The Effect of Multipoles on the Elasto-Plastic Properties of a Crystal: Theory and Three-Dimensional Dislocation Dynamics Modeling

Dislocations are a type of defect in crystalline materials. They are labeled as “line defects.” In a typical crystal, many dislocations can exist and the dislocation density and structure will have an effect on the physical properties of the material. When two or more dislocations interact closely from same/different slip planes, they are labeled as “multipoles.” Multipoles can be of edge or screw type. Depending on the proximity of the dislocations in the multipoles, interactions among the dislocations can affect the elastoplastic properties [1]. A material can be hardened or softened as a consequence of these interactions.

Stokes and Olsen [2] and Kroupa [3] described the mechanism of dipole formation for both screw and edge dislocations. Gilman [1] and Neumann [4] studied the interaction between dislocations and dislocation dipoles, tripoles, and quadrupoles. In the two-dimensional study by Neumann [4], it was found that decomposition, i.e., splitting of the dipole usually occurs more often than trapping of the approaching dislocations.

To understand dislocation phenomena in the plastic regime, researchers and scientists developed several codes that can simulate the dynamic behavior of dislocation interactions under simple to complex configurations, which are often not captured in real experiments. Kubin et al. [5] introduced a basic framework for three-dimensional (3D) dislocation dynamics simulations and various features were introduced in more recent dislocation dynamics codes. In the current study, the simulations are performed on a dislocation dynamics code developed by Refs. [6–14]. The first work to look at dipole interactions using discrete dislocation dynamics (DDD) simulations is by Ref. [15]. In such work, it showed dynamic zipping and unzipping, i.e., dynamic coupling/de-coupling of dipole dislocations. Such dipole sources were initially remote from one another unlike the initial equilibrium stacking studied herein.

The nature (stable/unstable) of the equilibrium points in a multipole configuration plays a vital role in the elastoplastic behavior. The authors of this article present an analytical/theoretical model for stability analysis for any number of dislocations in a multipole configuration. This is something that was lacking from the literature.

All the analytical/theoretical model cases presented in this article consider the dislocations as infinitely long. Although such studies can provide an understanding of multipole stability, the infinite length of the dislocation preclude the line tension effect, and the interplay between line tension and remote interactions, on elastoplasticity. To address this lacking knowledge, we therefore also show DDD simulation results of staggered Frank–Read (FR) sources [16] (pinned dislocation segments on different slip plane) in a 3D finite volume. These represent multipoles in real-world problems since both the medium and the dislocations are finite. Such simulations capture both the line tension and the interaction effects on a material's elastoplasticity.

Detailed development of the interaction equations among the dislocations in a multipole configuration is shown in the Theory section, and the stability analysis is shown in the Stability of Multipoles section. The simulation setup is discussed in the Simulation Methodology section for the reproducibility of the results. A comprehensive discussion of the simulation results is presented in the Results and Discussion section.

Tripoles can be formed in different configurations. Figure 3 shows two tripole arrangements where the dislocations lie on different slip planes.

For a quadrupole and pentapole configurations, these are provided in the  Appendix (the configuration, the glide force, and the glide force versus x).

Figure 4 shows the glide forces F_x for dipoles, as a function of x, and the equilibrium points (where F_x = 0) [18,19]. For a dipole having same-sign dislocations (Fig. 1), there exists two unstable equilibrium points (at x = ±d) and one stable (middle) equilibrium point at the origin (at x = 0). The two unstable equilibrium points occur when the relative angle between the two dislocations is $45deg$⁠, and the stable equilibrium point occurs when the relative angle between the two dislocations is $0deg$ (i.e., they are lined up vertically).

For the dipole with two oppositely signed dislocations (Fig. 2), the dipole has two stable equilibrium points (at x = ±d) and one unstable (middle) equilibrium point at the origin (at x = 0). The two stable equilibrium points occur when the relative angle between the two dislocations is $45deg$ and the unstable equilibrium point occurs when the relative angle between two dislocations is $0deg$ (i.e., they are lined up vertically).

For any perturbation from the equilibrium points, the glide force is directed toward the stable equilibrium points and away from the unstable equilibrium points. In other words, when any dislocation in a dipole configuration moves a little away from the stable equilibrium position, it feels a force that tends to bring it back to that stable equilibrium position. For an unstable equilibrium point, the opposite phenomenon occurs.

