In the previous sections, we studied the CADD3d method with a single dislocation, and validated it with two different, yet simple, dislocation shapes, the straight line and the loop. In this section, we will introduce more dislocations, in order to stress the coupling with regard to dislocations interaction. Consequently, a hybrid dislocation loop is first inserted at the center of the domain as shown in Figure 7.9. Then two fixed ellipsoidal discrete dislocation loops are set to surround the hybrid dislocation. Because the Burgers vectors of the fixed loops are identical to the one of the hybrid dislocation loop, all the loops repel each other. The interaction force between the DDD nodes is computed by DDD engine while the interaction between atoms and the fixed loops are enforced thanks to CADD3d. More precisely, when the displacement field of the pad atoms is evaluated, the contribution of the fixed DDD loops will be considered through the linear elasticity displacement field. Such a pad configuration will generate repelling forces acting on the MD part of the hybrid dislocation loop.
A shear loading of 400MPa is applied to drive the hybrid dislocation motion. The two DDD end nodes are connected by using a straight line since this simpler strategy was shown to be accurate enough in sub-Section 7.2.2. We synchronize the coupling boundary conditions every 50 steps (tupd at e= 50Δt) with the the core templates (cut-off radius rc= 10|b|).
Figure 7.8 (d) shows simulation snapshots at 0ps, 1ps, 4.5ps and 5.5ps from a top view of the slip plane. As the hybrid dislocation expands and approaches the fixed DDD loops, the repelling forces increase and the hybrid dislocation completely stops in front of the fixed loops.
The displacements and velocities as a function of time are shown in Figure 7.10. We find that the two sub-parts of the hybrid dislocation travel as one single dislocation. Even the velocity
7.4. Contraction of hybrid dislocation loop
time (ps) time (ps)
1 2 3 4 5 6 7 1 2 3 4 5 6 7
connection with
MD detection a straight line
Multiple connection with
dislocation loops
Figure 7.10 – Displacement and velocity evolutions of the dislocation loops with the three different simulation setting. The first two problems are the hybrid dislocation loops with the two different connection strategies. The third problem is the hybrid dislocation loop with the two fixed DDD networks.
profiles are similar which proves the quality of the CADD3d approach. The CADD3d method was shown to be valid also in the case where multiple dislocation interactions are considered.
Contraction of hybrid dislocation loop
The last test problem considers a hybrid dislocation loop under the shear loading 70MPa, which is smaller than the stress value maintaining static dislocation loop (370MPa). Such shear loading lets the hybrid dislocation loop contract under the action of self stresses. Fig-ure 7.11 shows a schematic view of the expected hybrid dislocation loop dynamics. As the loop contracts, we observe that the curvature increases and the distance between the two core templates (shown with the red hatching) decreases. As discussed in Section 5.6, this situation does not fulfill the condition in which the core template approximation is applicable.
Figure 7.12 shows the displacement evolution of MD and DDD parts of the hybrid dislocation loop along the x= −z axis. At the beginning of the simulation, the two sub-parts contract with the same speed, but they behave differently when the loop curvature increases.
This test-problem result shows that, to provide a correct CADD3d-coupling prediction the assumption of low curvature must be satisfied.
Summary
In this chapter, we presented various 3d dislocation dynamic simulations using the proposed CADD3d method. The first considered problem introduced a hybrid straight dis-location, made partly with atoms and partly with discrete dislocation dynamics segments.
The simulation was demonstrated to accurately model the dynamical evolution of a single
Chapter 7. Validation of CADD3d: Hybrid Dislocations
x y
z b
Δt
b
Figure 7.11 – Contraction of the hybrid dislocation loop. As the loop contracts, the curvature increases, and the spacing between the two core templates reduces.
time (ps)
displacemen t(
˚ A)
MD DDD
Figure 7.12 – Displacement evolution of the MD and DDD parts of the dislocation loop. As the loop contracts, the difference in behavior between the two sub-parts of the loop increases with time.
7.5. Summary
hybrid straight dislocation such that it behaves as one single dislocation. The sensitiveness of the coupling scheme has been analyzed with respect to three of the coupling parameters (core templates, mobility law and update step), and we showed that they have to be adjusted correctly in order to achieve accuracy. The second elected problem involved a hybrid dislo-cation loop. If compared to the first problem, a major difficulty arose from the geometry of the domain: the core structure needed at the coupling interface can be of multiple arbitrary character angles. Nevertheless CADD3d successfully preserved the curved dislocation loop.
Furthermore, one have shown that the coupling also works as expected in the case where dif-ferent dislocation loops interact through the coupling. Lastly, we have analyzed the limitation of the core template application by studying the contraction of the hybrid dislocation loop with a small shear loading. In consequence, we can conclude that the proposed CADD3d method is a valid approach for coupling MD with DDD domains, which contain dislocations with curved shapes and low curvatures in FCC aluminum.