Mathematical modeling of one-dimensional behavior of shape memory alloys

A general framework to model the thermomechanical behavior of a 1D shape memory alloy (SMA) body based on semi-empirical macroscale kinetics and a phenomenological constitutive law with martensite fraction as an internal variable is developed. As part of this framework a consistent mathematical description (global kinetic law) of martensite fraction evolution during SMA phase transformation is constructed. The global kinetic law is based on an empirical stress-temperature phase diagram and transformation functions for a 1D SMA body and a novel vector hysteresis model. The global kinetic law provides the phase fraction history given a thermomechanical loading path on the phase diagram and an initial value of the phase fraction. The developed procedure can be used to model a variety of different SMA behavior depending on the choice of transformation functions and local kinetic law algorithms. The symbolic representation and geometrical interpretation of the kinetic law are developed and its memory properties are discussed. The phase fraction evolution is examined in a number of characteristic examples, including cyclic loading inside and between the transformation strips which results in oscillatory transformation paths and internal loops of partial transformation with associated attractor loops. The simulation results using a cosine transformation function are in excellent agreement with experimental data.; Within the developed framework a model of the thermo-induced transformation of a prestressed semi-infinite 1D SMA polycrystalline body cooled from the boundary is constructed. The mathematical model is based on the nonstationary energy equation, the quasistationary approximation for the momentum equation and utilizes the developed kinetic law, the constitutive law and an incompressibility constraint. To close the system of equations the internal energy of austenite-martensite mixture in the two-phase zone is heuristically derived. The formulated initial-boundary value problem for the system is solved numerically on a fixed domain using the enthalpy method and compared to analytical results for a simplified model. Depending on the width of the transformation strip and initial stress two types solutions are obtained: (1) a diffuse, two-phase zone solution; (2) a localized, interface solution which leads to a surprising conclusion that a phase diagram with the transformation strips of finite width can support interface type solutions. The results of calculations emphasize the significance of geometry of the phase diagram as well as the stress dependence in the kinetic law and the transformation heat to the progress of transformation.