Viscoelasticity and Thermoviscoelasticity

  • Dallas N. LittleEmail author
  • David H. Allen
  • Amit Bhasin


This chapter presents an overview of the development of the three-dimensional theories of viscoelasticity and thermoviscoelasticity. An understanding of these models is essential to the ability to predict the response of flexible pavements containing asphalt binder, as well as rate-dependent base materials.


Viscoelasticity Thermoviscoelasticity Initial boundary value problem Thermodynamic constraints Direct analytic method Correspondence principle Direct method Collocation method Creep tests Ramp tests Relaxation tests Accelerated characterization tests Frequency sweeps Complex modulus Time–temperature superposition Mechanical analogs Prony series Nonlinear viscoelasticity 


  1. Allen, D., & Haisler, W. (1985). Introduction to aerospace structural analysis. Wiley.Google Scholar
  2. Alfrey, T. (1944). Nonhomogeneous stresses in viscoelastic media. Quarterly of Applied Mathematics, 2, 113.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Ban, H., Im, S., & Kim, Y. (2013). Nonlinear viscoelastic approach to model damage-associated performance behavior of asphaltic mixture and pavement structure. Canadian Journal of Civil Engineering, 40, 313.CrossRefGoogle Scholar
  4. Biot, M. (1954). Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. Journal of Applied Physics, 25, 1385.CrossRefzbMATHGoogle Scholar
  5. Biot, M. (1965). Mechanics of incremental deformations: Theory of elasticity and viscoelasticity of initially stressed solids and fluids, including thermodynamic foundations and applications to finite strains. Wiley.Google Scholar
  6. Boltzmann, L. (1874). Zur theorie der elastischen nachwirkungen. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche, 70, 275–306.Google Scholar
  7. Christensen, R., & Naghdi, P. (1967). Linear non-isothermal viscoelastic solids. Acta Mechanica, 3, 1.CrossRefzbMATHGoogle Scholar
  8. Christensen, R. (1982). Theory of viscoelasticity—An introduction (2nd ed.). New York: Academic Press.Google Scholar
  9. Coleman, B. (1964). Thermodynamics of materials with memory. Journal of Chemical Physics, 47, 597.CrossRefGoogle Scholar
  10. Coulombe, C. (1784). Recherches théoretiques et experimentales sur la force de torsion et sur l’élasticité des fils de metal. Mémoires de l’Académie Royale des Sciences.Google Scholar
  11. Day, W. (1972). The thermodynamics of simple materials with fading memory. Springer.Google Scholar
  12. Debye, P. (1913). Ver. Deut. Phys. Gesell., 15, 777, New York.Google Scholar
  13. Ferry, J. (1980). Viscoelastic properties of polymers (3rd ed.). New York: Wiley.Google Scholar
  14. Flugge, W. (1975). Viscoelasticity (Second Revised ed.). Berlin: Springer.CrossRefzbMATHGoogle Scholar
  15. Golden, H., Strganac, T., & Schapery, R. (1999). An approach to characterize nonlinear viscoelastic material behavior using dynamic mechanical tests and analyses. Journal of Applied Mechanics, 66, 872.CrossRefGoogle Scholar
  16. Green, A., & Rivlin, R. (1957). The mechanics of non-linear materials with memory; Part I. Archive for Rational Mechanics and Analysis, 1, 1.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Green, A., Rivlin, R., & Spencer, A. (1959). The mechanics of non-linear materials with memory: Part II. Archive for Rational Mechanics and Analysis, 3, 82.Google Scholar
  18. Greenburg, M. (1978). Foundations of applied mathematics. Prentice Hall.Google Scholar
  19. Huang, C., Abu Al-Rub, R., & Masad, E. (2010). Journal of Materials in Civil Engineering, 23, 56.Google Scholar
  20. Kim, R., & Little, D. (1990). One-dimensional constitutive modeling of asphalt concrete. Journal of Engineering Mechanics, 116, 751.CrossRefGoogle Scholar
