By using an analog of the δc -model, we reduce the problem of stressed state and limit equilibrium of an inhomogeneous shell of revolution weakened by an internal crack of any configuration with plastic strains developed on the continuation of the crack in the form of a narrow strip to an elastic problem. The indicated elastic problem is then reduced to a system of singular integral equations with unknown limits of integration and discontinuous functions on the right-hand sides. We propose an algorithm for the numerical solution of these systems with regard for the conditions of plasticity of thin shells and the conditions of boundedness for stresses. The effects of loading, geometric parameters, and mechanical characteristics on the crack-opening displacement and the sizes of plastic zones are investigated for cylindrical and spherical shells made of functionally graded materials and containing internal parabolic cracks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. A. Ambartsumyan, General Theory of Anisotropic Shells [in Russian], Nauka, Moscow (1974).
R. M. Kushnir, M. M. Nykolyshyn, and V. A. Osadchuk, Elastic and Elastoplastic Limit State of Shells with Defects [in Ukrainian], Spolom, Lviv (2003).
R. M. Kushnir, T. M. Nykolyshyn, and M. I. Rostun, “Limiting equilibrium of a spherical shell nonuniform across the thickness and containing a surface crack,” Fiz.-Khim. Mekh. Mater., 43, No. 3, 5–11 (2007); English translation: Mater. Sci., 43, No. 3, 291–299 (2007); https://doi.org/https://doi.org/10.1007/s11003-007-0034-z.
M. M. Nykolyshyn, “Stress state of a multilayered cylindrical shell containing a system of parallel cuts,” Mat. Met. Fiz.-Mekh. Polya, 22, 85–89 (1985).
M. Nykolyshyn and S. Zaryts’kyi, “An algorithm for the numerical investigation of the limit equilibrium of a conic shell containing a crack,” in: I. O. Lukovs'kyi, H. S. Kit, and R. M. Kushnir (editors), Mathematical Problems in the Mechanics of Inhomogeneous Structures [in Ukrainian], Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv (2010), pp. 131–132.
T. M. Nykolyshyn and M. Yo. Rostun, “Stress state and limiting equilibrium of a spherical shell nonuniform across the thickness and containing two surface cracks,” Mat. Met. Fiz.-Mekh. Polya, 52, No. 4, 166–172 (2009); English translation: J. Math. Sci., 174, No. 3, 331–340 (2011); https://doi.org/https://doi.org/10.1007/s10958-011-0302-1.
V. V. Panasyuk, Mechanics of Quasibrittle Fracture of Materials [in Russian], Naukova Dumka, Kiev (1991).
Ya. S. Pidstryhach and S. Ya. Yarema, Temperature Stresses in Shells [in Ukrainian], Akademiya Nauk Ukr. SSR, Kyiv (1961).
Ya. S. Pidstryhach, V. A. Osadchuk, E. M. Fedyuk, and M. M. Nykolyshyn, “Method of distortions in the theory of thin shells with cracks,” Mat. Met. Fiz.-Mekh. Polya, 1, 29–41 (1975).
W. Prager, Probleme der Plastizitätstheorie, Birkhäuser, Basel (1955).
K. S. Kim and N. Noda, “Green’s function approach to unsteady thermal stresses in an infinite hollow cylinder of functionally graded material,” Acta Mech., 156, No. 3-4, 145–161 (2002); https://doi.org/https://doi.org/10.1007/BF01176753.
M. Koizumi, “The concept of FGM,” in: Ceramic Transactions, Vol. 34, J. B. Holt, M. Koizumi, T. Hirai, and Z. A. Munir (editors), Functionally Gradient Materials, American Ceramic Society, Westerville (1993), pp. 3–10.
R. M. Kushnir, M. M. Nykolyshyn, and M. Yo. Rostun, “Limit equilibrium of inhomogeneous shells of revolution with internal cracks,” in: E. E. Gdoutos (editor), Proc. of the 14th Internat. Conf. on Fracture, (Rhodes, Greece, June 18–23, 2017), ICF14. Vol. 1, pp. 316–317; https://www.icfweb.org/Procf/ICF14/Vol1/316.
Y. Obata and N. Noda, “Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material,” J. Therm. Stresses, 17, No. 3, 471–487 (1994); https://doi.org/https://doi.org/10.1080/01495739408946273.
M. Ruhi, A. Angoshtari, and R. Naghdabadi, “Thermoelastic analysis of thick-walled finite-length cylinders of functionally graded materials,” J. Therm. Stresses, 28, No. 4, 391–408 (2005); https://doi.org/https://doi.org/10.1080/01495730590916623.
Z. S. Shao, L. F. Fan, and T. J. Wang, “Analytical solutions of stresses in functionally graded circular hollow cylinder with finite length,” Key Eng. Mater., 261–263, 251–256 (2004); https://doi.org/https://doi.org/10.4028/www.scientific.net/KEM.261-263.651.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 61, No. 4, pp. 56–65, October–December, 2018.
Rights and permissions
About this article
Cite this article
Kushnir, R.M., Nykolyshyn, M.M. & Rostun, M.Y. Elastoplastic Limit State of Inhomogeneous Shells of Revolution with Internal Cracks. J Math Sci 256, 426–438 (2021). https://doi.org/10.1007/s10958-021-05436-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05436-w