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Higher-order theories are required to analyse high-frequency shells' vibrations with boundary layers appearing near faces and to model waveguide dynamics. The hierarchy of refined shells' models can be constructed on the groundwork of the formal approach so-called ''dimensional reduction'' using the series expansion of the displacement, see [1, 2]. The hierarchical approach offers such features as the possibility of three-dimensional solutions' approximation in various norms and allows one the efficient use of various-order shell models together with the finite element simulation. On the other hand, it is well known that the formal reduction results the first-order approximation that has nothing to do with the asymptotically consistent classical Kirchhoff theory, moreover the bending stiffness become overestimated if the plane stress assumption is not introduced [1]. Thus, the formally obtained shell models' hierarchy is incomplete, but this drawback of the formal reduction approach can be eliminated if the boundary conditions on the shell faces are satisfied in the two-dimensional model. Indeed, for shell theories formulated in terms of two-dimensional manifolds on the basis of the lagrangian formalism the boundary conditions translated from the faces to the base surface and expressed through the set of expansion coefficients become the constraint equations for the variational problem [3]. It can be shown that the first-order theory accounting the constraints transforms immediately to the Kirchhoff theory with no one supplementary assumption.
Here the arbitrary-order model of an elastic shell is formulated as a two-dimensional lagrangian continuum system defined within a set of field variables, the surface density of the lagrangian, and the constraint equations. The field variables of the first kind are the biorthogonal expansion coefficients of the displacement, and the dynamic equations are derived using the Lagrange multipliers method. The normal waves in the plane layer and the hollow cylinder are considered to investigate the approximation of the three-dimensional spectral problem solution by the hierarchy of two-dimensional solutions [4] obtained on the groundwork of as well extended shell theories [3] as elementary ones [2]. The propagating modes with negative group velocity are considered as a very interesting particular case. Two types of base functions are used, the Legendre polynomials resulting classical shell models and compact basis resulting the finite element ones. It is shown that the constraints following from the boundary conditions on the faces effect qualitatively on the shell models properties. Edge waves modeling is also shown.

[1] Vekua I. N., Shell Theory: General Methods of Construction, Pitman Adv. Publ. Program, Boston, 1985.
[2] Zhavoronok S. I., A Vekua-type linear theory of thick elastic shells, ZAMM 2014, 94(1-2):164-184.
[3] Zhavoronok S. I., On the variational formulation of the extended thick anisotropic shells theory of I. N. Vekua type, Procedia
Engineering 2015, 111: 868-895.
[4] Egorova O. V., Zhavoronok S. I., and Kurbatov A. S., An application of various N’th order shell theories to normal waves
propagation problems, PNPU Mech. Bull. 2015, No. 2: 36-59.