Mechanics of Time-Dependent Materials

, Volume 18, Issue 1, pp 275–291 | Cite as

Quasi-static analysis of multilayered domains with viscoelastic layer using incremental-layerwise finite element method



This paper presents a layerwise finite element formulation for quasi-static analysis of laminated structures with embedded viscoelastic material whose constitutive behavior is represented by the Prony series. To account the time dependence of the constitutive relations of linear viscoelastic materials, the incremental formulation in the temporal domain is used. This approach avoids the use of relaxation functions and mathematical transformations. A computer code based on the presented formulation has been developed to provide the numerical results. The high accuracy of the method is exhibited by comparing the results with existing solutions in the literature and also with those obtained using the ABAQUS software. Finally, and as an application of the presented formulation, the effects of time and load rate on the quasi-static structural response of asphalt concrete (AC) pavements are studied.


Layerwise finite element Linear viscoelastic materials Laminated structures Pavements Prony series 


  1. Bahia, H.U., Anderson, D.A.: Development of the bending beam rheometer: basics and critical evaluation of the rheometer. In: Proc. ASTM Physical Properties of Asphalt Cement Binders Conf., vol. 1241, pp. 28–50 (1995) CrossRefGoogle Scholar
  2. Benedetto, H.D., Olard, F., Sauzéat, C., Delaporte, B.: Linear viscoelastic behavior of bituminous materials: from binders to mixes. Road Mater, Pavement Des. 5, 163–202 (2004) CrossRefGoogle Scholar
  3. Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1–27 (1996) CrossRefGoogle Scholar
  4. Boltzmann, L.: Zur theorie der elastischen nachwirkung sitzungsber. Math.-Nat. Kl. Kais. Akad. Wiss. 70, 275 (1878) Google Scholar
  5. Bozza, A., Gentili, G.: Inversion and quasi-static problems in linear viscoelasticity. Meccanica 30, 321–335 (1995) CrossRefMATHMathSciNetGoogle Scholar
  6. Chazal, C., Pitti, R.M.: Viscoelastic incremental formulation using creep and relaxation differential approaches. Mech. Time-Depend. Mater. 14, 173–190 (2010) CrossRefGoogle Scholar
  7. Chazal, C., Pitti, R.M.: Incremental constitutive formulation for time dependent materials: creep integral approach. Mech. Time-Depend. Mater. 15, 239–253 (2011) CrossRefGoogle Scholar
  8. Chen, E.Y., Pan, G.E., Green, R.: Surface loading of a multilayered viscoelastic pavement: semi-analytical solution. J. Eng. Mech. 35, 517–528 (2009) CrossRefGoogle Scholar
  9. Christensen, R.M.: Theory of Viscoelasticity: An Introduction. Academic Press, New York (1971) Google Scholar
  10. Chupin, O., Chabot, A., Piau, J.M., Duhamel, D.: Influence of sliding interfaces on the response of a layered viscoelastic medium under a moving load. Int. J. Solids Struct. 47, 3435–3446 (2010) CrossRefMATHGoogle Scholar
  11. Drozdov, A.D., Dorfmann, A.: A constitutive model in finite viscoelasticity of particle-reinforced rubbers. Meccanica 39, 245–270 (2004) CrossRefMATHGoogle Scholar
  12. Dubois, F., Chazal, C., Petit, C.: A finite element analysis of creep-crack growth in viscoelastic media. Mech. Time-Depend. Mater. 2, 269–286 (1999) CrossRefGoogle Scholar
  13. Dubois, F., Moutou-Pitti, R., Picoux, B., Petit, C.: Finite element model for crack growth process in concrete bituminous. Adv. Eng. Softw. 44, 35–43 (2012) CrossRefGoogle Scholar
  14. Elseifi, M.A., Al-Qadi, I.L., Yoo, P.J.: Viscoelastic modeling and field validation of flexible pavement. J. Eng. Mech. 132, 172–178 (2006) CrossRefGoogle Scholar
  15. Ghazlan, G., Caperaa, S., Petit, C.: An incremental formulation for the linear analysis of thin viscoelastic structures using generalized variables. Int. J. Numer. Methods Eng. 38, 3315–3333 (1995) CrossRefMATHGoogle Scholar
  16. Gibson, N.H., Schwartz, C.W., Schapery, R.A., Witczak, M.W.: Viscoelastic, viscoplastic, and damage modeling of asphalt concrete in unconfined compression. Transp. Res. Rec. 1860, 3–15 (2003) CrossRefGoogle Scholar
  17. Grassia, L., D’Amore, A.: The relative placement of linear viscoelastic functions in amorphous glassy polymers. J. Rheol. 53, 339–356 (2009) CrossRefGoogle Scholar
  18. Grassia, L., D’Amore, A., Simon, S.L.: On the viscoelastic Poisson’s ratio in amorphous polymers. J. Rheol. 54, 1009–1022 (2010) CrossRefGoogle Scholar
  19. Hu, S., Fujie, Z.: Development of a new interconversion tool for hot mix asphalt (HMA) linear viscoelastic functions. Can. J. Civ. Eng. 37, 1071–1081 (2010) CrossRefGoogle Scholar
  20. Huang, Y.H.: Pavement Analysis and Design, 2nd edn., p. 77. Pearson/Prentice Hall, Upper Saddle River (2004) Google Scholar
