Comparison of Nonsingular Dislocation Theory with Results of Numerically Smeared Cores

Abstract

Analytically smearing of the dislocation core to remove the singularity inherent in theory of infinitely thin dislocations results in the errors of the same order of magnitude in those that arise when the dislocation core is smeared numerically.

Methodology

Dependence of Error on Core Size Parameter

Both analytical and numerical smearing of the core yield an error in the stress field outside of the physical core region. These errors have the same order of magnitude. However, the error in the nonsingular theory is systematically larger than the error in the numerical solution.
The analytical and numerical stress fields have comparable errors. Here the physical core radius is fixed at 4 and the grid resolution is fixed at 0.25.

Increasing Error when Physical Core and Core Size Paramter are Equal

The choice of taking the physical core size to be the same as the core size parameter leads to an error in the stress fields around the dislocation core that increases with decreasing core size. This effect is present in both the analytical and numerical solutions.
Error shows a roughly inverse dependence on the numerical core radius. Here the grid resolution is fixed at 0.25.

Comparison of Difference Between Nonsingular Theory and Numerical Solution

The difference between the nonsingular theory is nontrivial but decreases as we shrink both the numerical core radius and the core radius parameter in the nonsingular theory. It appears that using a smaller core radius parameter in the nonsingular theory decreases the difference between the nonsingular theory and the numerical solutions.
Comparison of the difference between the stress fields in the nonsingular theory and the numerically computed stress fields. Here the physical core radius is fixed at 4.

Summary/Conclusions

At distances near the physical core size, any theory of dislocations that distributes the core over a finite region that is on the order of the physical core size leads to significant deviations from the stress fields for infinitely thin dislocation lines. For the nonsingular theory developed by Cai et. al., the deviations have roughly the same scaling and same order of magnitude as the errors in the numerical solution.

Any dislocation dynamics simulations that removes the singularity by smearing the dislocation core to a width on the order of the "physical core size" will show the same problems as our level set based simulation.

References