Method to Study the Creep of Complex-Shaped Functionally-Graded Bodies

Journal Journal of Mechanical Engineering
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 23, no. 1, 2020 (March)
Pages 38-45
Cited by J. of Mech. Eng., 2020, vol. 23, no. 1, pp. 38-45



Serhii M. Sklepus, A. Podgorny Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0002-4119-4310



The creep problem of complex-shaped functionally-graded bodies of revolution is considered. For the variational statement of the problem, the Lagrange functional is used, defined at kinematically possible displacement rates. A numerical-analytical method is developed for solving a non-linear initial-boundary creep problem. It is based on the combined use of the R-functions, Ritz and Runge-Kutta-Merson methods. The advantages of the proposed method include: exact consideration of the geometric information about the boundary-value problem at the analytical level, without any approximation thereof; representation of an approximate solution to the problem in an analytical form; exact satisfaction of boundary conditions; automatic time step selection. Solved are the problems of creep both for a hollow straight cylinder and a complex-shaped body of revolution (a cylinder with a rectangular cut-out on the outer surface), both cylinders being loaded with a constant inner pressure, made of the functionally graded material (FGM) based on SiC particle-reinforced aluminium. The creep of the material is described by Norton’ law. Both Young’s modulus and creep characteristics of the material depend on the volume part of the reinforcing material. Both ends of the cylinder are free of external load, and are fixed in such a way that the radial displacements are equal to zero. A corresponding partial solution structure is constructed that satisfies the boundary conditions for displacement rates. The calculations were performed for cylinders of two different composite materials: a material with a uniform distribution of SiC particles and an FGM with a difference in the volume content of reinforcing particles along the radius, with the average volumetric content of reinforcing SiC particles in the two cases being the same. The influence of both the gradient properties of the material and geometric shape on the stress-strain state (SSS) under creep conditions was investigated. The presence of a rectangular cut-out on the outer surface of a cylinder in all cases leads to an increase in displacements and stresses. Moreover, the degree of influence of the geometric shape on the SSS during creep substantially depends on the gradient properties of the material. For a cut-out cylinder made of the material with a uniform distribution of SiC particles, there is a significant increase in displacements and stresses after 100 hours of creep compared with a straight cylinder. For bodies of revolution made of a functionally graded material, the cut-out effect on the SSS is less pronounced.


Keywords: functionally graded material, body of revolution, creep, R-functions method.


Full text: Download in PDF



  1. Chen, J. J. (2007). Creep analysis for a functionally graded cylinder subjected to internal and external pressure. J. Strain Analysis, vol. 42, pp. 69–77.
  2. Nejad, M. Z. & Kashkoli, M. D. (2014). Time-dependent thermo-creep analysis of rotating FGM thick-walled cylindrical pressure vessels under heat flux. Int. J. Eng. Sci., vol. 82, pp. 222–237.
  3. Shrikant, K. V. (2012). Creep analysis of an isotropic functionally graded cylinder. J. Advances in Sci. and Techn., vol. 3, no. 4, pp. 1–9.
  4. Singh, T. & Gupta, V. K. (2011). Effect of anisotropy on steady state creep in functionally graded cylinder. Composite Structures, vol. 93, no. 2, pp. 747–758.
  5. Chen, J. J., Yoon, K. B., & Tu, S. T. (2011). Creep behavior of pressurized tank composed of functionally graded materials. J. Pressure Vessel Techn., vol. 133, pp. 69–77.
  6. Kashkoli, M. D. & Nejad, M. Z. (2015). Time dependent thermo-elastic creep analysis of thick-walled spherical pressure vessels made of functionally graded materials. J. Theoret. and Appl. Mech., vol. 53, no. 4, pp. 1053–1065.
  7. Breslavskyi, D. V., Korytko, Yu. M., & Tatarinova, O. A. (2017). Proektuvannia ta rozrobka skinchennoelementnoho prohramnoho zabezpechennia [Design and development of finite element software]. Kharkiv: «Pidruchnyk NTU «KhPI», 232 p. (in Ukrainian).
  8. Breslavsky, D., Kozlyuk, A., & Tatatarinova, O. (2018). Numerical simulation of two-dimensional problems of creep crack growth with material damage consideration. Eastern-European Journal of Enterprise Technologies, no. 2/7 (92), pp. 27–33.
  9. Rvachev, V. L. (1982). Teoriya R-funktsiy i nekotoryye yeye prilozheniya [The R-functions theory and some of its applications]. Kiyev: Naukova Dumka, 552 p. (in Russian).
  10. Rvachev, V. L. & Sinekop, N. S. (1990). Metod R-funktsiy v zadachakh teorii uprugosti i plastichnosti [The R-functions method in problems of the theory of elasticity and plasticity]. Kiyev: Naukova dumka, 216 p. (in Russian).
  11. Rodichev, Y. M., Smetankina, N. V., Shupikov, O. M., & Ugrimov, S. V. (2018). Stress-strain assessment for laminated aircraft cockpit windows at static and dynamic loads. Strength of Materials, vol. 50, iss. 6, pp. 868–873.
  12. Smetankina, N. V. (2015). Modeliuvannia kolyvan sharuvatykh tsylindrychnykh obolonok skladnoi formy pry udar-nomu navantazhenni [Modeling of oscillations of layered cylindrical shells of complex shape at impact loading]. Visn. Zaporiz. nats. un-tu. Fizyko-matematychni naukyBulletin of Zaporizhzhya National University. Physical and mathematical sciences, no. 1, pp. 162–170 (in Ukrainian).
  13. Zolochevsky, A., Sklepus, S., Galishin, A., Kühhorn, A., Kober, M. (2014). A comparison between the 3D and the Kirchhoff-Love solutions for cylinders under creep-damage conditions. Techn. Mechanik, vol. 34, no. 2, pp. 104–113.
  14. Singh, S. B. & Ray, S. (2003). Creep analysis in an isotropic FGM rotating disc of Al–SiC composite. J. Materials Proc. Techn., vol. 143–144, pp. 616-622.
  15. Ogierman, W. & Kokot, G. (2013). Mean field homogenization in multi-scale modelling of composite materials. J. Achievements in Materials Manufacturing Eng. (JAMME), vol. 61, iss. 2, pp. 343–348.
  16. Singla, A., Gard, M., & Gupta, V. K. (2015). Steady state creep behaviour of functionally graded composite by using analytical method. Intern. J. Computer Appl., no. 8, pp. 13–17.


Received 10 February 2020