To understand the motion of dislocations, one needs to understand the mechanics of strain. Suppose one has a volume V with boundary surface A deep inside a material. If the content exerts a force per unit volume on the boundary A, then the total force for each component from will equal . If the volume V is static, one must apply an qual and opposite force on boundary A. Let this opposite force per unit area have components . Then the total force applied to the surface is
where is the normal to the surface at the point of integration. It must be equal to the total force in the volume and by Green’s integral theorem, one can replace the circle integral with a volume integral giving
Now consider the work done by deforming the volume during a thermodynamically reversible process. If the displacement is , then the work done is
by partial integration. If the deformation is local, the first boundary term will vanish as the surface is taken large/far enough. Now the variation is essentially a differentiation in an undetermined parameter so we assume it can be interchanged with the differentiation . Instead of summing over , one can sum over and take halve its value, so
Introducing the stress tensor (see next paragraph for a motivation), one arrives at . From thermodynamics,
with F the free energy. Physically, it is clear that if there is no stress, there is no force per area needed to counter the stress. On the other hand, if the stress is big, the stress should also be big. Hence one can expand
From the discussion above,
therefore . One can assume Hooke`s law, making the force dependent on stress or deviation from equilibrium upto linear order. Then . Inserting the definition of the stress tensor back in the equation, combining with
and adopting the Einsitein summation convention in order to drop the sum signs, one obtains
If is constant, we have the static case and we can solve the poisson equation. If we want dynamics, we set the force per unit volume equal to with the mass density giving the equivalent of newton’s equation of motion. The resulting equation is a wave equation.
Now a short explanation about the stress tensor. Suppose that without strain/no deformation, we can define a coordinate system . After applying strain, the material is deformed and the coordinate axes shift so that for any position . Define . Then for any infinitesimal displacement in the old coordinates, the new displacement is
and if the total distance used to be , the new distance will be
Now we can play the same trick with the summation again summing
The stress tensor is thus the first-order correction in the length of a line between two points in the material after deformation.