  21. Kohlrausch, F. (1863). Pogg. Annals of Physics, 4, 337.CrossRefGoogle Scholar
  22. Kreyszig, E. (2006). Advanced engineering mathematics. Wiley.Google Scholar
  23. Leaderman, H. (1943). Elastic and creep properties of filamentous materials and other High polymers. Washington, D.C.: Textile Foundation. 175.Google Scholar
  24. Lee, H., Daniel, J., & Kim, Y. (2000). Continuum damage mechanics-based fatigue model of asphalt concrete. Journal of Materials in Civil Engineering, 12, 105.CrossRefGoogle Scholar
  25. Lou, Y., & Schapery, R. (1971). Viscoelastic characterization of a nonlinear fiber-reinforced plastic. Journal of Composite Materials, 5, 208.CrossRefGoogle Scholar
  26. Masad, E., Huang, C., Airey, A., & Muliana, A. (2008). Nonlinear viscoelastic analysis of unaged and aged asphalt binders. Construction and Building, 22, 2170.CrossRefGoogle Scholar
  27. Maxwell, J. (1867). On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London, 157, 49–88.CrossRefGoogle Scholar
  28. Maxwell, J. (1875). On the dynamical evidence of the molecular constitution of bodies. Journal of the Chemical Society London, 28, 493–508.Google Scholar
  29. Park, S., Kim, R., & Schapery, R. (1996). A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete. Mechanics of Materials, 24, 241.CrossRefGoogle Scholar
  30. Reiner, M. (1964). The Deborah number. Physics Today, 17, 62.CrossRefGoogle Scholar
  31. Saadeh, S., Masad, E., & Little, D. (2007). Characterization of asphalt mix response under repeated loading using anisotropic nonlinear viscoelastic-viscoplastic model. Journal of Materials in Civil Engineering, 19, 912.CrossRefGoogle Scholar
  32. Schapery, R. (1962). Approximate methods of transform inversion for viscoelastic stress analysis. In Proceedings of 4th U.S. National Congress of Applied Mechanics (Vol. 1075).Google Scholar
  33. Schapery, R. (1962). Irreversible thermodynamics and variational principles with applications to viscoelasticity. Cal Tech Thesis.Google Scholar
  34. Schapery, R. (1964). Application of thermodynamics to thermomechanical, fracture, and birefringent phenomena in viscoelastic media. Journal of Applied Physics, 35, 1451.MathSciNetCrossRefGoogle Scholar
  35. Schapery, R. (1966). An engineering theory of nonlinear viscoelasticity with applications. International Journal of Solids and Structures, 2, 407.CrossRefGoogle Scholar
  36. Schapery, R. (1969). On the characterization of nonlinear viscoelastic materials. Polymer Engineering & Science, 9, 295.CrossRefGoogle Scholar
  37. Schapery, R. (1997). Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics. Mechanics of Time-Dependent Materials, 1, 209.Google Scholar
  38. Truesdell, C., Noll, W., & Antman, S. (2004). The non-linear field theories of mechanics. Springer.Google Scholar
  39. Tschoegl, N. (1989). The phenomenological theory of linear viscoelastic behavior an introduction. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  40. Volterra, V. (1928). Sur la théorie mathématique des phénomènes héréditaires. Journal de Mathématiques Pures et Appliquées, 7, 249–298.zbMATHGoogle Scholar
  41. Volterra, V. (1959). Theory of functionals and of integral and integro-differential equations. Dover.Google Scholar
  42. Weber, W. (1835). Über die elastizität der seidenfäden. Annalen der Physik und Chemie Poggendorf’s, 4, 247–257.Google Scholar
  43. Weber, W. (1841). Poggendorf Annalen der Physik, 24, 1–26.Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Dallas N. Little
    • 1
    Email author
  • David H. Allen
    • 1
  • Amit Bhasin
    • 2
  1. 1.Texas A&M UniversityCollege StationUSA
  2. 2.The University of Texas at AustinAustinUSA

Personalised recommendations