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 o nly approximations occur in the conservation of energy and in the consequen t constitutive relations, which include expressions for the shell velocity and spin. No thickness expansions or kinematic hypotheses are made. The int roduction of a dynamic mixed-energy density avoids ill-conditioning associa ted with near inextensional bending or negligible rotational momentum. It i s shown that a variable scalar rotary inertia coefficient exists that minim izes the difference between the exact kinetic-energy density and that deliv ered by shell theory. Finally, it is shown how specialization of the dynami c mixed-energy density provides a simple and logical way to introduce a con stitutive form of the Kirchhoff hypothesis, thus avoiding certain unnecessa ry constraints (such as no thickness changes) imposed by the classical kine matic Kirchoff hypothesis.