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

Abstract

Dislocation dynamics has been an intensive research subject in materials science and engineering due to the significant roles it plays in plastic deformation and the hardening of metals, fracture mechanics, and the fabrication of semiconductor thin films. However, a long-standing problem from the three-dimensional dislocation dynamics is that the motion and interaction of dislocation loops heavily depend on the loop-segment sizes, which substantially reduces the accuracy of simulation. We herein propose a new three-dimensional dislocation dynamics model together with its physical background. The proposed model incorporates the inherent interactions among differential dislocation segments. The simulation results on motion of Frank–Read sources demonstrate that the proposed model can resolve the paradoxical segment-dependent phenomenon in dislocation dynamics.

1. Introduction

Metallic materials contain crystal defects in their microstructures, which can be point, line, surface, or volume defects. These defects locally disturb the regular arrangement of atoms. Their presence, especially the presence of the line defects (dislocation loops), essentially determines the strength and plastic deformation of crystalline solids.

Partly because the macroscopic plastic responses of metals are primarily controlled by the nucleation, motion, annihilation, and interaction of dislocations in them, dislocation dynamics has been of great interest to researchers and engineers in the past several decades. Analytical solutions to this problem are impossible in most cases due to the inherent complexity of dislocation loops. Various discretization techniques are widely employed instead, which are known as discrete dislocation dynamics. With the rapid development of computational technology, a lot of important research work on dislocation dynamics has been made since the 1980s. Dislocation dynamics was first applied to two-dimensional (2D) straight, infinitely long dislocations [1,2,3,4], and then to the much more complicated three-dimensional (3D) dislocation loops in the infinite domain [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].

While computational methods have been proposed to model the motion, evolution, and interactions of 3D dislocation arrays, an open question associated with these methods is how to discretize dislocation loops, since inappropriate selection of dislocation segments causes unrealistic responses, as indicated by Gómez-García, Devincre, and Kubin [21]. Ghoniem et al. [11] also showed a strange “velocity jump” phenomenon, illustrated in Figure 1a, in which the movements near the two pinning points of a straight dislocation loop can jump as the number of line segments increases. Such an unreasonable phenomenon contradicts with the numerical approaches that finer discretized meshes employ in order to obtain more accurate solutions. As a matter of fact, simulation results are influenced by the artificial selection of the number of segments and meshes, as shown in Figure 1b, which casts doubts on the reliability of simulations in dislocation dynamics. The current paper aims to investigate the reasons behind such abnormal appearances and to develop a novel model of dislocation motion and dynamics.

2. Methodology

With a given three-dimensional dislocation loop, the conventional dislocation-dynamics methods are based on the following governing equation [22]:

$$B_{ij}{V}_j=F_i$$

(1)

where F_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_ijV_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:

$$F_i=-b_j{\sigma }_{jk}{n}_k{t}_i$$

(2)

where ${\sigma }_{jk}$ is the second-rank stress tensor, $b_j$ the Burgers vector of the dislocation, $n_k$ the unit normal to the slip plane, and $t_i$ the in-plane unit normal to the dislocation loop. It is noted that the Peach–Koehler formula-based glide force F_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.

Mathematically, the combination of Equations (1) and (2) indicates that the dislocation velocity V_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.

From a physical point of view, the abnormal phenomenon that dislocation motion is dependent on segment size is due to the fact that only a minor portion of the interaction between adjacent segments is taken into consideration in Equation (1). In the mechanics theory of dislocations, a dislocation is considered as a continuous tube. Because of the stress field external to the tube, the elastic interaction between adjacent dislocation segments is represented by the contribution of self-stresses in Equation (2). However, as illustrated in Figure 2a, the local interaction due to the core structure of the tube is not taken into account in Equation (2). Such local interactions have been exemplified by the Finnis–Sinclair interatomic potential [28] and the interatomic bondings in dislocation cores [29], implying that the local interactions be taken into account in the framework of dislocation dynamics.

Based on the above arguments, the local interaction can be assumed to adhere to the viscous law as follows:

$$Q_i(s)={\eta }_{ij}\frac{\partial V_j(s)}{\partial s}$$

(3)

where Q_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:

$$B_{ij}{V}_j(s)-{\eta }_{ij}\frac{{\partial }^2{V}_j(s)}{\partial s^2}=F_i(s)$$

(4)

It is noted that the above governing equation considers both the elastic and local interactions between dislocation segments.

