Poisson's ratio in cubic materials

Andrew N Norris

Abstract

Expressions are given for the maximum and minimum values of Poisson's ratio ν for materials with cubic symmetry. Values less than −1 occur if and only if the maximum shear modulus is associated with the cube axis and is at least 25 times the value of the minimum shear modulus. Large values of Embedded Image occur in directions at which the Young modulus is approximately equal to one half of its 111 value. Such directions, by their nature, are very close to 111. Application to data for cubic crystals indicates that certain Indium Thallium alloys simultaneously exhibit Poisson's ratio less than −1 and greater than +2.

Keywords:

1. Introduction

The Poisson's ratio ν is an important physical quantity in the mechanics of solids, arguably second only in significance to the Young modulus. It is strictly bounded between −1 and 1/2 in isotropic solids, but no such simple bounds exist for anisotropic solids, even for those closest to isotropy in material symmetry: cubic materials. In fact, Ting & Chen (2005) demonstrated that arbitrarily large positive and negative values of Poisson's ratio could occur in solids with cubic material symmetry. The key requirement is that the Young modulus in the 111-direction is very large (relative to other directions), and as a consequence the Poisson's ratio for stretch close to but not coincident with the 111-direction can be large, positive or negative. Ting & Chen's result replaces conventional wisdom (e.g. Baughman et al. 1998) that the extreme values of ν are associated with stretch along the face diagonal (110-direction). Boulanger & Hayes (1998) showed that arbitrarily large values of Embedded Image are possible in materials of orthorhombic symmetry. Both pairs of authors analytically constructed sets of elastic moduli, which show the unusual properties while still physically admissible. The dependence of the large values of Poisson's ratio on elastic moduli and the related scalings of strain are discussed by Ting (2004) for cubic and more anisotropic materials.

To date there is no anisotropic elastic symmetry for which there are analytic expressions of the extreme values of Poisson's ratio for all materials in the symmetry class, although bounds may be obtained for some specific pairs of directions for certain material symmetries. For instance, Lempriere (1968) considered Poisson's ratios for stretch and transverse strain along the principal directions, and showed that it is bounded by the square root of the ratio of principal Young's moduli, Embedded Image (in the notation defined below). Gunton & Saunders (1975) performed some numerical searches for the extreme values of ν in materials of cubic symmetry. However, the larger question of what limits on ν exist for all possible pairs of directions remains open, in general. This paper provides an answer for materials of cubic symmetry. Explicit formulae are obtained for the minimum and maximum values of ν which allow us to examine the occurrence of the unusually large values of Poisson's ratio and the conditions under which they appear. Conversely, we can also define the range of material parameters for which the extreme values are of ‘standard’ form, i.e. associated with principal pairs of directions such as Embedded Image for stretch and measurement along the two face diagonals. For instance, we will see that a necessary condition that one or more of the extreme values of Poisson's ratio is not associated with a principal direction is that Embedded Image must be less than −1/2. The general results are also illustrated by application to a wide variety of cubic materials, and it will be shown that values of Embedded Image and ν>2 are possible for certain stretch directions in existing solids.

We begin in §2 with definitions of moduli and some preliminary results. An important identity is presented which enables us to obtain the extreme values of both the shear modulus and Poisson's ratio for a given choice of the extensional direction. Section 3 considers the central problem of obtaining extreme values of ν for all possible pairs of orthogonal directions. The solution requires several new quantities, such as the values of ν associated with principal direction pairs. Section 4 describes the range of possible elastic parameters consistent with positive definite strain energy. The explicit formulae, the global extrema, are presented and their overall properties are discussed in §5. It is shown that certain Indium Thallium alloys simultaneously display values of ν below −1 and above +2.

