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Extension of non-linear beam models with deformable cross sections

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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.

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Notes

  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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Acknowledgments

The research was partially supported by the Heart-e-Gel consortium, has received funding from the European Union Seventh Framework Program FP7/2007-2013 under Grant Agreement No. 258909, and partial support from the Diane and Arthur B. Belfer Chair in Mechanics and Biomechanics.

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  1. School of Mechanical Engineering, Tel Aviv University, Ramat-Aviv, 69978, Tel Aviv, Israel

    I. Sokolov, S. Krylov & I. Harari

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  1. I. Sokolov
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Correspondence to I. Harari.

Appendix 1: The tangent stiffnesses

Appendix 1: The tangent stiffnesses

1.1 Appendix 1.1: The material tangent stiffness

The material tangent stiffness is defined through the tangent elasticity matrix, see Eq. (88)

$$\begin{aligned} {\varvec{C}}^*= & {} \begin{bmatrix} \displaystyle \frac{\partial {\varvec{Q}}}{ \partial {\varvec{\varGamma }}}&\quad \displaystyle \frac{\partial {\varvec{Q}}}{ \partial {\varvec{\varOmega }}}&\quad \displaystyle \frac{\partial {\varvec{Q}}}{ \partial {\varvec{d}}}&\quad \displaystyle \frac{\partial {\varvec{Q}}}{ \partial {\varvec{d}}'}&\quad \displaystyle \frac{\partial {\varvec{Q}}}{ \partial {\varvec{l}}}&\quad \displaystyle \frac{\partial {\varvec{Q}}}{ \partial {\varvec{l}}'} \\ \displaystyle \frac{\partial {\varvec{M}}}{ \partial {\varvec{\varGamma }}}&\quad \displaystyle \frac{\partial {\varvec{M}}}{ \partial {\varvec{\varOmega }}}&\quad \displaystyle \frac{\partial {\varvec{M}}}{ \partial {\varvec{d}}}&\quad \displaystyle \frac{\partial {\varvec{M}}}{ \partial {\varvec{d}}'}&\quad \displaystyle \frac{\partial {\varvec{M}}}{ \partial {\varvec{l}}}&\quad \displaystyle \frac{\partial {\varvec{M}}}{ \partial {\varvec{l}}'} \\ \displaystyle \frac{\partial {\varvec{\rho }}}{ \partial {\varvec{\varGamma }}}&\quad \displaystyle \frac{\partial {\varvec{\rho }}}{ \partial {\varvec{\varOmega }}}&\quad \displaystyle \frac{\partial {\varvec{\rho }}}{ \partial {\varvec{d}}}&\quad \displaystyle \frac{\partial {\varvec{\rho }}}{ \partial {\varvec{d}}'}&\quad \displaystyle \frac{\partial {\varvec{\rho }}}{ \partial {\varvec{l}}}&\quad \displaystyle \frac{\partial {\varvec{\rho }}}{ \partial {\varvec{l}}'} \\ \displaystyle \frac{\partial {\varvec{\beta }}}{ \partial {\varvec{\varGamma }}}&\quad \displaystyle \frac{\partial {\varvec{\beta }}}{ \partial {\varvec{\varOmega }}}&\quad \displaystyle \frac{\partial {\varvec{\beta }}}{ \partial {\varvec{d}}}&\quad \displaystyle \frac{\partial {\varvec{\beta }}}{ \partial {\varvec{d}}'}&\quad \displaystyle \frac{\partial {\varvec{\beta }}}{ \partial {\varvec{l}}}&\quad \displaystyle \frac{\partial {\varvec{\beta }}}{ \partial {\varvec{l}}'} \\ \displaystyle \frac{\partial {\varvec{\pi }}}{ \partial {\varvec{\varGamma }}}&\quad \displaystyle \frac{\partial {\varvec{\pi }}}{ \partial {\varvec{\varOmega }}}&\quad \displaystyle \frac{\partial {\varvec{\pi }}}{ \partial {\varvec{d}}}&\quad \displaystyle \frac{\partial {\varvec{\pi }}}{ \partial {\varvec{d}}'}&\quad \displaystyle \frac{\partial {\varvec{\pi }}}{ \partial {\varvec{l}}}&\quad \displaystyle \frac{\partial {\varvec{\pi }}}{ \partial {\varvec{l}}'} \\ \displaystyle \frac{\partial {\varvec{\alpha }}}{ \partial {\varvec{\varGamma }}}&\quad \displaystyle \frac{\partial {\varvec{\alpha }}}{ \partial {\varvec{\varOmega }}}&\quad \displaystyle \frac{\partial {\varvec{\alpha }}}{ \partial {\varvec{d}}}&\quad \displaystyle \frac{\partial {\varvec{\alpha }}}{ \partial {\varvec{d}}'}&\quad \displaystyle \frac{\partial {\varvec{\alpha }}}{ \partial {\varvec{l}}}&\quad \displaystyle \frac{\partial {\varvec{\alpha }}}{ \partial {\varvec{l}}'} \end{bmatrix} \end{aligned}$$
(107)

