Abstract
This paper revisits a well-studied anti-plane shear deformation problem formulated by Knowles in 1976. It shows that a homogenous hyper-elasticity for general anti-plane shear deformation must be governed by a generalized neo-Hookean model. Based on minimum total potential principle, a well-determined fully nonlinear system is obtained for isochoric deformation, which admits nontrivial states of finite anti-plane shear without ellipticity constraint. By a pure complementary energy principle, a complete set of analytical solutions is obtained, both global and local extremal solutions are identified by a triality theory. It is proved that the Legendre condition (i.e., the strong ellipticity) does not necessary to guarantee a unique solutions. The uniqueness depends not only on the stored energy, but also on the external force. Knowles’ over-determined system is simply due to a pseudo-Lagrange multiplier \(p(x_1,x_2)\) and two self-balanced equilibrium equations in the plane. The constitutive condition in his theorems is naturally satisfied with \(b = \lambda /2\).
Keywords
- Anti-plane Shear Deformations
- Isochoric Deformation
- Pure Complementary Energy Principle
- Over-determined Problem
- Canonical Duality
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Acknowledgements
Insightful discussions with Professor C. Horgan from University of Virginia and Professor Martin Ostoja-Starzewski from University of Illinois are sincerely acknowledged. Reviewer’s important comments and constructive suggestions are sincerely acknowledged. The research was supported by US Air Force Office of Scientific Research (AFOSR FA9550-10-1-0487).
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Gao, D.Y. (2017). Remarks on Analytic Solutions and Ellipticity in Anti-plane Shear Problems of Nonlinear Elasticity. In: Gao, D., Latorre, V., Ruan, N. (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-58017-3_4
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