zbMATH — the first resource for mathematics

Neutral stability of compression solitons at bending of nonlinear elastic rod. (Russian, English) Zbl 1177.74213
Prikl. Mat. Mekh. 72, No. 3, 466-476 (2008); translation in J. Appl. Math. Mech. 72, No. 3, 323-330 (2008).
For the description of interaction of longitudinal and bending waves in a rod a system of equations of isotropic elasticity theory is applied with allowance for nonlinear corrections with regard to the interaction. This system of equations describes long longitudinal bending waves of small but finite amplitude. It is shown that there exist captured bending moda propagating together with the compression soliton. It is established that these moda, being the less stable ones, do not grow with time.
74J35 Solitary waves in solid mechanics
74K05 Strings
[1] Ostrovskii, L. A.; Sutin, A. M.: Non-linear elastic waves in rods, Prikl mat mekh 41, No. 3, 531-537 (1977)
[2] Ostrovskii, L. A.; Potapov, A. I.: Introduction to the theory of modulated waves, (2003)
[3] Pego R, Weinstein M. On the strong spectral stability of some Boussinesq solitary waves//Proc. IUTAM/ISIMM Symp. structure and Dynamics of Nonlinear Waves in Fluids. Singapore: World Sci 1995:370 – 82. · Zbl 0872.76042
[4] Bona, J. L.; Sachs, R. L.: Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Communs math phys, No. 1, 15-29 (1988) · Zbl 0654.35018
[5] Fu, Y.; Ogden, R. W.: Nonlinear elasticity: theory and applications, (2001) · Zbl 0962.00003
[6] Beliaev, A.; Il’ichev, A.: Conditional stability of solitary waves propagating in elastic rods, Physica D 90, No. 1 – 2, 107-118 (1996) · Zbl 0899.73282
[7] Dichmann, D. J.; Maddocks, J. H.; Pego, R. L.: Hamiltonian dynamics of ans elastica and the stability of solitary waves, Arch ration mech anal 135, No. 4, 347-396 (1996) · Zbl 0861.73036
[8] Il’ichev, A. T.: Stability of solitons in non-linear composite media, Zh eksper tekhn fiz 118, No. 3, 720-729 (2000)
[9] Il’ichev, A.: Stability of solitary waves in nonlinear composite media, Physica D 150, No. 3 – 4, 264-277 (2001) · Zbl 1013.74044
[10] Il’ichev, A. T.: Instability of solitons in in-extensible rods, Dokl ross akad nauk 397, No. 3, 304-307 (2004)
[11] Il’ichev AT. Theory of stability of an ”Euler loop” in elastic non-extensible rods. Tr Mat Inst im V A Steklova 2005; 251: 154 – 72.
[12] Il’ichev, A.: Instability of solitary waves on Euler’s elastica, Zamp 57, No. 4, 547-566 (2006) · Zbl 1112.74026
[13] Bakholdin, I. B.; Il’ichev, A. T.: Instability of solitary waves in non-linear composite media, Prikl mat mekh 65, No. 6, 1008-1016 (2001) · Zbl 1036.74029
[14] Bakholdin, I.; Il’ichev, A.; Tomashpol’skii, V.: Stability, instability and interaction of solitary pulses in a composite medium, Eur J mech A solids 21, No. 2, 333-346 (2002) · Zbl 1062.74026
[15] Kovrigin, D. A.; Potapov, A. I.: Non-linear resonance interactions of longitudinal and flexural waves in a ring, Dokl akad nauk SSSR 305, No. 4, 803-807 (1989)
[16] Colleman, B. D.; Dill, E. H.; Swigon, D.: On the dynamics of flexure amd stretch in the theory of elastic rods, Arch ration mech anal 129, No. 2, 147-174 (1995) · Zbl 0872.73020
[17] Potapov AI. Non-Linear Deformation Waves in Rods and Plates. Gor’kii:Izd Gor’k Univ im N I Lobachevskogo;1985.
[18] Bland, D.: Nonlinear dynamic elasticity, (1969) · Zbl 0236.73035
[19] Kulikovskii, A. G.; Sveshchnikova, Y. E. I.: Non-linear waves in elastic media, (1998)
[20] Guz’, A. N.: Elastic waves in bodies with initial stresses. Vols. 1, 2, (1986) · Zbl 0631.73084
[21] Love, A.: A treatise on the mathematical theory of elasticity, (1944) · Zbl 0063.03651
[22] Abramson HN, Plass HJ, Ripperger EA. Stress wave propagation in rods and beams. Advances in Applied Mechanics. New York: Acad. Press, 1958, V. 5, 111 – 194. · Zbl 0091.41301
[23] Zarembo, L. K.; Krasil’nikov, V. A.: Non-linear phenomena in the propagation of elastic waves in solid’s, Uspekhi fiz nauk 102, No. 4, 549-586 (1970)
[24] Grillakis, M.; Shatah, J.; Strauss, W.: Stability theory of soliatary waves in the presence of symmetry, J funct anal 74, No. 1, 160-197 (1987) · Zbl 0656.35122
[25] Pego, R.; Winstein, M.: Asympototic stability of solitary waves, Communs math phys 64, No. 2, 305-349 (1994) · Zbl 0805.35117
[26] Landau, L. D.; Lifshitz, E. M.: Quantum mechanics, (1989) · Zbl 0081.22207
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.