according to the relation \(\varDelta \varvec{R} = {\varvec{C}}^* \varDelta \varvec{\mathcal {E}}\) with \({\varvec{\mathcal {E}}} = \{ {\varvec{\varGamma }}, {\varvec{\varOmega }}, {\varvec{d}}, {\varvec{d}}' , {\varvec{l}}, {\varvec{l}}' \}^T\).

To derive the components of the tangent elasticity matrix \({\varvec{C}}^*\) we use the chain rule by differentiation of the reference force, moment, bi-shear and bi-moment resultants (\(\mathcal {R}_A\) components, Eq. (83b)) with respect to the reference local strain vectors \(\varvec{A}_I\) and beam strain and kinematic quantities of \(\varvec{\mathcal {E}}\) as follows

$$\begin{aligned} {C}^*_{AB}= & {} \frac{\partial {{\mathcal {R}}}_{A}}{\partial \varvec{A}_{I}} \frac{\partial \varvec{A}_{I}}{\partial {\mathcal {E}}_{B}}, \quad A,B=1,2,3,4,5, 6 \end{aligned}$$
(108)

The product of the differentiation of the reference local traction vectors with respect to the reference local strain vectors \( \varvec{A}_J\) is denoted as

$$\begin{aligned} \varvec{D}_{IJ} = \frac{\partial \left( \varvec{S}_{I} + \varvec{\varSigma }_{I}\right) }{\partial \varvec{A}_J} \end{aligned}$$
(109)

The derivatives of the reference local strain vectors \({\varvec{A}}_I\) with respect to the stain and kinematic quantities of the finite strain beam \(\varvec{\mathcal {E}}_B\) are

$$\begin{aligned} {\varvec{A}}_{\alpha ,\varvec{\varGamma }} = \frac{\partial {\varvec{A}}_{\alpha }}{\partial \varvec{\varGamma } }= & {} {\varvec{0}}, \quad {\varvec{A}}_{3,\varvec{\varGamma } } = \frac{\partial {\varvec{A}}_{3}}{\partial \varvec{\varGamma } } = {\varvec{I}} \end{aligned}$$
(110)
$$\begin{aligned} \varvec{A}_{\alpha ,\varvec{\varOmega } } = \frac{\partial {\varvec{A}}_{\alpha }}{\partial \varvec{\varOmega } }= & {} {\varvec{0}}, \quad {\varvec{A}}_{3,\varvec{\varOmega } } = \frac{\partial {\varvec{A}}_{3}}{\partial \varvec{\varOmega } } = - \widetilde{ \varvec{Z}} \end{aligned}$$
(111)
$$\begin{aligned} \varvec{A}_{\alpha ,\varvec{d} } = \frac{\partial {\varvec{A}}_{\alpha }}{\partial \varvec{d} }= & {} \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta ,\alpha }, \quad {\varvec{A}}_{3,\varvec{d} } = \frac{\partial {\varvec{A}}_{3}}{\partial \varvec{d} } = \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{\beta } \otimes {\varvec{\varpi }}_{\beta } \right) \nonumber \\ \end{aligned}$$
(112)
$$\begin{aligned} \varvec{A}_{\alpha ,\varvec{d'} } = \frac{\partial {\varvec{A}}_{\alpha }}{\partial \varvec{d'} }= & {} \varvec{0}, \quad {\varvec{A}}_{3,\varvec{d'} } = \frac{\partial {\varvec{A}}_{3}}{\partial \varvec{d'} } = \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta } \end{aligned}$$
(113)
$$\begin{aligned} \varvec{A}_{\alpha ,\varvec{l} } = \frac{\partial {\varvec{A}}_{\alpha }}{\partial \varvec{l} }= & {} \varvec{E}_{3} \otimes \varvec{\varrho }_{,\alpha }, \quad {\varvec{A}}_{3,\varvec{l} } = \frac{\partial {\varvec{A}}_{3}}{\partial \varvec{l} } = \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \nonumber \\ \end{aligned}$$
(114)
$$\begin{aligned} \varvec{A}_{\alpha ,\varvec{l'} } = \frac{\partial {\varvec{A}}_{\alpha }}{\partial \varvec{l'} }= & {} \varvec{0}, \quad {\varvec{A}}_{3,\varvec{l'}} = \frac{\partial {\varvec{A}}_{3}}{\partial \varvec{l'} } = \varvec{E}_{3} \otimes \varvec{\varrho } \end{aligned}$$
(115)

