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Mechanical metamaterials with negative compressibility transitions


When tensioned, ordinary materials expand along the direction of the applied force. Here, we explore network concepts to design metamaterials exhibiting negative compressibility transitions, during which a material undergoes contraction when tensioned (or expansion when pressured). Continuous contraction of a material in the same direction of an applied tension, and in response to this tension, is inherently unstable. The conceptually similar effect we demonstrate can be achieved, however, through destabilizations of (meta)stable equilibria of the constituents. These destabilizations give rise to a stress-induced solid–solid phase transition associated with a twisted hysteresis curve for the stress–strain relationship. The strain-driven counterpart of negative compressibility transitions is a force amplification phenomenon, where an increase in deformation induces a discontinuous increase in response force. We suggest that the proposed materials could be useful for the design of actuators, force amplifiers, micromechanical controls, and protective devices.

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Figure 1: Negative compressibility contrasted with other effects.
Figure 2: Constituents, metamaterial, and resulting hysteresis loops.
Figure 3: Effective energy barrier and the temperature dependence of the transitions.
Figure 4: Physical examples of stress-induced negative compressibility and strain-induced force amplification transitions.


  1. Veselago, V. G. The electrodynamics of substances with simultaneously negative values of ɛ and μ. Sov. Phys. Usp. 10, 509–514 (1968) Original Publication in Russian, 1967.

    Article  Google Scholar 

  2. Smith, D. R., Padilla, W. J., Vier, D. C., Nemat-Nasser, S. C. & Schultz, S. Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 84, 4184–4187 (2000).

    Article  CAS  Google Scholar 

  3. Shelby, R. A., Smith, D. R. & Schultz, S. Experimental verification of a negative index of refraction. Science 292, 77–79 (2001).

    Article  CAS  Google Scholar 

  4. Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000).

    Article  CAS  Google Scholar 

  5. Schurig, D. et al. Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977–980 (2006).

    Article  CAS  Google Scholar 

  6. Zhang, S., Yin, L. & Fang, N. Focusing ultrasound with an acoustic metamaterial network. Phys. Rev. Lett. 102, 194301 (2009).

    Article  Google Scholar 

  7. Baughman, R. H., Stafström, S., Cui, C. & Dantas, S. O. Materials with negative compressibilities in one or more dimensions. Science 279, 1522–1524 (1998).

    Article  CAS  Google Scholar 

  8. Lee, Y., Vogt, T., Hriljac, J. A., Parise, J. B. & Artioli, G. Pressure-induced volume expansion of zeolites in the natrolite family. J. Am. Chem. Soc. 124, 5466–5475 (2002).

    Article  CAS  Google Scholar 

  9. Vakarin, E. V., Duda, Y. & Badiali, J. P. Negative linear compressibility in confined dilatating systems. J. Chem. Phys. 124, 144515 (2006).

    Article  CAS  Google Scholar 

  10. Grima, J. N., Attard, D. & Gatt, R. Truss-type systems exhibiting negative compressibility. Phys. Status Solidi B 245, 2405–2414 (2008).

    Article  CAS  Google Scholar 

  11. Gatt, R. & Grima, J. N. Negative compressibility. Phys. Status Solidi RRL 2, 236–238 (2008).

    Article  CAS  Google Scholar 

  12. Liu, Z. et al. Locally resonant sonic materials. Science 289, 1734–1736 (2000).

    Article  CAS  Google Scholar 

  13. Fang, N. et al. Ultrasonic metamaterials with negative modulus. Nature Mater. 5, 452–456 (2006).

    Article  CAS  Google Scholar 

  14. Sheng, P., Xiao, R-F., Wen, W-J. & Zheng, Y. L. Composite materials with negative elastic constants. US patent 6,576,333 (2003).

  15. Lakes, R. S., Lee, T., Bersie, A. & Wang, Y. C. Extreme damping in composite materials with negative-stiffness inclusions. Nature 410, 565–567 (2001).

    Article  CAS  Google Scholar 

  16. Jaglinski, T., Kochmann, D., Stone, D. & Lakes, R. S. Composite materials with viscoelastic stiffness greater than diamond. Science 315, 620–622 (2007).

