Finite-element analysis and simulation of polymers: a bibliography (1976 - 1996)

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Modelling Simul. Mater. Sci. Eng.5 (1997) 615–650. Printed in the UK PII: S0965-0393(97)86644-2

BIBLIOGRAPHY

Finite-element analysis and simulation of polymers:a bibliography (1976–1996)

Jaroslav MackerleLinkoping Institute of Technology, Department of Mechanical Engineering, S-581 83 Linkoping,Sweden

Received 6 July 1997, accepted for publication 29 July 1997

Abstract. This paper gives a bibliographical review of the finite-element methods applied tothe analysis and simulation of polymers. The bibliography at the end of the paper containsreferences to papers, conference proceedings and theses/dissertations on the subject that werepublished between 1976–1996. The following topics are included: polymer flow and mixingsimulation; polymer processing; thermal analysis of polymers; fracture mechanics of polymers;modelling polymer behaviours and their mechanical properties; practical polymer applicationsin engineering; other topics.

1. Introduction

Polymers are a wide class of natural or synthetic substances composed of very largemolecules, and having a wide range of mechanical, physical and chemical properties. Theyhave good thermal and electrical insulation properties, low density and high resistance tochemicals but are mechanically weaker and exhibit a lower elastic moduli than metals.The advantage of polymers is that they can be easily manufactured into complicatedshapes. The basic manufacturing processes are extrusion, moulding, casting and forming ofsheets.

Plastics are composed of a mixture of polymeric materials and additives, and areusually classified into thermoplastics and thermosets. A main difference between bothcategories is in their intermolecular bonding. Thermoplastics have little or no bondingbetween molecules, thermosets have a strong intermolecular bonding. Rubbers are similarto plastics, the differences are mainly in their large extensibility and stretching. Polymersare rheologically characterized as viscoelastic materials.

Recent progress in computer hardware and software enables us to numerically studyindustrial polymer forming processes. The most frequently used numerical method is thefinite-element method, as can be seen from many papers published on the subject andlisted in the appendix. Finite-element analyses can help us to understand the interplay ofphysical phenomena in complicated material processing situations where many factors asthe nonlinear interactions between field and interfacial phenomena have to be taken intoaccount. The accuracy and reliability of these studies depend first of all on the materialproperties used in the analysis. The most important is to obtain the flow stress of thematerial in the whole strain range which appears in the process. Numerical studies areunsuccessful when the analysis is applied to an inappropriate geometry or when materialdata are not correct.

0965-0393/97/060615+36$19.50c© 1997 IOP Publishing Ltd 615

616 J Mackerle

This paper gives a review of published papers dealing with finite-element methodsapplied to polymer analysis and polymer process simulation. For a more efficientinformation retrieval, the lists of references of papers published between 1976–1996 aredivided into the following topics:

• polymer processing in general: polymer flow and mixing simulation• polymer processing• thermal analysis of polymers• fracture mechanics of polymers• modelling polymer behaviours and their mechanical properties• practical polymer applications in engineering• other topics.

This paper is organized into two parts. In the first one each topic is handled andcurrent trends in application of finite-element techniques are mentioned. The secondpart, an appendix, lists papers published in the open literature for the period 1976–1996 on the subjects presented above. References have been retrieved from the author’sdatabase, MAKEBASE [1, 2]. Readers interested in the finite-element literature ingeneral are referred to [3] or to the author’s Internet Finite-Element Book Bibliography(http://ohio.ikp.liu.se/fe/index.html). Basic information about manufacturing processes canbe found in [4, 5].

2. Polymer processing in general: polymer flow and mixing simulation

This section handles the development and application of existing algorithms for the solutionof general polymer processing problems, while the next section is devoted to specificindustrial polymer processing problems.

The development of new polymers is time consuming and expensive project. Newmaterials with the required properties may sometimes be simulated experimentally by thecompounding, blending, reinforcing and reactive processing of existing polymers. A partof this process can numerically be simulated by the finite-element method. Analyses of themixing of polymer melts, where the effects of operational and geometrical variables can bestudied, are helpful in determining the overall average progression of dispersion but theyare not suitable for the study of the dispersion mechanism. To study the mixing process it isimportant to know how a fluid particle will experience a deformation while it moves insidethe extruder. As a result the distribution of velocity and stress components, temperature andpressure fields is obtained. The tracking of individual particles can also be done. There aredifferent mixing measures—these can be investigated by the finite-element method, by 2Dor 3D flow approximations.

