We construct a system of differential equations aimed at the description of elastic deformations of thin anisotropic shells of revolution and solved with respect to the first-order partial derivatives with respect to the meridional coordinate. These equations are obtained by the {m,n}-approximation method. The approximations of unknown functions are in good agreement with force boundary conditions imposed on the front faces.
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A. E. Alekseev, “Two-parameter family of successive (M, N)-approximations of the equations of an elastic layer of variable thickness,” Prikl. Mekh. Tekh. Fiz., 37, No. 3, 133–144 (1996); English translation: J. Appl. Mech. Tech. Phys., 37, No. 3, 411–420 (1996).
N. S. Bondarenko and A. S. Gol’tsev, “Stress intensity factors for the thermoelastic bending of isotropic plates with heat-insulated notches in the case of symmetric heat exchange,” Visn. Donets’k. Nats. Univ., Ser. A: Pryrodnych. Nauky, No. 2, 20–26 (2013).
N. S. Bondarenko, “Isotropic plate with heat-permeable notch under the action of temperature gradients responsible for bending,” Probl. Obchysl. Mekh. Mitsn. Konstrukts., Issue 23, 52–63 (2014).
I. N. Vekua, Some General Methods of Construction of Various Versions of the Theory of Shells [in Russian], Nauka, Moscow (1982).
S. K. Godunov, “On the numerical solution of boundary-value problems for systems of linear ordinary differential equations,” Usp. Mat. Nauk, 16, No. 3 (99), 171–174 (1961).
É. I. Grigolyuk and G. M. Kulikov, Multilayer Reinforced Shells: Numerical Analyses of Pneumatic Tires [in Russian], Mashinostroenie, Moscow (1988).
É. I. Grigolyuk and V. I. Shalashilin, Problems of Nonlinear Deformation: A Method for the Continuation of Solution with Respect to the Parameter in the Nonlinear Problems of Mechanics of Deformable Bodies [in Russian], Nauka, Moscow (1988).
Ya. M. Grigorenko and A. Ya. Grigorenko, “Static and dynamic problems for anisotropic inhomogeneous shells with variable parameters and their numerical solution (review),” Prikl. Mekh., 49, No. 2, 3–70 (2013); English translation: Int. Appl. Mech., 49, No. 2, 123–193 (2013).
Ya. M. Grigorenko, “Some approaches to studying deformation of flexible shells,” Mat. Metody Fiz.-Mekh. Polya, 51, No. 2, 152–165 (2008); English translation: J. Math. Sci., 162, No. 2, 187–204 (2009).
V. A. Laz’ko, “Stress-strain state of laminar anisotropic shells with the presence of zones of nonideal layer contact. 2. Generalized equations of orthotropic laminar shells subjected to discontinuous displacements at the interface,” Mekh. Kompozit. Mater., No. 1, 77–84 (1982); English translation: Mech. Composit. Mater., 18, No. 1, 63–69 (1982).
A. Maksymuk and N. Shcherbyna, “Improved determination of the stressed state of finite shells with regard for residual strains,” Visn. L’viv. Univ., Ser. Prykl. Mat. Informat., Issue 7, 193–200 (2003).
M. V. Marchuk and M. M. Khom’yak, “Mixed finite-element method for the improved statement of the problem of interlayer contact. Part 1. Principal relations and the general procedure of construction of computational schemes in the {m,n}-approximation,” Probl. Prochn., No. 2, 118–127 (2000); English translation: Strength Mater., 32, No. 2, 187–194 (2000).
M. V. Marchuk and M. M. Khom’yak, “Mixed finite-element method for the improved statement of the problem of interlayer contact. Part 2. Numerical analysis of contact bending stresses in layered plates,” Probl. Prochn., No. 2, 128–135 (2000); English translation: Strength Mater., 32, No. 2, 195–200 (2000).
B. L. Pelekh, A. V. Maksimuk, and I. M. Korovaichuk, Contact Problems for Layered Structural Elements and Bodies with Coatings [in Russian], Naukova Dumka, Kiev (1988).
B. L. Pelekh and M. A. Sukhorol’skii, Contact Problems of the Theory of Elastic Anisotropic Shells [in Russian], Naukova Dumka, Kiev (1980).
B. L. Pelekh and M. A. Sukhorol’skii, “On one method of approximation of functions and their first derivative by Legendre polynomials and its applications,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 3, 26–29 (1980).
B. L. Pelekh and M. A. Sukhorol’skii, “On one new approach to the construction of the theory of shells with regard for the boundary conditions on the surfaces,” Dopov. Akad. Nauk Ukr. RSR, Ser. A, No. 5, 441–444 (1978).
R. B. Rikards, Finite-Element Method in the Theory of Shells and Plates [in Russian], Zinatne, Riga (1988).
M. A. Sukhorol’s’kyi, Functional Sequences and Series [in Ukrainian], Rastr-7, Lviv (2010).
I. Yu. Khoma, Generalized Theory of Anisotropic Shells [in Russian], Naukova Dumka, Kiev (1986).
G. Jaiani, “A model of layered prismatic shells,” Continuum Mech. Thermodynam., 28, No. 3, 765–784 (2016).
T. Meunargia, “On the refined theories of elastic plates,” Appl. Math. Inf. Mech., 19, No. 1, 53–67 (2014).
S. I. Zhavoronok, “On the variational formulation of the extended thick anisotropic shells theory of I. N. Vekua type,” Procedia Eng., 111, 888–895 (2015).
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 58, No. 3, pp. 43–56, July–September, 2015.
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Тuchapskyy, R.І. Equations of Thin Anisotropic Elastic Shells of Revolution in the {m, n}-Approximation Method. J Math Sci 226, 52–68 (2017). https://doi.org/10.1007/s10958-017-3518-x
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DOI: https://doi.org/10.1007/s10958-017-3518-x