Architectured Materials with Inclusions Having Negative Poisson’s Ratio or Negative Stiffness

  • E. Pasternak
  • A. V. DyskinEmail author
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 282)


Architectured materials with negative Poisson’s ratio (auxetic materials) have been subject of interest for quite some time. The effect of negative Poisson’s ratio is achieved macroscopically through various types of microstructure made of conventional materials. There also exist (unstable) microstructures that, under certain boundary conditions, exhibit negative stiffness. In this chapter we review the microstructures that which generate macroscopic negative Poisson’s ratio and negative stiffness, determine their effective moduli and discuss general properties of the materials with such microstructures. We then consider hybrid materials consisting of conventional (positive Poisson’s ratio and positive moduli) matrix and randomly positioned inclusions having either negative Poisson’s ratio or a negative stiffness (one of the moduli being negative). We use the differential scheme of the self-consisting method to derive the effective moduli of such hybrids keeping in the framework of linear time-independent theory. We demonstrate that the inclusions of both types can, depending on their properties, either increase or decrease the effective moduli.


  1. 1.
    K.L. Alderson, K.E. Evans, The fabrication of microporous polyethylene having a negative Poisson’s ratio. Polymer 33(20), 4435–4438 (1992)CrossRefGoogle Scholar
  2. 2.
    K.L. Alderson, K.E. Evans, Modelling concurrent deformation mechanism in auxetic microporous polymers. J. Mater. Sci. 32, 2797–2809 (1997)CrossRefGoogle Scholar
  3. 3.
    K.L. Alderson, K.E. Evans, Auxetic materials: the positive side of being negative. Eng. Sci. Educ. J. 9(4), 148–154 (2000)CrossRefGoogle Scholar
  4. 4.
    A. Alderson, K.E. Evans, Rotation and dilation deformation mechanism for auxetic behaviour in the α-cristolobite tetrahedral framework structure. Phys. Chem. Miner. 28(10), 711–718 (2001)CrossRefGoogle Scholar
  5. 5.
    A. Alderson, K.E. Evans, Molecular origin of auxetic behavior in tetrahedral framework silicates. Phys. Rev. Lett. 89, 225503 (2002)CrossRefGoogle Scholar
  6. 6.
    A. Alderson, K.L. Alderson, Auxetic materials. Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 221(4), 565–575 (2007)CrossRefGoogle Scholar
  7. 7.
    K.L. Alderson, V.R. Simkins, V.L. Coenen, P.J. Davies, A. Alderson, K.E. Evans, How to make auxetic fibre reinforced composites. Phys. Status Solidi B 242(3), 509–518 (2005)CrossRefGoogle Scholar
  8. 8.
    K.L. Alderson, V.L. Coenen, The low velocity impact response of auxetic carbon fibre laminates. Phys. Status Solidi B 245(3), 489–496 (2008)CrossRefGoogle Scholar
  9. 9.
    K.L. Alderson, A.P. Pickles, P.J. Neale, K.E. Evans, Auxetic polyethylene: the effect of a negative Poisson’s ratio on hardness. Acta Metall. Mater 42(7), 2261–2266 (1994)CrossRefGoogle Scholar
  10. 10.
    K.L. Alderson, A. Fitzgerald, K.E. Evans, The strain dependent indentation resilience of auxetic microporous polyethylene. J. Mater. Sci. 35, 4039–4047 (2000)CrossRefGoogle Scholar
  11. 11.
    K.L. Alderson, R.S. Webber, A.P. Kettle, K.E. Evans, Novel fabrication route for auxetic polyethylene. Part 1. Process. Microstruct. Polym. Eng. Sci. 45(4), 568–578 (2005)CrossRefGoogle Scholar
  12. 12.
    K. Alderson, A. Alderson, N. Ravirala, V. Simkins, P. Davies, Manufacture and characterisation of thin flat and curved auxetic foam sheets. Phys. Status Solidi (B) 249(7), 1315–1321 (2012). Scholar
  13. 13.
    R.F. Almgren, An isotropic three-dimensional structure with Poisson’s ratio = −1. J. Elast. 15, 427–430 (1985)CrossRefGoogle Scholar
  14. 14.
    O. Andersen, U. Waag, L. Schneider, G. Stephani, B. Kieback, Novel metallic hollow sphere structures. Adv. Eng. Mater. 2(4), 192–195 (2000)CrossRefGoogle Scholar
  15. 15.
    M.F. Ashby, Y.J.M. Bréchet, Designing hybrid materials. Acta Mater. 51, 5801–5821 (2003)CrossRefGoogle Scholar
  16. 16.
    M. Assidi, J.-F. Ganghoffer, Composites with auxetic inclusions showing both an auxetic behaviour and enhancement of their mechanical properties. Compos. Struct. 94, 2373–2382 (2012)CrossRefGoogle Scholar
  17. 17.
    D. Attard, J.N. Grima, Modelling of hexagonal honeycombs exhibiting zero Poisson’s ratio. Phys. Status Solidi B 248(1), 52–59 (2011)CrossRefGoogle Scholar
  18. 18.
    D. Attard, E. Manicaro, J.N. Grima, On rotating rigid parallelograms and their potential for exhibiting auxetic behaviour. Phys. Status Solidi B 246(9), 2033–2044 (2009)CrossRefGoogle Scholar
  19. 19.
    D. Attard, E. Manicaro, R. Gatt, J.N. Grima, On the properties of auxetic rotating stretching squares. Phys. Status Solidi B 246(9), 2045–2054 (2009)CrossRefGoogle Scholar
  20. 20.
    K.M. Azzopardi, G.-P. Brincat, J.N. Grima, R. Gatt, Advances in the study of deformation mechanism of stishovite. Phys. Status Solidi B 252, 1486–1491 (2015)CrossRefGoogle Scholar
  21. 21.
    S. Babaee, J. Shim, J.C. Weaver, E.R. Chen, N. Patel, K. Bertoldi, 3D soft metamaterials with negative Poisson’s ratio. Adv. Mater. 25, 5044–5049 (2013)CrossRefGoogle Scholar
  22. 22.
    E. Bafekrpour, A.V. Dyskin, E. Pasternak, A. Molotnikov, Y. Estrin, Internally architectured materials with directionally asymmetric friction. Sci. Rep. 5, 10732 (2015)CrossRefGoogle Scholar
  23. 23.
    G.D. Barrera, J.A.O. Bruno, T.H.K. Barron, N.L. Allan, Negative thermal expansion: a topical review. J. Phys.: Condens. Matter 17, R217–R252 (2005)Google Scholar
  24. 24.
    R. Bathurst, L. Rothenburg, Note on a random isotropic granular material with negative Poisson’s ratio. Int. J. Eng. Sci. 26, 373–383 (1988)CrossRefGoogle Scholar
  25. 25.
    R.H. Baughman, D.S. Galvao, Crystalline networks with unusual predicted mechanical and thermal properties. Nature 365, 735–737 (1993)CrossRefGoogle Scholar
  26. 26.
    R.H. Baughman, J.M. Shacklette, A.A. Zakhidov, S. Stafström, Negative Poisson’s ratios as a common feature of cubic metals. Nature 392, 362–365 (1998)CrossRefGoogle Scholar
  27. 27.
    R.H. Baughman, S. Stafström, C. Cui, S.O. Dantas, Materials with negative compressibilities in one or more dimensions. Science 279, 1522–1524 (1998)CrossRefGoogle Scholar
  28. 28.
    R.H. Baughman, S.O. Socrates, S. Stafström, A.A. Zakhidov, T.B. Mitchell, D.H.E. Dubin, Negative Poisson’s ratios for extreme states of matter. Science 288, 2018–2022 (2000)CrossRefGoogle Scholar
  29. 29.
    Z.P. Bažant, L. Cedolin, Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories (Oxford University Press, Oxford, 1991)Google Scholar
  30. 30.
    K. Bertoldi, P.M. Reis, S. Willshaw, T. Mullin, Negative Poisson’s ratio behavior induced by an elastic instability. Adv. Mater. 22, 361–366 (2010)CrossRefGoogle Scholar
  31. 31.
    A. Bezazi, F. Scarpa, Tensile fatigue of conventional and negative Poisson’s ratio open cell PU foams. Int. J. Fatigue 31, 488–494 (2009)CrossRefGoogle Scholar
  32. 32.
