Rotary inertia in the classical nonlinear theory of shells and the constitutive (non-kinematic) Kirchhoff hypothesis

A general nonlinear theory of isothermal shells is presented in which the only approximations occur in the conservation of energy and in the consequent constitutive relations, which include expressions for the shell velocity and spin. No thickness expansions or kinematic hypotheses are made. The introduction of a dynamic mixed-energy density avoids ill-conditioning associated with near inextensional bending or negligible rotational momentum. It is shown that a variable scalar rotary inertia coefficient exists that minimizes the difference between the exact kinetic-energy density and that delivered by shell theory. Finally, it is shown how specialization of the dynamic mixed-energy density provides a simple and logical way to introduce a constitutive form of the Kirchhoff hypothesis, thus avoiding certain unnecessary constraints (such as no thickness changes) imposed by the classical kinematic Kirchoff hypothesis.