Genuinely resultant shell finite elements accounting for geometric and material non‐linearity

Abstract

SUMMARYA general theoretical framework is presented for the fully non-linear analysis of shells by the finite element method. The governing equations are derived exclusively in terms of resulting quantities through a logical and straightforward descent from three-dimensional continuum mechanics without appealing to simplifying assumptions (hence the name genuinely resultant). As a result, the underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. The underlying mathematical structure and the variational formulation of the theory are examined. This appears to be crucial for the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method. The treatment of finite rotations through an arbitrary parametrization of the rotation group and the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis are other issues studied in this paper. A numerical analysis is presented in order to assess the effectiveness of the proposed formulation. Small strain problems as well as finite strain deformation of rubber-like shells undergoing finite rotations are considered. Special attention is devoted to the assessment of the relevance of the drilling degree-of-freedom and highly non-uniform through-the-thickness deformation in the case of shells made of incompressible material.

Computational shell mechanics by helicoidal modeling, II: Shell element

“…Some elements are built on mixed or full three-field variational formulations [Sansour and Bufler 1992;Wagner and Gruttmann 2005;Klinkel et al 2008]. Other elements (for example, [Chróścielewski et al 1992;Arciniega and Reddy 2007]) rely on high-order interpolants in order to avoid, or mitigate, shear and membrane locking. Others are based on particular techniques, such as reduced integration [Wriggers and Gruttmann 1993;Hauptmann et al 2000;Cardoso and Yoon 2005], discrete Kirchhoff-Love constraints [Areias et al 2005], or incompatible modes [Ibrahimbegović and Frey 1994].…”

“…Cylindrical roof under point load. The buckling problem of the shallow cylindrical panel hinged along two generatrices and subjected to a central point load has been considered by several authors; refer to [Simo et al 1990;Chróścielewski et al 1992;Gruttmann et al 1992;Sansour and Bufler 1992], and to most of the more recent works cited so far. The problem data and the undeformed configuration with a 4 × 4 mesh are shown in Figure 15 (owing to symmetry, one quarter of the panel is modeled).…”

“…Channel-section cantilever. The buckling of the channel-section cantilever, with the data introduced by [Chróścielewski et al 1992], has been considered by several authors to test shell elements in folded or intersecting structures, with either elastic or elastoplastic materials, for example, [Ibrahimbegović and Frey 1994;Eberlein and Wriggers 1999;Fontes Valente et al 2005;Chróścielewski and Witkowski 2006;Klinkel et al 2008]. In the present test, only the elastic case is considered; the problem data and a postbuckling configuration of the model, made of a regular mesh of (4 + 12 + 4) × 36 HSEs, are shown in Figure 19.…”

Enhanced transverse shear strain shell formulation applied to large elasto‐plastic deformation problems

“…The geometry is according to Figure 1, where values for length, width, height and thickness of the beam follow those originally proposed in Reference [75]. Among well-known works following with the treatment of this problem within shell formulations are, for instance, References [76][77][78][79].…”

The resultant linear six‐field theory of elastic shells: What it brings to the classical linear shell models?

“…In order the conditions (3) and (4) be satisfied, the resultant fields n α and m α require a unique 2D shell kinematics associated with the shell base surface M. Applying the virtual work identity Libai and Simmonds [13,22], Chróścielewski et al [14,23], and Eremeyev and Pietraszkiewicz [19] proved that such 2D kinematics consists of the translation vector u and the proper orthogonal (rotation) tensor Q, both describing the gross deformation (work-averaged through the shell thickness) of the shell cross section, such that…”