An Integrated Approach to the Optimization of Plates in Plane Stress State Operated at High Temperatures

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 24, no. 3, 2021 (September)
Pages 52-60
Cited by J. of Mech. Eng., 2021, vol. 24, no. 3, pp. 52-60



Мark M. Fridman, Kryvyi Rih Metallurgical Institute of the National Metallurgical Academy of Ukraine (5, Stephan Tilha str., Kryvyi Rih, 5006, Ukraine), e-mail:, ORCID: 0000-0003-3819-2776



Many critical elements of building and machine-building structures during their operation are in difficult operating conditions (high temperature, aggressive environment, etc.). In this case, they can be subject to a double effect: corrosion and material damage. Corrosion leads to a decrease in the cross-section of a structure, resulting in stress increase therein. In turn, damage to the material is accompanied by the appearance of microcracks and voids therein, due to inelastic deformation (creep), leading to a deterioration in its physical properties (for example, the elastic modulus) and a sharp decrease in the stress values at which the structure is destroyed. This article continues the study in the field of the optimal design of structures subject to the aforementioned double effect by the example of the optimization of plates with holes in the plane stress state, exposed to high temperatures (in previous works, the use of this approach was demonstrated in the optimization of the bending elements of rectangular and I-sections). Used as a corrosion equation is the modified Dolinsky mode, which takes into account the (additional) effect of the protective properties of an anticorrosive coating on the corrosion kinetics. Taken as a kinetic equation describing the change in material damage, is Yu. N. Rabotnov’s model, which enables to determine the duration of the incubation period of the beginning of the tangible process of material damage. To study the stress state of a plate, the finite element method is used. With a given contour of the plate, found is the optimal distribution of the thickness of the finite elements into which the given plate is divided. Acting as a constraint of the optimization problem is the parameter of damage to the plate material. The approach proposed in this work can be used to solve similar problems of the optimal design of structures operating under conditions of corrosion and material damage, using both analytical solutions and numerical methods.


Keywords: corrosion, material damage, optimization.


