Closed-form solutions for inhomogeneous states of a slender 3-D SMA cylinder undergoing stress-induced phase transitions

This paper considers the isothermal stress-induced phase transitions of shape memory alloys (SMAs), based on a two variant model with an internal variable. More specifically, we study analytically the inhomogeneous states of a slender SMA cylinder under uniaxial tension/extension in a three-dimensional (3-D) setting. By utilizing two small quantities, the complex 3-D mechanical system is reduced to three linear systems for three different regions (austenite, martensite and phase transition regions). Then, we concentrate on the inhomogeneous states with only one phase transition region which is far away from the two ends. For the case of planar and general interfaces among three regions, we construct closed-form solutions and semi-analytical solutions, respectively. Theoretically, the present analytical study generalizes the classical Ericksen's 1-D results. First, unlike the 1-D case, for which there are infinitely many equally stable states, the current 3-D problem has only one optimal state (by the energy criterion). Second, in the optimal state the average axial stress on the cross section is still the 1-D Maxwell stress, although there is a nonuniform distribution. Physically, the analytical solutions clearly reveal the roles of material and geometrical parameters. The localized inhomogeneous deformation is due to the structural effect and the strain softening behavior, and an explicit formula for the width of the transition region is deduced. We also make a comparison of our 3-D analytical results with those of a 1-D gradient model, which leads to a plausible choice of the gradient parameter.