Rocha da silva, Luisa Alexandra (2004) Viscoelastic Compressible Flow and Applications in 3D Injection Molding Simulation. PhD thesis Mécanique numérique, ENSMP - CEMEF Centre de Mise en Forme des Matériaux, ENSMP.

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Abstract

This work concerns simulation of viscoelastic compressible flows and applications in injection molding. Compressibility has been integrated in REM3D by considering a density evolution for the material, function of the pressure and temperature through the Tait law. Conservation of mass is written in terms of velocity, pressure and temperature, using the isothermal compressibility and dilatation material parameters. We are led to the Stokes compressible problem, which is solved using the Mixed Finite Element method. The linear system arising from the problem is non-linear and non-symmetrical. Thermal coupling is also taken into account, being the energy problem solved through the Space-Time Discontinuous Galerkin method. Extension of the compressible Stokes problem to a multi-domain configuration is given through the characteristic functions, computed by advection equations. Viscoelasticity is introduced using the Pom-Pom model. The extra-stress is considered function of microscopic properties of the material, such as molecular orientation and chain stretch. Elasticity is a perturbation in the mechanical problem. The presence of a solvent viscosity guarantees the ellipticity of the problem, and a stabilization scheme of the DEVSS type is adopted. Orientation and stretch are obtained solving evolution equations, of the hyperbolic type, through the Space-Time Discontinuous Galerkin method. Finally, a temperature equation for viscoelastic compressible models is derived. In what concerns injection molding applications, REM3D covers now all the main stages of the injection molding process, from filling to part ejection. Solidification and behavior of the polymer at the solid state are taken into account considering a very simple evolution. Compensation of the material's shrinkage is done using the material's compressible character. Anisotropy in the internal stresses remaining in the part affects its final mechanical, optical or dimensional properties and induce warpage once the part ejected. Comparison with the literature and experiments is performed, mainly to validate the post-filling stage, showing a good agreement.

References

[Agassant et al., 1986] Agassant, J.-F., Avenas, P., Sergent, J.-P., Vergnes, B., and Vincent, M. (1986). La mise en forme des matières plastiques. Lavoisier, Paris, 3rd edition.
[Ammar, 2001] Ammar, A. (2001). Modélisation numérique de la cristallisation induite par l'écoulement d'un thermoplastique. PhD thesis, école Normale Supérieure de Cachan.
[Arnold et al., 1984] Arnold, D., Brezzi, F., and Fortin, M. (1984). A stable finite element for Stokes equations. Calcolo, 21:337-344.
[Baaijens, 1991] Baaijens, F. (1991). Residual stresses in injection molded products. Rheologica Acta, 30(3):284-299.
[Baaijens, 1998] Baaijens, F. (1998). Mixed finite element method for viscoelastic flow analysis: a review. Journal of Non-Newtonian Fluid Mechanics, 79:361-386.
[Baaijens et al., 1998] Baaijens, F., Selen, S., Baaijens, H., and Peters, G. (1998). Viscoelastic flow past a confined cylinder of a low density polyethylene. Journal of Non-Newtonian Fluid Mechanics, 68:173-203.
[Batkam et al., 2003] Batkam, S., Bruchon, J., and Coupez, T. (2003). A space-time discontinuous Galerkin method for convection and diffusion in injection moulding. International Journal of Forming Processes, Accepted.
[Beraudo et al., 1998] Beraudo, C., Fortin, A., Coupez, T., Demay, Y., Vergnes, B., and Agassant, J.(1998). A finite element method for computing the flow of multi-mode viscoelastic fluids: Comparison with experiments. Journal of Non-Newtonian Fluid Mechanics, 75:1-23.
[Beris and Edwards, 1994] Beris, A. and Edwards, B. (1994). Thermodynamic of flowing systems. Oxford, New York.
[Bigot and Coupez, 2000] Bigot, E. and Coupez, T. (2000). Capture of 3d moving free surfaces and material interfaces by mesh deformation. ECCOMAS 2000, Barcelona, 11-14 September 2000.
[Bikard et al., 2004] Bikard, J., Bruchon, J., Coupez, T., and Silva, L. (2004). Polyurethane foam expansion macro-modelling. International Journal for Numerical Methods in Fluids, Submitted.
[Blackwell et al., 2000] Blackwell, R., McLeish, T., and Harlen, O. (2000). Molecular drag-strain coupling in branched polymer melts. Journal of Rheology, 44:121-136.