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  1. Kachanov, L. M. (1974). Osnovy mekhaniki razrusheniya [Fundamentals of fracture mechanics]. Moscow: Nauka, 308 p. (in Russian).
  2. Kachanov, L. M. (1985). O vremeni razrusheniya v usloviyakh polzuchesti [On the time of fracture under creep conditions]. Izv. AN SSSR. Otd. tekhn. nauk – Proceedings of the USSR Academy of Sciences. Department of Technical Sciences, no. 8, pp. 26–31 (in Russian).
  3. Rabotnov, Yu. N. (1966). Polzuchest elementov konstruktsiy [Creep of structural elements]. Moscow: Nauka, 752 p. (in Russian).
  4. Lemaitre, J. (1984). How to use damage mechanics. Nuclear Engineering and Design, vol. 80, iss. 2, pp. 233–245.
  5. Chaboche, J.-L. (1981). Continuous damage mechanics – a tool to describe phenomena before crack initiation. Nuclear Engineering and Design, vol. 64, iss. 2, pp. 233–247.
  6. Golub, V. P. (1996). Non-linear one-dimensional continuum damage theory. International Journal of Mechanical Sciences, vol. 38, iss. 10, pp. 1139–1150.
  7. Sosnovskiy, L. A. & Shcherbakov, S. S. (2011). Kontseptsii povrezhdennosti materialov [Concepts of material damage]. Vestnik TNTU – Scientific journal of TNTU, Special Issue (1), pp. 14–23 (in Russian).
  8. Travin, V. Yu. (2014). Otsenka povrezhdennosti materiala pri raschete prochnosti i dolgovechnosti elementov korpusnykh konstruktsiy [Assessment of material damage in calculating the strength and durability of elements of hull structures]. Izv. Tul. un-ta. Tekhn. naukiIzvestiya Tula State University. Series: Technical science, iss. 10, part 1, pp. 128–132 (in Russian).
  9. Volegov, P. S., Gribov, D. S., & Trusov, P. V. (2017). Damage and fracture: Classical continuum theories. Physical Mesomechanics, vol. 20, iss. 2, pp. 157–173.
  10. Kostyuk, A. G. (1953). Opredeleniye profilya vrashchayushchegosya diska v usloviyakh polzuchesti [Determination of the profile of a rotating disk under creep conditions]. Prikl. matematika i mekhanika – Journal of Applied Mathematics and Mechanics, vol. 17, iss. 5, pp. 615–618 (in Russian).
  11. Reitman, M. I. (1967). Theory of the optimum design of plastics structures with allowance for the time factor. Polymer Mechanics, vol. 3, iss. 2, pp. 243–244.
  12. Prager, W. (1968). Optimal structural design for given stiffness in stationary creep. Journal of Applied Mathematics and Physics (ZAMP), vol. 19, iss. 2, pp. 252–256.
  13. Nemirovskii, Yu. V. (1971). Design of optimum disks in relation to creep. Strength of Materials, vol. 3, iss. 8, pp. 891–894.
  14. Zyczkowski, M. (1971). Optimal structural design in rheology. Journal of Applied Mechanics, vol. 38, iss. 1, pp. 39–46.
  15. Pronina, Yu. & Sedova, O. (2021). Analytical solution for the lifetime of a spherical shell of arbitrary thickness under the pressure of corrosive environments: The effect of thermal and elastic stresses. Journal of Applied Mechanics, vol. 88, iss. 6, 061004.
  16. Pronina, Yu., Maksimov, A., & Kachanov, M. (2020). Crack approaching a domain having the same elastic properties but different fracture toughness: Crack deflection vs penetration. International Journal of Engineering Science, vol. 156, 103374.
  17. Pronina, Yu., Sedova, O., Grekov, M., & Sergeeva, T. (2018). On corrosion of a thin-walled spherical vessel under pressure. International Journal of Engineering Science, vol. 130, pp. 115–128.
  18. Pronina, Y. (2019). Design of pressurised pipes subjected to mechanochemical corrosion. In book: Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications. London: Taylor & Francis, pp. 644–649.
  19. Pronina, Y. G. (2017). An analytical solution for the mechanochemical growth of an elliptical hole in an elastic plane under a uniform remote load. European Journal of Mechanics – A/Solids, vol. 61, pp. 357–363.
  20. Pochtman, Yu. M. & Fridman, M. M. (1997). Metody rascheta nadezhnosti i optimalnogo proyektirovaniya konstruktsiy, funktsioniruyushchikh v ekstremalnykh usloviyakh [Methods for calculating the reliability and optimal design of structures operating in extreme conditions]. Dnepropetrovsk: Nauka i obrazovaniye, 134 p. (in Russian).
  21. Fridman, M. M. & Elishakoff, I. (2020). Optimal thickness of a spherical shell subjected to double-sided corrosion. International Journal of Sustainable Materials and Structural Systems (IJSMSS), vol. 4, no. 2/3/4, pp. 158–170.
  22. Fridman, M. M. (2017). Optimalnoye proyektirovaniye konstruktsiy pri kombinirovannom podkhode k uchetu korrozii i zashchitnykh svoystv antikorrozionnykh pokrytiy [Optimal design of structures with a combined approach to accounting for corrosion and protective properties of anti-corrosion coatings]. Problemy mashinostroyeniya – Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 20, no. 3, pp. 64–68 (in Russian).
  23. Fridman, M. (2018). Stepwise optimization of I-section flexible elements under a fuzzy approach to taking into account corrosion and protective properties of anticorrosive coating. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 21, no. 3, pp. 58–64.
  24. Fridman, М. M. (2019). Optimal Design of Bending Elements in Conditions of Corrosion and Material Damage. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 22, no. 3, pp. 63–69.
  25. Fridman, М. M. (2020). Optimization of Bendable I-Section Elements Subject to Corrosion and Material Damage. Journal of Mechanical Engineering – Problemy Mashynobuduvannia, vol. 23, no. 3, pp. 60–67.
  26. Dolinskii, V. M. (1967). Calculations on loaded tubes exposed to corrosion. Chemical and Petroleum Engineering, vol. 3, iss. 2, pp. 96–97.
  27. Karyakina, M. I. (1980). Fiziko-khimicheskiye osnovy protsessov formirovaniya i stareniya pokrytiy [Physical and chemical foundations of the formation and aging of coatings]. Moscow: Khimiya, 198 p. (in Russian).
  28. Ovchinnikov, I. G. & Pochtman, Yu. M. (1995). Tonkostennyye konstruktsii v usloviyakh korrozionnogo iznosa: raschet i optimizatsiya [Thin-walled structures in conditions of corrosive wear: Calculation and optimization]. Dnepropetrovsk: Dnepropetrovsk University, 190 p. (in Russian).
  29. Gurvich, I. B., Zakharchenko, B. G., & Pochtman, Yu. M. (1979). Randomized algorithm to solve problems of nonlinear programming. Izvestiya Akademii nauk SSSR. Tekhnicheskaya kibernetikaBulletin of the USSR Academy of Sciences. Engineering Cybernetics, no. 5, pp. 15–17 (in Russian).
  30. Odgvist, F. K. G. (1966). Mathematical theory of creep and creep rupture. Oxford Math. Mon., Clarendon Press, 234 p.


Received 04 June 2021

Published 30 September 2021