## 3 Classical NR Shell Theory for Moderately Thick, Linear Elastic, Shells
### Reduction from 3D to 2D Shell Elasticity by Means of the Principle of Virtual Power
#### 3.2.1 Effective Internal Stress Force and Stress Couple Resultants
Those effective efforts are linked in principle, in a less trivial way not described here, to the resultants and moments of the edge stresses acting on an surface strip element of area \(d\mathcal{S}_{\ell}=B_{\ell}d\bar{s}_{\ell}da_{3}\) of the lateral surface \(\mathcal{S}_{\ell}\):  
\[\sigma_{\
u\
u}\stackrel{{\rm def}}{{=}}\mathbf{v}\cdot\mathbf{\sigma} \cdot\mathbf{v}\quad,\quad\sigma_{\
u\tau}\equiv\mathbf{v}\cdot\mathbf{\sigma}\cdot\mathbf{ \tau}\quad\text{and}\quad\sigma_{\
u 3}\stackrel{{\rm def}}{{=}}\mathbf{v}\cdot\mathbf{ \sigma}\cdot\mathbf{n}\quad,\quad\text{for}\ \mathbf{r}(\mathbf{\alpha})\in\mathcal{S}_{\ell} \tag{42}\]  
while performing the calculus with the differential length \(B_{\ell}d\bar{s}_{\ell}\) (defined with the _total geodetic curvature_\(B_{\ell}\stackrel{{\rm def}}{{=}}\sqrt{B_{\tau}^{2}+B_{\
u}^{2}}\) of the curve \(\mathcal{C}\) locally located at a signed distance \(\alpha_{3}\mathbf{n}\) from \(\overline{\mathcal{C}}\)) in place of \(|\epsilon_{IL}|\left(1+\frac{\alpha_{3}}{R_{n_{L}}}\right)d\bar{s}_{L}\). Note that the previous formulas apply to shells made of any \"_simple_\", but not necessarily elastic, material. The class of simple materials contains all elastic materials, the linearly viscous fluid, and many of the other materials commonly studied in the applications. Moreover, those equations reduce to the Mindlin-Reissner moderately thick plate theory when the curvatures (\(\min_{l=1,2}\{R_{n_{l}}\}=\infty\)) are set to zero. **Matrix formulation of the generalized internal force relations of the NR shell model. ** For our PHS variational purposes, the bulk Equation (40) over \(\overline{\mathcal{S}}\) can be rewritten in matrix form as  
\[\mathcal{Q}_{\mathbf{q}/\overline{\mathcal{S}}}^{\text{int}}[\mathbf{q}]=\overline{ \mathbb{D}}\,\overline{\mathbf{F}}_{/\overline{\mathcal{S}}}^{\text{int}}[\bm {q}]\,\] (43a) with the following ( \[5\times 10\] ) local differential operator matrix (with possibly non-constant coefficients) \[\begin{array}{  
Now, it is also natural (but it may not always be) to additionally introduce the energy dual-column matrix for \(\mathfrak{q}_{/\overline{c}}\) in Equation (24) on any curvilinear portion of contour length \(d\tilde{s}_{\ell}\):  
\[\overline{\mathbf{F}}_{/\overline{c}}^{\text{int}\text{ def}}\left\{ \begin{array}{l}N_{\text{vT}}\\ M_{\text{vT}}\\ N_{\text{vT}}\\ M_{\text{vT}}\\ Q_{\text{v}}\end{array} \right\}\equiv-\mathcal{Q}_{\mathfrak{q}_{/\overline{c} }}^{\text{int}\text{ def}}=-\left\{ \begin{array}{l}\mathcal{Q}_{\mathfrak{a}_{ \overline{c}}}^{\text{int}}\\ \mathcal{Q}_{\mathfrak{a}_{\overline{c}}}^{\text{int}}\\ \mathcal{Q}_{\mathfrak{a}_{\overline{c}}}^{\text{int}}\\ \mathcal{Q}_{\mathfrak{a}_{3}}^{\text{int}}\end{array} \right\}. \tag{44}\]  
**Approximate effective stress-strain constitutive laws. ** Regarding the previous (generalized) internal forces and moments in Equation (35), the final non-trivial task would merely be to compute them if the stress tensor distribution \(\sigma\) was already sthenically known through the shell thickness. As usual, this is performed by default by using some constitutive equations, which, definitive for the shell 2D-formulation, relate the prior stress resultants and stress couples (which have a more complex form than in the plate theory), instead of merely stresses, to the corresponding strains and curvatures (and therefore, express the effective internal forces and moments in terms of the displacements \((\tilde{\mathbf{u}},\theta)\)). In this work, we were interested only in isotropic linear elastic shells (i.e., whose material properties have, therefore, no directional preference in the tangent plane of the middle surface) in small displacements and transversely homogeneous through the shell thickness.