# Forest hardening

It has been mentioned that two slip systems are activated during stage ll of the deformation of a single crystal. However, the plastic flow of the crystal is mainly governed by the primary slip system (i.e. it is an active slip system), where the primary dislocations can move freely. The dislocations in the other slip system, however, are immobile and are termed the **forest dislocations**.

Forest hardening is the dominant mechanism in stage ll of the single crystal deformation. The active dislocations gliding in the primary slip plane get stuck at obstacles when they intersect with the forest dislocations. These obstacles, or pinning points, are either jogs (when dislocations intersect each other) or Lomer locks (when dislocations react together). During stage ll, the number of fixed obstacles will increase as more Frank-Read sources are operated, which leads to an increase in the number of forest dislocations.

## Deriving an expression for the constant hardening rate

It has been mentioned that the hardening rate of stage ll is constant, with a typical value of G/200. We now aim to derive an expression for this hardening rate by considering the movement of an active dislocation in the primary slip system. Consider a segment of dislocation of length l pinned at both ends. The mean free path of the dislocation segment is λ. The change in the dislocation density with respect to the strain is:

\[ \frac{{{\rm{d}}\rho }}{{{\rm{d}}\gamma }} = \frac{{{\rm{d}}l}}{{b{\rm{d}}a}} \]where dl is the change in the line length of the dislocation segment and da is the area swept by the dislocation as it moves.

From Figure 10, we have dl = l and da = λl, hence:

\[ \frac{{{\rm{d}}\rho }}{{{\rm{d}}\gamma }} = \frac{1}{{b\lambda }} \]Differentiate the Taylor equation:

\[ \frac{{{\rm{d}}\tau }}{{{\rm{d}}\gamma }} = \frac{1}{2}\;\alpha \;b\;G\;{\rho ^{ - \frac{1}{2}}}\;\frac{{{\rm{d}}\rho }}{{{\rm{d}}\gamma }} \] \[ \frac{{{\rm{d}}\tau }}{{{\rm{d}}\gamma }} = \frac{1}{2}\;\alpha \;b\;G\;{\rho ^{ - \frac{1}{2}}}\;\frac{1}{{b\lambda }} \] \[ \frac{{{\rm{d}}\tau }}{{{\rm{d}}\gamma }} = \frac{{\alpha \;\;G}}{{2\;\lambda \;\sqrt \rho }} \]The mean free path λ is usually a small multiple, of order 10, of the mean dislocation spacing \( \frac{1}{{\sqrt \rho }} \) and so it follows that the expression above gives a reasonable value for the hardening rate of about G/200.

## Dislocation dynamics

Dislocation dynamics aims to simulate the dynamic, collective behavior of individual dislocations and their interactions. The following video uses dislocation dynamics to simulate the behavior of dislocations when an fcc single crystal is deformed. Although it is a single crystal, it does not exhibit any easy glide and multiple slips are initiated at the start.