"On Constructing the Unique Solution for the Phase Transition in a Hyperelastic Rod"

Abstract

We study the problem of the large axially symmetric deformations of a rod composed of an incompressible Ogden's hyperelastic material subject to a tensile force when its two ends are fixed to rigid bodies. The attention is on the class of energy functions for which the stress-strain curve in the case of the uniaxial tension has a peak and valley combination. Phase plane analysis is introduced to study the qualitative behaviour of the solutions and a few theorems are then presented to show the types of the critical points and their dependence on the physical parameters. Transition boundaries are given to divide the physical parametric plane into different regions corresponding to qualitatively different phase planes.

Then, by using the non-deforming boundary conditions at two ends, the solutions corresponding to trajectories in different phase planes are obtained and the associated graphic results are presented.

An important and difficult issue in phase transitions is the nonuniqueness of solutions. Here, by considering the effects of the end boundary layers (which arise due to the nontrivial boundary conditions imposed), our results show that the domain in which the multiple solutions arise can be much more reduced. Further, by converting the problem into a displacement-controlled problem, the unique solution is obtained. The engineering strain and engineering stress curve plotted from our solution exhibits the two well-known phenomena observed in experiments: (i) After the stress reaches the peak value there is a sudden stress drop; (ii) Afterwards it is followed by a stress plateau. Mathematical explanations for these two phenomena are then given from our model.

"Dynamical Phase Transitions in Slender Elastic Cylinders: A Nonlinear Wave Approach"

Abstract

In the literature, many people have used one-dimensional stress theory to study boundary-value and/or initial-value problems of phase-transforming materials. Here, we shall show that these pure one-dimensional theories may have some essential defects. More specifically, we reveal that for these materials, both from theoretical considerations and experimental observations, the radial deformation and traction-free boundary conditions play some essential roles. From the point of view of nonlinear waves, this implies that one needs to take into account the dispersion for the propagation of the phase boundary in a slender cylinder. Here, we shall use two approaches to establish the proper model equations. One is based on the Whitham's theory for nonlinear dispersive waves and another is to use a novel asymptotic and series expansions to directly derive the model equation from the three-dimensional field equations. The nonlinear dispersive equation obtained by us shows that the problem is a singular perturbation one. The model equations used in the literature are only the leading order equations valid in the outer regions. By using our model equation and matching its travelling wave solution to those in the outer regions, we obtain three relations for three unknowns across the phase boundary, which provide the uniqueness conditions for solutions. For a particular form of the stress function, we also provide the explicit solution for the two-phase deformation.