2. Definitions and preliminary results

The fourth order tensors of compliance and stiffness for a cubic material, Embedded Image and Embedded Image, may be written (Walpole 1984) in terms of three moduli κ, Embedded Image and Embedded Image,Embedded Image(2.1)Here, Embedded Image is the fourth order identity, Embedded Image, andEmbedded Image(2.2)The isotropic tensor Embedded Image and the tensors of cubic symmetry Embedded Image and Embedded Image are positive definite (Walpole 1984), so the requirement of positive strain energy is that κ, Embedded Image and Embedded Image are positive. These three parameters, called the ‘principal elasticities’ by Kelvin (Thomson 1856), can be related to the standard Voigt stiffness notation: Embedded Image, Embedded Image and Embedded Image. Alternatively, Embedded Image, Embedded Image and Embedded Image in terms of the compliance.

Vectors, which are usually unit vectors, are denoted by lowercase boldface, e.g. Embedded Image. The triad Embedded Image represents an arbitrary orthonormal set of vectors. Directions are also described using crystallographic notation, e.g. Embedded Image is the unit vector Embedded Image. The summation convention on repeated indices is assumed.

(a) Engineering moduli

The Young modulus Embedded Image sometimes written Embedded Image, shear modulus Embedded Image and Poisson's ratio Embedded Image are (Hayes 1972)Embedded Image(2.3)where Embedded Image, Embedded Image and Embedded Image. Thus, Embedded Image and Embedded Image are defined by the axial and orthogonal strains in the Embedded Image- and Embedded Image-directions, respectively, for a uniaxial stress in the Embedded Image-direction. E and G are positive, while ν can be of either sign or zero. A material for which ν<0 is called auxetic, a term apparently introduced by K. Evans in 1991. Gunton & Saunders (1975) provide an earlier informative historical perspective on Poisson's ratio. Love (1944) reported a Poisson's ratio of ‘nearly −1/7’ in Pyrite, a cubic crystalline material.

The tensors Embedded Image and Embedded Image are isotropic, and consequently the directional dependence of the engineering quantities is through Embedded Image. Thus,Embedded Image(2.4)Embedded Image(2.5)Embedded Image(2.6)whereEmbedded Image(2.7)We note for future reference the relationsEmbedded Image(2.8)

(b) General properties of E, G and related moduli

Although interested primarily in the Poisson's ratio, we first discuss some general results for E, G and related quantities in cubic materials: the area modulus A, and the traction-associated bulk modulus K, defined below. The extreme values of E and G follow from the fact that Embedded Image and Embedded Image (Walpole 1986; Hayes & Shuvalov 1998). Thus, Embedded Image, Embedded Image and Embedded Image for Embedded Image, with the values reversed for Embedded Image (Hayes & Shuvalov 1998). As noted by Hayes & Shuvalov (1998), the difference in extreme values of E and G are related byEmbedded Image(2.9)The extreme values also satisfyEmbedded Image(2.10)The shear modulus G achieves both minimum and maximum values if Embedded Image is directed along face diagonals, that is, Embedded Image for Embedded Image.

The area modulus of elasticity Embedded Image for the plane orthogonal to Embedded Image is the ratio of an equibiaxial stress to the relative area change in the plane in which the stress acts (Scott 2000). Thus, Embedded Image. Using the equations above it may be shown that, for a cubic material,Embedded Image(2.11)The averaged Poisson's ratio Embedded Image is defined as the average over Embedded Image in the orthogonal plane, or Embedded Image. The following result, apparently first obtained by Sirotin & Shaskol'skaya (1982), follows from the relations (2.8),Embedded Image(2.12)Equation (2.12) indicates that the extrema of Embedded Image and Embedded Image coincide. The traction-associated bulk modulus Embedded Image, introduced by He (2004), relates the uniaxial stress in the Embedded Image-direction to the relative change in volume in anisotropic materials. It is defined by Embedded Image, and for cubic materials is simply Embedded Image. It is interesting to note that the relations (2.10)–(2.12) have the same form as for isotropic materials, for which E, G, ν, A and K are constants. Equations (2.4)–(2.6) imply other identities, e.g. that the combination Embedded Image is constant.

