Application of fractional calculus to modeling the dynamic mechanical analysis of a NiTi SMA ribbon

Internal damping process in a ribbon of NiTi shape memory alloy (NiTi SMA) during a reverse transformation, from low temperature phase (martensite) to high temperature phase (austenite), has been studied by dynamic mechanical analysis (DMA) and using fractional calculus for the experimental data analysis. A mechanical fractional model for the description of dynamic modulus, E∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{*}$$\end{document}, has been developed for this purpose. This fractional model takes into account the relaxation peak associated with internal damping, and its corresponding differential equation has derivatives of fractional order between 0 and 1. By applying Fourier transform to fractional differential equation and considering cooperative or noncooperative atomic mobility, the real and imaginary parts of E∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{*}$$\end{document} have been computed. An agreement between experimental data and theoretical results of fractional model has been achieved. The fractional order parameters of the model are related to atomic mobility associated with internal damping of NiTi SMA ribbon.