## 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
Besides, we also have the (\(5\times 5\)) symmetric (and positive definite) matrices of the surface mass and inertia moment densities:  
\[\mathbb{M}_{/\overline{S}}\stackrel{{\mathrm{def}}}{{=}}\rho h \begin{pmatrix}\frac{\dot{\mathfrak{q}}_{1}}{12}&\dot{\mathfrak{q}}_{1}&\dot{ \mathfrak{q}}_{2}&\dot{\mathfrak{q}}_{3}\\ \frac{12+h^{2}\mathcal{G}}{12}&0&0&\frac{h^{2}\mathcal{H}}{6}&0\\ 0&\frac{h^{2}}{12}+\frac{h^{4}\mathcal{G}}{80}&-\frac{h^{2}\mathcal{H}}{6}&0 &0\\ 0&-\frac{h^{2}\mathcal{H}}{6}&\frac{12+h^{2}\mathcal{G}}{12}&0&0\\ \frac{h^{2}\mathcal{H}}{6}&0&0&\frac{h^{2}}{12}+\frac{h^{4}\mathcal{G}}{80}&0\\ 0&0&0&0&\frac{12+h^{2}\mathcal{G}}{12}\end{pmatrix}\begin{pmatrix}\mathfrak{p}_ {a_{\tau}}\\ \mathfrak{p}_{a_{\tau}}\\ \mathfrak{p}_{a_{\tau}}\\ \mathfrak{p}_{a_{\tau}}\\ \mathfrak{p}_{a_{\tau}}\\ \mathfrak{p}_{a_{\upsilon}}\\ \mathfrak{p}_{a_{\upsilon}}\end{pmatrix} . \tag{71b}\]  
As already expected and observed with the former simplifications that arise in the case of thin shells, where \(h|\mathcal{H}|\ll 1\) and \(h^{2}|\mathcal{G}|\ll 1\), the translational and rotational inertial contributions are also uncoupled at the first approximation order in \(h/\max\limits_{l=1,2}\{R_{n_{l}}\}\). As for the elastic energy, the complementary kinetic energy introduced in Equation (68) (with also, in fact, a necessary condition of maximization) also results from the Legendre-Fenchel transformation:  
\[\overline{\mathcal{K}}_{c}[\mathbf{\pi}]\overset{\text{def}}{=}\sup_{ \text{k.a.}\,\mathfrak{q}}\left\{\int_{\mathcal{S}}\mathfrak{q}^{\text{T}}\, \mathbf{\pi}\,d\overline{\mathcal{S}}+\oint_{\overline{\mathcal{C}}}\mathfrak{q}^{ \text{T}}\,\mathbf{\pi}\,d\bar{s}_{\ell}-\overline{\mathcal{K}}[\mathfrak{q}, \mathfrak{q}]\right\}. \tag{72}\]  
Moreover, when \(\mathfrak{q}\equiv\mathfrak{q}\), then Equation (68) provides the (real) kinetic power due to the accelerations:  
\[\overline{\mathcal{P}}^{\text{acc}}[\mathfrak{q},\mathfrak{q}]\equiv \overline{\dot{\mathcal{K}}}[\mathfrak{q},\mathfrak{q}]\quad(\equiv \overline{\dot{\mathcal{K}}}_{c}[\mathbf{\pi}]\,,\text{ as }\mathbf{\pi}_{/\overline{\mathcal{S}}}[\mathfrak{q}]\equiv\mathfrak{p}_{/ \overline{\mathcal{S}}}\equiv\mathbb{M}_{/\overline{\mathcal{S}}}\mathfrak{q} _{/\overline{\mathcal{S}}}\text{ and }\mathbf{\pi}_{/\overline{\mathcal{C}}}[\mathfrak{q}] \equiv\mathbf{0}_{5\times 1})\,. \tag{73}\]  
For the sake of simplicity, we shall consider hereafter the initial conditions for the displacement, velocity, and internal stress fields to be zero (i.e., rest configurations), in addition to having no impacting loads. Otherwise, in agreement with the physical principles of causality, the nonzero initial conditions for the displacement and velocity fields must be enforced variationally in terms of singularly time-localized inertial loading forces and couples following, for instance, the ideas in [91; 92; 93; 94] and also the [PERSON]' viewpoint adopted in [95; 96]; with such a description, they can be considered as already included in the external virtual power \(\overline{\mathcal{P}}^{\text{ext}}[\mathfrak{q},\mathfrak{q}]\).