Behavior of solution of the elasticity problem for a radial inhomogeneous cylinder with small thickness
Keywords:non-axisymmetric problem, radial inhomogeneous cylinder, asymptotic integration method, homogeneous solutions, boundary layer
A non-axisymmetric problem of the theory of elasticity for a radial inhomogeneous cylinder of small thickness is studied. It is assumed that the elastic moduli are arbitrary positive piecewise continuous functions of a variable along the radius.
Using the method of asymptotic integration of the equations of the theory of elasticity, based on three iterative processes, a qualitative analysis of the stress-strain state of a radial inhomogeneous cylinder is carried out. On the basis of the first iterative process of the method of asymptotic integration of the equations of the theory of elasticity, particular solutions of the equilibrium equations are constructed in the case when a smooth load is specified on the lateral surface of the cylinder. An algorithm for constructing partial solutions of the equilibrium equations for special types of loads, the lateral surface of which is loaded by forces polynomially dependent on the axial coordinate, is carried out.
Homogeneous solutions are constructed, i.e., any solutions of the equilibrium equations that satisfy the condition of the absence of stresses on the lateral surfaces. It is shown that homogeneous solutions are composed of three types: penetrating solutions, solutions of the simple edge effect type, and boundary layer solutions. The nature of the stress-strain state is established. It is found that the penetrating solution and solutions having the character of the edge effect determine the internal stress-strain state of a radial inhomogeneous cylinder. Solutions that have the character of a boundary layer are localized at the ends of the cylinder and exponentially decrease with distance from the ends. These solutions are absent in applied shell theories.
Based on the obtained asymptotic expansions of homogeneous solutions, it is possible to carry out estimates to determine the range of applicability of existing applied theories for cylindrical shells. Based on the constructed solutions, it is possible to propose a new refined applied theory.
- Birman, V., Byrd, L. W. (2007). Modeling and Analysis of Functionally Graded Materials and Structures. Applied Mechanics Reviews, 60 (5), 195–216. doi: https://doi.org/10.1115/1.2777164
- Tokovyy, Y., Ma, C.-C. (2019). Elastic Analysis of Inhomogeneous Solids: History and Development in Brief. Journal of Mechanics, 35 (5), 613–626. doi: https://doi.org/10.1017/jmech.2018.57
- Akhmedov, N. K., Ustinov, Yu. A. (1988). On St. Venant's principle in the torsion problem for a laminated cylinder. Journal of Applied Mathematics and Mechanics, 52 (2), 207–210. doi: https://doi.org/10.1016/0021-8928(88)90136-0
- Ahmedov, N. K. (1997). Analiz pogranichnogo sloya v osesimmetrichnoy zadache teorii uprugosti dlya radial'no-sloistogo tsilindra i rasprostraneniya osesimmetrichnyh voln. Prikladnaya matematika i mekhanika, 61 (5), 863–872.
- Akhmedov, N., Akbarova, S., Ismayilova, J. (2019). 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 (98)), 13–19. doi: https://doi.org/10.15587/1729-4061.2019.162153
- Ismayilova, J. (2019). Studying of elastic equilibrium of a small thickness isotropic cylinder with variable elasticity module. Transactions of NAS of Azerbaycan, ISSUE Mechanics, 39 (8), 17–23. Available at: http://transmech.imm.az/upload/articles/v-40/Jalala_Ismayilova_Trans_Mech_Vol_39_8_2019.pdf
- Ismayilova, J. J. (2017). The problem of torsion of a radially inhomogeneous cylinder. Bulletin of NTU “KhPI”. Series: Mechanical-technological systems and complexes, 16 (1238), 82–87. Available at: http://repository.kpi.kharkov.ua/handle/KhPI-Press/30089
- Ismayilova, D. D. (2017). Analysis of a problem of torsion of a cylinder with variable shear modulus with fastened lateral surface. Scientific works, 1, 88–93. Available at: http://www.aztu.edu.az/azp//elmi_tedqiqat/scientific_research_2/az/files/jurnal_2017_1/14.pdf
- Akhmedov, N. K., Akperova, S. B. (2011). Asymptotic analysis of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder. Mechanics of Solids, 46 (4), 635–644. doi: https://doi.org/10.3103/s0025654411040133
- Huang, C. H., Dong, S. B. (2001). Analysis of laminated circular cylinders of materials with the most general form of cylindrical anisotropy. International Journal of Solids and Structures, 38 (34-35), 6163–6182. doi: https://doi.org/10.1016/s0020-7683(00)00374-7
