Emergence and role of dipolar dislocation patterns in discrete and continuum formulations of plasticity

The plasticity transition, at the yield strength of a crystal, typically signifies the tendency of dislocation defects towards relatively unrestricted motion. An isolated dislocation moves in the slip plane with velocity proportional to the shear stress, while dislocation ensembles move towards satisfying emergent collective elastoplastic modes through the long-range interactions. Collective dislocation motions are discussed in terms of the elusively defined back stress. In this paper, we present a stochastic continuum model that is based on a two-dimensional continuum dislocation dynamics theory that clarifies the role of back stress and demonstrates precise agreement with the collective behavior of its discrete counterpart as a function of applied load and with only three essential free parameters. The main ingredients of the continuum theory are the evolution equations of statistically stored and geometrically necessary dislocation densities, which are driven by the long-range internal stress; a stochastic yield stress; and, finally, two local “diffusion”-like terms. The agreement is shown primarily in terms of the patterning characteristics that include the formation of dipolar dislocation walls.

Figure 1

A gradient in the geometrically necessary dislocation density is one form of spatial correlations that may affect local strength (through the back stress ${\tau }_b$) and lead to different yielding thresholds for (a) and (b). The configuration of (b) is stronger (weaker) if $D<0$ ($D>0$). The dashed lines denote spatial discretization, which is necessary to define continuous densities.

Figure 2

Dislocation pattern evolution in DDD simulations. Left column: Dislocation configurations obtained at different strain $\gamma$ values. Middle and right columns: Same-sign and opposite-sign spatial correlation functions ($d_{++}$ and $d_{+-}$, respectively) of discrete dislocations.

Figure 3

Stress fields in a model of a dipolar wall in the SCDD. (a) and (b) Total and GND densities of a dipolar wall placed in a constant dislocation density background. (c) Self-consistent field ${\tau }_{\text{sc}}$ of the dipolar wall. (d) Gradient stresses (${\tau }_b+{\tau }_d$) acting on a positive sign ($s=1$) dislocation. Notice the development of a back stress on the left side of the dipolar wall. The model parameters used are the same as in Fig. 4.

Figure 4

Dislocation pattern evolution in SCDD simulations with $D=A=0.25$, $\alpha =1.0$, and $a=2$. Left two columns: Total and GND density maps obtained at different strain $\gamma$ values. Right two columns: Same-sign and opposite-sign spatial correlation functions ($d_{++}$ and $d_{+-}$, respectively) of dislocation densities.

Figure 5

Correlation functions averaged along the DDW direction $C_{+-}$ for the two applied models: (a) DDD and (b) SCDD. Notice the asymmetry developing upon increasing strain. The simulation parameters are the same as in Figs. 3 and 4.

Figure 6

Dislocation pattern evolution in SCDD simulations for different parameters at $\gamma =1$ and $a=2$. Left two columns: Total and GND density maps obtained at different strain $\gamma$ values. Right two columns: Same-sign and opposite-sign spatial correlation functions ($d_{++}$ and $d_{+-}$, respectively) of dislocation densities.