Formulation of discrete balance laws of single phase in lattice scale for recrystallization

A crystal lattice in a metal during recrystallization process is modeled as an elastic bar element subject only to stretch and its kinematics is discussed. The balance laws of mass, momentum, angular momentum and energy of the lattice element are formulated. These laws are summed up over a phase in a representative volume element (RVE) and averaged in the RVE so as to prepare to develop macroscopic balance laws for a continuum mixture consisting of several phases. When the RVE converges on a material point at the final procedure of 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 simple body. During the averaging process, two useful theorems are proposed for averaging terms associated with mass source and then these theorems are verified. Moreover, defining the representative lengths both in macroscopic and microscopic scales and performing an order-estimation for the balance law of angular momentum, this law can be separated into the bulk and lattice parts. The former results in the usual form, so that the Cauchy stress keeps symmetric even though the spin angular momentum of crystal lattice is taken into account. On the other hand, the latter corresponds to the evolution equation of crystal orientation of KWC type phase-field model.