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A Three-Dimensional Continuum Simulation Method for Grain Boundary Motion Incorporating Dislocation Structure

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Abstract

We develop a continuum model for the dynamics of grain boundaries in three dimensions that incorporates the motion and reaction of the constituent dislocations. The continuum model is based on a simple representation of densities of curved dislocations on the grain boundary. Illposedness due to nonconvexity of the total energy is fixed by a numerical treatment based on a projection method that maintains the connectivity of the constituent dislocations. An efficient simulation method is developed, in which the critical but computationally expensive long-range interaction of dislocations is replaced by another projection formulation that maintains the constraint of equilibrium of the dislocation structure described by the Frank’s formula. This continuum model is able to describe the grain boundary motion and grain rotation due to both coupling and sliding effects, to which the classical motion by mean curvature model does not apply. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of evolutions of low angle grain boundaries and their dislocation structures.

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Data Availability Statement

The datasets generated in study are available upon reasonable request.

Notes

  1. In fact, the net dislocation flux across the boundary of any region \(\Omega \) on the grain boundary is \(\int _{\partial \Omega }\nabla _S \eta _j\cdot d{\mathbf {r}}=\int _\Omega \left( \frac{\partial \eta _{jv}}{\partial u}-\frac{\partial \eta _{ju}}{\partial v}\right) dudv=0\) using this condition.

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Acknowledgements

This work was supported by the Hong Kong Research Grants Council General Research Fund 16301720 and 16302818.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Xiaoxue Qin & Yang Xiang

  2. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

    Luchan Zhang

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A Derivation of the Formula for Misorientation Angle \(\theta \) in (30)

A Derivation of the Formula for Misorientation Angle \(\theta \) in (30)

Substituting \({\mathbf {V}}_1= {\mathbf {r}}_u\) and \({\mathbf {V}}_2= \mathbf{r}_v\) into Frank’s formula in Eq. (27), we have

$$\begin{aligned} \theta ({\mathbf {r}}_u\times {\mathbf {a}})-\sum _{j=1}^{J}{\mathbf {b}}^{(j)}\eta _{ju}=&0, \end{aligned}$$
(64)
$$\begin{aligned} \theta (\mathbf {r}_v\times {\mathbf {a}})-\sum _{j=1}^{J}{\mathbf {b}}^{(j)}\eta _{jv}=&0. \end{aligned}$$
(65)

Here we have used \(\nabla _S \eta _j\varvec{\cdot }{\mathbf {r}}_u=\eta _{ju}\) and \(\nabla _S \eta _j\varvec{\cdot }{\mathbf {r}}_v=\eta _{jv}\). Adding the two equations (64) and (65), multiplying both size of the summation by \(( {\mathbf {r}}_u+ {\mathbf {r}}_v) \times {\mathbf {a}} \), we have

$$\begin{aligned} \theta \Vert ( {\mathbf {r}}_u+ {\mathbf {r}}_v) \times {\mathbf {a}}\Vert ^2= \sum _{j=1}^J ( \eta _{ju}+ \eta _{jv})( {\mathbf {r}}_u + {\mathbf {r}}_v )\times {\mathbf {a}}{\cdot }{\mathbf {b}}^{(j)}. \end{aligned}$$
(66)

Integrating over the entire grain boundary S, we obtain the formula of \(\theta \) in Eq. (30).

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Qin, X., Zhang, L. & Xiang, Y. A Three-Dimensional Continuum Simulation Method for Grain Boundary Motion Incorporating Dislocation Structure. J Sci Comput 90, 3 (2022). https://doi.org/10.1007/s10915-021-01694-7

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