DIGIMU® CONTINUES TO EVOLVE

Introduction

Several phenomena can lead to grain size evolution in a metallic material during hot forming processes and heat treatments. When a material accumulates plastic deformation, dislocation networks develop within the grains. Subsequently, several thermal activated processes tend to reduce the energy of the system. These processes include recovery, that annihilates and rearranges the dislocations¹, grain boundary migration due to capillary effects (the so-called grain growth) and stored energy gradient (also called strain-induced boundary migration), and recrystallization (when the stored energy promotes the formation of new grains which nucleate mainly at grain boundaries).

Different criteria must be locally achieved to promote the evolution of a cluster of atoms into a nucleus. The first is the formation of a high-angle mobile grain boundary during the nucleation process. The second is the presence of a strong gradient of stored energy across the interface, which provides sufficient driving pressure to counteract the capillary effects applied to the nucleus.

DIGIMU^® relies on a Finite-Elements – Level-Set (FE-LS) formalism² for modelling microstructural evolutions at the polycrystal level. Grain boundary kinetics are piloted by a thermo-dependent grain boundary mobility M_b, the capillary pressure depending on the grain boundary energy , local grain boundary curvatures and stored energy gradients [ΔE]. The normal velocity of a grain boundary can therefore be expressed as:

where is the outward normal unit vector. Since in DIGIMU^® the energies are averaged per grain, an additional parameter  is introduced to take into account the fact that the gradients at grain boundaries may be higher than those computed with constant grain energy jump at grain boundaries.

Discontinuous dynamic recrystallization (DRX) is modelled using a critical nucleation energy . As the density of dislocations is considered constant in the grains, the prediction of the exact location of nucleation sites is not possible. We therefore consider that new nuclei appear at the boundaries of the grains that have reached the critical energy. This assumption is relevant, since based on the critical energy required for nucleation, it has been shown that the most favourable nucleation sites are the grain corners (triple junctions), followed by the grain edges, then the grain faces, while the inside of the grain (homogeneous nucleation) is the least energetically favourable site^3,4. 

The nuclei are created with the critical nucleus radius r*of the « Bailey-Hirsch » criterion⁵, which is required to compensate for capillary effects exerted by neighbouring grains. This condition is approximated by this criterion, such that: 

where represents the line energy of the dislocations, >1 is a safety factor guaranteeing that the nucleus formed has the driving force necessary for its growth, dim designates the spatial dimension, linked to the fact that the curvature of a sphere is twice that of a circle of the same radius, and is the energy of the grain boundary.

Furthermore, the nucleation rate is calculated according to a variant of the « Peczak & Luton »⁶ proportional nucleation model. At a given time  , the volume of inserted nuclei is  where is the volume of nuclei per unit time and per grain boundary area having reached nucleation conditions .

Although this model effectively describes DRX, the fact that for a given temperature and strain rate, all nuclei appear of the same radius have shown some limits for post-dynamic recrystallization (PDRX) evolutions. If nuclei sizes are too similar in size, capillary effects are compensated between neighbouring nuclei and grain growth is slowed down. Experimental micrographs show clearly a size dispersion around the theoretical value, due to local intragranular energy distributions (note that most of these micrographs are in 2D, meaning the observed sizes correspond to lateral sections of the nuclei). Taking that into account in the simulation maintains a certain competition between the nuclei and generates some grain growth due to capillary effects⁷.

This article demonstrates how integrating a nuclei size dispersion in DIGIMU^® will maintain this competition between them, and lead to more realistic grain size evolutions and distributions during PDRX.

Introducing a nucleus size distribution

In a study conducted in collaboration with AUBERT & DUVAL, a comparison of experimentally measured grain size distribution, in an Inconel 718 alloy following DRX/PDRX, with that obtained by DIGIMU^® for a constant nucleus radius, showed a slight divergence (see Figure 3). To remedy this discrepancy, we considered a log-normal distribution of the nuclei size around the calculated Bailey-Hirsch value and studied the impact of such a distribution on the resulting microstructure at the end of DRX/PDRX. Four different distributions, shown in Figure 1, were tested in our calculations to identify the one closest to the experimental results.

Figure 1: Representation of the tested lognormal nucleus size distributions. The value 1 in the x-axis represents the mean nucleus size (calculated with the Bailey-Hirsch criterion).

By following the microstructure evolution for each case, at the end of DRX and during PDRX, we compared the average equivalent diameter (D_eq) and the recrystallized fraction (F_RX) of the grains. We observed that the broader the nucleus size distributions, the larger the grain size at the end of DRX and the lower the recrystallized fraction. The results are summarized in the Figure 2, at the beginning and after 300 s of PDRX. A broadening of D_eq values is observed compared with the constant radius case, particularly towards larger values around 20 µm. Among the different distributions, the 4^th distribution appears to be the closest to the experimental result. However, a discrepancy in the recrystallized fraction, at the beginning of the PDRX, emerges between the simulation with constant nucleus size (100%) and the 4^th distribution (73%). This is an undesired effect, since the F_RX values obtained with the constant nucleus size were calibrated in previous studies performed on the same alloy, resulting in satisfactory outcomes.