Further discussion of the extremal properties of G and ν requires knowledge of how they vary with Embedded Image for given Embedded Image, and in particular, the extreme values as a function of Embedded Image for arbitrary Embedded Image, considered in §2c. Note that, Embedded Image and Embedded Image are the only directions for which Embedded Image and Embedded Image are independent of Embedded Image. It will become evident that Embedded Image is a critical direction, and we therefore rewrite E and ν in forms emphasizing this direction:Embedded Image(2.13)where Embedded Image, Embedded Image and Χ (Hayes & Shuvalov 1998) areEmbedded Image(2.14)Both Embedded Image and Embedded Image are independent of Embedded Image. The fact that Embedded Image with equality for Embedded Image implies that this is the only stretch direction for which E, and hence ν, are independent of Embedded Image. Equations (2.13) indicate that Embedded Image and Embedded Image depend on Embedded Image at any point in the neighbourhood of 111, with particularly strong dependence if Embedded Image is small. This singular behaviour is the reason for the extraordinary values of ν discovered by Ting & Chen (2005) and will be discussed at further length below after we have determined the global extrema for ν.

(c) Extreme values of G and ν for fixed Embedded Image

For a given Embedded Image, consider the defined vectorEmbedded Image(2.15)with ρ chosen to make Embedded Image a unit vector. Requiring Embedded Image implies that Embedded Image is orthogonal to Embedded Image ifEmbedded Image(2.16)i.e. if λ is a root of the quadraticEmbedded Image(2.17)It is shown in appendix A that the extreme values of Embedded Image for fixed Embedded Image coincide with these roots, which are non-negative, and that the corresponding unit Embedded Image vectors provide the extremal lateral directions. The basic result is described next.

(i) A fundamental result

Let Embedded Image be the roots of (2.17) and Embedded Image the associated vectors from (2.15), i.e.Embedded Image(2.18a)Embedded Image(2.18b)Embedded Image(2.18c)The extreme values of D for a given Embedded Image are Embedded Image associated with the orthonormal triad Embedded Image, i.e.Embedded Image(2.19)The extreme values of G and ν for fixed Embedded Image follow from equations (2.5) and (2.6).

The above result also implies that the extent of the variation of the shear modulus and the Poisson's ratio for a given stretch direction Embedded Image areEmbedded Image(2.20a)Embedded Image(2.20b)where Embedded Image is, see figure 1,Embedded Image(2.21)

Figure 1

The function H of equation (2.21) plotted versus Embedded Image and Embedded Image for the region of solid angle depicted in figure 2. Vertices Embedded Image, 110 and 001 are indicated. H vanishes at 111 and 001 and is positive elsewhere, with maximum of 1/4 along Embedded Image (face diagonals).

3. Poisson's ratio

We now consider the global extrema of Embedded Image over all directions Embedded Image and Embedded Image. Two methods are used to derive the main results. The first uses general equations for a stationary value of ν in anisotropic media to obtain a single equation which must be satisfied if the stationary value lies in the interior of the triangle in figure 2. It is shown that this condition, which is independent of material parameters, is not satisfied, and hence all stationary values of ν in cubic materials lie on the edges of the triangle. This simplifies the problem considerably, and permits us to deduce explicit relations for the stationary values. The second method, described in appendix B, confirms the first approach by a comprehensive numerical test of all possible material parameters.

Figure 2

The irreducible 1/48th of the cube surface is defined by the isosceles triangle with edges 1, 2 and 3. The vertices opposite these edges correspond to, Embedded Image, 110 and 001, respectively. Note that the edge 3′ is equivalent to 3 (which is used in appendix B).

