Full length articleA mean-field model of static recrystallization considering orientation spreads and their time-evolution
Graphical abstract
Introduction
In most models of static and dynamic recrystallization, recrystallized grains arise from a competitive growth of subgrains or cells pre-existing in the deformed microstructure [1], [2], [3], [4], [5], [6], [7], [8], [9]. In high stacking fault energy materials, the force driving the growth of subgrains comes almost entirely from the interfacial tension of the subgrain network (e.g. aluminium alloys [5]), while the energy stored in tangled dislocations plays a more important role as the stacking fault energy decreases (e.g. silver and nickel [2], copper [2], [6], [8]). While the micro-mechanisms at the origin of recrystallization are well known, the conditions leading to the development of recrystallized grains of particular orientations, and their incidence on the kinetics, remain difficult to identify.
This challenge is to a great extent due to the large number of features involved in recrystallization. Deformed grains contain of the order of 105 subgrains [5], out of which a handful turn into recrystallized grains during annealing. State-of-the-art full-field models (e.g. phase field, Vertex dynamics, level-set) can simulate this many subgrains [10], [11], but this is still insufficient to confidently predict recrystallization kinetics, grain size and crystallographic texture. As a result, the most significant applications of full-field models to recrystallization remain restricted to comparison with analytical model predictions [4], [7] or to parametric studies on the role of some initial microstructure parameters [3], [12].
Mean-field models are computationally more efficient than full-field models, but are limited by additional assumptions. In the early model of Bailey and Hirsch [1], [2], a subgrain is considered as a potential recrystallized grain when its radius exceeds the value where its inward capillary pressure is overcome by the outwards pressure induced by its neighbours. This model was extended by Zurob et al. [6] to predict the incubation period during which future recrystallized grains grow normally compared to the rest of the microstructure. This approach, however, misses the fact that every growing subgrain satisfies the Bailey-Hirsch criterion [1], [2]. Meeting the Bailey-Hirsch criterion is necessary but insufficient for a subgrain to become a grain in the recrystallized state. In two separate publications, Humphreys [13], and Rollett and Mullins [14] proposed an approach that considers that a recrystallized grain forms when the growth rate of a subgrain relative to the average is positive. Notably, the model highlights the role of heterogeneous subgrain size and boundary properties on the onset of recrystallization. Despite a few interesting applications to experimental cases [5], [15], and comparisons to full-field simulations [4], [16], this approach remains much less popular than those relying on the Bailey-Hirsch criterion (e.g. [8], [9], [17], [18]).
As the microstructural heterogeneities giving rise to recrystallization develop during prior deformation, substantial efforts have also been made to simulate recrystallization from outputs of crystal plasticity models. In these cases, heterogeneities of subgrain size and disorientation have been attributed to inter-granular contrast of slip activity (estimated by Taylor factors) [19], resolved shear stress [20], and intragranular disorientation levels [21]. These approaches generally focus on predicting the texture out of these heterogeneities while ignoring the recrystallization kinetics.
In this paper, we propose an extended mean-field model that builds on the approaches described above. In our approach, a discrete population of subgrains evolves according to classic cellular growth laws, with a time-integration scheme implemented to update the microstructural parameters. The recrystallized grains are identified based on a size threshold. The model extends beyond classic mean-field approaches by accounting for the variation of subgrain properties with crystallographic orientation by tracking the moments of several boundary property distributions. As a result, recrystallization kinetics and recrystallized grain orientations are predicted together. This model is tested against full-field vertex simulations of subgrain growth and its extension to predicting experimental results is discussed.
The paper starts by briefly introducing the methodology used for Vertex simulations. This serves to also familiarize the reader with the topology of the microstructures investigated. Next, the mean-field model is introduced. In the following sections, the ability of the mean-field model to reproduce the full-field simulations is shown, with a discussion on the strengths, weaknesses and areas for further improvement.
Section snippets
Full-field simulations
The conditions simulated in this work by the full-field model can be viewed as the recrystallization of a deformed grain in a high stacking fault energy material (e.g. an aluminium alloy or a ferritic steel). These will provide a means to validate the mean-field model in a configuration where the boundary properties and the topology of the microstructure are very well known. Yet, some differences with experiments will be noticed: (i) the dimensionality of the microstructure, (ii) the absence of
The mean-field model of cellular growth
Following the approach of Humphreys [13] and Rollett and Mullins [14], the microstructure is considered in the mean-field model as a set of grains and subgrains embedded in a homogeneous medium representing the average properties of the microstructure. Growth rates of grains and subgrains are calculated from classic capillary growth laws, and a time-integration scheme is used to update the microstructure. At each time step, the mean boundary energies and mobilities required to compute growth
Results
In this section, the mean-field model predictions are compared to a full-field simulation of recrystallization realized with an initial orientation spread of =3.5∘. This value is in the range of experimental measurements in deformed polycrystalline materials [38], [39]. The initial subgrain number density is denoted ρ0. This parameter is used as a normalizing factor in much of the subsequent analysis.
To highlight the role of the different components of the mean-field model to the
Comments on the prediction of recrystallization kinetics
Fig. 3 has shown that the prediction of recrystallization kinetics by the mean-field model is particularly sensitive to the definition of boundary properties. Kinetics are overpredicted when considering only the mean boundary disorientation angles to calculate the mean boundary mobilities and energies, in agreement with the previous attempt of Hurley and Humphreys [40]. The mean-field model prediction reaches a good agreement with the full-field simulation only by including the contribution of
Conclusion
A mean-field model was developed to simulate the time evolution of microstructures during static recrystallization. This model essentially simulates the growth of a population of subgrains contained in well recovered deformed grains, and identifies subgrains above a size threshold as recrystallized grains. At each time increment, the subgrain growth rates are calculated from classical cellular growth laws. The mean subgrain boundary energy and mobility are estimated statistically from knowledge
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors express their thanks to Jean-Denis Mithieux and Francis Chassagne for numerous and fruitful discussions on the topic of recrystallization. © Her Majesty the Queen in Right of Canada, as represented by the Minister of Natural Resources, 2020.
