Analytical study on the stress-induced phase or variant transformation in slender shape memory alloy samples

In this paper, the stress-induced phase or variant transformations in slender shape memory alloy (SMA) samples are studied. The constitutive models are proposed at the continuum level for both the single- and poly-crystalline SMAs. Based on a two-dimensional setting and by considering the features of the different kinds of materials, the governing PDE systems are formulated, which are composed of the mechanical field equations and some transformation criteria. By using the coupled series-asymptotic expansion method and some other manipulations, the governing systems are reduced into the simple asymptotic ODEs. Through some comparisons, it is found that the ODE systems obtained for the single- and poly-crystalline SMA samples are in fact consistent. The bifurcation problem of the asymptotic ODE systems can be easily discussed and the post-bifurcation solutions can be constructed. To show the validity of the asymptotic ODEs, some analytical results are derived, which can provide a comprehensive description of the stress-induced phase or variant transformation process in slender SMA samples.