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  4. To modeling the auxetic materials: some fundamental aspects

To modeling the auxetic materials: some fundamental aspects

MMC.
2014;
Volume 1, Number 1
: pp. 97-108
https://doi.org/10.23939/mmc2014.01.097
Received: May 31, 2014

Math. Model. Comput. Vol. 1, No. 1, pp. 97-108 (2014)

Authors:
  1. J. Rushchitsky
1
S. P. Timoshenko Institute of Mechanics

The auxetic materials are considered from the point of view of correspondence to the classical theory of elasticity. It is shown that some classical postulates relative to the elastic constants should be refined. Three cases of description of auxetic materials — by the model of linear elastic isotropic body, by the model of linear elastic transversally isotropic body, by the nonlinear elastic isotropic body (Murnaghan potential) — are analyzed shortly. The initial assumption on positivity of internal energy of deformation is saved and then the uniform stress states (unilateral tension, omnilateral    compression, pure shear) are used to analyze the elastic constants. This allows to describe the new mechanical effects: expansion of the standard sample-rod-prism under unilateral tension and expansion of the standard sample-cube under hydrostatic compression as well as an existence of the arbitrary negative values of Poisson ratios, what is accompanied by the negative values of the Lame $\lambda$, Young $E$  and compression $k$ moduli, for the linear isotropic case and some elastic constants in the linear transversely isotropic case. The case of nonlinear description shows that the auxetic materials should be defined by the primary physical effect — observation in the standard for mechanics of materials experiment of longitudinal tension of a prism that the transverse deformation of prism is positive (a material as if swells) in contrast to the classical materials, where it is negative.

auxetic material
isotropic material
anisotropic material
negative values of elastic moduli
restriction on elastic moduli
  1. Cattani C., Rushchitsky J. J. Wavelet and wave analysis as applied to materials with micro or nanostructures. Singapore-London: World Scientific Publishing Co. Pte. Ltd. (2007).
  2. Lur’e A. I. Theory of Elasticity. Springer, Berlin (1999).
  3. Ogden R. W.The Nonlinear Elastic Deformations. Dover, New York (1997).
  4. Holzapfel G. A. Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Wiley, Chichester (2006).
  5. Rushchitsky J. J. Nonlinear Elastic Waves in Materials. Springer, Heidelberg (2014).
  6. Lekhnitsky S. G. Theory of elasticity of anisotropic elastic body. Holden Day, San Francisco (1963).
  7. Dielesaint E., Royer D. Ondes elastiques dans les solides. Application au traitement du signal. Masson et Cie, Paris (1956) (in French).
  8. Guz I. A., Rodger A. A., Guz A. N., Rushchitsky J. J. Predicting the properties of micro and nanocomposites: from the micro whiskers to bristled nanocentipedes. Phil. Trans. of the Royal Soc. A: Мathematical, Physical and Engineering Sciences. 365, N1860, 3233–3239 (2008).
  9. Guz A. N., Rushchitsky J. J., Guz I. A. Introduction to Mechanics of Nanocomposites. Akademperiodika, Kiev (2010) (in Russian).
  10. Guz .A., Rushchitsky J. J., Guz A. N. Mechanical Models in Nanomaterials. Chapter 24, Vol.1. Handbook of Nanophysics. Principles and Methods. In 7 vols. Ed. K. D. Sattler. Taylor and Francis Publisher (CRC Press) Boca Raton, 24, 1–24, 18 (2010).
  11. Truesdell C. A first course in rational continuum mechanics. The John Hopkins University, Baltimore (1972).
  12. Leibensohn L. S. Short Course of the Theory of Elasticity. Gostechizdat, Moscow-Leningrad (1942) (in Russian).
  13. Love A. E. H. The Mathematical Theory of Elasticity. 4th edition. Dover Publications, New York (1944).
  14. Timoshenko S. P., Goodier J. N. Theory of Elasticity. 3rd ed., McGraw Hill, New York (1970).
  15. Nowacki W. Teoria sprezystosci. PWN, Warszawa (1970) (in Polish).
  16. Landau L. D., Lifshits E. M. Theoretical Physics. In 10 vols. Vol.VII. Theory of Elasticity. Moscow, Nauka, 1987.
  17. Germain P. Cours de mechanique des milieux continus. Tome 1. Theorie generale. Masson et Cie, Editeurs, Paris (1973).
  18. Hahn H. G. Elastizitatstheorie. Grundlagen der linearen Theorie und Anwendungen auf eindimensionale, ebene und raumliche Probleme. B. G. Teubner, Stuttgart (1985).
  19. Pravoto Y. Seeing auxetic materials from the mechanics point of view: A structural review on the negative Poisson’s ratio. Computational Materials Science 58, 140–153 (2012).
  20. Carneiro V. H., Meireles J., Puga H. Auxetic Materials – A Review. Materials Science – Poland 31 (4), 561–571 (2013).
  21. Gibson L. J., Ashby M. F., Schayer G. S., Robertson C. I. The Mechanics of Two-Dimensional Cellular Materials. Proc. Roy. Soc. Lond. A 382, 25–42 (1982).
  22. Gibson L. J., Ashby M. F. The Mechanics of Three-Dimensional Cellular Materials. Proc. Roy. Soc. Lond. A 382, 43–59 (1982).
  23. Lakes R. S. Foam Structures with a Negative Poisson’s Ratio. Science. 235, 1038–1040 (1987).
  24. Wojciechowski K. W. Constant thermodynamic tension Monte-Carlo studies of elastic properties of a two-dimensional system of hard cyclic hexameters. Molecular Physics. 61, 1247–1258 (1987).
  25. Evans K. E. Auxetic polymers: a new range of materials. Endeavour 15, 170–174 (1991).
  26. Flugge’s Encyclopedia of Physics. Vol. VIa/I. Mechanics of Solids. Springer-Verlag, Berlin (1973).
  27. Yao Y. T., Uzun M., Patel I. Working of auxetic nano-materials. J. of Achievement in Materials and Manufacturing Engineering. 49, N2, 585–593 (2011).
  28. Rushchitsky J. J. Certain class of nonlinear hyperelastic waves: classical and novel models, wave equations, wave effects. Int. J. of Applied Mathematics and Mechanics. 9, N6, 1–48 (2013).