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Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod. (English) Zbl 0910.73036
The analysis of the title problem is based on a set of highly nonlinear coupled equations depending on the longitudinal coordinate and time. The main progress is made by the study of the following particular cases: the linearized case which provides the existence of travelling waves of arbitrary shape, the finite-amplitude long-wave case, the far-field solution, and the finite-amplitude finite-wavelength case.

74H45Vibrations (dynamical problems in solid mechanics)
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
74B20Nonlinear elasticity
Full Text: DOI
[1] Green, W. A.: Dispersion relations for elastic waves in bars. In: Progress in solid mechanics, Vol. I (Sneddon, I. N., Hill, R., eds.), pp. 128-153. Amsterdam: North-Holland 1960.
[2] Achenbach, J. D.: Wave propagation in elastic solids. Amsterdam: North-Holland 1973. · Zbl 0268.73005
[3] Wright, T. W.: Nonlinear waves in rods. In: Proceedings of the IUTAM Symposium on Finite Elasticity (Carlson, D. E., Shields, R. T., eds.), pp. 423-443. The Hague: Martinus Nijhoff 1982. · Zbl 0536.73020
[4] Wright, T. W.: Nonlinear waves in rods: results for incompressible materials. Stud. Appl. Math.72, 149-160 (1985). · Zbl 0558.73020
[5] Coleman, B. D., Newman, D. C.: On waves in slender elastic rods. Arch. Rat. Mech. Anal.109, 39-61 (1990). · Zbl 0716.73013 · doi:10.1007/BF00377978
[6] Nariboli, G. A.: Nonlinear longitudinal dispersive waves in elastic rods. J. Math. Phys. Sci.4, 64-73 (1970). · Zbl 0222.73032
[7] Soerensen, M. P., Christiansen, P. L., Lomdahl, P. S.: Solitary waves on nonlinear rods I. J. Acoust. Soc. Am.78, 871-879 (1984). · Zbl 0564.73035 · doi:10.1121/1.391312
[8] Soerensen, M. P., Christiansen, P. L., Lomdahl, P. S., Skovgaard, O.: Solitary waves on nonlinear rods II. J. Acoust. Soc. Am.81, 1718-1722 (1987). · doi:10.1121/1.394786
[9] Clarkson, P. A., LeVeque, R. J., Saxton, R.: Solitary-wave interaction in elastic rods. Stud. Appl. Math.75, 95-122 (1986). · Zbl 0606.73028
[10] Cohen, H., Dai, H.-H.: Nonlinear axisymmetric waves in compressible hyperelastic rods: long finite amplitude waves. Acta Mech.100, 223-239 (1993). · Zbl 0788.73025 · doi:10.1007/BF01174791
[11] Cohen, H.: A non-linear theory of elastic directed curves. Int. J. Eng. Sci.4, 511-524 (1966). · doi:10.1016/0020-7225(66)90013-9
[12] Green, A. E., Naghdi, P. M., Wenner, M. L.: On the theory of rods. Part I. Derivations from three-dimensional equations. Proc. R. Soc. London Ser. A337, 451-483 (1974). · Zbl 0325.73053 · doi:10.1098/rspa.1974.0061
[13] Green, A. E., Naghdi, P. M., Wenner, M. L.: On the theory of rods. Part II. Developments by direct approach. Proc. R. Soc. London Ser. A337, 485-507 (1974). · Zbl 0327.73044 · doi:10.1098/rspa.1974.0062
[14] Antman, S. S.: The theory of rods. In: Handbuch der Physik, Bd. VI a/2. Berlin G?ttingen Heidelberg: Springer 1972.
[15] Benjamin, T. B., Bona, J. L., Mahony, J. J.: Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. Soc. London Ser. A227, 47-78 (1972). · Zbl 0229.35013
[16] Ciarlet, P. G.: Mathematical elasticity. Berlin Heidelberg New York: Springer 1987. · Zbl 0612.73060
[17] Truesdell, C., Noll, W.: Non-linear field theories of mechanics. In: Handbuch der Physik, Bd. III/3. Berlin Heidelberg New York: Springer 1965. · Zbl 0779.73004
[18] Ogden, R. W.: Non-linear elastic deformations. New York Chichester Toronto: Ellis Horwood 1984. · Zbl 0541.73044
[19] Jeffrey, A., Kawahara, T.: Asymptotic methods in nonlinear wave theory. London: Pitman 1982. · Zbl 0473.35002
[20] Drazin, P. G., Johnson, R. S.: Solitons: an introduction. Cambridge New York: Cambridge University Press 1989. · Zbl 0661.35001
[21] Olver, P. J.: Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Phil. Soc.85, 143-160 (1979(. · Zbl 0387.35050 · doi:10.1017/S0305004100055572
[22] Peregrine, D. H.: Calculations of the development of an undular bore. J. Fluid Mech.25, 321-330 (1966). · doi:10.1017/S0022112066001678
[23] Eilbeck, J. C., McGuire, G. R.: Numerical study of the regularized long wave equation. Part I. Numerical methods. J. Comput. Phys.19, 43-57 (1975). · Zbl 0325.65054 · doi:10.1016/0021-9991(75)90115-1
[24] Bona, J. L., Pritchard, W. G., Scott, L. R.: Numerical schemes for a model for nonlinear dispersive waves. J. Comput. Phys.60, 167-186 (1985). · Zbl 0578.65120 · doi:10.1016/0021-9991(85)90001-4
[25] Guo, B.-Y., Manoranjah, V. S.: A spectral method for solving the RLW equation. IMA J. Numer. Anal.5, 307-318 (1985). · Zbl 0577.65106 · doi:10.1093/imanum/5.3.307