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ISSN Печать: 2152-2057
ISSN Онлайн: 2152-2073
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ANALYTICAL SOLUTION OF THE PROBLEM ON THE THERMALLY STRESSED STATE OF FUNCTIONALLY GRADED PLATES BASED ON THE 3D ELASTICITY THEORY
Краткое описание
Two approaches to studying the thermally stressed state of layered functionally graded plates in the three-dimensional (3D) formulation have been developed. In the first approach, using the Reissner variational principle, a system of integrodifferential equilibrium equations and the corresponding boundary conditions are obtained without introducing an approximation. In the second approach, to construct differential equations of 3D thermoelastic equilibrium, a polynomial approximation of sought-for functions across the structure thickness is used. Its salient feature is assignment of the functions to the outer surfaces of layers, which allows splitting the layers into sublayers with a corresponding increase in the accuracy of calculation results. For the particular case of hinge-supported plates with a thermal load distributed according to a trigonometric law, the system of integrodifferential equilibrium equations obtained for the first approach and the differential equations of the second approach allow an analytical implementation. Then, equations of the first approach are transformed into a system of ordinary differential equations for the distribution of required functions across the plate thickness, with an analytical search for the roots of characteristic equations and the corresponding eigenvectors. In the second approach, the system of differential equations is converted to a system of algebraic equations. The assignment of sought-for functions to the outer surfaces of layers allows one to split the layers into sublayers and thus to reduce the approximation error. The same result obtained by the two methods may point to its reliability.
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Adineh, M. and Kadkhodayan, M., Three-Dimensional Thermo-Elastic Analysis and Dynamic Response of a Multi-Directional Functionally Graded Skew Plate on Elastic Foundation, Compos. Part B: Eng., vol. 125, pp. 227-240, 2017.
-
Ambartsumian, S.A., On a General Theory of Anisotropic Shells, J. Appl. Math. Mech., vol. 22, no. 2, pp. 305-319, 1958.
-
Bogdanovich, A.E. and Sierakowski, P.L., Composite Materials and Structures: Science, Technology and Applications. A Compendium of Books, Review, Papers, and Other Sources of Information, Appl. Mech. Rev, vol. 52, no. 12, pp. 351-366, 1999.
-
Brischetto, S., Leetsch, R., Carrera, E., Wallmersperger, T., and Kroplin, B., Thermo-Mechanical Bending of Functionally Graded Plates, J. Therm. Stresses, vol. 31, no. 3, pp. 268-308, 2008.
-
Cheng, Z.-Q. and Batra, R.C., Three-Dimensional Thermoelastic Deformations of a Functionally Graded Elliptic Plate, Compos. PartB: Eng., vol. 31, no. 2, pp. 97-106, 2000.
-
Demirbas, M.D., Thermal Stress Analysis of Functionally Graded Plates with Temperature-Dependent Material Properties Using Theory of Elasticity, Compos. PartB: Eng., vol. 131, pp. 100-124, 2017.
-
Dimitrienko, Yu.I. and Yakovlev, D.O., The Asymptotic Theory of Thermoelasticity of Multilayer Composite Plates, Compos.: Mech., Comput., Appl. An Int. J., no. 1, pp.13-51, 2015.
-
Formalev, V.F., Kolesnik, S.A., and Kuznetsova, E.L., Analytical Solution-Based Study of the Nonstationary Thermal State of Anisotropic Composite Materials, Compos.: Mech., Comput., Appl. An Int. J., vol. 9, no. 3, pp. 223-237, 2018.
-
Galerkin, B.G., Collection of Works [in Russian], vol. 1, M., Izd. AN SSSR (1952).
-
Gevorgyan, R.S., Asymptotic Solutions of Coupled Dynamic Problems of Thermoelasticity for Isotropic Plates, J. Appl. Math. Mech, vol. 72, no. 1, pp. 87-91, 2008.
-
Grigorenko, J.M., Vasilenko, A.T., and Pankratova, N.D., Zadachi of the Theory of Elasticity of Non-Uniform ph., K.: Naukova Dumka,1991.
-
Horechko, N.O. and Kushnir, R.M., Thermostressed State of a Composite Plate with Heat Exchange under the Action of a Uniformly Distributed Heat Source, J. Math. Sci., vol. 183, no. 2, pp. 177-189, 2012.
