## 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  
As before, starting from the standard 3D definition of the AVP:  
\[\mathcal{Q}^{\text{acc}}[\mathbf{u},\underline{\mathbf{u}}](t)\stackrel{{ \text{def}}}{{=}}\int_{\mathcal{V}}\rho\bar{\mathbf{u}}(\mathbf{\alpha},t)\cdot\underline {\mathbf{\pi}}(\mathbf{\alpha})\,d\mathcal{V} \tag{62}\]  
where \(\rho(\bar{\mathbf{\alpha}})\) represents the 3D mass density, we obtain then the 2D expression of the acceleration virtual power:\[\overline{\mathcal{P}}^{\rm acc}[\mathfrak{q},\mathfrak{q}] \stackrel{{\rm def}}{{=}} \int_{\overline{S}}\int_{-h/2}^{h/2}\rho\Big{[}\big{(}\ddot{u}_{ I}+\alpha_{3}\ddot{\beta}_{I}\big{)}\big{(}\ddot{u}_{I}+\alpha_{3}\dot{\beta}_{I} \big{)}+\ddot{u}_{3}\ddot{u}_{3}\Big{]}\Big{[}1+2\alpha_{3}\mathcal{H}+\alpha_{ 3}^{2}\mathcal{G}\Big{]}da_{3}d\overline{S} \tag{63}\] \[\equiv \int_{\overline{S}}\rho h\bigg{[}\bigg{(}1+\frac{h^{2}}{12} \mathcal{G}\bigg{)}\ddot{u}_{i}\dot{u}_{i}+\frac{h^{2}}{6}\mathcal{H}\Big{(} \ddot{u}_{i}\dot{\beta}_{I}+\ddot{\beta}_{I}\dot{\beta}_{I}\Big{)}\] \[\qquad\quad+\frac{h^{2}}{4}\bigg{(}\frac{1}{3}+\frac{h^{2}}{2} \mathcal{G}\bigg{)}\ddot{\beta}_{I}\dot{\beta}_{I}\bigg{]}d\overline{S}\] \[\equiv \int_{\overline{S}}\rho h\bigg{\{}\bigg{[}\bigg{(}1+\frac{h^{2}}{ 12}\mathcal{G}\bigg{)}\ddot{u}_{i}+\epsilon_{iI3}\frac{h^{2}\mathcal{H}}{6} \dot{\beta}_{I}\bigg{]}\dot{u}_{i}\] \[\qquad\quad+\bigg{[}\epsilon_{I}\frac{h^{2}\mathcal{H}}{6}\ddot{ u}_{I}+\bigg{(}\frac{h^{2}}{12}+\frac{h^{4}\mathcal{G}}{80}\bigg{)}\ddot{ \theta}_{I}\bigg{]}\dot{\theta}_{I}\bigg{\}}d\overline{S}\,.\]  
Here, we used the fact that \(\beta_{I}=\epsilon_{Ij}\theta_{I}\equiv\epsilon_{IJ3}\theta_{J}\), while introducing the [PERSON]'s 3D permutation (or alternating) symbol:  
\[\epsilon_{ijk} \stackrel{{\rm def}}{{=}} \det\left( \begin{array}{ccc}\delta_{i1}&\delta_{i2}&\delta_{i3} \\ \delta_{j1}&\delta_{j2}&\delta_{j3}\\ \delta_{k1}&\delta_{k2}&\delta_{k3}\end{array} \right) \tag{64}\] \[\equiv\left\{ \begin{array}{ccc}1&,&\text{if }(i,j,k)\text{ are an even permutation of }(1,2,3)\\ -1&,&\text{if }(i,j,k)\text{ are an odd permutation of }(1,2,3)\\ 0&,&\text{if any two indices are the same number}\end{array} \right.\,.\]  
We subsequently inferred as the D'Alembert-Lagrange generalized acceleration forces in Equation (29) related on \(\overline{S}\) to the _virtual kinematic variable_\(q\):  
\[\mathcal{Q}_{\dot{u}_{i}}^{\rm acc} = \rho h\bigg{[}\bigg{(}1+\frac{h^{2}}{12}\mathcal{G}\bigg{)} \ddot{u}_{i}+\epsilon_{iI3}\frac{h^{2}}{6}\mathcal{H}\dot{\theta}_{I}\bigg{]}\,\text{ for }i=1,2,3 \tag{65a}\] \[\mathcal{Q}_{\theta_{I}}^{\rm acc} = \frac{\rho h^{2}}{12}\bigg{[}2\epsilon_{I1}\mathcal{H}\ddot{u}_{I }+\bigg{(}1+\frac{3h^{2}}{20}\mathcal{G}\bigg{)}\ddot{\theta}_{I}\bigg{]}\,\text{ for }I=1,2\,. \tag{65b}\]  
On the other hand, there is no surface contribution for the AVP of the NR model, i.e.,  
\[\overline{\mathcal{P}}^{\rm acc}[\mathfrak{q},\mathfrak{q}]=\overline{ \mathcal{P}}^{\rm acc}\overline{\mathcal{P}}^{\rm acc}[\mathfrak{q},\mathfrak{ q}]\text{ as }\overline{\mathcal{P}}^{\rm acc}[\mathfrak{q},\mathfrak{ q}]=0\]  
and so  
\[\mathcal{Q}_{q,\varphi}^{\rm acc}[\mathfrak{q}]=0\text{ for }q\in\{\ddot{u}_{\tau}, \theta_{\tau},\ddot{u}_{\tau},\theta_{\
u},\ddot{u}_{3}\}\text{ on }\overline{\mathcal{C}}. \tag{66}\]  
Lately, as the 3D model that is related to a well-known potential (of inertial forces) named kinetic co-energy, the 2D shell also derives from an analogous one that reads as  
\[\overline{\mathcal{K}}\big{[}\mathfrak{q},\mathfrak{q}\big{]} \equiv \frac{1}{2}\int_{\overline{S}}\rho h\bigg{[}\bigg{(}1+\frac{h^{2} }{12}\mathcal{G}\bigg{)}\dot{u}_{i}\dot{u}_{i}+\frac{h^{2}}{6}\mathcal{H}\dot{ u}_{i}\dot{\beta}_{I}+\frac{h^{2}}{4}\bigg{(}\frac{1}{3}+\frac{h^{2}}{20} \mathcal{G}\bigg{)}\dot{\beta}_{I}\dot{\beta}_{I}\bigg{]}d\overline{S}\,. \tag{67}\]  
**Matrix formulation of the acceleration power and the kinetic co-energy and energy.