Abstract
The 7-parameter shell model as proposed by Büchter and Ramm (1992) is an extension of conventional shear deformation theories with five degrees of freedom. Application of this model is especially sensible if complete three-dimensional constitutive laws ought to be applied allowing also to solve problems involving large deformations and large strains. Based on a mixed formulation the seventh degree of freedom can be eliminated on the element level. Thus, the numerical effort is only slightly larger compared to `usual' shell elements. The model can consequently also be utilized for analyses of such shell structures, where a conventional shell formulation would be sufficient.
Büchter and Ramm (1992) describe the 7-parameter shell model along with a finite element formulation. Here, the seventh degree of freedom is introduced on the element level by means of a hybrid-mixed formulation. In the present work the 7-parameter model is derived independent from a finite element formulation. In this context it can be interpreted as semi-discretization of the shell continuum in thickness direction. The decisive difference to most of the conventional shell theories is that this discretization is based on a multifield variational formulation. By this procedure a system of partial differential equations can be obtained, describing the 7-parameter model as a two-dimensional, continuous theory with seven kinematical degrees of freedom per material point of the reference surface.
It is also intended to give a physical interpretation of the kinematic and static variables appearing in the model. Here, the main emphasis is put on those quantities, which do not show up in a conventional 5-parameter formulation. The investigations provide some insight into the mechanical behavior of the model. For the linear part of the transverse shear strains a new shear correction factor is proposed, which can reduce the error with respect to the three-dimensional solution in membrane dominated situations. It is shown that the 7-parameter model is `optimal' concerning the number of kinematic and static variables involved. This means that exactly those components are involved which are necessary for a complete three-dimensional material description.
Finally, a concept for the formulation of efficient finite elements for the 7-parameter model is presented. Here, established methods from the literature are combined with own developments. A special feature of the proposed concept is that a unified formulation for triangular and rectangular elements of arbitrary polynomial order is obtained. In addition, an improvement in the treatment of shells with kinks is proposed. In the course of the present study the concept is realized for linear and quadratic triangular and rectangular elements. The features of the proposed elements are investigated in numerical experiments, including both linear and geometrically and materially non-linear problems.
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