Journal of Mathematical Sciences, Vol. 273, No. 6, July, 2023 THREE-DIMENSIONAL DYNAMIC ANALYSIS OF LAYERED ELASTIC SHELLS L. A. Aghalovyan, 1 L. G. Ghulghazaryan, 1,2 J. D. Kaplunov, 3 and D. A. Prikazchikov 3,4 UDC 539.3 Three-dimensional dynamic problem for a layered orthotropic elastic shell with free upper face is con- sidered. The interfaces between the layers are assumed to be in perfect contact and the displacements of one of the interfaces are prescribed. A long-wave asymptotic solution is constructed and the thickness resonances are determined. The obtained results can be applied in the evaluation of some parameters of the earthquakes. Keywords: 3D elastodynamics, layered shell, asymptotic method. Introduction Mathematical modeling of thin elastic multilayered solids with various boundary conditions on the faces is of significant importance for numerous applications. Among these applications, we can mention the theories related to the earthquake prediction [19, 22, 25] relying on the data of displacements measured at certain points of the region under investigation. It is emphasized that restoration of the associated dynamic parameters of the stress-strain state on the basis of the measured discrete data by taking into account possible curvature of the lay- ers is of crucial importance for seismological theories. Problems of the mechanics of multilayered elastic plates and shells with nonclassical boundary conditions, i.e., conditions imposed not only on the stress-tensor component, were studied in various publications, see, e.g., the monograph [6], as well as journal papers including the analyses of both free [1, 7, 8] and forced vibrations [2, 9, 14]. We also mention important contributions to the study of free vibrations of single-layer plates and shells in the case where one or both faces are fixed [15, 17, 20, 23, 26]. The associated long-wave high- frequency motions investigated in these papers were also thoroughly studied in the case of classical boundary conditions (formulated in terms of stresses); see [3, 5, 21]. The present paper is devoted to the subsequent development of the results obtained in the above-mentioned works. Our approach relies on the asymptotic method widely used in the statics and dynamics of thin-walled elastic structures, see, e.g., [4, 16], as well as recent monographs [10, 25], and publications [11 13, 18, 24] to name a few, that take into account the effects of prestressing, nonlocality, high contrast, and also in the contact problems for coated solids. The asymptotic technique employed in our paper starts with a scaling typical of nonclassical face boundary conditions. We consider 3D dynamic problems for two- and three-layered elastic orthotropic shells with trac- 1 Institute of Mechanics of NAS RA, Armenia. 2 Kh. Abovyan Armenian State Pedagogical University, Armenia. 3 School of Computing and Mathematics, Keele University, Keele, UK. 4 Corresponding author; e-mail: d.prikazchikov@keele.ac.uk. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 4, pp. 96 108, October December, 2020. Original article submitted December 15, 2020. 1072-3374/23/2736 0999 © 2023 Springer Nature Switzerland AG 999 DOI 10.1007/s10958-023-06560-5