Full field model describing phase front propagation, transformation strains, chemical partitioning and diffusion in solid-solid phase transformations

Abstract

A novel mathematical formulation is presented for describing growth of phase in solid-to-solid phase transformations and it is applied for describing austenite to ferrite transformation. The formulation includes the effects of transformation eigenstrains, the local strains, as well as partitioning and diffusion. In the current approach the phase front is modelled as diffuse field, and its propagation is shown to be described by the advection equation, which reduces to the level-set equation when the transformation proceeds only to the interface normal direction. The propagation is considered as thermally activated process in the same way as in chemical reaction kinetics. In addition, connection to the Allen-Cahn equation is made. Numerical tests are conducted to check the mathematical model validity and to compare the current approach to sharp interface partitioning and diffusion model. The model operation is tested in isotropic two-dimensional plane strane condition for austenite to ferrite transformation, where the transformation produces isotropic expansion, and also for austenite to bainite transformation, where the transformation causes invariant plane strain condition. Growth into surrounding isotropic austenite, as well as growth of the phase which has nucleated on a grain boundary are tested for both ferrite and bainite formation.

Numerical experiments on the solution of advection equation for moving phase interface: Encountered problems and their solutions

“…The shape-preserving propagation of the phase interface towards the positive 𝑥 direction as function of time at speed 𝑐 is then described by the function 𝑓 𝑥 − 𝑐𝑡, 𝑡 . As shown in [9], this description generalizes to three dimensions, which in that case leads to level-set equation in the case that the phase front propagates only in to the interface normal direction. In the current article the focus is on one-dimensional case, which is examined in detail.…”

“…As described in [9], the advection equation can be applied to describe phase front propagation. Assume that a given function, 𝑓 𝑥 , describes the change of phase to another as function of the spatial coordinate so that 𝑓 𝑥 = 1 when 𝑥 belongs to transformed phase and 𝑓 𝑥 = 0 when 𝑥 belongs to the other phase.…”

“…Furthermore, due to its effect on the mechanical properties, the microstructure development during processing [3][4][5][6] needs to be understood in detail in order to control it effectively. The continuous development of accurate mean field models [7,8] also requires input from detailed microstructure formation models [9], where the phase front propagation needs to be explicitly modeled.…”

“…Therefore, it is very important to understand how the microstructure develops in order to control it in production. Newly developed physics based numerical simulations [9] provide the necessary tools for both the detailed understanding of the associated phenomena, their interactions, and therefore could be expected to achieve the predictive capability which is needed in virtual experimentation of novel processing routes. While this approach is the route to the complete understanding to the materials processing phenomena (in contrast to the solely data based models), it is not an easy one.…”

“…In order to obtain the capability to numerically simulate the development of phase morphology, which occurs in solid-to solid phase transformation, a mathematical description for the phase front propagation is required. It has been shown [9] that the advection equation can be used for this purpose. Unfortunately, the simple Euler forward time stepping in standard explicit finite difference framework is known [10] to yield unstable solution for the advection equation.…”