  We develop a global bifurcation theory for two classes of nonlinear elastic
materials. It is supposed that they are subjected to anti-plane shear
deformation and occupy an infinite cylinder in the reference configuration.
Curves of solutions to the corresponding elastostatic problem are constructed
using analytic global bifurcation theory. The curve associated with first class
is shown to exhibit broadening behavior, while for the second we find that the
governing equation undergoes a loss ellipticity in the limit. A sequence of
solutions undergoes broadening when their effective supports grow without
bound. This phenomena has received considerable attention in the context of
solitary water waves; it has been predicted numerically, yet it remains to be
proven rigorously. The breakdown of ellipticity is related to cracks and
instability making it an important aspect of the theory of failure mechanics.
