Geodesic
Static and Dynamics of a Rod at the Longitudinal Loading
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 7 (2014) no. 1, pp. 76-89 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A short review of works about static and dynamical stability of a thin rod under axial compression is given. By using linear static approach a critical compression has been found by L. Euler. In the paper of M. A. Lavrentiev and A. Y. Ishlinsky it has been established that at intensive loading which essentially exceeds the Eulerian one, the maximum growth of the lateral deflection corresponds to the mode with a large number of waves in the longitudinal direction. The following researches are connected with the longitudinal waves influence. The conditions of parametric resonances appearing and also the cases of stability loss under load less than the Eulerian one are found. Under quasi-linear approach the beating effect with energy transition from longitudinal vibrations into transversal ones and vice versa is established. At a long-time action of the load exceeding the Eulerian one both linear and quasi-linear approaches do not lead to finite values of transversal amplitude. That is why the non-linear approach is used and the growth of the post-critical deformations of the rod is studied. The connection of the deformation picture with the effect discovered by M. A. Lavrentiev and A. Y. Ishlinsky with the Eulerian elastics is marked.
Keywords: stability of rod; parametric resonance; beatings; Eulerian elastics. 
@article{VYURU_2014_7_1_a6,
     author = {N. F. Morozov and P. E. Tovstik and T. P. Tovstik},
     title = {Static and {Dynamics} of a {Rod} at the {Longitudinal} {Loading}},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {76--89},
     year = {2014},
     volume = {7},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2014_7_1_a6/}
}
TY  - JOUR
AU  - N. F. Morozov
AU  - P. E. Tovstik
AU  - T. P. Tovstik
TI  - Static and Dynamics of a Rod at the Longitudinal Loading
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2014
SP  - 76
EP  - 89
VL  - 7
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VYURU_2014_7_1_a6/
LA  - ru
ID  - VYURU_2014_7_1_a6
ER  - 
%0 Journal Article
%A N. F. Morozov
%A P. E. Tovstik
%A T. P. Tovstik
%T Static and Dynamics of a Rod at the Longitudinal Loading
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2014
%P 76-89
%V 7
%N 1
%U http://geodesic.mathdoc.fr/item/VYURU_2014_7_1_a6/
%G ru
%F VYURU_2014_7_1_a6
N. F. Morozov; P. E. Tovstik; T. P. Tovstik. Static and Dynamics of a Rod at the Longitudinal Loading. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 7 (2014) no. 1, pp. 76-89. http://geodesic.mathdoc.fr/item/VYURU_2014_7_1_a6/

[1] Euler L., Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes Sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti, Opera omnia: Opera mathematica, Springer, 1952

[2] W. J. Hutchinson, B. Budiansky, “Dynamic Buckling Estimates”, AIAA J., 4:3 (1966), 527–530 | MR

[3] W. G. Knauss, K. Ravi-Chandar, “Some Basic Problems in Stress Wave Dominated Fracture”, Intern. J. Fracture, 27:3–4 (1985), 127–143 | DOI

[4] Morozov N. F., Petrov Ju. V., Problems of destruction dynamics, St. Petersburg Univ. Press, St. Petersburg, 1997

[5] V. A. Bratov, N. F. Morozov, Yu. V. Petrov, Dynamic Strength of Continuum, St. Petersburg Univ. Press, SPb, 2009

[6] Bolotin V. V., Dynamical Stability of Elastic Systems, Nauka, M., 1956

[7] Vol'mir A. S., “Stability of Compressed Rod at Dynamical Loading”, Stroitelnaya mekhanica i raschet sooruzhenii, 1960, no. 1, 6–9

[8] Lavrent'ev M. A., Ishlinsky A. Ju., “Dynamical Modes of Stability Loss of Elastic Systems”, Doklady Physics, 64:6 (1949), 776–782 | MR

[9] Panovko Ya. G., Gubanova I. I., Stability and Vibrations of Elastic Systems, Nauka, M., 1987 | MR | Zbl

[10] Vol'mir A. S., Stability of Elastic Systems, GITTL, M., 1962 ; т. 2, 1953 | MR

[11] Bolotin V. V., Transversal Vibrations and Critical Velocities, v. 1, Izd. AN USSR, 1951; v. 2, 1953

[12] Morozov N. F., Tovstik P. E., “Dynamics of Rod at a Longitidinal Impact”, Vestnik St. Petersburg Univ. Ser. 1, 2009, no. 2, 105–111

[13] Belyaev A. K., Il'in D. N., Morozov N. F., “Dytamical Approach to the Ishlinsky–Lavrent'ev Problem”, Mech. of Solids, 48:5 (2013), 504–508 | DOI | MR | MR

[14] Morozov N. F., Tovstik P. E., “The Rod Dynamics under a Short-Term Longitudinal Impact”, Vestnik St. Petersburg Univ. Ser. 1, 2013, no. 3, 131–141

[15] Morozov N. F., Tovstik P. E., “Transverse Rod Vibrations under a Short-Term Longitudinal Impact”, Doklady Physics, 58:9 (2013), 387–391 | DOI | DOI | MR

[16] Palmov V. A., Vibrations of Elasto-Plastic Bodies, Nauka, M., 1976

[17] Lyapunov A. M., General Problem of a Motion Stability, Gostekhizdat, M.–L., 1950 | Zbl

[18] Jakubovich V. A., Sterzhinsky V. M., Linear Differential Equations with Periodic Coefficients and Their Applications, Nauka, M., 1972 | MR

[19] Ilukhin A. A., Space Problems of the Linear Theory of Elastic Rods, Naukova Dumka, Kiev, 1979 | MR