Inhomogeneous Systems and the Role of Locality

Perhaps the most elementary application of this idea that we have used was in writing the total elastic strain energy of a system as an integral of the form 

As noted above the spirit of locality is that the energy is reckoned point by point with an energy density of the form W(e) = CijiciCijeia. We note that the domain of validity of a locality assumption of this kind is based upon measuring the wavelength of the relevant elastic distortions against the lattice parameter. As noted 

Like with its linear elastic counterpart, the stored energy density W(F) at a given point depends only upon the deformation gradient F at that point. 

In considering the energetics of extended defects, we have repeatedly resorted to locality assumptions as well. In particular, in the context of dislocations we have invoked the line tension approximation to assign an energy of configuration to a dislocation of the form 

Whereas the line tension was invoked as a way to capture the self-energy of dislocations from an elastic perspective, there are also ways of capturing core effects on the basis of locality assumptions. Recall that in our treatment of dislocation cores we introduced the Peierls-Nabarro model (see section 8.6.2) in which the misfit energy associated with slip displacements across the slip plane is associated with an energy penalty of the form 

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