The Extreme Values of Young’s Modulus and the Negative Poisson’s Ratios of Rhombic Crystals

: The extreme values of Young’s modulus for rhombic (orthorhombic) crystals using the necessary and sufﬁcient conditions for the extremum of the function of two variables are analyzed herein. Seven stationary expressions of Young’s modulus are obtained. For three stationary values of Young’s modulus, simple analytical dependences included in the sufﬁcient conditions for the extremum of the function of two variables are revealed. The numerical values of the stationary and extreme values of Young’s modulus for all rhombic crystals with experimental data on elastic constants from the well-known Landolt-Börnstein reference book are calculated. For three stationary values of Young’s modulus of rhombic crystals, a classiﬁcation scheme based on two dimensionless parameters is presented. Rhombic crystals ((CH 3 ) 3 NCH COO (CH) 2 (COOH) 2 , I, SC(NH 2 ) 2 , (CH 3 ) 3 NCH 2 COO · H 3 BO 3 , Cu-14 wt%Al, 3.0wt%Ni, NH 4 B 5 O 8 · 4H 2 O, NH 4 HC 2 O 4 · 1/2H 2 O, C 6 N 2 O 3 H 6 and CaSO 4 ) having a large difference between maximum and minimum Young’s modulus values were revealed. The highest Young’s modulus among the rhombic crystals was found to be 478 GPa for a BeAl 2 O 4 crystal. More rigid materials were revealed among tetragonal (PdPb 2 ; maximum Young’s modulus, 684 GPa), hexagonal (graphite; maximum Young’s modulus, 1020 GPa) and cubic (diamond; maximum Young’s modulus, 1207 GPa) crystals. The analytical stationary values of Young’s modulus for tetragonal, hexagonal and cubic crystals are presented as special cases of stationary values for rhombic crystals. It was found that rhombic, tetragonal and cubic crystals that have large differences between their maximum and minimum values of Young’s modulus often have negative minimum values of Poisson’s ratio (auxetics). We use the abbreviated term auxetics instead of partial auxetics, since only the latter were found. No similar relationship between a negative Poisson’s ratio and a large difference between the maximum and minimum values of Young’s modulus was found for hexagonal crystals.


Introduction
Anisotropic materials occupy an important place in modern technical applications. While the description of the linear elastic properties of isotropic media requires only two independent elastic constants, the number of important elastic constants increases with decreasing symmetry of materials. The deformation of anisotropic crystalline bodies depends not only on the locations of external forces in relation to the body, but also on the orientation of the crystallographic axes inside it. In addition, restrictions on such important elastic engineering characteristics (combinations of elastic constants), such as Young's moduli, Poisson's ratios and shear moduli, are reduced. In particular, if Poisson's ratios in isotropic media have restrictions of −1 below and 0.5 above, then for crystals of all seven symmetry systems, including the most symmetric cubic system, there are no general restrictions on the values and signs of Poisson's ratios [1].
An analysis of the variability of Poisson's ratios and Young's moduli of a large number of real crystals of all seven crystal systems (cubic, hexagonal, rhombohedral, tetragonal, rhombic, monoclinic and triclinic) was carried out in [2,3], based on extensive information combination of nanochannels and nanolayers can lead, with a sufficiently large size of spherical inclusions, to the absence of auxetic properties of the composite.
Another line of research into the mechanism of auxeticity was undertaken by J.N. Grima, K.E. Evans and A. Alderson et al. . The concepts of the mechanism of auxeticity of materials in these articles were based on rotations of simplified 2D geometric structures from triangular, square, rectangular and rhombic forms (etc.). Following A. Alderson and K.E. Evans, the auxetic nature of the deformation of a number of crystals (zeolites, silicates, α-crystobalite and β-crystobalite in particular) was associated with rotation and dilation of 3D tetrahedral and rotating 3D cuboidal microstructures [59][60][61][62][63][64][65][66][67][68]. It was shown in [69] that the auxeticity and negative linear compressibility of Boron Arsenate arise mainly due to deformations of framework tetrahedra. In [70], the manifestation of auxeticity and negative linear compressibility was discussed in the case of the formation of a 3D microstructure of a metamaterial due to stretching in out-of-plane direction of the original 2D "rotating squares". In [71], the possibility of auxeticity for a broad range of loading directions and negative linear compressibility for a small number of such directions was discussed for a 3D metamaterial composed of arrowhead-like structural units. In [72], the role of the rearrangement of the 3D microstructure of boron arsenanite under shear deformation in auxeticity and negative linear compressibility was discussed. An important feature of the shearing deformation of tetrahedra on the projection planes is the distortion of the rotating squares.
Auxetic materials are often found among natural anisotropic materials. There are particularly many of them (about three hundred) among highly symmetric cubic crystals . Since the negativity of Poisson's ratio usually corresponds to the selected directions of crystal orientation [7,11], in this case we actually focus on partial auxetics. Fewer auxetics are found among crystals of lower symmetry.
In this article, we consider the problem of stationary and extreme values of Young's modulus, and the question of the relationship between the extrema of Young's modulus and the value and sign of Poisson's ratio. Section 2 begins with a presentation of Young's modulus versus crystal orientation angles. Then six anisotropy coefficients are introduced as linear combinations of the compliance triples that disappear in the isotropic limit. Anisotropy coefficients for 18 crystals are shown in Table 1. A more complete list is provided in the Supplemental Material. In Section 3, the analysis of the second derivatives made it possible to find the extrema of Young's modulus for 140 rhombic crystals, shown partially in Table 2 and completely in the Supplementary Material. The dependence of three stationary values of Young's modulus on the anisotropy coefficients is presented in the form of a classification scheme. An analysis of the extrema of Poisson's ratios showed that more than 50 rhombic crystals are auxetic; about 30 of them correspond to the ratio E max /E min > 3 (Table 3 and Supplementary Material). In Section 4, the stationary values of Young's modulus for cubic, hexagonal and tetragonal crystals are discussed briefly as special cases of the rhombic system. In Section 5, conclusions are given.

