Formulation of balance laws as mixture theory for nuclei and matrix in recrystallization

In the previous work, the authors formulated the balance laws of mass, momentum, angular momentum and energy of the lattice element used for recrystallization. These laws were summed up over a phase in a representative volume element (RVE) and averaged in the RVE so as to develop the discrete balance laws for single phase. Furthermore, the balance law of angular momentum was separated into a bulk and a lattice parts through the orderestimation with the representative lengths both in macroscopic and microscopic scales. In this paper, the RVE converges on a material point so that the laws are rewritten in the integration form. When the laws are summed up all over the phases and averaged in them, the balance laws of mass, momentum, angular momentum and energy for nuclei and matrix as mixture are formulated, using an useful theorem proposed for the mixing summation of unsteady terms. At this time, the macroscopic part of the balance law for angular momentum results in the usual equation of angular momentum, so that the stress tensor keeps symmetry even if the lattice rotation is considered. While, the microscopic one is localized as an equation of spin angular momentum for lattice, which is suggested to be equivalent to the evolution equation of crystal orientation in KWC type phase-field model. Moreover, the increase law of entropy for mixture is also formulated. During this process, the entropy flux is defined by use of relative mass flux and chemical potential of phase transformation.