We formulate the problem of viscoplastic dynamic deformation of metallic composite shells of layered-fibrous structure. We have developed an original numerical method for the integration of the posed initial boundary-value problem based on the successive discretization of the domain of definition of the solution first with respect to time and then with respect to space variables.
Similar content being viewed by others
References
N. A. Abrosimov and V. G. Bazhenov, Nonlinear Problems of Dynamics of Composite Structures [in Russian], Nizhny Novgorod State University, Nizhny Novgorod (2002).
L. Ya. Ainola and U. K. Nigul, “Wave processes of deformation of elastic plates and shells,” Izv. Akad. Nauk Eston. SSR, Ser. Fiz.-Mat. Tekhn. Nauk, 14, No. 1, 3–63 (1965).
A. N. Andreev and Yu. V. Nemirovskii, Multilayered Anisotropic Shells and Plates. Bending, Stability, and Vibrations [in Russian], Nauka, Novosibirsk (2001).
V. G. Bazhenov, V. K. Lomunov, and S. L. Osetrov, “Analysis of the applicability of a rigid-plastic model in problems of static and dynamic bending of plates,” Vestn. Chuvash. Gos. Ped. Univ. Im. I. Ya. Yakovleva. Ser. Mekh. Predel’n. Sostoyan., No. 2 (8), 64–70 (2010).
V. G. Bazhenov, V. K. Lomunov, M. V. Petrov, and A. G. Ugodchikov, “Investigation of large viscoplastic strains of cylindrical shells using the magnetic-pulse method of loading,” Mashinovedenie, No. 5, 73–80 (1983).
V. Z. Vlasov and N. N. Leont’ev, Beams, Plates, and Shells on an Elastic Base [in Russian], Fizmatlit, Moscow (1960).
H. Hopkins and W. Prager, “On the dynamics of plastic circular plates,” Z. Angew. Math. Mekh., 5, No. 4, 317–330 (1954).
A. Dzhuraev, Systems of Equations of Composite Type [in Russian], Nauka, Moscow (1972).
M. I. Erkhov, Theory of Ideally Plastic Bodies and Structures [in Russian], Nauka, Moscow (1978).
V. G. Zubchaninov, Mechanics of Processes of Plastic Media [in Russian], Fizmatlit, Moscow (2010).
L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).
T. D. Karimbaev, B. M. Myktybekov, and I. M. Panova, “Mathematical models of nonlinear deformation of unidirectionally reinforced composite materials,” Trudy Tsentr. Instit. Aviats. Motorostroen., No. 1334, TIAM, Moscow (2005).
V. V. Karpov, Strength and Stability of Stiffened Shells of Revolution, Vol. 1: Models and Algorithms of the Investigation of the Strength and Stability of Stiffened Shells of Revolution [in Russian], Fizmatlit, Moscow (2010).
L. M. Kachanov, Fundamentals of the Theory of Plasticity [in Russian], Nauka, Moscow (1969).
L. M. Kachanov, Theory of Creep [in Russian], Fizmatlit, Moscow (1960).
A. G. Kobelev, V. I. Lysak, V. N. Chernyshev, A. A. Bykov, and V. P. Vostrikov, Production of Metallic Layered Composite Materials [in Russian], Intermet Inzhiniring, Moscow (2002).
K. L. Komarov and Yu. V. Nemirovskii, Dynamics of Rigid-Plastic Structural Elements [in Russian], Nauka, Novosibirsk (1984).
S. G. Mikhlin, Numerical Realization of Variational Methods [in Russian], Nauka, Moscow (1966).
Kh. M. Mushtari, Nonlinear Theory of Shells [in Russian], Nauka, Moscow (1990).
Yu. V. Nemirovskii, “On the elastoplastic behavior of a reinforced layer,” Prikl. Mekh. Tekhn. Fiz., No. 6, 81–89 (1969).
