# Poisson's ratio in linear isotropic classical elasticity

### Negative Poisson mainMeaning of Poisson's ratio

Summary
We consider Poisson's ratio for a linear isotropic elastic homogeneous solid. Effects of introducing heterogeneity or anisotropy or nonlinearity or viscoelasticity upon Poisson's ratio are presented. Consistency studies to evaluate isotropic elasticity are reviewed.

Range for Poisson's ratio
The theory of isotropic linear elasticity (Sokolnikoff, 1983) allows Poisson's ratios in the range from -1 to 1/2 for an object with free surfaces with no constraint. This range is derived from concepts of stability. For an object without constraint to be stable, the elastic moduli (measures of material stiffness) must be positive. Poisson's ratio is interrelated with the moduli; the usual interrelations depend on the assumption of isotropy, linearity and elasticity. A positive bulk modulus implies Poisson's ratio greater than -1. A positive shear modulus implies Poisson's ratio less than 1/2. The allowable range is populated by various materials; see pages on Negative Poisson's ratio and Poisson's ratio introduction.

Material heterogeneity
The original negative Poisson's ratio foams (Lakes, 1987) had a coarse structure size, with cells about a millimeter across. Microstructure size is not pertinent to the effect, only the shape. Poisson's ratio is a continuum concept. The classical theory of elasticity has no length scale. Generalized continuum theories such as Cosserat elasticity account for effects of structural heterogeneity; such theories do have a characteristic length scale, but in isotropic Cosserat elasticity the range for Poisson's ratio is the same as that for classical elasticity (e. g. Eringen, 1968). Negative Poisson's ratio is not due to Cosserat elasticity (Lakes (1987b)). Negative Poisson's ratio is attainable in classical elasticity and does not require the characteristic length scale present in Cosserat or micropolar elasticity. The range of Poisson's ratio allowed by energy considerations of stability is the same in Cosserat elasticity as in classical elasticity: -1 to 0.5 for isotropic solids.
As for experimental examples of small structure size, negative Poisson's ratio foams have been made using microcellular foams with small cells. Negative Poisson's ratio has been observed in isotropic polycrystalline materials (Dong et al., 2010, McKnight et al. 2008) and in gels (Hirotsu, 1990) undergoing phase transformation as reviewed by Greaves et al. (2011); Li et al. (2012) observed negative Poisson's ratio over a range of temperature in alloy near a morphotropic phase transition. The pertinent heterogeneity size is on the atomic or molecular scale. All physical solids have such heterogeneity.

Anisotropy
In anisotropic solids, the allowable range of Poisson's ratio is expanded: values from minus infinity to plus infinity are possible. For orthotropic symmetry the Poisson's ratio is |ν21| < (E22 / E11)1/2 with E as Young's modulus, so if there is a large ratio in modulus with direction, Poisson's ratio of large magnitude is possible (Christensen 1979). Large magnitudes of Poisson's ratio occur in oriented honeycomb (Gibson and Ashby, 1997) and are also known in single crystals.

Nonlinearity
If strain is sufficiently large, material properties such as moduli and Poisson's ratio can depend on strain. Poisson's ratio should then be described as a function (Beatty and Stalnaker, 1986). The definition of Poisson's ratio as a material constant is valid for small strain; one can choose to apply a small strain to a flexible material such as rubber or polymer foam. For large strain, there is a strain dependence even in materials such as rubber.

Viscoelasticity
In viscoelastic materials, the Poisson's ratio is not a material constant but can depend upon time (Lakes, 1992; Lakes and Wineman, 2006). The viscoelastic Poisson's ratio can increase or decrease with time, can change sign with time, and it need not be monotonic with time. Viscoelasticity does not expand or constrict the range of Poisson's ratio.

Verification: how do we know if it is isotropic?
Perhaps the most elegant way to demonstrate a material is linearly isotropic elastic is via resonant ultrasound spectroscopy (RUS). In RUS a compact specimen such as a cube, short cylinder, or rectangular prism is subject to vibration at low amplitude (linear range) by ultrasonic transducers. The vibration mode structure is determined and the elastic constants inferred. The mode structure allows one to distinguish isotropic from anisotropic materials and to determine the degree of anisotropy if any. Torsion modes are sensitive to the shear modulus, bending modes to Young's modulus, breathing modes primarily sensitive to bulk modulus, and axial modes to Poisson's ratio and Young's modulus. This sensitivity allows consistency checks. Such experiments have been done by Demarest (1971) on fused quartz with a Poisson's ratio of 0.17 and by McKnight et al. (2008) on polycrystalline quartz in which Poisson's ratio varies from positive (+0.25) to negative values (-0.3) during a phase transformation.
It is also possible to test for isotropy via wave ultrasound. Measure the propagation velocity of longitudinal waves in each orthogonal principal direction and also in at least one oblique direction. If the material is isotropic, the velocity in each direction will be equal. From the velocity one can calculate the constrained C modulus. To obtain the shear modulus, transmit shear waves in various directions as was done for the longitudinal waves.

Tuning of Poisson's ratio
Tuning of Poisson's ratio may be done via temperature, pressure, or composition in materials for which the Poisson's ratio varies via a phase transformation. Tuning may also be achieved in foam or honeycomb by preparing materials with different cell geometry.

References
Demarest, H. H. Jr., "Cube resonance method to determine the elastic constants of solids", J. Acoust. Soc. Am. 49, 768-775 (1971).
Dong, L. Stone, D. S., Lakes, R. S., "Softening of bulk modulus and negative Poisson's ratio in barium titanate ceramic near the Curie point", Philosophical Magazine Lett. 90, 23-33, (2010). Get pdf.
Eringen, A.C., "Theory of micropolar elasticity", In Fracture Vol. 1, 621-729 (1968).
Gibson L.J. and Ashby, M., Cellular solids, 2nd ed. Cambridge, (1997).
Greaves, G. N., Greer, A. L., Lakes, R. S., and Rouxel, T., "Poisson's Ratio and Modern Materials", Nature Materials, 10, 823-837 (2011). get pdf.
Hirotsu,S., "Elastic anomaly near the critical point of volume phase transition in polymer gels", Macromolecules, 23, 903-905, (1990).
Lakes, R. S.,"Foam structures with a negative Poisson's ratio", Science, 235 1038-1040 (1987). Get pdf
Lakes, R. S., "Negative Poisson's ratio materials", Science, 238, 551 (1987b). Get pdf or gif
Lakes, R. S., "The time dependent Poisson's ratio of viscoelastic cellular materials can increase or decrease", Cellular Polymers, 11, 466-469, (1992). pdf
Lakes, R. S. and Wineman, A., "On Poisson's ratio in linearly viscoelastic solids", Journal of Elasticity, 85, 45-63 (2006). get pdf
Li, D., Jaglinski, T. M., Stone, D. S. and Lakes, R. S., "Temperature insensitive negative Poisson's ratios in isotropic alloys near a morphotropic phase boundary, Appl. Phys. Lett, 101, 251903, (2012). get pdf
McKnight, R. E., Moxon, A. T., Buckley, A., Taylor, P. A., Darling, T. W. and Carpenter, M. A., "Grain size dependence of elastic anomalies accompanying the alpha-beta phase transition in polycrystalline quartz", J. Phys. Cond. Mat. 20, 075229 (2008).
Sokolnikoff, S., Mathematical theory of elasticity. Krieger, Malabar FL, second edition, (1983).
Beatty, M. F., Stalnaker, D. O., "The Poisson function of finite elasticity", J. Appl. Mech. 53 807-13 (1986).

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