The Anti-Plane Shear Problem in Nonlinear Elasticity Revisited
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.
KeywordsAnti-plane shear Compatibility problem Isotropic incompressible elasticity
Mathematics Subject Classification (2010)74B20 35N10
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.