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New principles for auxetic periodic design. (English) Zbl 1372.52028

Summary: We show that, for any given dimension \(d\geq 2\), the range of distinct possible designs for periodic frameworks with auxetic capabilities is infinite. We rely on a purely geometric approach to auxetic trajectories developed within our general theory of deformations of periodic frameworks.

MSC:

52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
74N10 Displacive transformations in solids
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References:

[1] K. Bertoldi, P.M. Reis, S. Willshaw, and T. Mullin, {\it Negative Poisson’s ratio behavior induced by an elastic instability}, Adv. Mater., 22 (2010), pp. 361-366.
[2] C. S. Borcea and I. Streinu, {\it Periodic frameworks and flexibility}, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), pp. 2633-2649, . · Zbl 1211.82053
[3] C. S. Borcea and I. Streinu, {\it Kinematics of expansive planar periodic mechanisms}, in Advances in Robot Kinematics, J. Lenarcic and O. Khatib, eds., Springer-Verlag, New York, 2014, pp. 395-408.
[4] C. S. Borcea and I. Streinu, {\it Expansive periodic mechanisms}, in IMA Conference on Mathematics of Robotics, 2015; preprint, , 2015. · Zbl 1319.52030
[5] C. S. Borcea and I. Streinu, {\it Liftings and stresses for planar periodic frameworks}, Discrete Comput. Geom., 53 (2015), pp. 747-782. · Zbl 1319.52030
[6] C. S. Borcea and I. Streinu, {\it Geometric auxetics}, Proc. A, 471 (2015), 20150033, . · Zbl 1371.82115
[7] L. Cabras and M. Brun, {\it Auxetic two-dimensional lattices with Poisson’s ratio arbitrarily close to \(-1\)}, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140538, . · Zbl 1371.74074
[8] J. C. A. Elipe and A. D. Lantada, {\it Comparative study of auxetic geometries by means of computer-aided design and engineering}, Smart Materials Structures, 21 (2012), 105004.
[9] K. E. Evans, M. A. Nkansah, I. J. Hutchinson, and S. C. Rogers, {\it Molecular network design}, Nature, 353 (1991), pp. 124-125.
[10] G. N. Greaves, {\it Poisson’s ratio over two centuries: Challenging hypotheses}, Notes Records Roy. Soc. London, 67 (2013), pp. 37-58.
[11] G. N. Greaves, A. I. Greer, R. Lakes, and T. Rouxel, {\it Poisson’s ratio and modern materials}, Nature Materials, 10 (2011), pp. 823-837.
[12] J. Grima, A. Alderson, and K. E. Evans, {\it Auxetic behavior from rotating rigid units}, Phys. Status Solidi B, 242 (2005), pp. 561-575.
[13] R. Lakes, {\it Foam structures with a negative Poisson’s ratio}, Science, 235 (1987), pp. 1038-1040.
[14] J.-H. Lee, J. P. Singer, and E. L. Thomas, {\it Micro-/nanostructured mechanical metamaterials}, Adv. Materials, 24 (2012), pp. 4782-4810.
[15] G. W. Milton, {\it Composite materials with Poisson’s ratios close to -1}, J. Mech. Phys. Solids, 40 (1992), pp. 1105-1137. · Zbl 0780.73047
[16] H. Mitschke, V. Robins, K. Mecke, and G. E. Schroeder-Turk, {\it Finite auxetic deformations of plane tessellations}, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20120465. · Zbl 1371.74115
[17] H. Mitschke, F. Schury, K. Mecke, F. Wein, M. Stingl, and G. E. Schröder-Turk, {\it Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids}, Internat. J. Solids Structures, 100-101 (2016), pp. 1-10.
[18] P. M. Reis, H. M. Jaeger, and M. van Hecke, {\it Designer matter: A perspective}, Extreme Mech. Lett., 5 (2015), pp. 25-29.
[19] H. Tanaka, {\it Bi-stiffness property of motion structures transformed into square cells}, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20130063, .
[20] F. Wang, O. Sigmund, and J. S. Jensen, {\it Design of materials with prescribed nonlinear properties}, J. Mech. Phys. Solids, 69 (2014), pp. 156-174.
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