Acta Mechanica

, Volume 213, Issue 1–2, pp 71–96 | Cite as

A general inelastic internal state variable model for amorphous glassy polymers

  • J. L. BouvardEmail author
  • D. K. Ward
  • D. Hossain
  • E. B. Marin
  • D. J. Bammann
  • M. F. Horstemeyer


This paper presents the formulation of a constitutive model for amorphous thermoplastics using a thermodynamic approach with physically motivated internal state variables. The formulation follows current internal state variable methodologies used for metals and departs from the spring-dashpot representation generally used to characterize the mechanical behavior of polymers like those used by Ames et al. in Int J Plast, 25, 1495–1539 (2009) and Anand and Gurtin in Int J Solids Struct, 40, 1465–1487 (2003), Anand and Ames in Int J Plast, 22, 1123–1170 (2006), Anand et al. in Int J Plast, 25, 1474–1494 (2009). The selection of internal state variables was guided by a hierarchical multiscale modeling approach that bridged deformation mechanisms from the molecular dynamics scale (coarse grain model) to the continuum level. The model equations were developed within a large deformation kinematics and thermodynamics framework where the hardening behavior at large strains was captured using a kinematic-type hardening variable with two possible evolution laws: a current method based on hyperelasticity theory and an alternate method whereby kinematic hardening depends on chain stretching and material plastic flow. The three-dimensional equations were then reduced to the one-dimensional case to quantify the material parameters from monotonic compression test data at different applied strain rates. To illustrate the generalized nature of the constitutive model, material parameters were determined for four different amorphous polymers: polycarbonate, poly(methylmethacrylate), polystyrene, and poly(2,6-dimethyl-1,4-phenylene oxide). This model captures the complex character of the stress–strain behavior of these amorphous polymers for a range of strain rates.


Amorphous Polymer Helmholtz Free Energy Glassy Polymer Internal State Variable Entanglement Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahzi S., Makradi A., Gregory R.V., Edie D.D.: Modeling of deformation behavior and strain-induced crystallization in poly(ethylene terephthalate) above the glass transition temperature. Mech. Mater. 35, 1139–1148 (2003)CrossRefGoogle Scholar
  2. 2.
    Anand L.: On H. Hencky’s approximate strain-energy function for moderate deformations. ASME J. Appl. Mech. 46, 78–82 (1979)zbMATHGoogle Scholar
  3. 3.
    Anand L.: Moderate deformations in extension–torsion of incompressible isotropic elastic materials. J. Mech. Phys. Solids 34, 293–304 (1986)CrossRefGoogle Scholar
  4. 4.
    Anand L., Gu C.: Granular materials: constitutive equations and strain localization. J. Mech. Phys. Solids 48, 1710–1733 (2000)MathSciNetGoogle Scholar
  5. 5.
    Anand L., Gurtin M.E.: A theory of amorphous solids undergoing large deformations, with application to polymeric glasses. Int. J. Solids Struct. 40, 1465–1487 (2003)zbMATHCrossRefGoogle Scholar
  6. 6.
    Anand L., Ames N.M.: On modeling the micro-indentation response of an amorphous polymer. Int. J. Plast. 22, 1123–1170 (2006)zbMATHCrossRefGoogle Scholar
  7. 7.
    Anand L., Ames N.M., Srivastava V., Chester S.A.: A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part I: formulation. Int. J. Plast. 25, 1474–1494 (2009)zbMATHCrossRefGoogle Scholar
  8. 8.
    Ames N.M., Srivastava V., Chester S.A., Anand L.: A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: applications. Int. J. Plast. 25, 1495–1539 (2009)zbMATHCrossRefGoogle Scholar
  9. 9.
    Argon A.S.: A theory for the low temperature plastic deformation of glassy polymers. Philos. Mag. 28, 839–865 (1973)CrossRefGoogle Scholar
  10. 10.
    Arruda E.M., Boyce M.C.: Evolution of plastic anisotropy in amorphous polymers during finite straining. Int. J. Plast. 9, 697–720 (1993)CrossRefGoogle Scholar
  11. 11.
    Arruda E.M., Boyce M.C., Jayachandran R.: Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers. Mech. Mater. 19, 193–212 (1995)CrossRefGoogle Scholar
  12. 12.
    Bammann D.J.: Internal variable model of viscoplasticity. Int. J. Eng. Sci. 22, 1041–1053 (1984)zbMATHCrossRefGoogle Scholar
  13. 13.
