# Dislocation Dynamics Model to Simulate Motion of Dislocation Loops in Metallic Materials

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{i}(i = 1, 2, 3) represents the local glide force per unit length exerting on the dislocation loop, B

_{ij}(i, j = 1, 2, 3) is the second-rank tensor of dislocation drag coefficients, and V

_{j}(j = 1, 2, 3) is the local velocity of the dislocation. Furthermore, B

_{ij}V

_{j}is also called the drag force of the dislocation. It is noted that all equations are written with tensorial indicial notation (i.e., the free indices take values from 1 to 3 and the repeated indices are summed from 1 to 3 unless explicit statements are indicated [23,24]). From the Peach–Koehler formula [25], glide force F

_{i}can be expressed as follows:

_{i}is always in the same direction of ${t}_{i}$, perpendicular to the dislocation loop. Contributions to the stresses ${\sigma}_{jk}$ may come from the applied stresses and the self-stresses due to the dislocation loop itself. The self-stresses induced by a dislocation itself is best found using Brown’s formula [26] or Gavazza and Barnett’s equation [27] to avoid singularity issues.

_{j}is determined by the glide force F

_{i}, as long as the size of the dislocation and applied loads are given. Further investigation shows that the glide force F

_{i}is completely derived, independent of the velocity variation along the dislocation line. Therefore, Equation (1) is essentially an algebraic equation rather than differential equation, from which the velocity V

_{j}can be calculated without the need of discretization processes. If the entire dislocation loop is segmented into many pieces of lines, the nodes at the ends of segments may jump because of the certain averaging approaches involved. That is the reason why the abnormal appearance occurs in Figure 1a causing the dislocation positions in Figure 1b dependent on the number and size of segments during its motion processes.

_{i}signifies the shear force between adjacent differential segments, η

_{ij}denotes the second-rank tensor of the local interaction of dislocation segments, and s represents a local one-dimensional coordinate along the dislocation loop. From the free-body diagram for a tubed dislocation segment, as shown in Figure 2b, a revised governing equation of dislocation dynamics can be derived as follows:

## 3. Numerical Implementation

_{i}is the coordinate of a point on the segment, r

_{j}

^{s}is the node coordinate of the segment, V

_{j}

^{s}is the velocity vector of the segment, and N

_{ij}(u) is the interpolation function dependent on the parameter $u\left(-1\le u\le 1\right)$. The weak form of Equation (4) for each segment can then be rewritten as follows:

_{p}is the total number of nodes of the dislocation loop, and ${K}_{ij}^{\eta}$ and ${K}_{ij}^{B}$ are obtained by assembling ${K}_{ij}^{s\eta}$ and ${K}_{ij}^{sB}$, respectively. Solving Equation (7) with appropriate constraints on ${V}_{j}\left(j=1,2,\dots ,3{M}_{p}\right)$, the node velocities, as well as the velocity of the entire dislocation loop, can be computed.

_{j}at time step t

_{i}, say, ${\left\{{V}_{j}\right\}}_{{t}_{i}}$, has already been obtained by solving Equation (7), then the dislocation position at the next time step t

_{i}

_{+1}can be obtained as follows:

## 4. Results and Discussion

## 5. Conclusions

_{ij}and local interaction tensor η

_{ij}of the dislocation, provided that they vary within a reasonable range. For example, it is ${10}^{-5}\sim {10}^{-3}$ Pa·S for B

_{ij}and ${10}^{-24}\sim {10}^{-19}$ N·S for η

_{ij}in our simulation of the Frank–Read source. In contrast, the evolution profiles of the dislocation from the conventional models are heavily dependent on the values of B

_{ij}. Under the action of applied stresses larger than, but comparable to, the critical stress, the dislocation expansion speed varies greatly in the multiplication process. However, the dislocation velocity does not vary appreciably if the applied stress is sufficiently large. It is interesting that if we divide $\eta ={10}^{-20}$ N·S by the cross-sectional area of the dislocation tube, we can find $\eta /\pi {r}_{c}^{2}=3.89\times {10}^{-2}$ Pa·S, which falls into the normal range of material viscosity coefficients. It can be surmised that the intrinsic local interaction within a dislocation loop may be determined with the molecular dynamics methods. Yet, experimental approaches to determining the interaction parameter η

_{ij}would surely be of significant benefit. Finally, it should be noted that the new dislocation dynamics model applies not only to the Frank–Read sources but also to general dislocation dynamics problems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Simulation results from a classical dislocation dynamics model for a straight dislocation loop: (

**a**) the dislocation segment jumps near the pinning points in the first time-step as the number of segment increases; (

**b**) dislocation positions are dependent on the number of segments during the dislocation loop’s motion and evolution processes.

**Figure 2.**(

**a**) Interactions of adjacent differential dislocation segments can be divided into two parts. (

**b**) Free-body diagram of a differential dislocation segment with forces acting on it.

**Figure 3.**Comparison between the current model and classical dislocation dynamics simulation at (

**a**) the first-time step of a straight dislocation loop and at (

**b**) the 200th time-step during the evolution process of the dislocation.

**Figure 4.**Simulation results for equilibrium bowing-out of the Frank–Read source and its motion and evolution processes: (

**a**) Equilibrium positions of an initial edge dislocation of relative length L/b = 1000 under different applied stresses; (

**b**) Evolution of the dislocation with strong local interaction under the action of an applied stress of $2.0$ μb/L; (

**c**) Equilibrium positions of the dislocation of relative length L/b = 500 under different applied stresses; (

**d**) Evolution of the dislocation with weak local interaction under the action of an applied stress of $2.0$ μb/L. The time between neighboring evolution lines is $\delta t=2.0\times {10}^{-10}$ S.

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**MDPI and ACS Style**

Tan, X.; Tan, E.; Sun, L.
Dislocation Dynamics Model to Simulate Motion of Dislocation Loops in Metallic Materials. *Metals* **2022**, *12*, 1804.
https://doi.org/10.3390/met12111804

**AMA Style**

Tan X, Tan E, Sun L.
Dislocation Dynamics Model to Simulate Motion of Dislocation Loops in Metallic Materials. *Metals*. 2022; 12(11):1804.
https://doi.org/10.3390/met12111804

**Chicago/Turabian Style**

Tan, Xinze, Enhui Tan, and Lizhi Sun.
2022. "Dislocation Dynamics Model to Simulate Motion of Dislocation Loops in Metallic Materials" *Metals* 12, no. 11: 1804.
https://doi.org/10.3390/met12111804