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


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.


Auxetic materials Elastic restraint Longitudinal waves Plates 

List of symbols


Cross-sectional area in unstressed portion of the plate


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


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


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


Side boundary parameter


Longitudinal wave speed


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






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


Displacement parallel to X-axis



  1. 1.
    Fung, Y.C.: Foundations of Solid Mechanics, 2nd edn. Prentice-Hall, New Jersey (1965)Google Scholar
  2. 2.
    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 2nd edn. Pergamon Press, Oxford (1970)zbMATHGoogle Scholar
  3. 3.
    Wojciechowski, K.W.: Constant thermodynamic tension Monte-Carlo studies of elastic properties of a two-dimensional system of hard cyclic hexamers. Mol. Phys. 61(5), 1247–1258 (1987)CrossRefGoogle Scholar
  4. 4.
    Lakes, R.: Foam structures with negative Poisson’s ratio. Science 235(4792), 1038–1040 (1987)CrossRefGoogle Scholar
  5. 5.
    Evans, K.E.: Auxetic polymers: a new range of materials. Endeavour 15(4), 170–174 (1991)CrossRefGoogle Scholar
  6. 6.
    Alderson, A.: A triumph of lateral thought. Chem. Ind. 10, 384–391 (1999)Google Scholar
  7. 7.
    Alderson, A., Alderson, K.L.: Auxetic materials. J. Aerosp. Eng. 221(4), 565–575 (2007)MathSciNetGoogle Scholar
  8. 8.
    Liu, Y., Hu, H.: A review on auxetic structures and polymeric materials. Sci. Res. Essays 5(10), 1052–1063 (2010)Google Scholar
  9. 9.
    Greaves, G.N., Greer, A.L., Lakes, R.S., Rouxel, T.: Poisson’s ratio and modern materials. Nat. Mater. 10(11), 823–837 (2011)CrossRefGoogle Scholar
  10. 10.
    Jiang, J.W., Kim, S.Y., Park, H.S.: Auxetic nanomaterials: recent progress and future development. Appl. Phys. Rev. 3(4), 041101 (2016)CrossRefGoogle Scholar
  11. 11.
    Saxena, K.K., Das, R., Calius, E.P.: Three decades of auxetics research—materials with negative Poisson’s ratio: a review. Adv. Eng. Mater. 18(11), 1847–1870 (2016)CrossRefGoogle Scholar
  12. 12.
    Park, H.S., Kim, S.Y.: A perspective on auxetic nanomaterials. Nano Converg. 4, 10 (2017)CrossRefGoogle Scholar
  13. 13.
    Lim, T.C.: Analogies across auxetic models based on deformation mechanism. Phys. Status Solidi RRL 11(6), 1600440 (2017)CrossRefGoogle Scholar
  14. 14.
    Lakes, R.S.: Negative-Poisson’s-ratio materials: auxetic solids. Ann. Rev. Mater. Res. 47, 63–81 (2017)CrossRefGoogle Scholar
  15. 15.
    Ren, X., Das, R., Tran, P., Ngo, T.D., Xie, Y.M.: Auxetic metamaterials and structures: a review. Smart Mater. Struct. 27(2), 023001 (2018)CrossRefGoogle Scholar
  16. 16.
    Lim, T.C.: Auxetic Materials and Structures. Springer, Singapore (2015)CrossRefGoogle Scholar
  17. 17.
    Lipsett, A.W., Beltzer, A.I.: Reexamination of dynamic problems of elasticity for negative Poisson’s ratio. J. Acoust. Soc. Am. 84(6), 2179–2186 (1988)CrossRefGoogle Scholar
  18. 18.
