Applications of Wavelets in the Representation and Prediction of Transformation in Shape-Memory Polycrystals

In recent years, new developments in materials characterization techniques have led to a vast amount of data on the microstructure of polycrystals. Simultaneously, improvements in computational capabilities have enabled accurate full-field simulations for the micro-mechanical fields developing in polycrystalline aggregates. These show that in phenomena including phase transformation, localized bands of deformation percolate in a complex way across various grains. Our objective is to develop a methodology for analyzing, storing and representing microstructure data and, in turn, to identify the relevant information dictating the macroscopic behavior in superelastic polycrystals. To this end, wavelets are used in a case study of a polycrystalline aggregate in anti-plane shear. It is demonstrated how the transformation fields developing within the material can be efficiently represented by thresholding their wavelet expansion, maintaining more than 90% of the L_2 norm of the original field, while using approximately 10% of the number of terms in the original data. The macroscopic stress-strain relation resulting from solving the governing equations using a thresholded transformation strain is shown to be in a good agreement with the exact relation. Finally, the set of the functions retained in the expansion after thresholding was found to be similar in adjacent loading steps. Motivated by these observations, we propose a new wavelet-based algorithm for calculating the developing fields in phase transforming polycrystals