We also define the director across the shell surface as $$\begin{equation} \mathbf{d}\left(\xi_{\alpha}\right)=\mathbf{x}\left(\xi_{1},\xi_{2},1\right)-\mathbf{x}\left(\xi_{1},\xi_{2},-1\right)=\sum\limits _{b=1}^{m}M_{b}\left(\xi_{\alpha}\right)\mathbf{d}_{b}\,,\label{eq384} \end{equation}$$ where $$\begin{equation} \mathbf{d}_{b}=\mathbf{x}_{b+m}-\mathbf{x}_{b},\quad b=1-m\label{eq385} \end{equation}$$ are the nodal directors. Note that the magnitude of the nodal director represents the shell thickness, $h\left(\xi_{\alpha}\right)=\left\Vert \mathbf{d}\left(\xi_{\alpha}\right)\right\Vert$ and the shell thicknesses at the nodes are $h_{b}=\left\Vert \mathbf{d}_{b}\right\Vert$ . With these definitions we find that the interpolation across the parametric space of the shell element is $$\begin{equation} \mathbf{x}\left(\xi_{i}\right)=\mathbf{\hat{x}}\left(\xi_{\alpha}\right)+\frac{1}{2}\xi_{3}\mathbf{d}\left(\xi_{\alpha}\right)=\sum\limits _{b=1}^{m}M_{b}\left(\xi_{\alpha}\right)\left(\mathbf{\hat{x}}_{b}+\frac{1}{2}\xi_{3}\mathbf{d}_{b}\right)\,.\label{eq386} \end{equation}$$ From this relation we can obtain the covariant basis vectors as $$\begin{equation} \begin{aligned}\mathbf{g}_{\alpha}\left(\xi_{i}\right) & =\frac{\partial\mathbf{x}}{\partial\xi_{\alpha}}=\sum\limits _{b=1}^{m}\frac{\partial M_{b}}{\partial\xi_{\alpha}}\left(\mathbf{\hat{x}}_{b}+\frac{1}{2}\xi_{3}\mathbf{d}_{b}\right)\\ \mathbf{g}_{3}\left(\xi_{i}\right) & =\frac{\partial\mathbf{x}}{\partial\xi_{3}}=\frac{1}{2}\sum\limits _{b=1}^{m}M_{b}\left(\xi_{\alpha}\right)\mathbf{d}_{b} \end{aligned} \,,\label{eq387} \end{equation}$$ from which we may evaluate the contravariant basis vectors $\mathbf{g}^{j}$ using the identity $\mathbf{g}_{i}\cdot\mathbf{g}^{j}=\delta_{i}^{j}$ . Then, the gradients of the shape functions are given by $$\begin{equation} \grad M_{b}=\frac{\partial M_{b}}{\partial\xi_{\alpha}}\mathbf{g}^{\alpha},\quad\grad\left(\frac{1}{2}\xi_{3}M_{b}\right)=\frac{1}{2}\left(\xi_{3}\grad M_{b}+M_{b}\mathbf{g}^{3}\right)\,.\label{eq388} \end{equation}$$ It follows from (4.2.1-7) that the virtual displacement is $$\begin{equation} \delta\mathbf{u}\left(\xi_{i}\right)=\sum\limits _{a=1}^{m}M_{a}\left(\xi_{\alpha}\right)\left(\delta\mathbf{\hat{u}}_{a}+\frac{1}{2}\xi_{3}\delta\mathbf{d}_{a}\right)\,,\label{eq389} \end{equation}$$ and the incremental displacement is $$\begin{equation} \Delta\mathbf{u}\left(\xi_{i}\right)=\sum\limits _{b=1}^{m}M_{b}\left(\xi_{\alpha}\right)\left(\Delta\mathbf{\hat{u}}_{b}+\frac{1}{2}\xi_{3}\Delta\mathbf{d}_{b}\right)\,.\label{eq390} \end{equation}$$

In FEBio, for historical reasons, the nodal director $\mathbf{d}_{b}$ is currently called rotation. This is a misnomer and users should treat this rotation as the vector $\mathbf{d}_{a}$ whose components have units of length. Thus, fixing or prescribing rotation components in the input file effectively places these constraints on the components of the nodal director; similarly, requesting rotation in the output files will produce the components of the director.

When this type of shell is connected face-to-face with a solid element, the nodes located at $\mathbf{\hat{x}}_{b}$ automatically share their displacement degrees of freedom $\mathbf{u}_{b}$ with the corresponding nodes from the face of the solid element. However, no continuity is enforced between the directors $\mathbf{d}_{b}$ and the solid element deformation. One consequence of this condition is that a shell sandwiched between two solid elements will not detect out-of-plane shear and normal stresses transmitted by the solid element(s). Another consequence is that bending of the solid element(s) will not produce a bending moment in the shell. Therefore, these shell elements are best used as shell-only structures.