    M. Bianchi, F. Scarpa, M. Banse, C.W. Smith, Novel generation of auxetic open cell foams for curved and arbitrary shapes. Acta Mater. 59, 686–691 (2011)CrossRefGoogle Scholar
  33. 33.
    R. Blumenfeld, Auxetic strains—insight from iso-auxetic materials. Mol. Simul. 31(13), 867–871 (2005)CrossRefGoogle Scholar
  34. 34.
    R. Blumenfeld, S.F. Edwards, Theory of strains in auxetic materials. J. Supercond. Novel Magn. 25(3), 565–571 (2012)CrossRefGoogle Scholar
  35. 35.
    A.C. Brańka, K.W. Wojciechowski, Auxeticity of cubic materials: the role of repulsive core interaction. J. Non-Cryst. Solids 354, 4143–4145 (2008)CrossRefGoogle Scholar
  36. 36.
    A.C. Brańka, D.M. Heyes, K.W. Wojciechowski, Auxeticity of cubic materials under pressure. Phys. Status Solidi B 248, 96–104 (2011)CrossRefGoogle Scholar
  37. 37.
    W.M. Bruner, Comment on ‘Seismic velocities in dry and saturated cracked solids’ by Richard J. O’Connell and Bernard Budiansky. J. Geophys. Res. 81(14), 2573–2576 (1976)CrossRefGoogle Scholar
  38. 38.
    B. Budiansky, On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13(4), 223–227 (1965)CrossRefGoogle Scholar
  39. 39.
    B. Budiansky, R.J. O’Connell, Elastic moduli of a cracked solid. Int. J. Solid Struct. 12(2), 81–97 (1976)CrossRefGoogle Scholar
  40. 40.
    L. Cabras, M. Brun, Auxetic two-dimensional lattices with Poisson’s ratio arbitrarily close to −1. Proc. R. Soc. London. Sect. B 470, 20140538 (2014)CrossRefGoogle Scholar
  41. 41.
    L. Cabras, M. Brun, Effective properties of a new auxetic triangular lattice: an analytical approach. Frattura ed Integrità Strutturale 29, 9–18 (2014)CrossRefGoogle Scholar
  42. 42.
    B.D. Caddock, K.E. Evans, Microporous materials with negative Poisson’s ratios. I. Microstructure and mechanical properties. J. Phys. D Appl. Phys. 22, 1877–1882 (1989)CrossRefGoogle Scholar
  43. 43.
    F. Cardin, M. Favretti, Dynamics of a chain of springs with non-convex potential energy. Math. Mech. Solids 8, 651–668 (2003)CrossRefGoogle Scholar
  44. 44.
    A. Carrella, T.P. Brennan, T.P. Waters, Demonstrator to show the effects of negative stiffness on the natural frequency of a simple oscillator. Proc. IMechE Part C J. Mech. Eng. Sci. 222, 1189–1192 (2008)CrossRefGoogle Scholar
  45. 45.
    A.R. Champneys, G.W. Hunt, J.M.T. Thompson, Localisation and Solitary Waves in Solid Mechanics (World Scientific, Singapore, 1984), pp. 1–28Google Scholar
  46. 46.
    N. Chan, K.E. Evans, Fabrication methods for auxetic foams. J. Mater. Sci. 32, 5945–5953 (1997)CrossRefGoogle Scholar
  47. 47.
    L. Chen, C. Liu, J. Wang, W. Zhang, C. Hu, S. Fan, Auxetic materials with large negative Poisson’s ratios based on highly oriented carbon nanotube structures. Appl. Phys. Lett. 94, 253111 (2009)CrossRefGoogle Scholar
  48. 48.
    Y.J. Chen, F. Scarpa, I.R. Farrow, Y.J. Liu, J.S. Leng, Composite flexible skin with large negative Poisson’s ratio range: numerical and experimental analysis. Smart Mater. Struct. 22, 045005 (2013)CrossRefGoogle Scholar
  49. 49.
    J.B. Choi, R.S. Lakes, Nonlinear properties of metallic cellular materials with a negative Poisson’s ratio. J. Mater. Sci. 27(19), 5375–5381 (1992)CrossRefGoogle Scholar
  50. 50.
    J.B. Choi, R.S. Lakes, Nonlinear properties of polymer cellular materials with a negative Poisson’s ratio. J. Mater. Sci. 27(19), 4678–4684 (1992)CrossRefGoogle Scholar
  51. 51.
    J.B. Choi, R.S. Lakes, Fracture toughness of re-entrant foam materials with a negative Poisson’s ratio: experiment and analysis. Int. J. Fract. 80, 73–83 (1996)CrossRefGoogle Scholar
  52. 52.
    A. Choi, T. Sim, J.H. Mun, Quasi-stiffness of the knee joint in flexion and extension during the golf swing. J. Sports Sci. 33(16), 1682–1691 (2015)CrossRefGoogle Scholar
  53. 53.
    V.L. Coenen, K.L. Alderson, Mechanisms of failure in the static indentation resistance of auxetic carbon fibre laminates. Phys. Status Solidi B 248(1), 66–72 (2011)CrossRefGoogle Scholar
  54. 54.
    N.G.W. Cook, The failure of rock. Int. J. Rock Mech. Min. Sci. 2, 389–403 (1965)CrossRefGoogle Scholar
  55. 55.
    D.M. Correa, T.D. Klatt, S. Cortes, M. Haberman, D. Kovar, C. Seepersad, Negative stiffness honeycombs for recoverable shock isolation. Rapid Prototyping J. 21(2), 193–200 (2015)CrossRefGoogle Scholar
  56. 56.
    D.M. Correa, C. Seepersad, M. Haberman, Mechanical design of negative stiffness honeycomb materials. Integrating Mater. Manuf. Innov. 4(1), 1–11 (2015)CrossRefGoogle Scholar
  57. 57.
    S. Czarnecki, P. Wawruch, The emergence of auxetic material as a result of optimal isotropic design. Phys. Status Solidi B 252, 1620–1630 (2015)CrossRefGoogle Scholar
  58. 58.
    L. Dong, R.S. Lakes, Advanced damper with negative structural stiffness elements. Smart Mater. Struct. 21(7), 075026 (2012)CrossRefGoogle Scholar
  59. 59.
    L. Dong, R.S. Lakes, Advanced damper with high stiffness and high hysteresis damping based on negative structural stiffness. Int. J. Solids Struct. 50(14–15), 2416–2423 (2013)CrossRefGoogle Scholar
  60. 60.
    L. Dong, D.S. Stone, R.S. Lakes, Anelastic anomalies and negative Poisson’s ratio in tetragonal BaTiO3 ceramics. Appl. Phys. Lett. 96, 141904 (2010)CrossRefGoogle Scholar
  61. 61.
    J.P. Donoghue, K.L. Alderson, K.E. Evans, The fracture toughness of composite laminates with a negative Poisson’s ratio. Phys. Status Solidi B 246, 2011–2017 (2009)CrossRefGoogle Scholar
  62. 62.
    W.J. Drugan, Elastic composite materials having a negative stiffness phase can be stable. Phys. Rev. Lett. 98(5), 055502 (2007)CrossRefGoogle Scholar
  63. 63.
    A.V. Dyskin, E. Pasternak, Effective anti-plane shear modulus of a material with negative stiffness inclusions, in 9th HSTAM10, Limassol, Cyprus 12–14 July, 2010, Vardoulakis mini-symposia—Wave Propagation, paper 116, ed. by P. Papanastasiou, E. Papamichos, A. Zervos, M. Stavropoulou (2010), pp. 129–136Google Scholar
  64. 64.
    A.V. Dyskin, E. Pasternak, Friction and localisation associated with non-spherical particles, in Advances in Bifurcation and Degradation in Geomaterials. Proceedings of the 9th International Workshop on Bifurcation and Degradation in Geomaterials, ed. by S. Bonelli, C. Dascalu, F. Nicot (Springer, 2011), pp. 53–58. ISBN/ISSN 978-94-007-1420-5Google Scholar
  65. 65.
    A.V. Dyskin, E. Pasternak, Elastic composite with negative stiffness inclusions in antiplane strain. Int. J. Eng. Sci. 58, 45–56 (2012)CrossRefGoogle Scholar
  66. 66.
