# 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.

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.

# Grain boundaries

Many important aspects of the macroscopic material behavior in metals are controlled by the microlevel grain structure. Strength and ductility depends largely on the size and distribution of the grains. Especially the presence of grain boundaries plays an important role, not least in the macroscopic deformation hardening of the material. Grain boundaries pose obstacles to slip deformation by preventing dislocation motion, resulting in localized dislocation storage and heterogeneous deformation fields within the grains. This is closely related to the classical Hall-Petch relation, where the macroscopic flow stress is porportional to the inverse of the square root of the average grain size. More fine-grained materials will thus exhibit a higher flow stress than more coarse-grained materials.

The Hall-Petch effect will, however, tend to reverse as very small grains are considered. At a grain sizes of approximately 10 nm or below, other mechanisms such as grain boundary sliding usually cause a softening effect instead of the common Hall-Petch hardening.

Grain boundaries also contribute to the material behavior in many other aspects. They are often sites for initiation of defects such as cracks, they influence the material's susceptibility to chemical attack such as through corrosion and they influence the electrical conductivity of the material. In addition, migrating grain boundaries tend to interact with second-phase particles and the presence and arrangement of grain boundaries influence both the electrical conductivity and the diffusion properties of the material.