To numerically simulate a viscoelastic flow at an entry region is still a great challengebecause of the high dependency of the flow on viscoelastic properties. The obtained resultshave a poor accuracy and there is a problem with convergence.

Topics included: polymer process modelling; free and moving boundary problems;isothermal and non-isothermal viscoelastic flow; steady and unsteady viscoelastic flow;plastic flow analysis; viscoplastic fluid flow; elasto-viscoplastic fluid flow; nonlinearviscoelastic fluid flow; contraction flow; multilayered flow; viscous flow incl. heat transfer;resin flow; polymer solidification; solidification of semicrystalline polymers; polymercrystallization; droplet deformation; non-Newtonian jets; viscous mixing; chaotic mixing;studies of mixing efficiency; flow in a die; packing stage of viscoelastic melts; die swell

Finite-element analysis and simulation of polymers 617

of polymer melts; polymer stretching; diffusion in polymers; 2D and 3D analyses. Rate-dependent constitutive equations for polymers.

Finite-elements methods included: generalized Eulerian–Lagrangian method; U-ALEformulation; collocation and Galerkin methods; semi-implicit Taylor–Galerkin method;Eulerian based FEM; FEM with moving grid; penalty finite elements; FEM with controlvolume technique; streamline-upwind Petrov–Galerkin FEM.

Materials included: polymers; anisotropic polymers; liquid-crystalline polymers;semicrystalline polymers; particle reinforced polymer composites; glassy polymers; glass;thermosetting polymers; thermoplastic polymers; polyethylene; polycarbonate; rubber.

3. Polymer processing

The basic manufacturing processes for polymers are moulding, extrusion, casting andsheet forming. The references listed in the appendix are sorted according to thesesubheadings.

The moulding process includes the following steps: plastics are heated above thesoftening point and thereafter the molten material is forced into a mould determining thedimensions of the part. The final step is that the part is allowed to harden by freezing orcuring. Some typical moulding processes are: compression moulding, transfer moulding,and injection moulding.

To simulate the moulding process is not easy. With regard to the mould filling it isnecessary to track the molten material moving front in the time and space. In the finite-element simulation several methods are known to solve such a problem, i.e. using an adaptiveor a fixed mesh. It is also possible to use the particle method within the FEM.

Injection moulding is one of the most important manufacturing processes for producingthermoplastic products. The injection moulding process involves filling, packing andhoulding, and solidification. Most computer simulations have concentrated on the fillingstage but some papers have dealt with the post-filling stages. Some problems in connectionto injection moulding can be caused by flow-induced and thermally-induced residual stressesoccurring during the filling stage. They can be predicted by numerical simulations as canthe shrinkage, warpage and sink marks—the other main problems in polymer injectionmoulding. The time- and temperature-dependent constitutive equations of an amorphouspolymer are needed for the finite-element analysis so that the warpage and residual stressescan be analysed.

During the filling of complicated cavities of moulding dies, extensional deformationoccurs as the melt is convected through the cavity and accelerates in the direction of itsmotion. The main issue in the analysis of filling is the tracking of the free surface betweenthe filling material and the escaping gas (air) which was present in the initially emptymould. The simulation of the packing stage is difficult due to the complexity of dealingwith compressible fluid flows. Many factors play an important role such as thermodynamicproperties, the rheology of the resin and eventually the kinetics of crystallization. Somemultidimensional finite-element models for the packing stage have been developed, basedfor example on an augmented Lagrangian approach.

Blow moulding, originally used for the production of small bottles, has been employed inthe production of hollow plastic articles (i.e. containers) as used in automotive, medical andelectrical engineering. Several technologies are known including extrusion blow mouldingand injection blow moulding. The finite-element method is capable of predicting the finalwall thickness distributions. This wall thickness distribution is influenced by the materialbehaviour, deformation history, melt temperature, inflation pressure, etc. The geometrical

618 J Mackerle

shapes of blow moulded parts are often complicated, therefore the use of preforms can helpcreate blow-moulded parts with the required thickness distribution.