The region in space where a perturbation of the dislocation position tends to return it to an equilibrium point is termed a stable region, whereas a region in space where a perturbation of the dislocation position tends to move it away from an equilibrium point is termed an unstable region. The same-sign dipole has stable and unstable regions (the stable region is |x| < d and the unstable regions are |x| > d) but the opposite sign dipole has a stable region only (the stable region is −∞ < x < +∞). Therefore, the opposite sign dipole should be resistant to dislocation flow on an extended range whereas the same-sign dipole is only resistant to flow on a shorter range, and after that, it would not resist but rather assist in the dislocation flow.

Figure 5 shows the glide forces F_x for tripoles, as a function of x, and the equilibrium points (where F_x = 0). For a tripole having same-sign dislocations (Fig. 3(a)), there exists two unstable equilibrium points and one stable (middle) equilibrium point at the origin. This is true for both the top (or bottom) dislocation and the middle dislocation although both have a relatively smaller region of stability. For Fig. 3(b), there are also three equilibrium points with two being stable but with an infinite stability region. This is true for both the top/bottom and middle dislocations. Note that the force/stability curves for the tripole are more complex than for the dipole. For one reason, there are more of them to consider. Also for the tripole (or higher-order poles), the curves are not always anti-symmetric which requires careful identification of peak or absolute peak values as shown in the  Appendix.

From the stability figures (Fig. 5 and the  Appendix), it is evident that the top or outer dislocation in the multipole configuration is weakly coupled with other dislocations in the multipole and therefore subject to flowing first, i.e., detaching away first from the multipole. In other words, it is the weakest link in the multipole. The middle dislocation on the other hand seems to be more stable. With focus thus on the top/outer dislocation, Figs. 6 and 7 show surface plots based on Eqs. (10) and (13). Each surface point shows the PK force on the top dislocation of the multipole configurations. The force is a function of n and x. The figures show the complex dependence on n and x.

Figure 8 shows the maximum absolute PK force values acting on the top and middle dislocation. These maximum points are evaluated numerically, and each point shows the absolute maximum force irrespective of the x-location of this maximum point on the force curve. To get a curve for a middle dislocation at least three dislocation are required in a multipole. As seen in this figure, all the curves seem to plateau to some limiting value because the dislocation interaction has an inverse relationship to the separation distance between dislocations in the multipole configuration, i.e., the distances between the top dislocation and other dislocations will increase as n increases. Note in this figure that only when n = 2 or n = 3 that the absolute value of the glide force is the same whether one considers a same-sign multipole configuration (all vertically aligned) or a vertical zigzag configuration with alternating dislocation signs. For any other n, the glide forces are not exactly the same. This figure shows that the middle dislocation could be twice as stable (factor of 2) as the top dislocation with this factor depending on n.

The first group of simulations is conducted by changing the separation distance between active slip planes for the same-signed dislocations in the multipole (dipole, tripole, quadrupole, and pentapole) configurations. Initially, all FR sources are placed on different slip planes as they make $0deg$ angle with each other with respect to the vertical axis. This is a stable position for such multipoles, as mentioned in the Stability of Multipoles section.

Then in the second group of simulations, the dislocation sign was alternated between neighboring dislocations while the dislocations themselves are arranged in a vertical zigzag pattern. This is done for different multipoles (dipole, triple, quadrupole, and pentapole). To reverse the dislocation sign one can take opposite Burgers vectors for the neighboring dislocations, i.e., [0 1 0] and $[01¯0]$⁠. Here also, the separation between the dislocations on the different slip planes is varied and the result of this is obtained and studied. In other words, for the zigzag pattern each dislocation is making a $45deg$ angle with respect to the vertical axis with the neighboring dislocations. See Fig. 3(b) and the quadrupole and pentapole figures in the  Appendix.

At the start of each simulation, i.e., at time zero, the glide force F_y experienced by any dislocation, or dislocation segment, in the above multipole configuration can be obtained using Eq. (1). For this simulation setup, F_y = σ_yzb since $e^=[100]$⁠. Of course after the bowing of the FR sources commence, each dislocation segment can experience both glide components of the PK force, i.e., F_x and F_y.