    A.V. Dyskin, E. Pasternak, Mechanical effect of rotating non-spherical particles on failure in compression. Phil. Mag. 92(28–30), 3451–3473 (2012)CrossRefGoogle Scholar
  67. 67.
    A.V. Dyskin, E. Pasternak, Rock mass instability caused by incipient block rotation, in Harmonising Rock Engineering and the Environment, Proceedings of 12th International Congress on Rock Mechanics, ed. by Q. Qian, Y. Zhou (CRC Press, Balkema, 2012c), pp. 201–204Google Scholar
  68. 68.
    A.V. Dyskin, E. Pasternak, Rock and rock mass instability caused by rotation of non-spherical grains or blocks, in Rock Engineering & Technology for Sustainable Underground Construction. Proceedings of Eurock 2012, paper 102P (2012d)Google Scholar
  69. 69.
    A.V. Dyskin, E. Pasternak, Bifurcation in rolling of non-spherical grains and fluctuations in macroscopic friction. Acta Mech. 225(8), 2217–2226 (2014)CrossRefGoogle Scholar
  70. 70.
    A.V. Dyskin, E. Pasternak, Negative stiffness: Is thermodynamics defeated? in Proceedings of 8th Australasian Congress on Applied Mechanics: ACAM 8 (Melbourne, Australia, 2014)Google Scholar
  71. 71.
    B. Ellul, M. Muscat, J.N. Grima, On the effect of the Poisson’s ratio (positive and negative) on the stability of pressure vessel heads. Phys. Status Solidi (B) 246(9), 2025–2032 (2009). Scholar
  72. 72.
    M. Esin, E. Pasternak, A. Dyskin, Stability of chains of oscillators with negative stiffness normal, shear and rotational springs. Int. J. Eng. Sci. 108, 16–33 (2016)CrossRefGoogle Scholar
  73. 73.
    M. Esin, E. Pasternak, A.V. Dyskin, Stability of 2D discrete mass-spring systems with negative stiffness springs. Phys. Status Solidi (B) Basic Res. 253(7), 1395–1409 (2016b)Google Scholar
  74. 74.
    U.E. Essien, A.O. Akankpo, M.U. Igboekwe, Poisson’s ratio of surface soils and shallow sediments determined from seismic compressional and shear wave velocities. Int. J. Geosci. 5(12), 1540–1546 (2014)CrossRefGoogle Scholar
  75. 75.
    Y. Estrin, A.V. Dyskin, E. Pasternak, S. Schaare, S. Stanchits, A.J. Kanel-Belov, Negative stiffness of a layer with topologically interlocked elements. Scripta Mater. 50(2), 291–294 (2003)CrossRefGoogle Scholar
  76. 76.
    J.S.O. Evans, T.A. Mary, A.W. Sleight, Negative thermal expansion materials. Phys. B 241–243, 311–316 (1998)Google Scholar
  77. 77.
    K.E. Evans, K.L. Alderson, The static and dynamic moduli of auxetic microporous polyethylene. J. Mater. Sci. 11, 1721–1724 (1992)Google Scholar
  78. 78.
    K.E. Evans, A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking! Adv. Mater. 12(9), 617–628 (2000)CrossRefGoogle Scholar
  79. 79.
    K.E. Evans, B.D. Caddock, Microporous materials with negative Poisson’s ratios: II. Mechanisms and interpretation. J. Phys. D: Appl. Phys. 22(12), 1883–1887 (1989)CrossRefGoogle Scholar
  80. 80.
    K.E. Evans, M.A. Nkansah, I.J. Hutchinson, S.C. Rogers, Molecular network design. Nature 353, 124 (1991)CrossRefGoogle Scholar
  81. 81.
    B.A. Fulcher, D.W. Shahan, M.R. Haberman, C.C. Seepersad, P.S. Wilson, Analytical and experimental investigation of buckled beams as negative stiffness elements for passive vibration and shock isolation systems. J. Vibr. Acoust.-Trans. ASME 136(3), 31009 (2014)CrossRefGoogle Scholar
  82. 82.
    C.J. Gantes, J.J. Connor, L.D. Logcher, Y. Rosenfeld, Structural analysis and design of deployable structures. Comput. Struct. 32(3–4), 661–669 (1989)CrossRefGoogle Scholar
  83. 83.
    N. Gaspar, X.J. Ren, C.W. Smith, J.N. Grima, K.E. Evans, Novel honeycombs with auxetic behaviour. Acta Mater. 53, 2439–2445 (2005)CrossRefGoogle Scholar
  84. 84.
    N. Gaspar, C.W. Smith, K.E. Evans, Auxetic behaviour and anisotropic heterogeneity. Acta Mater. 57, 875–880 (2009)CrossRefGoogle Scholar
  85. 85.
    N. Gaspar, A granular material with a negative Poisson’s ratio. Mech. Mater. 42, 673–677 (2010)CrossRefGoogle Scholar
  86. 86.
    N. Gaspar, C.W. Smith, A. Alderson, J.N. Grima, K.E. Evans, A generalised three-dimensional tethered-nodule model for auxetic materials. J. Mater. Sci. 46, 372–384 (2011)CrossRefGoogle Scholar
  87. 87.
    R. Gatt, L. Mizzi, K.M. Azzopardi, J.N. Grima, A force-field based analysis of the deformation mechanism in α-cristobalite. Phys. Status Solidi B 252, 1479–1485 (2015)CrossRefGoogle Scholar
  88. 88.
    L.N. Germanovich, A.V. Dyskin, Virial expansions in problems of effective characteristics. Part I. General concepts. J. Mech. Compos. Mater. 30(2), 222–237 (1994)CrossRefGoogle Scholar
  89. 89.
    L.N. Germanovich, A.V. Dyskin, Virial expansions in problems of effective characteristics. Part II. Anti-plane deformation of fibre composite. Analysis of self-consistent methods. J. Mech. Compos. Mater. 30(2), 234–243 (1994)CrossRefGoogle Scholar
  90. 90.
    C.M. Gerrard, Elastic models of rock masses having one, two and three sets of joints. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 19(1), 15–23 (1982). Scholar
  91. 91.
    G.N. Greaves, A.L. Greer, R.S. Lakes, T. Rouxel, Poisson’s ratio and modern materials. Nat. Mater. 10, 823–837 (2011)CrossRefGoogle Scholar
  92. 92.
    J.N. Grima, K.E. Evans, Auxetic behavior from rotating triangles. J. Mater. Sci. 41, 3193–3196 (2006)CrossRefGoogle Scholar
  93. 93.
    J.N. Grima, R. Gatt, Perforated sheets exhibiting negative Poisson’s ratios. Adv. Eng. Mater. 12(6), 460–464 (2010). Scholar
  94. 94.
    J.N. Grima, R. Jackson, A. Alderson, K.E. Evans, Do zeolites have negative Poisson’s ratios? Adv. Mater. 12(24), 1912–1918 (2000)CrossRefGoogle Scholar
  95. 95.
    J.N. Grima, A. Alderson, K.E. Evans, Auxetic behaviour from rotating rigid units. Phys. Status Solidi B 242(3), 561–575 (2005)CrossRefGoogle Scholar
  96. 96.
    J.N. Grima, R. Gatt, N. Ravirala, A. Alderson, K.E. Evans, Negative Poisson’s ratios in cellular foam materials. Mater. Sci. Eng., A 423(1–2), 214–218 (2006)CrossRefGoogle Scholar
  97. 97.
    J.N. Grima, P.S. Farrugia, R. Gatt, V. Zammit, A system with adjustable positive or negative thermal expansion. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 463, 1585–1596 (2007)CrossRefGoogle Scholar
  98. 98.
    J.N. Grima, R. Caruana-Gauci, M.R. Dudek, K.W. Wojciechowski, R. Gatt, Smart metamaterials with tunable auxetic and other properties. Smart Mater. Struct. 22, 084016 (2013)CrossRefGoogle Scholar
  99. 99.
    J.N. Grima, B. Ellul, R. Gatt, D. Attard, Negative thermal expansion from disc, cylindrical, and needle shaped inclusions. Phys. Status Solidi B 250, 2051–2056 (2013)CrossRefGoogle Scholar
  100. 100.
    J.N. Grima, R. Cauchi, R. Gatt, D. Attard, Honeycomb composites with auxetic out-of-plane characteristics. Compos. Struct. 106, 150–159 (2013)CrossRefGoogle Scholar
  101. 101.
