Optimal Design of Bending Elements in Conditions of Corrosion and Material Damage

DOI https://doi.org/10.15407/pmach2019.03.063
Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 22, no. 3, 2019 (September)
Pages 63-69
Cited by J. of Mech. Eng., 2019, vol. 22, no. 3, pp. 63-69



Мark M. Fridman, Kryvyi Rih Metallurgical Institute of the National Metallurgical Academy of Ukraine (5, Stephan Tilho Str., Kryvyi Rih, 5006, Ukraine), e-mail: mark17@i.ua, ORCID: 0000-0003-3819-2776



During operation, many of the critical elements of building and engineering structures are in difficult operating conditions (high temperature, aggressive environment, etc.). In this case, they may be subject to a double effect: corrosion and material damage. Corrosion leads to a reduction in the cross-section of a structure, resulting in stress increase therein. In turn, the damage to the material is accompanied by the appearance of microcracks and voids therein due to inelastic deformation (creep), which leads to a deterioration of physical characteristics of the material (for example, elastic modulus) and a sharp decrease in the stress values at which the structure is destroyed. This paper considers the optimization of bending rectangular cross-section elements operated in conditions conducive to the appearance of both corrosion and material damage. As the equation of corrosion, the model of V. M. Dolinsky is taken. This model takes into account the effect of stresses on the corrosion wear of structures. As a kinetic equation describing the change in material damage, the model of Yu. N. Rabotnov is used. The optimality criterion is the minimum mass of the structure. The height of the rectangular cross-section bending element along its length is optimized using the principle of equal damage at the final moment of the lifetime of the structure. The proposed approach can be used to solve similar problems of the optimal design of structures operating in conditions of corrosion and material damage with the use of both analytical solutions and numerical methods.


Keywords: corrosion, material damage, optimization.


Full text: Download in PDF



  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. https://doi.org/10.1016/0029-5493(84)90169-9
  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. https://doi.org/10.1016/0029-5493(81)90007-8
  6. Golub, V. P. (1996). Non-linear one-dimensional continuum damage theory. International Journal of Mechanical Sciences, vol. 38, iss. 10, pp. 1139–1150. https://doi.org/10.1016/0020-7403(95)00106-9
  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.
  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. https://doi.org/10.1134/S1029959917020060
  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. https://doi.org/10.1007/BF00858872
  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. https://doi.org/10.1007/BF01601470
  13. Nemirovskii, Yu. V. (1971). Design of optimum disks in relation to creep. Strength of Materials, vol. 3, iss. 8, pp. 891–894. https://doi.org/10.1007/BF01527642
  14. Zyczkowski M. (1971). Optimal structural design in rheology. Journal of Applied Mechanics, vol. 38, iss. 1, pp. 39–46. https://doi.org/10.1115/1.3408764
  15. 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.
  16. Fridman, M. M. & Zyczkowski, M. (2001). Structural optimization of elastic columns under stress corrosion conditions. Structural and Multidisciplinary Optimization, vol. 21, iss. 3, pp. 218–228. https://doi.org/10.1007/s001580050186
  17. Fridman, M.M. & Elishakoff, I. (2013). Buckling optimization of compressed bars undergoing corrosion. Ocean Systems Engineering, vol. 3, iss. 2, pp. 123–136. https://doi.org/10.12989/ose.2013.3.2.123
  18. Fridman, M. M. & Elishakoff, I. (2015). Design of bars in tension or compression exposed to a corrosive environment. Ocean Systems Engineering, vol. 5, iss. 1, pp. 21–30. https://doi.org/10.12989/ose.2015.5.1.021
  19. Fridman, M. M. (2016). Optimalnoye proyektirovaniye trubchatykh sterzhnevykh konstruktsiy, podverzhennykh korrozii [Optimal design of tubular bar structures subject to corrosion]. Problemy mashinostroyeniyaJournal of Mechanical Engineering, vol. 19, no. 3, pp. 37–42 (in Russian). https://doi.org/10.15407/pmach2016.03.037
  20. Dolinskii, V. M. (1967). Calculations on loaded tubes exposed to corrosion. Chemical and Petroleum Engineering, vol. 3, iss. 2, pp. 96–97. https://doi.org/10.1007/BF01150056
  21. Gurvich, I. B., Zakharchenko, B. G., & Pochtman, Yu. M. (1979). Randomized algorithm to solve problems of nonlinear programming. Izv. Ac. Sci. USSR. Engineering Cybernetics, no. 5, pp. 15–17 (in Russian).


Received 17 April 2019

Published 30 September 2019