We study the elastic fields in a planar sheet that includes a circular hole, an elliptical hole or a finite crack at the vicinity of an isolated topological defect (e.g. a conical disclination). The problem of a crack-defect forms a representative model for the interaction between cracks and other mechanical objects, and it uncover the significant complexity inherent to such problems.
Our analysis combines two distinct methods: (i) The traditional method of stress functions expressed in terms of Muskhelishvili’s formalism of complex analysis (ii) A geometrical approach to nonlinear elasticity expressed in terms of elastic charges. The resulting formalism provides new insights to complex problems in cracks mechanics, as well as new computational tools for calculating the nonlinear fields and interactions between cracks and defects.