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

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Abstract

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.

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Notes

  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.

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Acknowledgments

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

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Authors and Affiliations

  1. Department of Mechanical Engineering, Texas A & M University, College Station, TX, 77843-3123, United States

    P. Alagappan & K. R. Rajagopal

  2. Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India

    K. Kannan

Authors
  1. P. Alagappan
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  2. K. R. Rajagopal
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  3. K. Kannan
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Corresponding author

Correspondence to K. R. Rajagopal.

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Funding

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.

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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). https://doi.org/10.1007/s11012-016-0539-3

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