On a continuum theory of dislocation equilibrium

A continuum theory of dislocations is suggested which is capable of predicting the equilibrium distributions of a large number of screw dislocations in anisotropic beams with arbitrary cross-section. The theory leads to a boundary value problem with unknown boundary. The problem is solved for isotropic beams with circular cross-sections and thin rectangular cross-sections. The solution for circular cross-sections is compared with the results of numerical simulations by [Weinberger, C. R., 2011. The structure and energetics of, and the plasticity caused by, Eshelby dislocations, International Journal of Plasticity, 27, 1391–1408]. An extension of the theory to equilibrium of edge dislocations is discussed.