Stress–strain diagrams are generated for each 3D DDD simulation. Although these diagrams were generated for different dipole, tripole, quadrupole, and pentapole configurations, they are only shown here for the pentapole configurations for brevity and as exemplary figures of the results of the DDD simulations. Figures 10 and 11 show the stress–strain responses of the RVE for different slip plane separations (normalized d) for pentapole configurations with the same and oppositely alternating dislocation signs, respectively. All the strain–stress diagrams are compared with a single FR source (labeled as “One FR”) to capture any occurring hardening or softening phenomena associated with the different multipole configurations. The proportional limit (PL) (stress value at the end point of the elastic region) and the flow stress (FS) (averaged stress values over strain after any initial drop in stress beyond the elastic regime) are recorded for each simulation. Note that in Fig. 10, some stress–strain diagrams show a clear drop in stress value right after the elastic regime. This sudden drop is similar to the Yield Point Phenomenon (YPP) observed in low carbon steels, although the initial locking of dislocations here is due to the other dislocations in the multipole, unlike YPP in steels which is due to the solute carbon atoms.

Figures 12–15 show the proportional limit and flow stress for different multipoles as the separation distance d between any two slip planes is varied. The minimum d/b considered is 50. All the diagrams show both hardening and softening effects from the elastic interaction between two or more FR sources having the same sign.

This result shows good agreement with the information in the Theory/Stability of Multipoles sections regarding small source separations. For small separations, the sources bow together, trying to hold each other in equilibrium as they bow (Fig. 16). This raises the proportional limit (i.e., the critical bowing of source(s)) and thus the flow stress above a one source. However, as the distance between the sources increases more and more, the interaction effect of the sources on one another become less and less during bowing. This results in the proportional limit and flow stress eventually dropping to the same level as that of one source. For this kind of interaction, two parameters play an essential role in conjunction with one another: one is the separation distance between any two slip planes, and the other is the PK force due to the self-stress field created by the dislocations, i.e., the line tension of a dislocation. This force, with perturbations in the self-stress of the dislocations, can cause the same-signed dislocation sources to either take longer to critically bow (i.e., a hardening or strengthening effect) or bow critically quickly, i.e., at lowered strain value (i.e., a softening effect as seen in a stress–strain diagram). The reason for this softening effect at very small d values, is that the interaction force is very high and any perturbations in the bowing between the sources can at one point flip a stable position between any two nearby dislocations to an unstable position pushing the dislocations away from each other instead of pulling them to stay together. That will accelerate the production of strain at the same applied load value which is the exact definition of softening.

Another result from the simulations is that for any number n ≥ 3, the DDD simulations show that the outer FR sources bow out critically first before the middle ones. Figure 16 shows a snapshot of a quadruple were the outer sources are starting to take off, i.e., break away from the pack, before the middle ones. This is in line with the stability discussions above regarding infinite-length dislocations that the outer dislocations are the least stable. This figure also shows how all the dislocations in a same-sign multipole initially bend/bow together while trying to maintain their vertical originally stable configuration. This continuous attempt to align the dislocations in a vertical configuration produces resistance to applied loading and hence a hardening effect that reflects itself on the PL results in Figs. 12–15.

Figures 17–20 show different results for the oppositely-signed multipole configurations (in a zigzag pattern) compared with the same-signed multipoles having the same number n (Figs. 12–15). Here also, the minimum d/b considered is 50. For smaller separation distances in the oppositely-signed multipole configurations, both the proportional limit and the flow stress are significantly higher than for a single FR source. It can be as high as a factor of 3. For these multipole configurations, the entire slip plane acts as a basin of attraction for all the Frank–Read sources back to the equilibrium points, as shown in Figs. 4 and 5 (see the  Appendix for higher-order multipoles). Thus one should not expect any softening effect from this configuration because this continuing stability of the configuration and continuous attraction to the equilibrium position hinders plastic flow. Figures 17–20 also show a good agreement with the study hypothesis: as the separation distance increases, the stress–strain response mimics that of a single Frank–Read source response. This is true in the limit of high d values.

It is important to note that in such simulations the ensuing dislocation microstructures can be complex as they evolve with strain. This complexity is a function of the source vertical separation (d/b) and the number of sources (n). To give one example of this complexity, consider Fig. 21. In this figure, a pentapole is shown at one strain value and as can be seen from the differently colored (or grey-shaded in the print version) dislocations/sources, their interaction is complex. This snapshot shows, for example, the formation of multiple dynamic dipoles at that instant of time. All such dynamics dipoles represent temporary pinning of dislocation motion/glide and thus cause an increase to the applied stress value needed to maintain plastic flow.

