## 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  
The IVP (see [84; 85]) due to the stresses within the moderately thick shell:  
\[ \begin{split}\mathcal{P}^{\text{int}}[\boldsymbol{u},\underline{ \boldsymbol{u}}]&(t)\stackrel{{\text{def}}}{{=}}- \int_{\mathcal{V}}\boldsymbol{\sigma}(\boldsymbol{\alpha},t):\boldsymbol{ \varepsilon}^{s}[\underline{\boldsymbol{u}}(\boldsymbol{\alpha})]d\mathcal{V} \,\\ \text{where }\boldsymbol{\varepsilon}^{s}[\underline{\boldsymbol{u}}]& \stackrel{{\text{def}}}{{=}}\frac{1}{2}\Big{[}\
abla \boldsymbol{u}+(\
abla\boldsymbol{u})^{\text{T}}\Big{]}\text{ and }\boldsymbol{\sigma}^{\text{T}}=\boldsymbol{\sigma}\end{split}  \tag{32}\]  
relies on the ensuing (Cauchy) classical (non-polar) continuum stress tensor (in absence of bulk concentrated loading couples):  
\[\boldsymbol{\sigma}=\sigma_{II}\boldsymbol{t}_{I}\otimes\boldsymbol{t}_{I}+ \sigma_{I3}\boldsymbol{t}_{I}\otimes\boldsymbol{n}+\sigma_{31}\boldsymbol{n} \otimes\boldsymbol{t}_{I}+\sigma_{33}\boldsymbol{n}\otimes\boldsymbol{n}\,. \tag{33}\]  
Here, \(\sigma_{ii}\) (for \(i\in\{1,2,3\}\)) represent (respectively, tangent and transverse) normal stress components (respectively, for \(i\in\{1,2\}\) and \(i=3\)); \(\sigma_{II}\equiv\sigma_{II}\) (for \(I\
eq J\), \(I\), \(\in\{1,2\}\)) are the membrane (or tangent) shear stresses; \(\sigma_{31}\equiv\sigma_{I3}\) are the transverse shear stresses. Taking the symmetries \(\sigma_{ij}=\sigma_{ii}\) into account, which again is valid for non-polar media and in absence of bulk concentrated couples, as well as the symmetries of the linear virtual strain fields \(\varepsilon^{s}_{ij}[\underline{\boldsymbol{u}}]\equiv\varepsilon^{s}_{ji}[ \underline{\boldsymbol{u}}]\), the well-known IVP due to the stresses generated within a thin shell can be expressed and provides then the requested IVP expression for the 2D shell modeling:  
\[ \begin{split}\overline{\mathcal{P}}^{\text{int}}[\mathfrak{q}, \mathfrak{q}]&\stackrel{{\text{def}}}{{=}}-\int_{ \overline{\mathcal{S}}}\!\Big{[}N_{II}\bar{\theta}_{I||I}+|\epsilon_{JK}|M_{ IJK}\bar{\theta}_{K||I}+Q_{I}\Big{(}\bar{\theta}_{3||I}+\epsilon_{IK}\bar{ \theta}_{K}\Big{)}\Big{]}d\overline{\mathcal{S}}\,\\ \text{with}&|\epsilon_{JK}|M_{IJK}\bar{\theta}_{K| I}\equiv|\epsilon_{IJ}|\Big{(}M_{IJI}\bar{\theta}_{I||I}+M_{II}\bar{ \theta}_{I||I}\Big{)}\.\end{split}  \tag{34}\]  
Accordingly, with [PERSON]'s shell modeling expositions and terminology (cf. the generalization in [72], Section 11, pages 512-515), this functional--wherein the variations with respect to \(\alpha_{3}\) are completely eliminated (by integration) to give a 2D theory of thin shells--involves all the possible, cross-sectional, and internal force and moment resultants (per infinitesimal unit of length \(|\epsilon_{IL}|d\bar{s}_{L}=|\epsilon_{IL}|A_{L}d\alpha_{L}\) of the neutral surface \(\overline{\mathcal{S}}\)) for the shell, with, for \(I,J\in\{1,2\}\),  
\[ \begin{split} N_{IJ}&\stackrel{{\text{def}}}{{= }}\int_{-h/2}^{h/2}\sigma_{II}|\epsilon_{IL}|\Big{(}1+\frac{\alpha_{3}}{R_{ \alpha_{L}}}\Big{)}d\alpha_{3}\approx\int_{-h/2}^{h/2}\sigma_{IJ}d\alpha_{3}+ \mathcal{O}\!\left(\max_{L=1,2}\!\Big{\{}\big{|}\frac{h}{R_{\alpha_{L}}} \big{|}\Big{\}}\right)\\ Q_{I}&\stackrel{{\text{def}}}{{=}}\int_ {-h/2}^{h/2}\sigma_{I3}|\epsilon_{IL}|\Big{(}1+\frac{\alpha_{3}}{R_{\alpha_{L }}}\Big{)}d\alpha_{3}\approx\int_{-h/2}^{h/2}\sigma_{I3}d\alpha_{3}+\mathcal{O }\!\left(\max_{L=1,2}\!\Big{\{}\big{|}\frac{h}{R_{\alpha_{L}}}\big{|}\Big{\}} \right)\\ \epsilon_{JK}M_{IJK}&\stackrel{{\text{def}}}{{= }}\int_{-h/2}^{h/2}\sigma_{IJ}\alpha_{3}|\epsilon_{IL}|\Big{(}1+\frac{ \alpha_{3}}{R_{\alpha_{L}}}\Big{)}d\alpha_{3}\approx\int_{-h/2}^{h/2}\sigma_{ IJ}\alpha_{3}d\alpha_{3}+\mathcal{O}\!\left(\max_{L=1,2}\!\Big{\{}\big{|}\frac{h ^{2}}{R_{\alpha_{L}}}\big{|}\Big{\}}\right)\end{split}