  In this paper, we introduce a methodology applicable to a wide range of
localized two-dimensional sources of stress. This methodology is based on a
geometric formulation of elasticity. Localized sources of stress are viewed as
singular defects---point charges of the curvature associated with a reference
metric. The stress field in the presence of defects can be solved using a
scalar stress function that generalizes the classical Airy stress function to
the case of materials with nontrivial geometry. This approach allows the
calculation of interaction energies between various types of defects. We apply
our methodology to two physical systems: shear-induced failure of amorphous
materials and the mechanical interaction between contracting cells.