Recall

$$\begin{aligned} {\varvec{Z}} = \varvec{\varLambda }^T {\varvec{z}} \end{aligned}$$
(116)

Inserting the above expressions in the definition of the force, moment and bi-force resultants we obtain explicitly the components of the tangent elasticity tensor as follows

$$\begin{aligned} \varvec{C}^*_{{\varvec{Q}} {\varvec{\varGamma }}}= & {} \int _A \varvec{D}_{33} dA \end{aligned}$$
(117)
$$\begin{aligned} \varvec{C}^*_{{\varvec{Q}} {\varvec{\varOmega }}}= & {} - \int _A \varvec{D}_{33} \widetilde{\varvec{Z}} dA \end{aligned}$$
(118)
$$\begin{aligned} \varvec{C}^*_{{\varvec{Q}} {\varvec{d}}}= & {} \int _A \left[ \varvec{D}_{3\alpha } \left( \varvec{E}_{\beta } \otimes {\varvec{\varpi }}_{\beta ,\alpha } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{33} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{\beta } \otimes {\varvec{\varpi }}_{\beta } \right) \right) \right] dA \end{aligned}$$
(119)
$$\begin{aligned} \varvec{C}^*_{{\varvec{Q}} {\varvec{d}}'}= & {} \int _A \varvec{D}_{33} \left( \varvec{E}_{\beta } \otimes {\varvec{\varpi }}_{\beta } \right) dA \end{aligned}$$
(120)
$$\begin{aligned} \varvec{C}^*_{{\varvec{Q}} {\varvec{l}}}= & {} \int _A \left[ \varvec{D}_{3\alpha } \left( \varvec{E}_{3} \otimes {\varvec{\varrho }}_{,\alpha } \right) + \varvec{D}_{33} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{3} \otimes {\varvec{\varrho }} \right) \right) \right] dA\nonumber \\ \end{aligned}$$
(121)
$$\begin{aligned} \varvec{C}^*_{{\varvec{Q}} {\varvec{l}}'}= & {} \int _A \varvec{D}_{33} \left( \varvec{E}_{3} \otimes {\varvec{\varrho }} \right) dA \end{aligned}$$
(122)
$$\begin{aligned} \varvec{C}^*_{{\varvec{M}} {\varvec{\varGamma }}}= & {} \int _A \widetilde{\varvec{Z}}\varvec{D}_{33} dA \end{aligned}$$
(123)
$$\begin{aligned} \varvec{C}^*_{{\varvec{M}} {\varvec{\varOmega }}}= & {} - \int _A \widetilde{\varvec{Z}} \varvec{D}_{33} \widetilde{\varvec{Z}} dA \end{aligned}$$
(124)
$$\begin{aligned} \varvec{C}^*_{{\varvec{M}} {\varvec{d}}}= & {} \int _A \widetilde{\varvec{Z}} \left[ \varvec{D}_{3\alpha } \left( \varvec{E}_{\beta } \otimes {\varvec{\varpi }}_{\beta ,\alpha } \right) \right. \nonumber \\&\left. +\, \varvec{D}_{33} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{\beta } \otimes {\varvec{\varpi }}_{\beta } \right) \right) \right] dA \nonumber \\&+\, \int _A \left( \widetilde{\varvec{E}}_{\beta } \left( {\varvec{S}}_3 + \varvec{\varSigma }^0_{3} \right) \right) \otimes {\varvec{\varpi }}_{\beta } \,dA \end{aligned}$$
(125)
$$\begin{aligned} \varvec{C}^*_{{\varvec{M}} {\varvec{d}}'}= & {} \int _A \widetilde{\varvec{Z}} \varvec{D}_{33} \left( \varvec{E}_{\beta } \otimes {\varvec{\varpi }}_{\beta } \right) dA \end{aligned}$$
(126)
$$\begin{aligned} \varvec{C}^*_{{\varvec{M}} {\varvec{l}}}= & {} \int _A \widetilde{\varvec{Z}} \left[ \varvec{D}_{3\alpha } \left( \varvec{E}_{3} \otimes {\varvec{\varrho }}_{,\alpha } \right) + \varvec{D}_{33} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{3} \otimes {\varvec{\varrho }} \right) \right) \right] \,dA \nonumber \\&+ \, \int _A \left( \widetilde{ \varvec{E}_{3}} \left( {\varvec{S}}_3 + \varvec{\varSigma }^0_{3} \right) \right) \otimes {\varvec{\varrho }} \,dA \end{aligned}$$
(127)
$$\begin{aligned} \varvec{C}^*_{{\varvec{M}} {\varvec{l}}'}= & {} \int _A \widetilde{\varvec{Z}} \varvec{D}_{33} \left( \varvec{E}_{3} \otimes {\varvec{\varrho }} \right) dA \end{aligned}$$
