On the dynamics of shells in the theory of Kirchhoff and Love

The non-linear theory of thin shells is examined from a new point of view that regards the classical Kirchhoff-Love hypotheses as constitutive assumptions reflecting the presence of internal constraints in the material of which the body is formed; the consequent splitting of the stress into an active and a reactive part together with the choice of a response mapping compatible with the assumed internal constraints make the Kirchhoff-Love theory fully consistent with the three-dimensional theory. The restrictions on the constitutive relations due to the presence of the internal constraints are discussed and the materials compatible with the considered constraints are determined. A derivation of the displacement field corresponding to the Kirchhoff-Love assumptions is given and the two-dimensional equations of motion, with the accompanying boundary conditions, are obtained by integrating over the thickness the conditions for a stationary value of the three-dimensional Hamiltonian functional associated with the motion of the shell.