## 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
#### 3.2.3 D'Alembert-Lagrange's Generalized Acceleration (Inertial) Forces and Couple Resultants
** The equivalent matrix-based forms that will be convenient for our PHS purposes read as follows:  
\[\overline{\mathcal{P}}^{\rm acc}_{/\overline{S}}[\mathfrak{q}, \mathfrak{q}] \equiv \int_{\overline{S}}\mathfrak{q}^{\rm T}\,\dot{\boldsymbol{r}}[ \mathfrak{q}]\,d\overline{S}\,\text{ with }\] \[\pi_{/\overline{S}}[\mathfrak{q}]\equiv \begin{pmatrix}\pi_{ \theta_{1}}\\ \pi_{\theta_{1}}\\ \pi_{\theta_{2}}\\ \pi_{\theta_{2}}\\ \pi_{\theta_{3}}\end{pmatrix} \stackrel{{\rm def}}{{=}}\delta_{ \mathfrak{q}/\overline{S}}\overline{\mathcal{T}}[\mathfrak{q},\mathfrak{q}]\,\ \dot{\pi}_{/\overline{S}}[\mathfrak{q}]\equiv\mathcal{Q}_{q/ \overline{S}}^{\rm acc}[\mathfrak{q}]=\mathbb{M}\ddot{\mathfrak{q}}\,;\]\[ \begin{array}{rcl}\overline{\mathcal{P}}^{\mathrm{acc}}_{/\overline{C}}[\mathfrak{q}, \mathfrak{g}]&\equiv&\oint_{\overline{C}}\mathfrak{q}^{\mathrm{T}}\,\dot{\pi}[ \dot{\mathfrak{q}}]\,ds_{\ell}\;,\;\mathrm{with}\\ &&\pi_{/\overline{C}}[\dot{\mathfrak{q}}]\equiv\begin{cases}\pi_{a_{\tau}}\\ \pi_{a_{\tau}}\\ \pi_{a_{\upsilon}}\\ \pi_{a_{\upsilon}}\\ \pi_{a_{\upsilon}}\end{cases}\stackrel{{\mathrm{def}}}{{=}} \delta_{\dot{\mathfrak{q}}/\overline{C}}\overline{\mathcal{K}}[\mathfrak{q}, \dot{\mathfrak{q}}]\;,\;\dot{\pi}_{/\overline{C}}[\dot{\mathfrak{q}}]\equiv \,Q^{\mathrm{acc}}_{\mathfrak{q}/\overline{C}}[\mathfrak{q}]=\,0_{5\times 1}\,;\\ \\ \overline{\mathcal{K}}[\mathfrak{q},\dot{\mathfrak{q}}]&\equiv&\frac{1}{2} \int_{\overline{S}}\mathfrak{q}^{\mathrm{T}}\mathbb{M}\dot{\mathfrak{q}}\,d \overline{S}\equiv\overline{\mathcal{K}}_{c}[\pi]\equiv\frac{1}{2}\int_{ \overline{S}}\mathfrak{p}^{\mathrm{T}}\mathbb{M}^{-1}\mathfrak{p}\,d \overline{S}\,,\;\mathrm{with}\\ &&\mathfrak{p}\stackrel{{\mathrm{def}}}{{=}}\frac{1}{2}\frac{ \partial}{\partial\dot{\mathfrak{q}}}(\dot{\mathfrak{q}}^{\mathrm{T}}\mathbb{M }\dot{\mathfrak{q}})\;\mathrm{on}\;\overline{S}\cup\overline{C}\,.\end{array}  \tag{68c}\]  
For more generality (which is necessary for inertially enhanced models), we distinguish here two different notions of momenta, which fortuitously turn out here to be equal up to time constant on \(\overline{S}\), \(\pi_{/\overline{S}}[\dot{\mathfrak{q}}]\equiv\mathfrak{p}_{/\overline{S}} \equiv\mathbb{M}_{/\overline{S}}\mathfrak{q}_{/\overline{S}}\): the first _Newtonian kinematic_ (or _kinetic_) one \(\mathfrak{p}\) provides the _linear momenta_ defined in classical (Euler-Newton) sense:  
\[\mathfrak{p}\equiv \begin{cases}\mathfrak{p}_{a_{1}}\\ \mathfrak{p}_{\theta_{1}}\\ \mathfrak{p}_{\theta_{2}}\\ \mathfrak{p}_{\theta_{3}}\end{cases} \mathrm{on}\;\overline{S}\;\;\mathrm{and}\; \;\mathfrak{p}\equiv \begin{cases}\mathfrak{p}_{a_{\tau}}\equiv\dot{\tau}_{ \mathrm{f}}\mathfrak{p}_{\theta_{I}}\\ \mathfrak{p}_{\theta_{\tau}}\equiv\dot{\tau}_{\mathrm{f}}\mathfrak{p}_{a_{I} }\\ \mathfrak{p}_{\theta_{\tau}}\equiv\dot{\tau}_{\mathrm{f}}\mathfrak{p}_{\theta_{ \tau}}\\ \mathfrak{p}_{\theta_{\tau}}\equiv\mathfrak{p}_{\theta_{\tau}}\end{cases}  \mathrm{with}\bigg{\{}\begin{array}{rcl}\mathfrak{p}_{a_{\tau}}\dot{a}_{ \tau}+\mathfrak{p}_{\theta_{\tau}}\dot{a}_{\upsilon}&\equiv&\mathfrak{p}_{a_{ \bar{I}}}\dot{a}_{I}\\ \mathfrak{p}_{\theta_{\tau}}\dot{a}_{\tau}+\mathfrak{p}_{\theta_{\tau}}\dot{a} _{\upsilon}&\equiv&\mathfrak{p}_{\theta_{\bar{I}}}\dot{a}_{I}\\ \mathfrak{p}_{\theta_{\bar{I}}}\dot{a}_{\eta_{3}}\end{cases}\;\mathrm{on}\; \overline{C}; \tag{69}\]  
the second _Lagrangian_ one \(\pi[\dot{\mathfrak{q}}]\) provides the _generalized/canonical momenta_ that come from the (Euler-Lagrange) variational calculus as the conjugate to the generalized velocity \(\dot{\mathfrak{q}}\). One observes that their counterparts on the boundary differ for the NR shell model, as  
\[\pi_{/\overline{C}}[\dot{\mathfrak{q}}]\equiv 0_{5\times 1}\
eq\mathfrak{p}_{/ \overline{C}}\dot{\mathfrak{q}}_{/\overline{C}}\;, \tag{70}\]  
owing to the fact that, in general, on \(\overline{C}\), the displacements introduced in Equation (24) are \(\mathfrak{q}_{/\overline{C}}\
ot\equiv 0\), and their time variations \(\dot{\mathfrak{q}}_{/\overline{C}}\
ot\equiv 0\).