Cauchy elasticity with dislocations in the small strain assumption

In this letter it is shown how the singularities created by dislocations in an elastic body must not preclude from a linear approach. Cauchy elasticity is considered and hence no variational approach is needed and the displacement field only appears in a second step, as an interpretation of the one of the two model variables, arisen from a Beltrami decomposition of the strain tensor. The second model variable obeys nonclassical PDEs relying on the incompatibility operator, and whose well posedness is here addressed.