An analytical study of the instability of a superelastic shape memory alloy cylinder subjectto practical boundary conditions

Abstract

In this paper, we study phase transitions in a slender circular cylinder composed of acompressible hyperelastic material with a non-convex strain energy function. We aim toconstruct asymptotic solutions based on an axisymmetrical three-dimensional setting anduse the results to describe the key features observed in the experiments by others. Theproblem of the solution bifurcations of the governing nonlinear partial differential equations(PDEs) is solved through a novel approach involving coupled series–asymptotic expansions.We derive the normal form equation of the original complicated system of nonlinear PDEs.By writing the normal form equation into a first-order dynamical system andwith a phase-plane analysis, we deduce the global bifurcation properties andsolve the boundary-value problem analytically. The asymptotic solutions in termsof integrals are obtained. The engineering stress–strain curve plotted from theasymptotic solutions can capture some key features of the curve measured in theexperiments. It appears that the asymptotic solutions obtained shed certain light on theinstability phenomena associated with phase transitions in a cylinder. Also, animportant feature of this work is that we consider the clamped end conditions,which are more practical but rarely used in the literature for phase transitionproblems.