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Abstract

For the case of plane strain and in the absence of body forces an exact solution that describes the straightening of annular cylindrical sectors composed of compressible Hencky materials is obtained and, using the energy criterion, its stability is investigated. The most general class of isotropic, compressible, hyperelastic materials for which a solution of this type is possible is also determined. Under certain restrictions, an inequality that may be regarded as a universal relation is shown to hold for all materials in this class as well as for all isotropic, compressible, hyperelastic materials that satisfy the Baker-Ericksen inequality. Some other inequalities, showing that certain deformations of bodies composed of Hencky materials are necessarily accompanied by an overall volume increase (which may also be viewed as universal relations for the Hencky materials), are established.

Keywords

Finite elastic deformations Hencky materials

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References

Hencky, H. Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen . Zeitschrift für Technische Physik, 9, 215-220, 457-457 (1928).

Bonet, J. and Wood, R. D. Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, Cambridge , 1997.

Anand, L. On Hencky’s approximate strain-energy function for moderate deformations . Transactions of the ASME, Journal of Applied Mechanics, 46, 78-82 (1979).

Bruhns, O. T. , Xiao, H. and Meyers, A. Hencky’s elasticity model with logarithmic strain measure: a study on Poynting effect and stress response in torsion of tubes and rods . Archives of Mechanics, 52, 489-509 (2000).

Bruhns, O. T. , Xiao, H. and Meyers, A. Finite bending of a rectangular block of an elastic Hencky material . J. Elasticity, 66, 237-256 (2002).

Bruhns, O. T. , Xiao, H. and Meyers, A. Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky’s logarithmic strain tensor . Proceedings of the Royal Society of London A, 2207-2226 (2001).

Truesdell, C. and Noll, W. The non-linear field theories of mechanics, in Handbuch der Physik III/3, ed., S. Flügge , Springer, Berlin , 1965.

Aron, M. Some remarks concerning a boundary-value problem in nonlinear elastostatics . J. Elasticity, 60, 165-172 (2000).

Krawietz, A. A comprehensive constitutive inequality in finite elastic strain . Archive for Rational Mechanics and Analysis, 58, 127-149 (1975).

10.

Ogden, R. W. Non-linear Elastic Deformations, Ellis-Horwood, Chichester , 1984.

11.

Aron, M. On global volume changes accompanying certain plane-strain nonlinear elastic deformations . International Journal of Solids and Structures, 34, 2803-2813 (1997).

12.

Murphy, J. G. Strain energy functions for a Poisson power low function in simple tension of compressible hyperelastic materials . J. Elasticity, 60, 151-164 (2000).

13.

Carroll, M. M. and Horgan, C. O. Finite strain solutions for a compressible elastic solid . Quarterly Applied Mathematics, 48, 767-780 (1990).

14.

Aron, M. , Christopher, C and Wang, Y. On the straightening of compressible, nonlinearly elastic, annular cylindrical sectors . Mathematics and Mechanics of Solids, 3, 131-145 (1998).

15.

Beatty, M. F. and Stalnaker, D. O. The Poisson function of finite elasticity . Transactions of the ASME, Journal of Applied Mechanics, 53, 807-813 (1986).

16.

Knowles, J. K. and Sternberg, E. On the failure of ellipticity of the equations for finite elastostatic plane strain . Archive for Rational Mechanics and Analysis, 63, 321-335 (1977).

17.

Hardy, G. H. , Littlewood, J. E. and Piola, G. Inequalities, Cambridge University Press, Cambridge , 1952.

18.

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

19.

Ogden, R. W. Inequalities associated with the inversion of elastic stress-deformation relations and their implications . Mathematical Proceedings of the Cambridge Philosophical Society, 81, 313-324 (1977).

On Certain Deformation Classes of Compressible Hencky Materials

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