    Chen, C.P., Lakes, R.S.: Dynamic wave dispersion and loss properties of conventional and negative Poisson’s ratio polymeric cellular materials. Cell. Polym. 8, 343–369 (1989)Google Scholar
  19. 19.
    Chen, C.P., Lakes, R.S.: Micromechanical analysis of dynamic behavior of conventional and negative Poisson’s ratio foams. J. Eng. Mater. Technol. 118, 285–288 (1996)CrossRefGoogle Scholar
  20. 20.
    Ruzzene, M., Scarpa, F.: Control of wave propagation in sandwich beams with auxetic core. J. Intell. Mater. Syst. Struct. 14(7), 443–453 (2003)CrossRefGoogle Scholar
  21. 21.
    Ruzzene, M., Scarpa, F., Soranna, F.: Wave beaming effects in two-dimensional cellular structures. Smart Mater. Struct. 12(3), 363–372 (2003)CrossRefGoogle Scholar
  22. 22.
    Malischewsky, P.G.: Comparison of approximated solutions for the phase velocity of Rayleigh waves. Nanotechnology 16(6), 995–996 (2005)CrossRefGoogle Scholar
  23. 23.
    Vinh, P.C., Malischewsky, P.G.: An approach for obtaining approximate formulas for the Rayleigh wave velocity. Wave Motion 44(7–8), 549–562 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Malischewsky, P.G., Tuan, T.T.: A special relation between Young’s modulus, Rayleigh-wave velocity, and Poisson’s ratio. J. Acoust. Soc. Am. 126(6), 2851–2853 (2009)CrossRefGoogle Scholar
  25. 25.
    Malischewsky, P.G., Lorato, A., Scarpa, F., Ruzzene, M.: Unusual behaviour of wave propagation in auxetic structures: P-waves on free surface and S-waves in chiral lattices with piezoelectrics. Phys. Status Solidi B 249(7), 1339–1346 (2012)CrossRefGoogle Scholar
  26. 26.
    Trzupek, D., Trarog, D., Zielinski, P.: Surface dynamics and phononic properties of 2D field tunable auxetic crystal. Acta Phys. Polonica A 115(2), 579–582 (2009)CrossRefGoogle Scholar
  27. 27.
    Trzupek, D., Zielinski, P.: Isolated true surface wave in a radiative band on a surface of a stressed auxetic. Phys. Rev. Lett. 103(7), 075504 (2009)CrossRefGoogle Scholar
  28. 28.
    Zielinski, P., Trarog, D., Trzupek, D.: On surface waves in materials with negative Poisson ratio. Acta Phys. Polonica A 115(2), 513–515 (2009)CrossRefGoogle Scholar
  29. 29.
    Scarpa, F., Malischewsky, P.G.: Some new considerations concerning the Rayleigh-wave velocity in auxetic materials. Phys. Status Solidi B 245(3), 578–583 (2008)CrossRefGoogle Scholar
  30. 30.
    Remillat, C., Wilcox, P., Scarpa, F.: Lamb wave propagation in negative Poisson’s ratio composites. Proc. SPIE 6935, 69350C (2008)CrossRefGoogle Scholar
  31. 31.
    Tee, K.F., Spadoni, A., Scarpa, F., Ruzzene, M.: Wave propagation in auxetic tetrachiral honeycombs. J. Vib. Acoust. 132(3), 031007 (2010)CrossRefGoogle Scholar
  32. 32.
    Koenders, M.A.: Wave propagation through elastic granular and granular auxetic materials. Phys. Status Solidi B 246(9), 2083–2088 (2009)CrossRefGoogle Scholar
  33. 33.
    Maruszewski, B., Drzewiecki, A., Starosta, R.: Magnetoelastic surface waves in auxetic structure. IOP Conf. Ser. Mater. Sci. Eng. 10, 012160 (2010)CrossRefGoogle Scholar
  34. 34.
    Kolat, P., Maruszewski, B.T., Tretiakov, K.V., Wojciechowski, K.W.: Solitary waves in auxetic rods. Phys. Status Solidi B 248(1), 148–157 (2011)CrossRefGoogle Scholar
  35. 35.
