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Stress-softening Effects in the Transverse Vibration of a Non-Gaussian Rubber String

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Abstract

The Mullins effect in the small amplitude transverse vibration of a rubber cord is investigated. The fundamental frequency is determined for a specific class of stress-softening materials. Analytical relations for the cord vibration frequency are illustrated graphically for three phenomenological models. These results demonstrate the role of the material parameters and exhibit response characteristic of those reported in experiments by others and subsequently described here in new experiments. Frequency versus stretch results for two kinds of non-Gaussian molecular network models for rubber elasticity are compared with experimental data for four varieties of rubber cords, for each of which only three experimentally determined material constants are needed. It is shown that the theoretical predictions stand in excellent agreement with test data.

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References

  1. Mullins, L., 'Effect of stretching on the properties of rubber', J. Rubber Res. 16 (1947) 275-289.

    Google Scholar 

  2. Ogden, R.W. and Roxburgh, D.G., 'A pseudo-elastic model for the Mullins effect in filled rubber', Proc. R. Soc. Lond. A 455 (1999) 2861-2878.

    Google Scholar 

  3. Beatty, M.F. and Krishnaswamy, S., 'A theory of stress-softening in incompressible isotropic materials', J. Mech. Phys. Solids 48 (2000) 1931-1965.

    Google Scholar 

  4. Zúñiga, A.E. and Beatty, M.F., 'A new phenomenological model for stress-softening in elastomers', J. Appl. Math. Phys. (ZAMP) 53 (2002) 794-814.

    Google Scholar 

  5. Beatty, M.F., 'The Mullins effect in the transverse vibration of a rubber cord', in: Proceedings of the 2nd Canadian Conference on Nonlinear Solid Mechanics, Simon Fraser University, Vancouver, British Columbia, June 19–23, 2002 357-367.

    Google Scholar 

  6. Johnson, M.A. and Beatty, M.F., 'The Mullins effect in uniaxial extension and its influence on the transverse vibration of a rubber string', Cont. Mech. Therm. 5 (1993) 83-115.

    Google Scholar 

  7. Beatty, M.F. and Chow, A.C., 'On the transverse vibration of a rubber string', J. Elasticity 13 (1983) 317-344.

    Google Scholar 

  8. Gent, A.N., 'A new constitutive relation for rubber', Rubber Chem. Technol. 69 (1996) 59-61.

    Google Scholar 

  9. Fung, Y.C.B., 'Elasticity of soft tissues in simple extension', Am. J. Physiol. 213 (1967) 1532-1544.

    Google Scholar 

  10. Horgan, C.O. and Saccomandi, G., 'Anti-plane shear deformations for non-Gaussian isotropic, incompressible hyperelastic materials', Proc. R. Soc. Lond. A 457 (2001) 1999-2017.

    Google Scholar 

  11. Horgan, C.O. and Saccomandi, G., 'Constitutive modelling of rubber-like and biological materials with limiting chain extensibility', Math. Mech. Solids 7 (2002) 1-19.

    Google Scholar 

  12. De Souza Neto, E.A., Perić, D. and Owen, D.R.J., 'A phenomenological three-dimensional rate independent continuum model for highly filled polymers: formulation and computational aspects', J. Mech. Phys. Solids 42 (1994) 1533-1550.

    Google Scholar 

  13. Beatty, M.F., 'A stretch averaged full-network model for rubber elasticity', (submitted for publication).

  14. James, H.M. and Guth, E., 'Theory of the elastic properties of rubber', J. Chem. Phys. 10 (1943) 455-481.

    Google Scholar 

  15. De Simone, A., Marigo, J.-J. and Tersi, L., 'A damage mechanics approach to stress softening and its application to rubber', Eur. J. Mech. A/Solids 20 (2001) 873-892.

    Google Scholar 

  16. Gurtin, M.E. and Francis, E.C., 'Simple rate independent model for damage', J. Spacecraft 18 (1981) 285-286.

    Google Scholar 

  17. Truesdell, C., The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788. L. Euler Opera Omnia (2) 11, Part 2, Orell Füssli Turice, Switzerland, 1960.

    Google Scholar 

  18. Strutt, J.W., Lord Rayleigh, The Theory of Sound, Dover, New York, 1945.

    Google Scholar 

  19. Timoshenko, S., Vibration Problems in Engineering, 3rd edn, Van Nostrand, New York, 1955.

    Google Scholar 

  20. Wang, M.C. and Guth, E., 'Statistical theory of networks of non-Gaussian flexible chains', J. Chem. Phys. 20 (1952) 1144-1157.

    Google Scholar 

  21. Arruda, E.M. and Boyce, M.C., 'A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials', J. Mech. Phys. Solids 41 (1993) 389-412.

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Ingeniería Mecánica, Instituto Technologico y de Estudios Superiores de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, N.L., Mexico

    Alex ElÍas-Zúñiga

  2. Department of Engineering Mechanics, University of Nebraska-Lincoln, P.O. Box 910215, Lexington, KY, 40591-0215, U.S.A

    Millard F. Beatty

Authors
  1. Alex ElÍas-Zúñiga
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  2. Millard F. Beatty
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ElÍas-Zúñiga, A., Beatty, M.F. Stress-softening Effects in the Transverse Vibration of a Non-Gaussian Rubber String. Meccanica 38, 419–433 (2003). https://doi.org/10.1023/A:1024636319510

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