## 3 Classical NR Shell Theory for Moderately Thick, Linear Elastic, Shells
### Standard Infinitesimal Deformation Kinematics (For Transversely Rigid Shells)
 \tag{19}\]  
This decomposition emphasizes separately the _membrane_ (or _in-plane, tangent_ in reference to the local tangent plane \(\mathcal{T}_{\overline{\mathcal{P}}}\overline{\mathcal{S}}\)) _strains_\(\varepsilon^{s}_{IJ}\) (with \(I=J\) for the _in-plane normal strains_ and \(I\
eq J\) for the _tangential shear_ ones, while \(I,J\in\{1,2\}\)):  
\[\varepsilon^{s}_{IJ}[\tilde{\mathbf{u}}]\stackrel{{\text{ def}}}{{=}}\tilde{u}_{I;I} \equiv \frac{1}{B_{I}}\bigg{[}\frac{\partial\tilde{u}_{I}}{\partial a_{I }}+\tilde{u}_{K}\frac{\epsilon_{IK}A_{I}}{R_{gl}}+\tilde{u}_{3}\frac{A_{I}}{R _{n_{I}}}\bigg{]}\, \tag{20a}\] \[2\varepsilon^{s}_{IJ}[\tilde{\mathbf{u}}]\stackrel{{ \text{def}}}{{=}}\tilde{u}_{I;J}+\tilde{u}_{J;I} \equiv \frac{1}{B_{J}}\bigg{[}\frac{\partial\tilde{u}_{I}}{\partial a_{J }}+\tilde{u}_{J}\frac{\epsilon_{IJ}A_{I}}{R_{gl}}\bigg{]}+\frac{1}{B_{I}}\bigg{[} \frac{\partial\tilde{u}_{I}}{\partial a_{I}}+\tilde{u}_{I}\frac{\epsilon_{IJ }A_{I}}{R_{gl}}\bigg{]}\, \tag{20b}\]  
the _change-in-curvature strains_\(\kappa^{s}_{IJ}\) (with \(I=J\) for _bending_ and \(I\
eq J\) for _torsion_ or _twist_, while \(I,J\in\{1,2\}\)):  
\[\kappa^{s}_{IJ}[\mathbf{\theta}]\stackrel{{\text{def}}}{{=}}\beta_{ I;I} \equiv \frac{1}{B_{I}}\bigg{[}\frac{\partial\beta_{I}}{\partial a_{I}}+ \beta_{K}\frac{\epsilon_{IK}A_{I}}{R_{gl}}\bigg{]}\, \tag{20c}\] \[2\kappa^{s}_{IJ}[\mathbf{\theta}]\stackrel{{\text{def}}}{{=}}\beta_{ I;J}+\beta_{I;J} \equiv \frac{1}{B_{I}}\bigg{[}\frac{\partial\beta_{I}}{\partial a_{I}}+ \beta_{I}\frac{\epsilon_{IJ}A_{I}}{R_{gl}}\bigg{]}+\frac{1}{B_{I}}\bigg{[} \frac{\partial\beta_{I}}{\partial a_{I}}+\beta_{I}\frac{\epsilon_{IJ}A_{I}}{R _{gl}}\bigg{]}\, \tag{20d}\]  
and finally, the _transverse_ (or _out-of-plane_) _shear strains_\(\gamma_{I}\) (while \(I\in\{1,2\}\)):  
\[2\varepsilon^{s}_{I3}[\mathbf{u}]\equiv\gamma_{I}[\mathbf{u}]\stackrel{{ \text{def}}}{{=}}u_{I;3}+u_{3;I} \equiv \frac{1}{B_{I}}\bigg{[}\frac{\partial\tilde{u}_{3}}{\partial a_{I }}+A_{I}\bigg{(}\beta_{I}-\frac{\tilde{u}_{I}}{R_{n_{I}}}\bigg{)}\bigg{]}\,. \tag{20e}\]  
When evaluating these expressions on \(\overline{\mathcal{S}}\) (where, therefore, \(\alpha_{3}=0\) and \(B_{I}=A_{I}\)), we shall denote them as (while \(I,J\in\{1,2\}\  
with therefore  
\[\theta_{J||I}=\frac{1}{A_{I}}\frac{\partial\theta_{I}}{\partial\alpha_{I}}-\theta _{I}\frac{\epsilon_{IJ}}{R_{SI}}+\theta_{K}\frac{\delta_{IJ}\epsilon_{IK}}{R_{ SI}}. \tag{22}\]  
The transverse shear strain \(\gamma_{I}\) in Equation (21) is merely expressed here as a function of a column matrix \(\mathfrak{q}\) formed from the couple \((\bar{n},\mathbf{\theta})\) as, hereafter, in Equation (23). **Matrix formulation of the linear strain-displacement relations of the NR shell model. ** Of great interest for the PHS formulation is that the following \"surface strain-based\" matrix:  
\[\overline{\mathbf{E}}[\mathfrak{q}]\stackrel{{\text{def}}}{{= }}\overline{\mathbb{B}}\mathfrak{q}\,\ \text{ over }\overline{\mathcal{S}}\] (23a) which is locally related, by means of the (10 \[\times\] 5) differential operator matrix \[\overline{\mathbb{B}}\stackrel{{\text{def}}}{{=}}