    J.N. Grima, S. Winczewski, L. Mizzi, M.C. Grech, R. Cauchi, R. Gatt, Tailoring graphene to achieve negative Poisson’s ratio properties. Adv. Mater. 27, 1455–1459 (2015)CrossRefGoogle Scholar
  102. 102.
    J.N. Grima, L. Mizzi, K.M. Azzopardi, R. Gatt, Auxetic perforated mechanical metamaterials with randomly oriented cuts. Adv. Mater. 28(2), 385–389 (2015)CrossRefGoogle Scholar
  103. 103.
    J.N. Grima, D. Attard, R. Gatt, R.N. Cassar, A novel process for the manufacture of auxetic foams and for their re-conversion to conventional form. Adv. Eng. Mater. 11(7), 533–535 (2009)CrossRefGoogle Scholar
  104. 104.
    J.N. Grima, R.N. Cassar, R. Gatt, On the effect of hydrostatic pressure on the auxetic character of NAT-type silicates. J. Non-Cryst. Solids 355, 1307–1312 (2009)CrossRefGoogle Scholar
  105. 105.
    J.N. Grima, B. Ellul, R. Gatt, D. Attard, Negative thermal expansion from disc cylindrical, and needle shaped inclusions. Phys. Status Solidi B 250(10), 2051–2056 (2012)Google Scholar
  106. 106.
    J.N. Grima, E. Chetcuti, E. Manicaro, D. Attard, M. Camilleri, R. Gatt, K.E. Evans, On the auxetic properties of generic rotating rigid triangles. Proc. R. Soc. A 468, 810–830 (2012)CrossRefGoogle Scholar
  107. 107.
    J.N. Grima, E. Manicaro, D. Attard, Auxetic behaviour from connected different-sized squares and rectangles. Proc. R. Soc. A 467, 439–458 (2010)CrossRefGoogle Scholar
  108. 108.
    D.J. Gunton, G.A. Saunders, The Young’s modulus and Poisson’s ratio of arsenic, antimony and bismuth. J. Mater. Sci. 7, 1061–1068 (1972)CrossRefGoogle Scholar
  109. 109.
    Y. Guo, W.A. Goddard III, Is carbon nitride harder than diamond? No, but its girth increases when stretched (negative Poisson ratio). Chem. Phys. Let. 237, 72–76 (1995)CrossRefGoogle Scholar
  110. 110.
    C.S. Ha, E. Hestekin, J. Li, M.E. Plesha, R.S. Lakes, Controllable thermal expansion of large magnitude in chiral negative Poisson’s ratio lattices. Phys. Status Solidi B 252, 1431–1434 (2015)CrossRefGoogle Scholar
  111. 111.
    L.J. Hall, V.R. Coluci, D.S. Galvão, M.E. Kozlov, M. Zhang, S.O. Dantas, R.H. Baughman, Sign change of Poisson’s ratio for carbon nanotube sheets. Science 320, 504–507 (2008)CrossRefGoogle Scholar
  112. 112.
    Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2), 127–140 (1963)CrossRefGoogle Scholar
  113. 113.
    E.W. Hawkes, E.V. Eason, A.T. Asbeck, M.R. Cutkosky, IEEE/ASME Trans. Mechatron. 18, 518 (2013)Google Scholar
  114. 114.
    G.F. Hawkins, M.J. O’Brien, C.Y. Tang, Proceedings of SPIE. Smart Mater. III 5648, 37 (2004)CrossRefGoogle Scholar
  115. 115.
    T.A.M. Hewage, K.L. Alderson, A. Alderson, F. Scarpa, Double-negative mechanical metamaterials displaying simultaneous negative stiffness and negative Poisson’s ratio properties. Adv. Mater. 28, 10323–10332 (2016)CrossRefGoogle Scholar
  116. 116.
    R. Hill, The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc. (Lond.) A65, 349–354 (1952)CrossRefGoogle Scholar
  117. 117.
    R. Hill, Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)CrossRefGoogle Scholar
  118. 118.
    S. Hirotsu, Softening of bulk modulus and negative Poisson’s ratio near the volume phase transition of polymer gels. J. Chem. Phys. 94(5), 3949–3957 (1991). Scholar
  119. 119.
    D.T. Ho, H. Kim, S.-Y. Kwon, S.Y. Kim, Auxeticity of face-centered cubic metal (001) nanoplates. Phys. Status Solidi B 252, 1492–1501 (2015)CrossRefGoogle Scholar
  120. 120.
    X. Hou, H. Hu, V. Silberschmidt, A novel concept to develop composite structures with isotropic negative Poisson’s ratio: effects of random inclusions. Compos. Sci. Technol. 72, 1848–1854 (2012)CrossRefGoogle Scholar
  121. 121.
    X. Hou, H. Hu, V. Silberschmidt, Numerical analysis of composite structure with in-plane isotropic negative Poisson’s ratio: effects of materials properties and geometry features of inclusions. Compos.: Part B 58, 152–159 (2014)CrossRefGoogle Scholar
  122. 122.
    T.P. Hughes, A. Marmier, K.E. Evans, Auxetic frameworks inspired by cubic crystals. Int. J. Solids Struct. 47, 1469–1476 (2010)CrossRefGoogle Scholar
  123. 123.
    G.W. Hunt, H.-B. Mühlhaus, A.I.M. Whiting, Folding processes and solitary waves in structural geology, in Localisation and Solitary Waves in Solid Mechanics, ed. by A.R. Champneys, G.W. Hunt, J.M.T. Thompson (World Scientific, Singapore, 1984), pp. 332–348Google Scholar
  124. 124.
    T. Jaglinski, R.S. Lakes, Anelastic instability in composites with negative stiffness inclusions. Philos. Mag. Lett. 84(12), 803–810 (2004)CrossRefGoogle Scholar
  125. 125.
    T. Jaglinski, D. Kochmann, D. Stone, R.S. Lakes, Composite materials with viscoelastic stiffness greater than diamond. Science 315, 620–622 (2007)CrossRefGoogle Scholar
  126. 126.
    T. Jaglinski, P. Frascone, B. Moore, D.S. Stone, R.S. Lakes, Internal friction due to negative stiffness in the indium–thallium martensitic phase transformation. Phil. Mag. 86(27), 4285–4303 (2006)CrossRefGoogle Scholar
  127. 127.
    T. Jaglinski, D. Stone, R.S. Lakes, Internal friction study of a composite with a negative stiffness constituent. J. Mater. Res. 20(09), 2523–2533 (2005)CrossRefGoogle Scholar
  128. 128.
    S. Jayanty, J. Crowe, L. Berhan, Auxetic fibre networks and their composites. Phys. Status Solidi B 248(1), 73–81 (2011)CrossRefGoogle Scholar
  129. 129.
    H. Kalathur, R.S. Lakes, Column dampers with negative stiffness: high damping at small amplitude. Smart Mater. Struct. 22(8), 084013 (2013)CrossRefGoogle Scholar
  130. 130.
    H. Kalathur, T.M. Hoang, R.S. Lakes, W.J. Drugan, Buckling mode jump at very close load values in unattached flat-end columns: theory and experiment. J. Appl. Mech.-Trans. ASME 81(4), 41010 (2014)CrossRefGoogle Scholar
  131. 131.
    P. Kanouté, D.P. Boso, J.L. Chaboche, B.A. Schrefler, Multiscale methods for composites: a review. Arch. Comput. Methods Eng. 16, 31–75 (2009)CrossRefGoogle Scholar
  132. 132.
    N. Keskar, J.R. Chelikowsky, Negative Poisson’s ratio in crystalline SiO2 from first-principles calculations. Nature 358, 222–224 (1992)CrossRefGoogle Scholar
  133. 133.
    D.M. Kochmann, W.J. Drugan, Dynamic stability analysis of an elastic composite material having a negative-stiffness phase. J. Mech. Phys. Solids 57(7), 1122–1138 (2009)CrossRefGoogle Scholar
  134. 134.
    D.M. Kochmann, G.W. Milton, Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases. J. Mech. Phys. Solids 71, 46–63 (2014)CrossRefGoogle Scholar
  135. 135.
    D.M. Kochmann, G.N. Venturini, Homogenized mechanical properties of auxetic composite materials in finite-strain elasticity. Smart Mater. Struct. 22, 084004–084011 (2013)CrossRefGoogle Scholar
  136. 136.
