Phase transitions in a slender cylinder composed of an incompressible elastic material. II. Analytical solutions for two boundary-value problems

In this sequel paper, we solve two boundary-value problems for the asymptotic model equation derived in the first paper. One of the purposes is to compare the analytical solutions with the experimental results obtained by a number of other authors. With the help of a phase-plane analysis on this model, first we manage to solve a force-controlled problem. We find that it is necessary to divide the external stress into seven intervals. Analytical solutions in all intervals are obtained. An interesting finding is that the value of the radius-length ratio has a great influence on the solutions. In particular, it influences the number of non-trivial solutions. In this regard, the potential application of this result to the cold drawing industry is pointed out. Our results also yield the possible nucleation stress which turns out to be dependent on the radius-length ratio a. A simple approximate formula for the nucleation stress is given, which shows that it is a decreasing function of a. This seems to capture the size effect observed in experiments: the smaller the diameter is, the larger the nucleation stress is. We also obtain the analytic solutions for a displacement-controlled problem with the help of those of a force-controlled problem. The engineering stress-strain curve plotted from the solution of the preferred configurations seems to capture the key features of the curve measured in a few experiments. © 2005 The Royal Society.