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Abstract

Rivlin’s exact solution for finite bending of a rectangular block of incompressible isotropic elastic material into a circular cylindrical sector is specialized to the case of complete bending, in which two ends of the block are brought together. These ends may be glued together to form a circular cylindrical tube without introducing any stress discontinuity. Several boundary value problems that admit exact solutions, radial inflation or compaction, eversion, steady rotation, radial oscillation, torsion, azimuthal shearing and telescopic shearing, for a natural (unstressed) tube admit similar exact solutions for the pre-stressed tube. Except for the last two shearing deformations, these solutions for both the natural tube and the pre-stressed tube are independent of the strain energy. The availability of similar exact solutions for a naturally circular tube and for a pre-stressed tube that was originally a rectangular block should provide useful information about the effects of pre-stress on non-linearly elastic response.

Keywords

complete bending exact solutions finite elasticity pre-stress universal solutions.

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References

Rivlin, R.S. Large elastic deformations of isotropic materials, VI. Further results in the theory of torsion, shear and flexure. Philosophical Transactions of the Royal Society London A, 242, 178-195 ( 1949).

Barenblatt, G. I. and Joseph, D. D. (eds.). Collected papers of R. S. Rivlin, Volume 1, Springer, New York, 1997.

Green, A.E. and Shield, R.T. Finite elastic deformation of incompressible elastic bodies. Proceedings of the Royal Society London A , 202, 407-419 (1950).

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Carroll, M.M. Non-isochoric bending and shearing. Mathematics and Mechanics of Solids, 10, 691-793 (2005).

Finite Deformations and Motions of a Pre-stressed Incompressible Elastic Tube

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