    D.M. Kochmann, W.J. Drugan, Infinitely stiff composite via a rotation-stabilized negative-stiffness phase. Appl. Phys. Lett. 99, 011909 (2011)CrossRefGoogle Scholar
  137. 137.
    A.G. Kolpakov, Determination of the average characteristics of elastic frameworks. Appl. Math. Mech. 49, 739–745 (1985)CrossRefGoogle Scholar
  138. 138.
    R.S. Lakes, Cellular solids with tunable positive or negative thermal expansion of unbounded magnitude. Appl. Phys. Lett. 90, 221905 (2007)CrossRefGoogle Scholar
  139. 139.
    R.S. Lakes, Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987)CrossRefGoogle Scholar
  140. 140.
    R.S. Lakes, Deformation mechanisms of negative Poisson’s ratio materials: structural aspects. J. Mater. Sci. 26, 2287–2292 (1991)CrossRefGoogle Scholar
  141. 141.
    R.S. Lakes, Design considerations for materials with negative Poisson’s ratios. J. Mech. Des. 115(4), 696–700 (1993). Scholar
  142. 142.
    R.S. Lakes, Extreme damping in composite materials with a negative stiffness phase. Phys. Rev. Let. 86(13), 2897–2900 (2001)CrossRefGoogle Scholar
  143. 143.
    R.S. Lakes, Extreme damping in compliant composites with a negative-stiffness phase. Phil. Mag. Lett. 81(2), 95–100 (2001)CrossRefGoogle Scholar
  144. 144.
    R.S. Lakes, K. Elms, Indentability of conventional and negative Poisson’s ratio foams. J. Compos. Mater. 27, 1193–1202 (1993)CrossRefGoogle Scholar
  145. 145.
    R.S. Lakes, W.J. Drugan, Dramatically stiffer elastic composite materials due to a negative stiffness phase? J. Mech. Phys. Solids 50, 979–1009 (2002)CrossRefGoogle Scholar
  146. 146.
    R.S. Lakes, A. Lowe, Negative Poisson’s ratio foam as seat cushion material. Cell. Polym. 19, 157–167 (2000)Google Scholar
  147. 147.
    R.S. Lakes, P. Rosakis, A. Ruina, Microbuckling instability in elastomeric cellular solids. J. Mat. Sci 28, 4667–4672 (1993)CrossRefGoogle Scholar
  148. 148.
    R.S. Lakes, T. Lee, A. Bersie, Y.C. Wang, Extreme damping in composite materials with negative stiffness inclusions. Nature 410, 565–567 (2001)CrossRefGoogle Scholar
  149. 149.
    R.S. Lakes, K.W. Wojciechowski, Negative compressibility, negative Poisson’s ratio, and stability. Phys. Status Solidi 245, 545–551 (2008)CrossRefGoogle Scholar
  150. 150.
    L.D. Landau, E.M. Lifshitz, Theory of Elasticity (Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 1959)Google Scholar
  151. 151.
    U.D. Larsen, O. Sigmund, S. Bouwstra, Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio. J. Microelectromech. Systems 6, 99–106 (1997)CrossRefGoogle Scholar
  152. 152.
    M.L. Latash, V.M. Zatsiorsky, Joint stiffness: myth or reality? Hum. Mov. Sci. 12(6), 653–692 (1993)CrossRefGoogle Scholar
  153. 153.
    I.M. Lifshits, L.N. Rosentsveig, Zur Theorie der elastischen Eigenschaften yon Polykristallen. Zh. Eksp. Teor. Fiz. 16, 967–975 (1946)Google Scholar
  154. 154.
    T.-C. Lim, Out-of-plane modulus of semi-auxetic laminates. Eur. J. Mech. A/Solids 28, 752–756 (2009)CrossRefGoogle Scholar
  155. 155.
    T.-C. Lim, Coefficient of thermal expansion of stacked auxetic and negative thermal expansion laminates. Phys. Status Solidi B 248(1), 140–147 (2011)CrossRefGoogle Scholar
  156. 156.
    T.-C. Lim, Thermal stresses in thin auxetic plates. J. Therm. Stresses 36(11), 1131–1140 (2012)CrossRefGoogle Scholar
  157. 157.
    T.-C. Lim, Negative thermal expansion structures constructed from positive thermal expansion trusses. J. Mater. Sci. 47, 368–373 (2012)CrossRefGoogle Scholar
  158. 158.
    T.-C. Lim, U. Rajendra Acharya, Counterintuitive modulus from semi-auxetic laminates. Phys. Status Solidi B 248(1), 60–65 (2011)CrossRefGoogle Scholar
  159. 159.
    C. Lira, P. Innocenti, F. Scarpa, Transverse elastic shear of auxetic multi re-entrant honeycombs. Compos. Struct. 90, 314–322 (2010)CrossRefGoogle Scholar
  160. 160.
    C. Lira, F. Scarpa, M. Olszewska, M. Celuch, The SILICOMB cellular structure: mechanical and dielectric properties. Phys. Status Solidi B 246, 2055–2062 (2009)CrossRefGoogle Scholar
  161. 161.
    Y. Liu, H. Hu, A review on auxetic structures and polymeric materials. Sci. Res. Essays 5(10), 1052–1063 (2010)Google Scholar
  162. 162.
    A.E.H. Love, A Treatise on Mathematical Theory of Elasticity (Dover, New York, 1927)Google Scholar
  163. 163.
    P. Martin, A.D. Mehta, A.J. Hudspeth, Negative hair-bundle stiffness betrays a mechanism for mechanical amplification by the hair cell. Proc. Natl. Acad. Sci. U.S.A. 97(22), 12026–12031 (2000)CrossRefGoogle Scholar
  164. 164.
    E.O. Martz, R.S. Lakes, J.B. Park, Hysteresis behaviour and specific damping capacity of negative Poisson’s ratio foams. Cell. Polym. 15, 349–364 (1996)Google Scholar
  165. 165.
    B.T. Maruszewski, A. Drzewiecki, R. Starosta, On effective Young’s modulus and Poisson’s ratio of the auxetic thermoelastic material. Comput. Methods Sci. Technol. 22, 233–237 (2016)CrossRefGoogle Scholar
  166. 166.
    T.A. Mary, J.S.O. Evans, T. Vogt, A.W. Sleight, Negative thermal expansion from 0.3 to 1050 K in ZrW2O8. Science 272, 90–92 (1996)CrossRefGoogle Scholar
  167. 167.
    D.M. McCutcheon, J.N. Reddy, M.J. O’Brien, T.S. Creasy, G.F. Hawkins, Damping composite materials by machine augmentation. J. Sound Vib. 294(4–5), 828–840 (2006). Scholar
  168. 168.
    R. McLaughlin, A study of the differential scheme in composite materials. Int. J. Eng. Sci. 15, 237–244 (1977)CrossRefGoogle Scholar
  169. 169.
    P. Michelis, V. Spitas, Numerical and experimental analysis of a triangular auxetic core made of CFR-PEEK using the directionally reinforced integrated single-yarn (DIRIS) architecture. Compos. Sci. Technol. 70, 1064–1071 (2010)CrossRefGoogle Scholar
  170. 170.
    W. Miller, C.W. Smith, D.S. Mackenzie, K.E. Evans, Negative thermal expansion: a review. J. Mater. Sci. 44(20), 5441–5451 (2009)CrossRefGoogle Scholar
  171. 171.
    W. Miller, P.B. Hook, C.W. Smith, X. Wanga, K.E. Evans, The manufacture and characterisation of a novel, low modulus, negative Poisson’s ratio composite. Compos. Sci. Technol. 69, 651–655 (2009)CrossRefGoogle Scholar
  172. 172.
    W. Miller, C.W. Smith, K.E. Evans, Honeycomb cores with enhanced buckling strength. Compos. Struct. 93, 1072–1077 (2011)CrossRefGoogle Scholar
  173. 173.
    F. Milstein, K. Huang, Existence of a negative Poisson’s ratio in fcc crystals. Phys. Rev. B 19, 2030–2033 (1979)CrossRefGoogle Scholar
  174. 174.
    G.W. Milton, Complete characterization of the macroscopic deformations of periodic unimode metamaterials of rigid bars and pivots. J. Mech. Phys. Solids 61(7), 1543–1560 (2013)CrossRefGoogle Scholar
  175. 175.
