Tag Archives: Finite Element Method

Phase transformations

Phase transformations in metallic materials have a major impact on vital engineering aspects of the material behavior such as ductility, strength and formability. Some phase transformations, such as the formation of pearlite and bainite, occur through diffusion-based processes where the constituents in the microstructure are redistributed. Being based on diffusion, these kinds of phase transformations tend to be relatively slow. On the other hand, phase transformations can also proceed by pure displacements in the crystal lattice structure. This is typical for the very rapid and diffusionless formation of martensite in austenitic steels.

Distribution of martensite (blue is austenite, red is martensite) in an austenitic metal sheet at three stages during a deep-drawing process at 213K.
Distribution of martensite (blue is austenite, red is martensite) in an austenitic metal sheet at three stages during a deep-drawing process at 213K.

Specifically, the latter kind of materials, undergoing microstructural changes in terms of austenite-martensite transformation, have in recent years gained increasing attention in relation to shape memory alloys (SMAs) and alloys exhibiting transformation-induced plasticity (TRIP steels).

Description of phase transformations is further involved due to the strong temperature-dependence of the process. Combined with significant differences in mechanical properties between the phases and the volumetric deformations accompanying e.g. martensitic phase transformations, strongly thermo-mechanically coupled phenomena arise.

The presence of martensite also changes the fracture behavior of a material since the martensite is considerably harder than the more ductile austenite parent phase. This influences e.g. initiation and propagation of crack and may become detrimental to metal forming and forging processes.

Continuum scale modeling of recrystallization

Adopting a continuum mechanical approach, recrystallization can be modeled using an internal variable representation of the pertaining quantities, such as the average grain size and the dislocation density. The macro-scale material behavior will in this way be based on parameters related to the evolving microstructure.

An example of continuum-scale modeling and simulation of ECAP-processing of Aluminum is given in Hallberg et al. (2010). Some results on the distributions of grain size and dislocation density in the work piece are shown in the figures below.

Results from simulations of ECAP-processing. The top figure shows the distribution of average grain size after and two ECAP-passes, respectively. The bottom figure shows the distribution of normalized dislocation density, also after one and two ECAP-passes, respectively. Note that after two passes, both grain size and dislocation density remain at relatively constant levels all through the specimen along the indicated lines.

Results from simulations of ECAP-processing. The top figure shows the distribution of average grain size after and two ECAP-passes, respectively. The bottom figure shows the distribution of normalized dislocation density, also after one and two ECAP-passes, respectively. Note that after two passes, both grain size and dislocation density remain at relatively constant levels all through the specimen along the indicated lines.
Results from simulations of ECAP-processing. The top figure shows the distribution of average grain size after and two ECAP-passes, respectively. The bottom figure shows the distribution of normalized dislocation density, also after one and two ECAP-passes, respectively. Note that after two passes, both grain size and dislocation density remain at relatively constant levels all through the specimen along the indicated lines.

 

Mesoscale modeling of recrystallization

Several numerical algorithms have been employed in recrystallization modeling. Some techniques, suitable for mesoscale modeling of recrystallization, are summarized below. A more extensive discussion is given in Hallberg (2011).

Monte Carlo Potts models

Historically, Monte Carlo Potts (MCP) algorithms have been used to simulate recrystallization - mainly static recrystallization - in both 2D and 3D on fixed computational grids. The system energy is minimized through probabilistic changes to the state variables, defined at the grid points. Physical time is not available in MCP models where “Monte Carlo steps” are used as a measure of time. This makes comparison between MCP results and experimental results cumbersome.

 MCP models are relatively computationally efficient and can be used in both 2D and 3D. The method is also suited for computer parallelization

Cellular automata

An alternative method is given by cellular automata (CA) which have been employed frequently in studies on both static and dynamic recrystallization. As with MCP models, CA models are also usually defined on fixed grids, but use physical cell state switching conditions, based on recrystallization kinetics defined in physical time. The cell state switching can be performed as either deterministic or probabilistic. Cellular automata are attractive since high spatial and temporal resolution can be achieved at the grain-scale, cf. Hallberg et al. (2010c) and Hallberg et al. (2014).

Examples from 3D cellular automaton simulations of dynamic recrystallization in copper at different temperatures.
Examples from 3D cellular automaton simulations of dynamic recrystallization in copper at different temperatures.

The curvature of interfaces is an important aspect of grain boundary migration kinetics but being based on discrete grids with no direct representation of interface curvature, CA algorithms have shortcomings in this respect. The choice of grid type (square, hexagonal etc.) also influences the grain growth kinetics and may be detrimental to the simulation results. As with MCP, CA are computationally efficient and 3D implementations are straight-forward. CA methods also scale well when subject to computer parallelization.

Level set models

The level set formulation was introduced as a numerical tool to trace the spatial and temporal evolution of single interfaces. The method was later also extended to consider interfaces with multiple junctions. Standard level set formulations do not correctly capture the interaction between multiple grains, occurring for example at grain boundary triple junctions, and corrections have to be implemented to remedy this shortcoming. Level set modeling of recrystallization is discussed in Hallberg 2013 and of grain growth in Hallberg 2014.

A level set representation of a polycrystal with the finite element mesh discretization shown to the right.
A level set representation of a polycrystal with the finite element mesh discretization shown to the right.

Front tracking or vertex methods

Interface migration can also be described by front tracking or vertex models where the migration kinetics of grain boundary triple junctions are considered. The topology of the grain structure is represented by nodes placed at the triple junctions, interconnected by grain boundary line segments. The representation of curvature, however, comes with additional computational cost as intermediate nodes have to be introduced between the triple junctions. In addition, the extension of the method to 3D is not easily realized since it requires surface tesselations to be performed. Also, topological changes to the grain structure require dealing with different transformation conditions. It can also be noted that usual formulations of front tracking models do not hold information related to the grain interiors.

A combined crystal plasticity and vertex model is established in Mellbin et al. 2015, to allow concurrent modeling of large deformations, texture development and recrystallization.

Phase field models

The Phase-field method (PF) has received significant interest in recent years in simulations of a broad spectrum of physical processes, including recrystallization. In PF models of recrystallization, the grain microstructure is described by phase field variables. These are functions that are continuous in space and a distinction is made between conserved and non-conserved variables. A conserved variable is typically a measure of the local composition whereas a non-conserved variable contains information on the local structure and could represent for example the crystallographic orientation. Within a single grain, a phase field variable maintains a nearly constant value that corresponds to the properties of that grain. Grain boundaries are represented as interfaces where the value of the phase field variable gradually varies between the values in the neighboring grains on opposing sides of the grain boundary. Grain boundaries are hence described as diffuse transition regions of the phase field variables. The computational effort in treating the rapidly changing fields across diffuse interfaces can be considerable and the formulation of the energy densities to capture physical microstructure features is not trivial. In addition, topological changes such as nucleation of new grains are not easily handled.

Animation showing a 2D level set simulation of Dynamic Discontinuous Recrystallization (DDRX)

A simple 2D simulation of dynamic discontinuous recrystallization (DDRX) in pure Cu at an elevated temperature. The simulation is based on level sets in a finite element setting. Adaptive remeshing is performed in each step. The animation speed is increased compared to actual time.