Unequally and Non-linearly Weighted Averaging Operators as a General Homogenization Approach for Phase Field Modeling of Phase Transforming Materials

Abstract

The phase field method has been shown to have tremendous potential to serve as a continuum modeling approach of microstructural evolution mechanisms in many contexts, such as alloy solidification, fracture, and chemo-mechanics. By replacing sharp interfaces between phases with a diffuse representation, additional degrees of freedom, namely order parameters, enter the continuum model, in order to describe the current phase state at each material point. Single-phase properties thus need to be interpolated carefully within diffuse interface regions by applying mixture rules subject to specific, microscopic constraints in an underlying homogenization framework. However, there exists a variety of well-established nonlinear interpolation schemes—especially incorporating symmetric or hyperspherical order parameters—for which it turns out that they cannot consistently be described within conventional homogenization theories. To overcome this problem, an extension toward unequally, non-linearly weighted averaging operators is presented, in which conventional, unweighted homogenization represents a special case. The embedding of Reuss–Sachs, Taylor–Voigt, and rank-one convexification models—extended by nonlinear interpolation—within the proposed framework is demonstrated by identifying necessary constraints on corresponding weighting functions. Since this concept establishes a generalization of conventional homogenization, the following question arises: Could any effective property interpolation within the diffuse interface fit into the proposed framework by choosing appropriate weighting functions, and if so, under which microscopic constraints? To this end, the concepts of macroscopic links and domain relations are introduced and applied for conventional homogenization schemes in phase field modeling. Important, yet often subtle, implications of such theoretical considerations on the prediction of microstructure formation and evolution by means of phase field modeling are the focus of discussion in this contribution.