## 2 The Geometrically Exact Beam Model
### Beam Statics and Its Linearization  
According to continuum mechanics, the Green-Lagrangian strain tensor \(\mathbf{E}(\xi_{1},\xi_{2},\xi_{3})\) is defined as:  
\[E_{ij}=\frac{1}{2}\Big{(}F_{ik}^{\mathrm{T}}F_{kj}-\delta_{ij}\Big{)}=\frac{1} {2}\Bigg{(}\frac{\partial\mathbf{x}_{p}^{\mathrm{T}}}{\partial\xi_{i}}\frac{ \partial\mathbf{x}_{p}}{\partial\xi_{j}}-\frac{\partial\mathbf{x}_{p}^{ \mathrm{0T}}}{\partial\xi_{i}}\frac{\partial\mathbf{x}_{p}^{\mathrm{0}}}{ \partial\xi_{j}}\Bigg{)} \tag{6}\]  
in which \(F_{kj}\) is the deformation gradient, and \(\delta_{ij}\) is the Kronecker delta. The derivatives of the displacement field in Equation (5) with respect to \(\xi_{i}\) (\(i=1,2,3\)) are given by:  
\[\left\{ \begin{array}{l}\frac{\partial\mathbf{x}_{p}(\xi_{1},\xi_{2},\xi_{3} )}{\partial\xi_{1}}=\mathbf{x}^{{}^{\prime}}+\mathbf{R}^{{}^{\prime}}\mathbf{ y}_{p}\\ \frac{\partial\mathbf{x}_{p}(\xi_{1},\xi_{2},\xi_{3})}{\partial\xi_{2}}= \mathbf{RA}_{2}\\ \frac{\partial\mathbf{x}_{p}(\xi_{1},\xi_{2},\xi_{3})}{\partial\xi_{3}}= \mathbf{RA}_{3}\end{array} \right. \tag{7}\]  
where \((\cdot)^{\prime}\) denotes the derivative relative to arc-length parameter \(S\). In the classical geometrically exact beam theory, the strain measures \(\varepsilon\) can be represented as:  
\[\varepsilon=\left[ \begin{array}{c}\mathbf{\gamma}\\ \mathbf{\kappa}\end{array} \right]=\left[ \begin{array}{c}\mathbf{R}^{\mathrm{T}} \mathbf{x}^{{}^{\prime}}-\mathbf{R}^{\mathrm{T}}\mathbf{x}^{{}^{\prime}}_{0} \\ \mathbf{T}\mathbf{\Theta}^{{}^{\prime}}-\mathbf{T}\mathbf{\Theta}^{{}^{\prime}}_{0} \end{array} \right] \tag{8}\]  
For an initially straight beam, the strain measures \(\varepsilon\) can be thus written as:  
\[\varepsilon=\left[ \begin{array}{c}\mathbf{\gamma}\\ \mathbf{\kappa}\end{array} \right]=\left[ \begin{array}{c}\mathbf{R}^{\mathrm{T}} \mathbf{x}^{{}^{\prime}}-\mathbf{A}_{1}\\ \mathbf{T}\mathbf{\Theta}^{{}^{\prime}}\end{array} \right] \tag{9}\]  
in which \(\mathbf{\gamma}=[\gamma_{1},\,\gamma_{2},\,\gamma_{3}]^{\mathrm{T}}\) quantifies the beam axis extension and two shearing; \(\mathbf{\kappa}=[\kappa_{1},\,\kappa_{2},\,\kappa_{3}]^{\mathrm{T}}\) measures its twist and two curvatures in local frame. The tangent operator can be obtained from [10], giving:  
\[\mathbf{T}=\mathbf{I}_{3\times 3}-a_{2}(\mathbf{\Theta})\mathbf{\tilde{\Theta}}+a_{3}( \mathbf{\Theta})\mathbf{\tilde{\Theta}}\mathbf{\tilde{\Theta}} \tag{10}\]  
with the coefficients:  
\[a_{3}(\mathbf{\Theta})=\frac{\|\mathbf{\Theta}\|-\sin(\|\mathbf{\Theta}\|)}{\|\mathbf{\Theta} \|^{3}}\hskip 28.452756 pta_{4}(\mathbf{\Theta})=\frac{1}{\|\mathbf{\Theta}\|^{2}} \Big{(}1-\frac{\|\mathbf{\Theta}\|}{2}\cot(\frac{\|\mathbf{\Theta}\|}{2})\Big{)} \tag{11}\]Consequently, based on the rigid perimeter and small strain assumptions, the non-vanishing terms in Equation (6) can be given as:  
\[\mathbf{E}=\left[ \begin{array}{ccc}\gamma_{1}+\kappa_{3}\xi_{2}-\kappa_{2}\xi_{ 3}&\frac{1}{2}(\gamma_{2}+\kappa_{1}\xi_{3})&\frac{1}{2}(\gamma_{3}-\kappa_{1} \xi_{2})\\ \frac{1}{2}(\gamma_{2}+\kappa_{1}\xi_{3})&0&0\\ \frac{1}{2}(\gamma_{3}-\kappa_{1}\xi_{2})&0&0\end{array} \right] \tag{12}\]  
Assuming a constitutive law of isotropic linear elastic materials, the second Piola-Kirchhoff stress tensor \(\mathbf{S}=\left[S_{11},\ S_{12},\ S_{13}\right]^{\mathrm{T}}\) refers to:  
\[\mathbf{S}=\mathbf{DE},\qquad\mathbf{D}=\left[ \begin{array}{ccc}E&0&0\\ 0&G&0\\ 0&0&G\end{array} \right] \tag{13}\]  
in which \(E\), and \(G\) are the elasticity modulus and shear modulus, respectively. Based on the Green-Lagrange strain tensor, the internal virtual work can be computed as:  
\[\mathbf{\delta\mathcal{W}_{\mathrm{int}}}=\int_{V}\delta\mathbf{E}^{\mathrm{T} }\mathbf{S}\mathrm{d}V=\int_{l}\delta\mathbf{E}\ \mathbf{DE}\delta_{1} \tag{14}\]  
with \(\mathbf{\tilde{E}}=\mathbf{\varepsilon}\) and the elastic matrix:  
\[\mathbf{\tilde{D}}=\left[ \begin{array}{cccccc}EA&0&0&0&0&0\\ &\tau GA&0&0&0&0\\ &&\tau GA&0&0&0\\ &&GI_{0}&0&0\\ &&\mathrm{Sym}&EI_{33}&-EI_{32}\\ &&&&&EI_{22}\end{array}