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 modeled 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 2D plane strain 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.

Application of two-grid interpolation to enhance average gradient method for solving partial differential equations

“…In the physical science based numerical full field modelling of mesoscale phenomena, it is often necessary to solve partial differential equations describing the physical phenomena such as phase transformations [5,6,7], deformation [8,7], diffusion [5,7], fluid flow [9,10], recrystallization [11,12] etc. During processing of materials, deformation of the material is often necessary in order to obtain desired shape.…”

“…In addition, the deformation can have several beneficial effects to the material properties by affecting the internal microstructure and its development. Also, microscopic phenomena, such as phase transformations, can introduce deformations to the material microstructure [13,8,7] due to the strains caused by the transformations. For these reasons, it is desirable to apply numerical solvers that can handle both regular and deformed numerical grids so that the physical phenomena can be simulated during and after the material deformation.…”

Moving iso-contour method for solving partial differential equations

“…In the previous study [1], the movement of a phase interface was simulated with level-set type method using a physical science based model which takes into account the transformation strains and carbon partitioning and diffusion. In that study the connection to the Allen-Cahn equation [1] was made, which connected it's solution to the level-set approach.…”

“…In the previous study [1], the movement of a phase interface was simulated with level-set type method using a physical science based model which takes into account the transformation strains and carbon partitioning and diffusion. In that study the connection to the Allen-Cahn equation [1] was made, which connected it's solution to the level-set approach. As a extension of this idea, the connection between a general partial differential equation (PDE) with first order time derivative and the level set formulation is investigated in the current study.…”

Application of two-grid interpolation to enhance average gradient method for solving partial differential equations

“…In the physical science based numerical full field modelling of mesoscale phenomena, it is often necessary to solve partial differential equations describing the physical phenomena such as phase transformations [5,6,7], deformation [8,7], diffusion [5,7], fluid flow [9,10], recrystallization [11,12] etc. During processing of materials, deformation of the material is often necessary in order to obtain desired shape.…”