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Nonlinear wave propagation in micropolar media. I: The general theory. (English) Zbl 0694.73006

Summary: The plane wave propagation in nonlinear micropolar solids is asymptotically investigated. Micropolar theory in the linear approximation predicts a dispersive optical (high-frequency) mode as well as a dispersive acoustical (low-frequency) mode for the harmonic waves in an unbounded medium. The acoustical mode has a weakly dispersive region when the wave number is small. If nonlinearity is also present in this weakly dispersive region and if both effects are small but finite, it may be expected that nonlinearity and dispersive effects can balance each other, and the wave propagation can be asymptotically governed by a nonlinear evolution equation which admits a solitary wave type solution. Using the reductive perturbation method to examine the plane wave propagation in a general nonlinear polar solid, it is found that far- field approximation of wave motion is governed by coupled modified Korteweg-de Vries equations.

MSC:

74B20 Nonlinear elasticity
74J20 Wave scattering in solid mechanics
35Q99 Partial differential equations of mathematical physics and other areas of application
74H45 Vibrations in dynamical problems in solid mechanics
74A35 Polar materials

Citations:

Zbl 0694.73007
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References:

[1] Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley: Wiley New York · Zbl 0373.76001
[2] Eringen, A. C.; Şuhubi, E. S., Elastodynamics. I, Finite Motions (1974), Academic Press: Academic Press New York · Zbl 0138.21202
[3] Jeffrey, A.; Kawahara, T., Asymptotic Methods in Nonlinear Wave Theory (1981), Pitman: Pitman Boston
[4] Jeffrey, A.; Kakutani, T., SIAM Rev., 14, 582 (1972)
[5] Maugin, G. A.; Miled, A., Int. J. Engng Sci, 24, 1477 (1986)
[6] Kafadar, C. B.; Eringen, A. C., Int. J. Engng Sci., 9, 271 (1971)
[7] Taniuti, T.; Wei, C. C., J. Phys. Soc. Jpn, 24, 941 (1968)
[8] Eringen, A. C.; Şuhubi, E. S., Int. J. Engng Sci., 2, 189 (1964)
[9] Şuhubi, E. S.; Eringen, A. C., Int. J. Engng Sci., 2, 389 (1964)
[10] Eringen, A. C., (Liebowitz, H., Fracture II (1968), Academic Press: Academic Press New York)
[11] Parfitt, V. R.; Eringen, A. C., General Technology Corporation Report No. 8-3 (1966)
[12] Gardner, C. S.; Morikawa, G. K., Courant Inst. Math. Sci. Report No. NYO-9082 (1960)
[13] Karney, C. F.F.; Sen, A.; Chu, F. Y.F., (Plasma Physics Laboratory Report PPPL-1452 (1978), Princeton University)
[14] Karney, C. F.F.; Sen, A.; Chu, F. Y.F., Solitons and Condensed Matter Physics, (Bishop, A. K.; Schneider, T. (1978), Springer: Springer Berlin)
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