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Computational Mechanics
Erschienen in: Computational Mechanics 6/2015

01.12.2015 | Original Paper

Extension of non-linear beam models with deformable cross sections

verfasst von: I. Sokolov, S. Krylov, I. Harari

Erschienen in: Computational Mechanics | Ausgabe 6/2015

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Abstract

Geometrically exact beam theory is extended to allow distortion of the cross section. We present an appropriate set of cross-section basis functions and provide physical insight to the cross-sectional distortion from linear elastostatics. The beam formulation in terms of material (back-rotated) beam internal force resultants and work-conjugate kinematic quantities emerges naturally from the material description of virtual work of constrained finite elasticity. The inclusion of cross-sectional deformation allows straightforward application of three-dimensional constitutive laws in the beam formulation. Beam counterparts of applied loads are expressed in terms of the original three-dimensional data. Special attention is paid to the treatment of the applied stress, keeping in mind applications such as hydrogel actuators under environmental stimuli or devices made of electroactive polymers. Numerical comparisons show the ability of the beam model to reproduce finite elasticity results with good efficiency.
Vorheriger Artikel Variational theory of complex rays applied to shell structures: in-plane inertia, quasi-symmetric ray distribution, and orthotropic materials
Nächster Artikel An effective 3D leapfrog scheme for electromagnetic modelling of arbitrary shaped dielectric objects using unstructured meshes

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Fußnoten
1
The term geometrically exact with reference to shells was introduced by [59] and used for the first time for the beams by [65].
 
2
The deformation map in this case is constrained to have a form \(\varvec{x} = {\varvec{\varphi }} (\varvec{X}) = {\varvec{r}} + \hat{\varvec{z}} \big ( \{\varvec{e}_I\}, \varvec{d}, \varvec{l} \big ) \) where \(\hat{\varvec{z}}\) is a prescribed function. Richer theories can be developed using the extension to a high-order form, see [3].
 
3
Note the relation \(\left( \varvec{a} \otimes \varvec{b} \right) \varvec{c} = \left( \varvec{b} \cdot \varvec{c} \right) \varvec{a}\).
 
4
Describing beam measures, we consider two representations of vectors and tensors, designated as forward-rotated (spatial) and back-rotated (material) forms. These forms relate in such a way that components of the forward-rotated form measured in the reference frame \(\{\varvec{E}_I\}\) have the same components as the back-rotated form measured in the current frame \(\{\varvec{e}_{I}\}\).
 
5
For instance, in the finite element model all nodes with the same reference \(X_3\)-coordinate (nodes that lying in the ”beam“ cross-section plane) are constrained to the rigid body motion of a corresponding single node with \(X_1=X_2=0\) coordinate. Thus all these nodes move as a rigid ”beam“ section.
 
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Metadaten
Titel
Extension of non-linear beam models with deformable cross sections
verfasst von
I. Sokolov
S. Krylov
I. Harari
Publikationsdatum
01.12.2015
Verlag
Springer Berlin Heidelberg
Erschienen in
Computational Mechanics / Ausgabe 6/2015
Print ISSN: 0178-7675
Elektronische ISSN: 1432-0924
DOI
https://doi.org/10.1007/s00466-015-1215-5

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