## 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
 \tag{35}\]  
Equation (35) allows introducing the following effective internal force tensor for later use:  
\[\boldsymbol{\Sigma}\stackrel{{\text{def}}}{{=}}N_{IJ}\boldsymbol{t} _{I}\otimes\boldsymbol{t}_{I}+Q_{I}\boldsymbol{t}_{I}\otimes\boldsymbol{n}. \tag{36}\]The latter contains no contribution, like \(N_{33}\mathbf{n}\otimes\mathbf{n}\), that would be related to the energetically work-less or negligible normal stress \(\sigma_{33}\mathbf{n}\otimes\mathbf{n}\). The tensor \(\mathbf{\Sigma}\) includes, however, other contributions providing various stress resultants that act as illustrated in Figure 4. More explicitly, \(N_{II}t_{I}\) and \(|\epsilon_{IJ}|N_{II}t_{J}\) for \(I=1,2\) are, respectively, the \"_membrane_\" (or \"in-plane\") _normal_ and _shear_ stress resultants (or force densities) per unit length \(|\epsilon_{II}|dS_{L}\) of neutral surface \(\overline{\mathcal{S}}\), acting on an surface strip element of area \(d\mathcal{S}_{I}=|\epsilon_{IL}|B_{L}da_{L}da_{3}\equiv|\epsilon_{IL}|\bigg{(} 1+\frac{\alpha_{3}}{R_{n_{L}}}\bigg{)}d\varepsilon_{L}da_{3}\) and oriented as \(\epsilon_{IL}t_{L}=\mathbf{n}\times t_{I}\). Besides, \(Q_{I}\mathbf{n}\) for \(I=1,2\) in Equation (36) represent the _transverse shear_ stress resultants per unit length \(|\epsilon_{IL}|dS_{L}\) of \(\overline{\mathcal{S}}\), due to possibly nonzero _transverse shear_ stresses \(\sigma_{I3}\) acting on the prior surface strip element. Equation (35) provides as well a stress couple (pseudo) tensor made of different contributions as  
\[\mathcal{M}\overset{\text{def}}{=}|\epsilon_{IK}|M_{IJK}\mathbf{t}_{I}\otimes\mathbf{ t}_{K}\equiv|\epsilon_{II}|\big{[}M_{IJI}\mathbf{t}_{I}\otimes\mathbf{t}_{I}+M_{III} \mathbf{t}_{I}\otimes\mathbf{t}_{J}\big{]} \tag{37}\]  
but without the component on \(\mathbf{t}_{I}\otimes\mathbf{n}\) (for \(I=1,2\)). This is due to the fact that the transverse differential element of the shell along \(\mathbf{n}\) has negligible side dimensions with respect to \(h\) and can sustain then no moment action about \(\mathbf{n}\). In the \"membrane\" tensor \(\mathcal{M}\), the terms \(|\epsilon_{II}|M_{II}t_{J}\) and \(|\epsilon_{IJ}|M_{IJI}t_{I}\) for \(I=1,2\) represent, respectively, (as illustrated in Figure 4) the _bending_ and _twisting_ stress couples per unit length \(|\epsilon_{IL}|d\overline{s}_{L}\) of \(\overline{\mathcal{S}}\), and that results from the stress acting about the neutral surface \(\overline{\mathcal{S}}\) on the aforementioned surface strip element at the local material point \(\overline{P}\in\overline{\mathcal{S}}\). **Remark on the geometric shape parameter of intrinsic coupling modes. ** The approximations of the foregoing effective stress forces and stress couples at first order in \(\alpha_{3}\) in Equation (35) correspond to the standard NR first-order shell case; they notably imply the following standard relations of symmetry:  
\[N_{12}=N_{21}\text{ and }M_{121}=-M_{212}\,,\text{ as }\sigma_{12}=\sigma_{21}\,,\]  
which obviously hold only when the second term \(\frac{\alpha_{3}}{R_{n_{L}}}\) in parentheses can be neglected (as for the shallow shell and flat plate cases) or \(R_{n1}\equiv R_{n2}\) (as for the spherical shell case) [76]. Nevertheless, these symmetries do not necessarily arise otherwise if higher orders of approximation are considered; instead, we have, in general,  
\[N_{12}-N_{21}\equiv\frac{M_{121}}{R_{n_{1}}}+\frac{M_{212}}{R_{n_{2}}}\equiv \varrho\int_{-h/2}^{h/2}\sigma_{12}\frac{12\alpha_{3}}{h}\,d\alpha_{3}\,, \text{ as }\sigma_{12}=\sigma_{21}\,, \tag{38}\]  
which must be viewed as stress field constraints when the shell modeling must be interpreted as resulting from an asymptotic dimensional reduction process.