The effect of a heat flux of constant intensity on a circular three-layer plate, thermally insulated along the contour and lower surface, is considered. A solution of the problem of thermal conductivity of the plate with thickness-averaged thermophysical parameters of materials is presented. The nonstationary temperature field is nonuniform over the thickness of the plate. It is shown that on instantaneous drop, the heat flux can cause sagging and free vibrations of the three-layer plate.
The kinematics of the plate package obeys the broken line hypothesis. After applying a load, the normal in thin load-bearing layers does not change its length and remains perpendicular to the middle surface of the layer. In a relatively thick filler, the deformed normal retains its length and straightness, but is rotated by a small additional angle, i.e., the shift is taken into account. The formulation of the corresponding initial boundary-value problem is given. The equations of motion were obtained using the variational method with account for the transverse forces of inertia. The boundary pivot conditions are accepted on the contour of the plate. Radial movements in the layers are expressed through three sought functions: plate sagging, shear, and radial movement of the middle plane of the filler. It is shown that these sought functions satisfy the inhomogeneous system of three differential equations. To solve the system, the method of series expansion in the constructed fundamental system of eigenorthonormal functions was used. A transcendental equation is written out to obtain the corresponding eigenvalues. A numerical parametric analysis of the solution was carried out depending on the geometric and thermophysical characteristics of the layer materials and the time of exposure to the heat flux.
Similar content being viewed by others
References
E. I. Starovoitov and F. B. Nagiyev, Foundations of the Theory of Elasticity, Plasticity, and Viscoelasticity, Apple Academic Press, Toronto, New Jersey (2012).
L. Aghalovyan, Asymptotic Theory of Anisotropic Plates and Shells, World Scientific Publishing, Singapore–London (2015).
E. Carrera, F. A. Fazzolari, and M. Cinefra, Thermal Stress Analysis of Composite Beams, Plates and Shells: Computational Modelling and Applications, Academic Press (2016).
É. I. Starovoitov, D. V. Leonenko, and D. V. Tarlakovskii, Thermal-force deformation of a physically nonlinear three-layer stepped rod, J. Eng. Phys. Thermophys., 89, No. 6, 1582–1590 (2016).
É. I. Starovoitov and D. V. Leonenko, Effect of heat flow on the stressed state of a three-layer rod, J. Eng. Phys. Thermophys., 92, No. 1, 60–72 (2019); doi: https://doi.org/10.1007/s10891-019-01907-9.
É. I. Starovoitov, Yu. M. Pleskachevskii, D. V. Leonenko, and D. V. Tarlakovsky, Deformation of a step composite beam in a temperature field, J. Eng. Phys. Thermophys., 88, No. 4, 1023–1029 (2015).
E. I. Starovoitov, D. V. Leonenko, and A. V. Yarovaya, Elastoplastic bending of a sandwich bar on an elastic foundation, Int. Appl. Mech., 43, No. 4, 451–459 (2007).
M. Pradhan, P. R. Dash, and P. K. Pradhan, Static and dynamic st ability analysis of an asymmetric sandwich beam resting on a variable Pasternak foundation subjected to thermal gradient, Meccanica, 51, No. 3, 725–739 (2016); doi: https://doi.org/10.1007/s11012-015-0229-6.
B. V. Nerubailo, Toward numerical solution of the problem on stressed state of thermoelastic physically orthotropic cylindrical shells, J. Eng. Phys. Thermophys., 93, No. 1, 241–246 (2020).
V. S. Surov, Towards elastoplastic deformation of a solid body by the hybrid Godunov method and by the multidimensional nodal method of characteristics, J. Eng. Phys. Thermophys., 95, No. 3, 830–845 (2022).
A. V. Yarovaya, Thermoelastic bending of a sandwich plate on a deformable foundation, Int. Appl. Mech., 42, 206–213 (2006).
E. I. Starovoitov and D. V. Leonenko, Impact of thermal and ionizing radiation on a circular sandwich plate on an elastic foundation, Int. Appl. Mech., 47, No. 5, 580–589 (2011).
A. P. Yankovskii, Refined modeling of flexural deformation of layered plates with a regular structure made from nonlinear hereditary materials, Mech. Compos. Mater., 53, No. 6, 1015–1042 (2017).
A. M. Zenkour and N. A. Alghamdi, Bending analysis of functionally graded sandwich plates under the effect of mechanical and thermal loads, Mech. Adv. Mater. Struct., 17, No. 6, 419–432 (2010).