    Bammann D.J.: Modeling temperature and strain rate dependent large deformations of metals. Appl. Mech. Rev. 1, 312–318 (1990)CrossRefGoogle Scholar
  14. 14.
    Bamman D.J., Chiesa M.L., Johnson G.C.: Modeling large deformation and failure in manufacturing processes. In: Tatsumi, T., Wanatabe, E., Kambe, T. (eds) Theoretical and Applied Mechanics, pp. 359–376. Elsevier Science, USA (1996)Google Scholar
  15. 15.
    Bardenhagen S.G., Stout M.G., Gray G.T.: Three-dimensional finite deformation viscoplastic constitutive models for polymeric materials. Mech. Mater. 25, 235–253 (1997)CrossRefGoogle Scholar
  16. 16.
    Bouvard, J.L., Ward, D.K., Hossain, D., Nouranian, S., Marin, E.B., Horstemeyer, M.F.: Review of hierarchical multiscale modeling to describe the mechanical behavior of amorphous polymers. JEMT. doi: 10.1115/1.3183779 (2009)
  17. 17.
    Bouvard, J.L., Bouvard, C., Tyson, M., Fletcher, S., Tucker, M., Wang, P.: Model for predicting the strain rate dependence-Impact performance of plastic components: phase I, CAVS Report MSU.CAVS.CMD.2009-R0020 (2009)Google Scholar
  18. 18.
    Boyce M.C., Parks D.M., Argon A.S.: Large inelastic deformation of glassy deformation of glassy polymers part I : rate dependent constitutive model. Mech. Mater. 7, 15–33 (1988)CrossRefGoogle Scholar
  19. 19.
    Boyce M.C., Weber G.G., Parks D.M.: On the kinematics of finite strain plasticity. J. Mech. Phys. Solids 37, 647–665 (1989)zbMATHCrossRefGoogle Scholar
  20. 20.
    Christensen R.M.: Theory of Viscoelasticty: an Introduction. Academic Press, New York (1982)Google Scholar
  21. 21.
    Coleman B., Gurtin M.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)CrossRefGoogle Scholar
  22. 22.
    Elias-Zuniga A., Beatty M.F.: Constitutive equations for amended non-Gaussian network models of rubber elasticity. Int. J. Eng. Sci. 40, 2265–2294 (2002)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Eyring H.: Viscosity, plasticity, and diffusion as examples of absolute reaction rates. J. Chem. Phys. 4, 283–291 (1936)CrossRefGoogle Scholar
  24. 24.
    Fotheringham D.G., Cherry B.W.: Comment on the compression yield behaviour of polymethyl methacrylate over a wide range of temperatures and strain-rates. J. Mater. Sci. 11, 1368–1370 (1976)CrossRefGoogle Scholar
  25. 25.
    Fotheringham D.G., Cherry B.W.: The role of recovery forces in the deformation of linear polyethylene. J. Mater. Sci. 13, 951–964 (1978)CrossRefGoogle Scholar
  26. 26.
    Gent A.N.: A new constitutive relation for rubber. Rubber Chem. Technol. 69, 59–61 (1996)MathSciNetGoogle Scholar
  27. 27.
    Ghorbel E.: A viscoplastic constitutive model for polymeric material. Int. J. Plast. 24, 2032–2058 (2008)zbMATHCrossRefGoogle Scholar
  28. 28.
    Govaert L.E., Timmermans P.H.M., Brekelmans W.A.M.: The influence of intrinsic strain softening on strain localization in polycarbonate: modeling and experimental validation. J. Eng. Mater. Technol. 122, 177–185 (2000)CrossRefGoogle Scholar
  29. 29.
    Gurtin M.: An Introduction to Continuum Mechanics. Academic Press, New York (1981)zbMATHGoogle Scholar
  30. 30.
    Gurtin M.E., Anand L.: The decomposition F = FeFp, material symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous. Int. J. Plast. 21, 1686–1719 (2005)zbMATHCrossRefGoogle Scholar
  31. 31.
    Hasan O.A., Boyce M.C.: A constitutive model for the nonlinear viscoelastic viscoplastic behavior of glassy polymers. Polym. Eng. Sci. 35, 331–344 (1995)CrossRefGoogle Scholar
  32. 32.
    Haupt P., Lion A., Bachaus E.: On the dynamic behaviour of polymers under finite strains: constitutive modelling and identification of parameters. Int. J. Solids Struct. 37, 3633–3646 (2000)zbMATHCrossRefGoogle Scholar
  33. 33.
