  This paper revisits a well-studied anti-plane shear deformation problem
formulated by Knowles in 1976 and analytical solutions in general nonlinear
elasticity proposed by Gao since 1998. Based on minimum potential principle, a
well-determined fully nonlinear system is obtained for isochoric deformation,
which admits non-trivial states of finite anti-plane shear without ellipticity
constraint. By using canonical duality theory, a complete set of analytical
solutions are obtained for 3-D finite deformation problems governed by
generalized neo-Hookean model. Both global and local extremal solutions to the
nonconvex variational problem are identified by a triality theory. Connection
between challenges in nonconvex analysis and NP-hard problems in computational
science is revealed. It is proved that the ellipticity condition for general
fully nonlinear boundary value problems depends not only on differential
operators, but also sensitively on the external force field. The homogenous
hyper-elasticity for general anti-plane shear deformation must be governed by
the generalized neo-Hookean model. Knowles' over-determined system is simply
due to a pseudo-Lagrange multiplier and two extra equilibrium conditions in the
plane. The constitutive condition in his theorems is naturally satisfied with
$b = \lambda/2$. His ellipticity condition is neither necessary nor sufficient
for general homogeneous materials to admit nontrivial states of anti-plane
shear.