References (58)
- et al.
On abnormal subgrain growth and the origin of recrystallization nuclei
Acta Mater.
(2003) - et al.
Modelling the recrystallization of single-phase aluminium
Acta Mater.
(2003) - et al.
Quantitative criterion for recrystallization nucleation in single-phase alloys: prediction of critical strains and incubation times
Acta Mater.
(2006) - et al.
Modeling the recrystallized grain size in single phase materials
Acta Mater.
(2011) - et al.
A review of dynamic recrystallization phenomena in metallic materials
Mater. Des.
(2016) On the growth of abnormal grains
Scr. Mater.
(1997)- et al.
A simple model for abnormal grain growth
ISIJ Int.
(2012) - et al.
Conditions for the occurrence of abnormal grain growth studied by a 3 d vertex dynamics model
(2012) - et al.
A deformation-based model for recrystallization of anisotropic materials
Acta Mater.
(1997) - et al.
Modelling recrystallization textures driven by intragranular fluctuations implemented in the viscoplastic self-consistent formulation
Acta Mater.
(2019)
Generalized vertex model of recrystallization – application to polycrystalline copper
Comput. Mater. Sci.
A combined crystal plasticity and graph-based vertex model of dynamic recrystallization at large deformations
Modell. Simul. Mater. Sci. Eng.
Introduction to Texture Analysis: Macrotexture, Microtexture, and Orientation Mapping
On the theory of normal grain growth
Acta Metall.
Retrieving orientation correlations in deformation structures from orientation maps
Mater. Sci. Technol.
Scaling of dislocation cell structures: diffusion in orientation space
Proc. R. Soc. Lond. Series A: Math. Phys. Eng. Sci.
A theory of texture controlled grain growth–i. derivation and general discussion of the model
Acta Metall.
Quantitative characterization of the orientation spread within individual grains in copper after tensile deformation
Int. J. Mater. Res.
Contribution of intragranular misorientations to the cold rolling textures of ferritic stainless steels
Acta Mater.
Recovery and recrystallization in commercial purity aluminum cold rolled to an ultrahigh strain
Acta Mater.
On the activation of recrystallization nucleation sites in Cu and Fe
Mater. Sci. Eng.: A
Orientation pinning during growth
Grain Growth in Polycrystalline Materials 3
Current issues in recrystallization: a review
Mater. Sci. Eng.: A
Nucleation problems in metallurgy of the solid state: recent developments and open questions
Comptes Rendus de Physique
Spatial characterisation of the orientation distributions in a stable plane strain-compressed cu crystal: a statistical analysis
Acta Mater.
Computer simulation of recrystallization in non-uniformly deformed metals
Acta Metall.
A statistical ensemble cellular automaton microstructure model for primary recrystallization
Acta Mater.
Inferential statistics of electron backscatter diffraction data from within individual crystalline grains
J. Appl. Cryst.
Review grain and subgrain characterisation by electron backscatter diffraction
J. Mater. Sci.
Cited by (11)
Initial grain orientation controls static recrystallization outcomes in cold-worked iron: Insight from coupled crystal plasticity/vertex dynamics modeling: Initial grain orientation controls static recrystallization outcomes in cold-worked iron
2023, Acta MaterialiaCitation Excerpt :Hence, the ‘nuclei’ which give rise to the new recrystallized grains are not formed during annealing; they are already present in the deformed state. It is thought that deformation structures with high local orientation gradients, such as transition and shear bands, become the origin of these nuclei [18–23]. However, this has not been established conclusively, and the identification of recrystallization nuclei remains a crucial aspect of SRX studies.
Static recovery of A5083 aluminum alloy after a small deformation through various measuring approaches
2022, Journal of Materials Science and TechnologyCitation Excerpt :Electron backscatter diffraction (EBSD) characterization can provide not only information on the microstructure morphology, such as the grain size, but also more detailed information on grains, such as grain orientation spread and grain average misorientation (GAM), which can be used to monitor the microstructural changes. It is worth noting that these two approaches have been quite popular in studies on recrystallization in recent years [24,25], but have not been used much (although a few [26]) in studies on recovery. The objective of this study was to investigate the kinetics of SRV through various tests under various experimental conditions, including different temperatures, pre-strains, and strain rates, but also with and without unloading during annealing.
Effects of dislocation boundary spacings and stored energy on boundary migration during recrystallization: A phase-field analysis
2021, Acta MaterialiaCitation Excerpt :Computer simulations can play an important role in performing virtual experiments where important parameters can be studied individually or collectively. Various simulation methods are available [15–24] to study grain boundary migration during recrystallization. However, several of these cannot be easily extended to reproduce the complex nature of deformation fields observed in experimental studies.
Modelling the relationship between deformed microstructures and static recrystallization textures: Application to ferritic stainless steels
2021, Acta MaterialiaCitation Excerpt :Another possibility would be to use the outcomes of mean-field crystal plasticity simulations to estimate the subgrains properties. As discussed previously [16], this is made possible by the fact that the outputs of mean-field crystal-plasticity models are very much similar to the input parameters of the recrystallization model. For example, distributions of disorientation angles can be generated from the intragranular orientation spreads calculated by the model of Zecevic et al. [13,55].
Computationally Efficient Cellular Automata-Based Full-Field Models of Static Recrystallization: A Perspective Review
2023, Steel Research International