirectional fiber orientations. A stress-optic law was formulated, based on the concept that the birefringence components contributed by each component of plane stress are combined according to a Mohr circle of birefringence. Applying this concept, the difference of the physical and optical principal directions was accounted for, and a general method of photoelastic solution for the plane-stress problem in orthotropic sheets was developed. The method of analysis is little more complex than the well-known procedures for isotropic materials, but at least three experimental measurements are required to characterize the optical response of the material to plane stress. Partial confirmation of the proposed stress-optic law was obtained by comparison of the theory to limited experimental data obtained in uniaxial-stress samples. It remains to establish a more positive verification by experiments in a variety of biaxial-stress conditions.     

聚合物注射成型流动残余应力的数值分析

建立了可压缩黏弹性聚合物熔体在薄壁型腔中充模/保压过程中非等温、非稳态流动 的数学模型,用数值方法实现了注射成型过程中流动应力和取向建立及松弛过程的模拟,研 究了熔体温度、模具温度和注射速率等工艺条件对分子冻结取向的影响,取得了与实验相符 的结果.     

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.     

Modelling inelastic behaviour of orthotropic metals in a unique alignment of deviatoric plane within the stress space

A finite strain constitutive model to predict the deformation behaviour of orthotropic metals is developed in this paper. The important features of this constitutive model are the multiplicative decomposition of the deformation gradient and a new Mandel stress tensor combined with the new stress tensor decomposition generalized into deviatoric and spherical parts. The elastic free energy function and the yield function are defined within an invariant theory by means of the structural tensors. The Hill’s yield criterion is adopted to characterize plastic orthotropy, and the thermally micromechanical-based model, Mechanical Threshold Model (MTS) is used as a referential curve to control the yield surface expansion using an isotropic plastic hardening assumption. The model complexity is further extended by coupling the formulation with the shock equation of state (EOS). The proposed formulation is integrated in the isoclinic configuration and allows for a unique treatment for elastic and plastic anisotropy. The effects of elastic anisotropy are taken into account through the stress tensor decomposition and plastic anisotropy through yield surface defined in the generalized deviatoric plane perpendicular to the generalized pressure. The proposed formulation of this work is implemented into the Lawrence Livermore National Laboratory-DYNA3D code by the modification of several subroutines in the code. The capability of the new constitutive model to capture strain rate and temperature sensitivity is then validated. The final part of this process is a comparison of the results generated by the proposed constitutive model against the available experimental data from both the Plate Impact test and Taylor Cylinder Impact test. A good agreement between experimental and simulation is obtained in each test.