Density Function Optimization for the Homogenized Energy Model of Shape Memory Alloys

In this paper, we present two methods for optimizing the density functions in the homogenized energy model (HEM) of shape memory alloys (SMA). The density functions incorporate the polycrystalline behavior of SMA by accounting for material inhomogeneities and localized interaction effects. One method represents the underlying densities for the relative stress and interaction stress as log-normal and normal probability density functions, respectively. The optimal parameters in the underlying densities are found using a genetic algorithm. A second method represents the densities as a linear parameterization of log-normal and normal probability density functions. The optimization algorithm determines the optimal weights of the underlying densities. For both cases, the macroscopic model is integrated over the localized constitutive behavior using these densities. In addition, the estimation of model parameters using experimental data is described. Both optimized models accurately and efficiently quantify the SMA’s hysteretic dependence on stress and temperature, making the model suitable for use in real-time control algorithms.