    Haward, R.N., Thackray, G.: The use of a mathematical model to describe isothermal stress–strain curves in glassy thermoplastics. In: Proceedings of the Royal Society of London, vol. 302, pp. 453–472 (1968)Google Scholar
  34. 34.
    Hencky H.: The elastic behavior of vulcanized rubber. J. Appl. Mech. 1, 45–53 (1933)Google Scholar
  35. 35.
    Holzapfel G.A., Simo J.C.: A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. Int. J. Solids Struct. 33, 3019–3034 (1996)zbMATHCrossRefGoogle Scholar
  36. 36.
    Hossein, D., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F.: Atomistic Exploration of Amorphous Glassy Polymers. CAVS Internal Report (2009)Google Scholar
  37. 37.
    Hoy, R.S., Robbins, M.O.: Strain hardening of polymer glasses: limitations of network models. Phys. Rev. Lett. doi: 10.1103/PhysRevLett.99.117801 (2007)
  38. 38.
    Hoy R.S., Robbins M.O.: Strain hardening of polymer glasses: entanglements, energetics, and plasticity. Phys. Rev. E 77, 031801 (2008)CrossRefGoogle Scholar
  39. 39.
    Hoover W.G.: Canonical dynamics: equilibrium phase-space distributions. Phys. Rev., A31, 1695–1697 (1985)Google Scholar
  40. 40.
    James H.M., Guth E.: Theory of elastic properties of rubber. J. Chem. Phys. 11, 455–481 (1943)CrossRefGoogle Scholar
  41. 41.
    Khan A.S., Zhang H.: Finite deformation of a polymer and constitutive modeling. Int. J. Plast. 17, 1167–1188 (2001)zbMATHCrossRefGoogle Scholar
  42. 42.
    Khan A.S., Lopez-Pamies O., Kazmi R.: Thermo-mechanical large deformation response and constitutive modeling of viscoelastic polymers over a wide range of strain rates and temperatures. Int. J. Plast. 22, 581–601 (2006)zbMATHCrossRefGoogle Scholar
  43. 43.
    Krempl E.: The overstress dependence of the inelastic rate of deformation inferred from transient tests. Mater. Sci. Res. Int. 1, 3–10 (1995)Google Scholar
  44. 44.
    Krempl E.: A small strain viscoplasticity theory based on overstress. In: Krausz, A., Krausz, K. (eds) Unified Constitutive Laws of Plastic Deformation, pp. 281–318. Academic Press, San Diego (1996)CrossRefGoogle Scholar
  45. 45.
    Krempl E., Ho K.: An overstress model for solid polymer deformation behavior applied to Nylon 66. ASTM STP 1357, 118–137 (2000)Google Scholar
  46. 46.
    Krempl E., Khan F.: Rate (time)-dependent deformation behavior: an overview of some properties of metals and solid polymers. Int. J. Plast. 19, 1069–1095 (2003)zbMATHCrossRefGoogle Scholar
  47. 47.
    Kröner E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Ration. Mech. Anal. 4, 273–334 (1960)zbMATHCrossRefGoogle Scholar
  48. 48.
    Lee E.H.: Elastic plastic deformation at finite strain. ASME J. Appl. Mech. 36, 1–6 (1969)zbMATHGoogle Scholar
  49. 49.
    Leonov A.I.: Nonequilibrium thermodynamics and rheology of viscoelastic polymer media. Rheol. Acta 15, 85–98 (1976)zbMATHCrossRefGoogle Scholar
  50. 50.
    Lion A.: On the large deformation behaviour of reinforced rubber at different temperatures. J. Mech. Phys. Solids 45, 1805–1834 (1997)CrossRefGoogle Scholar
  51. 51.
    Lubarda V.A., Benson D.J., Meyers M.A.: Strain-rate effects in rheological models of inelastic response. Int. J. Plast. 19, 1097–1118 (2003)zbMATHCrossRefGoogle Scholar
  52. 52.
    Makradi A., Ahzi S., Gregory R.V., Edie D.D.: A two-phase self-consistent model for the deformation and phase transformation behavior of polymers above the glass transition temperature: application to PET. Int. J. Plast. 21, 741–750 (2005)zbMATHCrossRefGoogle Scholar
  53. 53.
    Mayo S.L., Olafson B.D., Goddard W.A. III: Dreiding: a generic force field for molecular simulations. J. Phys. Chem. 94, 8897–8909 (1990)CrossRefGoogle Scholar
  54. 54.