Gas-assisted injection moulding is a novel manufacturing process where the mouldcavity is partially filled with polymer melt followed by the injection of inert gas into thecore of the polymer melt. The required injection pressure is reduced and therefore warpageand residual stresses are minimized. An algorithm based on a control volume/finite-elementmethod combined with a particle-tracing scheme has been used to solve two-componentflow-front advancements.

To realistically model resin flow through fibrous media is an important phenomenon inthe polymer composite manufacturing process such as resin transfer moulding, autoclaveprocessing or press laminating. The particular features of each composite manufacturingprocess have to be taken into account when developing finite-element models.

Extrusion produces continuous shapes from the raw material. This material is fed intoa screw chamber where it is preheated, compressed and forced through a heated die. Apolymer under processing passes through an extruder at least once in its path from thepolymerization reactor to the finished product. When leaving the die it has to be rapidlycooled to maintain the extruded shape. Extrusion produces sheets, films, tubes, pipes, rods,profiles, etc. Uniform mixtures produced in an extruder can be used in other manufacturingprocesses such as injection moulding, compression moulding and extrusion forming.

The transporting elements in the extrusion process are important when considering theaction of the pressure pushing the polymer through the kneading elements and the heatingof polymers. The metering section of a single-screw extruder controls the throughput andthe exit melt temperature of the extruder. There are many 2D and 3D studies dealing withthe transport phenomena under screw pumping. In the twin-screw extruder the channelof the transporting elements is interrupted by the flights of the other screw. This zone isimportant in the mixing. We study the relationship between the screw speed, throughputand the pressure gradient along the down-channel direction. Screw extruders are often usedfor blending or compounding to produce uniform mixtures of polymers. Various finite-element studies of polymer flow in intermeshing co-rotating and counter-rotating twin-screwextruders can be find in the appendix.

In an extrusion die the cross section of the flow channel changes from the circularentrance cross section to the requested exit profile. The die swelling and cooling shrinkagein the extrudate are important factors in the design of extrusion dies. Determining thepressure distribution in the die for the specified flow rate is very important—studies ofthe pressure drop across extrusion dies have been done by, for example, univariant andmultivariant finite elements.

Casting is used to form parts which are difficult to produce by the other named methods.Polymers can be cast under gravity in rigid or flexible moulds. This process is cheap butslow. Rotational casting permits the manufacture of completely enclosed hollow mouldings.

The casting is an important process in polymer film forming. The molten polymer isextruded through a die and then is taken up by a rotating chill roll. The velocity of the chillroll is greater than the extrusion velocity at the die, so the molten film is stretched. Thereare some problems in the manufacturing: the film width decreases relative to the die width(neck-in), and the film thickness is thicker at the edge than in the middle of the cross section(edge bead). The operational conditions can minimize these phenomena. Numerical simu-lations can provide a powerful tool for analysing the fluid flow and heat transfer in polymercasting and can be used to examine the process over a wide range of operational conditions.

Thermoforming is the shaping process applied to previously shaped plastics with thehelp of heat. This term is usually limited to sheet forming; thermoforming of tubes is called

Finite-element analysis and simulation of polymers 619

blow moulding. Papers dealing with thermoforming are included in section A.2Othermanufacturing processes; blow moulding is also included in the subsectionMoulding insection A.2.