(128)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\rho }} {\varvec{\varGamma }}}= & {} \int _A \left[ \left( {\varvec{\varpi }}_{\beta ,\alpha } \otimes \varvec{E}_{\beta } \right) \varvec{D}_{\alpha 3} \right. \nonumber \\&\left. + \, \left( {\varvec{\varpi }}_{\beta } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\beta } \right) \right) \varvec{D}_{3 3} \right] dA \end{aligned}$$
(129)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\rho }} {\varvec{\varOmega }}}= & {} - \int _A \left[ \left( {\varvec{\varpi }}_{\beta ,\alpha } \otimes \varvec{E}_{\beta } \right) \varvec{D}_{\alpha 3} \right. \nonumber \\&\left. + \, \left( {\varvec{\varpi }}_{\beta } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\beta } \right) \right) \varvec{D}_{3 3} \right] \widetilde{\varvec{Z}} \,dA \nonumber \\&+ \, \int _A {\varvec{\varpi }}_{\beta } \otimes \left( \widetilde{\varvec{E}_{\beta }} \left( {\varvec{S}}_3 + \varvec{\varSigma }^0_{3} \right) \right) \,dA \end{aligned}$$
(130)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\rho }} {\varvec{d}}}= & {} \int _A \left( \varvec{\varpi }_{\beta ,\alpha } \otimes \varvec{E}_{\beta } \right) \left[ \varvec{D}_{\alpha \delta } \left( \varvec{E}_{\gamma } \otimes \varvec{\varpi }_{\gamma ,\delta } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{\alpha 3} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta } \right) \right) \right] dA \nonumber \\&+ \, \int _A \left( \varvec{\varpi }_{\alpha } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\alpha } \right) \right) \left[ \varvec{D}_{3 \delta } \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta ,\delta } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{3 3} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta } \right) \right) \right] dA \end{aligned}$$
(131)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\rho }} {\varvec{d}}'}= & {} \int _A \left[ \left( \varvec{\varpi }_{\beta ,\alpha } \otimes \varvec{E}_{\beta } \right) \varvec{D}_{\alpha 3} \left( \varvec{E}_{\gamma } \otimes \varvec{\varpi }_{\gamma } \right) \right. \nonumber \\&\left. + \, \left( \varvec{\varpi }_{\alpha } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\alpha } \right) \right) \varvec{D}_{3 3} \left( \varvec{E}_{\gamma } \otimes \varvec{\varpi }_{\gamma } \right) \right] dA \end{aligned}$$
(132)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\rho }} {\varvec{l}}}= & {} \int _A \left( \varvec{\varpi }_{\beta ,\alpha } \otimes \varvec{E}_{\beta } \right) \left[ \varvec{D}_{\alpha \delta } \left( \varvec{E}_{3} \otimes \varvec{\varrho }_{,\delta } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{\alpha 3} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \right) \right] dA \nonumber \\&+\, \int _A \left( \varvec{\varpi }_{\alpha } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\alpha } \right) \right) \left[ \varvec{D}_{3 \delta } \left( \varvec{E}_{3} \otimes \varvec{\varrho }_{,\delta } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{3 3} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \right) \right] dA \end{aligned}$$
(133)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\rho }} {\varvec{l}}'}= & {} \int _A \left[ \left( \varvec{\varpi }_{\beta ,\alpha } \otimes \varvec{E}_{\beta } \right) \varvec{D}_{\alpha 3} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \right. \nonumber \\&\left. + \, \left( \varvec{\varpi }_{\alpha } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\alpha } \right) \right) \varvec{D}_{3 3} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \right] dA \end{aligned}$$
