Elastic interactions between two-dimensional geometric defects

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.