Thermoelastic Shear of Rubberlike Solid Cylindrical Tubes

Abstract

The condition of incompressibility, which is acceptable in the theory of large homothermal, elastic deformation of materials such as rubbers, is questionably so when the deformation gradient tensor is distortional, i.e., not spherical, and accompanied by large temperature variations. A form of the two-parameter Mooney-Rivlin model, modified by Chadwick to take account of compressibility and successfully matched by him to experiment by suitable choice of the material constants, is described as an example of a more general theory of rubberlike materials. This specific model is used to analyze by numerical methods the boundary value problem of azimuthal shear of cylindrical tubes, which are subject to a temperature gradient in the radial direction. It is found that the physical component B _<12> of the left Cauchy-Green strain tensor, which corresponds to a constant S_<12> component of the distortional stress tensor additional to the pressure, differs markedly from the value obtained from the incompressible theory when the temperature gradient is high and the inner radius of the tube is large in comparison to its thickness. The difference is even more marked if the tube of compressible material is allowed to expand in thickness by a small percentage.

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References

[1] Rivlin, R. S. : Large elastic deformations of isotropic materials. IV: Further developments of the general theory. Philosophical Transactions of the Royal Society of London, A 241, 379-348 (1948).

[2] Rivlin, R. S. : A note on the torsion of an incompressible highly elastic cylinder. Proceedings of the Cambridge Philosophical Society, 45, 485-487 (1948).

[3] Rivlin, R. S. : Large elastic deformations of isotropic materials. V: The problem of flexure. Proceedings of the Royal Society of London, A195, 463-475 (1949).

[4] Ericksen, J. L. : Deformations possible in every isotropic, incompressible, perfectly elastic body. Zeitschrift fur Angewandte Mathematik und Physik, 5, 466-468 (1954).

[5] Truesdell, C. and Noll, W. : Non-Linear Field Theories of Mechanics, Handbuch der Physik, 111/3, ed. S. Flügge , Springer-Verlag, Berlin-Heidelberg-New York, 1965.

[6] Rivlin, R. S. and Saunders, D. W. : Large elastic deformations of isotropic materials. VII: Experiments on the deformation of rubber. Philosophical Transactions of the Royal Society of London, A243, 251-288 (1951).

[7] Treloar, L.R.G. : The Physics of Rubber Elasticity, 3rd ed., Clarendon, Oxford, UK, 1975.

[8] Pipkin, A. C. and Rivlin, R. S. : The Formulation of Constitutive Equations in Continuum Physics, Division of Applied Mathematics Report, Brown University, 1958.

[9] Petroski, H. J. and Carlson, D. E. : Controllable states of elastic heat conductors. Archive for Rational Mechanics and Analysis, 31, 127-150 (1968).

10.

[10] Chadwick, P. : Thermomechanics of rubberlike materials. Philosophical Transactions of the Royal Society of London, A276, 371-402 (1974).

11.

[11] Spencer, A.J.M. : The static theory of finite elasticity. Journal of the Institute for Mathematics and Applications, 6, 164-200 (1970).

12.

[12] Chadwick, P. and Creasy, C.F.M. : Modified entropic elasticity of rubberlike materials. Journal of the Mechanics and Physics of Solids, 32, 337-357 (1984).

13.

[13] Besseling, J. F. and Voetman, H. H. : Thermo-elastic effects in rubber. Archiwum Mechaniki Stosowanej, 20, 189-202 (1968).

14.

[14] Allen, G. , Bianchi, U. , and Price, C. : Thermodynamics of elasticity of natural rubber. Transactions of the Faraday Society, 59, 2493-2502 (1963).

15.

[15] Ogden, R. W. : Large deformation isotropic elasticity: On the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London, A328, 567-583 (1972).

16.

[16] Wood, L. A. and Martin, G. M. : Compressibility of natural rubber at pressures below 500 kg/cm2. Journal of Research of the National Bureau of Standards, 68A, 259-268 (1964).

17.

[17] Chadwick, P. and Ogden, R. W. : A theorem of tensor calculus and its application to isotropic elasticity. Archive for Rational Mechanics and Analysis, 44, 54-68 (1971).

18.

[18] Ogden, R. W. : Large deformation isotropic elasticity—On the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London, A326, 565-584 (1972).

19.

[19] Dillon, O. W. : A non-linear thermoelasticity theory. Journal of the Mechanics and Physics of Solids, 10, 123-131 (1962).

20.

[20] Jiang, X. and Ogden, R. W. : On azimuthal shear of a circular cylindrical tube of compressible elastic material. Quarterly Journal Mechanics and Applied Mathematics, 51, 143-158 (1998).

21.

[21] Ogden, R. W. and Chadwick, P. : On the deformation of solid and tubular cylinders of incompressible isotropic elastic material. Journal of the Mechanics and Physics of Solids, 20, 77-90 (1972).

22.

[22] Baker, M. and Ericksen, J. L. : Inequalities restricting the form of the stress-deformation relations for isotropic solids and Reiner-Rivlin fluids. Journal of the Washington Academy of Science, 44, 33-35 (1954).

23.

[23] Rajagopal, K. R. : Boundary layers in finite thermoelasticity. Journal of Elasticity, 36, 271-301 (1995).

24.

[24] Ghoneimy, A. E. and Maneschy, C. E. : Steady state shearing deformation of a generalized neo-Hookean Material. Mechanics Research Communications, 21, 501-508 (1994).