Archive
Special Issues

Volume 4, Issue 6, December 2016, Page: 269-275
The Asymptotic Analysis of the Solution of an Elasticity Theory Problem for a Transversely Isotropic Hollow Cylinder with Mixed Boundary Conditions on the Side Surface
Magomed Farman Mekhtiyev, Department of Applied Mathematics, Institute of Mathematics and Mechanics of National Academy of Sciences of Azerbaijan, Baku, Azerbaijan
Nina Ilyinichna Fomina, Faculty of Applied Mathematics and Cybernetics, Baku State University, Baku, Azerbaijan
Nazaket Boyukaga Mammadova, Faculty of Applied Mathematics and Cybernetics, Baku State University, Baku, Azerbaijan
Received: Sep. 23, 2016;       Accepted: Oct. 7, 2016;       Published: Nov. 3, 2016
Abstract
The problem of elasticity theory for the transversely isotropic hollow cylinder with mixed conditions on the side surface is considered in the paper. Transcendental equations are obtained regarding the eigenvalues of the problem. The roots of the characteristic equations are studied thoroughly. The study of the eigenvalues allowed to establish the essential characteristics of the stress-strain state of an anisotropic shell in comparison with isotropic shells. Homogeneous solutions were built here.
Keywords
Theory of Elasticity, Transversely Isotropic Hollow Cylinder, Side Surface, Mixed Boundary Conditions, Stress-Strain State, Eigenvalues, Transcendental Equation, Anisotropic Shell
Magomed Farman Mekhtiyev, Nina Ilyinichna Fomina, Nazaket Boyukaga Mammadova, The Asymptotic Analysis of the Solution of an Elasticity Theory Problem for a Transversely Isotropic Hollow Cylinder with Mixed Boundary Conditions on the Side Surface, Science Journal of Applied Mathematics and Statistics. Vol. 4, No. 6, 2016, pp. 269-275. doi: 10.11648/j.sjams.20160406.14
Reference
[1]
Goldenveizer A. L. The theory of thin shells. M.: Nauka, 1976, 512 p.
[2]
Goldenveizer A. L., Lidskiy V. B., Tovstik P. E. Free vibrations of thin elastic shells. M.: Nauka, 1979, 383 p.
[3]
Novojilov V. V. Theory of elasticity. L.: Sudopromgiz, 1958, 370 p.
[4]
Chernykh K. F. Linear shell theory. Part I, L.: LSU, 1962, 274 p.
[5]
Chernykh K. F. Linear shell theory. Part II, L.: LSU, 1964, 395 p.
[6]
Fridrichs K. O. Kirchoff’s boundary conditions and the edge effect for elastic plates. // Proc. Simpos. Appl.. Mech., 1950.
[7]
Fridrichs K. O. and Dressler R. F. A boundary layer theory for elastic plates.// Commun. Pure Appl. Math., 1961, vol. XIV, N1.
[8]
Vorovich I. I. Some mathematical questions of the theory of plates and shells. // Proc.of the II-nd All Union Congress on theoretical and applied mechanics, 1964. Review reports , issue 3, M.: Nauka, 1966, pp. 116-136.
[9]
Vorovich I. I., Kovalchuk V. Ye. On the basic properties of one system of homogeneous solutions. // Appl. math. and mech., 1967, vol. 31, issue 5, pp.861-869.
[10]
Vorovich I. I. Some results and problems of asymptotic theory of plates and shells. // Proc. of I All Union school on theory and numerical methods of calculations of plates and shells. Tbilisi, 1975, pp. 51-150.
[11]
Kolos A. V. On elaboration of classic theory of the bend of round plates. // Appl. math. and mech., 1964, vol.28, issue 3, pp.325-333.
[12]
Kolos A. V. Methods of elaboration of classic theory of bend and stretching of plates. // Appl. math. and mech., 1965, vol.29, issue 4, pp.250-310.
[13]
Agalovyan L. A. Application of the method of asymptotic integrating to the construction of the approximations of the theory of anisotropic shells. // Appl. math. and mech., 1966, vol.30, issue 2, pp. 388-393.
[14]
Aksentyan O. K. On concentration of strains in thick plates. // Appl. math. and mech., 1966, vol.30, issue 5, pp. 963-970.
[15]
Aksentyan O. K. Asymptotic analysis of the solutions to elasticity problems for plates at mixed boundary conditions. // Proc. of VIII All Union Conference on the theory of shells and plates. M., Nauka, 1973, pp.17-19.
[16]
Malkina O. S. On the error in the definition of strain concentration on a free hole by the methods of plane elasticity theory. // Appl. math. and mech., 1968, vol.32, issue 6, pp.314-322.
[17]
Malkina O. S. Strain-stress state of a thick plate on loading symmetric with respect to the median plane. Dissert. of the cand. of phys.-math. sci., Rostov-na-Donu, 1968, 171 p.
[18]
Bazarenko N. A., Vorovich I. I. Asymptotic behaviour of the solution to an elasticity problem for a hollow cylinder of a finite length at a small thickness. // Appl.math. and mech., 1965, vol. 29, issue 6, pp.1035-1052.