    Miehe C., Goktepe S., Mendez Diez J.: Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space. Int. J. Solids Struct. 46, 181–202 (2008)CrossRefGoogle Scholar
  55. 55.
    Nose S.: A molecular dynamics method for simulations in the canonical ensemble 1. Mol. Phys. 50, 255–268 (1984)CrossRefGoogle Scholar
  56. 56.
    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, vol. A326, pp. 565–584 (1972)Google Scholar
  57. 57.
    Perzyna P.: Fundamental problems in viscoplasticity. Adv. Appl. Mech. 9, 243–377 (1966)CrossRefGoogle Scholar
  58. 58.
    Plimpton S.J.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995)zbMATHCrossRefGoogle Scholar
  59. 59.
    Prantil V.C., Jenkins J.T., Dawson P.R.: An analysis of texture and plastic spin for planar polycrystals. J. Mech. Phys. Solids 41, 1357–1382 (1993)zbMATHCrossRefGoogle Scholar
  60. 60.
    Reese S., Govindjee S.: A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35, 3455–3482 (1998)zbMATHCrossRefGoogle Scholar
  61. 61.
    Richeton J., Ahzi S., Daridon L., Remond Y.: A formulation of the cooperative model for the yield stress of amorphous polymers for a wide range of strain rates and temperatures. Polymer 46, 6035–6043 (2006)CrossRefGoogle Scholar
  62. 62.
    Richeton J., Ahzi S., Vecchio K.S., Jiang F.C., Makradi A.: Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates. Int. J. Solids Struct. 44, 7938–7954 (2007)zbMATHCrossRefGoogle Scholar
  63. 63.
    Rivlin R.S., Saunders D.W.: Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philos. Trans. R. Soc. Lond. A 243, 251–2881 (1951)CrossRefGoogle Scholar
  64. 64.
    Robbins M.O., Hoy R.S.: Scaling of the strain hardening modulus of glassy polymers with the flow stress. J. Polym. Sci. B 47, 1406 (2009)CrossRefGoogle Scholar
  65. 65.
    Shepherd J.E., McDowell D.L., Jacob K.I.: Modeling morphology evolution and mechanical behavior during thermo-mechanical processing of semi-crystalline polymers. J. Mech. Phys. Solids 54, 467–489 (2006)zbMATHCrossRefGoogle Scholar
  66. 66.
    Shepherd, J.E.: Multiscale modeling of the deformation of semi-crystalline polymers. Ph. D. thesis, Georgia Institute of Technology, Atlanta, GA (2006)Google Scholar
  67. 67.
    Tomita Y.: Constitutive modeling of deformation behavior of glassy polymers and applications. Int. J. Mech. Sci. 42, 1455–1469 (2000)zbMATHCrossRefGoogle Scholar
  68. 68.
    Tervoort T.A., Smit R.J.M., Brekelmans W.A.M., Govaert L.E.: A constitutive equation for the elasto-viscoplastic deformation of glassy polymers. Mech. Time Depend. Mater. 1, 269–291 (1998)CrossRefGoogle Scholar
  69. 69.
    Tervoort T.A., Govaert L.E.: Strain-hardening behavior of polycarbonate in the glassy state. J. Rheol. 44, 1263–1277 (2000)CrossRefGoogle Scholar
  70. 70.
    Van der Sluis O., Schreurs P.J.G., Meijer H.E.H.: Homogenisation of structured elastoviscoplastic solids at finite strains. Mech. Mater. 33, 499–522 (2001)CrossRefGoogle Scholar
  71. 71.
    Yeoh O.H.: Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber Chem. Technol. 63, 792–805 (1990)Google Scholar
  72. 72.
    Wendlandt M., Tervoort T.A., Suter U.W.: Nonlinear, rate dependent strain-hardening behavior of polymer glasses. Polymer 46, 11786–11797 (2005)CrossRefGoogle Scholar
  73. 73.
    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, 427–456 (1993)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • J. L. Bouvard
    • 1
    Email author
  • D. K. Ward
    • 1
    • 2
  • D. Hossain
    • 1
  • E. B. Marin
    • 1
  • D. J. Bammann
    • 1
  • M. F. Horstemeyer
    • 1
  1. 1.Center for Advanced Vehicular SystemsMississippi State UniversityStarkvilleUSA
  2. 2.Sandia National LaboratoriesLivermoreUSA

Personalised recommendations