  Strain fields, dislocations and defects may be used to control electronic
properties of graphene. By using advanced imaging techniques with
high-resolution transmission electron microscopes, we have measured the strain
and rotation fields about dislocations in monolayer graphene with single-atom
sensitivity. These fields differ qualitatively from those given by conventional
linear elasticity. However, atom positions calculated from two dimensional (2D)
discrete elasticity and three dimensional discrete periodized F\"oppl-von
K\'arm\'an equations (dpFvKEs) yield fields close to experiments when
determined by geometric phase analysis. 2D theories produce symmetric fields
whereas those from experiments exhibit asymmetries. Numerical solutions of
dpFvKEs provide strain and rotation fields of dislocation dipoles and pairs
that also exhibit asymmetries and, compared with experiments, may yield
information on out-of-plane displacements of atoms. While discrete theories
need to be solved numerically, analytical formulas for strains and rotation
about dislocations can be obtained from 2D Mindlin's hyperstress theory. These
formulas are very useful for fitting experimental data and provide a template
to ascertain the importance of nonlinear and nonplanar effects. Measuring the
parameters of this theory, we find two characteristic lengths between three and
four times the lattice spacings that control dilatation and rotation about a
dislocation. At larger distances from the dislocation core, the elastic fields
decay to those of conventional elasticity. Our results may be relevant for
strain engineering in graphene and other 2D materials of current interest.
