Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936

Russian Russian English English About Journal | Issues | Guidelines | Editorial Board | Contact Us

IPMech RASWeb hosting is provided
by the Ishlinsky Institute for
Problems in Mechanics
of the Russian
Academy of Sciences
IssuesArchive of Issues2017-4pp.364-377

Archive of Issues

Total articles in the database: 10671
In Russian (. . ): 7873
In English (Mech. Solids): 2798

<< Previous article | Volume 52, Issue 4 / 2017 | Next article >>
A.K. Belyaev, P.E. Tovstik, and T.P. Tovstik, "Thin Rod under Longitudinal Dynamic Compression," Mech. Solids. 52 (4), 364-377 (2017)
Year 2017 Volume 52 Number 4 Pages 364-377
DOI 10.3103/S0025654417040021
Title Thin Rod under Longitudinal Dynamic Compression
Author(s) A.K. Belyaev (Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Bol'shoy pr. 61, St. Petersburg, 199078 Russia)
P.E. Tovstik (Saint-Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg, 199034 Russia,
T.P. Tovstik (Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, Bol'shoy pr. 61, St. Petersburg, 199078 Russia)
Abstract The paper contains a short survey of the papers on the static and dynamic longitudinal compression of a thin rod initiated by Morozov and and carried out in 2009-2016 with his direct participation. We consider linear and nonlinear problems related to the propagation of longitudinal waves in a rod and the transverse vibrations generated by these waves; parametric resonances; beating due to energy exchange between longitudinal and transverse vibrations; the rod shape evolution as the load exceeds the Euler critical value; the possibility of buckling of the rod rectilinear shape under a load less than the Euler load; and the rod dynamics at the initial stage of motion. The prospects of further investigations related to the complication of the models are considered, in particular, the problem of longitudinal impact by a body on a rod and the transverse vibrations generated by it.
Keywords rod, longitudinal impact, transverse bending, parametric resonances, beating, Euler elastics
1.  L. Euler, Method for Determining Curves with the Maximum or Minimum Property (GTTI, Moscow-Leningrad, 1934) [in Russian].
2.  A. S. Vol'mir, Stability of Elastic Systems (GITTL, Moscow, 1962) [in Russian].
3.  Ya. G. Panovko and I. I. Gubanova, Stability and Vibrations of Elastic Systems (Nauka, Moscow, 1987) [in Russian].
4.  M. A. Lavrentiev and A. Yu. Ishlinskii, "Dynamic Buckling Modes of Elastic Systems," Dokl. Akad. Nauk SSSR 64 (6), 779-782 (1949).
5.  A. S. Vol'mir, "Stability of Compressed Rods under Dynamic Loading," Stroit. Mekh. Rashch. Sooruzh, No. 1, 6-9 (1960).
6.  V. V. Bolotin, Transverse Vibrations and Critical Velocities, Vols. 1 and 2 (Izdat. AN SSSR, Moscow, 1951, 1953) [in Russian].
7.  W. J. Hutchinson and B. Budiansky, "Dynamic Buckling Estimates," AIAA Journal 4 (3), 527-530 (1966).
8.  W. G. Knauss and K. Ravi-Chandar, "Some Basic Problems in Stress Wave Dominated Fracture," Int. J. Fract. 27 (3-4), 127-143 (1985).
9.  N. F. Morozov and Yu. V. Petrov, Dynamics of Fracture (Izdat. SPbGU, St.Petersburg, 1997; Springer, Berlin-Heidelberg-New York, 2000).
10.  N. F. Morozov and P. E. Tovstik, "Dynamics of a Rod on Longitudinal Impact," Vestnik St. Peterzburg. Univ. Ser. I. Mat. Mekh. Astr., No. 2, 105-111 (2009).
11.  A. K. Belyaev, D. N. Il'in, and N. F. Morozov "Dynamic Approach to the Ishlinsky-Lavrent'ev Problem," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 28-33 (2013) [Mech. Solids (Engl. Transl.) 48 (5), 504-508 (2013)].
12.  N. F. Morozov and P. E. Tovstik, "Dynamics of a Rod on Short-Time Longitudinal Impact," Vestnik St. Peterzburg. Univ. Ser. I. Mat. Mekh. Astr., No. 3, 131-141 (2013).
13.  N. F. Morozov and P. E. Tovstik, "The Rod Dynamics under Longitudinal Impact," in Book of Abstracts of International Conference on Nonlinear Dynamics in Engineering: Modeling, Analysis, and Applications, August 21-23, 2013, Aberdeen, UK, Ed. by J. Ing, Y. Liu, E. Pavlovskaya, A. Postnikov, and M. Wiercigroch (Aberdeen, 2013), p. 73.
14.  N. F. Morozov and P. E. Tovstik, "Transverse Rod Vibrations under a Short-Term Longitudinal Impact," Dokl. Ross. Akad. Nauk 452 (1), 37-41 (2013) [Dokl. Phys. (Engl. Transl.) 58 (9), 387-391 (2013)].
15.  A. K. Belyaev, N. F. Morozov, and P. E. Tovstik, "On Static and Dynamic Instability of Thin Rods," in Proc. 7 All-Russia Conf. "Mechanics of Deformable Solid" (Izdat. YuFU, Rostov-on-Don, 2013), pp. 80-84 [in Russian].
16.  N. F. Morozov, P. E. Tovstik, and T. P. Tovstik, "Statics and Dynamics of a Rod under Axial Compression," in ICNAAM 2014, AIP Conference Proc. (2014).
