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Archive of Applied Mechanics

, Volume 89, Issue 4, pp 659–668 | Cite as

Longitudinal wave speed in auxetic plates with elastic constraint in width direction

  • Teik-Cheng LimEmail author
Original
  • 67 Downloads

Abstract

This paper evaluates the longitudinal wave speed through a plate in which its two opposing sides are elastically restrained in the width direction, taking into consideration the material auxeticity and strain as well as changes to the density and cross-sectional area. Apart from the known role of Young’s modulus and density, the present results reveal that the wave speed can be enhanced by increasing the width elastic restraint. In the case of high elastic restraint, the speed of both tensile and compressive waves can be minimized by selecting plate materials with Poisson’s ratio of low magnitude. In the case of low elastic restraint, the speed of tensile and compressive waves can be greatly reduced by selecting plate materials with large positive and large negative Poisson’s ratio, respectively. For the special case of negligible strain, the longitudinal wave speed reduces to the elementary wave speed in prismatic rods and in plates of infinite width when the width elastic restraint stiffness approaches zero and infinity, respectively. The obtained results not only avail more parameters for adjusting the longitudinal waves in plates, but also identify the differing methods of effectively controlling the wave speed between tensile and compressive waves when the strain magnitude is non-negligible.

Keywords

Auxetic materials Elastic restraint Longitudinal waves Plates 

List of symbols

\(A_0\)

Cross-sectional area in unstressed portion of the plate

A

Cross-sectional area in stressed portion of the plate at distance x from origin

\(A+\mathrm{d}A\)

Cross-sectional area in stressed portion of the plate at distance \(x+\mathrm{d}x\) from origin

\(\bar{{A}}\)

Mean cross-sectional area in stressed portion of the plate between x and \(x+\mathrm{d}x\)

b

Side boundary parameter

c

Longitudinal wave speed

E

Young’s modulus of plate material

\(\varepsilon _x\)

Longitudinal strain at distance x from origin

\(\varepsilon _x +\mathrm{d}\varepsilon _x\)

Longitudinal strain at distance \(x+\mathrm{d}x\) from origin

f

\(v(1-bv)/(1-v)\)

g

\(E(1-bv^{2})/(1-v^{2})\)

v

Poisson’s ratio of plate material

\(\rho _0\)

Density in unstressed portion of the plate

\(\rho \)

Mean plate density between x and \(x+\mathrm{d}x\)

\(\sigma _x\)

Longitudinal stress at distance x from origin

\(\sigma _x +\mathrm{d}\sigma _x\)

Longitudinal stress at distance \(x+\mathrm{d}x\) from origin

\(u_x\)

Displacement parallel to X-axis

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Science and TechnologySingapore University of Social SciencesSingaporeSingapore

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