Journal of Applied and Computational Mechanics

Axisymmetric Problem of the Elasticity Theory for the Radially ‎Inhomogeneous Cylinder with a Fixed Lateral Surface

Document Type : Research Paper

Author

Department of Mathematics and Statistics, Azerbaijan State University of Economics (UNEC), Baku, Azerbaijan

Abstract

By the method of the asymptotic integration of the equations of elasticity theory, the axisymmetric problem of elasticity theory is studied for a radially inhomogeneous cylinder of small thickness. It is considered that the elasticity moduli are arbitrary positive continuous functions of the radius of the cylinder. It is also assumed that the lateral surface of the cylinder is fixed, and stresses are imposed at the end faces of the cylinder, which leave the cylinder in equilibrium. The analysis is carried out when the cylinder thickness tends to zero. It is shown that solutions corresponding to the first and second iterative processes that determine the internal stress-strain state of the radially inhomogeneous cylinder with a fixed surface do not exist. The third iterative process defines solutions that have the boundary layer character equivalent to the Saint-Venant end effect on the theory of inhomogeneous plates. The stresses determined by the third iterative process are localized at the ends of the cylinder and decrease exponentially with distance from the ends. The asymptotic integration method is used to study the problem of torsion of the radially inhomogeneous cylinder of small thickness. The nature of the stress-strain state is analyzed.

