Abstract
Rods are simple structures that are often regarded as one-dimensional continua, but because of finite transverse dimensions, propagating waves are subject to dispersion, which may mask other effects. A one-dimensional continuum theory with one internal, scalar variable can be used to model a solid rod with longitudinal waves. In this paper the material is assumed to be homogeneous, isotropic, and hyperelastic. First, the linear theory is reviewed to exhibit clearly the multiple wave hierarchy that exists in a rod. Next, expansion and scaling techniques are used in the fully nonlinear case to examine the main pulse. Finally, steady nonlinear waves are examined and both solitary waves and periodic waves are found to exist. Results are compared to the simple wave solutions available from the most elementary one-dimensional nonlinear theory for rods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. F. Bell, The Experimental Foundations of Solid Mechanics, Handbuch der Physik, Vol. VIa/1, Springer-Verlag, New York, 1973.
G. E. Hauver, Penetration with instrumented rods, Int. J. Eng. Sci. 16 (1978), 871–877.
R. Skalak, Longitudinal impact of a semi-infinite circular elastic bar, J. Appl. Mech. 24 (1957), 59–64.
W. A. Green, Dispersion relations for elastic waves in bars, in Progress in Solid Mechanics, Vol. I (I. N. Sneddon and R. Hill, eds.), North-Holland, Amsterdam, 1960.
J. H. Shea, Propagation of plastic strain pulses in cylindrical lead bars, J. Appl. Phys. 39 (1968), 4004–4011.
G. P. DeVault, The effect of lateral inertia on the propagation of plastic strain in a cylindrical rod, J. Mech. Phys. Sol. 13 (1965), 55–68.
S. S. Antman, The Theory of Rods, Handbuch der Physik, Vol. VIa/2, Springer-Verlag, New York, 1972.
J. W. Nunziato and E. K. Walsh, On the influence of void compaction and material non-uniformity on the propagation of one-dimensional acceleration waves in granular materials, Arch. Rat. Mech. Anal. 64 (1977), 299–316 and Addendum, Arch. Rat. Mech. Anal. 67 (1977), 395–397.
J. W. Nunziato and E. K. Walsh, One-dimensional shock waves in uniformly distributed granular materials, Int. J. Solids and Structures 14 (1978), 681–689.
D. F. Parker and B. R. Seymour, Finite amplitude one-dimensional pulses in an inhomogeneous granular material, Arch. Rat. Mech. Anal. 72 (1980), 265–284.
M. F. McCarthy, private communication.
S. S. Antman, Qualitative theory of the ordinary differential equations of nonlinear elasticity, in Mechanics Today, Vol. 1 (S. Nemat-Nasser, ed.), Pergamon, New York, 1972.
R. D. Mindlin and G. Herrmann, A one-dimensional theory of compressional waves in an elastic rod, Proceedings of the First U.S. National Congress of Applied Mechanics, 1950, pp. 187–191.
G. B. Whitham, Some comments on wave propagation and shock wave structure with applications to magnetohydrodynamics, Comm. on Pure and Appl. Math. 12 (1959), 113–158.
G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1974.
T. T. Wu, A note on the stability condition for certain wave propagation problems, Comm. on Pure and Appl. Math. 14 (1961), 745–747.
F. Bowman, Introduction to Bessel Functions, Dover, New York, 1958.
M. Abromowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
J. W. Nunziato and E. K. Walsh, Small-amplitude wave behavior in one-dimensional granular solids, J. Appl. Mech. 44 (1977), 559–564.
E. Varley and E. Cumberbatch, Non-linear high frequency sound waves, J. Inst. Maths. Applics. 2 (1966), 133–143.
S. Leibovich and A. R. Seebass, Examples of dissipative and dispersive systems leading to the Burgers and the Korteweg-de Vries equations, in Nonlinear Waves (S. Leibovich and A. R. Seebass, eds.), Cornell University Press, Ithaca, N.Y., 1974.
G. A. Nariboli and A. Sedov, Burger’s-Korteweg-De Vries equation for viscoelastic rods and plates, J. Math. Anal. and Appl. 32 (1970), 661–667.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 Martinus Nijhoff Publishers, The Hague/Boston/London
About this paper
Cite this paper
Wright, T.W. (1981). Nonlinear Waves in Rods. In: Carlson, D.E., Shield, R.T. (eds) Proceedings of the IUTAM Symposium on Finite Elasticity. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7538-5_27
Download citation
DOI: https://doi.org/10.1007/978-94-009-7538-5_27
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-7540-8
Online ISBN: 978-94-009-7538-5
eBook Packages: Springer Book Archive