-
Houari, M.S.A., Tounsi, A., and Beg, O.A. Thermoelastic Bending Analysis of Functionally Graded Sandwich Plates Using a New Higher Order Shear and Normal Deformation Theory, Int. J. Mech. Sci., vol. 76, pp. 102-111, 2013.
-
Kit, H.S. and Andriichuk, R.M., Influence of a Stationary Heat Source on the Stress State of a Half Space with Rigidly, Smoothly, or Flexibly Fastened Boundary, J. Math. Sci., vol. 228, no. 2, pp. 91-104, 2018.
-
Marchuk, A.V., Three-Dimensional Analytic Solution for a Hinged Slab on an Elastic Half-Space, Int. Appl. Mech, vol. 33, no. 10, pp. 794-798, 1997.
-
Marchuk, A.V. and Piskunov, V.G., Statics, Vibrations and Stability of Composite Panels with Gently Curved Orthotopic Layers. 1. Statics and Vibrations,Mech. Compos. Mater., vol. 35, no. 4, pp. 285-292, 1999.
-
Marchuk, A.V. and Putvinskayte, Yu.K., Analytical Solution of the Problem on the Thermally Stressed State of Composite Plates with Rigid and Sliding Contacts Between Layers Based on the 3D Elasticity Theory, Mechanics of Composite Materials, vol. 55, no. 2, pp. 155-170, 2019.
-
Nemirovskii, Yu.V. and Yankovskii, A.P., A Method of Asymptotic Expansions of the Solutions of the Steady Heat Conduction Problem for Laminated Non-Uniform Anisotropic Plates, J. Appl. Math. Mech, vol. 72, no. 1, pp. 92-101, 2008.
-
Noor, K. and Burton, W.S., Three-Dimensional Solutions for Antisymmetrically Laminated Anisotropic Plates, J. Appl. Mech. ASME., vol. 57, pp. 182-187, 1990.
-
Pagano, N.J., Exact Solutions for Rectangular Bidirectional Composites Sandwich Plates, J. Compos. Mater, vol. 4, no. 1, pp. 20-34, 1970.
-
Prusakov, A.P., Bondarenko, V.D., and Prusakov, V.A., Bending of Freely Supported Three-Layer Plates of Asymmetric Structure Under the Action of a Transverse Sinusoidal Load (Exact Solution), Dep. Ukr. NIINTI 26.11.86, N2366-UK.86.
-
Reddy, J.N. and Cheng, Z.-Q., Three-Dimensional Thermomechanical Deformations of Functionally Graded Rectangular Plates, Eur. J. Mech. A Solids, vol. 20, no. 5, pp. 841-855, 2001.
-
Savooia, M. and Reddy, J.N., A Variational Approach to Three-Dimensional Elasticity Solutions of Laminated Composite Plates, J. Appl. Mech. ASME., vol. 59, pp. 166-175, 1992.
-
Timoschenko, S.P. and Goodier, J.N., Theory of Elasticity, 3rd. ed., New York: McGraw-Hill, 1970.
-
Tokovyy, Y. and Ma, C.-C., Analytic Solutions to 2D Elasticity and Thermoelasticity Problems for Inhomo-geneous Planes and Half-Planes, Arch. Appl. Mech., vol. 79, no. 5, pp. 441-456, 2009.
-
Tuimetov, Sh., Statics and Thermoelasticity Layered Orthotopic Plates Hingely Supported along their Contour, Cand. Dissert. of Phys.-Math. Sci., Kiev, 1987.
-
Vel, S. and Batra, R.C., Three-Dimensional Analysis of Transient Thermal Stresses in Functionally Graded Plates, Int. J. Solids Struct., vol. 40, no. 25, pp. 7181-7196, 2003.
-
Vlasov, B.F., On Case of Bending of a Rectangular Thick Plate, Vest. MGU, no. 2, pp. 25-34, 1957.
-
Yankovsky, A.P., A Comparative Analysis Thermoelasticoplastic Bending Deformations of Reinforced Plates, Prikl. Mat. Mekh, vol. 82, no. 1, pp. 58-83, 2018.
-
Marchuk A. V., Shevchuk L. O., Free and forced vibrations of functionally graded shallow shells based on the 3D elasticity theory, Acta Mechanica, 2022. Crossref