[Bogaerds et al., 1999] Bogaerds, A., Verbeeten, W., Peters, G., and Baiijens, F. (1999). 3d viscoelastic analysis of a polymer solution in a complex flow. Computer Methods in Applied Mechanics and Engineering, 180:413-430.
[Bourrigaud et al., 2003] Bourrigaud, S., Marin, G., and Poitou, A. (2003). Shear modification of long-chain branched polymers: A theoretical approach using the pom-pom model. Macromolecules, 36:1388-1394.
[Brezzi and Fortin, 1991] Brezzi, F. and Fortin, M. (1991). Mixed and hybrid finite element methods. Springer-Verlag, Berlin.
[Bristeau et al., 1990] Bristeau, M., Glowinski, R., Dutto, L., Periaux, J., and Roge, G. (1990). Compressible viscous flow calculations using compatible finite element approximations. International Journal for Numerical Methods in Fluids, 11:719-749.
[Broyer et al., 1974] Broyer, E., Guttfinger, C., and Tadmor, Z. (1974). Flow-analysis network (fan) method for solving flow problems in polymer processing. Polymer Engineering and Science, 14:660.
[Bruchon and Coupez, 2003] Bruchon, J. and Coupez, T. (2003). The foam structure prediction by the 3d calculation of the bubble growth in a molten polymer. Journal of Non-Newtonian Fluid Mechanics, Submitted.
[Brujan, 2001] Brujan, E. (2001). The equation of bubble dynamics in a compressible linear viscoelastic liquid. Fluid Dynamics Research, 298:287-294.
[Chang, 1994] Chang, M. (1994). On the study of surface defects in the injection moulding of rubbermodified thermoplastics. ANTEC94, pages 360-367.
[Chiang et al., 1991] Chiang, H., Hieber, C., and Wang, K. (1991). A unified simulation of filling and post-filling stages in injection molding. part 1: formulation. Polymer Engineering and Science, 31(2):116-140.
[Chinesta et al., 2000] Chinesta, F., Poitou, A., and Torres, R. (2000). A semi-lagrangian strategy to predict the fiber orientation in the steady flows of reinforced thermoplastics. Computer Methods in Applied Mechanics and Engineering, 189:233-247.
[Clemeur et al., 2002] Clemeur, N., Rutgers, R., and Debbaut, B. (2002). On the evaluation of some differential formulations for the 'pom-pom' constitutive model. Rheologica Acta, 14(1):25-32.
[Cormenzana et al., 2001] Cormenzana, J., Ledda, A., Laso, M., and Debbaut, B. (2001). Calculation of free surface flows using CONNFESSIT. Journal of Rheology, 45(1):237-258.
[Coupez, 1996] Coupez, T. (1996). Stable-stabilized finite element for 3d forming calculation. CEMEF, ENSMP, internal report.
[Crochet and Keunings, 1982] Crochet, M. and Keunings, R. (1982). Finite element analysis of die swell of a highly elastic fluid. Journal of Non-Newtonian Fluid Mechanics, 10:339-356.
[Cruickshank and Munson, 1981] Cruickshank, J. and Munson, B. (1981). Viscous buckling of plane and axisymetric jets. Journal of Fluid Mechanics, 13:221-239.
[Davis, 1983] Davis, G. D. (1983). Natural convection of air in a square cavity: a benchmark numerical solution. International Journal for Numerical Methods in Fluids, 3:249-264.
[Dee and Walsh, 1988] Dee, G. and Walsh, D. (1988). Equations of state for polymer liquids. Macromolecules, 21:815-817.
[des Cloizeaux, 1989] des Cloizeaux, J. (1989). Relaxation of entangled polymers in melt. Macromolecules, 23(17):3992-4006.
[Dietz et al., 1978] Dietz, W., White, J., and Clark, E. (1978). Orientation development and relaxation in injection molding of amorphous polymers. Polymer Engineering and Science, 18:273.
[Digonnet and Coupez, 2003] Digonnet, H. and Coupez, T. (2003). Object-oriented programming for " fast-and-easy " development of parallel applications in forming processes simulation. In K.J.Bathe, editor, Second MIT Conference on Computational Fluid and Solid Mechanics, pages 1922-1924. Massachussets Institute of Technology, Elsevier.
[Doi and Edwards, 1986] Doi, M. and Edwards, S. (1986). The theory of polymer dynamics. Oxford University Press, Oxford.
