## 1 Introduction
Currently, what it is accepted in the engineering and scientific community as key starting points of the theories of thin and moderately thick shells are the equations derived by [PERSON](-[PERSON]) and [PERSON] for the thin shell category [74] (see also [73; 75]) and [PERSON] and [PERSON] (NR) for the moderately thick shell category [72; 76; 77; 78; 79; 80]. [PERSON]'s extrapolated [PERSON]'s plate assumptions to shells of a generic geometry and developed what is commonly referred as \"the first order approximation shell theory\", the local kinematics of which is based on three displacement fields of translation and their gradients. Currently, most mathematical models and finite elements [55] usually considered for shell structures are based on the NR kinematical assumption, which locally is kinematically based on five displacement fields of translation and rotation. Limiting our present PHS analysis to this type of shell modeling, in order to keep the essential ideas in focus, attention will be further restricted hereafter to the transversely homogeneous, and tangentially isotropic, material case. Nevertheless, we still leave the stored (kinetic and elastic) energy density dependent only on the reference positions of the material particles on the mid-surface. At the end of the aforementioned reduction process, the resulting partial differential equations of the linear elastic NR shell theory in a port-Hamiltonian setting consider as well some intrinsic higher-order geometric coupling effects that are often overlooked. **Organization of this manuscript. ** This article is composed as follows. We begin in Section 2 with some brief, but fundamental concepts, such as the shell initial geometry and the curvilinear surface description. To keep the essential ideas in focus, we restricted our attention to the formulation in the principal curvature coordinates. We remind then in Section 3 about the main geometrical and kinematic concepts required to describe the NR shell structure models. Section 3.1 introduces notably the NR-constrained displacement-strain relationship. The essential features of the classical 2D shell NR formulations are then obtained in Section 3.2 by using D'Alembert-Lagrange's virtual power formalism. The latter provides a most expedient and precise reduction route to the fundamental dynamic equations of the evolution and edge (boundary) conditions, while exhibiting the main sthenical variables. In order to obtain their expressions in terms of the displacement fields, the standard (but modified) [PERSON]'s linear isotropic elastic law for the stress-strain relationship is adopted for the 2D shell constitutive material. For completeness, a short discussion highlights both its well-known and possibly less-well-known limitations. Throughout our development, we provide the mechanical and geometrical interpretation of the 2D shell variables such as the strain, stress, and external loads applied on the 2D shell model in correlation with their 3D counterparts. In addition to the specific features of the NR's (mixed) first-order shell model, some remarks regarding the (mixed) higher-order shell theories are also provided to briefly assess their coupling or uncoupling effects. The essential variational features of the PHS formulation of the NR shell model are then introduced in Section 4. The equations are formally re-exposed via the Hamilton(-Lagrange-D'Alembert) generalized variational principle with the intent to add physical (engineering) insights to the fundamental formulation of the PHS of the 2D shell and surface theories. Then, we discuss the relevance of these partially or totally variational PHS formalisms, based on some works already existing in the literature. The last Section 5 concludes this communication with some remarks related to our future applications. **Notation and conventions. ** Before proceeding, we summarize some notational conventions.