    Hou, X., Deng, Z., Zhou, J.: Symplectic analysis for the wave propagation properties of conventional and auxetic cellular structures. Int. J. Num. Anal. Model. (Ser. B) 2(4), 298–314 (2011)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Bianchi, M., Scarpa, F.: Vibration transmissibility and damping behaviour for auxetic and conventional foams under linear and nonlinear regimes. Smart Mater. Struct. 22(8), 084010 (2013)CrossRefGoogle Scholar
  37. 37.
    Lim, T.C.: Stress wave transmission and reflection through auxetic solids. Smart Mater. Struct. 22(8), 084002 (2013)CrossRefGoogle Scholar
  38. 38.
    Lim, T.C., Cheang, P., Scarpa, F.: Wave motion in auxetic solids. Phys. Status Solidi B 251(2), 388–396 (2014)CrossRefGoogle Scholar
  39. 39.
    Lim, T.C., Alderson, A., Alderson, K.L.: Experimental studies on the impact properties of auxetic materials. Phys. Status Solidi B 251(2), 307–313 (2014)CrossRefGoogle Scholar
  40. 40.
    Goldstein, R.V., Gorodtsov, V.A., Lisovenko, D.S.: Rayleigh and Love surface waves in isotropic media with negative Poisson’s ratio. Mech. Solids 49(4), 422–434 (2014)CrossRefGoogle Scholar
  41. 41.
    Boldrin, L., Hummel, S., Scarpa, F., Di Maio, D., Lira, C., Ruzzene, M., Remillat, C.D.L., Lim, T.C., Rajasekaran, R., Patsias, S.: Dynamic behaviour of auxetic gradient composite hexagonal honeycombs. Compos. Struct. 149, 114–124 (2016)CrossRefGoogle Scholar
  42. 42.
    Reda, H., Rahali, Y., Ganghoffer, J.F., Lakiss, H.: Wave propagation in 3D viscoelastic auxetic and textile materials by homogenized continuum micropolar models. Compos. Struct. 141, 328–345 (2016)CrossRefGoogle Scholar
  43. 43.
    He, J.H., Huang, H.H.: Tunable acoustic wave propagation through planar auxetic metamaterial. J. Mech. 34(2), 113–122 (2018)CrossRefGoogle Scholar
  44. 44.
    Lim, T.C.: Plane waves of dilatation in auxetic bulk solids. Mater. Sci. Forum 866, 206–210 (2016)CrossRefGoogle Scholar
  45. 45.
    Lim, T.C.: Longitudinal wave velocity in auxetic rods. J. Eng. Mater. Technol. 137(2), 024502 (2015)CrossRefGoogle Scholar
  46. 46.
    Ruzzene, M., Mazzarella, L., Tsopelas, P., Scarpa, F.: Wave propagation in sandwich plates with periodic auxetic core. J. Intell. Mater. Syst. Struct. 13(9), 587–597 (2002)CrossRefGoogle Scholar
  47. 47.
    Kolat, P., Maruszewski, B.T., Wojciechowski, K.W.: Solitary waves in auxetic plates. J. Non-Cryst. Solids 356(37–40), 2001–2009 (2010)CrossRefGoogle Scholar
  48. 48.
    Sobieszczyk, P., Majka, M., Kuźma, D., Lim, T.C., Zieliński, P.: Effect of longitudinal stress on wave propagation in width-constrained elastic plates with arbitrary Poisson’s ratio. Phys. Status Solidi B 252(7), 1615–1619 (2015)CrossRefGoogle Scholar
  49. 49.
    Lim, T.C.: Longitudinal wave motion in width-constrained auxetic plates. Smart Mater. Struct. 25(5), 054008 (2016)CrossRefGoogle Scholar
  50. 50.
    Kolsky, H.: Stress Waves in Solids. Dover Publications, New York (1963)zbMATHGoogle Scholar
  51. 51.
    Graff, K.F.: Wave Motion in Elastic Solids. Clarendon Press, Oxford (1975)zbMATHGoogle Scholar

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

Personalised recommendations