Metallurgical Phase Transformations
Definitions
The material consists of a number of metallurgical phases. The fraction of each phase i is denoted ξi. There are in general N phases, where
The initial phase fraction must be defined for each phase, and the sum of initial phase fractions should be one. At the onset of an analysis, some phases may not be present, and have zero initial phase fraction.
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The phase fractions are defined in the material frame.
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Because of the employed weighting scheme for the effective mass density of the compound material, the phase fractions become algebraic fractions.
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See Compound Material Properties.
Each phase transformation describes how a source phase s transforms into a destination phase d. A phase transformation is formally defined by the rate at which the destination phase d forms at the expense of the source phase s. This can be expressed as
(3-1)
Note that this equation describes only a single contribution to the total rates at which the destination phase forms, and the source phase decomposes. With several simultaneous phase transformations, some phases may receive more than one contribution. As an example, consider the case of three phases and two phase transformations, where phase 1 transforms into phases 2 and 3. Using the terminology above, the total rate equations for the three phases can be expressed as
Note that these rate equations satisfy
In COMSOL Multiphysics, a weak contribution is generated for each phase i:
where the summation is done over every phase transformation for which phase j transforms into phase i. The exception to this is when all phases share the same selection as the physics interface, in which case one of the phase fractions is algebraically prescribed by the phase fractions of the other phases, and the requirement that the phase fractions add to unity.
In some situations, you may want to lump several phase transformations into one, so that several source phases decompose using the same phase transformation kinetics. One example is a simplified description of austenitization, if you are not concerned with the exact order in which phases like martensite and pearlite transform into austenite. Here, the destination phase is taken to form according to
The negative of this rate is taken to be the sum of the rates of the (now several) source phases so that
where the summation is taken over the participating source phases. The contribution of each source phase to the total rate is arbitrary, and a modeling decision has to be made with regard to this indeterminacy. In COMSOL Multiphysics, the use of multiple source phases in a phase transformation is handled by assuming that each source phase rate is proportional to through its own current phase fraction divided by the phase fraction sum of all participating source phases. For each source phase, the rate contribution is therefore
(3-2)
By using Equation 3-2, the correct sum of the source phase rates is ensured, and the indeterminacy of their respective contributions is eliminated. Moreover, Equation 3-2 reduces to the standard form given by Equation 3-1 in the case of a single source phase.