(134)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\beta }} {\varvec{\varGamma }}}= & {} \int _A \left( \varvec{\varpi }_{\alpha } \otimes \varvec{E}_{\alpha } \right) \varvec{D}_{33} \;dA \end{aligned}$$
(135)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\beta }} {\varvec{\varOmega }}}= & {} - \int _A \left( \varvec{\varpi }_{\alpha } \otimes \varvec{E}_{\alpha } \right) \varvec{D}_{33} \widetilde{\varvec{Z}} \;dA \end{aligned}$$
(136)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\beta }} {\varvec{d}}}= & {} \int _A \left( \varvec{\varpi }_{\gamma } \otimes \varvec{E}_{\gamma } \right) \left[ \varvec{D}_{3 \alpha } \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta , \alpha } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{3 3} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta } \right) \right) \right] dA \end{aligned}$$
(137)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\beta }} {\varvec{d}}'}= & {} \int _A \left( \varvec{\varpi }_{\gamma } \otimes \varvec{E}_{\gamma } \right) \varvec{D}_{3 3} \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta } \right) \;dA \end{aligned}$$
(138)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\beta }} {\varvec{l}}}= & {} \int _A \left( \varvec{\varpi }_{\gamma } \otimes \varvec{E}_{\gamma } \right) \left[ \varvec{D}_{3 \alpha } \left( \varvec{E}_{3} \otimes \varvec{\varrho }_{, \alpha } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{3 3} \left( \widetilde{\varvec{\varOmega }} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \right) \right] dA \end{aligned}$$
(139)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\beta }} {\varvec{l}}'}= & {} \int _A \left( \varvec{\varpi }_{\gamma } \otimes \varvec{E}_{\gamma } \right) \varvec{D}_{3 3} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \;dA \end{aligned}$$
(140)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\pi }} {\varvec{\varGamma }}}= & {} \int _A \left[ \left( \varvec{\varrho }_{,\alpha } \otimes \varvec{E}_{3} \right) \varvec{D}_{\alpha 3} + \varvec{\varrho } \otimes \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \varvec{D}_{3 3} \right] dA \end{aligned}$$
(141)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\pi }} {\varvec{\varOmega }}}= & {} - \int _A \left[ \left( \varvec{\varrho }_{,\alpha } \otimes \varvec{E}_{3} \right) \varvec{D}_{\alpha 3} + \left( \varvec{\varrho } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \right) \right) \varvec{D}_{3 3} \right] \widetilde{\varvec{Z}} \,dA \nonumber \\&+\, \int _A \varvec{\varrho } \otimes \left( \widetilde{\varvec{E}_3} \left( \varvec{S}_3 + \varvec{\varSigma }_3^0 \right) \right) \,dA \end{aligned}$$
(142)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\pi }} {\varvec{d}}}= & {} \int _A \left( \varvec{\varrho }_{,\alpha } \otimes \varvec{E}_{3} \right) \left[ \varvec{D}_{\alpha \gamma } \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta ,\gamma } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{\alpha 3} \left( \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\beta } \right) \otimes \varvec{\varpi }_{\beta } \right) \right] dA \nonumber \\&+ \, \int _A \left( \varvec{\varrho } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \right) \right) \left[ \varvec{D}_{3 \gamma } \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta ,\gamma } \right) \right. \nonumber \\&\left. + \,\varvec{D}_{3 3} \left( \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\beta } \right) \otimes \varvec{\varpi }_{\beta } \right) \right] dA \end{aligned}$$
