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.
Recrystallization (RX) is one of the main mechanism to control the evolution of grain microstructures. RX is generally accepted to be defined as the formation of a new grain structure in a cold-worked material and occurs through the formation and migration of high-angle boundaries. The grain boundary migrations are primarily driven by stored energy reduction and minimization of grain boundary surface energy.
As a metallic materials is deformed through plastic slip, energy will be accumulated in the material. This energy is to a large extent expended as heat while the remainder is stored in the material microstructure through the generation and redistribution of imperfections, mainly dislocations. By this process, the material becomes increasingly thermodynamically unstable. During subsequent annealing of the material, reduction of the stored energy can take place through relatively slow recovery or by more rapid static recrystallization (SRX). While the recovery proceeds as a continuous process, SRX is discontinuous. During thermomechanical processing of the material, i.e. when the material is exposed to plastic deformation at elevated temperatures, stored energy generation through dislocation accumulation and stored energy reduction through nucleation of new grains work in parallel. This process is commonly labeled dynamic recrystallization (DRX). The latter process of DRX may be further subdivided into a relatively slow continuous dynamic recrystallization (CDRX) or a more rapidly progressing discontinuous dynamic recrystallization (DDRX).
In materials of high stacking-fault energy, such as aluminum, dynamic recovery is significant and recrystallization occurs mainly by CDRX. In this case, subgrains with low-angle boundaries are formed from dislocation networks. With progressing plastic deformation, misorientation is increased until enough energy is achieved and the initially mobile subgrain walls have become immobilized, allowing new grains to be separated by subgrain growth. In materials of low stacking-fault energy, such as copper, dynamic recovery processes such as cross slip and climb are less influential and the recrystallization is dominated by DDRX during which new grains are nucleated as regions of low dislocation density grow to consume more dislocation-dense surroundings. RX nuclei are commonly accepted to form from subgrains and DDRX will be most significant in the microstructure regions having the highest dislocation density, primarily at grain boundary triple junctions, secondly along grain boundaries and at inclusions and with lesser probability in the grain interiors.
Processing conditions, such as temperature and strain rate, as well as material purity will influence the recrystallization process. This allows some control to be exerted over the resulting microstructure. Simulation models can provide the means for design and processing of materials through 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
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).
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.
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.