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In this paper, we study phase transitions in a slender circular cylindercomposed of a compressible hyperelastic material with a non-convex strainenergy function. We aim to construct the asymptotic solutions based on anaxisymmetrical three-dimensional setting and use the results to describe thekey features (in particular, instability phenomena) observed in the experimentsby others. The difficult problem of the solution bifurcations of the governingnonlinear partial differential equations (PDE's) is solved through a novelapproach. By using a methodology involving coupled series-asymptoticexpansions, we derive the normal form equation of the original complicatedsystem of nonlinear PDE's. By writing the normal form equation into afirst-order dynamical system and with a phase-plane analysis, we manage todeduce the global bifurcation properties and to solve the boundary-valueproblem analytically. The asymptotic solutions (including post-bifurcationsolutions) in terms of integrals are obtained. The engineering stress-straincurve plotted from the asymptotic solutions can capture the key features of thecurve measured in a few experiments (e.g., the stress drop, the stress plateau,and the small stress valley). It appears that the asymptotic solutions obtainedshed certain light on the instability phenomena associated with phasetransitions in a cylinder, in particular the role played by the radius-lengthratio. Also, an important feature of this work is that we consider the clampedend conditions, which are more practical but rarely used in literature forphase transition problems.

On Constructing the Asymptotic Solutions for Phase Transitions in a  Slender Cylinder Composed of a Compressible Hyperelastic Material with  Clamped End Conditions