Skip to main content

Deformations of infinite slabs of non-linear viscoelastic solids containing an elliptic hole


In this paper we study the state of stress and strain in infinite elastic slabs of nonlinear viscoelastic solids containing elliptic holes subject to an uni-axial as well as a bi-axial state of stress. The geometry affords one to get some inkling concerning the states of stress and strain in bodies containing a crack by obtaining the limit of the solutions as the aspect ratio (in this case the ratio of the minor axis to the major axis) of the ellipse tends to zero. We consider two classes of non-linear viscoelastic bodies, the classical incompressible Kelvin–Voigt solid (Thomson in R Soc Lond 14:289–297, 1865; Voigt in Ann Phys 283(12):671–693, 1892) and a generalization of a compressible model due to Gent (Rubber Chem Technol 69(1):59–61, 1996). We also study for the sake of comparison the case of a nonlinear neo-Hookean elastic solid with an elliptic hole.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15


  1. Most of these studies appeal to ideas that are take offs used in the theory of linearized elasticity for conservative systems, and we shall not discuss them here.

  2. To our knowledge even this problem concerning a neo-Hookean has not been solved in either an infinite plate or a finite plate with an elliptic hole.

  3. Henceforth we suppress ’t’ from the subscript.

  4. To our knowledge, there is no experimental support that an elastic material that is described by such a constitutive equation exists. It seems that the assumption was made so that the governing equations would reduce to a form that is amenable to analysis in view of the mathematical results available.


  1. Alagappan P, Kannan K, Rajagopal KR (2016) On a possible methodology for identifying the initiation of damage of a class of polymeric materials. Proc R Soc A. doi:10.1098/rspa.2016.0231

    Google Scholar 

  2. Gent AN (1996) A new constitutive relation for rubber. Rubber Chem Technol 69(1):59–61

    MathSciNet  Article  Google Scholar 

  3. Oldroyd JG (1950) On the formulation of rheological equations of state. Proc R Soc Lond A Math Phys Eng Sci 200(1063):523–541

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. Rajagopal KR, Srinivasa AR (1998) Mechanics of the inelastic behavior of materials. part ii: inelastic response. Int J Plast 14(10):969–995

    Article  MATH  Google Scholar 

  5. Rajagopal KR, Srinivasa AR (2000) A thermodynamic frame work for rate type fluid models. J Non-Newton Fluid Mech 88(3):207–227

    Article  MATH  Google Scholar 

  6. Rajagopal KR, Srinivasa A (2004) On the thermomechanics of materials that have multiple natural configurations. Zeitsch Angew Math Phys ZAMP 55(6):1074–1093

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. Rajagopal KR (1995) Multiple Congurations in continuum mechanics. Technical report, report 6, Institute for computational and applied mechanics, University of Pittsburgh

  8. Thomson W (1865) On the elasticity and viscosity of metals. Proc R Soc Lond 14:289–297

    Article  Google Scholar 

  9. Varley E, Cumberbatch E (1977) The finite deformation of an elastic material surrounding an elliptical hole. Finite elasticity, pp. 41–64

  10. Voigt W (1892) Ueber innere reibung fester körper, insbesondere der metalle. Ann Phys 283(12):671–693

    Article  MATH  Google Scholar 

Download references


K. R. Rajagopal thanks the National Science Foundation and the Office of Naval Research for support of this work.

Author information

Authors and Affiliations


Corresponding author

Correspondence to K. R. Rajagopal.

Ethics declarations


We thank the National Science Foundation (Award Number:1028894) and the Office of Naval Research (Award Number:C1200292) for support of this work.

Conflict of interest

There is no conflict of interest, no other funding or support was received for this work.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alagappan, P., Rajagopal, K.R. & Kannan, K. Deformations of infinite slabs of non-linear viscoelastic solids containing an elliptic hole . Meccanica 51, 3067–3080 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Viscoelastic
  • Elliptic hole
  • Stress concentration
  • Large deformation
  • Plate