Discrete conservation laws based on micropolar theory considering lattice-scale-director during phase transformation

In this paper, the conventional conservation laws are formulated by modeling the lattice behavior during phase transformation as the rotation of a director. More precisely, a crystal lattice in a metal is modeled during the recrystallization process as an elastic bar element subject to stretching. Using this model, the discrete conservation laws for micropolar theory are finally derived. These conservation laws are the basis of the governing equations of Kobayashi–Warren–Carter (KWC)-type phase-field models. Hence, the derivation of this theory is significant in gaining a deeper comprehension of KWC-type phase-field models. First, balance laws for the mass, momentum, angular momentum, and energy of a lattice element are formulated. These laws are summed over a phase in a representative volume element (RVE) and averaged over the RVE. This enables the development of macroscopic balance laws for a continuum mixture consisting of several phases. When the RVE is reduced to a material point in the final formulation, the present model can be regarded as a director model whose direction vector expressing the crystal orientation is attached to a material point of a simple body. By performing an order estimation, the balance law of angular momentum can be separated into bulk and lattice parts. The bulk part results in the usual form and the latter corresponds to the evolution equation of the crystal orientation in a KWC-type phase-field model.