Mechanics of Time-Dependent Materials

, Volume 1, Issue 2, pp 209–240 | Cite as

Nonlinear Viscoelastic and Viscoplastic Constitutive Equations Based on Thermodynamics

  • R.A. Schapery


An approach to modeling the mechanical behavior of fiber reinforced and unreinforced plastics with an evolving internal state is described. Intrinsic nonlinear viscoelastic and viscoplastic behavior of the resin matrix is taken into account along with growth of damage. The thermodynamic framework of the method is discussed first. The Gibbs free energy is expressed in terms of stresses, internal state variables (ISVs), temperatureand moisture content. Simplifications are introduced based on physical models for evolution of the ISVs and on experimental observations of thedependence of strain state on stress state and its history. These simplifications include use of master creep functions that account for multiaxial stresses, environmental factors and aging in a reduced time and other scalars. An explicit representation of the strains follows, which isthen specialized to provide three-dimensional homogenized constitutiveequations for transversely isotropic, fiber composites. Experimentalsupport for these equations is briefly reviewed. Finally, physicalinterpretation of some of the constitutive functions is discussed usingresults from a microcracking model as well as molecular rate process andfree volume theories. It is shown that the present thermodynamicformulation leads to a generalized rate process theory that accounts for abroad distribution of thermally activated transformations in polymers.