Topics included:moulding—compression moulding; injection compression moulding;injection moulding; reaction injection moulding; gas-assisted injection moulding; sequentialinjection moulding; blow moulding; injection blow moulding; stretch-blow moulding;powder injection moulding; extrusion blow moulding; preform for blow moulding; injectionmoulding die design; mould filling; resin transfer mould filling; linear and nonlinear bucklingof moulded components; shrinkage and warpage analysis; residual stresses;extrusion—single-screw extruders; twin-screw extruders; co-rotating twin-screw extruders; counter-rotating twin-screw extruders; hydrostatic extrusion; viscoelastic extrusion; profile extrusioncoating; mixing in extrusion; flow in extrusion dies; profile extrusion dies;casting—cast filmprocess; tape casting; centrifugal casting; cast filling;sheet forming—sheet co-extrusion;thermoforming of polymer sheets; blow stretching and vacuuming process; flat sheet dies;sheet extrusion die design;other manufacturing processes—thermoforming; cold drawingprocess; stretching process; forging; solid state forming; magnetic pulse deformation;pultrusion process; plug-assist forming; vacuum forming; coating process; wire coating;filament winding/spinning; woven glass fabrics; glass fibre fabric shaping process.

Materials: polymers; thermoplastics; thermosets; plastics; polyester; polymercomposites; crystalline polymers; glassy polymers; PVC; rubber; silicon rubber.

4. Thermal analysis of polymers

The deformation of glassy polymers in their processing (extrusion, blow moulding,stretching) involves large strain deformations at elevated temperatures and rapid strain rates,and produces anisotropically strengthened components. How these properties depend onprocessing parameters (temperature, pressure, strain rate) can be investigated by the helpof numerical methods. Finite-element thermomechanical modelling as well as heat transferproblems are the subjects of this section. Finite-element studies provide insight into theeffects of strain rate and temperature sensitivities and thermomechanical coupling on thedeformation behaviour of polymers.

Polymer composites exposed to extreme temperatures can undergo thermochemicaldecomposition (charring). Numerical methods have been developed to predict the thermalresponse of polymeric composites during thermochemical decomposition.

Topics included: heat transfer in polymers; heat transfer in single- and twin-screwmixing; polymer flow and heat transfer; heat transfer in non-isothermal resin transfermoulding; heat transfer in extrusion; heat transfer in injection moulding; polymer meltflow simulation; injection mould cooling; transient mould cooling; cooling in blowmoulding; cooling in plastic pipe manufacturing; curing processes; temperature-dependentdynamic behaviour of viscoelastic materials; thermal properties of polymers; thermo-electric field analysis; thermal oxidation; thermomechanical coupling; thermoviscoelasticresponse; thermomechanical decomposition; thermal stresses in polymers; thermal stressesin polymer composites; thermal deformations in polymers; hygrothermal stress analysis;thermomechanical contact problems; thermal mixing effects; residual thermally inducedstresses; drying of thin films; thermal analysis of polymer insulators.

Materials: polymers; plastics; thermosets; thermoplastics; glassy polymers;semicrystalline polymers; polyurethane; polyethylene; coated polymers; polymer concrete;polymer composites.

620 J Mackerle

5. Fracture mechanics of polymers

The fracture mechanics of polymers and finite-element methods are considered in thissection, where compliance, strain-energy release rates, stress intensity factors, etc, forpolymers are studied.

The study of the crack-tip plastic deformations in polymers is not easy because polymersexhibit a different and more complicated plastic behaviour than metals. The plasticity inamorphous polymers typicallly manifested by softening immediately after yield, followedby progressive strain hardening at large deformations. Another well known phenomenonis the neck propagation in polymers. The coupling between the constitutive equations ofpolymers and the crack-tip plastic deformation also needs more research.

A common problem in manufacturing polymer matrix composites is that finished partscontain voids. The presence of voids can reduce the interlaminar shear and flexural strengthof the transverse modulus of polymer unidirectional composites. A finite-element methodcan be used to analyse the effect of the gap closure rate on the void formation mechanism.

Transverse failure initiation and interfacial debonding in polymer composites can bestudied by finite-element parametric studies where the effects of constituent properties,interphase properties and thickness are included.

Topics: linear and nonlinear fracture analysis of polymers; analysis of dynamicfracture events; failure in polymer composites; fracture of metal–polymer structures;damage models for polymers; local damage in polymers; voids distribution in polymercomposites; progressive fracture in polymer composites; propagation of delamination;indentation debonding for polymer layers; necking crazing and crack growth in polymers;vibration of cracked polymers; rapid crack propagation; crack-tip fields; bond strength;impact strength; fracture analysis of protective coatings; fracture analysis of ductilepolymers; brittle failure in ductile polymers; crack-inclusion interaction; moisture-assistedcrack growth; micromechanical modelling of crack-tip in polymers; mechanical testing ofpolymers.

