  Classical elasticity is concerned with bodies that can be modeled as smooth
manifolds endowed with a reference metric that represents local equilibrium
distances between neighboring material elements. The elastic energy associated
with a configuration of a body in classical elasticity is the sum of local
contributions that arise from a discrepancy between the actual metric and the
reference metric. In contrast, the modeling of defects in solids has
traditionally involved extra structure on the material manifold, notably
torsion to quantify the density of dislocations and non-metricity to represent
the density of point defects. We show that all the classical defects can be
described within the framework of classical elasticity using tensor fields that
only assume a metric structure. Specifically, bodies with singular defects can
be viewed as affine manifolds; both disclinations and dislocations are captured
by the monodromy that maps curves that surround the loci of the defects into
affine transformations. Finally, we show that two dimensional defects with
trivial monodromy are purely local in the sense that if we remove from the
manifold a compact set that contains the locus of the defect, the punctured
manifold can be isometrically embedded in Euclidean space.