nonlinear viscoelasticity viscoplasticity constitutive equations thermodynamics free volume durability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Boyce, M.C., Parks, D.M. and Argon, A.S., ‘Large inelastic deformation of glassy polymers. Part I. Rate dependent constitutive model’, Mech. Mat. 7, 1988, 15–33.Google Scholar
  2. Brouwer, R., ‘Nonlinear viscoelastic characterization of transversely isotropic fibrous composites under biaxial loading’, Ph.D. Thesis, Free University of Brussels, 1986.Google Scholar
  3. Crissman, J.M., ‘Creep and recovery behavior of a linear high density polythelene and an ethylenehexene copolymer in the region of small uniaxial deformations’, Polymer Eng. Sci. 26, 1986, 1050–1059.Google Scholar
  4. Emri, I. and Pavsek, V., ‘On the influence of moisture on the mechanical properties of polymers’, Materials Forum 16, 1993, 123–131.Google Scholar
  5. Ferry, J.D., Viscoelastic Properties of Polymers, 3rd edn, John Wiley & Sons, New York, 1980.Google Scholar
  6. Fillers, R.W. and Tschoegl, N.W., ‘The effect of pressure on the mechanical properties of polymers’, Trans. Soc. Rheol. 21(1), 1977, 51–100.Google Scholar
  7. Findley, W.N., Reed, R.M. and Stern, P., ‘Hydrostatic creep of solid plastics’, J. App. Mech. 34, 1967, 895–904.Google Scholar
  8. Fung, Y.C., Fundamentals of Solid Mechanics, PrenticeHall, Englewood Cliffs, NJ, 1965.Google Scholar
  9. Gates, T.S. and Sun, C.T., ‘Elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites’, AIAA J. 29, 1991, 457–463.Google Scholar
  10. Hasan, O.A. and Boyce, M.C., ‘A constitutive model for the nonlinear viscoelastic viscoplastic behavior of glassy polymers’, Polymer Eng. Sci. 35, 1995, 331–344.Google Scholar
  11. Hashin, Z., ‘Failure criteria for unidirectional fiber composites’, J. Appl. Mech. 47, 1980, 329–334.Google Scholar
  12. Hult, J.A.H., Creep in Engineering Structures, Blaisdell, Waltham, MA, 1966.Google Scholar
  13. Kachanov, M., ‘Elastic solids with many cracks and related problems’, in Advances in Applied Mechanics, Vol. 30, J. Hutchinson and T. Wu (eds), Academic Press, New York, 1993, 259–455.Google Scholar
  14. Knauss, W.G. and Emri, I., ‘Volume change and the nonlinearly thermoviscoelastic constitution of polymers’, Polymer Eng. Sci. 27, 1987, 86–100.Google Scholar
  15. Lai, J., ‘Non-linear time-dependent deformation behavior of high density polyethylene’, Ph.D. Thesis, Delft University Press, Delft, 1995.Google Scholar
  16. Lai, J. and Bakker, A., ‘An integral constitutive equation for nonlinear plasto-viscoelastic behavior of high-density polyethylene’, Polymer Eng. Sci. 35, 1995, 1339–1347.Google Scholar
  17. Lamborn, M.J. and Schapery, R.A., ‘An investigation of the existence of a work potential for fiberreinforced plastic’, J. Comp. Mat. 27(4), 1993, 352–382.Google Scholar
  18. Liang, Y.M. and Liechti, K.M., ‘On the large deformation and localization behavior of an epoxy resin under multiaxial stress states’, Int. J. Solids Structures 33, 1996, 1479–1500.Google Scholar
  19. Losi, G.U. and Knauss, W.G., ‘Free volume theory and nonlinear thermoviscoelasticity’, Polymer Eng. Sci. 32(8), 1992, 542–557.Google Scholar
  20. Lou, Y.C. and Schapery, R.A., ‘Viscoelastic characterization of a nonlinear fiber-reinforced plastic’, J. Comp. Mat. 5, 1971, 208–234.Google Scholar
  21. McClintock, F.A. and Argon, A.S., Mechanical Behavior of Materials, Addison-Wesley, Reading, MA, 1966.Google Scholar
  22. McKenna, G.B., ‘On the physics required for prediction of long term performance of polymers and their composites’, J. Research of the National Institute of Standards and Technology 99, 1994, 169–189.Google Scholar
  23. Mignery, L.A. and Schapery, R.A., ‘Viscoelastic and nonlinear adherend effects in bonded composite joints’, J. Adhesion 34, 1991, 17–40.Google Scholar
  24. Onat, E.T. and Leckie, F.A., ‘Representation of mechanical behavior in the presence of changing internal structure’, J. Appl. Mech. 55, 1988, 1–10.Google Scholar
  25. Papka, S.D. and Kyriakides, S. ‘In-plane crushing of a polycarbonate honeycomb’, Int. J. Solids Structures, 1997 (in press).Google Scholar
  26. Park, S. and Schapery, R.A., ‘A viscoelastic constitutive model for particulate composites with growing damage’, Int. J. Solids Structures 34, 1997, 931–947.Google Scholar
  27. Reddy, J.N. and Rasmussen, M.L., Advanced Engineering Analyis, John Wiley & Sons, New York, 1982.Google Scholar
  28. Rice, J.R., ‘Inelastic contitutive relations for solids: An internal variable theory and its application to metal plasticity’, J. Mech. Phys. Solids 19, 1971, 433–455.Google Scholar
  29. Schapery, R.A., ‘Application of thermodynamics to thermomechanical, fracture and birefringent phenomena in viscoelastic media’, J. Appl. Phys. 35, 1964, 1451–1465.Google Scholar
