|Figure 1. A dislocation dynamics simulation of strain hardening in bcc molybdenum shows a fragment of the dislocation network. Binary junctions connect the mass of dislocation lines (grey). Multinodes (red) complicate the topology by forming a strong skeleton throughout the network.|
Since the invention of the transmission electron microscope (TEM), materials scientists and physicists have investigated the structure of dislocation networks in crystalline materials to understand the origins of their unique mechanical strength. Dislocations—the carriers of plastic deformation in crystalline materials—often increase in density by several orders of magnitude during the deformation process. The interaction among dislocations and the topological rearrangements that occur in dislocation collisions are believed to be responsible for strain hardening, a remarkable property of metals in which a material’s strength increases as deformation increases.
Two colliding dislocations may partially merge, or zip, into a junction bounded by two nodes. The behavior of dislocation nodes (where three or more dislocation lines connect) and the constraints they place on the motion of their lines are believed to strongly influence the heterogeneous microstructures that develop with large deformations. Unfortunately, the dislocations in these microstructures become so severely entangled that they can no longer be individually distinguished by TEM. With the development of the Parallel Dislocation Simulator (ParaDiS) code at LLNL, the elements of these highly entangled dislocation microstructures can now be investigated, providing a means for in situ computational microscopy.
Lawrence Livermore’s stockpile stewardship mission requires an understanding of the thermo-mechanical behavior of metals that are subject to extreme loading conditions and high rates of strain. Under these conditions, dislocations behave in ways that experiments are unable to assess and the existing theory is unable to foresee. ParaDiS allows us to investigate these behaviors in microscopic detail to understand how the dislocation microstructure forms and how it affects the mechanical behavior of the material. This new understanding is used to develop predictive models of materials performance under extreme deformation conditions.
|Figure 2. Stress–strain curves from a simulation on bcc molybdenum show that the formation of multijunctions significantly increases the hardening rate (line slope). Multijunctions that form in large numbers result in a well-defined slope (black) and are contrasted with the nearly flat lines when only few (red) or no (green) multijunctions are formed. (Simulation image courtesy of M. Tang.)|
The classic dislocation theory is an elegant approach resulting in useful analytical solutions for the energy and stress associated with dislocations. However, the solutions are singular and undefined on the dislocations themselves, making numerical calculations difficult. In previous attempts to develop a non-singular theory, either the theory was too complicated to be useful for a numerical implementation or it was simple but mathematically inconsistent. In response, we have developed ParaDiS, a massively parallel dislocation dynamics code that can follow large numbers of dislocation dynamics for a long enough time.
ParaDiS incorporates a new, non-singular continuum theory of dislocations
that is both mathematically rigorous and analytically simple. In one of its
early applications, the new theory provided an accurate description of the
physics of dislocation nodes.
The new theory was instrumental in our recent discovery of multijunctions and
multinodes that resulted from collisions of three or more dislocation lines.
Multijunctions are significant in that they tie together three and, possibly,
more lines into very tight knots that serve as strong anchors for the whole
network. A multinode connects four (possibly more) dislocation lines together.
Figure 1 shows a fragment of the dislocation network created in a very large
dislocation dynamics simulation. We found that this ternary interaction in
body-centered-cubic (bcc) metals is three to four times stronger than any binary
interaction in the same metal and strong enough to pin dislocation lines indefinitely.
|Figure 3. (a) An atomistic simulation based on an interatomic potential for molybdenum shows the formation of a multijunction. (b) A transmission electron microscopy image shows a symmetric 4-node (numbered 1–4) and verifies the existence of multijunctions and multinodes.|
Because the multijunctions in dislocation networks are nearly indestructible, their existence is predicted to greatly affect the mechanical strength of metals. In particular, the existence of this element of dislocation microstructure offers an explanation to the unexplained large directional variation of strength observed in bcc single crystals.
In a series of large-scale dislocation dynamics simulations, we observed that the rate at which bcc metals harden during plastic straining is defined, to a large extent, by the presence or absence of the multijunctions (Figure 2). The existence of multijunctions was later confirmed by atomistic calculations and ultimately by experiments. Figure 3 shows an unmistakable signature of a multinode observed using TEM in a bcc single crystal of molybdenum deformed to 1 percent strain. The newly explained large variation in strength results in the propensity of many bcc alloys to localize their deformation in bands and affects their ultimate failure.
V. Bulatov, et al., “Scalable Line Dynamics in ParaDiS,” (pdf) Supercomputing 2004, Pittsburg, PA
W. Cai, et al., “A Non-Singular Continuum Theory of Dislocations,” J. Mech. Phys. Solids, in press.