In this paper we discuss and compare three types of 4-node and 9-node ®nite elements for a recently formulated ®nite deformation shell theory with seven degrees of freedom. The shell theory takes thickness change into account and circumvents the use of a rotation tensor. It allows for the applicability of three-dimensional constitutive laws and equipes the con®guration space with the structure of a vector space. The ®nite elements themselves are based either on a hybrid stress functional, on a hybrid strain functional, or on a nonlinear version of the enhanced strain concept. As independent variables either the normal and shear resultants, the strain tensor related to the deformation of the midsurface, or the incompatible enhanced strain ®eld are taken as independent variables. The ®elds of equivalence of these different formulations, their limitations as well as possible improvements are discussed using different numerical examples.
A non-linear shell theory, including transverse shear strains, with exact description of the kinematical fields is developed. The strain measures are derived via the polar decomposition theorem allowing for an explicit use of a three parametric rotation tensor. Thus in-plane rotations, also called drilling degrees of freedom, are included in a natural way. Various alternatives of the theory are derived. For a special version of the theory, with altogether six kinematical fields, different mixed variational principles are given. A hybrid finite element formulation, which does not exhibit locking phenomena, is developed. Numerical examples of shell deformation at finite rotations, with excellent element performance, are presented. Comparison with results reported in the literature demonstrates the features of the theory as well as the proposed finite element formulation.
Received 22 November I990Revised 7 March I991
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C. SANSOUR AND H. BUFLERKirchhofS-Love assumption was the most preferred choice. With the work of sander^,^' Koiter," Budiansky' and Pietraszkie~icz~O the theory was fully developed. In spite of this, computational requirements motivated the development of new shell theories to allow the use of Co-shape functions within the finite element computations by taking into account transverse shear strains. Further, they should be capable of calculating finite deformations. Within these theories the exact description of the kinematics of the shell plays an important role. The assumed inextensibility of fibres perpendicular to the midsurface gives cause to describe the kinematics of the normal vector of the midsurface by a two parametric rotation tensor. This fact was recognized by Kratzig et ~1 . ,~l although not explicitly used, and exploited in computations by Simo and and Simo et ul.,42,43 where the strains chosen were those of Green. The use of such a rotation tensor suffers from two serious shortcomings.
The paper is concerned with large viscoplastic deformations of shells when the constitutive model is based on the concept of uni®ed evolution equations. Speci®cally the model due to Bodner and Partom is modi®ed so as to ®t in the frame of multiplicative viscoplasticity. Although the decomposition of the deformation gradient in elastic and inelastic parts is employed, no use is made of the concept of the intermediate con®guration. A logarithmic elastic strain measure is used. An algorithm for the evaluation of the exponential map for nonsymmetric arguments as well as a closed form of the tangent operator are given. On the side of the shell theory itself, the shell model is chosen so as to allow for the application of a three-dimensional constitutive law. The shell theory, accordingly, allows for thickness change and is characterized by seven parameters. The constitutive law is evaluated pointwise over the shell thickness to allow for general cyclic loading. An enhanced strain ®nite element method is given and various examples of large shell deformations including loading-unloading cycles are presented.
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