A novel homogenization method for phase field approaches based on partial rank-one relaxation

Abstract

This paper deals with the analysis of homogenization assumptions within phase field theories in a finite strain setting. Such homogenization assumptions define the average bulk׳s energy within the diffusive interface region where more than one phase co-exist. From a physical point of view, a correct computation of these energies is essential, since they define the driving force of material interfaces between different phases. The three homogenization assumptions considered in this paper are: (a) Voigt/Taylor model, (b) Reuss/Sachs model, and (c) Khachaturyan model. It is shown that these assumptions indeed share some similarities and sometimes lead to the same results. However, they are not equivalent. Only two of them allow the computation of the individual energies of the co-existing phases even within the aforementioned diffusive interface region: the Voigt/Taylor and the Reuss/Sachs model. Such a localization of the averaged energy is important in order to determine and to subsequently interpret the driving force at the interface. Since the Voigt/Taylor and the Reuss/Sachs model are known to be relatively restrictive in terms of kinematics (Voigt/Taylor) and linear momentum (Reuss/Sachs), a novel homogenization approach is advocated. Within a variational setting based on (incremental) energy minimization, the results predicted by the novel approach are bounded by those corresponding to the Voigt/Taylor and the Reuss/Sachs model. The new approach fulfills equilibrium at material interfaces (continuity of the stress vector) and it is kinematically compatible. In sharp contrast to existing approaches, it naturally defines the mismatch energy at incoherent material interfaces. From a mathematical point of view, it can be interpreted as a partial rank-one convexification.