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Longitudinal Pochhammer — Chree Waves in Mild Auxetics and Non-Auxetics

Published online by Cambridge University Press:  02 July 2018

A. V. Ilyashenko  and
S. V. Kuznetsov
A. V. Ilyashenko
Affiliation:
Moscow State University of Civil EngineeringMoscow, Russia
S. V. Kuznetsov*
Affiliation:
Institute for Problems in MechanicsMoscow, Russia
*
*Corresponding author (kuzn-sergey@yandex.ru)
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Abstract

The exact solutions of Pochhammer — Chree equation for propagating harmonic waves in isotropic elastic cylindrical rods, are analyzed. Spectral analysis of the matrix dispersion equation for the longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of the wave polarization due to variation of Poisson’s ratio for mild auxetics (Poisson’s ratio is greater than -0.5) is analyzed and compared with the non-auxetics. It is observed that polarization of the waves for both considered cases (auxetics and non-auxetics) exhibits abnormal behavior in the vicinity of the bulk shear wave speed.

Keywords

AuxeticsDispersionPochhammer — Chree wavePolarization

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Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 3 , June 2019 , pp. 327 - 334
Copyright
© The Society of Theoretical and Applied Mechanics 2018 

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References

REFERENCES

Pochhammer, L., “Ueber die Fortpflanzungsgeschwindigkeiten Kleiner Schwingungen in Einem Unbegrenzten Isotropen Kreiscylinder,” Journal für die Reine und Angewandte Mathematik, 81, pp. 324–336 (1876).Google Scholar
Chree, C., “Longitudinal Vibrations of a Circular Bar,” The Quarterly Journal of Pure and Applied Mathematics, 21, pp. 287–298 (1886).Google Scholar
Chree, C., “The Equations of An Isotropic Elastic Solid in Polar and Cylindrical Coordinates, Their Solutions and Applications,” Transactions of the Cambridge Philosophical Society, 14, pp. 250–309 (1889).Google Scholar
Field, G. S., “Velocity of Sound in Cylindrical Rods,” Canadian Journal of Research, 5, pp. 619–624 (1931).Google Scholar
Field, G. S., “Longitudinal Waves in Cylinders of Liquid, in Hollow Tubes and in Solid Rods,” Canadian Journal of Research, 11, pp. 254–263 (1934).Google Scholar
Field, G. S., “Dispersion of Supersonic Waves in Cylindrical Rods,” Physical Review, 57, 1188 (1940).Google Scholar
Shear, S. K. and Focke, A. B., “The Dispersion of Supersonic Waves in Cylindrical Rods of Polycrystalline Silver, Nickel, and Magnesium,” Physical Review, 57, pp. 532–537 (1940).Google Scholar
Bancroft, D., “The Velocity of Longitudinal Waves in Cylindrical Bars,” Physical Review, 59, pp. 588–593 (1941).Google Scholar
Hudson, G. E., “Dispersion of Elastic Waves in Solid Circular Cylinders,” Physical Review, 63, pp. 46–51 (1943).Google Scholar
Holden, A. H., “Longitudinal Modes of Elastic Waves in Isotropic Cylinders and Slabs,” The Bell System Technical Journal, 30, pp. 956–969 (1951).Google Scholar
Adem, J., “On the Axially-Symmetric Steady Wave Propagation in Elastic Circular Rods,” Quarterly of Applied Mathematics, 12, pp. 261–275 (1954).Google Scholar
Redwood, M. and Lamb, J., “On Propagation of High Frequency Compressional Waves in Isotropic Cylinders,” Proceedings of the Physical Society of London, 70, pp. 136–143 (1957).Google Scholar
Mindlin, R. D. and McNiven, H. D., “Axially Symmetric Waves in Elastic Rods,” Transactions of ASME, Journal of Applied Mechanics, 27, pp. 145–151 (1960).Google Scholar
McNiven, H. D. and Perry, D. C., “Axially Symmetric Waves in Infinite, Elastic Rods,” Journal of the Acoustical Society of America, 34, pp. 433–437 (1962).Google Scholar
Onoe, M., McNiven, H. D. and Mindlin, R. D., “Dispersion of Axially Symmetric Waves in Elastic Rods,” Transactions of ASME, Journal of Applied Mechanics, 29, pp. 729–734 (1962).Google Scholar
Meeker, T. R. and Meitzler, A. H., “Guided Wave Propagation in Elongated Cylinders and Plates,” In, Physical Acoustics, Principles and Methods, Academic Press, N.Y., 1A, pp. 111–167 (1964).Google Scholar
Kolsky, H., “Stress Waves in Solids,” Journal of Sound and Vibration, 1, pp. 88–110 (1964).Google Scholar
Hutchinson, J. R. and Percival, C. M., “Higher Modes of Longitudinal Wave Propagation in Thin Rod,” Journal of the Acoustical Society of America, 44, pp. 1204–1210 (1968).Google Scholar
Zemanek, J., “An Experimental and Theoretical Investigation of Elastic Wave Propagation in a Cylinder,” Journal of the Acoustical Society of America, 51, pp. 265–283 (1972).Google Scholar
Thurston, R. N., “Elastic Waves in Rods and Clad Rods,” Journal of the Acoustical Society of America, 64, pp. 1–37 (1978).Google Scholar
Pao, Y.-H. and Mindlin, R. D., “Dispersion of Flexural Waves in An Elastic, Circular Cylinder,” Transactions of ASME, Journal of Applied Mechanics, 27, pp. 513–520 (1960).Google Scholar
Valsamos, G., Casadei, F. and Solomos, G., “A Numerical Study of Wave Dispersion Curves in Cylindrical Rods with Circular Cross-Section,” Applied and Computational Mechanics, 7, pp. 99–114 (2013).Google Scholar
Kuznetsov, S. V., “Lamb Waves in Anisotropic plates (Review),” Acoustical Physics, 60, pp. 95–103 (2014).Google Scholar
Kuznetsov, S. V., “Love Waves in Nondestructive Diagnostics of Layered Composites. Survey,” Acoustical Physics, 56, pp. 877–892 (2010).Google Scholar
Tyutekin, V. V. and Boiko, A. I., “Helical Normal Waves Near a Cylindrical Cavity in An Elastic Medium,” Acoustical Physics, 56, pp. 141–144 (2010).Google Scholar
Pavić, G., Chevillotte, F. and Heraud, J., “Dynamics of Large-Diameter Water Pipes in Hydroelectric Power Plants,” Journal of Physics, Conference Series, 813, pp. 1–5 (2017).Google Scholar
Sharma, G. S., Skvortsov, A., MacGillivray, I. and Kessissoglou, N., “Acoustic Performance of Gratings of Cylindrical Voids in a Soft Elastic Medium with a Steel Backing,” Journal of the Acoustical Society of America, 141, pp. 4694–4704 (2017).Google Scholar
Altman, W. and De Oliveira, A. M., “Physical Components of Tensors,” CRC Press, 2014. ISBN: 9781482263824.Google Scholar