Dislocation and grain boundary interaction

Mesoscale models of microstructure evolution allow studying of heterogeneous dislocation density distributions and related gradient effects. This is closely connected to the well-renowned Hall-Petch effect, stating a proportionality between the yield stress \sigma_{\mathrm{y}} and the inverse of the square-root of the average grain size d according to

\sigma_{\mathrm{y}}\propto\frac{1}{\sqrt{d}}

The interaction between dislocation motion and grain boundaries can be modeled by different approaches. Some examples are given below.

In Hallberg and Ristinmaa (2013) (also discussed in this conference presentation), a hybrid finite difference/cellular automaton model is established where dislocation density gradient are modeled in a reaction-diffusion system. This approach results in the expected Hall-Petch behavior of the macroscopic flow stress – without including an explicit dependence on the grain size – in addition to providing a physically sound dislocation density distribution, indicated in the figure below.

Simulated distribution of total dislocation density in an artificial polycrystal. Due to the reaction-diffusion modeling of dislocation density evolution and the presence of grain boundaries, dislocation pile-ups are formed along the grain boundaries and particularly at triple junctions.
Simulated distribution of total dislocation density in an artificial polycrystal. Due to the reaction-diffusion modeling of dislocation density evolution and the presence of grain boundaries, dislocation pile-ups are formed along the grain boundaries and particularly at triple junctions.

By this modeling approach, the dislocation density will be concentrated at grain boundaries and particularly at triple junctions, directly providing the sites for nucleation of recrystallization. This is in contrast to the common modeling approach where nuclei are placed manually at appropriate sites in the microstructure.

Another approach to modeling of dislocation and grain boundary interaction is taken in Hallberg (2013) where a level set formulation is used to model polycrystal grain structures.

Dislocation density distribution influenced by the presence of grain boundaries. The microstructure model is based on level sets, representing the individual grains. The extent of dislocation accumulation at grain boundaries can be controlled by the parameter w, as illustrated in images a-d.
Dislocation density distribution influenced by the presence of grain boundaries. The microstructure model is based on level sets, representing the individual grains. The extent of dislocation accumulation at grain boundaries can be controlled by the parameter w, as illustrated in images a-d.