We obtain the structure theorem for a differentiable shape functional defined in a domain including curved cracks. The theorem is an extension of the structure theorem given eg. in [15] in the case of smooth domains. Four examples of applications of our result are given for shape functionals defined for elliptic boundary value problems: to nonlinear elliptic boundary value problems, to optimal control problems and to the shape differentiability of the first eigenvalue of the Laplacian.