    H. Mitschke, V. Robins, K. Mecke, G.E. Schröder-Turk, Finite auxetic deformations of plane tessellations. Proc. R. Soc. Lond A: Math. Phys. Eng. Sci. 469(2149), 20120465Google Scholar
  176. 176.
    H. Mitschke, J. Schwerdtfeger, F. Schury, M. Stingl, C. Körner, R.F. Singer et al., Finding auxetic frameworks in periodic tessellations. Adv. Mater. 23(22–23), 2669–2674 (2011)CrossRefGoogle Scholar
  177. 177.
    L. Mizzi, R. Gatt, J.N. Grima, Non-porous grooved single-material auxetics. Phys. Status Solidi B 252, 1559–1564 (2015)CrossRefGoogle Scholar
  178. 178.
    J.W. Narojczyk, K.W. Wojciechowski, Elastic properties of degenerate f.c.c. crystal of polydisperse soft dimers at zero temperature. J. Non-Cryst. Solids 356, 2026–2032 (2010)CrossRefGoogle Scholar
  179. 179.
    J.W. Narojczyk, A. Alderson, A.R. Imre, F. Scarpa, K.W. Wojciechowski, Negative Poisson’s ratio behavior in the planar model of asymmetric trimers at zero temperature. J. Non-Cryst. Solids 354, 4242–4248 (2008)CrossRefGoogle Scholar
  180. 180.
    F. Nazare, A. Alderson, Models for the prediction of Poisson’s ratio in the ‘α-cristobalite’ tetrahedral network. Phys. Status Solidi B 252, 1465–1478 (2015)CrossRefGoogle Scholar
  181. 181.
    N. Novak, M. Vesenjak, Z. Ren, Auxetic cellular materials—a review. Strojniski Vestnik-J. Mech. Eng. 62(9), 485–493 (2016)CrossRefGoogle Scholar
  182. 182.
    W. Nowacki, The Linear Theory of Micropolar Elasticity (Springer, Wien, New York), pp. 1–43Google Scholar
  183. 183.
    R.J. O’Connell, B. Budiansky, Seismic velocities in dry and saturated cracked solids. J. Geophys. Res. 79, 5412–5426 (1974)CrossRefGoogle Scholar
  184. 184.
    S.-T. Park, T-T. Luu, Techniques for optimizing parameters of negative stiffness. Proc. IMechE Part C: J. Mech. Eng. Sci. 221, 505–511 (2007)Google Scholar
  185. 185.
    E. Pasternak, A.V. Dyskin, Multiscale hybrid materials with negative Poisson’s ratio, in IUTAM Symposium on Scaling in Solid Mechanics, ed. by F. Borodich (Springer, 2008a), pp. 49–58Google Scholar
  186. 186.
    E. Pasternak, A.V. Dyskin, Materials with Poisson’s ratio near −1: properties and possible realisations, in ICTAM 2008, XXII International Congress of Theoretical and Applied Mechanics, paper 11982, CD-ROM Proceedings, August 24–29, 2008, ed. by J. Denier, M.D. Finn, T. Mattner (Adelaide, 2008b), 2 p. ISBN 978-0-9805142-1-6Google Scholar
  187. 187.
    E. Pasternak, A.V. Dyskin, Materials and structures with macroscopic negative Poisson’s ratio. Int. J. Eng. Sci. 52, 103–114 (2012)CrossRefGoogle Scholar
  188. 188.
    E. Pasternak, E., A.V. Dyskin, Instability and failure of particulate materials caused by rolling of non-spherical particles, in Proceedings of the 13th International Conference on Fracture (Beijing, China, 2013)Google Scholar
  189. 189.
    E. Pasternak, H.-B. Mühlhaus, Cosserat continuum modelling of granulate materials, in Computational Mechanics—New Frontiers for New Millennium, ed. by S. Valliappan, N. Khalili (Elsevier, 2001), pp. 1189–1194Google Scholar
  190. 190.
    E. Pasternak, H.-B. Mühlhaus, Generalised homogenisation procedures for granular materials. Eng. Math. 52, 199–229 (2005)CrossRefGoogle Scholar
  191. 191.
    E. Pasternak, A.V. Dyskin, Dynamic instability in geomaterials associated with the presence of negative stiffness elements, in Bifurcation and Degradation of Geomaterials in the New Millennium, ed. by K.-T. Chau, J. Zhao (Springer, 2015), pp. 155–160Google Scholar
  192. 192.
    E. Pasternak, A.V. Dyskin, I. Shufrin, Negative Poisson’s ratio materials’ design principles and possible applications, in Proceedings of the 6th Australasian Congress on Applied Mechanics, ACAM 6, 12–15 December 2010, Perth, Paper 1266, ed. by K. Teh, I. Davies, I. Howard (2010), 10 ppGoogle Scholar
  193. 193.
    E. Pasternak, A.V. Dyskin, G. Sevel, Chains of oscillators with negative stiffness elements. J. Sound Vib. 333(24), 6676–6687 (2014)CrossRefGoogle Scholar
  194. 194.
    E. Pasternak, I. Shufrin, A.V. Dyskin, Thermal stresses in hybrid materials with auxetic inclusions. Composite Structures 138, 313–321 (2016)CrossRefGoogle Scholar
  195. 195.
    E. Pasternak, A.V. Dyskin, M. Esin, Wave propagation in materials with negative Cosserat shear modulus. Int. J. Eng. Sci. 100, 152–161 (2016)CrossRefGoogle Scholar
  196. 196.
    S. Pellegrino, Deployable structures in engineering, in Deployable structures, ed. by S. Pellegrino (Springer, Wien GmbH, 2014)Google Scholar
  197. 197.
    N. Phan-Thien, B.L. Karihaloo, Materials with negative Poisson’s ratio: a qualitative microstructural model. J. Appl. Mech. Trans. ASME 61, 1001–1004 (1994)CrossRefGoogle Scholar
  198. 198.
    P.V. Pikhitsa, M. Choi, H.-J. Kim, S.-H. Ahn, Auxetic lattice of multipods. Phys. Status Solidi B 246, 2098–2101 (2009)CrossRefGoogle Scholar
  199. 199.
    D.L. Platus, Negative-stiffness-mechanism vibration isolation systems, in SPIE Conference on Current Developments in Vibration Control 98 for Optomechanical Systems, Denver, Colorado, July 1999, SPIE, vol. 3786 (1999), pp. 98–105Google Scholar
  200. 200.
    M.E. Pontecorvo, S. Barbarino, G.J. Murray, F.S. Gandhi, Bistable arches for morphing applications. J. Intell. Mater. Syst. Struct. 24(3), 274–286 (2013)CrossRefGoogle Scholar
  201. 201.
    A.A. Pozniak, J. Smardzewski, K.W. Wojciechowski, Computer simulations of auxetic foams in two dimensions. Smart Mater. Struct. 22, 084009 (2013)CrossRefGoogle Scholar
  202. 202.
    Y. Prawoto, Seeing auxetic materials from the mechanics point of view: a structural review on the negative Poisson’s ratio. Comput. Mater. Sci. 58, 140–153 (2012)CrossRefGoogle Scholar
  203. 203.
    G. Puglisi, L. Truskinovsky, Mechanics of a discrete chain with bi-stable elements. J. Mech. Phys. Solids 48, 1–27 (2000)CrossRefGoogle Scholar
  204. 204.
    N. Ravirala, A. Alderson, K.L. Alderson, Interlocking hexagons model for auxetic behaviour. J. Mater. Sci. 42, 7433–7445 (2007)CrossRefGoogle Scholar
  205. 205.
    N. Ravirala, K.L. Alderson, P.J. Davies, V.R. Simkins, A. Alderson, Negative Poisson’s ratio polyester fibers. Text. Res. J. 76, 540–546 (2006)CrossRefGoogle Scholar
  206. 206.
    B.W. Rosen, Z. Hashin, Effective thermal expansion coefficients and specific heats of composite materials. Int. J. Eng. Sci. 8, 157–173 (1970)CrossRefGoogle Scholar
  207. 207.
    L. Rothenburg, A.L. Berlin, R. Bathurst, Microstructure of isotropic materials with negative Poisson’s ratio. Nature 325, 470–472 (1991)Google Scholar
  208. 208.
    A.L. Roytburd, Deformation through transformations. J. Phys. IV C1, 11–25 (1996). Scholar
  209. 209.
