# An Investigation of Spiral Dislocation Sources Using Discrete Dislocation Dynamics (DDD) Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

^{−1}]. The minimum segment length is equal to 300b. The total source length (L) is equal to 30,000b. The minimum time step is equal to 10

^{−12}[s]. The result of this simulation is presented in Figure 3. It is observed in the figure that the stress–strain curve starts linear with elastic deformation and then reaches a proportional limit (initial yielding [23]) followed eventually by a steady state of plasticity generation (the average stress value of which is termed the “flow stress” or ${\sigma}_{f}$).

## 3. Simulation Results and Discussion

#### 3.1. Spiral Dislocation Configuration in a Large Simulation Box (without Surface Effect)

#### 3.2. DDD Simulations for the Study of Size–Scale Effect (without Surface Effect)

^{−1}$]$, a minimum segment length equal to 300b, a total spiral source length (L) equal to 30,000b = S1/2, and a minimum time step equal to 10

^{−12}s.

#### 3.3. Simulations for Multiple Spiral Dislocation Sources (without Surface Effect)

#### 3.3.1. Frank–Read Source Generation from Spiral Dislocation Sources

#### 3.3.2. Simulations for Multipoles

#### 3.4. Simulations for a Single Spiral Dislocation with Surface Effect

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Configuration of a traditional FR source. (

**b**) Configuration of a spiral dislocation source.

**Figure 4.**(

**a**,

**c**,

**e**) (dark blue) A spiral dislocation source with one end fixed at the origin of the coordinate system by a pinning point while the other end is free; (

**b**,

**d**,

**f**) (red) a spiral dislocation source with one end fixed at the origin of the coordinate system by an extended dislocation while the other end is free (simulation box dimensions: S1 = S2 = S3 = 600,000b). Figures (

**a**,

**b**) are the initial configurations. Figures (

**c**,

**d**) show the initial bowing of the source. Figures (

**e**,

**f**) show multiple loops of the dislocation source.

**Figure 5.**Stress–strain diagram for DDD simulations using various S2 values (other simulation box dimensions: S1 = S3 = 60,000b). Here, no surface effect is accounted for.

**Figure 6.**Flow stress (mean values of the steady-state portions of the Stress–strain curves in Figure 5) vs. S2 (without surface effect).

**Figure 7.**(

**a**) A spiral dislocation source in a simulation box with dimensions S1 = S2 = 60,000b and S3 = 40,000b; (

**b**) a spiral dislocation source in a simulation box with dimensions S1 = 60,000b, S3 = 40,000b, and S2 = 6000b (without surface effect).

**Figure 8.**DDD simulation for two spiral dislocations (red lines in the figures) initially with edge character at different time steps (without surface effect).(

**a**) Initial configuration of two edge spiral dislocations; (

**b**) Operation of the edge spiral dislocations under a constant shear strain rate; (

**c**) Formation of traditional screw FR source; (

**d**–

**f**) Operation of the traditional screw FR source under a constant shear strain rate.

**Figure 9.**DDD simulation for two spiral dislocations initially with screw character at different time steps (without surface effect). (

**a**) Initial configuration of two screw spiral dislocations; (

**b**) Operation of the screw spiral dislocations under a constant shear strain rate; (

**c**) Formation of traditional edge FR source; (

**d**–

**f**) Operation of the traditional edge FR source under a constant shear strain rate.

**Figure 10.**Simulation data for two edge spiral dislocations and a Frank–Read screw dislocation source (Simulation box dimensions: S1 = S2 = 60,000b and S3 = 40,000b (without surface effect)).

**Figure 11.**Spiral dislocation source multipoles of edge character, with constant separation 1000b along the z-axis. (

**a**) Dipole, (

**b**) tripole, and (

**c**) quadrupole (no surface effect).

**Figure 12.**Single and multipoles (see Figure 11) in action. They act as a fan with multiple blades. (

**a**) Single, (

**b**) dipole, (

**c**) tripole, and (

**d**) quadrupole (no surface effect). (Different color lines in the figures represent the spiral dislocation sources separated along the z-axis.)

**Figure 13.**(

**a**) Stress–strain curves for multipoles simulations. (

**b**) Flow stress vs. the number of spiral sources of the multipoles.

**Figure 14.**The DDD simulation for a spiral dislocation (purple line in the figures) near a free surface with surface effect activated (Top view). (

**a**) Initial configuration of the simulation; (

**b**,

**c**) The spiral dislocation is attracted towards the free surface under the effect of the image stress and almost touches the surface in (

**c**); (

**d**) The spiral dislocation eventually vanishes at the free surface with its two ends still in the crystal.

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**MDPI and ACS Style**

Li, L.; Khraishi, T.
An Investigation of Spiral Dislocation Sources Using Discrete Dislocation Dynamics (DDD) Simulations. *Metals* **2023**, *13*, 1408.
https://doi.org/10.3390/met13081408

**AMA Style**

Li L, Khraishi T.
An Investigation of Spiral Dislocation Sources Using Discrete Dislocation Dynamics (DDD) Simulations. *Metals*. 2023; 13(8):1408.
https://doi.org/10.3390/met13081408

**Chicago/Turabian Style**

Li, Luo, and Tariq Khraishi.
2023. "An Investigation of Spiral Dislocation Sources Using Discrete Dislocation Dynamics (DDD) Simulations" *Metals* 13, no. 8: 1408.
https://doi.org/10.3390/met13081408