Speaker
Description
The homogenization of solids with a fixed microstructure has been extensively studied over the last decades for a variety of material systems. However, homogenizing phase-transforming materials with an evolving microstructure still remains an unresolved challenge, even from a methodological perspective. Specifically, the conventional definition of a microscopic length scale is based on the wavelength of fluctuations in material properties --- such as density, mechanical stiffness, or heat conductivity --- as determined by the physical modeling problem. In this regard, the ratio of micro- and macroscopic scales characterizes the scale separation and therefore determines the mathematical structure of the homogenized continuum theory. For phase transforming materials, however, these fluctuation's lengths change over time. This makes it almost impossible to properly define representative volume elements, as the microstructure evolves heterogeneously, in general. Consequently, conventional two-scale homogenization approaches, such as the FE2-method, cannot be consistently applied when considering phase transformation mechanisms with complex interface topologies.
To address these issues, the concept of variable scale separations was recently developed in [1] as a fundamental homogenization framework for phase transforming solids. In the present work, this rather theoretical approach is applied to homogenize the phase state of an Allen-Cahn-type system for different spatial scale separations. More precisely, the ratio between micro- and macroscopic scales is now defined through a scale separation factor, which is independent of any wavelength of property fluctuations, and can therefore be chosen arbitrarily. The procedure for deriving the macroscopic phase evolution equations is motivated by spatial regularization methods known from non-local damage theories and micromorphic continua, see [2] and [3], respectively. Whereas the micro-macro transition is formulated only based on the unweighted average of the system's energy within conventional homogenization schemes, the outlined approach incorporates arbitrary measures of effectiveness. As an illustrative example, the martensite laminate orientation is homogenized as the result of spatially and temporally resolved two-dimensional finite element simulations of microstructure formation in a multigrain structure, demonstrating the method's applicability to systems with complex phase topologies.
References:
[1] von Oertzen, V.~and Kiefer, B., 2024, J. Mech. Phys. 195, 105961
[2] Peerlings, R. H. J., De Borst, R., Brekelmans, W. A. M. and De Vree, J. H. P., 1996, Int. J. Numer. Methods Eng. 39, 3391--3403
[3] Forest, S. and Trinh, D. K., 2011, Z. Angew. Math. Mech. 91, 90--109
