On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation

In recent years, considerable attention has been given to the development of higher order plate and shell models. These models are able to approximately represent three-dimensional e€ects, while pertaining the eciency of a twodimensional formulation due to pre-integration of the structural sti€ness matrix across the thickness. Especially, the possibility to use unmodi®ed, complete three-dimensional material laws within shell analysis has been a major motivation for the development of such models. While the theoretical and numerical formulation of so-called 7-parameter shell models, including a thickness stretch of the shell, has been discussed in numerous papers, no thorough investigation of the physical signi®cance of the additional kinematic and static variables, coming along with the extension into three dimensions, is known to the authors. However, realization of the mechanical meaning of these quantities is decisive for both a proper modeling of shell structures, e.g. concerning loading and kinematic boundary conditions, and a correct interpretation of the results. In the present paper, the signi®cance of kinematic and static variables, appearing in a 7-parameter model proposed by Buchter and Ramm (1992a) are discussed. It is shown, how these quantities Ôre®neÕ the model behavior and how they can be related to the ÔclassicalÕ variables, such as ÔcurvaturesÕ and Ôstress resultantsÕ. Furthermore, the special role of the material law within such a formulation is addressed. It is pointed out that certain requirements must hold for the variation of kinematic and static variables across the thickness, to ensure correct results. In this context it is found, that the considered 7-parameter model can be regarded as ÔoptimalÕ with respect to the number of degrees of freedom involved