Abstract
A classical problem in the framework of nonlinear elasticity theory is the characterization of the materials that may sustain a pure state of anti-plane shear in the absence of body forces. This problem has been solved by Knowles and by Hill in the framework of isotropic and incompressible elasticity in the seventies. Here we provide a simpler and shorter proof of these classical results. Moreover, we extend these results to nonlinear elastodynamics and we provide some new special solutions.
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Notes
The existence of such multipliers is a classical result in algebraic and differential geometry. See exercise 2.33 in the first edition of [18]. (The same exercise is renumbered 2.35 in the second edition of the book).
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Acknowledgements
The research is partially supported by PRIN-2009 project Matematica e meccanica dei sistemi biologici e dei tessuti molli and GNFM of Italian INDAM. We thank you Michel Destrade, Roger Fosdick, Jeremiah Murphy, Ray Ogden and two anonymous referees for providing constructive comments and help in improving the contents of this paper.
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Pucci, E., Saccomandi, G. The Anti-Plane Shear Problem in Nonlinear Elasticity Revisited. J Elast 113, 167–177 (2013). https://doi.org/10.1007/s10659-012-9416-z
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DOI: https://doi.org/10.1007/s10659-012-9416-z