Link Statistics of Dislocation Network during Strain Hardening

TL;DR

This study reveals that dislocation link lengths on individual slip systems in copper during strain hardening follow distinct exponential forms: active-slip links display a double-exponential distribution with a stress-driven long tail, while inactive-slip links are well described by a single exponential. By analyzing Discrete Dislocation Dynamics data across 118 loading directions, the authors link the tail strength to slip-system activity and loading orientation, illustrating bowing-out of long links under stress. They introduce a generalized Poisson process with a length-dependent growth term, showing that constant growth yields single-exponential behavior, whereas super-linear growth beyond a threshold reproduces the observed double-exponential form. The findings provide mechanistic insight into dislocation microstructure evolution and offer a pathway toward physics-based crystal plasticity models that connect link-length statistics to slip rates and macroscopic behavior.

Abstract

Dislocations are line defects in crystals that multiply and self-organize into a complex network during strain hardening. The length of dislocation links, connecting neighboring nodes within this network, contains crucial information about the evolving dislocation microstructure. By analyzing data from Discrete Dislocation Dynamics (DDD) simulations in face-centered cubic (fcc) Cu, we characterize the statistical distribution of link lengths of dislocation networks during strain hardening on individual slip systems. Our analysis reveals that link lengths on active slip systems follow a double-exponential distribution, while those on inactive slip systems conform to a single-exponential distribution. The distinctive long tail observed in the double-exponential distribution is attributed to the stress-induced bowing out of long links on active slip systems, a feature that disappears upon removal of the applied stress. We further demonstrate that both observed link length distributions can be explained by extending a one-dimensional Poisson process to include different growth functions. Specifically, the double-exponential distribution emerges when the growth rate for links exceeding a critical length becomes super-linear, which aligns with the physical phenomenon of long links bowing out under stress. This work advances our understanding of dislocation microstructure evolution during strain hardening and elucidates the underlying physical mechanisms governing its formation.

Figures (11)

• Figure 1: Schematic illustration for accumulating the link length statistics based on the loading direction $[0.03,\ 0.05,\ 0.99]$. (a) The resolved shear stress-strain curve. The strain region $\gamma_d\in[0.9,\ 1.05]\%$, over which the link length statistics are averaged, are shown in the shaded area. The dislocation structures corresponding to the start and end of the averaging window are shown in (b) and (c), respectively. The dislocation lines are colored based on their character angle (red for screw and blue for edge orientations).
• Figure 2: The stereographic triangle with symbols corresponding to loading orientations. Circles correspond to the loading orientations of the 118 DDD simulations used in this analysis. Other symbols correspond to reference orientations used to define the loading orientations for the DDD simulations. The four shaded regions correspond to areas where typical behaviors are reported in Sections 3.1 to 3.4.
• Figure 3: Link length distribution of dislocations on individual slip systems corresponding to a DDD simulation along $[0.03,\,0.05,\,0.99]$ loading orientation, averaged over the strain window of $\gamma_d \in 0.9-1.05\%$. For active slip systems, the red line represents the best fit to the DDD data using two exponentials as shown in Eq. (\ref{['eq:2Expo_i']}), the green line indicates the fit corresponding to the first term in the double-exponential distribution; For inactivate slip systems, the green line is the single exponential function shown in Eq. (\ref{['eq:1Expo_i']}).
• Figure 4: Link length distribution of dislocations on individual slip systems corresponding to a relaxed configuration taken at $\gamma_d=0.9\%$ under $[0.03,\,0.05,\,0.99]$ loading orientation. Single exponential is shown by green lines. The fitting coefficient of the single exponential is imposed to be 1.
• Figure 5: Link length distribution of dislocations on individual slip systems corresponding to a DDD simulation along $[0.00,\,0.65,\,0.76]$ loading orientation.