    M.D.G. Salamon, Stability, instability and design of pillar workings. Int. J. Rock Mech. Min. Sci. 7, 613–631 (1970)CrossRefGoogle Scholar
  210. 210.
    R.L. Salganik, Mechanics of bodies with many cracks. Mech. Solids 8, 135–143 (1973)Google Scholar
  211. 211.
    A.A. Sarlis, D.T.R. Pasala, M.C. Constantinou, A.M. Reinhorn, S. Nagarajaiah, D.P. Taylor, Negative stiffness device for seismic protection of structures. J. Struct. Eng. 139(7), 1124–1133 (2013)CrossRefGoogle Scholar
  212. 212.
    F. Scarpa, G. Tomlinson, Sandwich structures with negative Poisson’s ratio for deployable structures, in IUTAM-IASS Symposium on Deployable Structures: Theory and Applications (2000), pp. 335–343Google Scholar
  213. 213.
    F. Scarpa, P.J. Tomlin, On the transverse shear modulus of negative Poisson’s ratio honeycomb structures. Fatigue Fract. Eng. Mater. Struct. 23(8), 717–720 (2000)CrossRefGoogle Scholar
  214. 214.
    F. Scarpa, S. Adhikari, C.Y. Wang, Nanocomposites with auxetic nanotubes. Int J. Smart Nano Mater. 1(2), 83–94 (2010)CrossRefGoogle Scholar
  215. 215.
    F. Scarpa, P. Pastorino, A. Garelli, S. Patsias, M. Ruzzene, Auxetic compliant flexible PU foams: static and dynamic properties. Phys. Status Solidi B 242(3), 681–694 (2005)CrossRefGoogle Scholar
  216. 216.
    F. Scarpa, J.W. Narojczyk, K.W. Wojciechowski, Unusual deformation mechanisms in carbon nanotube heterojunctions (5,5)–(10,10) under tensile loading. Phys. Status Solidi B 248(1), 82–87 (2011)CrossRefGoogle Scholar
  217. 217.
    S. Schaare, A.V. Dyskin, Y. Estrin, E. Pasternak, A. Kanel-Belov, Point loading of assemblies of interlocked cube-shaped elements. Int. J. Eng. Sci. 46, 1228–1238 (2008). Scholar
  218. 218.
    M.M. Shokrieh, A. Assadi, Determination of maximum negative Poisson’s ratio for laminated fiber composites. Phys. Status Solidi B 248(5), 1237–1241 (2011)CrossRefGoogle Scholar
  219. 219.
    I. Shufrin, E. Pasternak, A.V. Dyskin, Symmetric structures with negative Poisson’s ratio, in Australian and New Zealand Industrial and Applied Mathematics Conference, ANZIAM—2010, Queenstown, New Zealand (2010)Google Scholar
  220. 220.
    I. Shufrin, E. Pasternak, A.V. Dyskin, Planar isotropic structures with negative Poisson’s ratio. Int. J. Solids Struct. 49(17), 2239–2253 (2012)CrossRefGoogle Scholar
  221. 221.
    I. Shufrin, E. Pasternak, A.V. Dyskin, Negative Poisson’s ratio in hollow sphere materials. Int. J. Solids Struct. 54, 192–214 (2015)CrossRefGoogle Scholar
  222. 222.
    I. Shufrin, E. Pasternak, A.V. Dyskin, Hybrid materials with negative Poisson’s ratio inclusions. Int. J. Eng. Sci. 89, 100–120 (2015)CrossRefGoogle Scholar
  223. 223.
    I. Shufrin, E. Pasternak, A.V. Dyskin, Deformation analysis of reinforced-core auxetic assemblies by close-range photogrammetry. Phys. Status Solidi (B) Basic Res. 253(7), 1342–1358 (2016)Google Scholar
  224. 224.
    O. Sigmund, Tailoring materials with prescribed elastic properties. Mech. Mater. 20, 351–368 (1995)CrossRefGoogle Scholar
  225. 225.
    V.R. Simkins, A. Alderson, P.J. Davies, K.L. Alderson, Single fibre pullout tests on auxetic polymeric fibres. J. Mater. Sci. 40, 4355–4364 (2005)CrossRefGoogle Scholar
  226. 226.
    A. Slan, W. White, F. Scarpa, K. Boba, I. Farrow, Cellular plates with auxetic rectangular perforations. Phys. Status Solidi B 252, 1533–1539 (2015)CrossRefGoogle Scholar
  227. 227.
    C.W. Smith, J.N. Grima, K.E. Evans, A novel mechanism for generating auxetic behaviour in reticulated foams: missing rib foam model. Acta Mater. 48, 4349–4356 (2000)CrossRefGoogle Scholar
  228. 228.
    A. Sogame, H. Furuya, Conceptual study on cylindrical deployable space structures, in IUTAM-IASS Symposium on Deployable Structures: Theory and Applications (2000), pp 383–392Google Scholar
  229. 229.
    A. Spadoni, M. Ruzzene, Elasto-static micropolar behaviour of a chiral auxetic lattice. J. Mech. Phys. Solids. 60, 156–171 (2012)Google Scholar
  230. 230.
    P.J. Stott, R. Mitchell, K. Alderson, A. Alderson, A growing industry. Mater. World 8, 12–14 (2000)Google Scholar
  231. 231.
    T. Strek, H. Jopek, Effective mechanical properties of concentric cylindrical composites with auxetic phase. Phys. Status Solidi B 249(7), 1359–1365 (2012)CrossRefGoogle Scholar
  232. 232.
    K. Takenaka, Negative thermal expansion materials: technological key for control of thermal expansion’. Sci. Technol. Adv. Mater. 13, 013001–013012 (2012)CrossRefGoogle Scholar
  233. 233.
    C.Y. Tang, M.J. O’Brien, G.F. Hawkins, Embedding simple machines to add novel dynamic functions to composites. JOM 57, 32 (2005)CrossRefGoogle Scholar
  234. 234.
    M. Tatlier, L. Berhan, Modelling the negative Poisson’s ratio of compressed fused fibre networks. Phys. Status Solidi B 246, 2018–2024 (2009)CrossRefGoogle Scholar
  235. 235.
    P.S. Theocaris, G.E. Stavroulakis, P.D. Panagiotopoulos, Negative Poisson’s ratios in composites with star-shaped inclusions: a numerical homogenization approach. Arch. Appl. Mech. 67, 274–286 (1997)CrossRefGoogle Scholar
  236. 236.
    J.M.T. Thompson, G.W. Hunt, A General Theory of Elastic Stability (Wiley, London, 1973)Google Scholar
  237. 237.
    K. Toru, M. Yoshitaka, Nanoscale mechanics of carbon nanotube evaluated by nanoprobe manipulation in transmission electron microscope. Japanese Journal of Applied (2006)Google Scholar
  238. 238.
    K.V. Tretiakov, Negative Poisson’s ratio of two-dimensional hard cyclic tetramers. J. Non-Cryst. Solids 355, 1435–1438 (2009)CrossRefGoogle Scholar
  239. 239.
    K.V. Tretiakov, K.W. Wojciechowski, Poisson’s ratio of simple planar ‘isotropic’ solids in two dimensions. Phys. Status Solidi B 244(3), 1038–1046 (2007)CrossRefGoogle Scholar
  240. 240.
    K.V. Tretiakov, K.W. Wojciechowski, Elastic properties of soft sphere crystal from Monte Carlo simulations. J. Phys. Chem. B 112, 1699–1705 (2009)CrossRefGoogle Scholar
  241. 241.
    K.V. Tretiakov, K.W. Wojciechowski, Elastic properties of fcc crystals of polydisperse soft spheres. Phys. Status Solidi B, 1–10 (2013)Google Scholar
  242. 242.
    A.S. Vavakin, R.L. Salganik, Effective characteristics of nonhomogeneous media with isolated nonhomogeneities. Mech. Solids 10(3), 58–66 (1975)Google Scholar
  243. 243.
    A.S. Vavakin, R.L. Salganik, Effective elastic characteristics of bodies with isolated cracks, cavities, and rigid nonhomogeneities. Mech. Solids 13, 87–97 (1978)Google Scholar
  244. 244.
    P. Verma, M.L. Shofner, A. Lin, K.B. Wagner, A.C. Griffin, Inducing out-of-plane auxetic behaviour in needle-punched nonwovens. Phys. Status Solidi B 252, 1455–1464 (2015)CrossRefGoogle Scholar
  245. 245.