[Dressler et al., 1999] Dressler, M., Edwards, B., and Ottinger, H. (1999). Macroscopic thermodynamics of flowing polymeric liquids. Rheologica Acta, 38:117-136.
[Flaman, 1990] Flaman, A. (1990). Built-up and relaxation of molecular orientation in injection molding. PhD thesis, Technische Universiteit Eindhoven.
[Flory et al., 1964] Flory, P., Orwoll, R., and Vrij, A. (1964). Statistical thermodynamics of chain molecule liquids. i: the equation of state for paraffin hydrocarbons. Journal of American Chemical Society, 86:3507-3514.
[Fulchiron, 2002] Fulchiron, R. (2002). Comportement des matériaux sous-pression. ´ Ecole thématique CNRS, Injection des polymères.
[Gennes, 1970] Gennes, P. D. (1970). Reptation of a polymer chain in the presence of fixed obstacles. Journal of Chemical Physics, 55:572-579.
[Graham et al., 2001] Graham, R., McLeish, T., and Harlen, O. (2001). Using the pom-pom equations to analyze polymer melts in exponential shear. Journal of Rheology, 45(1):27-52.
[Greener and Pearson, 1983] Greener, J. and Pearson, G. (1983). Orientation residual stresses and birefringence in injection molding. Journal of Rheology, 27:115.
[Guenette and Fortin, 1995] Guenette, R. and Fortin, M. (1995). A new finite element method for computing viscoelastic flows. Journal of Non-Newtonian Fluid Mechanics, 60:275-290.
[Hammer, 2001] Hammer, C. (2001). Calcul des contraintes et deformations residuelles lors de la mise en oeuvre de verre ophtalmique en polymère thermodurcissable. Ecole Nationale Supérieure des Mines de Paris.
[Harlow and Amsden, 1971] Harlow, F. and Amsden, A. (1971). A numerical fluid dynamics calculation method for all flow speeds. Journal of Computational Physics, 8:197-213.
[Harry and Parrot, 1970] Harry, D. and Parrot, R. (1970). Numerical simulation of injection mold filling. Polymer Engineering and Science, 10:209-.
[Hauke and Hughes, 1994] Hauke, G. and Hughes, T. (1994). A unified approach to compressible and incompressible flows. Computer Methods in Applied Mechanics and Engineering, 113:389-395.
[Hieber and Shen, 1980] Hieber, C. and Shen, S. (1980). A finite element/finite difference method of the injection mold-filling process. Journal of Non-Newtonian Fluid Mechanics, 7:1-32.
[Hoff, 1995] Hoff, D. (1995). Global solutions of the navier-stokes equations for multidimensional compressible flow with discontinuous initial data. Journal of Differential Equations, 120(1):215-254.
[Illinca and Hetu, 2001] Illinca, F. and Hetu, J.-F. (2001). Three dimensional filling and post-filling of polymer injection molding. International Polymer Processing, 16:291-301.
[Inkson et al., 1999] Inkson, N., McLeish, T., Harlen, O., and Groves, D. (1999). Predicting low density polyethylene melt rheology in elongational and shear flows with 'pom-pom' constitutive equations. Journal of Rheology, 43:873-869.
[Isayev, 1983] Isayev, A. (1983). Orientation development in the injection molding of amorphous polymers. Polymer Engineering and Science, 23:271.
[Isayev, 1987] Isayev, A. (1987). Injection and Compression Molding Fundamentals. Marcel Dekker, New York.
[Isayev and Hieber, 1980] Isayev, A. and Hieber, C. (1980). Towards viscoelastic modelling of the injection molding of polymers. Rheologica Acta, 19:168-182.
[Issa et al., 1986] Issa, R., Gosman, A., and Watkins, A. (1986). The computation of compressible and incompressible free recirculating flows by a non-iterative implicit scheme. Journal of Computational Physics, 62:66-82.
[Janeschitz, 1977] Janeschitz, H. (1977). Injection molding of plastics: some ideas about the relationship between mold filling and birefringence data. Rheologica Acta, 16:327.
[Kamal et al., 1986] Kamal, M., Chiu, E., Lafleur, P., and Ryan, M. (1986). Computer simulation of injection mold filling for viscoelastic melts with fountain flow. Polymer Engineering and Science, 26:190-196.
[Kamal and Kenig, 1972] Kamal, M. and Kenig, S. (1972). The injection molding of thermoplastics. Polymer Engineering and Science, 12:294-308.
[Kanel, 1968] Kanel, Y. (1968). A model system for the one-dimensional motion of a gas. Differencial'nya Uravenenija, 4:721-734.
