## 1 Introduction
We can identify then the mid-surface unit normal vector \(\mathbf{n}\equiv\mathbf{t}_{1}\times\mathbf{t}_{2}\) and choose (as illustrated in Figure 2) the geodesic normal direction of each curve \(\mathcal{L}_{l}\) as \(\mathbf{g}_{I}\stackrel{{\mathrm{def}}}{{=}}\mathbf{n}\times\mathbf{t}_{I} \equiv\mathbf{\epsilon}_{IJ}\mathbf{t}_{J}\). The Darboux-Ribaucour formula for each principal curvature line \(\mathcal{L}_{l}\) of \(\overline{\mathcal{S}}\) reads then  
\[\frac{\partial}{\partial\overline{s}_{I}}\left\{ \begin{array}{c}\mathbf{t}_{I} \\ \mathbf{g}_{I}\\ \mathbf{n}\end{array} \right\}=\left( \begin{array}{ccc}0&\frac{-1}{R_{g_{I}}}& \frac{-1}{R_{n_{I}}}\\ \frac{1}{R_{g_{I}}}&0&\frac{-1}{T_{g_{I}}}\\ \frac{1}{R_{n_{I}}}&\frac{1}{T_{g_{I}}}&0\end{array} \right)\left\{ \begin{array}{c} \mathbf{t}_{I}\\ \mathbf{g}_{I}\\ \mathbf{n}\end{array} \right\}\text{with }\frac{1}{T_{g_{I}}}=0\,,\text{ for }I=1,2\,, \tag{4}\]  
while denoting the radius of principal normal curvature as \(|R_{n_{I}}|\), the radius of geodesic curvature as \(|R_{g_{I}}|\), and the geodesic (or geodetic) torsion as \(|T_{g_{I}}|\), which is infinite along each principal curvature line \(\mathcal{L}_{l}\). For completeness, the following simplified tensor expressionsof the Weingarten's and [PERSON] formulas are also mentioned with the mid-surface-covariant gradient operator \(\overline{\
abla}\) introduced hereafter in Equation (18):  
\[\overline{\
abla}\mathbf{n}=\frac{\mathbf{t}_{I}\otimes\mathbf{t}_{I}}{R_{n_{ I}}}\] (Weingarten) \[\overline{\
abla}\mathbf{t}_{I}=\epsilon_{II}\bigg{[}\frac{\mathbf{t}_{I} \otimes\mathbf{t}_{I}}{R_{Sj}}+\frac{\mathbf{t}_{I}\otimes\mathbf{t}_{J}}{R_{Sl}}\bigg{]} -\frac{\mathbf{t}_{I}\otimes\mathbf{n}}{R_{n_{I}}}\,\ \text{for}\ J=1,2\quad\text{(Gauss)}. \tag{5}\]  
Naturally, those expressions are defined at regular material points \(\overline{P}\in\overline{\mathcal{S}}\). Singular cases where the normal \(\mathbf{n}\) (and the director \(\mathbf{n}^{\prime}\)) can be discontinuous, as at shell junctions/intersections, have been considered in other works; see e.g., [64]. With respect to the previous description and Figure 1, the three coordinates \(\mathbf{r}(\mathbf{\alpha})\) of any material point \(P\) of the shell in its reference configuration \(\mathcal{V}\) can be defined then with the two curvilinear surface coordinates and the normal to the surface coordinate as  
\[\mathbf{r}(\mathbf{\alpha})=\bar{\mathbf{r}}(\bar{\mathbf{\alpha}})+\alpha_{3}\ \mathbf{n}(\bar{\mathbf{\alpha}})\,. \tag{6}\]  
The following Lame (metric) coefficients (with respect to the _principal_ coordinate system) are used on the parallel surface passing by that point \(P\):  
\[B_{I}\stackrel{{\text{def}}}{{=}}\|\frac{\partial\mathbf{r}}{ \partial\mathbf{t}_{I}}\|\equiv A_{I}\bigg{(}1+\frac{\alpha_{3}}{R_{n_{I}}}\bigg{)} \ \text{for}\ I=1,2\ ;\quad\text{and}\ B_{3}\stackrel{{\text{def}}}{{=}}\|\frac{ \partial\mathbf{r}}{\partial\alpha_{3}}\|\equiv 1\,. \tag{7}\]  
**Darboux-Ribaucour's geodesic frame on the lateral surface \(\mathcal{S}_{\ell}\)**. A complete analysis of the non-closed shell dynamics also requires a detail geometric description of the contour \(\overline{\mathcal{C}}=\overline{\mathcal{S}}_{\ell}\cap\overline{\mathcal{S}}\) of the reference mid-surface boundary (cf. Figure 1). Similar to [76] (page 39), that contour \(\overline{\mathcal{C}}\) that consists of a finite set of piecewisely smooth curves will be described with another orthonormal basis \((\bar{\mathbf{\tau}},\bar{\mathbf{v}},\bar{\mathbf{n}})\) formed with a couple \((\bar{\mathbf{\tau}},\bar{\mathbf{v}})\) of unit-tangential and outward-pointing-normal basis vectors (both being tangent though the mid-surface \(\overline{\mathcal{S}}\) and existing almost everywhere along the Lipschitz-continuous contour \(\overline{\mathcal{C}}\)). These provide the directional cosines \(\mathbf{\tau}_{I}\equiv\bar{\mathbf{\tau}}\cdot\mathbf{t}_{I}\) and \(\bar{\mathbf{v}}_{I}\equiv\bar{\mathbf{v}}\cdot\mathbf{t}_{I}\).