Dislocation patterning in a two-dimensional continuum theory of dislocations

Understanding the spontaneous emergence of dislocation patterns during plastic deformation is a long standing challenge in dislocation theory. During the past decades several phenomenological continuum models of dislocation patterning were proposed, but few of them (if any) are derived from microscopic considerations through systematic and controlled averaging procedures. In this paper we present a two-dimensional continuum theory that is obtained by systematic averaging of the equations of motion of discrete dislocations. It is shown that in the evolution equations of the dislocation densities diffusionlike terms neglected in earlier considerations play a crucial role in the length scale selection of the dislocation density fluctuations. It is also shown that the formulated continuum theory can be derived from an averaged energy functional using the framework of phase field theories. However, in order to account for the flow stress one has in that case to introduce a nontrivial dislocation mobility function, which proves to be crucial for the instability leading to patterning.

Figure 1

The $M\left(x\right)$ mobility function.

Figure 2

The ${\lambda }_{+}(k_x,k_y)$ function at $A=1, D=1$, and $\beta =-1$. The function is positive within the region marked by the contour line ${\lambda }_{+}(k_x,k_y)=0$.

Figure 3

The ${\lambda }_{+}(k_x,0)$ function at $A=1, D=1$, and $\beta =-1$.