    Article  CAS  Google Scholar 

  17. Reichl, L. E. A Modern Course in Statistical Physics (Wiley, 2009).

    Google Scholar 

  18. Lakes, R. S. Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987).

    Article  CAS  Google Scholar 

  19. Baughman, R. H., Shacklette, J. M., Zakhidov, A. A. & Stafström, S. Negative Poisson’s ratio as a common feature of cubic metals. Nature 392, 362–365 (1998).

    Article  CAS  Google Scholar 

  20. Janmey, P. A., McCormick, M. E., Rammensee, S., Leight, J. L., Georges, P. C. & MacKintosh, F. C. Negative normal stress in semiflexible biopolymer gels. Nature Mater. 6, 48–51 (2007).

    Article  CAS  Google Scholar 

  21. Moore, B., Jaglinski, T., Stone, D. S. & Lakes, R. S. Negative incremental bulk modulus in foams. Phil. Mag. Lett. 86, 651–659 (2006).

    Article  CAS  Google Scholar 

  22. Goodwin, A. L., Keen, D. A. & Tucker, M. G. Large negative linear compressibility of Ag3[Co(CN)6]. Proc. Natl Acad. Sci. USA 105, 18708–18713 (2008).

    Article  CAS  Google Scholar 

  23. Fortes, D. A., Suard, E. & Knight, K. S. Negative linear compressibility and massive anisotropic thermal expansion in methanol monohydrate. Science 331, 742–746 (2011).

    Article  CAS  Google Scholar 

  24. Braess, D., Nagurney, A. & Wakolbinger, T. On a paradox of traffic planning. Transp. Sci. 39, 446–450 (2005) Original Publication in German, 1968.

    Article  Google Scholar 

  25. Roughgarden, T. Selfish Routing and the Price of Anarchy (MIT Press, 2005).

    Google Scholar 

  26. Beckmann, M. J., McGuire, C. B. & Winsten, C. B. Studies in the Economics of Transportation (Yale Univ. Press, 1956).

    Google Scholar 

  27. Pigou, A. C. The Economics of Welfare (Macmillan, 1920).

    Google Scholar 

  28. Cohen, J. E. & Horowitz, P. Paradoxical behaviour of mechanical and electrical networks. Nature 352, 699–701 (1991).

    Article  Google Scholar 

  29. Mishima, O. & Stanley, H. E. The relationship between liquid, supercooled and glassy water. Nature 396, 329–335 (1998).

    Article  CAS  Google Scholar 

  30. Otsuka, K. & Wayman, C. M. Shape Memory Materials (Cambridge Univ. Press, 1998).

    Google Scholar 

  31. Abeyaratne, R. & Knowles, J. K. Evolution of Phase Transitions: A Continuum Theory (Cambridge Univ. Press, 2006).

    Book  Google Scholar 

  32. Puglisi, G. & Truskinovsky, L. Mechanics of a discrete chain with bi-stable elements. J. Mech. Phys. Solids 48, 1–27 (2000).

    Article  Google Scholar 

  33. Mallikarachchi, H. M. Y. C. & Pellegrino, S. Quasi-static folding and deployment of ultrathin composite tape-spring hinges. J. Spacecr. Rockets 48, 187–198 (2011).

    Article  Google Scholar 

  34. Andersen, H. C. Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys. 72, 2384–2393 (1980).

    Article  CAS  Google Scholar 

  35. Martyna, G. J., Klein, M. L. & Tuckerman, M. Nosé–Hoover chains: The canonical ensemble via continuous dynamics. J. Chem. Phys. 97, 2635–2643 (1992).

    Article  Google Scholar 

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This study was supported by the Materials Research Science and Engineering Center at Northwestern University through Grant No. DMR-0520513 (Z.G.N.), the National Science Foundation Grants No. DMS-0709212 (Z.G.N. and A.E.M.) and No. DMS-1057128 (A.E.M.), a National Science Foundation Graduate Research Fellowship (Z.G.N.) and a Sloan Research Fellowship (A.E.M.).

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Z.G.N. and A.E.M. designed the research. Z.G.N. performed the numerical and analytical calculations. A.E.M. supervised the research and the analysis of the results. Both authors contributed to the preparation of the manuscript.

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Correspondence to Adilson E. Motter.

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The authors declare no competing financial interests.

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Nicolaou, Z., Motter, A. Mechanical metamaterials with negative compressibility transitions. Nature Mater 11, 608–613 (2012).

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