[Kellog and Liu, 1996] Kellog, R. and Liu, B. (1996). A finite element method for the compressible navier-stokes equations. SIAM Journal of Numerical Analysis, 33:780-788.
[Kellog and Liu, 1997] Kellog, R. and Liu, B. (1997). A penalized finite element method for a compressible stokes systems. SIAM Journal of Numerical Analysis, 34:1093-1105.
[Keshtiban et al., 2004] Keshtiban, I., Belbidlia, F., and Webster, M. (2004). Numerical simulation of compressible viscoelastic liquids. Journal for Non-Newtonian Fluid Mechanics, 122:131-146.
[Keunings, 1986] Keunings, R. (1986). An algorithm for the simulation of transient viscoelastic flows with free surfaces. Journal of Computational Physics, 62:199-220.
[Keunings and Bousfield, 1987] Keunings, R. and Bousfield, D. (1987). Analysis of surface tension driven levelling in horizontal viscoelastic films. Journal of Non-Newtonian Fluid Mechanics, 22:219-233.
[Khazikov and Shelukh, 1977] Khazikov, A. and Shelukh, V. (1977). Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Journal of Applied Mathematics in Mechanics, 41:273-282.
[Khazikov and Weigant, 1995] Khazikov, A. and Weigant, V. (1995). On the existence of global solutions of two-dimensional navier-stokes equations of a compressible viscous fluids. Journal of Siberian Mathematics, 36:1108-1141.
[Kim et al., 1999] Kim, I., Park, S., Chung, J., and Kwon, T. (1999). Numerical modeling of injection/compression molded center-gated disk. Polymer Engineering and Science, 39:1930-1942.
[Kuo and Kamal, 1976] Kuo, Y. and Kamal, M. (1976). The fluid mechanics and heat transfer of injection mold filling of thermoplastics materials. A.I.C.H.E. Journal, 22:661-669.
[Kuo and Kamal, 1977] Kuo, Y. and Kamal, M. (1977). Flow of thermoplastics in the filling and packing stages of injection molding. In International Conference on Polymer Processing, pages 329-348. Cambridge, Massachussets.
[Kwon, 1996] Kwon, Y. (1996). On hadamard stability for compressible viscoelastic constitutive equation. Journal of Non-Newtonian Fluid Mechanics, 65:151-163.
[Lee et al., 2001] Lee, K., Mackley, M., MacLeish, T., Nicholson, T., and Harlen, O. (2001). Experimental observation and numerical simulation of transient "stress fangs" within flowing molten polyethylene. Journal of Rheology, 45(6):1261-1277.
[Lefebvre, 1993] Lefebvre, L. (1993). Numerical simulation of the flow of polyurethane foams. PhD thesis, Louvain-la-Neuve.
[Leonov, 1976] Leonov, A. (1976). Non-equilibrium thermodynamics and rheology of compressible viscoelastic media. Rheologica Acta, 15:85-98.
[Lions, 1993] Lions, P. (1993). Existence globale de solutions pour les equations de navier-stokes compressibles isentropiques. Comptes-Rendus de l'Académie des Sciences, 316:1335-1340.
[Lord, 1979] Lord, H. (1979). Flow of polymers with pressure dependent viscosity. Polymer Engineering and Science, 19:469-473.
[Mahishi, 1998] Mahishi, M. (1998). Material characterization for thin wall molding simulation. In ANTEC1998, pages 547-551.
[Maillot, 1993] Maillot, I. (1993). Simulation du remplissage/compactage pour la mise en forme des thermoplastiques par injection. Applications industrielles. PhD thesis, Université Joseph Fourier, Grenoble.
[Marchal and Crochet, 1987] Marchal, J. and Crochet, M. (1987). A new mixed finite element for calculating viscoelastic flow. Journal of Non-Newtonian Fluid Mechanics, 26:77-114.
[Marrucci, 1983] Marrucci, G. (1983). Testing of a constitutive equation for entangled networks by elongational and shear data of polymer melts. Rheologica Acta, 12:269.
[Marrucci, 1985] Marrucci, G. (1985). Relaxation by reptation and tube enlargement: a model for polydisperse polymers. Journal of Polymer Science: Polymer Physics, 23:159-177.
[Matsumura and Nishida, 1979] Matsumura, A. and Nishida, T. (1979). The initial value problem for the equations of motion of compressible viscous and heat conductive fluids. Proceedings of the Japan Academy, 55:337-342.