    Y.-C. Wang, Influences of negative stiffness on a two-dimensional hexagonal lattice cell. Philos. Mag. 87(24), 3671–3688 (2007)CrossRefGoogle Scholar
  246. 246.
    M. Wang, N. Pan, Predictions of effective physical properties of complex multiphase materials. Mat. Sci. Eng. R 63, 1–30 (2008)CrossRefGoogle Scholar
  247. 247.
    Y.C. Wang, R.S. Lakes, Extreme thermal expansion, piezoelectricity, and other coupled field properties in composites with a negative stiffness phase. J. Appl. Phys. 90, 6458–6465 (2001)CrossRefGoogle Scholar
  248. 248.
    Y.C. Wang, R.S. Lakes, Extreme stiffness systems due to negative stiffness elements. Am. J. Phys. 72, 40–50 (2004)CrossRefGoogle Scholar
  249. 249.
    Y.-C. Wang, R.S. Lakes, Stable extremely-high-damping discrete viscoelastic systems due to negative stiffness elements. Appl. Phys. Lett. 84(22), 4451–4453 (2004)CrossRefGoogle Scholar
  250. 250.
    Y.-C. Wang, R.S. Lakes, Composites with inclusions of negative bulk modulus: extreme damping and negative Poisson’s ratio. J. Compos. Mater. 39(18), 1645–1657 (2005)CrossRefGoogle Scholar
  251. 251.
    Y.-C. Wang, C.-C. Ko, K.-W. Chang, Anomalous effective viscoelastic, thermoelastic, dielectric and piezoelectric properties of negative-stiffness composites and their stability. Phys. Status Solidi B 252, 1640–1655 (2015)CrossRefGoogle Scholar
  252. 252.
    Y.-C. Wang, M. Ludwigson, R.S. Lakes, Deformation of extreme viscoelastic metals and composites. Mater. Sci. Eng. A 370, 41–49 (2004)CrossRefGoogle Scholar
  253. 253.
    Y.C. Wang, J.G. Swadener, R.S. Lakes, Two-dimensional viscoelastic discrete triangular system with negative-stiffness components. Philos. Mag. Lett. 86(2), 99–112 (2006)CrossRefGoogle Scholar
  254. 254.
    Y.-C. Wang, J.G. Swadener, R.S. Lakes, Anomalies in stiffness and damping of a 2D discrete viscoelastic system due to negative stiffness components. Thin Solid Films 515(6), 3171–3178 (2007)CrossRefGoogle Scholar
  255. 255.
    Z.G. Wang, C.K. Kim, P. Martin, A.D. Mehta, A.J. Hudspeth, Negative hair-bundle stiffness betrays a mechanism for mechanical amplification by the hair cell. Proc. Natl. Acad. Sci. U.S.A 97, 12026–12031 (2000)CrossRefGoogle Scholar
  256. 256.
    R.S. Webber, K.L. Alderson, K.E. Evans, A novel fabrication route for auxetic polyethylene, part 2: mechanical properties. Polym. Eng. Sci. 48(7), 1351–1358 (2008). Scholar
  257. 257.
    G. Wei, S.F. Edwards, Poisson ratio in composites of auxetics. Phys. Rev. E 58(5), 6173–6181 (1998)CrossRefGoogle Scholar
  258. 258.
    G. Wei, S.F. Edwards, Effective elastic properties of composites of ellipsoids (I). Nearly spherical inclusions. Phys. A 264, 388–403 (1999)CrossRefGoogle Scholar
  259. 259.
    G. Wei, S.F. Edwards, Effective elastic properties of composites of ellipsoids (II). Nearly disk and needle-like inclusions. Phys. A 264, 404–423 (1999)CrossRefGoogle Scholar
  260. 260.
    J.J. Williams, C.W. Smith, K.E. Evans, Z.A.D. Lethbridge, R.I. Walton, An analytical model for producing negative Poisson’s ratios and its application in explaining off-axis elastic properties of the NAT-type zeolites. Acta Mater. 55, 5697–5707 (2007)CrossRefGoogle Scholar
  261. 261.
    K. Wohlhart, Double-chain mechanisms, in IUTAM-IASS Symposium on Deployable Structures: Theory and Applications (2000), pp. 457–466Google Scholar
  262. 262.
    K.W. Wojciechowski, Two-dimensional isotropic system with a negative Poisson’s ratio. Phys. Lett. A 137, 60–64 (1989)CrossRefGoogle Scholar
  263. 263.
    K.W. Wojciechowski, Non-chiral, molecular model of negative Poisson ratio in two dimensions. J. Phys. A: Math. Gen. 36, 11765–11778 (2003)CrossRefGoogle Scholar
  264. 264.
    K.W. Wojciechowski, A.C. Brańka, Elastic moduli of a perfect hard disc crystal in two dimensions. Phys. Lett. A 134, 314–318 (1988)CrossRefGoogle Scholar
  265. 265.
    K.W. Wojciechowski, A.C. Brańka, Negative Poisson’s ratio in a two-dimensional “isotropic” solid. Phys. Rev. A 40, 7222–7225 (1989)CrossRefGoogle Scholar
  266. 266.
    K.W. Wojciechowski, K.V. Tretiakov, M. Kowalik, Elastic properties of dense solid phases of hard cyclic pentamers and heptamers in two dimensions. Phys. Rev. E 67, 036121 (2003)CrossRefGoogle Scholar
  267. 267.
    J.R. Wright, M.R. Sloan, K.E. Evans, Tensile properties of helical auxetic structures: a numerical study. J. Appl. Phys. 108, 044905 (2010)CrossRefGoogle Scholar
  268. 268.
    W. Yang, Z.-M. Li, W. Shi, B.-H. Xie, M.-B. Yang, Review on auxetic materials. J. Mater. Sci. 39(10), 3269–3279 (2004)CrossRefGoogle Scholar
  269. 269.
    Z. Yang, H.M. Dai, N.H. Chan, G.C. Ma, P. Sheng, Acoustic metamaterial panels for sound attenuation in the 50–1000 Hz regime. Appl. Phys. Lett. 96, 041906 (2010)CrossRefGoogle Scholar
  270. 270.
    Y.T. Yao, A. Alderson, K.L. Alderson, Can nanotubes display auxetic behaviour? Phys. Status Solidi B 245, 2373–2382 (2008)CrossRefGoogle Scholar
  271. 271.
    H.W. Yap, R.S. Lakes, R.W. Carpick, Mechanical instabilities of individual multiwalled carbon nanotubes under cyclic axial compression. Nano Lett. 7(5), 1149–1154 (2007)CrossRefGoogle Scholar
  272. 272.
    H.W. Yap, R.S. Lakes, R.W. Carpick, Negative stiffness and enhanced damping of individual multiwalled carbon nanotubes. Phys. Rev. B 77, 045423 (2008)CrossRefGoogle Scholar
  273. 273.
    A. Yeganeh-Haeri, D.J. Weidner, J.B. Parise, Elasticity of α-cristobalite: a silicon dioxide with a negative Poisson’s ratio. Science 257, 650–652 (1992)CrossRefGoogle Scholar
  274. 274.
    V.Y. Zaitsev, A.V. Radostin, E. Pasternak, A.V. Dyskin, Extracting real-crack properties from nonlinear elastic behavior of rocks: abundance of cracks with dominating normal compliance and rocks with negative Poisson’s ratio. Nonlinear Process. Geophys. (NPG) 24, 543–551 (2017)CrossRefGoogle Scholar
  275. 275.
    V.Y. Zaitsev, A.V. Radostin, E. Pasternak, A.V. Dyskin, Extracting properties of crack-like defects from pressure dependences of elastic-wave velocities using an effective-medium model with decoupled shear and normal compliances of cracks. Int. J. Rock Mech. Min. Sci. 97, 122–133 (2017)CrossRefGoogle Scholar
  276. 276.
    R. Zhang, H.-L. Yeh, H.-Y. Yeh, A discussion of negative Poisson’s ratio design for composites. J. Reinf. Plast. Compos. 18, 1546–1556 (1999)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe University of Western AustraliaPerthAustralia
  2. 2.Department of Civil, Environment and Mining EngineeringThe University of Western AustraliaPerthAustralia

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