6. Modelling polymer behaviours and their mechanical properties

Polymeric materials subjected to large strains undergo an evolution in molecular orientationand the corresponding strengthening is highly dependent on the state of strain. Linearand nonlinear viscoelastic and viscoplastic finite-element models have been used to studythe time-dependent deformation or failure of polymers, as well as the thermomechanicalbehaviour of anisotropic polymers under large deformations. Constitutive models withproperties corresponding to the necking and strain-hardening behaviour of polymers havebeen derived and the relevant papers are listed in the appendix.

Epoxy resins are frequently toughened by the addition of rubber particles and thesetwo-phase polymers are important as adhesives or as matrices for fibre and particulatecomposites. The finite-element method has been used to study stress distributions in andaround a rubber particle/void in a matrix of epoxy polymer.

Topics included: modelling of the mechanical/thermomechanical properties; time-dependent behaviour of polymers; dynamic behaviour and viscoelastic damping of polymers;parameter identification; material properties for microelectronics; linear and nonlineardeformation behaviour of polymers; viscoelastic and relaxation properties; viscoelasticityand creep recovery in thin films; compressive deformation and mechanical properties inpolymer fibres; influence of polymer matrix on the mechanical response of composites;modelling of the anisotropy; micromechanics of polymer–matrix composites; modelling

Finite-element analysis and simulation of polymers 621

of the toughening mechanisms; aging process in polymers; interfacial properties; contactimpedance of polymers; polymeric material testing; indentation process.

The material and mechanical properties of the following materials are considered:polymers; plastics; thermoplastics; thermosets; glassy polymers; polymer composites;semicrystalline polymers; polymer concrete; polymer foam materials; smart polymers;cellular materials; polyimide; polycarbonate; polyprophylene; polystyrene; polyethylene;polysilicon; epoxy resins; rubber toughened polymers.

7. Practical polymer applications in engineering

The increasing use of polymers in different fields of engineering can be explained on thebasis of performance or economics. The performance advantage of polymers is light weight,corrosion resistance, low coefficient of friction, low thermal/electrical conductivity, opticalproperties, etc. The ease of manufacturing and no need for finishing further adds to theireconomic advantages. Examples where polymers are successfully used in various practicalapplications are given in section A.6.

Some of the major application areas for polymer usage are in electrical and electronicproducts (polymeric insulators, sensors/actuators, electronic packaging, electrooptics,waveguides, etc), automotive engineering (windows, belts, hubs, connectors, compressorvalves, etc), biomechanics (hip joint prosthesis, knee prosthesis, orthotic devices,microcatheters, acetabular cups, dental applications, etc), geomechanics (retaining walls,geogrids, etc), civil and building engineering (pipes, buried pipes, polymer concrete, etc),packaging, textiles, household appliances etc.

8. Other topics

In this category the following topics are included: fastening and joining; contact problems;visualization of mixing; polymer cutting; description of FE software for polymer processing.

Acknowledgment

The bibliography presented in the appendix is by no means complete but it gives acomprehensive representation of different finite-element applications on the subjects. Theauthor wishes to apologize for the unintentional exclusion of missing references and wouldappreciate receiving comments and pointers to other relevant literature for a future update.

References

[1] Mackerle J 1989 MAKEBASE, an information retrieval system in structural mechanics for main-frames andpersonal computersEng. Comput.6 178–85

[2] Mackerle J 1993 An information retrieval system for finite element and boundary element literature andsoftwareEng. Anal. Boundary Elem.11 177–87

[3] Mackerle J 1991Finite Element Methods, A Guide to Information Sources(Amsterdam: Elsevier)[4] Farag M M 1989 Selection of Material and Manufacturing Processes for Engineering Design(New York:

Prentice-Hall)[5] Tadmor Z and Gogos C G 1979Principles of Polymer Processing(New York: Wiley)

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Appendix. A Bibliography (1976–1996)

This bibliography provides a list of literature references on finite-element analysis and thesimulation of polymers. Presented listings contain papers published in scientific journals,conference proceedings, and theses/dissertations retrospectively to 1976. References havebeen retrieved from the author’s database, MAKEBASE. The COMPENDEX database andApplied Mechanics Reviewshave also been checked. The references presented are groupedunder the same section headings as listed in the first part of this paper.