17.  N. F. Morozov, P. E. Tovstik, and T. P. Tovstik, "Statics and Dynamics of a Rod under Longitudinal Loading," Vestnik Yuzhno-Ural Univ. Ser. Mat. Model. Progr. 7 (1), 76-89 (2014).
18.  N. F. Morozov and P. E. Tovstik, "Dynamic Buckling of a Rod under Longitudinal Load Lower Than the Eulerian Load," Dokl. Ross. Akad. Nauk 453 (3), 282-285 (2014) [Dokl. Phys. (Engl. Transl.) 58 (11), 510-513 (2013)].
19.  A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and T. P. Tovstik, "Buckling Problem for a Rod Longitudinally Compressed by a Force Smaller Than the Euler Critical Force," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 28-39 (2016) [Mech. Solids (Engl. Transl.) 51 (3), 263-272 (2016)].
20.  A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and T. P. Tovstik "Beating in the Problem of Longitudinal Impact on a Thin Rod," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 112-125 (2015) [Mech. Solids (Engl. Transl.) 50 (4), 451-462 (2015)].
21.  A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, "Parametric Resonances in the Problem of Longitudinal Impact on a Thin Rod," Vestnik SPbGU, Ser. 1, No. 1, 77-94 (2016).
22.  A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, "Statics and Dynamics of a Rod in Longitudinal Compression," in 7th Polyakhov Readings, Theses (St. Petersburg, 2015), p. 9 [in Russian].
23.  A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, "Ishlinskii-Lavrentiev Problem. Development of the Idea," in Proc. Meeting in Fundamental Problems of Theoretical and Appllied Mechanics, Kazan, August 20-24. 2015 (KFU, Kazan, 2015), pp. 2636-2638 [in Russian].
24.  B. A. Gordienko, "Buckling of Rods under Impact Loading," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 1, 185-188 (1969).
25.  A. S. Vol'mir, Nonlinear Dynamics of Plates and Shells (Nauka, Moscow, 1972) [in Russian].
26.  M. A. Il'gamov, "Dependence of Dynamic Buckling of a Rod on the Initial Conditions," Dokl. Ross. Akad. Nauk 457 (6), 656-659 (2014) [Dokl. Phys. (Engl. Transl.) 59 (8), 385-388 (2014)].
27.  N. F. Morozov, A. K. Belyaev, P. E. Tovstik, and T. P. Tovstik, "The Ishlinskii-Lavrent'ev Problem at the Initial Stage of Motion," Dokl. Ross. Akad. Nauk 463 (5), 543-546 (2015) [Dokl. Phys. (Engl. Transl.) 60 (8), 368-371 (2015)].
28.  N. F. Morozov, A. K. Belyaev, P. E. Tovstik, and T. P. Tovstik, "Initial Stage of Motion in the Lavrent'ev-Ishlinskii Problem on Longitudinal Shock on a Rod," Dokl. Ross. Akad. Nauk 465 (3), 302-306 (2015) [Dokl. Phys. (Engl. Transl.) 60 (11), 519-523 (2015)].
29.  A. K. Belyaev, N. F. Morozov, P. E. Tovstik, and T. P. Tovstik, "The Lavrentiev-Ishlinsky Problem at the Initial Stage of Motion," Int. J. Engng Sci. 98, 92-98 (2016).
30.  N. F. Morozov, P. E. Tovstik, and P. E. Tovstik, "Again on the Ishlinskii-Lavrentyev problem," Dokl. Ross. Akad. Nauk 455 (4), 412-415 (2014) [Dokl. Phys. (Engl. Transl.) 59 (4), 189-192 (2014)].
31.  N. F. Morozov, P. E. Tovstik, and T. P. Tovstik, "Stability of a Rod under Long-Term Axial Compression," Probl. Prochn. Plastichn. 77 (1), 40-48 (2015).
32.  N. F. Morozov, A. K. Belyaev, P. E. Tovstik, and T. P. Tovstik, "Dynamic Behavior of a Thin Elastic Rod under the Long-Term Longitudinal Compression," COMPDYN 2015.
33.  A. K. Belyaev, Ch. Ch. Ma, and A. O. Shurpatov, "Semianalytic, Finite-Element, and Experimental Determination of Contact Force of Axial Collision of a Rod and a Punch," Nauch. Tekhn. Vedomosti SPbGTU. Fiz.-Mat. Nauki (2017) (in Press).
34.  V. A. Palmov, Vibrations of Elastoplastic Bodies (Nauka, Moscow, 1976) [in Russian].
35.  A. M. Lyapunov, General Problem of Stability of Motion (GITTL, Moscow-Leningrad, 1950) [in Russian].
36.  V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients and Their Applications (Nauka, Moscow, 1972) [in Russian].
37.  N. N. Bogolyubov and Yu. A. Mitropolskii, Asymptotic Methods in Theory of Nonlinear Oscillations (Nauka, Moscow, 1969) [in Russian].
38.  B. D. St.-Venant, "Sur le choc longitudinal de deux barres élastiques de grosseur te de matières semblables uo differénts," J. Math. (Liouville) Ser. 2 12, 237-277 (1967).
39.  J. E. Sears, "On the Longitudinal Impact of Metal Rods with Rounded Ends," Proc. Camb. Phil. Soc. 14, 257-286 (1908).
40.  S. A. Zegzhda, Collision of Elastic Bodies (Izd. St. Petersb. Un-ta, St. Petersburg, 1997) [in Russian].
Received 19 February 2017
Link to Fulltext
<< Previous article | Volume 52, Issue 4 / 2017 | Next article >>
Orphus SystemIf you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter

101 Vernadsky Avenue, Bldg 1, Room 246, 119526 Moscow, Russia (+7 495) 434-3538
Founders: Russian Academy of Sciences, Branch of Power Industry, Machine Building, Mechanics and Control Processes of RAS, Ishlinsky Institute for Problems in Mechanics RAS
© Mechanics of Solids
Rambler's Top100