Failure of classical elasticity in auxetic foams

A recent derivation [P.H. Mott and C.M. Roland, Phys. Rev. B 80, 132104 (2009).] of the bounds on Poisson's ratio, v, for linearly elastic materials showed that the conventional lower limit, -1, is wrong, and that v cannot be less than 0.2 for classical elasticity to be valid. This is a significant result, since it is precisely for materials having small values of v that direct measurements are not feasible, so that v must be calculated from other elastic constants. Herein we measure directly Poisson's ratio for four materials, two for which the more restrictive bounds on v apply, and two having values below this limit of 0.2. We find that while the measured v for the former are equivalent to values calculated from the shear and tensile moduli, for two auxetic materials (v<0), the equations of classical elasticity give inaccurate values of v. This is experimental corroboration that the correct lower limit on Poisson's ratio is 0.2 in order for classical elasticity to apply.

(2), together with the requirement that these moduli are finite and positive, yields the wellknown "classical" bounds on Poisson's ratio for isotropic materials The left-hand-side of eq. (4) has the virtue of corresponding to data for real materials. This result was derived mathematically, using only the assumptions of classical elasticity. The implication is that when experiments yield values of Poisson's ratio smaller than 0.2, the equations of classical elasticity do not apply. This is not a trivial result, because the materials with  < 0.2 tend to be very hard (e.g., diamond [9], beryllium [10], and fused quartz [11]). This makes direct measurements difficult, whereby recourse is often made to using two elastic constants to calculate others. Given eq. (4), the fact that such calculations employ classical elasticity would make them invalid.
The purpose of this paper is to experimentally assess the applicability of classical elasticity whenever  < 0.2. We carry out elastic measurements on foams exhibiting isotropic auxeticity, originating in de-buckling of cell ribs which leads to large changes in transverse dimensions during longitudinal elongation [8,12]. We compare Poission's ratios measured directly on the foams,  exp , with values calculated from the classical expression where E is Young's modulus. Although these foams have negative , and thus deviate from eq.
(4), they comply with eq. (3); thus, a comparison of  exp and  calc provides an experimental test of the proposition that relations such as eq. (5) are erroneous whenever  < 0.2.
We find herein that measurements of the dimensional changes during elongation yield values of  that are significantly smaller than those calculated from eq. (5). The failure of this equation of classical elasticity confirms the prediction that classical elasticity is valid only when  > 0.2.

II. Experimental
The polyurethane ( where  e and  e are the respective engineering stress and strain.
The shear modulus was measured with a sandwich configuration, also using the Instron at a shear strain rate equal to 0.002 s -1 ; test samples were 50 mm long × 4 mm wide × 4 mm thick.
To verify these measurements, G was also determined using a torsion geometry on ring specimens (25.4 mm outer diameter and 11.7 mm inner diameter) with an ARES rheometer operating at the same low shear rate. The shear modulus is given by where  and f are the displacement, and force, respectively, and l, w, and t are the respective sample length, width, and thickness. The shear strain  =  / t.

Direct determination of elastic constants.
PU1 is transversely isotropic, having a modulus 60% higher in the third dimension.
Measurements were of the displacement of fiducial marks lying in the symmetric plane. The auxetic foams behave isotropically up to at least 5% strain. The deformation mechanism of the auxetic foams involves de-buckling of the cell ribs, which causes their modulus to be lower than that of the precursor material [8,12]. Figure 1 shows Poisson's ratio measured directly for the polyisoprene and the three foams.
The uncertainty in the data arises mainly from our ability to resolve the fiducial images, although for the foams inhomogeneity of the inherent cell structure may also contribute. The polyisoprene has a homogeneous structure, and typically elastomers have Poisson's ratio within the range 0.49 and 0.5 [13]; that is, near the upper bound on  in eqs.
(3) and (4). We find no systematic variation for the polyisoprene over our range of strain measurements, obtaining  exp = 0.496. The foams all show  exp that increases over the range of strains (ca. 1 -5% elongation). For the PU1 measured in the isotropic plane,  exp increases about 10% with strain, and linear extrapolation to zero strain gives  exp = 0.20. This is at the lower limit of the more restrictive range of eq. (4).
Over this same range of strains the auxetic foams show an increase in  exp of about 20%. We extrapolate to zero strain by a linear fit to the data, obtaining  exp = 0.70 and 0.65 for PU2 and PU3, respectively. The more compressed foam has a smaller (absolute value) of Poisson's ratio.
Similar results were reported previously for polyurethane foams [12]. For both auxetic materials herein Poisson's ratio is within the conventional limits of classical elasticity (eq. (3)), but beyond the more restrictive range (eq. (4)) posited in refs. [4,5]. All values of Poisson ratios determined by direct measurement of longitudinal and transverse deformations,  exp , are tabulated in Table 1, along with the uncertainties. tensile strain for all materials. Also shown are the shear moduli, which showed some dependence on strain. The shear moduli measured by torsional rheometry were consistent with these data.
Regression yields the zero strain values given in Table 1. Note that PU3, which has the greater volume compression, has a larger shear modulus than PU2. Since the mechanical response involves de-buckling of the foam, there is no certainty that the measured behaviour extrapolates smoothly to zero strain. Thus, the limit of error on G for the foams (Table 1) is taken as the difference between the value determined by extrapolation to zero strain and the value measured for the lowest strain.

Poisson's ratio calculated from elasticity equations
Poisson's ratio was calculated from the equation of classical elasticity (eq. (5)), using the values determined for the shear and Young's moduli. For polyisoprene and PU0, whose  exp fall at the upper and lower bounds of the more restrictive range (eq. (4)), the difference between the calculated and experimental values is less than 1%; that is, the agreement is well within the experimental uncertainties. This is expected, and does not prove the superiority of eq. (4) over eq. (3), but rather affirms the validity of our experimental methods.
The situation is different for the two auxetic foams. For both  exp < 0.2, so that equations such as eq.(5) are inaccurate if eq. (3) represents the range of validity of classical elasticity. We find for both PU2 and PU3, v calc underestimates the measured Poisson's ratio, by an amount (about 10%) that exceeds the experimental uncertainties (which are less than 3%).

IV. Summary
Classical elasticity applies to small deformations for which the mechanical response is linear (e.g., strain energy quadratic in the strain), and the behaviour is elastic     Table 1.