Variable Scale Separations in Morphology-Based Homogenization of Phase Transforming Solids
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The physical behavior of phase transforming materials strictly depends on characteristic microscopic evolution processes---such as twinning, reorientation, solidification or phase separation phenomena---during which the microstructure topology changes over time. In order to mathematically describe the associated transformation mechanisms between austenite and multi-variant martensite the phase-field method has manifested itself as a well-established modeling framework over the last decades, see [1] and the references therein. The majority of existing models thus relies on transient and spatially-regularized approaches, which therefore incorporate intrinsic time and length scales. Consequently, large-scale simulations are usually associated with enormous computational costs, since the involved temporal and spatial scales need to be resolved numerically with sufficient accuracy. The embedding of existing routines into modern two-scale approaches, such as the FE²-method, is thus immensely constrained by technical limitations. Moreover, a general notion for consistently transferring specific phase-field models between multiple, variable scales has still not been fully developed. To address these issues, the recently introduced concept of unequally and nonlinearly weighted averaging operators [2] provides a promising homogenization framework. Here, this rather general approach is applied to spatially homogenize displacive dual-phase systems between arbitrary scales---whereby the notion of variable scale separations is introduced as an additional theoretical concept. More specifically, it is mathematically demonstrated that macroscopic driving forces as well as effective mechanical properties can be expressed as functions of the average martensite volume fraction and, additionally, of a finite collection of weighted phase averages---so-called phase-morphologies. In this regard, the structure and number of the corresponding weighting functions, necessary to obtain such unique macroscopic relations vary based on the topology of the microstructure. The validity of these theoretical results is demonstrated through two- and three-dimensional finite element simulations of microstructure formation in several material systems. References: [1] N. Provatas and K. Elder, 2010, Wiley-VCH, Berlin. [2] V. von Oertzen and B. Kiefer, 2022, Shape Memory and Superelasticity 8, 425--437.