The emphasis of this bibliography is to list, first of all, papers published in variousinternational journals. There have been numerous national and international conferencesand symposia held worldwide but the conference proceedings are a source of never-endingbibliographical confusion. No review on conference proceedings can ever be comprehensivebut some conferences dealing fully/partially with the finite-element analyses of polymershave been examined and a selection of references is also presented in this paper. Themain criticism of conferences is that the material presented is often a repetition of what ispublished elsewhere in the literature, and also the complaint about the uneven quality ofpapers is often heard. Surveys have shown low usage of published conference proceedingsin practice. Also, many important conference papers are published afterwards in an editedversion in international journals.

References are not arranged chronologically but sorted in each category alphabeticallyaccording to the first author’s name. If a specific paper is relevant for several subjectcategories, the same reference can be listed under respective section headings.

A.1. Polymer processing in general: polymer flow and mixing simulation

Agassant J F and Espy M 1985 Theoretical and experimental study of the molten polymer flow in the calenderbankPolym. Eng. Sci.25 118–21

Agassant J Fet al 1992 Polymer processing modeling: simplifying assumptions and computation methodsNum.Meth. Ind. Form. Process.ed J L Chenot (Rotterdam: Balkema) pp 3–13

Ahmed A 1993 Generalized Eulerian–Lagrangian formulations in free and moving boundary problems: processingof viscoelastic polymers and fiber suspensionsPhD ThesisWorcester Polytechnic Institute

Akkerman R 1993 Euler–Lagrange simulations of nonisothermal viscoelastic flowsPhD ThesisUniversity ofTwente, Enschede, Netherlands

Akkerman Ret al 1992 Towards simulation of nonisothermal viscoelastic flows by finite elementsNum. Meth.Ind. Form. Process.ed J L Chenot (Rotterdam: Balkema) pp 329–34

Amoedo J 1990 Rate-dependent constitutive equations and process modeling of polymer materialsPhD ThesisRensselaer Polytechnic Institute

Andreucci Det al 1991 On a mathematical model for the crystallization of polymersProc. ECMI IVed H J Wackerand W Zulehner (Stuttgart: Teubner) pp 3–16

——1994 Numerical simulation of polymer crystallizationMath. Models Meth. Appl. Sci.4 135–45Armstrong R Cet al 1995 Finite element model to predict structure development in spinneret flow of liquid-

crystalline polymers53rd Ann. Tech. Conf. (Boston)pp 2590–4Baaijems F P T1993 An U-ALE formulation of 3-D unsteady viscoelastic flowInt. J. Num. Meth. Eng.36 1115–43Baliga B R and Pantakar S V 1983 A control volume finite element method for two-dimensional fluid flow and

heat transferNum. Heat Transf.6 245–61Baliga B R and Pham T T 1983 Solution of some two-dimensional incompressible fluid flow and heat transfer

problems, using a control volume finite element methodNum. Heat Transf.6 263–82Benard A and Advani S G 1995 Linking microscopic and macroscopic effects during solidification of

semicrystalline polymersNUMIFORM 95ed S F Shen and P Dawson (Rotterdam: Balkema) pp 231–5Bernstein B et al 1994 Steady flows of viscoelastic fluids in axisymmetric abrupt contraction geometry: a

comparison of numerical resultsJ. Rheol.38 53–71Boger D V 1987 Viscoelastic flows through contractionsAnn. Rev. Fluid Mech.19 157–82Boone Tet al 1995 Modelling of microcapsulated polymer shell solidification1994 MRS Fall Meet.372 193–8

Finite-element analysis and simulation of polymers 623

Boriek A M 1990 Modeling of setting stresses in particle reinforced polymer composites using finite elementanalysisPhD ThesisRice University