3. Numerical Implementation

Similar to the standard finite element method, the numerical approach applied herein discretizes the dislocation loop into M curved segments, each of which has $M^s$ nodes. The coordinates and velocities at any point on a segment are given as follows:

$$r_i={\displaystyle \sum _{j=1}^{3M^s}{N}_{ij}\left(u\right)r_j^s}{;}_{}\hspace{1em}{V}_i={\displaystyle \sum _{j=1}^{3M^s}{N}_{ij}{\left(u\right)}_{}}{V}_j^s$$

(5)

where r_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:

$${\displaystyle \sum _{j=1}^{3M^s}\left(K_{ij}^{s\eta }+K_{ij}^{sB}\right)V_j^s}=F_i^s\hspace{1em}\left(i=1,2,\dots ,3M^s\right)$$

(6)

where $K_{ij}^{s\eta }={\displaystyle {\int }_{-1}^1{\eta }_{lm}\frac{\partial N_{li}}{\partial s}\frac{\partial N_{mj}}{\partial s}du}$, $K_{ij}^{sB}={\displaystyle {\int }_{-1}^{ 1}{J}^2{B}_{lm}{N}_{li}{N}_{mj}du}$, $F_i^s={\displaystyle \sum _{k=1}^{3M^s}{\displaystyle {\int }_{-1}^{ 1}{J}^2{N}_{ji}{N}_{jk}{F}_k^{d}du}}$ with $J=‖\partial s/\partial u‖$ being the Jacobian and $F_k^d{}_{}\left(i=1,2,\dots ,3M^s\right)$ as the local force at the nodes. With the assembly processes, the corresponding global equation is obtained as follows:

$${\displaystyle \sum _{j=1}^{3M_P}\left(K_{ij}^{\eta }+K_{ij}^B\right)V_j}=F_i\hspace{1em}\left(i=1,2,\dots ,3M_P\right)$$

(7)

where M_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 $V_j\left(j=1,2,\dots ,3M_p\right)$, the node velocities, as well as the velocity of the entire dislocation loop, can be computed.

The time span is also discretized into many time steps. If the velocity V_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:

$${\left\{r_j\right\}}_{t_{i+1}}={\left\{r_j\right\}}_{t_i}+{\left\{V_j\right\}}_{t_i}\hspace{1em}\left(t_{i+1}-t_i\right)$$

(8)

4. Results and Discussion

The critical stress necessary to activate a Frank–Read source of dislocation and its evolution have been a major topic in the theory of dislocation dynamics. Foreman [30] was the first to employ computer technology to determine the equilibrium shape and critical stress of a Frank–Read source by continuously adjusting node positions until the dislocation reached an equilibrium state. This method, however, cannot simulate the evolution of a Frank–Read source with any great degree of precision. Devincre and Condat [31] applied a 3D dislocation dynamics model to simulate the evolution and evaluate the critical stress of a Frank–Read source. While various dislocation dynamics models have been actively researched since then, the ability to simulate the evolution of dislocations is still limited by the inherent complexity of this problem, as previously indicated in Section 1.

With the newly developed governing equations and numerical scheme derived in this paper, the motion, bowing, evolution, and critical stress of a Frank–Read source are examined herein. Without losing any generality, the parameters are selected as follows: the Burgers vector $b=2.86 \AA$, the distance between the two pinning points $L=1000b=2860 \AA$, shear modulus $\mu =71.0$ GPa, Poisson’s ratio $\nu =0.3333$, drag coefficient $B=1.0\times {10}^{-4}$ Pa·S, local interaction $\eta =1.0\times {10}^{-20}$ N·S, and time step $\Delta t=5.0\times {10}^{-12}$ s (second). Figure 3 shows the direct comparison between the current model and classical dislocation dynamics simulation. Figure 3a shows the shapes of a Frank–Read source after the first-time step simulated using the classical model and the current model, respectively, demonstrating that the current model is robust, while the classical simulation maintains a peculiar “velocity jump” phenomenon at the beginning steps. Also, Figure 3b shows that the current model quickly stabilizes and converges when the number of dislocation segments approaches 20; yet, for the classical model, dislocation positions are heavily dependent on the number of segments during the motion and evolution processes of the Frank–Read source.

Furthermore, Figure 4 illustrates the simulation results for the equilibrium bowing-out of the Frank–Read source and its motion and evolution processes. Specifically, Figure 4a demonstrates that, for an initial edge dislocation Frank–Read source subject to an applied stress no greater than the critical stress, the equilibrium positions are dependent on applied stresses. The results are in good agreement with those given by Foreman [30]. If the applied stress is greater than the critical stress, the annihilation mechanism of Frank–Read source is activated. The entire dislocation evolution of such a case is shown in Figure 4b, exhibiting that the dislocation decelerates, accelerates, and then remains at an almost constant speed during the entire expansion process. For a Frank–Read source with a smaller length, our model can still converge quickly and accurately predict the critical stress, as shown in Figure 4c. The effect of local interaction in a dislocation core is also investigated with the change of η values of the Frank–Read source, showing in Figure 4d that weak interaction yields early evolution and annihilation of the Frank–Read dislocation loop.

Figure 5 demonstrates the relationship between calculated critical stresses and segment numbers. The new model demonstrates excellent stability for various segment sizes. The calculated critical stresses quickly converge to the solution given by Foreman [30], while the results from the classical model heavily depend on the segment size and the artificial ‘manipulations’ during the simulation processes.

5. Conclusions

In our proposed model, we fully take into account the local interaction between dislocation segments to resolve the long-standing paradoxical problem mentioned in the literature [11,21]. As expected, the determinations of equilibrium shape and critical stress are insensitive to the values of drag coefficient tensor B_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.