(a) General conditions for stationary Poisson's ratio

General conditions can be derived which must be satisfied in order that Poisson's ratio is stationary in anisotropic elastic materials (Norris submitted). These areEmbedded Image(3.1)where the stretch is in the Embedded Image direction Embedded Image and Embedded Image is in the lateral direction Embedded Image. The conditions may be obtained by considering the derivative of ν with respect to rotation of the pair Embedded Image about an arbitrary axis. Setting the derivatives to zero yields the stationary conditions (3.1).

The only non-zero contributions to Embedded Image, Embedded Image, Embedded Image, Embedded Image and Embedded Image in a material of cubic symmetry come from Embedded Image. Thus, we may rewrite the conditions for stationary values of ν in terms of Embedded Image, etc. asEmbedded Image(3.2)The first is automatically satisfied by virtue of the choice of the direction-Embedded Image as either of Embedded Image. Regardless of which is chosen,Embedded Image(3.3)The final identity may be derived by first splitting each term into partial fractions and using the following (cf. appendix A):Embedded Image(3.4)

With no loss in generality, consider the specific case of Embedded Image, Embedded Image, where Embedded Image, and in either case, Embedded Image. It may be shown without much difficulty (appendix B) that Embedded Image for Embedded Image in the interior of the triangle of figure 2. It then follows that inside the triangle,Embedded Image(3.5)These identities may be obtained using partial fraction identities similar to those in equations (3.3) and (3.4). Equations (3.2)2 and (3.2)3 can be rewrittenEmbedded Image(3.6)However, using (3.5), the determinant of the matrix isEmbedded Image(3.7)which is non-zero inside the triangle of figure 2. This gives us the important result: there are no stationary values of ν inside the triangle of figure 2. Hence, the only possible stationary values are on the edges.

(b) Stationary conditions on the triangle edges

The analysis above for the three conditions (3.2) is not valid on the triangle edges in figure 2, because the quantities Embedded Image become zero and careful limits must be taken. We avoid this route by considering the conditions (3.2) afresh for Embedded Image directed along the three edges. We find, as before, that Embedded Image on the three edges, so that (3.2)1 always holds. Of the remaining two conditions, one is always satisfied, and imposing the other condition gives the answer sought.

The direction-Embedded Image can be parametrized along each edge with a single variable. Thus, Embedded Image, Embedded Image, on edge 1. Similarly, edges 2 and 3 are together covered by Embedded Image, with Embedded Image. In each case, we also need to consider the two possible values of Embedded Image, which we proceed to do, focusing on the conditions (3.2)2 and (3.2)3.

(i) Edge 1: Embedded Image, Embedded Image and Embedded Image or 001

For Embedded Image, we find that Embedded Image and Embedded Image. Hence, equation (3.2)2 is automatically satisfied, while equation (3.2)3 becomesEmbedded Image(3.8)Conversely, for Embedded Image it turns out that Embedded Image and Embedded Image. In this case, the only non-trivial equation from equations (3.2) is the second one,Embedded Image(3.9)Apart from the specific cases ν=0 or 1, equations (3.8) and (3.9) imply that stationary values of ν occur only at the end points p=0 and 1. Thus, Embedded Image, Embedded Image and Embedded Image are potential candidates for global extrema of ν.

(ii) Edges 2 and 3: Embedded Image, Embedded Image and Embedded Image

Proceeding as before, we find that Embedded Image, Embedded Image and Embedded Image. Hence, equation (3.2)3 is automatically satisfied, while equation (3.2)2 becomesEmbedded Image(3.10)The zero p=0 corresponds to Embedded Image which was considered above. Thus, all three conditions (3.2) are met if p is such thatEmbedded Image(3.11)

Further progress is made using the representation of equation (2.13) combined with the limiting values of D which can be easily evaluated. We findEmbedded Image(3.12a)Embedded Image(3.12b)Substituting for Embedded Image from equation (3.11) into (3.12) gives two coupled equations for Embedded Image and Embedded Image:Embedded Image(3.13)Eliminating Embedded Image yields a single equation for possible stationary values of Embedded Image,Embedded Image(3.14)We will return to this after considering the other possible Embedded Image vector.

