Abstract
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.
Highlights
Lightweight materials with high strength, ductility, and toughness have been actively researched to solve various environmental problems
An representative volume element (RVE) includes many lattice element (LE), and a material point in the macroscopic control volume contains the physical information of a representative volume
Lattice elements in materials were modeled during the recrystallization phenomenon and the kinetics of the lattice elements were discussed
Summary
Lightweight materials with high strength, ductility, and toughness have been actively researched to solve various environmental problems. There is no concept of fluxes in the energy principle; we cannot introduce the characteristics of the materials into models through constitutive flux equations To solve these problems, a KWC-type phase-field model must be developed based on the conservation laws. The aim of this study is to formulate conservation laws at the lattice scale that consider microscale phenomena so as to derive phase-field models of recrystallization. We divide the conservation law of angular momentum into a bulk part and a lattice part through an order estimation based on the characteristic lengths at the bulk scale and lattice scale as representative quantities Through this process, the Cauchy stress retains its symmetry, even when the angular momentum of the crystal lattice is considered. The law of entropy increase is formulated for a mixture
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