Behavior of solution of the elasticity problem for a radial inhomogeneous cylinder with small thickness

Authors

DOI:

https://doi.org/10.15587/1729-4061.2021.247500

Keywords:

non-axisymmetric problem, radial inhomogeneous cylinder, asymptotic integration method, homogeneous solutions, boundary layer

Abstract

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.

Author Biographies

Natik Akhmedov, Azerbaijan State Economic University (UNEC)

Doctor of Mathematics, Professor, Head of Department

Department of Mathematics and Statistics

Sevda Akbarova, Azerbaijan State Economic University (UNEC)

PhD, Associate Professor

Department of Mathematics and Statistics

References

  1. 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
  2. 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
  3. 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
  4. 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.
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. Lur'e, A. I. (1970). Teoriya uprugosti. Moscow: Nauka, 939. Available at: https://lib-bkm.ru/load/86-1-0-2388
  26. Ustinov, Yu. A. (2006). Matematicheskaya teoriya poperechno-neodnorodnyh plit. Rostov-na Donu: OOO’’TSVVR’’, 257.
  27. 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.
  28. 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
  29. Mekhtiev, M. F. (2018). Vibrations of hollow elastic bodies. Springer, 212. doi: https://doi.org/10.1007/978-3-319-74354-7
  30. Mekhtiev, M. F. (2019). Asymptotic analysis of spatial problems in elasticity. Springer, 241. doi: https://doi.org/10.1007/978-981-13-3062-9

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Published

2021-12-21

How to Cite

Akhmedov, N., & Akbarova, S. (2021). Behavior of solution of the elasticity problem for a radial inhomogeneous cylinder with small thickness. Eastern-European Journal of Enterprise Technologies, 6(7 (114), 29–42. https://doi.org/10.15587/1729-4061.2021.247500

Issue

Vol. 6 No. 7 (114) (2021): Applied mechanics

Section

Applied mechanics

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