(143)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\pi }} {\varvec{d}}'}= & {} \int _A \left( \varvec{\varrho }_{,\alpha } \otimes \varvec{E}_{3} \right) \varvec{D}_{\alpha 3} \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta } \right) \, dA \nonumber \\&+ \, \int _A \left( \varvec{\varrho } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \right) \right) \varvec{D}_{3 3} \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta } \right) \,dA \end{aligned}$$
(144)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\pi }} {\varvec{l}}}= & {} \int _A \left( \varvec{\varrho }_{,\alpha } \otimes \varvec{E}_{3} \right) \left[ \varvec{D}_{\alpha \gamma } \left( \varvec{E}_{3} \otimes \varvec{\varrho }_{,\gamma } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{\alpha 3} \left( \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \right) \otimes \varvec{\varrho } \right) \right] \,dA \nonumber \\&+ \, \int _A \left( \varvec{\varrho } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \right) \right) \left[ \varvec{D}_{3 \gamma } \left( \varvec{E}_{3} \otimes \varvec{\varrho }_{,\gamma } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{3 3} \left( \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \right) \otimes \varvec{\varrho } \right) \right] \,dA \end{aligned}$$
(145)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\pi }} {\varvec{l}}'}= & {} \int _A \left[ \left( \varvec{\varrho }_{,\alpha } \otimes \varvec{E}_{3} \right) \varvec{D}_{\alpha 3} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \right. \nonumber \\&\left. + \, \left( \varvec{\varrho } \otimes \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \right) \right) \varvec{D}_{3 3} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \right] \,dA \end{aligned}$$
(146)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\alpha }} {\varvec{\varGamma }}}= & {} \int _A \left( \varvec{\varrho } \otimes \varvec{E}_{3} \right) \varvec{D}_{33} \,dA \end{aligned}$$
(147)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\alpha }} {\varvec{\varOmega }}}= & {} - \int _A \left( \varvec{\varrho } \otimes \varvec{E}_{3} \right) \varvec{D}_{33} \widetilde{\varvec{Z}} \,dA \end{aligned}$$
(148)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\alpha }} {\varvec{d}}}= & {} \int _A \left( \varvec{\varrho } \otimes \varvec{E}_{3} \right) \left[ \varvec{D}_{3 \alpha } \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta , \alpha } \right) \right. \nonumber \\&\left. + \,\varvec{D}_{3 3} \left( \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{\beta } \right) \otimes \varvec{\varpi }_{\beta } \right) \right] \,dA \end{aligned}$$
(149)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\alpha }} {\varvec{d}}'}= & {} \int _A \left( \varvec{\varrho } \otimes \varvec{E}_{3} \right) \varvec{D}_{3 3} \left( \varvec{E}_{\beta } \otimes \varvec{\varpi }_{\beta } \right) \,dA \end{aligned}$$
(150)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\alpha }} {\varvec{l}}}= & {} \int _A \left( \varvec{\varrho } \otimes \varvec{E}_{3} \right) \left[ \varvec{D}_{3 \alpha } \left( \varvec{E}_{3} \otimes \varvec{\varrho }_{, \alpha } \right) \right. \nonumber \\&\left. + \, \varvec{D}_{3 3} \left( \left( \widetilde{\varvec{\varOmega }} \varvec{E}_{3} \right) \otimes \varvec{\varrho } \right) \right] \,dA \end{aligned}$$
(151)
$$\begin{aligned} \varvec{C}^*_{{\varvec{\alpha }} {\varvec{l}}'}= & {} \int _A \left( \varvec{\varrho } \otimes \varvec{E}_{3} \right) \varvec{D}_{3 3} \left( \varvec{E}_{3} \otimes \varvec{\varrho } \right) \,dA \end{aligned}$$
(152)

