Summary
For some equilibrium states of a finitely deformed elastic body, variational principles can be used to provide bounds on overall quantities of physical interest. The principles are applied to the problem of the allaround finite extension of a plane sheet with a circular hole, and accurate estimates for the stress resultant at the outer edge are obtained for various extensions. The finite extension and torsion of an elastic cylinder is considered and bounds on the strain energy per unit length are obtained for an elliptical cylinder of neo-Hookean material with axes in the ratios of 2:1 and 4:1. The bounds lead to reliable estimates for the twisting moment and axial force.
Zusammenfassung
Für gewisse Gleichgewichtszustände eines endlich deformierten elastischen Körpers können Variationsprinzipien verwendet werden, um Schranken für globale Grössen von physikalischem Interesse zu erhalten. Die Prinzipien werden auf das Problem der allseitigen endlichen Extension einer ebenen Scheibe mit kreisförmigem Loch angewendet, und es werden für verschiedene Extensionen gute Abschätzungen der Resultierenden am Aussenrand gewonnen. Ferner wird die endliche Zug- und Torsionsverformung eines elastischen Zylinders betrachtet, und es werden Schranken für die Verformungsenergie je Längeneinheit für einen elliptischen Zylinder aus Neo-Hookeschem Material für die Achsenverhältnisse 2:1 und 4:1 erhalten. Die Schranken liefern zuverlässige Schätzungen für das Torsionsmoment und die Axialkraft.
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References
S. J. Lee, andR. T. Shield,Variational Principles in Finite Elastostatics, Z. angew. Math. Phys.31, 437–453 (1980).
R. S. Rivlin, andA. G. Thomas,Large Elastic Deformations of Isotropic Materials, Part VIII, Phil. Trans. Roy. Soc., Ser. A.,243, 289–298 (1950).
F. S. Wong, andR. T. Shield,Large Plane Deformations of Thin Elastic Sheets of Neo-Hookean Material, Z. angew. Math. Phys.,20, 175–199 (1969).
A. E. Green, andR. T. Shield,Finite Extension and Torsion of Cylinders, Phil. Trans. Roy. Soc., Ser. A,244, 47–86 (1951).
S. J. Lee,Variational Principles in Finite Elasticity with Applications, Ph.D. Thesis, University of Illinois at Urbana-Champaign, December, 1979. T. & A.M. Report No. 437, December 1979.
R. T. Shield,On the Stability of Finitely Deformed Elastic Membranes, Part I, Z. angew. Math. Phys.,22, 1016–1028 (1971).
L. R. G. Treloar,Stress-Strain Data for Vulcanized Rubber under Various Types of Deformation, Trans. Faraday Soc.,40, 59–70 (1944).
R. T. Shield,An Energy Method for Certain Second-Order Effects with Application to Torsion of Elastic Bars under Tension, J. Appl. Mech.,47, 75–81 (1980).
A. E. Green,On Some General Formulae in Finite Elastostatics, Arch. Rat. Mech. Anal.,50, 73–80 (1973).
A. E. Green, andA. J. M. Spencer,The Stability of a Circular Cylinder under Finite Extension and Torsion, J. Math. and Phys.,37, 316–338 (1958).
R. S. Rivlin,Large Elastic Deformations of Isotropic Materials VI. Further Results in the Theory of Torsion, Shear and Flexure, Phil. Trans. Roy. Soc. A,242, 173–195 (1949); reprinted inProblems of Non-linear Elasticity, Gordon and Breach, New York (1965).
A. E. Green,A Note on Second-Order Effects in the Torsion of Incompressible Cylinders, Proc. Camb. Phil. Soc.,50, 488–490 (1954).
W. H. Pierce,Numerical Integration over the Planar Annulus, J. Soc. Indust. Appl. Math.,5, No. 2, 66–73 (1957).
R. T. Shield,Finite Extension and Torsion of Thin Elastic Strips, in preparation.
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Lee, S.J., Shield, R.T. Applications of variational principles in finite elasticity. Journal of Applied Mathematics and Physics (ZAMP) 31, 454–472 (1980). https://doi.org/10.1007/BF01590857
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DOI: https://doi.org/10.1007/BF01590857