A robust three-dimensional phenomenological model for polycrystalline SMAs: Analytical closed-form solutions

This paper presents a robust three-dimensional phenomenological model and analytical closed-form solutions to simulate self-accommodation, martensitic transformation and orientation/reorientation of martensite in polycrystalline shape memory alloys (SMAs). The model is developed within the classical framework of thermo-dynamics of irreversible processes and utilizes the volume fractions of self-accommodated and oriented martensite as scalar internal variables and the preferred direction of oriented martensite variants as tensorial internal variable. Linear and exponential interpolation functions are introduced which respectively result in coarse and smooth transitions in stress-induced martensitic transformation. A unified constitutive model is presented for both stress and strain control modes that has the property of completely decoupling the reorientation mechanism from the martensitic transformation mechanism. The time-discrete counterpart of the unified constitutive model is introduced, integrating the evolution equation of martensite reorientation using both implicit backward Euler and explicit forward Euler schemes. Analytical closed-form solutions are derived for the preferred direction of oriented martensite variants and the volume fractions of self-accommodated and oriented martensite. In order to examine capabilities of the developed SMA model as well as the proposed closed-form solutions, two boundary value problems are solved including a thin NiTi wire under combined tension-torsion non-proportional loadings and a thin-walled NiTi tube subjected to combined internal pressure-tension/compression/torsion-heating paths. In the first problem, the model predictions are compared with the experimental data that shows good correlations. Due to simplicity and accuracy, the model can be used as an efficient and analytic computational tool to analyze structures made of SMAs under multi-axial non-proportional loading histories.