Bifurcations in a boundary-value problem of a nonlinear model for stress-induced phase transitions

In this paper, some methodology in nonlinear dynamics is used to study a boundary-value problem of a nonlinear model arisen in phase transitions in a slender cylinder composed of a compressible hyperelastic material. We transform the original system of boundary-value problem to an initial-value (dynamical) problem of finding periodic solutions of coupled nonlinear autonomous oscillators in a four-dimensional space. Hopf-like bifurcation analysis of the periodic solutions of the system is studied. Both analytical and numerical solutions are obtained by using a nonlinear transformation formulation. The accuracy of analytical solutions is investigated by comparing with the numerical solutions. The engineering stress-strain curve is plotted and compared with that from the normal form equation, which is a simplification of the original system. © 2011 World Scientific Publishing Company.