Acta Mechanica Sinica

, Volume 34, Issue 5, pp 936–948 | Cite as

Novel material tailoring method for internally pressurized FG spherical and cylindrical vessels

  • Fayyaz Nosouhi Dehnavi
  • Ali ParviziEmail author
  • Karen Abrinia
Research Paper


A new material tailoring method for spherical and cylindrical vessels made of functionally graded materials (FGMs) is presented. It is assumed that the FG material is composed of an Al–SiC metallic-matrix composite. A uniform ratio of in-plane shear stress to yield strength [\(\varphi \left( r \right) \)] is used as the design criterion to utilize the maximum capacity of the vessel. The aim is to find a distribution of SiC particles in the radial direction, i.e., \(f\left( r \right) \), that achieves a uniform index \(\varphi \left( r \right) =\hbox {const}.\) through the wall thickness of the internally pressurized spherical or cylindrical vessel. Both the Mori–Tanaka and rule-of-mixtures homogenization schemes are used to express the effective elastic module and Poisson’s ratio. Moreover, the strength of the composite is expressed based on the rule of mixtures. Besides, finite element simulation is carried out to verify the accuracy of the analytical solution. The effects of input parameters such as the internal pressure, strength of the SiC particles, ratio of in-plane shear stress to effective yield strength, and choice of homogenization scheme on the tailored distribution of the SiC volume fraction in the radial direction are also investigated.


Material tailoring Sphere Cylinder FGM Mori–Tanaka Rule of mixtures 

List of symbols


Inner and outer radius of sphere or cylinder, respectively


Radial displacement


Constant value of in-plane shear stress divided by effective yield strength

\(f\left( r \right) \)

Radial distribution of SiC particles

\(N_1 ,\,N_2 \)

Constants related to material properties based on Mori–Tanaka homogenization

\(C_1-C_9 \)

Constants related to material properties based on Mori–Tanaka homogenization

\(Z_1-Z_9 \)

Constants related to material properties based on Mori–Tanaka homogenization

\(P_a ,P_b \)

Internal and external pressure, respectively


Elastic modulus


Bulk modulus


Yield strength

\(\mu \)

Shear modulus

\(\varphi \left( r \right) \)

In-plane shear stress divided by effective yield strength

\(\varepsilon _r ,\varepsilon _\theta ,\varepsilon _\varphi \)

Strains in radial, and first and second circumferential directions, respectively

\(\sigma _r ,\sigma _\theta \)

Radial and circumferential stress, respectively


Poisson’s ratio

\(\mathrm {p},\,\mathrm {m}\)

Subscripts denoting particle andmatrix, respectively


Subscript (abbreviation) for Mori–Tanaka homogenization scheme


Subscript (abbreviation) for rule-of-mixtures homogenization scheme



The work was supported by the Iran National Science Foundation (INSF).