[Matusu-Necasova et al., 1999] Matusu-Necasova, S., Sequeira, A., and Videman, J. (1999). Existence of classical solutions for compressible viscoelastic fluids of Oldroyd type past an obstacle. Mathematical Methods in Applied Sciences, 22:449-460.
[Mavridis et al., 1988] Mavridis, H., Hrymak, A., and Vlachopoulos, J. (1988). Finite element simulation of fountain flow in injection molding. Polymer Engineering and Science, 26:449.
[McLeish and Larson, 1998] McLeish, T. and Larson, R. (1998). Molecular constitutive equations for a class of branched polymers: the Pom-Pom polymer. Journal of Rheology, 42:81-110.
[Nigro et al., 1997] Nigro, N., Storti, M., and Idelsohn, S. (1997). A general algorithm for compressible and incompressible flow. stability analysis and explicit time integration. International Journal of Numerical Methods for Heat and Fluid Flow, 7(2/3):141-168.
[Nonaka and Nakayama, 1996] Nonaka, N. and Nakayama, T. (1996). A unified method for the numerical analysis of compressible and incompressible viscous flows. Computational Mechanics, 19:369-376.
[Ottinger, 2001] Ottinger, H. (2001). Thermodynamic admissibility of the pompon model for branched polymers. Rheologica Acta, 40(4):317-321.
[Pantani et al., 2004] Pantani, R., Sorrentino, A., Speranza, V., and Titomanlio, G. (2004). Molecular orientation in injection molding: experiments and analysis. Rheologica Acta, 43:109-118.
[Peters and Baaijens, 1997] Peters, G. and Baaijens, F. (1997). Modelling of non-isothermal viscoelastic flows. Journal of Non-Newtonian Fluid Mechanics, 168:205-224.
[Pichelin and Coupez, 1998] Pichelin, E. and Coupez, T. (1998). Finite element solution of the 3d mold filling problem for viscous incompressible fluid. Computer Methods in Applied Mechanics and Engineering, 163:359-371.
[Pontes, 2002] Pontes, A. (2002). Shrinkage and ejection forces in injection molded products. PhD thesis, Universidade do Minho.
[Rajagopalam et al., 1990] Rajagopalam, D., Armstrong, R., and Brown, R. (1990). Finite element method for steady calculation of viscoelastic flow using constitutive equations with a newtonian viscosity. Journal of Non-Newtonian Fluid Mechanics, 36:159-192.
[Redjeb, 2003] Redjeb, A. (2003). Simulation numérique de l'orientation de fibre en injection de thermoplastiques renforcés. Rapport d'avancement, CEMEF.
[Rodgers, 1993] Rodgers, P. (1993). PVT relationships for polymeric liquids: a review of equations of state and their characteristic parameters for 56 polymers. Journal of Applied Polymer Science, 48:1061-1080.
[Ryan and Chung, 1980] Ryan, M. and Chung, T. (1980). Conforming mapping analysis of injection mold filling. Polymer Engineering and Science, 20:642.
[Saez, 2003] Saez, E. (2003). ´ Etude numérique du remplissage 3D en fonderie. PhD thesis, école Nationale Supérieure des Mines de Paris.
[Shyu et al., 2003] Shyu, G., Isayev, A., and Lee, H. (2003). Numerical simulation of flow induced birefringence in injection molded disk. Korea-Australia Rheology Journal, 15:159-166.
[Simha and Somcynski, 1969] Simha, R. and Somcynski, T. (1969). On the statistical thermodynamics of spherical and chain molecule fluids. Macromolecules, 2:342-350.
[Simo, 1987] Simo, J. (1987). On a fully three dimensional finite strain viscoelastic damage model: formulation and computational aspects. Computer Methods in Applied Mechanics and Engineering, 60:153-173.
[Sirakov, 2000] Sirakov, I. (2000). Etude par éléments finis des écoulements viscoélastiques des polymères fondus dans des géometries complexes. Résultats numériques et expérimentaux. PhD thesis, école des Mines de Saint-Etienne.
[Smirnova et al., 2004] Smirnova, J., Silva, L., Monasse, B., Chenot, J., and Haudin, J. (2004). Structure development in injection molding. A 3d simulation with a differential formulation of the kinetics equations. International Polymer Processing, Submitted.
[Sun et al., 1996] Sun, J., Phan-Thien, N., and Tanner, R. (1996). An adaptive viscoelastic stress splitting scheme and its applications: Avss/si and avss/supg. Journal of Non-Newtonian Fluid Mechanics, 65:75-91.
