Energy-dissipation balance of a smooth moving crack

In this paper we provide necessary and sufficient conditions in order to guarantee the energy-dissipation balance of a Mode III crack, growing on a prescribed smooth path. Moreover, we characterize the singularity of the displacement near the crack tip, generalizing the result in [S.Nicaise, A.M.Sandig - $\mathit{\text{J.}}\mathit{\text{Math.}}\mathit{\text{Anal.}}\mathit{\text{Appl.}}$ 2007] valid for straight fractures.