Yu. V. Nemirovskii and A. P. Yankovskii, “Viscoplastic dynamics of layered-fibrous plates of variable thickness under the action of explosion-type loads,” Mekh. Kompoz. Mater. Konstruk., 12, No. 4, 484–501 (2006).
Yu. V. Nemirovskii and A. P. Yankovskii, “Dynamic viscoplastic bending of reinforced bars of variable cross-section,” Mat. Met. Fiz.-Mekh. Polya, 49, No. 1, 53–66 (2006).
Yu. V. Nemirovskii and A. P. Yankovskii, “Integration of the problem of dynamic viscoplastic deformation of isotropic cylindrical shells of revolution by a generalized Runge–Kutta method,” Vestn. Chuvash. Gos. Ped. Univ. Im. I. Ya. Yakovleva. Ser. Mekh. Predel’n. Sostoyan., No. 2 (5), 129–144 (2008).
Yu. V. Nemirovskii and A. P. Yankovskii, “Integration of the problem of dynamic elastoplastic bending of reinforced bars of varying cross-section by generalized Runge–Kutta methods,” Vychisl. Tekhnol., 9, No. 4, 77–95 (2004).
Yu. V. Nemirovskii and A. P. Yankovskii, “On some features of equations of shells reinforced by fibers of constant cross-section,” Mekh. Kompoz. Mater. Konstruk., 3, No. 2, 20–40 (1977).
Yu. V. Nemirovskii and A. P. Yankovskii, “Viscoplastic deformation of reinforced plates with varying thickness under explosive loads,” Prikl. Mekh., 44, No. 2, 85–98 (2008); English translation: Int. Appl. Mech., 44, No. 2, 188–199 (2008).
Yu. V. Nemirovskii and A. P. Yankovskii, “Steady-state creep of layered metallic composite plates with complex reinforcement structures in longitudinal-transverse bending,” Mekh. Kompoz. Mater. Konstruk., 15, No. 1, 59–82 (2009).
Yu. V. Nemirovskii and A. P. Yankovskii, “Numerical integration of initial boundary-value problems with large gradients of the solution by generalized Runge–Kutta methods,” Mat. Met. Fiz.-Mekh. Polya, 47, No. 1, 43–62 (2004).
V. V. Pikul’, Mechanics of Shells [in Russian], Dal’nauka, Vladivostok (2009).
Yu. N. Rabotnov, Creep of Structural Elements [in Russian], Fizmatgiz, Moscow (1966).
A. A. Samarskii, Theory of Difference Schemes [in Russian], Nauka, Moscow (1989).
Yu. P. Trykov, L. M. Gurevich, and V. G. Shmorgun, Layered Composites Based on Aluminum and Aluminum Alloys [in Russian], Metallurgizdat, Moscow (2004).
A. P. Yankovskii, “Steady-state creep of layered metallic composite shells with complex reinforcement structures,” Mekh. Kompoz. Mater. Konstruk., 16, No. 3, 400–420 (2010).
A. P. Yankovskii, “Numerical integration of the problem of viscoplastic dynamics of layered-fibrous rectilinear elongated plates,” in: V. M. Fomin (editor), Numerical Methods of the Solution of Problems of the Theory of Elasticity and Plasticity. Proceedings of the 19th All-Russian Conference (August 28–31, 2005, Biisk) [in Russian], Parallel,’ Novosibirsk (2005), pp. 290–297.
K.-J. Bathe, Finite Element Procedures, Prentice Hall, New Jersey (1996).
K. Dekker and J. G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam (1984).
O. C. Zeinkiewicz and R. L. Taylor, The Finite Element Method, Butterworth-Heinemann, Oxford (2000).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 55, No. 2, pp. 119–130, April–June, 2012
Rights and permissions
About this article
Cite this article
Yankovskii, A.P. Viscoplastic dynamics of metallic composite shells of layered-fibrous structure under the action of loads of explosive type. I. Statement of the problem and method for solution. J Math Sci 192, 623–633 (2013). https://doi.org/10.1007/s10958-013-1421-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-013-1421-7