Skip to main content
    • Aa
    • Aa

A Brief Review of Elasticity and Viscoelasticity for Solids

  • Harvey Thomas Banks (a1), Shuhua Hu (a1) and Zackary R. Kenz (a1)

There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.

    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A Brief Review of Elasticity and Viscoelasticity for Solids
      Available formats
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      A Brief Review of Elasticity and Viscoelasticity for Solids
      Available formats
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      A Brief Review of Elasticity and Viscoelasticity for Solids
      Available formats
Corresponding author
Corresponding author. URL: Email:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] H. T. Banks , J. B. Hood , N. G. Medhin and J. R. Samuels , A stick-slip/Rouse hybrid model for viscoelasticity in polymers, Technical Report CRSC-TR06-26, NCSU, November, 2006, Nonlinear. Anal. Real., 9 (2008), pp. 21282149.

[5] H. T. Banks , N. G. Medhin and G. A. Pinter , Multiscale considerations in modeling of nonlinear elastomers, Technical Report CRSC-TR03-42, NCSU, October, 2003, J. Comp. Meth. Engr. Sci. Mech., 8 (2007), pp. 5362.

[8] H. T. Banks and G. A. Pinter , A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue, Mult. Model. Sim., 3 (2005), pp. 395412.

[13] J. P. Bardet , A viscoelastic model for the dynamic behavior of saturated poroelastic soils, J. Appl. Mech. ASME., 59 (1992), pp. 128135.

[15] M. A. Biot , Theory of propagation of elastic waves in a fluid saturated porous solid, J. Acust. Soc. Am., 28 (1956), pp. 168191.

[16] R. M. Christensen , Theory of Viscoelasticity, 2nd ed., Academic Proess, New York, 1982.

[18] M. A. Del Nobile , S. Chillo , A. Mentana and A. Baiano , Use of the generalized Maxwell model for describing the stress relaxation behavior of solid-like foods, J. Food. Eng., 78 (2007), pp. 978983.

[20] C. S. Drapaca , S. Sivaloganathan and G. Tenti , Nonlinear constitutive laws in vis-coelasticity, Math. Mech. Solids., 12 (2007), pp. 475501.

[21] J. D. Ferry , E. R. Fitzgerald , L. D. Grandine and M. L. Williams , Temperature dependence of dynamic properties of elastomers: relaxation distributions, Ind. Engr. Chem., 44 (1952), pp. 703706.

[22] W. N. Findley and J. S. Y. Lai , A modified superposition principle applied to creep of nonlinear viscoelastic materials under abrupt changes in state of combined stress, Trans. Soc. Rheol., 11 (1967), pp. 361380.

[25] Y. C. Fung , Biomechanics: Mechanical Properties of Living Tissue, Springer-Verlag, Berlin, 1993.

[27] A. E. Green and R. S. Rivlin , The mechanics of non-linear materials with memory, Arch. Ration. Mech. An., 1 (1957), pp. 121.

[28] M. Gurtin and E. Sternberg , On the linear theory of viscoelasticity, Arch. Ration. Mech. An., 11 (1965), pp. 291365.

[30] Y. M. Haddad , Viscoelasticity of Engineering Materials, Chapman & Hall, Dordrecht, Netherlands, 1995.

[32] A. R. Johnson and R. G. Stacer , Rubber viscoelasticity using the physically constrained systems stretches as internal variables, Rubber. Chem. Technol., 66 (1993), pp. 567577.

[36] P. Le Tallec , C. Rahier and A. Kaiss , Three-dimensional incompressible viscoelasticity in large strains: formulation and numerical approximation, Comput. Meth. Appl. Mech. Eng., 109 (1993), pp. 233258.

[40] P. Michaels , Relating damping to soil permeability, Int. J. Geomech., 6 (2006), pp. 158165.

[43] J. G. Oakley , A. J. Giacomin and J. A. Yosick , Molecular origins of nonlinear viscoelas-ticity: fundamental review, Mikrochimica. Acta., 130 (1998), pp. 128.

[45] A. P. Penneru , K. Jayaraman and D. Bhattacharyya , Viscoelastic behaviour of solid wood under compressive loading, Holzforschung., 60 (2006), pp. 294298.

[46] A. C. Pipkin and T. G. Rogers , A non-linear integral representation for viscoelastic behaviour, J. Mech. Phys. Solids., 16 (1968), pp. 5972.

[48] R. S. Rivlin , Large elastic deformations of isotropic materials, I, II, III, Phil. Trans. Roy. Soc. A., 240 (1948), pp. 459525.

[49] P. E. Rouse Jr., A theory of the linear viscoelastic properties of dilute solutions of coiling polymers, J. Chem. Phys., 21 (1953), pp. 12721280.

[51] R. A. Schapery , On the characterization of nonlinear viscoelastic solids, Polymer. Eng. Sci., 9 (1969), pp. 295310.

[52] R. A. Schapery , Nonlinear viscoelastic and viscoplastic constitutive equations based on ther-modynamics, Mech. Time-Depend. Mat., 1 (1997), pp. 209240.

[53] R. A. Schapery , Nonlinear viscoelastic solids, Int. J. Solids. Struct., 37 (2000), pp. 359366.

[54] C. T. Schroder , W. R. Scott Jr., AND G. D. Larson , Elastic waves interacting with buried land mines: a study using the FDTD method, IEEE Trans. Geosci. Remote. Sens., 40 (2002), pp. 14051415.

[57] T. H. Shellhammer , T. R. Rumsey and J. M. Krochta , Viscoelastic properties of edible lipids, J. Food. Eng., 33 (1997), pp. 305320.

[58] J. Smart and J. G. Williams , A comparison of single integral non-linear viscoelasticity theories, J. Mech. Phys. Solids., 20 (1972), pp. 313324.

[60] C. Truesdell , Rational Thermodynamics, Springer, Berlin, 1984.

[61] A. Wineman , Nonlinear viscoelastic solids-a review, Math. Mech. Solids., 14 (2009), pp. 300366.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 160 *
Loading metrics...

Abstract views

Total abstract views: 131 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 27th July 2017. This data will be updated every 24 hours.