A continuum model for dislocation dynamics in three dimensions using the dislocation density potential functions and its application in understanding the micro-pillar size effect

In this paper, we present a dislocation-density-based three-dimensional continuum model, where the dislocation substructures are represented by pairs of dislocation density potential functions (DDPFs), denoted by $\phi$ and $\psi$. The slip plane distribution is characterized by the contour surfaces of $\psi$, while the distribution of dislocation curves on each slip plane is identified by the contour curves of $\phi$ which represents the plastic slip on the slip plane. By using DDPFs, we can explicitly write down an evolution equation system, which is shown consistent with the underlying discrete dislocation dynamics. The system includes i) A constitutive stress rule, which describes how the total stress field is determined in the presence of dislocation networks and applied loads; ii) A plastic flow rule, which describes how dislocation ensembles evolve. The proposed continuum model is validated through comparisons with discrete dislocation dynamics simulation results and experimental data. As an application of the proposed model, the "smaller-being-stronger" size effect observed in single-crystal micro-pillars is studied. A scaling law for the pillar flow stress $\sigma_{\text{flow}}$ against its (non-dimensionalized) size $D$ is derived to be $\sigma_{\text{flow}}\sim\log(D)/D$.