Boriek A M et al 1988 Setting stress distribution in particle reinforced polymer compositesJ. Composite Mater.22 986–1002

Chang P Wet al 1979 Collocation and Galerkin finite element methods for viscoelastic fluid flow. 1. Descriptionof method and problems with fixed geometryComput. Fluids7 267–83

——1979 Collocation and Galerkin finite element methods for viscoelastic fluid flow. 2. Die swell problems witha free surfaceComput. Fluids7 285–93

Charbonneaux T G and Macosko C W 1994 Modelling of laminar tube flow of thermosetting polymers—applicationof a finite element method with moving gridPolym. React. Eng.2 347–88

Chekhonin K A and Bulgakov V K 1991 Thermodynamics of flows of a polymerizing nonlinearly viscoplasticfluid having a free surfaceJ. Eng. Phys.59 1395–401

Chenot J Let al 1991 A method for determining free surfaces in steady state finite element computationsComp.Meth. Appl. Mech. Eng.92 245–60

Cohen Aet al 1988 Use of mathematical modeling to identify the material mechanisms involved in the fabricationand performance of plastic materials3rd Chemical Cong. North Amer. (Toronto)pp 142–7

Coyle D J and Perdikoulas J 1991 Flow simulation and visualization in spiral mandrel dies49th Ann. Tech. Conf.ANTEC ’91 (Montreal)pp 2445–7

Crochet M J 1986 Recent developments in the numerical simulation of viscoelastic flowNUMIFORM 86ed KMattiassonet al (Rotterdam: Balkema) pp 11–17

Crochet M J and Keunings R 1984 Numerical simulation of viscoelastic flow in some polymer processingapplicationsNum. Anal. Forming Process.(New York: Wiley) pp 239–67

Czyz J A and Szyszkowski W 1990 An effective method for non-linear viscoelastic structural analysisComput.Struct.37 637–46

David B and Tadmor Z 1993 Effect of mixing history on the morphology of immiscible polymer blendsPolym.Eng. Sci.33 227–39

Dawson P R and Thompson E G 1978 Finite element analysis of steady-state elasto-visco-plastic flow by the initialstress-rate methodInt. J. Num. Meth. Eng.12 47–57

Delaby Iet al 1994 Droplet deformation in polymer blends during uniaxial elongational flow: influence of viscosityratio for large capillary numbersJ. Rheol.38 1705–20

Derby J Jet al 1994 Large-scale numerical analysis of materials processing systems: high-temperature crystalgrowth and molten glass flowsComp. Meth. Appl. Mech. Eng.112 69–89

Dietsche L and Dooley J 1995 Numerical simulation of viscoelastic polymer flow—effect of secondary flows onmultilayer coextrusion53rd Ann. Tech. Conf. (Boston)pp 188–93

Dooley J 1990 Simulating the flow in a film die using finite element analysisPlastics Envir. Soc. Plast. Eng.168–71

Dooley J and Dietschke L 1996 Numerical simulation of viscoelastic polymer flow—effects of secondary flowson multilayer coextrusionPlast. Eng.52 37–9

Dumas C G and Dixit R S 1988 Finite element modeling of polymer flow and heat transfer in processing equipment3rd Chemical Cong. North Amer. (Toronto)pp 801–5

——1989 Finite element modeling of polymer flow and heat transfer in processing equipmentComp. Applic. Appl.Polym. Sci.ed T Provder (American Chemical Society) pp 521–36

Feigl K and Ottinger H C 1996 Numerical study of the flow of a low-density-polyethylene melt in a planarcontraction and comparison to experimentsJ. Rheol.40 21–35

Finlayson B A and McClelland M A 1985 Non-isothermal extrudate swell of elastic liquidsNum. Meth. HeatTransfervol III (New York: Wiley) pp 269–87

Georgiou G C and Crochet M J 1994 Compressible viscous flow in slits with slip at the wallJ. Rheol.38 639–54Ghoreishy M H R 1996 Transient finite element model of mixing rubber compounds in a Banbury mixerIranian

J. Polym. Sci. Tech.5 30–41Gupta M and Kwon T H 1990 3-D flow analysis of non-Newtonian viscous fluids using enriched finite elements

