# Modelling the Effect of Deformation on Discontinuous Precipitation in Magnesium—Aluminium Alloy

^{*}

## Abstract

**:**

## 1. Introduction

_{17}Al

_{12}. Despite the ability of these alloys to form a volume fraction of precipitate close to 15%, the strengthening response obtained is poor; less than half that achieved in some aluminium alloys that form a much lower fraction of precipitates [6]. There is thus considerable interest in improving the strengthening response of AZ type alloys during precipitation. One issue that contributes to the poor strengthening response of AZ alloys to ageing is the formation of DP. The DP lamellae are usually widely spaced and offer poor barriers to dislocation motion. In addition, since DP and CP are in competition, when DP forms, it suppresses CP, mainly by consuming the supersaturated matrix and reducing the volume available to CP [7]. Therefore, it is considered desirable to suppress DP and promote CP, and achieving this may lead to better strengthening behaviour in a temperature range that is amenable to commercial heat treatment (for example <24h ageing time).

## 2. Method

## 3. Modelling

- CP is modelling using a classical size-class model based on the Kampann and Wagner numerical (KWN) framework. Classical nucleation and growth controlled by aluminium is assumed. The particles are modelled as plates with a fixed aspect ratio.
- DP is modelled assuming nucleation occurs from a certain proportion of precipitates heterogeneously nucleated on grain boundaries [12]. Growth of the DP regions is predicted using a model developed by Klinger et al. [5] that provides a unique solution to the growth velocity, based on the assumption that concentration gradients control the growth of the interlamellar $\alpha $-matrix and gradients in interface curvature control growth of the $\beta $ lamellar.
- CP influences DP by two effects; the reduction in supersaturation due to CP (which is modelled using a mean-field approximation) and the pinning effect of CP on the migrating DP front. Of these effects, it has been shown that the reduction is supersaturation is by far the most important [7].
- DP influences CP by reducing the free volume available in which CP can occur. This is modelled using the classical method of relating extended and real transformed volumes (Johnson–Mehl–Avrami–Kolmogorov [1]).

#### Crystal Plasticity Modelling

## 4. Results

#### 4.1. Microstructural Observations

#### 4.2. Precipitation Model

#### 4.3. Deformation Model

## 5. Discussion

## 6. Conclusions

- Consistent with previous studies, cold rolling was found to strongly suppress DP and promote CP after aging under conditions that lead to a fully DP microstructure in the solution treated and aged condition.
- The model predicts that the dominant effect of deformation is to promote the nucleation of CP on dislocations, which rapidly reduces the supersaturation and therefore the driving force for DP, causing it to stall.
- Even after deformation, some regions of DP persist close to the original grain boundaries. Crystal plasticity modelling demonstrates that this corresponds to regions where high levels of strain localization are expected, which provides an enhanced driving force for DP at the exact location where the initial DP nodules nucleate. This may allow limited DP to occur in these regions before CP can become established.
- Twins also directly suppress DP by acting as hard obstacles that block the growth of DP nodules. However, the effect of twinning alone on suppressing DP is predicted to be less than the effect of the reduced supersaturation due to promotion of CP by deformation. By delaying the DP transformation, twins may also enable more CP nucleation and growth, which itself will further retard DP.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Optical micrographs showing (

**a**) hot rolled and solution treated condition (

**b**) cold rolled condition (

**c**) condition in (

**a**) after aging 220 ${}^{\circ}$C, 16 h, (

**d**) condition in (

**b**) after aging 220 ${}^{\circ}$C, 16 h.

**Figure 2.**EBSD maps (IPF colouring, normal direction) showing (

**a**) hot rolled and solution treated condition (

**b**) cold rolled condition. (

**c**) Pole figures for the hot rolled + solution treated and cold rolled material.

**Figure 3.**Backscattered SEM images showing the hot rolled, solution treated and aged condition at increasing magnification (

**a**,

**b**).

**Figure 4.**Backscattered SEM images showing the hot rolled, solution treated, cold rolled and aged condition at (

**a**) low magnification and (

**b**) higher magnification, indicating blocking of DP by twin boundaries (red dashed lines) and examples of precipitates on twin boundaries (arrows).

**Figure 5.**Predictions of the evolution of (

**a**) CP and (

**b**) DP volume fraction for aging at 220 ${}^{\circ}$C as a function of dislocation density.

**Figure 6.**Predicted transition from DP to CP with increasing dislocation density for aging at 220 ${}^{\circ}$C, 16 h.

**Figure 7.**Model predictions of CP number density (

**a**) and mean radius (

**b**) compared with experimental data from [10].

**Figure 8.**Predicted von-Mises equivalent strain distribution after cold rolling compression using crystal plasticity modelling (DAMASK).

**Figure 9.**Two dimensional slice through the representative volume element (

**a**) showing line along which local von-Mises equivalent strains are plotted in (

**b**).

**Figure 10.**Enhancement to the driving force for DP due to the energy stored in the dislocation sub-structure as a function of aging temperature.

**Figure 11.**Pressure driving the migration of the DP growth front considering a reduction in the local stored energy due to the dislocation sub-structure as a function of distance from the original boundary position.

**Figure 12.**Schematic showing how DP can evolve in a grain containing a twin assuming no competition from CP. DP regions are indicated with a shaded pattern. (

**a**,

**b**) Single nucleation site active, (

**c**,

**d**) multiple nucleation sites active.

**Figure 13.**Prediction of the effect of twin blocking on the kinetics of DP volume fraction evolution assuming no competition from CP.

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**MDPI and ACS Style**

Robson, J.; Guo, J.; Davis, A.
Modelling the Effect of Deformation on Discontinuous Precipitation in Magnesium—Aluminium Alloy. *Alloys* **2022**, *1*, 54-69.
https://doi.org/10.3390/alloys1010005

**AMA Style**

Robson J, Guo J, Davis A.
Modelling the Effect of Deformation on Discontinuous Precipitation in Magnesium—Aluminium Alloy. *Alloys*. 2022; 1(1):54-69.
https://doi.org/10.3390/alloys1010005

**Chicago/Turabian Style**

Robson, Joseph, Jiaxuan Guo, and Alec Davis.
2022. "Modelling the Effect of Deformation on Discontinuous Precipitation in Magnesium—Aluminium Alloy" *Alloys* 1, no. 1: 54-69.
https://doi.org/10.3390/alloys1010005