Keywords

Main Subjects

[1]   Birman, V., Byrd, L.W., Modeling and analysis of functionally graded materials and structures, Applied Mechanics Reviews, 60, 2007, 195-215. 
[2]   Ustinov, Yu.A., Mathematical Theory of Transversely Inhomogeneous Plates, TSBBP, Rostov-on-Don, 2006 (in Russian)
[3]   Mekhtiev, M.F., Asymptotic Analysis of Spatial Problems in Elasticity, Springer, 2019.
[4]   Bazarenko, N.A., Vorovich, I.I., Asymptotic behavior of the solution of the problem of the theory of elasticity for a hollow cylinder of finite length at a small thickness, Applied Mathematics and Mechanics, 29(6), 1965, 1035-1052 (in Russian)
[5]   Akhmedov, N.K., Ustinov, Yu.A., St. Venant’s principle in the torsion problem for a laminated cylinder, Applied Mathematics and Mechanics, 52(2), 1988, 207-210.
[6]   Akhmedov, N.K., Analysis of the boundary layer in the axisymmetric problem of the theory of elasticity for a radially layered cylinder and the propagation of axisymmetric waves, Applied Mathematics and Mechanics, 61(5), 1997, 863-872.
[7]   Huang, C.H., Analysis of laminated circular cylinders of materials with the most general form of cylindrical anisotropy: I. Axially symmetric deformations, International Journal of Solids and Structures, 38(34-35), 2001, 6163-6182.
[8]   Lin, H.C, Stanley, B., Dong, B., On the Almansi-Michell problems for an inhomo -geneous, anisotropic cylinder, Journal of Mechanics, 22(1), 2006, 51-57.
[9]   Horgan, C.O., Chan, A.M., The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials, Journal of Elasticity, 55(1), 1999, 43-59.
[10] Tarn, J.Q., Chang, H.H., Torsion of cylindrically orthotropic elastic circular bars with radial inhomogeneity: some exact solutions and effects, International Journal of Solids and Structures, 45, 2008, 303-319.
[11] Ieşan, D., Quintanilla, R., On the deformation of inhomogeneous orthotropic elastic cylinders, European Journal of Mechanics A/Solids, 26, 2007, 999-1015.
[12] Grigorenko, A.Ya., Yaremchenko S.N., Analysis of the stress-strain state of inhomogeneous hollow cylinders, International Applied Mechanics, 52(4), 2016, 342-349.
[13] Grigorenko, A.Ya., Yaremchenko, S.N., Three-dimensional analysis of the stress-strain state of inhomogeneous hollow cylinders using various approaches, International Applied Mechanics, 55(5), 2019 ,487-494.
[14] Tutuncu, M., Temel B., A novel approach to stress analysis of pressurized FGM cylinders, disk and spheres, Composite Structures, 91, 2009, 385-390.
[15] Zhang, X., Hasebe, M., Elasticity solution for a radially nonhomogeneous hollow circular cylinder, Journal of Applied Mechanics, 66, 1999, 598-606.
[16] Liew, K.M., Kitipornchai, S., Zhang, X.Z., Lim, C.W., Analysis of the thermal stress behaviour of functionally graded hollow circular cylinders, International Journal of Solids and Structures, 40, 2003, 2355-2380.
[17] Jabbari, M., Bahtui, A., Eslami, M.R., Axisymmetric mechanical and thermal stresses in thick long FGM cylinders, Journal Thermal Stresses, 29(7), 2006, 643-663.
[18] Jabbari, M., Mohazzab, A.H., Bahtui, A., Eslami, M.R., Analytical solution for three -dimensional stresses in a short length FGM hollow cylinder, ZAMM - Journal of Applied Mathematics and Mechanics,87(6), 2007, 413-429.
[19] Hosseini, K., Naghdabadi, R., Thermoelastic analysis of functionally graded cylinders under axial loading, Journal Thermal Stresses, 31(1), 2008, 1-17.
[20] Tarn, J.Q., Exact solutions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads, International Journal of Solids and Structures, 38, 2001, 8189-8206.
[21] Akhmedov, N.K., Akbarova S., Ismailova J., Analysis of axisymmetric problem from the theory of elasticity for an isotropic cylinder of small thickness with alternating elasticity modules, Eastern-European Journal of Enterprise Technologies, 2(7), 2019, 13-19.
[22] Akhmedov, N.K., Akperova, S.B., Asymptotic analysis of the three-dimensional problem of the theory of elasticity for a radially inhomogeneous transversely isotropic hollow cylinder, Bulletin of the Russian Academy of Sciences. Solid Mechanics, 4, 2011, 170-180 (in Russian).
[23] Tokovyy, Y., Ma, C.C., Elastic analysis of inhomogeneous solids: history and development in brief, Journal of Mechanics, 2019, 35(5), 613-626.
[24] Lurie, A.I., Theory of Elasticity, Nauka, Moscow, 1970 (in Russian).
[25] Goldenweiser, A.L., Construction of an approximate theory of shell bending using asymptotic integration of the equations of elasticity theory, Applied Mathematics and Mechanics, 27(4), 1963, 593-608.
[26] Akhmedov, N.K., Mekhtiev, M.F., Analysis of the three-dimensional problem of the theory of elasticity for an inhomogeneous truncated hollow cone, Applied Mathematics and Mechanics, 57(5), 1993, 113-119.
[27] Akhmedov, N.K., Mekhtiev, M.F., The axisymmetric problem of the theory of elasticity for an inhomogeneous plate of variable thickness, Applied Mathematics and Mechanics, 59(3), 1995, 518-523.
[28] Akhmedov, N.K., Sofiyev, A.N., Asymptotic analysis of three-dimensional problem of elasticity theory for radially inhomogeneous transversally-isotropic thin hollow spheres, Thin-Walled Structures, 139, 2019, 232-241.
[29] Vorovich, I.I., Kadomtsev, I.G., Ustinov, Yu.A., On the theory of plates non-uniform in thickness., Reports of USSR Academy of Sciences: Solid Mechanics, 3, 1975, 119-129 (in Russian).
[30] Ustinov, Yu.A., Yudovich, V.I., On the completeness of a system of elementary solutions of a biharmonic equation in a half-strip, Applied Mathematics and Mechanics, 37(4), 1973, 707-714
[31] Mikhlin, S.G., Partial Linear Equations, Visshaya Skola, Moscow, 1977 (in Russian).
Journal of Applied and Computational Mechanics
Volume 7, Issue 2
April 2021
Pages 598-610