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 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.
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 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, (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 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 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 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, and , may be written (Walpole 1984) in terms of three moduli κ, and ,(2.1)Here, is the fourth order identity, , and(2.2)The isotropic tensor and the tensors of cubic symmetry and are positive definite (Walpole 1984), so the requirement of positive strain energy is that κ, and are positive. These three parameters, called the ‘principal elasticities’ by Kelvin (Thomson 1856), can be related to the standard Voigt stiffness notation: , and . Alternatively, , and in terms of the compliance.
Vectors, which are usually unit vectors, are denoted by lowercase boldface, e.g. . The triad represents an arbitrary orthonormal set of vectors. Directions are also described using crystallographic notation, e.g. is the unit vector . The summation convention on repeated indices is assumed.
(a) Engineering moduli
The Young modulus sometimes written , shear modulus and Poisson's ratio are (Hayes 1972)(2.3)where , and . Thus, and are defined by the axial and orthogonal strains in the - and -directions, respectively, for a uniaxial stress in the -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 and are isotropic, and consequently the directional dependence of the engineering quantities is through . Thus,(2.4)(2.5)(2.6)where(2.7)We note for future reference the relations(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 and (Walpole 1986; Hayes & Shuvalov 1998). Thus, , and for , with the values reversed for (Hayes & Shuvalov 1998). As noted by Hayes & Shuvalov (1998), the difference in extreme values of E and G are related by(2.9)The extreme values also satisfy(2.10)The shear modulus G achieves both minimum and maximum values if is directed along face diagonals, that is, for .
The area modulus of elasticity for the plane orthogonal to is the ratio of an equibiaxial stress to the relative area change in the plane in which the stress acts (Scott 2000). Thus, . Using the equations above it may be shown that, for a cubic material,(2.11)The averaged Poisson's ratio is defined as the average over in the orthogonal plane, or . The following result, apparently first obtained by Sirotin & Shaskol'skaya (1982), follows from the relations (2.8),(2.12)Equation (2.12) indicates that the extrema of and coincide. The traction-associated bulk modulus , introduced by He (2004), relates the uniaxial stress in the -direction to the relative change in volume in anisotropic materials. It is defined by , and for cubic materials is simply . 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 is constant.
Further discussion of the extremal properties of G and ν requires knowledge of how they vary with for given , and in particular, the extreme values as a function of for arbitrary , considered in §2c. Note that, and are the only directions for which and are independent of . It will become evident that is a critical direction, and we therefore rewrite E and ν in forms emphasizing this direction:(2.13)where , and Χ (Hayes & Shuvalov 1998) are(2.14)Both and are independent of . The fact that with equality for implies that this is the only stretch direction for which E, and hence ν, are independent of . Equations (2.13) indicate that and depend on at any point in the neighbourhood of 111, with particularly strong dependence if 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
For a given , consider the defined vector(2.15)with ρ chosen to make a unit vector. Requiring implies that is orthogonal to if(2.16)i.e. if λ is a root of the quadratic(2.17)It is shown in appendix A that the extreme values of for fixed coincide with these roots, which are non-negative, and that the corresponding unit vectors provide the extremal lateral directions. The basic result is described next.
(i) A fundamental result
Let be the roots of (2.17) and the associated vectors from (2.15), i.e.(2.18a)(2.18b)(2.18c)The extreme values of D for a given are associated with the orthonormal triad , i.e.(2.19)The extreme values of G and ν for fixed 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 are(2.20a)(2.20b)where is, see figure 1,(2.21)
3. Poisson's ratio
We now consider the global extrema of over all directions and . 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.
(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 are(3.1)where the stretch is in the direction and is in the lateral direction . The conditions may be obtained by considering the derivative of ν with respect to rotation of the pair about an arbitrary axis. Setting the derivatives to zero yields the stationary conditions (3.1).
The only non-zero contributions to , , , and in a material of cubic symmetry come from . Thus, we may rewrite the conditions for stationary values of ν in terms of , etc. as(3.2)The first is automatically satisfied by virtue of the choice of the direction- as either of . Regardless of which is chosen,(3.3)The final identity may be derived by first splitting each term into partial fractions and using the following (cf. appendix A):(3.4)
With no loss in generality, consider the specific case of , , where , and in either case, . It may be shown without much difficulty (appendix B) that for in the interior of the triangle of figure 2. It then follows that inside the triangle,(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 rewritten(3.6)However, using (3.5), the determinant of the matrix is(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 become zero and careful limits must be taken. We avoid this route by considering the conditions (3.2) afresh for directed along the three edges. We find, as before, that 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- can be parametrized along each edge with a single variable. Thus, , , on edge 1. Similarly, edges 2 and 3 are together covered by , with . In each case, we also need to consider the two possible values of , which we proceed to do, focusing on the conditions (3.2)2 and (3.2)3.
(i) Edge 1: , and or 001
For , we find that and . Hence, equation (3.2)2 is automatically satisfied, while equation (3.2)3 becomes(3.8)Conversely, for it turns out that and . In this case, the only non-trivial equation from equations (3.2) is the second one,(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, , and are potential candidates for global extrema of ν.
(ii) Edges 2 and 3: , and
Proceeding as before, we find that , and . Hence, equation (3.2)3 is automatically satisfied, while equation (3.2)2 becomes(3.10)The zero p=0 corresponds to which was considered above. Thus, all three conditions (3.2) are met if p is such that(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 find(3.12a)(3.12b)Substituting for from equation (3.11) into (3.12) gives two coupled equations for and :(3.13)Eliminating yields a single equation for possible stationary values of ,(3.14)We will return to this after considering the other possible vector.