(iii) Edges 2 and 3: Embedded Image

In this case Embedded Image, Embedded Image and Embedded Image. Equation (3.2)2 holds, while equation (3.2)3 is zero if p=0, which is disregarded, or if p is such thatEmbedded Image(3.15)The Young modulus is independent of Embedded Image and given by (3.12a), while ν satisfiesEmbedded Image(3.16)Using the value of Embedded Image from (3.15) in equations (3.12a) and (3.16) yields another pair of coupled equations, for Embedded Image and Embedded Image,Embedded Image(3.17)These imply a single equation for possible stationary values of Embedded Image,Embedded Image(3.18)

(c) Definition of Embedded Image and Embedded Image

The analysis for the three edges gives a total of seven candidates for global extrema: Embedded Image, Embedded Image and Embedded Image from the endpoints of edge 1, and the four roots of equations (3.14) and (3.18) along edges 2 and 3. The latter are very interesting because they are the only instances of possible extreme values associated with directions other than the principal directions of the cube (axes, face diagonals). Results below will show that five of the seven candidates are global extrema, depending on the material properties. These are Embedded Image, Embedded Image, Embedded Image and the following two distinct roots of equations (3.14) and (3.18), respectively,Embedded Image(3.19a)Embedded Image(3.19b)The quantity Embedded Image has been replaced to emphasize the dependence upon the two parameters Embedded Image and the anisotropy ratio Embedded Image. The associated directions follow from equations (3.11) and (3.15),Embedded Image(3.20a)Embedded Image(3.20b)A complete analysis is provided in appendix B. At this stage, we note that Embedded Image is identical to the minimum value of ν deduced by Ting & Chen (2005), i.e. eqns (4.13) and (4.15) of their paper, with the minus sign taken in eqn (4.13).

4. Material properties in terms of Poisson's ratios

Results for the global extrema are presented after we introduce several quantities.

(a) Non-dimensional parameters

It helps to characterize the Poisson's ratio in terms of two non-dimensional material parameters which we select as Embedded Image and Embedded Image, whereEmbedded Image(4.1)That is, Embedded Image is the axial Poisson's ratio Graphic, independent of the orthogonal direction, and Embedded Image is the non-dimensional analogue of Χ. Thus,Embedded Image(4.2)a form which shows clearly that ν is negative (positive) for all directions if Embedded Image and Embedded Image (Embedded Image and Embedded Image). These conditions for cubic materials to be completely auxetic (non-auxetic) were previously derived by Ting & Barnett (2005). The extreme values of the Poisson's ratio for a given Embedded Image areEmbedded Image(4.3)where F is defined in (2.7) and H in (2.21). Thus, Embedded Image is the minimum (maximum) and Embedded Image the maximum (minimum) if Embedded Image Embedded Image, respectively.

The Poisson's ratio is a function of the direction pair Embedded Image and the material parameter pair Embedded Image, i.e. Embedded Image. The dependence upon Embedded Image has an interesting property: for any orthonormal triad,Embedded Image(4.4)This follows from (4.2) and the identities (2.8). Result (4.4) will prove useful later.

Several particular values of Poisson's ratio have been introduced: Embedded Image, Embedded Image associated with the two directions 001 and 111 for which ν is independent of Embedded Image. These are two vertices of the triangle in figure 2. At the third vertex (Embedded Image along the face diagonals), we have Embedded Image where, in the notation of (Milstein & Huang 1979) Embedded Image and Embedded Image. Three of these four values of Poisson's ratio associated with principal directions can be global extrema, and the fourth, Embedded Image plays a central role in the definition of Embedded Image and Embedded Image of (3.19). We therefore consider them in terms of the non-dimensional parameters Embedded Image and Embedded Image,Embedded Image(4.5)We return to Embedded Image and Embedded Image later.