The tangent tensor \(\varvec{D}_{IJ}\) of the St. Venant-Kirchhoff material, Eq. (109), is obtained by the derivation of the local traction vector, Eq. (95), with respect to the local strain vector

$$\begin{aligned} \varvec{D}_{IJ}= & {} \frac{\lambda }{2} \left[ \delta _{IJ} \left( \varvec{H}_K \cdot \varvec{H}_K \right) \varvec{1} + 2 \varvec{H}_I \otimes \varvec{H}_J - 3 \, \delta _{IJ} \varvec{1} \right] \nonumber \\&+\, \mu \left[ \delta _{IJ} \left( \varvec{H}_K \otimes \varvec{H}_K \right) + \varvec{H}_J \otimes \varvec{H}_I \right. \nonumber \\&\left. + \,\left( \varvec{H}_J \cdot \varvec{H}_I \right) \varvec{1} - \delta _{IJ} \varvec{1} \right] + \frac{\partial \varvec{\varSigma }_{I}}{\partial \varvec{A}_J} \end{aligned}$$
(153)

For the compressible neo-Hookean material the tangent tensor \(\varvec{D}_{IJ}\) is given by, see [14]

$$\begin{aligned} \varvec{D}_{IJ}= & {} \left[ \frac{\lambda }{2} \left( 1 + \frac{1}{J^2} \right) + \frac{\mu }{J^2} \right] \varvec{G}_I \otimes \varvec{G}_J + \mu \, \delta _{IJ} \varvec{1} \nonumber \\&- \,\frac{1}{J} \left[ \frac{\lambda }{2} \left( J^2 - 1 \right) - \mu \right] \epsilon _{IJK} \widetilde{ \varvec{H} }_K + \frac{\partial \varvec{\varSigma }_{I}}{\partial \varvec{A}_J} \end{aligned}$$
(154)

Here, \(\epsilon _{IJK}\) is the permutation symbol.

1.2 Appendix 1.2: The geometric tangent stiffness

The geometric tangent stiffness is defined through the geometric tangent matrix, see Eq. (88)

$$\begin{aligned} {\varvec{\mathcal {K}}}^G = \begin{Bmatrix} {\varvec{0}}&\quad {\varvec{0}}&\quad -{\varvec{\varLambda }} \widetilde{\varvec{Q}} \varvec{T}^T&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}} \\ {\varvec{0}}&\quad {\varvec{0}}&\quad \varvec{C}_2&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}} \\ {\varvec{T}} \widetilde{\varvec{Q}} \varvec{\varLambda }^T&\quad \varvec{C}_2^T&\quad {\varvec{\mathcal {K}}}^G_{33}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}} \\ {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}} \\ {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}} \\ {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}} \\ {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}}&\quad {\varvec{0}} \end{Bmatrix} \end{aligned}$$
(155)

Here, \({\varvec{\mathcal {K}}}^G_{33} = {\varvec{T}} \widetilde{\varvec{Q}} \widetilde{\left( \varvec{\varLambda }^T \varvec{r}'\right) } \varvec{T}^T + \varvec{C}_2( \widetilde{\varvec{Q}} \varvec{\varLambda }^T \varvec{r}', \varvec{\varTheta }) + \varvec{C}_3( \varvec{M},\varvec{\varTheta }',\varvec{\varTheta }) \). The tensors \(\varvec{C}_2\) and \(\varvec{C}_3\) are given in Appendix 2.

1.3 Appendix 2: Tensors \(\varvec{C}_1\), \(\varvec{C}_2\), \(\varvec{C}_3\)

The tensor \(\varvec{C}_1\), Eq. (81), and tensors \(\varvec{C}_2\) and \(\varvec{C}_3\), Appendix 1.2, are given here for the convenience. For details of the derivation, see [53] and [40].

