Poisson’s ratio for an anisotropic linear elastic material depends on two orthogonal directions n and m. Materials with negative Poisson’s ratios for all (n,m) pairs are called completely auxetic while those with positive Poisson’s ratios for all (n,m) pairs are called nonauxetic. Simple necessary and sufficient conditions on elastic compliances are derived to identify if any given material of cubic or hexagonal symmetry is completely auxetic or nonauxetic. When these conditions are not satisfied, the medium is auxetic for some (n,m) pairs. Several simple necessary conditions for completely auxetic or nonauxetic media are derived for a general anisotropic elastic material.

1.
Baughman
,
R. H.
,
Shacklette
,
J.
,
Zakhidov
,
A. A.
, and
Stafström
,
S.
, 1998, “
Negative Poisson’s Ratios as a Common Feature of Cubic Metals
,”
Nature (London)
0028-0836,
392
, pp.
362
365
.
2.
Baughman
,
R. H.
, 2003, “
Avoiding the Shrink
,”
Nature (London)
0028-0836,
425
, p.
667
.
3.
Lakes
,
R. S.
, 1993, “
Advances in Negative Poisson’s Ratio Materials
,”
Adv. Mater. (Weinheim, Ger.)
0935-9648,
5
, pp.
293
296
.
4.
Ting
,
T. C. T.
, and
Chen
,
T.
, 2005, “
Poisson’s Ratio for Anisotropic Elastic Materials can have no Bounds
,”
Q. J. Mech. Appl. Math.
0033-5614,
58
(
1
), pp.
73
82
.
5.
Ting
,
T. C. T.
, 2004, “
Very Large Poisson’s Ratio With a Bounded Transverse Strain in Anisotropic Elastic Materials
,”
J. Elast.
0374-3535,
77
(
2
), pp.
163
176
.
6.
Voigt
,
W.
, 1910,
Lehrbuch der Kristallphysik
,
Teubner
, Leipzig.
7.
Lekhnitskii
,
S. G.
, 1963,
Theory of Elasticity of an Anisotropic Body
,
Holden-Day
, San Francisco.
8.
Ting
,
T. C. T.
, 1996,
Anisotropic Elasticity: Theory and Applications
,
Oxford University Press
, New York.
9.
Simmons
,
G.
, 1961, “
Single Crystal Elastic Constants and Calculated Aggregate Properties
,”
Journal of the Graduate Research Center
, Vol.
XXXIV
, March 1961, Vols.
1
and
2
,
Southern Methodist University Press
, Dallas.
10.
Nye
,
J. F.
,
Physical Properties of Crystals: Their Representation by Tensors and Matrices
,
Clarendon Press
, Oxford.
11.
Sirotin
,
Yu. I.
, and
Shakol’skaya
,
M. P.
, 1982,
Fundamentals of Crystal Physics
,
MIR Publisher
, Moscow.
12.
Hayes
,
M.
, and
Shuvalov
,
A.
, 1998, “
On the Extreme Values of Young’s Modulus, the Shear Modulus, and Poisson’s Ratio for Cubic Materials
,”
ASME J. Appl. Mech.
0021-8936,
65
, pp.
786
787
.
13.
Landolt
,
H. H.
, and
Börnstein
,
R.
, 1979,
Numerical Data and Functional Relationships in Science and Technology
,
New Series
, editor-in-chief
K. H.
Hellwege
, Group III:
Crystal and Solid State Physics
, Vol.
11
, Revised and Extended Version of Vols. III∕1 and III∕2,
Elastic, Piezoelectric, Pyroelectric, Piezooptic, Electrooptic Constants and Non-Linear Dielectric Susceptibilities of Crystals
,
Springer-Verlag
, Berlin: Heidelberg, New York.
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