[Szadi et al., 1995] Szadi, M., Salmon, T., Liu, A., Bornside, D., and Armstrong, R. (1995). New mixed finite element method for viscoelastic flows governed by differential constitutive equations. Journal of Non-Newtonian Fluid Mechanics, 59:215-243.
[Tadmor, 1974] Tadmor (1974). Molecular orientation in injection molding. Journal of Applied Polymer Science, 18:1753.
[Takeshima and Funakoshi, 2001] Takeshima, M. and Funakoshi, N. (2001). Molecular orientation distribution in polycarbonate disks. Journal of Applied Polymer Science, 32:3457-3468.
[Tanner, 1970] Tanner, R. (1970). A theory of die swell. Journal of Polymer Science, 8:2067-2078.
[Titomanlio et al., 1980] Titomanlio, G., D.Acierno, and Mantia, F. L. (1980). Modelling of the filling and packing step in the injection molding of thermoplastic materials. Ing.Chim.It., 16:110.
[Valette, 2003] Valette, R. (2003). The matching of numerical simulation involving time dependent compressible viscoelastic flow with precise polyethylene processing data. AERC Conference, Portugal.
[Verbeeten et al., 2002] Verbeeten, W., Peters, G., and Baaijens, F. (2002). Viscoelastic analysis of complex polymer melt flows using the extended pompom model. Journal of Non-Newtonian Fluid Mechanics, 108:301-326.
[Wakashima and Saitoh, 2004] Wakashima, S. and Saitoh, T. (2004). Benchmark solutions for natural convection in a cubic cavity using the high-order space time method. International Journal of Heat and Mass Transfer, 47:853-864.
[Wales, 1976] Wales, J. (1976). The application of flow birefringence to rheological studies of polymer melts. PhD thesis, Delft University.
[Wang et al., 1986] Wang, K., Hieber, C., and Wang, K. (1986). Dynamic simulation and graphics for the injection molding of three dimensional thin parts. Journal of Polymer Engineering, 7:21-45.
[Wapperom, 1996] Wapperom, P. (1996). Non-isothermal flows of viscoelastic fluids. PhD thesis, Delft University.
[Wapperon and Hulsen, 1998] Wapperon, P. and Hulsen, M. (1998). Thermodynamics of viscoelastic fluids: the temperature equation. Journal of Rheology, 42(5):999-1019.
[Westover, 1992] Westover, R. (1992). Measuring polymer melt viscosities at high pressures: The hydrostatic pressure rheometer. Advances in Polymer Technology, 11(2):147-151.
[Williams and Lord, 1975] Williams, G. and Lord, H. (1975). Mold filling studies for the injection molding of thermoplastic materials. Polymer Engineering and Science, 15:553-268.
[Yabe and Wang, 1991] Yabe, T. and Wang, P. (1991). Unified numerical procedure for compressible and incompressible flows. Journal of the Physical Society of Japan, 60:2105-2108.
[Zienkiewicz and Wu, 1992] Zienkiewicz, O. and Wu, J. (1992). A general explicit or semi-explicit algorithm for compressible and incompressible flows. International Journal for Numerical Methods in Engineering, 35:475-479.

Table of content

1 Introduction
1.1 Injection molding
1.2 An unified model
1.3 Context, objectives and outline
2 Compressibility
2.1 Isothermal compressibility
2.2 Thermal compressibility
2.3 Polymers compressibility
2.4 Extension to compressible free surface flows
2.5 Applications to more complex systems
2.6 Conclusions
3 Viscoelasticity
3.1 Why considering viscoelasticity in injection molding flows ?
3.2 Polymer viscoelasticity and viscoelastic models
3.3 Isothermal viscoelasticity
3.4 Compressible viscoelasticity
3.5 Applications in viscoelastic free surface flows
3.6 Some remarks on thermal viscoelasticity
3.7 Conclusions
4 Applications in injection molding
4.1 The general injection molding problem
4.2 Comparison with the literature
4.3 Experimental results and comparison with simulation
4.4 Application in injection molding of a 3D complex geometry
4.5 Conclusion
5 Conclusion and perspectives
5.1 Synthesis and conclusion
5.2 Perspectives and improvements
A Numerical resolution of transport equations in REM3D
B Thermodynamics of viscoelastic compressible media
C Mathematical considerations on Stokes compressible flows

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