Polym. Eng. Sci.30 1420–30Han W H and Rey A 1993 Stationary bifurcations and tricriticality in a creeping nematic polymer flowJ. Non-

Newton. Fluid Mech.50 1–28Hartt W H and Baird D G 1996 The confined flow of polyethylene melts past a cylinder in a planar channel

J. Non-Newton. Fluid Mech.65 247–68Heng F L and Mitsoulis E 1988 Errors in omitting radial convection in polymer melt flow simulationNum. Heat

Transf.13 499–513Hua C C and Schieber J D 1996 Application of kinetic theory models in spatiotemporal flows for polymer solutions

liquid crystals and polymer melts using the CONNFFESSIT ApproachChem. Eng. Sci.51 1473–85

624 J Mackerle

Huang L M et al 1996 Numerical simulation of non-Newtonian jets emanating from a converging coneJ. Polym.Res.3 39–48

Huh J D 1985 A numerical simulation of the flow of viscoelastic materials in the die-entry regionPhD ThesisRice University

Ijam A Z 1977 Finite elements in incompressible viscous flow including heat transferPhD ThesisUniversity ofWales, Swansea

Isayev A I and Upadhyay R K 1985 Two-dimensional viscoelastic flows: experimentation and modelingJ. Non-Newton. Fluid Mech.19 135–60

Jiang Xet al 1995 A P1-P0 finite element method for a model of polymer crystallizationComp. Meth. Appl. Mech.Eng. 125 303–17

Kajiwara T et al 1990 Numerical simulation of die swell of polymer meltsTheor. Appl. Mech.39 337–43——1991 Numerical simulation of fiber orientation and dispersion for fiber-filled polymer meltsTrans. Japan.

Soc. Mech. Eng. Ser.B 57 2740–7——1993 Numerical simulation of converging flow of polymer melts through a tapered slit dieJ. Non-Newton

Fluid Mech.48 111–24Kiani A 1989 A Eulerian-based finite element method for mixing analysis in polymer processing machineryPhD

ThesisStevens Institute of TechnologyKiani A et al 1989 Three-dimensional computational analysis of fluted mixing devicesANTEC 89(New York:

SPE) pp 62–5Kim S J and Kwon T H 1996 Measures of mixing for extrusion by averaging conceptsPolym. Eng. Sci.36 1466–76Kwon T H and Ahn S Y 1994 Characterization of slip of powder/binder mixtures and its effects in pim filling

process analysisPowder Metall. World Cong. (Paris)pp 1093–6Lai J and Bakker A 1994 A 3-D plasto-viscoelastic representation for FEM calculation3rd World Cong. Comp.

Mech. (Chiba, Japan)E11–3Lawal A et al 1993 Computational study of chaotic mixing in co-rotating two-tipped kneading paddles: two-

dimensional approachPolym. Eng. Sci.33 140–8Lee H S 1989 Thin-cavity filling analysis using the finite element method with control volume techniquesPhD

ThesisRensselaer Polytechnic InstituteLi C S et al 1995 Finite element analysis for phase change problem in polymer processingInt. Commun. Heat

Mass Transf.22 167–78Li T and Manas-Zloczower I 1995 Flow field analysis of a multi-cut transfermix53rd Ann. Tech. Conf. (Boston)

pp 113–18Loh K W L et al 1996 Effect of constitutive models on the numerical simulation of viscoelastic flow at an entry

regionPolym. Eng. Sci.36 1990–2000Luo X L 1996 Control volume approach for integral viscoelastic models and its application to contraction flow of

polymer meltsJ. Non-Newton. Fluid Mech.64 173–89Luo X L and Mitsoulis E 1990 An efficient algorithm for strain history tracking in finite element computations of

non-Newtonian fluids with integral constitutive equationsInt. J. Num. Meth. Fluids11 1015–31——1990 A numerical study of the effect of elongational viscosity on vortex growth in contraction flows of

polyethylene meltsJ. Rheol.34 309–42Luo X L and Tanner R I 1986 A streamline element scheme for solving viscoelastic problems. I. Differential

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