Remarks on Analytic Solutions and Ellipticity in Anti-plane Shear Problems of Nonlinear Elasticity

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 $b = \lambda /2$.