Performance of Hyperelastic Material Laws in Simulating Biaxial Deformation Response of Polypropylene and High Impact Polystyrene

  • K. Y. TshaiEmail author
  • E. M. A. Harkin-Jones
  • P. J. Martin
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 35)


Free surface moulding processes such as thermoforming and blow moulding involve thermal and spatial varying rate dependent biaxial deformation of polymer. These processes are so rapid that the entire forming took place in a matter of seconds. As a result of the elevated rate of deformation, assumption that the deforming polymers experience no time dependent viscous dissipation or perfectly elastic up to large strain has became a common practice in numerical simulation. Following the above assumption, Cauchy’s elastic and hyperelastic theories, originally developed for vulcanised natural rubber has been widely used to represent deforming polymeric materials in free surface moulding processes. To date, various methodologies were applied in the development of these theories, the most significant are those develop purely based on mathematical interpolation (mathematical models) and a more scientific network theories that involves the interpretation of macro-molecular structure within the polymer. In this chapter, the most frequently quoted Cauchy’s elastic and hyperelastic theories, including Ogden, Mooney–Rivlin, neo-Hookean, 3-chain, 8-chain, Van der Waals full network, Ball’s tube model, Edwards–Vilgis crosslinks-sliplinks model and the elastic model of Sweeney–Ward are reviewed. These models were analysed and fitted to a series of experimental high strain rate, high temperature, biaxial deformations data of polypropylene (PP) and high impact polystyrene (HIPS). The performance and suitability of the various models in capturing the polymer’s complex deformation behaviour during free surface moulding processes is presented.


Hyperelastic model Biaxial deformation High strain rate High temperature Free surface moulding 