  1. 1.
    Loghman, A., Parsa, H.: Exact solution for magneto–thermo–elastic behaviour of double-walled cylinder made of an inner FGM and an outer homogeneous layer. Int. J. Mech. Sci. 88, 93–99 (2014)CrossRefGoogle Scholar
  2. 2.
    Jha, D., Kant, T., Singh, R.: A critical review of recent research on functionally graded plates. Compos. Struct. 96, 833–849 (2013)CrossRefGoogle Scholar
  3. 3.
    Rodrıguez-Castro, R., Wetherhold, R., Kelestemur, M.: Microstructure and mechanical behavior of functionally graded Al A359/SiC p composite. Mater. Sci. Eng. A 323, 445–456 (2002)CrossRefGoogle Scholar
  4. 4.
    Mahmoudi, T., Parvizi, A., Poursaeidi, E., et al.: Thermo-mechanical analysis of functionally graded wheel-mounted brake disk. J. Mech. Sci. Technol. 29, 4197–4204 (2015)CrossRefGoogle Scholar
  5. 5.
    Nosouhi Dehnavi, F., Parvizi, A.: Electrothermomechanical behaviors of spherical vessels with different configurations of functionally graded piezoelectric coating. J. Intel. Mat. Syst. Struct. 10.1177/1045389X17742737 (2017)Google Scholar
  6. 6.
    Sharma, D., Sharma, J., Dhaliwal, S., et al.: Vibration analysis of axisymmetric functionally graded viscothermoelastic spheres. Acta Mech. Sin. 30, 100–111 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kim, Y.W.: Effect of partial elastic foundation on free vibration of fluid-filled functionally graded cylindrical shells. Acta Mech. Sin. 31, 920–930 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dehnavi, F.N., Parvizi, A.: Investigation of thermo-elasto-plastic behavior of thick-walled spherical vessels with inner functionally graded coatings. Meccanica 52, 1–18 (2016)MathSciNetGoogle Scholar
  9. 9.
    Eraslan, A., Akis, T.: On the plane strain and plane stress solutions of functionally graded rotating solid shaft and solid disk problems. Acta Mech. 181, 43–63 (2006)CrossRefGoogle Scholar
  10. 10.
    Ozturk, A., Gulgec, M.: Elastic-plastic stress analysis in a long functionally graded solid cylinder with fixed ends subjected to uniform heat generation. Int. J. Eng. Sci. 49, 1047–1061 (2011)CrossRefGoogle Scholar
  11. 11.
    Keles, I., Conker, C.: Transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres with exponentially-varying properties. Eur. J. Mech. A Sol. 30, 449–455 (2011)CrossRefGoogle Scholar
  12. 12.
    Tutuncu, N.: Stresses in thick-walled FGM cylinders with exponentially-varying properties. Eng. Struct. 29, 2032–2035 (2007)CrossRefGoogle Scholar
  13. 13.
    Parvizi, A., Naghdabadi, R., Arghavani, J.: Analysis of Al A359/SiCp functionally graded cylinder subjected to internal pressure and temperature gradient with elastic–plastic deformation. J. Therm. Stress. 34, 1054–1070 (2011)CrossRefGoogle Scholar
  14. 14.
    Mirzaei, M., Kiani, Y.: Free vibration of functionally graded carbon nanotube reinforced composite cylindrical panels. Compos. Struct. 142, 45–56 (2016)CrossRefGoogle Scholar
  15. 15.
    Xin, L., Yang, S., Zhou, D., et al.: An approximate analytical solution based on the Mori-Tanaka method for functionally graded thick-walled tube subjected to internal pressure. Compos. Struct. 135, 74–82 (2016)CrossRefGoogle Scholar
  16. 16.
    Ebrahimi, F., Barati, M.R., Dabbagh, A.: A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. Int. J. Eng. Sci. 107, 169–182 (2016)CrossRefGoogle Scholar
  17. 17.
    Xin, L., Dui, G., Yang, S., et al.: Solutions for behavior of a functionally graded thick-walled tube subjected to mechanical and thermal loads. Int. J. Mech. Sci. 98, 70–79 (2015)CrossRefGoogle Scholar
  18. 18.
    Eslami, M., Babaei, M., Poultangari, R.: Thermal and mechanical stresses in a functionally graded thick sphere. Int. J. Press. Vessels Pip. 82, 522–527 (2005)CrossRefGoogle Scholar
  19. 19.
    Jabbari, M., Sohrabpour, S., Eslami, M.R.: Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int. J. Press. Vessels Pip. 79, 493–497 (2002)CrossRefGoogle Scholar
  20. 20.
