An internal-variable rod model for stress-induced phase transitions in a slender SMA layer. II. Analytical solutions for the outer loop and inner loops

In this sequel paper, the analytical solutions to the asymptotic rod equations (derived in part I of this series of papers) are constructed. More specifically, it is considered that the two ends of the layer satisfy the natural boundary conditions. By conducting a phase plane analysis, both a force-controlled problem and a displacement-controlled problem are studied. It is found that under certain conditions, both the constant solutions and the nontrivial solutions exist. The explicit expressions for the possible solutions are derived and the corresponding stress–strain curves are plotted. By comparing the total pseudo-elastic energies, the preferred solution in the displacement-controlled problem is determined. With the preferred solution obtained, the stress–strain curve and the profiles of the layer at the different stages are plotted, which can capture the key experimental features. Some further analysis has been given on the inner loops, where the reverse loading processes start just after the phase transformations are partially completed. By using the WKB method, the asymptotic solutions corresponding to the inner loops are constructed, which appear to provide the first analytical descriptions for the inner loops. The corresponding stress–strain curves and the graphes of the axial strain distribution of the layer are plotted. It is shown that the analytical solutions for the inner loops are also consistent with the experimental results.

► Phase transition in a slender SMA layer is studied in this paper.
► Some analytical solutions to the asymptotic rod equations are constructed.
► Both the outer- and inner-loops of the stress–strain responses can be modeled.
► The total pseudo-elastic energy is used to determine the preferred solution.
► The key features of the experimental results can be captured.