(b) Positive definiteness and Poisson's ratios

In order to summarize the global extrema of ν, we first need to consider the range of possible material parameters. It may be shown that the requirements for the strain energy to be positive definite: κ>0, Embedded Image and Embedded Image, can be expressed in terms of Embedded Image and Embedded Image asEmbedded Image(4.6)It will become evident that the global extrema for ν depend most simply on the two values for Embedded Image along a face diagonal: Embedded Image and Embedded Image. The constraints (4.6) becomeEmbedded Image(4.7)which define the interior of a triangle in the Embedded Image plane, see figure 3. This figure also indicates the lines Embedded Image and Embedded Image (isotropy). It may be checked that the four quantities Embedded Image are different as long Embedded Image, with the exception of Embedded Image and Embedded Image which are distinct if Embedded Image. Consideration of the four possibilities yields the orderingEmbedded Image(4.8a)Embedded Image(4.8b)Embedded Image(4.8c)Embedded Image(4.8d)Note that Embedded Image is never a maximum or minimum. We will see below that (4.8a) is the only case for which the extreme values coincide with the global extrema for ν. This is one of the reasons the classification of the extrema for ν is relatively complicated, requiring that we identify several distinct values. In particular, the global extrema depend upon more than Embedded Image and Embedded Image, but are best characterized by the two independent non-dimensional parameters Embedded Image and Embedded Image.

Figure 3

The interior of the triangle in the Embedded Image plane represents the entirety of possible cubic materials with positive definite strain energy. The vertices correspond to κ=0, Embedded Image and Embedded Image, as indicated. The edges of the triangle opposite the vertices are the limiting cases in which Embedded Image, Embedded Image and Embedded Image vanish, respectively. The dashed curves correspond to Embedded Image (vertical) and Embedded Image (diagonal) and the regions a, b, c and d defined by these lines coincide with the four cases in equation (4.8), respectively.

We are now ready to define the global extrema.

5. Minimum and maximum Poisson's ratio

Tables 1 and 2 list the values of the global minimum Embedded Image and the global maximum Embedded Image, respectively, for all possible combinations of elastic parameters. For table 1, Embedded Image, Embedded Image and Embedded Image are defined in (4.5), and Embedded Image and Embedded Image are defined in (3.19a) and (3.20a). For table 2, Embedded Image and Embedded Image are defined in (3.19b) and (3.20b). No second condition is necessary to define the region for case e, which is clear from figure 5. The data in tables 1 and 2 are illustrated in figures 4 and 5, respectively, which define the global extrema for every point in the interior of the triangle defined by (4.7). The details of the analysis and related numerical tests leading to these results are presented in appendix B.

View this table:
Table 1

The global minimum of Poisson's ratio for cubic materials.

View this table:
Table 2

The global maximum of Poisson's ratio.

Figure 4

The global minimum of Poisson's ratio based on table 1. The value of Embedded Image depends upon the location of the cubic material parameters in the four distinct regions a, b, c and d, defined by the heavy lines inside the triangle of possible materials. The diagonal dashed line delineates the region in which Embedded Image, from equation (5.1).

Figure 5

The global maximum of Poisson's ratio based on table 2. The value of Embedded Image depends upon the location of Embedded Image in five distinct regions defined by the heavy lines. The dashed line delineates the (small) region in which Embedded Image, from equation (5.1).

(a) Discussion

Conventional wisdom prior to Ting & Chen (2005) was that the extreme values were characterized by the face diagonal values Embedded Image and Embedded Image. But as equation (4.8) indicates, even these are not always extrema, since Embedded Image can be maximum or minimum under appropriate circumstances (equations (4.8c) and (4.8d), respectively). The extreme values in equation (4.8) are all bounded by the limits of the triangle in figure 3. Specifically, they limit the Poisson's ratio to lie between −1 and 2. Ting & Chen (2005) showed by explicit demonstration that this is not the case, and that values less than −1 and larger than 2 are feasible, and remarkably, no lower or upper limits exist for ν.