Figure 2: Simulation results obtained at the end of DRX, which marks the start of PDRX (t_PDRX=0), and after a duration of 300 s. The colored images show the microstructures during PDRX for the four nucleus size distributions, and the histograms represent the corresponding mean equivalent grain diameter distributions.

In order to make up for this discrepancy, we can adjust the parameters and K_g previously described. As explained above, these parameters directly affect the number and size of nuclei as well as their rate of appearance during nucleation and can be adjusted according to the studied material. An adjustment of and K_g up to 15% was performed, which leads to the matching of the F_RX of the simulation with the 4^th distribution and the one that considers a constant nucleus size.

To validate our improvements, the same material (Inconel 718 - 10/11 ASTM) was studied again in collaboration with AUBERT & DUVAL. The material was preheated (1000°C, 15 min), deformed (975°C, 5 min, 0.02 s^-1, ε = 1.3) and then cooled to 860°C at a rate of 140°C/min. The evolution of the microstructure was then examined at the end of DRX and during PDRX. The histograms in Figure 3 show the results obtained for the different simulations, in comparison with the experimental results.

Figure 3: Comparison between the grain size distribution (in µm and ASTM) in an Inconel 718 alloy measured experimentally (in orange) at the end of DRX/PDRX, and the one obtained by DIGIMU® for: a constant nucleus radius (in green) and the 4^th distribution with parameter adjustment (in blue).

First, we note that the simulation’s result without a nucleus size distribution reveals grain sizes centred around the mean value, in comparison with the experimental distribution which is much flatter. In fact, the simulation without a nucleus size distribution gives rise to a peak at 12 ASTM which is not present in the experimental result. This can be explained by an evolution of grain sizes in the experiment towards higher values (10-9 ASTM).

Nevertheless, for the lower classes (15-14 ASTM), we observe results relatively close to reality, although the peak at 15 ASTM is absent in the experimental distribution. This can be explained by the experimental characterization of the microstructure, which ignores smaller grains.

On the other hand, when we deploy DIGIMU^®'s new ability to insert nuclei following a size distribution, the final evolution of the microstructure shows a distribution closer to experimental observations. In fact, the resulting grain size distribution after DRX/PDRX is flatter than the one that does not consider a nucleus size distribution. This result can be explained by the difference in the size of the nuclei at their creation, which gives rise to different curvatures, thus generating a driving force that favours competition between nuclei and influences their subsequent evolution.

Note that the absence of 9 ASTM grains can be attributed to a slower evolution in the simulation. This could result from various factors, such as slightly reduced mobility, minor temperature fluctuations during the experiment (a few degrees below 980°C), or a combination of both.

To highlight the benefits of incorporating a nucleus size distribution, we present an animation below that compares the evolution of a microstructure with and without nucleus size distribution. In the upper simulation, which uses the 4^th distribution, the histogram reveals a broader spread of grain diameters and a more flattened profile. This highlights a heterogeneous grain size distribution that can be seen in the final microstructure. In contrast, the lower simulation, which does not account for nucleus size distribution, displays a homogeneous grain size with a histogram centred around the calculated Bailey-Hirsch value. This comparison clearly demonstrates how the adoption of a nucleus size distribution allows to obtain realistic microstructural simulations.

Figure 4: Animation comparing the evolution of a microstructure with and without the distribution of nuclei sizes.

CONCLUSION

In the latest updates of DIGIMU^®, exciting functionalities were introduced to refine microstructural simulations and align them closer with experimental results. Among the highlights is the incorporation of nucleus size distribution during nucleation, enhancing the accuracy of simulated final grain size distributions as demonstrated in the present article for an Inconel 718. Additional features include heterogeneous hardening and updates to material files. These tools are ready to help you unlock better predictions and optimize your material properties.

Take your simulations to the next level with DIGIMU^® 5.1! 

References

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2. Scholtes, B. et al. New finite element developments for the full field modeling of microstructural evolutions using the level-set method. Computational Materials Science 109, 388–398 (2015).
3. Huang, W. & Hillert, M. The role of grain corners in nucleation. Metallurgical Transactions, A 27, (1996).
4. Maire, L. et al. Modeling of dynamic and post-dynamic recrystallization by coupling a full field approach to phenomenological laws. Materials & Design 133, 498–519 (2017).
5. Bailey, J. & Hirsch, P. B. The recrystallization process in some polycrystalline metals. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 267, 11–30 (1962).
6. Peczak, P. & Luton, M. The effect of nucleation models on dynamic recrystallization I. Homogeneous stored energy distribution. Philosophical Magazine B 68, 115–144 (1993).
7. Roth, M. Extension d’un modèle de recristallisation dynamique discontinue à champ-moyen vers les hautes vitesses de déformation sur un acier 316L, Thèse de doctorat. (Université Paris sciences et lettres, Centre de mise en forme des matériaux, 2024).