The problem of equilibrium of a shallow Timoshenko-type shell containing a through-thickness crack

We study the nonlinear problem of the equilibrium of an elastic shallow Timoshenko-type shell containing a through-thickness crack. Boundary conditions in the form of inequalities are imposed on the curve defining the crack. We establish the unique solvability of the variational statement of the nonlinear problem of the equilibrium of the shell. We prove that, for sufficient smoothness of the solution, the initial variational statement is equivalent to the differential formulation of the problem. We deduce boundary conditions on the inner boundary that describes the crack. In the case of the zero disclosure of the crack, we prove the local infinite differentiability of the solution function with additional assumptions on the functions defining the curvatures of the shell and the external load.