  We present a field formulation for defects that draws from the classical
representation of the cores as force dipoles. We write these dipoles as
singular distributions. Exploiting the key insight that the variational setting
is the only appropriate one for the theory of distributions, we arrive at
universally applicable weak forms for defects in nonlinear elasticity.
Remarkably, the standard, Galerkin finite element method yields numerical
solutions for the elastic fields of defects that, when parameterized suitably,
match very well with classical, linearized elasticity solutions. The true
potential of our approach, however, lies in its easy extension to generate
solutions to elastic fields of defects in the regime of nonlinear elasticity,
and even more notably for Toupin's theory of gradient elasticity at finite
strains(Arch. Rat. Mech. Anal., 11, 385, 1962). In computing these solutions we
adopt recent numerical work on an isogeometric analytic framework that enabled
the first three-dimensional solutions to general boundary value problems of
Toupin's theory (Rudraraju et al. Comp. Meth. App. Mech. Engr., 278, 705,
2014). We first present exhaustive solutions to point defects, edge and screw
dislocations, and a study on the energetics of interacting dislocations. Then,
to demonstrate the generality and potential of our treatment, we apply it to
other complex dislocation configurations, including loops and low-angle grain
boundaries.