  21. Junior, P.C.A., Soares, J.B., Holanda, A.S., Junior, E.P., Junior, F.E.: Dynamic viscoelastic analysis of asphalt pavements using a finite element formulation. Road Mater, Pavement Des. 11, 409–433 (2010) CrossRefGoogle Scholar
  22. Kim, K.S., Sung Lee, H.: An incremental formulation for the prediction of two-dimensional fatigue crack growth with curved paths. Int. J. Numer. Methods Eng. 72, 697–721 (2007) CrossRefMATHGoogle Scholar
  23. Lee, H.J.: Uniaxial constitutive modeling of asphalt concrete using viscoelasticity and continuum damage theory. Ph.D. dissertation, North Carolina State University, Raleigh, NC (1996) Google Scholar
  24. Lee, H.J., Kim, Y.R.: A viscoelastic constitutive model for asphalt concrete under cyclic loading. J. Eng. Mech. 124, 32–40 (1998) CrossRefGoogle Scholar
  25. Lee, C.Y., Liu, D.: An interlaminar stress continuity theory for laminated composite analysis. Comput. Struct. 42, 69–78 (1992) CrossRefMATHGoogle Scholar
  26. Malekzadeh, P.: A two-dimensional layerwise-differential quadrature static analysis of thick laminated composite circular arches. Appl. Math. Model. 33, 1850–1861 (2009) CrossRefMATHGoogle Scholar
  27. Malekzadeh, P., Setoodehc, A.R., Barmshouri, E.: A hybrid layerwise and differential quadrature method for in-plane free vibration of laminated thick circular arches. J. Sound Vib. 315, 212–225 (2008) CrossRefGoogle Scholar
  28. Matsunaga, H.: Interlaminar stress analysis of laminated composite beams according to global higher-order deformation theories. Compos. Struct. 55, 105–114 (2002) CrossRefGoogle Scholar
  29. Olard, F., Benedetto, H.D., Dony, A., Vaniscote, J.C.: Properties of bituminous mixtures at low temperatures and relations with binder characteristics. Mater. Struct. 38, 121–126 (2005) CrossRefGoogle Scholar
  30. Park, S.W., Shapery, R.A.: Methods of interconversion between linear viscoelastic material functions. Part I: a numerical method based on Prony series. Int. J. Solids Struct. 36, 1653–1675 (1999) CrossRefMATHGoogle Scholar
  31. Pellinen, T.K., Witczak, M.W.: Use of stiffness of hot-mix asphalt as a simple performance test. Transp. Res. Rec. 1789, 80–90 (2002) CrossRefGoogle Scholar
  32. Reddy, J.N.: A generalization of two-dimensional theories of laminated composite plates. Commun. Appl. Numer. Methods 3, 173–180 (1987) CrossRefMATHGoogle Scholar
  33. Schapery, R.A.: A theory of crack initiation and growth in viscoelastic media: I. Theoretical development. Int. J. Fract. 11, 141–159 (1975) CrossRefGoogle Scholar
  34. Setoodeh, A.R., Karami, G.: Static, free vibration and buckling analysis of anisotropic thick laminated composite plates on distributed and point elastic supports using a 3-D layer-wise FEM. Eng. Struct. 26, 211–220 (2004) CrossRefGoogle Scholar
  35. Setoodeh, A.R., Malekzadeh, P., Nikbin, K.: Low velocity impact analysis of laminated composite plates using a 3D elasticity based layerwise FEM. Mater. Des. 30, 3795–3801 (2009) CrossRefGoogle Scholar
  36. Shen, Y.P., Hasebe, N., Lee, L.X.: The finite element method of three-dimensional nonlinear viscoelastic large deformation problems. Comput. Struct. 55, 659–666 (1995) CrossRefMATHGoogle Scholar
  37. Sorvari, J., Hämäläinen, J.: Time integration in linear viscoelasticity—a comparative study. Mech. Time-Depend. Mater. 14, 307–328 (2010) CrossRefGoogle Scholar
  38. Sungho, M., Goangseup, Z.: Modeling the viscoelastic function of asphalt concrete using a spectrum method. Mech. Time-Depend. Mater. 14, 191–202 (2010) CrossRefGoogle Scholar
  39. Tahani, M.: Analysis of laminated composite beams using layerwise displacement theories. Compos. Struct. 79, 535–547 (2007) CrossRefGoogle Scholar
  40. Taylor, R.L., Pister, K.S., Gourdreau, G.L.: Thermomechanical analysis of viscoelastic solids. Int. J. Numer. Methods Eng. 2, 45–59 (1970) CrossRefMATHGoogle Scholar
  41. Theocaris, P.S.: Creep and relaxation contraction ratio of linear viscoelastic materials. J. Mech. Phys. Solids 12, 125–138 (1964) CrossRefGoogle Scholar
  42. Zocher, M.A., Groves, S.E., Aellen, D.H.: A three-dimensional finite element formulation for thermoviscoelastic orthotropic media. Int. J. Numer. Methods Eng. 40, 2267–2288 (1997) CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Civil EngineeringIran University of Science and TechnologyTehranIran
  2. 2.Transportation Research InstituteIran University of Science and TechnologyTehranIran
  3. 3.Head of Center of Excellence of PMSTransportation and SafetyTehranIran
  4. 4.Department of Mechanical EngineeringPersian Gulf UniversityBushehrIran

Personalised recommendations