- Lin, H.-C., Dong, S. B. (2006). On the Almansi-Michell Problems for an Inhomogeneous, Anisotropic Cylinder. Journal of Mechanics, 22 (1), 51–57. doi: https://doi.org/10.1017/s1727719100000782
- Horgan, C. O., Chan, A. M. (1999). The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. Journal of Elasticity, 55, 43–59. doi: https://doi.org/10.1023/A:1007625401963
- Ieşan, D., Quintanilla, R. (2007). On the deformation of inhomogeneous orthotropic elastic cylinders. European Journal of Mechanics - A/Solids, 26 (6), 999–1015. doi: https://doi.org/10.1016/j.euromechsol.2007.03.004
- Grigorenko, A. Y., Yaremchenko, S. N. (2016). Analysis of the Stress–Strain State of Inhomogeneous Hollow Cylinders. International Applied Mechanics, 52 (4), 342–349. doi: https://doi.org/10.1007/s10778-016-0757-3
- Grigorenko, A. Y., Yaremchenko, S. N. (2019). Three-Dimensional Analysis of the Stress–Strain State of Inhomogeneous Hollow Cylinders Using Various Approaches. International Applied Mechanics, 55 (5), 487–494. doi: https://doi.org/10.1007/s10778-019-00970-2
- Tutuncu, N., Temel, B. (2009). A novel approach to stress analysis of pressurized FGM cylinders, disks and spheres. Composite Structures, 91 (3), 385–390. doi: https://doi.org/10.1016/j.compstruct.2009.06.009
- Zhang, X., Hasebe, N. (1999). Elasticity Solution for a Radially Nonhomogeneous Hollow Circular Cylinder. Journal of Applied Mechanics, 66 (3), 598–606. doi: https://doi.org/10.1115/1.2791477
- Tokovyy, Y., Ma, C.-C. (2016). Axisymmetric Stresses in an Elastic Radially Inhomogeneous Cylinder Under Length-Varying Loadings. Journal of Applied Mechanics, 83 (11). doi: https://doi.org/10.1115/1.4034459
- Liew, K. M., Kitipornchai, S., Zhang, X. Z., Lim, C. W. (2003). Analysis of the thermal stress behaviour of functionally graded hollow circular cylinders. International Journal of Solids and Structures, 40 (10), 2355–2380. doi: https://doi.org/10.1016/s0020-7683(03)00061-1
- Jabbari, M., Bahtui, A., Eslami, M. R. (2006). Axisymmetric Mechanical and Thermal Stresses in Thick Long FGM Cylinders. Journal of Thermal Stresses, 29 (7), 643–663. doi: https://doi.org/10.1080/01495730500499118
- Kordkheili, S. A. H., Naghdabadi, R. (2007). Thermoelastic Analysis of Functionally Graded Cylinders Under Axial Loading. Journal of Thermal Stresses, 31 (1), 1–17. doi: https://doi.org/10.1080/01495730701737803
- Tarn, J.-Q. (2001). Exact solutions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads. International Journal of Solids and Structures, 38 (46-47), 8189–8206. doi: https://doi.org/10.1016/s0020-7683(01)00182-2
- Zimmerman, R. W., Lut, M. P. (1999). Thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder. Journal of Thermal Stresses, 22 (2), 177–188. doi: https://doi.org/10.1080/014957399280959
- Tarn, J.-Q., Chang, H.-H. (2005). Extension, Torsion, Bending, Pressuring, and Shearing of Piezoelectric Circular Cylinders with Radial Inhomogeneity. Journal of Intelligent Material Systems and Structures, 16 (7-8), 631–641. doi: https://doi.org/10.1177/1045389x05048144
- Lur'e, A. I. (1970). Teoriya uprugosti. Moscow: Nauka, 939. Available at: https://lib-bkm.ru/load/86-1-0-2388
- Ustinov, Yu. A. (2006). Matematicheskaya teoriya poperechno-neodnorodnyh plit. Rostov-na Donu: OOO’’TSVVR’’, 257.
- Gol'denveyzer, A. L. (1963). Postroenie priblizhennoy teorii izgiba obolochki pri pomoschi asimptoticheskogo integrirovaniya uravneniy teorii uprugosti. Prikladnaya matematika i mekhanika, 27 (4), 593–608.
- Akhmedov, N. K., Sofiyev, A. H. (2019). Asymptotic analysis of three-dimensional problem of elasticity theory for radially inhomogeneous transversally-isotropic thin hollow spheres. Thin-Walled Structures, 139, 232–241. doi: https://doi.org/10.1016/j.tws.2019.03.022
- Mekhtiev, M. F. (2018). Vibrations of hollow elastic bodies. Springer, 212. doi: https://doi.org/10.1007/978-3-319-74354-7
- Mekhtiev, M. F. (2019). Asymptotic analysis of spatial problems in elasticity. Springer, 241. doi: https://doi.org/10.1007/978-981-13-3062-9
How to Cite
Copyright (c) 2021 Natik Akhmedov, Sevda Akbarova
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.
A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with PC TECHNOLOGY CENTER, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher PC TECHNOLOGY CENTER does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.