## 1 Introduction
The latter is defined with the direct orthonormal vector basis \((\mathbf{b}_{1},\mathbf{b}_{2},\mathbf{b}_{3})\) (cf. Figure 1) of the three-dimensional Euclidean space \(\mathbb{R}^{3}\). At equilibrium rest, the reference placements \(\mathcal{V}\) of the shell, of its mid-surface \(\overline{\mathcal{S}}\), and of its Lipschitz-continuous contour \(\overline{\mathcal{C}}\) are described as embedded in \(\mathbb{R}^{3}\) like  
\[ \begin{array}{rcl}\mathcal{V}&\stackrel{{\text{def}}}{{=}}& \big{\{}\mathbf{r}(\mathbf{\alpha})=\mathbf{x}(\mathbf{\alpha})=x^{i}(\mathbf{\alpha})\mathbf{b}_{i} \in\mathbb{R}^{3}\,\\ &&\mathbf{\alpha}\stackrel{{\text{def}}}{{=}}(\alpha_{1},\alpha_{2}, \alpha_{3})\in\Omega\times[-h/2\,,\,h/2]\big{\}}\\ \overline{\mathcal{S}}&\stackrel{{\text{def}}}{{=}}&\big{\{} \bar{\mathbf{r}}(\mathbf{\alpha})\stackrel{{\text{def}}}{{=}}\mathbf{r}( \mathbf{\alpha}_{1},\alpha_{2},0)\in\mathbb{R}^{3}\,,\bar{\mathbf{\alpha}}\stackrel{{ \text{def}}}{{=}}(\alpha_{1},\alpha_{2})\in\Omega\big{\}}\subset \mathcal{V}\\ \overline{\mathcal{C}}&\stackrel{{\text{def}}}{{=}}&\big{\{} \mathbf{r}(\bar{\mathbf{\alpha}}(\bar{\mathbf{z}}_{\ell}))\stackrel{{\text{ def}}}{{=}}\mathbf{r}(\mathbf{\alpha}_{1}(\bar{\mathbf{z}}_{\ell}),\alpha_{2}(\bar{\mathbf{z}}_{\ell}),0)\in \mathbb{R}^{3}\,\\ &&\bar{\mathbf{x}}(\bar{\mathbf{z}}_{\ell})\stackrel{{\text{def}}}{{=}}( \alpha_{1}(\bar{\mathbf{z}}_{\ell}),\alpha_{2}(\bar{\mathbf{z}}_{\ell}))\in\Omega \big{\}}\equiv\partial\overline{\mathcal{S}}\subset\partial\mathcal{V}\,.\end{array}  \tag{2}\]  
Here, in the first domain definition, the position of the suffix (superscript or subscript) in the coordinate \(x^{i}(\mathbf{\alpha})\) is, of course, irrelevant, since the basis \((\mathbf{b}_{1},\mathbf{b}_{2},\mathbf{b}_{3})\) is Cartesian; this will also be the case for the forthcoming orthogonal/principal basis \((\mathbf{t}_{1},\mathbf{t}_{2},\mathbf{n})\). Besides, \(\Omega\) is a bounded parametric domain in \(\mathbb{R}^{3}\) (respectively, \(\mathbb{R}^{2}\), spanned by the scalar variable pair of curvilinear coordinates \(\bar{\mathbf{x}}\) parameterizing \(\overline{\mathcal{S}}\), while \(\alpha_{3}\in[-h/2\,,\,h/2]\) denotes the shell through thickness coordinate in the shell transverse direction. The thickness of that latter is a relatively small (with respect to both the principal radius of curvature min \(|R_{n}|\) introduced hereafter and the shell in-plane characteristic lengths), but strictly positive--and possibly varying--function \(h(\bar{\mathbf{x}})\) of \(\bar{\mathbf{k}}\). The locations of a point \(P\) in the shell \(\mathcal{V}\) and a point \(\overline{P}\) on the  
Figure 1: Generic doubly curved shell model and its coordinate systems. A point \(P\) in the shell \(\mathcal{V}\) is defined as the intersection of three curves \(\mathscr{L}_{i}\) (with \(i=1,2,3\)) obtained by varying only the parameter \(\alpha_{i}\) at a time, along the principal curvature directions. The related point \(\overline{P}\) in the shell mid-surface \(\overline{\mathcal{S}}\) (corresponding to \(\alpha_{3}=0\)) is defined at the intersection of the two curves \(\mathscr{L}_{I}\) (with \(I=1,2\)) of the mid-surface \(\overline{\mathcal{S}}\). The (initial and rest) thickness \(h\) along the line \(\mathcal{L}_{3}\) is considered relatively (much) smaller than the two others (\(\mathscr{L}_{1}\) and \(\mathscr{L}_{2}\) ) or than the smaller principal radius of curvature min \(|R_{n}|\) (introduced hereafter) of \(\overline{\mathcal{S}}\). In this work, the local (unstretchable) director passing by the material points \(\overline{P}\) and \(P\) is merely considered collinear to the unit-normal vector \(\mathbf{n}\) of \(\overline{\mathcal{S}}\) in the rest reference state. neutral surface \(\overline{\mathcal{S}}\) are, respectively, expressed by vectors \(\mathbf{r}(\mathbf{\alpha})\) and \(\mathbf{r}(\bar{\mathbf{\alpha}})\).