    Arefi, M.: Nonlinear thermoelastic analysis of thick-walled functionally graded piezoelectric cylinder. Acta Mech. 224, 2771 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    You, L., Zhang, J., You, X.: Elastic analysis of internally pressurized thick-walled spherical pressure vessels of functionally graded materials. Int. J. Press. Vessels Pip. 82, 347–354 (2005)CrossRefGoogle Scholar
  22. 22.
    Atashipour, S.A., Sburlati, R., Atashipour, S.R.: Elastic analysis of thick-walled pressurized spherical vessels coated with functionally graded materials. Meccanica 49, 2965–2978 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang, Z., Zhang, Q., Xia, L., et al.: Thermomechanical analysis of pressure vessels with functionally graded material coating. J. Press. Vessel Technol. 138, 011205 (2016)CrossRefGoogle Scholar
  24. 24.
    Mohammadi, M., Saha, G.C., Akbarzadeh, A.H.: Elastic field in composite cylinders made of functionally graded coatings. Int. J. Eng. Sci. 101, 156–170 (2016)CrossRefGoogle Scholar
  25. 25.
    Parvizi, A., Alikarami, S., Asgari, M.: Exact solution for thermoelastoplastic behavior of thick-walled functionally graded sphere under combined pressure and temperature gradient loading. J. Therm. Stress. 39, 1152–1170 (2016)CrossRefGoogle Scholar
  26. 26.
    Alikarami, S., Parvizi, A.: Elasto–plastic analysis and finite element simulation of thick-walled functionally graded cylinder subjected to combined pressure and thermal loading. Sci. Eng. Compos. Mater. 24, 609–620 (2017)CrossRefGoogle Scholar
  27. 27.
    Cho, J.R., Ha, D.Y.: Volume fraction optimization for minimizing thermal stress in Ni–Al\(_2\)O\(_3\) functionally graded materials. Mat. Sci. Eng. A 334, 147–155 (2002)CrossRefGoogle Scholar
  28. 28.
    Taheri, A.H., Hassani, B., Moghaddam, N.Z.: Thermo-elastic optimization of material distribution of functionally graded structures by an isogeometrical approach. Int. J. Sol. Struct. 51, 416–429 (2014)CrossRefGoogle Scholar
  29. 29.
    Nemat-Alla, M.: Reduction of thermal stresses by composition optimization of two-dimensional functionally graded materials. Acta Mech. 208, 147–161 (2009)CrossRefGoogle Scholar
  30. 30.
    Zhang, X.D., Hong, Y.L.: Li, A.H.: Optimization of axial symmetrical FGM under the transient-state temperate field. Int. J. Miner. Metall. Mat. 19, 59–63 (2012)CrossRefGoogle Scholar
  31. 31.
    Wang, Z.W., Zhang, Q., Xia, L.Z., et al.: Stress analysis and parameter optimization of an FGM pressure vessel subjected to thermo-mechanical loadings. Procedia Eng. 130, 374–389 (2015)CrossRefGoogle Scholar
  32. 32.
    Nie, G., Batra, R.: Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness. Compos. Struct. 92, 720–729 (2010)CrossRefGoogle Scholar
  33. 33.
    Nie, G., Batra, R.: Material tailoring and analysis of functionally graded isotropic and incompressible linear elastic hollow cylinders. Compos. Struct. 92, 265–274 (2010)CrossRefGoogle Scholar
  34. 34.
    Nie, G., Zhong, Z., Batra, R.: Material tailoring for orthotropic elastic rotating disks. Compos. Sci. Technol. 71, 406–414 (2011)CrossRefGoogle Scholar
  35. 35.
    Nie, G., Zhong, Z., Batra, R.: Material tailoring for functionally graded hollow cylinders and spheres. Compos. Sci. Technol. 71, 666–673 (2011)CrossRefGoogle Scholar
  36. 36.
    Benveniste, Y.: A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech. Mat. 6, 147–157 (1987)CrossRefGoogle Scholar
  37. 37.
    Sadd, M.H.: Elasticity: Theory, Applications, and Numerics. Academic Press, Cambridge (2009)Google Scholar
  38. 38.
    Xin, L., Lu, W., Yang, S., et al.: Influence of linear work hardening on the elastic–plastic behavior of a functionally graded thick-walled tube. Acta Mech. 227, 2305–2321 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Fayyaz Nosouhi Dehnavi
    • 1
  • Ali Parvizi
    • 1
    Email author
  • Karen Abrinia
    • 1
  1. 1.School of Mechanical Engineering, College of EngineeringUniversity of TehranTehranIran

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