  In this paper, we study phase transitions in a slender circular cylinder
composed of a compressible hyperelastic material with a non-convex strain
energy function. We aim to construct the asymptotic solutions based on an
axisymmetrical three-dimensional setting and use the results to describe the
key features (in particular, instability phenomena) observed in the experiments
by others. The difficult problem of the solution bifurcations of the governing
nonlinear partial differential equations (PDE's) is solved through a novel
approach. By using a methodology involving coupled series-asymptotic
expansions, we derive the normal form equation of the original complicated
system of nonlinear PDE's. By writing the normal form equation into a
first-order dynamical system and with a phase-plane analysis, we manage to
deduce the global bifurcation properties and to solve the boundary-value
problem analytically. The asymptotic solutions (including post-bifurcation
solutions) in terms of integrals are obtained. The engineering stress-strain
curve plotted from the asymptotic solutions can capture the key features of the
curve measured in a few experiments (e.g., the stress drop, the stress plateau,
and the small stress valley). It appears that the asymptotic solutions obtained
shed certain light on the instability phenomena associated with phase
transitions in a cylinder, in particular the role played by the radius-length
ratio. Also, an important feature of this work is that we consider the clamped
end conditions, which are more practical but rarely used in literature for
phase transition problems.
