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Plastic deformation of crystals generally proceeds by the propagation of dislocations along the slip planes. In most brittle crystals and some metals, it is the changes in misfit energy as the dislocation moves that are the predominant obstacle to dislocation motion. These arise from atoms in the crystal lattice being displaced from their equilibrium positions, so this effect is known as lattice resistance.
Why do dislocations ever form? Deformation can be thought of a kinetic process since dislocation motion (the most common cause of plastic flow) requires the breaking and reforming of bonds, like in a chemical reaction.
In this TLP we will be setting up an atomistic model, rather than using continuum elasticity as in the Peierls Nabarro model, where it is assumed that in-plane strains do not change as the dislocation moves.
It can be shown that despite the general form of the expression it is the same as that derived by Peierls, but with a slightly lower magnitude.
The misfit energy is made up of in-plane strains and misalignment strains (across the slip plane). This energy changes as the dislocation moves. The misalignment energy increases as the dislocation width increases, so act to localise the misfit strains. Whereas the in-plane strain energy decreases as dislocation width increases, so acts to spread the misfit strain over a larger region. The final arrangement of atoms results in a minimisation of the overall misfit energy for a given dislocation width.
Considering the energy changes as the dislocation moves allows us to calculate the Peierls stress, required to move the dislocation, and see that it is exponentially dependent on the dislocation width. The Peierls stress depends on the atomic spacing both normal to the slip plane and parallel to it – in fact it is extremely sensitive to the ratio of lattice parameters b/d.