## 3 Geometrically Exact 3D Beam Model and Variational Form for Assigned Strains
### Kinematics of the 3D Beam Structural Model  
The 3D shear deformable beams represent the structural model under consideration, which is now briefly recalled also in order to introduce some notation. The beam axis in the initial reference configuration is defined by a curve \(s\mapsto\mathbf{r}_{0}(s)\in\mathbb{R}^{3}\) with the curvilinear abscissa \(s\in[0,\ell]\) and \(\ell\) the initial length of the beam axis. The description of the reference configuration is completed by a field of right-handed orthonormal triads\(s\mapsto\{\mathbf{G}_{1}(s),\mathbf{G}_{2}(s),\mathbf{G}_{3}(s)\}\) also known as material triads. The triad is attached to the beam cross-section, assumed to be rigid. The first vector of the triad is the unit tangent vector of the initial beam axis  
\[\mathbf{G}_{1}=\mathbf{r}_{0,s} \tag{7}\]  
where a comma stands for derivative, while \(\mathbf{G}_{2}\) and \(\mathbf{G}_{3}\) are aligned with the principal axes of the section. Letting \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}\) the orthonormal vectors of a fixed Cartesian system, the rotation from global frame to material local frame is defined by the rotation matrix \(\mathbf{R}_{0}(s)\):  
\[\mathbf{G}_{i}=\mathbf{R}_{0}\mathbf{e}_{i}\quad i=1\cdots 3\quad\Rightarrow \quad\mathbf{R}_{0}=\sum_{i=1}^{3}\mathbf{G}_{i}\otimes\mathbf{e}_{i}.\]  
where \(\otimes\) denotes the tensor product. The initial beam configuration is then defined by the pair \((\mathbf{r}_{0},\mathbf{R}_{0})\in\mathbb{R}^{3}\times SO(3)\). Similarly, we can define the current configuration using the current axis positions \(\mathbf{r}(s)=\mathbf{r}_{0}(s)+\mathbf{u}(s)\), with \(\mathbf{u}(s)\) the displacement of the beam axis, and the current orientations \(\{\mathbf{g}_{1},\mathbf{g}_{2},\mathbf{g}_{3}\}\) defined by the rotation \(\mathbf{R}(s)\):  
\[\mathbf{g}_{i}=\mathbf{R}\mathbf{G}_{i}=\mathbf{R}\mathbf{R}_{0}\mathbf{e}_{i},\quad i=1\cdots 3\quad\Rightarrow\quad\mathbf{R}=\sum_{i}\mathbf{g}_{i} \otimes\mathbf{G}_{i}.\]  
The material strain measure [18] for the beam is  
\[\mathbf{\varepsilon}(s)\equiv\mathbf{R}^{T}\mathbf{r}_{s}-\mathbf{G}_{1},\qquad \mathbf{\chi}(s)=\text{axial}\Big{(}\mathbf{R}^{T}\mathbf{R}_{s}\,\Big{)}. \tag{8}\]  
In terms of the rotation vector, the curvature can be written as  
\[\mathbf{\chi}(s)\equiv\text{axial}\Big{(}\mathbf{R}^{T}\mathbf{R}_{s}\,\Big{)}= \mathbf{T}(\mathbf{\varphi})^{T}\mathbf{\varphi}_{s} \tag{9}\]  
with tensor \(\mathbf{T}\) defined in Equation (2). Vectors \(\mathbf{\varepsilon}\) and \(\mathbf{\chi}\) collect strain variables work-conjugate to material resultant forces \(\mathbf{n}\) and moments \(\mathbf{m}\). The constitutive law can be expressed, for hyperelastic materials, as  
\[ \begin{bmatrix}\mathbf{n}(s)\\ \mathbf{m}(s)\end{bmatrix} = \begin{bmatrix}\mathbf{C}_{mn}&\mathbf{C}_{mn}\\ \mathbf{C}_{nm}^{T}&\mathbf{C}_{mm}\end{bmatrix}\begin{bmatrix}\mathbf{\varepsilon} (s)\\ \mathbf{\chi}(s)\end{bmatrix} . \tag{10}\]  
Defining  
\[\mathbf{\rho}= \begin{bmatrix}\mathbf{\varepsilon}\\ \mathbf{\chi}\end{bmatrix} \qquad\mathbf{\tau}= \begin{bmatrix}\mathbf{n}\\ \mathbf{m}\end{bmatrix} \qquad\mathbf{C}_{\rho}= \begin{bmatrix}\mathbf{C}_{mn}& \mathbf{C}_{nm}\\ \mathbf{C}_{mm}^{T}&\mathbf{C}_{mm}\end{bmatrix}