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On a stress resultant geometrically exact shell model. II: The linear theory; computational aspects. (English) Zbl 0724.73138
Summary: Computational aspects of a linear stress resultant (classical) shell theory obtained by systematic linearization of the geometrically exact nonlinear theory, considered in Part I of this work [the first two authors, Comput. Methods Appl. Mech. Eng. 72, No.3, 267-304 (1989; Zbl 0692.73062)], are examined in detail. In particular, finite element interpolations for the reference director field and the linearized rotation field are constructed such that the underlying geometric structure of the continuum theory is preserved exactly by the discrete approximation. A discrete canonical, singularity-free mapping between the five and the six degree of freedom formulation is constructed by exploiting the geometric connection between the orthogonal group (SO(3)) and the unit sphere \((S^ 2).\)
The proposed numerical treatment of the membrane and bending fields, based on a mixed Hellinger-Reissner formulation, provides excellent results for the 4-node bilinear isoparametric element. As an example, convergent results are obtained for rather coarse meshes in fairly demanding, singularity-dominated problems such as the classical rhombic plate test. The proposed theory and finite element implementation are evaluated through an extensive set of benchmark problems. The results obtained with the present approach exactly match previous solutions obtained with state-of-the-art implementations based on the so-called degenerated solid approach.

MSC:
74K15 Membranes
74S05 Finite element methods applied to problems in solid mechanics
74P99 Optimization problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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