$$\begin{aligned} \varvec{C}_1\left( \varvec{\varTheta }', \varvec{\varTheta } \varvec{}\right)= & {} c_1 \varvec{\varTheta }' \otimes \varvec{\varTheta } - c_2 \left( \widetilde{ \varvec{\varTheta } } \varvec{\varTheta }' \right) \otimes \varvec{\varTheta } \nonumber \\&+ \,c_3 \left( \varvec{\varTheta } \cdot \varvec{\varTheta }' \right) \varvec{\varTheta } \otimes \varvec{\varTheta } \nonumber \\&-\, c_4 \widetilde{ \varvec{\varTheta }' } + c_5 \left( \left( \varvec{\varTheta } \cdot \varvec{\varTheta }' \right) \varvec{1} + \varvec{\varTheta } \otimes \varvec{\varTheta }' \right) \end{aligned}$$
(156)
$$\begin{aligned} \varvec{C}_2\left( \varvec{V}, \varvec{\varTheta } \varvec{}\right)= & {} c_1 \varvec{V} \otimes \varvec{\varTheta } + c_2 \left( \widetilde{ \varvec{\varTheta } } \varvec{V} \right) \otimes \varvec{\varTheta } \nonumber \\&+\, c_3 \left( \varvec{\varTheta } \cdot \varvec{V} \right) \varvec{\varTheta } \otimes \varvec{\varTheta } \nonumber \\&+\, c_4 \widetilde{ \varvec{V} } + c_5 \left( \left( \varvec{\varTheta } \cdot \varvec{V} \right) \varvec{1} + \varvec{\varTheta } \otimes \varvec{V} \right) \end{aligned}$$
(157)
$$\begin{aligned} \varvec{C}_3 \left( \varvec{M}, \varvec{\varTheta }', \varvec{\varTheta }\right)= & {} \left( c_1 \left( \varvec{M} \cdot \varvec{\varTheta '} \right) + c_2 \left( \widetilde{ \varvec{M} } \varvec{\varTheta }' \cdot \varvec{\varTheta } \right) \right. \nonumber \\&\left. + \,c_3 \left( \varvec{\varTheta } \cdot \varvec{M} \right) \left( \varvec{\varTheta } \cdot \varvec{\varTheta }' \right) \right) \varvec{1} \nonumber \\&+\, 2 c_2 \left( \varvec{\varTheta } \otimes \widetilde{ \varvec{M} } \varvec{\varTheta }' \right) ^s \nonumber \\&+\, 2 c_3 \left( \varvec{\varTheta } \cdot \varvec{\varTheta }' \right) \left( \varvec{\varTheta } \otimes \varvec{M} \right) ^s \nonumber \\&+\, \frac{1}{\varTheta } \left( c_6 \left( \varvec{\varTheta }' \cdot \varvec{M} \right) + c_7 \left( \varvec{\varTheta } \cdot \widetilde{ \varvec{M} } \varvec{\varTheta }' \right) \right) \nonumber \\&+\, c_8 \left( \varvec{\varTheta } \cdot \varvec{\varTheta }' \right) \left( \varvec{M} \cdot \varvec{\varTheta } \right) \left( \varvec{\varTheta } \otimes \varvec{\varTheta } \right) \nonumber \\&+\, 2 c_3 \left( \varvec{M} \cdot \varvec{\varTheta } \right) \left( \varvec{\varTheta } \otimes \varvec{\varTheta }' \right) ^s + 2 c_5 \left( \varvec{\varTheta }' \otimes \varvec{M} \right) ^s \nonumber \\ \end{aligned}$$
(158)

where \(\varvec{V}\) is a constant vector and

$$\begin{aligned} c_1= & {} \frac{\varTheta \cos \varTheta - \sin \varTheta }{\varTheta ^3} \nonumber \\ c_2= & {} \frac{\varTheta \sin \varTheta +2 \cos \varTheta -2}{\varTheta ^4} \end{aligned}$$
(159)
$$\begin{aligned} c_3= & {} \frac{3\sin \varTheta -2 \varTheta -\varTheta \cos \varTheta }{\varTheta ^5} \qquad c_4 = \frac{\cos \varTheta -1}{\varTheta ^2} \end{aligned}$$
(160)
$$\begin{aligned} c_5= & {} \frac{\varTheta - \sin \varTheta }{\varTheta ^3} \nonumber \\ c_6= & {} \frac{3 \sin \varTheta - \varTheta ^2 \sin \varTheta - 3 \varTheta \cos \varTheta }{\varTheta ^4} \end{aligned}$$
(161)
$$\begin{aligned} c_7= & {} \frac{\varTheta ^2 \cos \varTheta - 5 \varTheta \sin \varTheta - 8 \cos \varTheta + 8}{\varTheta ^5} \end{aligned}$$
(162)
$$\begin{aligned} c_8= & {} \frac{ 7 \varTheta \cos \varTheta + 8 \varTheta + \varTheta ^2 \sin \varTheta -15 \sin \varTheta }{\varTheta ^6} \end{aligned}$$
(163)

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Sokolov, I., Krylov, S. & Harari, I. Extension of non-linear beam models with deformable cross sections. Comput Mech 56, 999–1021 (2015). https://doi.org/10.1007/s00466-015-1215-5

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