(iii) Edges 2 and 3:
In this case , and . Equation (3.2)2 holds, while equation (3.2)3 is zero if p=0, which is disregarded, or if p is such that(3.15)The Young modulus is independent of and given by (3.12a), while ν satisfies(3.16)Using the value of from (3.15) in equations (3.12a) and (3.16) yields another pair of coupled equations, for and ,(3.17)These imply a single equation for possible stationary values of ,(3.18)
(c) Definition of and
The analysis for the three edges gives a total of seven candidates for global extrema: , and 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 , , and the following two distinct roots of equations (3.14) and (3.18), respectively,(3.19a)(3.19b)The quantity has been replaced to emphasize the dependence upon the two parameters and the anisotropy ratio . The associated directions follow from equations (3.11) and (3.15),(3.20a)(3.20b)A complete analysis is provided in appendix B. At this stage, we note that 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 and , where(4.1)That is, is the axial Poisson's ratio , independent of the orthogonal direction, and is the non-dimensional analogue of Χ. Thus,(4.2)a form which shows clearly that ν is negative (positive) for all directions if and ( and ). 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 are(4.3)where F is defined in (2.7) and H in (2.21). Thus, is the minimum (maximum) and the maximum (minimum) if , respectively.
The Poisson's ratio is a function of the direction pair and the material parameter pair , i.e. . The dependence upon has an interesting property: for any orthonormal triad,(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: , associated with the two directions 001 and 111 for which ν is independent of . These are two vertices of the triangle in figure 2. At the third vertex ( along the face diagonals), we have where, in the notation of (Milstein & Huang 1979) and . Three of these four values of Poisson's ratio associated with principal directions can be global extrema, and the fourth, plays a central role in the definition of and of (3.19). We therefore consider them in terms of the non-dimensional parameters and ,(4.5)We return to and 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, and , can be expressed in terms of and as(4.6)It will become evident that the global extrema for ν depend most simply on the two values for along a face diagonal: and . The constraints (4.6) become(4.7)which define the interior of a triangle in the plane, see figure 3. This figure also indicates the lines and (isotropy). It may be checked that the four quantities are different as long , with the exception of and which are distinct if . Consideration of the four possibilities yields the ordering(4.8a)(4.8b)(4.8c)(4.8d)Note that 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 and , but are best characterized by the two independent non-dimensional parameters and .
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 and the global maximum , respectively, for all possible combinations of elastic parameters. For table 1, , and are defined in (4.5), and and are defined in (3.19a) and (3.20a). For table 2, and 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.
Conventional wisdom prior to Ting & Chen (2005) was that the extreme values were characterized by the face diagonal values and . But as equation (4.8) indicates, even these are not always extrema, since 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 and in figure 5 in the region . Using equation (3.19a), we can determine that is strictly less than −1 if . Similarly, equation (3.19b) implies that is strictly greater than 2 if . By converting these inequalities, we deduce(5.1a)(5.1b)The two sub-regions defined by the , inequalities are depicted in figures 4 and 5. They define neighbourhoods of the vertex, i.e. , where the extreme values of ν can achieve arbitrarily large positive and negative values. The condition for 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 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 is much less than either or κ. is either the maximum or minimum of G, and it is associated with directions pairs along orthogonal face diagonals, . Hence, the Ting & Chen effect requires that this shear modulus is much less than , and much less than the bulk modulus κ. In the limit of very small , equations (3.19) give . Ting (2004) found that the extreme values are for small values of their parameter δ. In current notation, this is , and replacing the two theories are seen to agree.
The implications of small for Young's modulus are apparent. Thus, , and equation (2.13)1 indicates that 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 for materials with very large values of (see table 3 below) look like very sharp starfish. Although, the directions at which and are large in magnitude are close to the 111-direction, the value of E in the stationary directions can be quite different from . The precise values of the Young modulus, and at the associated stretch directions are given by(5.2)These identities, which follow from equations (3.13) and (3.17), respectively, indicate that if or 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 occur in directions at which . Such directions, by their nature, are close to 111.
The appearance of in both figures 4 and 5 is not surprising if one considers that , and also occur in both the minimum and maximum. It can be checked that in the region where is the maximum value in figure 5, it satisfies . In fact, it is very close to but not equal to in this region, and numerical results indicate that in this small sector.
What is special about the transition values in figures 4 and 5: and ? Quite simply, they are the values of and as the stationary directions approach the face diagonal direction-110. Thus, and are both the continuation of the face diagonal value , 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 are as follows, with the coordinates 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 . The 16 materials with also have , 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 , . 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).
Materials with are listed in table 3. These all lie within the region where the minimum is , and of these, five materials are in the sub-region where the maximum is . Three materials are in the sub-regions with and . 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 vertex, with , and (!) 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 and ).
We note that the stretch directions for the extremal values of ν, defined by and , are distinct. As the materials approach the vertex, the directions coalesce as they tend towards the cube diagonal 111. The three materials in table 3 with and are close to the incompressibility limit, the line in figure 3. In this limit, the averaged Poisson's ratio is , and therefore those Poisson's ratios which are independent of tend to , i.e. . Also, and , with(5.3)These are reasonable approximations for the last three materials in table 3, which clearly satisfy and .
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.
Discussions with Prof. T.C.T. Ting are appreciated.