  30. Schapery, R.A., ‘A theory of nonlinear thermoviscoelasticity based on irreversible thermodynamics’, in Proc. 5th U.S. Nat. Cong. Appl. Mech., ASME, New York, 1966, 511–530.Google Scholar
  31. Schapery, R.A., ‘Further development of a thermodynamic constitutive theory: Stress formulation’, Purdue University Report No. AA & ES. 692, 1969a.Google Scholar
  32. Schapery, R.A., ‘On the characterization of nonlinear viscoelastic materials’, Polymer Eng. Sci. 9, 1969b, 295–310.Google Scholar
  33. Schapery, R.A., ‘Viscoelastic behavior and analysis of composite materials’, in Mechanics of Composite Materials, G.P. Sendeckyj (ed.), Academic Press, New York, 1974, 85–168.Google Scholar
  34. Schapery, R.A., ‘On viscoelastic deformation and failure behavior of composite materials with distributed flaws’, in Advances in Aerospace Structures and Materials, ASME, New York, AD01, 1981, 5–20.Google Scholar
  35. Schapery, R.A., ‘Correspondence principles and a generalized J integral for large deformation and fracture analysis of viscoelastic media’, Int. J. Fracture 25, 1984, 195–223.Google Scholar
  36. Schapery, R.A., ‘A theory of mechanical behavior of elastic media with growing damage and other changes in structure’, J. Mech. Phys. Solids 38(2), 1990a, 215–253.Google Scholar
  37. Schapery, R.A., ‘On some path independent integrals and their use in fracture of nonlinear viscoelastic media’, Int. J. Fracture 42, 1990b, 189–207.Google Scholar
  38. Schapery, R.A., ‘Simplifications in the behavior of viscoelastic composites with growing damage’, in IUTAM Symposium on Inelastic Deformation of Composite Materials, G.J. Dvorak (ed.), SpringerVerlag, New York, 1991, 193–214.Google Scholar
  39. Schapery, R.A., ‘On nonlinear viscoelastic constitutive equations for composite materials’, in Proceedings of VII International Congress on Experimental Mechanics, Las Vegas, 1992, 9–21.Google Scholar
  40. Schapery, R.A., ‘Nonlinear viscoelastic constitutive equations for composites based on work potentials’, Mechanics USA, A.S. Kobayashi (ed.), Appl. Mech. Rev. 47, 1994, S269–S275.Google Scholar
  41. Schapery, R.A., ‘Prediction of compressive strength and kink bands in composites using a work potential’, Int. J. Solids Structures 32(6/7), 1995, 739–765.Google Scholar
  42. Schapery, R.A., ‘Characterization of nonlinear, timedependent polymers and polymeric composites for durability analysis’, in Progress in Durability Analysis of Composite Systems, A.H. Cardon, H. Fukuda and K. Reifsnider (eds.), Balkema, Rotterdam, 1996, 21–38.Google Scholar
  43. Schapery, R.A., ‘Nonlinear viscoelastic and viscoplastic constitutive equations with growing damage’, University of Texas Report No. SSM-97-2, 1997.Google Scholar
  44. Schapery, R.A. and Sicking, D.L., ‘On nonlinear constitutive equations for elastic and viscoelastic composites with growing damage’, in Mechanical Behavior of Materials, A. Bakker (ed.), Delft University Press, Delft, 1995, 45–76.Google Scholar
  45. Shay, R.M. Jr. and Caruthers, J.M., ‘A new nonlinear viscoelastic constitutive equation for predicting yield in amorphous solid polymers’, J. Rheology 30, 1986, 781–827.Google Scholar
  46. Skontorp, A., ‘Isothermal high-temperature oxidation, aging and creep of carbonfiber/polyimide composites’, Ph.D. Dissertation, Department of Mechanical Engineering, University of Houston, TX, 1995.Google Scholar
  47. Spencer, A.J.M., ‘Theory of invariants’, in Continuum Physics Vol. I: Mathematics, A.C. Eringen (ed.), Academic Press, New York, 1971.Google Scholar
  48. Struik, L.C.E., Physical Aging in Amorphous Polymers and Other Materials, Elsevier, Amsterdam, 1978.Google Scholar
  49. Sullivan, J.L., ‘Creep and physical aging of composites’, Comp. Sci. Technol. 39, 1990, 207–232.Google Scholar
  50. Sun, C.T. and Chen, I.L., A simple flow rule for characterizing nonlinear behavior of fiber composites, J. Comp. Mat. 23, 1989, 1009–1020.Google Scholar
  51. Sun, C.T. and Rui, Y., ‘Orthotropic elasto-plastic behavior of AS4/PEEK thermoplastic composite in compression’, Mech. Mat. 10, 1990, 117–125.Google Scholar
  52. Tuttle, M.E., Pasricha, A. and Emery, A., ‘The nonlinear viscoelastic-viscoplastic behavior of IM7/5260 composite subjected to cyclic loading’, J. Comp. Mat. 29, 1995, 2025–2046.Google Scholar
  53. Weitsman, Y., ‘A continuum diffusion model for viscoelastic materials’, J. Phys. Chem. 94(2), 1990, 961–968.Google Scholar
  54. Yoon, K.J. and Sun, C.T., ‘Characterization of elastic-viscoplastic properties of an AS4/PEEK thermoplastic composite’, J. Comp. Mat. 25, 1991, 1277–1226.Google Scholar
  55. Zapas, L.J. and Crissman, J.M., ‘Creep and recovery behavior of ultra-high molecular weight polyethylene in the region of small uniaxial deformations’, Polymer 25, 1984, 57–62.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • R.A. Schapery
    • 1
  1. 1.Department of Aerospace Engineering and Engineering MechanicsThe University of TexasAustinU.S.A

Personalised recommendations