Three-dimensional dynamic problem for a layered orthotropic elastic shell with free upper face is considered. 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. A. Agalovyan, M. L. Agalovyan, and V. V. Tagvoryan, “On solutions of dynamic three-dimensional problems of the theory of elasticity for modeling earthquakes,” Izv. Nats. Acad. Nauk Armen.. Mekh., 71, No. 4, 17–29 (2018); https://doi.org/10.33018/71.4.2.
L. A. Agalovyan and L. G. Gulgazaryan, “Forced vibrations of orthotropic shells: nonclassical boundary-value problems,” Int. Appl. Mech., 45, No. 8, 888–904 (2009); https://doi.org/10.1007/s10778-009-0231-6.
V. L. Berdichevskii, “High-frequency long-wave vibrations of plates,” Sov. Phys. Dokl., 22, No. 10, 604–606 (1977).
A. L. Gol’denvejzer, Theory of thin elastic shells [in Russian], Nauka, Moscow (1976).
Yu. D. Kaplunov, “High-frequency stress-strain states of low variability in elastic thin shells,” Izv. Akad, Nauk SSSR. Мekh. Tverd. Tela, No. 5, 147–157 (1990).
L. A. Aghalovyan, Asymptotic Theory of Anisotropic Plates and Shells, World Scientific, Singapore–London (2015).
L. A. Aghalovyan, “On one class of three-dimensional problems of elasticity theory for plates,” Proc. Razmadze Math. Inst. [Georgia], 155, 1–8 (2011).
L. A. Aghalovyan and M. L. Aghalovyan, “Monitoring of stress-strain state of plate-like packet of base-foundation constructions on the base of the data of seismic stations and GPS systems,” Proc. of the Fifth Europ. Conf. Struct. Control, EACS–2012 (18–20 June 2012, Genoa, Italy), Paper No. 069 (2012), 8 p.
L. A. Aghalovyan and L. G. Ghulghazaryan, “Forced vibrations of a two-layered shell in the case of viscous resistance,” IOP J. Phys: Conf. Ser., 991, 012002 (2018), 10 p.; https://doi.org/10.1088/1742-6596/991/1/012002.
I. Argatov and G. Mishuris, Contact Mechanics of Articular Cartilage Layers, Cham, Springer (2015).
F. M. Borodich, B. A. Galanov, N. V. Perepelkin, and D. A. Prikazchikov, “Adhesive contact problems for a thin elastic layer: Asymptotic analysis and the JKR theory,” Math. Mech. Solids, 24, No. 5, 1405–1424 (2019); https://doi.org/10.1177/1081286518797378.
R. Chebakov, J. Kaplunov, and G. A. Rogerson, “Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects,” Proc. R. Soc. A: Math., Phys. Eng. Sci., 472, No. 2186, Art. 20150800 (2016); https://doi.org/10.1098/-rspa.2015.0800.
B. Erbaş, E. Yusufoğlu, and J. Kaplunov, “A plane contact problem for an elastic orthotropic strip,” J. Eng. Math., 70, No. 4, 399–409 (2011); https://doi.org/10.1007/s10665-010-9422-8.
L. G. Ghulghazaryan and L. V. Khachatryan, “Forced vibrations of a two-layer orthotropic shell with an incomplete contact between layers,” Mech. Compos. Mater., 53, No. 6, 821–826 (2018); https://doi.org/10.1007/s11029-018-9707-y.
J. D. Kaplunov, “Long-wave vibrations of a thin-walled body with fixed faces,” Quart. J. Mech. Appl. Math., 48, No. 3, 311–327 (1995); https://doi.org/10.1093/qjmam/48.3.311.
J. D. Kaplunov, L. Yu. Kossovich, and E. V. Nolde, Dynamics of Thin Walled Elastic Bodies, Acad. Press, San Diego (1998).
J. D. Kaplunov and E. V. Nolde, “Long-wave vibrations of a nearly incompressible isotropic plate with fixed faces,” Quart. J. Mech. Appl. Math., 55, No. 3, 345–356 (2002); https://doi.org/10.1093/qjmam/55.3.345.
J. Kaplunov, D. Prikazchikov, and L. Sultanova, “Justification and refinement of Winkler–Fuss hypothesis,” Z. Angew. Math. Phys., 69, No. 3, Art. 80 (2018); https://doi.org/10.1007/s00033-018-0974-1.
K. Kasahara, Earthquake Mechanics, Cambridge Univ. Press, Cambridge (1981).
M. I. Lashhab, G. A. Rogerson, and L. A. Prikazchikova, “Small amplitude waves in a pre-stressed compressible elastic layer with one fixed and one free face,” Z. Angew. Math. Phys., 66, No. 5, 2741–2757 (2015); https://doi.org/10.1007/s00033-015-0509-y.
C. Le Khanh, “High frequency vibrations and wave propagation in elastic shells: Variational-asymptotic approach,” Int. J. Solids Struct., 34, No. 30, 3923–3939 (1997); https://doi.org/10.1016/S0020-7683(97)00011-5.
X. Le Pichon, J. Francheteau, and J. Bonnin, Plate Tectonics, Elsevier, Amsterdam (1973).
E. V. Nolde and G. A. Rogerson, “Long wave asymptotic integration of the governing equations for a pre-stressed incompressible elastic layer with fixed faces,” Wave Motion, 36, No. 3, 287–304 (2002); https://doi.org/10.1016/S0165-2125(02)00017-3.
A. V. Pichugin and G. A. Rogerson, “An asymptotic membrane-like theory for long-wave motion in a pre-stressed elastic plate,” Proc. R. Soc. London. Ser. A: Math., Phys. Eng. Sci., 458, No. 2022, 1447–1468 (2002); https://www.jstor.org/stable/3067470.
T. Rikitake, Earthquake Prediction, Elsevier, Amsterdam (1976).
G. A. Rogerson, K. J. Sandiford, and L. A. Prikazchikova, “Abnormal long wave dispersion phenomena in a slightly compressible elastic plate with non-classical boundary conditions,” Int. J. Nonlin. Mech., 42, No. 2, 298–309 (2007); https://doi.org/10.1016/j.ijnonlinmec.2007.01.005.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 63, No. 4, pp. 96–108, October–December, 2020.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Aghalovyan, L.A., Ghulghazaryan, L.G., Kaplunov, J.D. et al. Three-Dimensional Dynamic Analysis of Layered Elastic Shells. J Math Sci 273, 999–1015 (2023). https://doi.org/10.1007/s10958-023-06560-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06560-5