In this talk, I present a variational characterization of mechanical equilibrium in the planar strain regime for elastic bodies in the presence of topological defects, such as dislocations and disclinations. By introducing an appropriate core–radius regularization of the underlying singular problem and considering non–simply connected domains, I show that the equilibrium equations can be reformulated as a well-posed minimization problem for the Airy stress potential, in which the relevant (in)compatibility conditions follow from the minimality condition.
I then demonstrate that this minimization problem can be effectively reduced to a finite-dimensional optimization problem involving cell formulas that, perhaps surprisingly, depend only on the geometry and topology of the domain, and not on the elastic material parameters or on the specific nature of the defects. A key practical consequence of this result is the emergence of a clear and efficient computational strategy for evaluating strain and stress fields for arbitrary defect configurations and crystalline materials, once the domain geometry and topology are fixed.
In the final part of the talk, I present several numerical simulations of interacting edge dislocations and wedge disclinations. The computations are carried out using a non-conforming finite element formulation for the numerical solution of fourth-order elliptic boundary value problems.