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Fig. 2 Axisymmetric shell mid-surface—coordinate systems
This and the next image are from:
Balázs Tóth and Dániel Burmeister (Institute of Applied Mechanics, University of Miskolc, Miskolc-Egyetemváros, Hungary),
“Dual-mixed hp-version axisymmetric shell finite element using NURBS mid-surface interpolation”, Acta Mechanica, Vol. 231, No. 6, pp 2457-2483, June 2020, https://doi.org/10.1007/s00707-020-02661-3
ABSTRACT: A new, general hp-version axisymmetric finite element is derived for the boundary value problems of thin linearly elastic shells of revolution, applying a complementary strain energy-based three-field dual-mixed variational principle. For the interpolation of the mid-surface geometry, non-uniform rational B-splines—NURBS—is used. The independent field variables of the weak formulation are the a priori non-symmetric stress tensor, the displacement vector, and the infinitesimal skew-symmetric rotation tensor. The theoretical model of the shell formulation is based on a consistent dimensional reduction process and a systematic variable-number reduction procedure. The inverse of the unvaried three-dimensional constitutive equation is employed since neither the classical kinematical assumptions nor the stress hypotheses are built in the mathematical model; namely, both the through-the-thickness variation and the normal stress to the shell mid-surface are not excluded. The new hp axisymmetric shell finite element is tested by a representative model problem for extremely thin and moderately thick, singly and doubly curved shells of negative and positive Gaussian curvature. Following from the numerical experiments, the constructed hp-shell finite element gives locking-free results not only for the displacement but also for the stresses.
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