The Ting & Chen ‘effect’ occurs in figure 4 in the region, where Embedded Image and in figure 5 in the region Embedded Image. Using equation (3.19a), we can determine that Embedded Image is strictly less than −1 if Embedded Image. Similarly, equation (3.19b) implies that Embedded Image is strictly greater than 2 if Embedded Image. By converting these inequalities, we deduceEmbedded Image(5.1a)Embedded Image(5.1b)The two sub-regions defined by the Embedded Image, Embedded Image inequalities are depicted in figures 4 and 5. They define neighbourhoods of the Embedded Image vertex, i.e. Embedded Image, where the extreme values of ν can achieve arbitrarily large positive and negative values. The condition for Embedded Image is independent of the bulk modulus κ. Thus, the occurrence of negative values of ν less than −1 does not necessarily imply that relatively large positive values (greater than 2) also occur, but the converse is true. This is simply a consequence of the fact that the dashed region near the tip Embedded Image in figure 5 is contained entirely within the dashed region of figure 4.

These results indicate that the necessary and sufficient condition for the occurrence of large extrema for ν is that Embedded Image is much less than either Embedded Image or κ. Embedded Image is either the maximum or minimum of G, and it is associated with directions pairs along orthogonal face diagonals, Embedded Image. Hence, the Ting & Chen effect requires that this shear modulus is much less than Embedded Image, and much less than the bulk modulus κ. In the limit of very small Embedded Image, equations (3.19) give Embedded Image. Ting (2004) found that the extreme values are Embedded Image for small values of their parameter δ. In current notation, this is Embedded Image, and replacing Embedded Image the two theories are seen to agree.

The implications of small Embedded Image for Young's modulus are apparent. Thus, Embedded Image, and equation (2.13)1 indicates that Embedded Image is small everywhere except near the 111-direction, at which it reaches a sharply peaked maximum. Cazzani & Rovati (2003) provide numerical examples illustrating the directional variation of E for a range of cubic materials, some of which are considered below. Their three-dimensional plots of Embedded Image for materials with very large values of Embedded Image (see table 3 below) look like very sharp starfish. Although, the directions at which Embedded Image and Embedded Image are large in magnitude are close to the 111-direction, the value of E in the stationary directions can be quite different from Embedded Image. The precise values of the Young modulus, Embedded Image and Embedded Image at the associated stretch directions are given byEmbedded Image(5.2)These identities, which follow from equations (3.13) and (3.17), respectively, indicate that if Embedded Image or Embedded Image become large in magnitude then the second term in the left member is negligible. The associated value of the Young modulus is approximately one half of the value in the 111-direction and consequently large values of Embedded Image occur in directions at which Embedded Image. Such directions, by their nature, are close to 111.

View this table:
Table 3

Properties of the 11 materials of cubic symmetry in figure 6 with Embedded Image. (The boldfaced numbers indicate Embedded Image and Embedded Image. Unless otherwise noted the data are from Landolt & Bornstein (1992). G&S indicates Gunton & Saunders (1975).)

The appearance of Embedded Image in both figures 4 and 5 is not surprising if one considers that Embedded Image, Embedded Image and Embedded Image also occur in both the minimum and maximum. It can be checked that in the region where Embedded Image is the maximum value in figure 5, it satisfies Embedded Image. In fact, it is very close to but not equal to Embedded Image in this region, and numerical results indicate that Embedded Image in this small sector.

What is special about the transition values in figures 4 and 5: Embedded Image and Embedded Image? Quite simply, they are the values of Embedded Image and Embedded Image as the stationary directions Embedded Image approach the face diagonal direction-110. Thus, Embedded Image and Embedded Image are both the continuation of the face diagonal value Embedded Image, but on two different branches. See appendix B for further discussion.