D. V. Leonenko and E. I. Starovoitov, Thermoplastic strain of circular sandwich plates on an elastic base, Mech. Solids, 44, No. 5, 744–755 (2009).
E. I. Starovoitov, Yu. M. Pleskachevskii, and D. V. Leonenko, Thermal force loading of a physically nonlinear threelayer circular plate, J. Eng. Phys. Thermophys., 93, No. 3, 554–564 (2020); https://doi.org/https://doi.org/10.1007/s10891-020-02150-3.
L. Yang, O. Harrysson, H. West, and D. A. Cormier, Comparison of bending properties for cellular core sandwich panels, Mater. Sci. Appl., 4, No. 8, 471–477 (2013).
E. I. Starovoitov and D. V. Leonenko, Deformation of a three-layer elastoplastic beam on an elastic foundation, Mech. Solids, 46, No. 2, 291–298 (2011).
V. N. Paimushin and R. K. Gazizullin, Static and monoharmonic acoustic impact on a laminated plate, Mech. Compos. Mater., 53, No. 3, 407–436 (2017).
H. V. Zadeh and M. Tahani, Analytical bending analysis of a circular sandwich plate under distributed load, Int. J. Recent Adv. Mech. Eng., 6, No. 1, 1–10 (2017); doi: https://doi.org/10.14810/ijmech.2017.6101.
G. I. Mikhasev and H. Altenbach, Free vibrations of elastic laminated beams, plates and cylindrical shells, in: Thin-Walled Laminated Structures, Springer, Cham (2019), pp. 157–198 (Adv. Struct. Mater., 106); doi: https://doi.org/10.1007/978-3-030-12761-9_4.
E. Yu. Mikhailova and G. V. Fedotenkov, Nonstationary axisymmetric problem of the impact of a spherical shell on an elastic half-space (Initial stage of interaction), Mech. Solids, 46, No. 2, 239–247 (2011).
M. A. Zhuravkov and A. V. Krupoderov, Dynamic load effect in the vicinity of goafs within rock masses, J. Mining Sci., 46, No. 3, 241–249 (2010).
K. T. Takele, Interfacial strain energy continuity assumption-based analysis of an orthotropicskin sandwich plate using a refined layer-by-layer theory, Mech. Compos. Mater., 54, No. 3, 419–444 (2018).
V. N. Bakulin, D. A. Boitsova, and A. Ya. Nedbai, Parametric resonance of a three layered cylindrical composite rib-stiffened shell, Mech. Compos. Mater., 57, No. 5, 623–634 (2021).
A. G. Gorshkov, É. I. Starovoitov, and A. V. Yarovaya, Harmonic vibrations of a viscoelastoplastic sandwich cylindrical shell, Int. Appl. Mech., 37, No. 9, 1196–1203 (2001).
E. I. Starovoitov, D. V. Leonenko, and D. V. Tarlakovsky, Resonance vibrations of circular composite plates on an elastic foundation, Mech. Compos. Mater., 51, No. 5, 561–570 (2015).
I. Ivañez, M. M. Moure, S. K. Garcia-Castillo, and S. Sanchez-Saez, The oblique impact response of composite sandwich plates, Compos. Struct., No. 133, 1127–1136 (2015).
E. I. Starovoitov and D. V. Leonenko, Vibrations of circular composite plates on an elastic foundation under the action of local loads, Mech. Compos. Mater., 52, No. 5, 665–672 (2016); doi: https://doi.org/10.1007/s11029-016-9615-y.
V. N. Bakulin and A. Y. Nedbai, Dynamic stability of a composite cylindrical shell with linear-variable thickness under pulsed external pressure, J. Eng. Phys. Thermophys., 94, No. 2, 525–533 (2021).
E. von Kamke, Differentialtechungen Lösungsmethoden und Lösungen [Russian translation], Nauka, Moscow (1976).
E. I. Starovoitov, Description of the thermomechanical properties of some structural materials, Strength Mater., 20, No. 4, 426–431 (1988); doi: https://doi.org/10.1007/BF01530849.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 96, No. 6, pp. 1445–1455, November–December, 2023.
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
Starovoitov, É.I., Pleskachevskii, Y.M., Leonenko, D.V. et al. Free Vibrations of a Three-Layer Plate Excited by a Heat Flux. J Eng Phys Thermophy 96, 1432–1442 (2023). https://doi.org/10.1007/s10891-023-02811-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10891-023-02811-z