The concept of unequally and nonlinearly weighted averaging operators as a fundamental homogenization framework in phase‐field modeling

Abstract

AbstractIn recent years, the phase‐field method has gained tremendous potential to serve as a continuum modeling approach of micro structural evolution mechanisms for multiple applications such as alloy solidification, phase transitions, and fracture. By replacing sharp interfaces between phases with a diffusive formulation, additional degrees of freedom, namely order parameters, enter the continuum model to locally describe the current phase composition at each material point. Single phase properties thus need to be interpolated carefully within diffuse interface regions by applying mixture rules subject to specific constraints in an underlying homogenization framework, c.f. [1,2]. However, there exists a variety of well established nonlinear interpolation schemes that can only partially, or not at all, comply with conventional homogenization approaches.To this end, an extension towards unequally and nonlinearly weighted averaging operators is presented, in which conventional unweighted homogenization represents a special case. The embedding of nonlinearly interpolated Reuss‐Sachs, Taylor‐Voigt and rank‐one homogenization models within the given framework is demonstrated by means of energy relaxation. Furthermore, upper and lower energetic bounds are compared between unweighted and nonlinearly weighted averages of order distributions, where the weighted case is shown to yield an improved effective energy description.