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Finite element analysis of hyperelastic thin shells with large strains. (English) Zbl 0880.73060
A theoretical model is proposed for the behaviour of isotropic hyperelastic shells which are capable of undergoing large deformations and large strains. The intention is that the model form the basis for a finite element analysis. Compressible and incompressible materials are accommodated in the model, though incompressible materials form the focus. The authors review the literature on nonlinear shell models, indicating the restrictions adopted in these cases.
A central feature of the model under consideration is the approximation of the deformed position vector by means of a quadratic function of the thickness coordinate. Thickness-stretching is decoupled from the director-like vector which specifies the deformed position of a material point relative to the deformed middle surface, by the introduction of an independent scalar variable $$\lambda$$ for this purpose. The quadratic term is important in compressible problems, if locking is to be avoided in finite element models. Then the models are distinguished by their use of the thickness stretching variable $$\lambda$$ and the higher-order displacement which enters the problem through the quadratic approximation.
Attention is restricted to thin shells, and, as a result, the transverse shear strains are neglected. The consequences of this restriction are then developed in detail. With the particular assumptions peculiar to this model, a considerable part of the article is devoted to the formulation of the problem in a form that is amenable to analysis. Thus there is much tensor manipulation. The rest of the article follows a standard procedure: a weak formulation in incremental form, the choice of specific strain energy functions (Mooney-Rivlin and neo-Hookean), and an appropriate choice of finite elements. Regarding the latter, the authors use triangular and rectangular elements, having respectively 54 and 48 degrees of freedom. The midsurface displacement field is approximated by quintic and Hermitian bicubic polynomials, for the two types of elements used.
The model is subjected to quite comprehensive numerical testing, through the consideration of various benchmark problems. The elements chosen appear to be free of locking, and are able to deal with genuine large strain problems. It is also observed that stress distributions are predicted with reasonable accuracy.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74K15 Membranes 74B20 Nonlinear elasticity
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