The peak (maximum) and minimum values for the PL and FS from Figs. 12–15, independent of separation distance d, are plotted in Fig. 22. Figure 22 shows how added dislocations in the multipole contribute to dislocation interactions that results in increased PL/FS with n. Given the prior discussion on the stabilizing effect of dislocations in a multipole, this result is not surprising. The curves in Fig. 22 seem to be reaching a plateau similar to what is seen in Fig. 6.

There are a few things to note about this figure. First, the dislocation/source interactions are contributing to higher FS and PL than a single FR source (last legend in the figure). Indeed, at the highest n (i.e., n = 5), this increase from a single FR source is about 55% or a factor of 1.52 higher to critically bow a pentapole than to critically bow a single FR. A higher factor is possible if n is higher. Note that this increase in PL is due to dislocation interaction between sources and not due to line tension (which is driving the base value for a single FR source). Hence, dislocation interactions are shown here as clearly significant for the elastoplasticity of a single crystal.

Another thing to note about Fig. 22 is that if one divides the increase in PL value from n = 2 to n = 5, the factor here will be 1.33 (16/12). However, from Fig. 8 for the top dislocation (the least stable), the ratio of the breakaway force for n = 5 compared with n = 2 is 1.76 (0.44/0.25). These values indicate the effect of dislocation interaction going from n = 2 to n = 5 and not the effect of line tension. Moreover, the reason the ratio from Fig. 8 is higher is that it considers infinite dislocations. Therefore, it appears that the theoretical analysis done in the Stability of Multipoles section is relevant to the analysis of finite-length sources arranged similarly as multipoles.

The peak (maximum) and minimum values for the PL and FS from Figs. 17–20, independent of separation distance d, are plotted in Fig. 23. Figure 23 shows how added dislocations in the multipole contribute to dislocation interactions boosting the PL/FS over a single FR source. However, unlike Fig. 22 for dislocations of the same sign, Fig. 23 for dislocations of alternating sign shows an oscillating behavior. These oscillations are similar to what is seen in Fig. 7 and should ultimately reach a limiting value with higher n. Notice that the PL and FS maximum values in Fig. 23 are higher than those in Fig. 22. The reason for this was discussed earlier. Note also that the minimum PL/FS values in this figure occur for large source separation d.

An important outcome of these studies is that there were significant changes in the elastoplastic properties of a crystal not from a change in the dislocation density of that crystal but rather from changes in the configuration and relative position of the dislocations in the crystal. This is an important outcome of this work as dislocation density is a major parameter used in continuum plasticity models.

The current study is limited to the hardening/softening of a crystalline material only due to the dislocation interactions in a multipole. Here dislocations are kept equally apart from each other in both same and opposite type multipole configurations. Moreover, the other aspects of material hardening processes (i.e., solid solution hardening and precipitation hardening) are not considered because the objective of this study is to understand the dislocation interactions in a multipole and its hardening or softening effect on the crystalline materials. In the analytical study, dislocations are considered to be infinitely long, so the line tension is neglected. Though the DDD simulations are done with finite dislocation sources considering line tension, the analytical prediction still reflects the results.

Again the unequal separation between dislocations in the multipole may somewhat alter the trend of the current results. Simulations are done for a certain value of mobility constant, and the effect of dislocation mobility is not studied in this research.

In this article, the authors presented a study on dislocation multipoles, and presented analytical stability diagrams/equations and 3D DDD simulation outcomes. The simulations incorporated line tension effects whereas the analytical models did not since they assume infinite length of dislocations. Although the simulations used finite-length dislocation sources, the analytical models were still helpful in understanding the simulations. The simulations showed that both hardening and softening behaviors in the material can occur. One of the main findings of this study is that zigzagged multipoles with alternating dislocation sign only cause hardening. On the contrary, same-sign multipoles can exhibit either hardening or softening behavior depending on the separation distance between slip planes. At a separation distance of 4500b, the sources act the same as a single FR source, i.e., the interaction between them diminishes. Another important finding is that the elastoplastic properties of a crystal are dependent not just on dislocation density but also on the configuration and relative position of the dislocations in the crystal. Yet another finding is that dislocation interactions significantly contribute to the proportional limit and flow stress of a crystal and not just the line tension of the dislocations. In future studies, one could consider the initial source length as a parameter in the multipole configurations.

This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.

Opposite burgers vector: For middle dislocation (third dislocation from bottom)