[19]
Bazarenko N. A., Vorovich I.I. Analysis of a three-dimensional strain and deformed state of round cylindrical shells. Construction of elaborated applied theories. // Appl.math. and mech., 1969, vol. 33, issue 3, pp.495-510.
[20]
Vilenskaya T. V., Vorovich I.I. Asymptotic behavior of the solution to an elasticity problem for a spherical shell of small thickness. // Appl.math. and mech., 1966, vol. 30, issue 2, pp. 278-295.
[21]
Ustinov Yu. A. On some features of asymptotic method on its application to the study of the vibrations of thin nonhomogeneous elastic plates.// Proc. of I All Union school on theory and numerical methods of calculation of shells and plates. Tbilisi, 1975, pp.395-403.
[22]
Ustinov Yu. A., Yudovich V. I. On completeness of a system of elementary solutions to a biharmonic equation in a semi-stripe. // Appl. math. and mech., 1973, vol.37, issue 4, pp.706-714.
[23]
Ustinov Yu. A. Boundary value problems and the problem of limit passage from three-dimensional elasticity problems to two-dimensional ones for nonhomogeneous plates. Dissert. of the doctor of phys.-math. sci., M., 1977, 355 p.
[24]
Mekhtiev M. F. Asymptotic analysis of some special elasticity problems for hollow solids, Baku, 2010, 316 p.
[25]
Mekhtiev M. F. Method of homogeneous solutions in anisotropic shell theory. Baku, 2009, 334 p.
[26]
Lidskiy V. B., Sadovnichiy V. A. Asymptotic formulas for the roots of one class of integer functions. Math.sbornik, 1968, N4, pp. 556-566.
[27]
Mekhtiev M. F. Construction of dynamic elaborated theory for a hollow cylinder. // Intern. sbornik “Applied mathematics”, Leningrad, LSU, 1988, pp. 207-211.
[28]
Akhmedov N. K., Mekhtiev M. F. Analysis of three-dimensional elasticity problems for nonhomogeneous cut cone. // Russian Academy of Sciences, Appl. math. and mech., 1993, vol. 57, issue 5, pp. 113-119.
[29]
Magsudov F. G., Mehdiyev M. F., Sadikhov P. M. The asymptotic theory for transversely isotropic hollow cylinder. // Proceedings of the V International Conference "Modern Problems of Continuum Mechanics", Rostov-on-Don, 2000, vol. 2, pp. 134-139.
[30]
Mekhtiev M. F., Bergman R. M. Asymptotic analysis of dynamic problem of the theory of elasticity for transversely isotropic hollow cylinder. // Journal of Sound and Vibration, London, 2001, N 2, pp. 177-194.
[31]
Mekhtiev M. F., Fomina N. I. Free vibrations of transversally isotropic hollow cylinders. Mechanics of Composite materials. New-York, 2002, vol.38, N1, pp. 55-67.
[32]
Maksudov F. G., Mekhtiev M. F., Sadikov P. M. Construction of homogeneous solutions for a transversally isotropic hollow cylinder. Proceedings of IMM of Azerbaijan AS, 1999, vol. X, pp. 199-209.
[33]
Lekhnitskiy S. G. Theory of elasticity of an anisotropic solid. M., Nauka, 1977, 415 p.
[34]
Ambartsumyan S. A. General theory of anisotropic shells. M., Nauka, 1974, 448 p.
[35]
Mekhtiyev M. F., Mardanov I. D., Amrahova A. R. Asymptotic analysis of bending problem for transversal-isotropic plate of variable thickness. Trans. of NAS of Azerbaijan, series of phys.-tech. and math. sciences, Baku, 2002, N4, p. 223-236.
[36]
Mekhtiyev M. F., Guseynov F. S. The construction of homogeneous solutions for transversally isotropic hollow sphere . Trans. of NAS of Azerbaijan, series of phys.-tech. and math. sciences, Baku, 2002, Vol. XXII, N1, p. 206-210.
[37]
Mekhtiyev M. F. Asymptotic analysis of anisotropic elasticity theory problem for finite length hollow cylinder. Trans. of NAS of Azerbaijan, series of phys.-tech. and math. sciences, Baku, 2003, Vol. XXIII, N1, p. 219-228.
[38]
Mekhtiyev M. F., Fomina N. I., Sardarli N. A. Torsion problem of transversally isotropic hollow cone of variable thickness. Proc. of IMM of NAS of Azerbaijan, XXIII, Baku, 2005, p. 199-206.
[39]
Mekhtiyev M. F. Construction of homogeneous solutions of a non-axially-symmetric tension problem of elasticity theory for transversally isotropic plates of variable thickness. Trans. of NAS of Azerbaijan, series of phys.-tech. and math. sciences, Baku, 2006, XXVI, N1, p. 177-185.
[40]
Mekhtiyev M. F. Non-axially-symmetric problem of elasticity theory for transversally isotropic hollow sphere. Trans. of NAS of Azerbaijan, series of phys.-tech. and math. sciences, Baku, 2007, XXVII, N1, p. 155-164.