(b) Application to cubic materials

We conclude by considering elasticity data for 44 materials with cubic symmetry, figure 6. The data are from Musgrave (2003) unless otherwise noted. The 17 cubic materials in the region, where Embedded Image are as follows, with the coordinates Embedded Image for each: GeTeSnTe1 (mol% GeTe=0) (0.01, 0.70), RbBr1 (0.06, 0.64), KI (0.06, 0.61), KBr (0.07, 0.59), KCl (0.07, 0.56), Nb1 (0.21, 0.61), AgCl (0.23, 0.61), KFl (0.12, 0.49), CsCl (0.14, 0.44), AgBr (0.26, 0.55), CsBr (0.16, 0.40), NaBr (0.15, 0.38), NaI (0.15, 0.38), NaCl (0.16, 0.37), CrV1 (Cr0.67 at.% V) (0.15, 0.35), CsI (0.18, 0.38), NaFl (0.17, 0.32). This lists them roughly in the order from top left to lower right. Note that all the materials considered have positive Embedded Image. The 16 materials with Embedded Image also have Embedded Image, so the coordinates of the above materials correspond to their extreme values of ν. The extreme values are also given by the coordinates in the region with Embedded Image, Embedded Image. The materials there are: Al (0.41, 0.27), diamond (0.12, 0.01), Si (0.36, 0.06), Ge (0.37, 0.02), GaSb (0.44, 0.03), InSb (0.53, 0.03), CuAu1 (0.73, 0.09), Fe (0.63, −0.06), Ni (0.64, −0.07), Au (0.88, −0.03), Ag (0.82, −0.09), Cu (0.82, −0.14), α-brass (0.90, −0.21), Pb1 (1.02, −0.20), Rb1 (1.15, −0.40), Cs1 (1.22, −0.46).

Figure 6

The 44 materials considered are indicated by dots on the chart showing the Embedded Image regions, cf. figure 4.

Materials with Embedded Image are listed in table 3. These all lie within the region where the minimum is Embedded Image, and of these, five materials are in the sub-region where the maximum is Embedded Image. Three materials are in the sub-regions with Embedded Image and Embedded Image. These indium thallium alloys of different composition and at different temperatures are close to the stability limit where they undergo a martensitic phase transition from face-centred cubic form to face-centred tetragonal. The transition is discussed by, for instance, Gunton & Saunders (1975), who also provide data on another even more auxetic sample: InTl (at 27% Tl, 125 K). This material is so close to the Embedded Image vertex, with Embedded Image, Embedded Image and Embedded Image (!) that we do not include it in the table or the figure for being too close to the phase transition, or equivalently, too unstable (it has Embedded Image and Embedded Image).

We note that the stretch directions for the extremal values of ν, defined by Embedded Image and Embedded Image, are distinct. As the materials approach the Embedded Image vertex, the directions coalesce as they tend towards the cube diagonal 111. The three materials in table 3 with Embedded Image and Embedded Image are close to the incompressibility limit, the line Embedded Image in figure 3. In this limit, the averaged Poisson's ratio is Embedded Image, and therefore those Poisson's ratios which are independent of Embedded Image tend to Embedded Image, i.e. Embedded Image. Also, Embedded Image and Embedded Image, withEmbedded Image(5.3)These are reasonable approximations for the last three materials in table 3, which clearly satisfy Embedded Image and Embedded Image.

6. Summary

Figures 4 and 5 along with tables 1 and 2 are the central results which summarize the extreme values of Poisson's ratio for all possible values of the elastic parameters for solids with positive strain energy and cubic material symmetry. The application of the related formulae to the materials in figure 6 shows that values less than −1 and greater than +2 are associated with certain stretch directions in some indium thallium alloys.

Acknowledgements

Discussions with Prof. T.C.T. Ting are appreciated.

Footnotes

References

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