  1. 1.
    Hooke, R., De Potentia Restitutiva or of Spring Explaining the Power of Springing Bodies, p. 23. London (1678)Google Scholar
  2. 2.
    Markovitz, H.: The emergency of rheology. Physics Today (American Institute of Physics) 21(4), 23–33 (1968)Google Scholar
  3. 3.
    Truesdell, C.A., Cauchy’s First Attempt at Molecular Theory of Elasticity, Bollettino di Storia delle Scienze Matematiche. Il Giardino di Archimede 1(2), 133–143 (1981)Google Scholar
  4. 4.
    Bogolyubov, A.N., Augustin Cauchy and His Contribution to Mechanics and Physics (Russian), Studies in the History of Physics and Mechanics, pp. 179–201. Nauka, Moscow (1988)Google Scholar
  5. 5.
    Dahan-Dalmédico, A., La Propagation Des Ondes En Eau Profonde Et Ses Développements Mathématiques (Poisson, Cauchy, 1815–1825), in The History of Modern Mathematics II, pp. 129–168. Boston, MA (1989)Google Scholar
  6. 6.
    Truesdell, C.A.: Cauchy and the modern mechanics of continua. Rev. Hist. Sci. 45(1), 5–24 (1992)CrossRefGoogle Scholar
  7. 7.
    Mooney, M.: A theory of large elastic deformation. J. Appl. Phys. 11(9), 582–592 (1940)CrossRefGoogle Scholar
  8. 8.
    Rivlin, R.S.: Large elastic deformations of isotropic materials, I, II, III, fundamental concepts. Philos. Trans. R. Soc. Lond. Ser. A 240, 459–525 (1948)CrossRefGoogle Scholar
  9. 9.
    Rivlin, R.S.: Large elastic deformations of isotropic materials, IV, further developments of the general theory. Philos. Trans. R. Soc. Lond. Ser. A 241(835), 375–397 (1948)CrossRefGoogle Scholar
  10. 10.
    Ogden, R.W., Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubberlike solids. In: Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, vol. 365, pp. 565–584 (1972)Google Scholar
  11. 11.
    Treloar, L.R.G.: Stress-strain data for vulcanized rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70 (1944)CrossRefGoogle Scholar
  12. 12.
    Ward, I.M., Mechanical Properties of Solid Polymers, 2nd edn. Wiley, Hoboken (1983)Google Scholar
  13. 13.
    Treloar, L.R.G.: The elasticity of a network of long chain molecules I. Trans. Faraday Soc. 39, 36–64 (1943)CrossRefGoogle Scholar
  14. 14.
    Treloar, L.R.G.: The elasticity of a network of long chain molecules II. Trans. Faraday Soc. 39, 241–246 (1943)CrossRefGoogle Scholar
  15. 15.
    Kuhn, W., Grun, F.: Beziehungen zwichen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer stoffe. Kolloideitschrift 101, 248–271 (1942)Google Scholar
  16. 16.
    James, H.M., Guth, E.: Theory of elastic properties of rubber. J. Chem. Phys. 11(10), 455–481 (1943)CrossRefGoogle Scholar
  17. 17.
    Wang, M.C., Guth, E.: Statistical theory of networks of non-gaussian flexible chains. J. Chem. Phys. 20, 1144–1157 (1952)CrossRefGoogle Scholar
  18. 18.
    Arruda, E.M., Boyce, M.C.: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41(2), 389–412 (1993)CrossRefGoogle Scholar
  19. 19.
    Treloar, L.R.G., Riding, G., A non-gaussian theory for rubber in biaxial strain I, mechanical properties. In: Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, vol. 369, pp. 261–280 (1979)Google Scholar
  20. 20.
    Wu, P.D., Van der Giessen, E.: On improved 3-D non-Gaussian network models for rubber elasticity. Mech. Res. Commun. 19(5), 427–433 (1992)CrossRefGoogle Scholar
  21. 21.
    Wu, P.D., Van der Giessen, E.: On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers. J. Mech. Phys. Solids 41(3), 427–456 (1993)CrossRefGoogle Scholar
  22. 22.
    Flory, P.J., Erman, B.: Theory of elasticity of polymer networks. Macromolecules 15(3), 800–806 (1982)CrossRefGoogle Scholar
  23. 23.
    Ball, R.C., Doi, M., Edwards, S.F., Warner, M.: Elasticity of entangled networks. Polymer 22(8), 1010–1018 (1981)CrossRefGoogle Scholar
  24. 24.
    Doi, M., Edwards, S.F.: The theory of polymer dynamics. Oxford University Press, Oxford (1986)Google Scholar
  25. 25.
    Kilian, H.G.: Equation of state of real networks. Polymer 22, 209–217 (1981)CrossRefGoogle Scholar
  26. 26.
    Kilian, H.G.: Energy balance in networks simply elongated at constant temperature. Colloid Polym. Sci. 259, 1084–1091 (1981)CrossRefGoogle Scholar
  27. 27.
    Vilgis, T.H., Kilian, H.G.: The van der waals-network—a phenomenological approach to dense networks. Polymer 25, 71–74 (1984)CrossRefGoogle Scholar
  28. 28.
    Kilian, H.G., Vilgis, T.H.: Fundamental aspects of rubber-elasticity in real networks. Colloid Polym. Sci. 262, 15–21 (1984)CrossRefGoogle Scholar
  29. 29.
    Kilian, H.G.: An interpretation of the strain-invariants in largely strained networks. Colloid Polym. Sci. 263, 30–34 (1985)CrossRefGoogle Scholar
  30. 30.
    Edwards, S.F., Vilgis, T.H.: The effect of entanglements in rubber elasticity. Polymer 27(4), 483–492 (1986)CrossRefGoogle Scholar
  31. 31.
    Edwards, S.F., Vilgis, T.H.: The stress-strain relationship in polymer glasses. Polymer 28(3), 375–378 (1987)CrossRefGoogle Scholar
  32. 32.
    Rouse, P.E.: A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21, 1272–1280 (1953)CrossRefGoogle Scholar
  33. 33.
    Flory, P.J.: Principles of Polymer Chemistry. Cornell University Press, New York (1966)Google Scholar
  34. 34.
    Flory, P.J.: Theory of Polymer Networks. The Effect of Local Constraints on Junctions. J. Chem. Phys. 66(12), 5720 (1977)CrossRefGoogle Scholar
  35. 35.
    Sweeney, J., Ward, I.M.: The modeling of multiaxial necking in polypropylene using a sliplink-crosslink theory. J. Rheol. 39(5), 861–872 (1995)CrossRefGoogle Scholar
  36. 36.
    Sweeney, J., Kakadjian, S., Craggs, G., Ward, I.M., The multiaxial stretching of polypropylene at high temperatures. In: ASME Mechanics of Plastics and Plastic Composites, MD vol. 68 / AMD vol. 215 pp. 357–364 (1995)Google Scholar
  37. 37.
    Sweeney, J., Ward, I.M.: A constitutive law for large deformations of polymers at high temperatures. J. Mech. Phys. Solids 44(7), 1033–1049 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • K. Y. Tshai
    • 1
    Email author
  • E. M. A. Harkin-Jones
    • 2
  • P. J. Martin
    • 2
  1. 1.Department of Mechanical, Materials and Manufacturing EngineeringUniversity of Nottingham Malaysia CampusSemenyihMalaysia
  2. 2.School of Mechanical and Aerospace EngineeringQueen’s University BelfastBelfastNorthern Ireland, UK

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