Young's Modulus
This expression for the reciprocal of Young's modulus is obtained as the ratio of the tensile force uniformly distributed over the transverse surface to the relative elongation using Hooke's law for an anisotropic material. Young's modulus E(n) for anisotropic materials depends on the tensor compliance coefficients s ijkl and direction of the axis of extension [104]: 1 E(n) = s ijkl n i n j n k n l .
Here n i are the components of the unit vector n, which is directed along the axis of extension. Rhombic crystals are characterized by nine independent matrix compliance coefficients s 11 , s 22 , s 33 , s 44 , s 55 , s 66 , s 12 , s 13 and s 23 [105]. The matrix of compliance coefficients is represented as follows Using the matrix compliance coefficients, the expression for Young's modulus of rhombic crystals can be written as E −1 (n) = s 11 n 4 1 + s 22 n 4 2 + s 33 n 4 3 + (2s 23 + s 44 )n 2 2 n 2 3 + (2s 13 + s 55 )n 2 1 n 2 3 + (2s 12 + s 66 )n 2 1 n 2 2 . (1) If the orientation of the crystalline rod in the crystallographic coordinate system is described with three Euler's angles ϕ, θ, ψ, then using the relationship between the unit vector n and Euler's angles ϕ, θ, the expression of Young's modulus E for rhombic crystals can be rewritten as follows.
It is convenient to introduce six anisotropy coefficients of rhombic crystals for analyzing the variability of Young's modulus: which disappear in the limit of an isotropic medium. The number of anisotropy coefficients of rhombic crystals is greater than those of the cubic, hexagonal and tetragonal crystals. The last crystals have one [105,106], two [8,12] and three [9] anisotropy coefficients, respectively. The values of the anisotropy coefficients for some rhombic crystals are given in Table 1, and in Table S1 from the Supplementary Material the values for all rhombic crystals from the reference book [4] are presented.
The second stationary value of Young's modulus is achieved at ϕ = 0, θ = π/2 and ϕ = π, θ = π/2. It corresponds to stretching in the [010] and [010] directions. The third value also has a simple form, and is achieved at θ = 0 and an arbitrary angle ϕ. This stationary value corresponds to stretching in the [001] direction. At ϕ = 0 the fourth stationary value of Young's modulus has the form at the limitations This value corresponds to stretching in the (100) plane. Young's moduli E 2 and E 3 also lie in the (100) plane. At ϕ = π/2 the fifth stationary value of Young's modulus has the form with the limitations This value corresponds to stretching in the (010) plane. Young's moduli E 1 and E 3 also lie in the (010) plane. At θ = π/2 the sixth stationary value of Young's modulus has the form with the limitation This value corresponds to stretching in the (001) plane. Young's moduli E 1 and E 2 also lie in the (001) plane. The seventh stationary value of Young's modulus has the form (2) with the constraints We further investigate these stationary points using the sufficient condition for the extremum of the function of two variables. If at the indicated stationary points from the second derivatives of Young's modulus, a combination is formed then at D > 0 extremes of Young's modulus are achieved at the corresponding stationary point (maximum at A < 0 and C < 0 or minimum at A > 0 and C > 0). In the case D < 0, extrema are absent at the stationary point, and at D = 0 additional analysis is required [107].
In the case of a stationary point ϕ = π/2, θ = π/2, we have E = E 1 and Then, according to the sufficient condition for the extremum of the function, the value of Young's modulus E 1 will be extremal if ∆ 1 ∆ 3 > 0. The value E 1 will be the maximum at ∆ 1 < 0 or ∆ 3 < 0 and the minimum at ∆ 1 > 0 or ∆ 3 > 0.
For the stationary values of Young's modulus-E 4 , E 5 , E 6 and E 7 -the second derivatives A, B, C and D have a cumbersome analytical form. Therefore, only numerical analysis of them for 142 rhombic crystals was carried out. The results of this analysis are presented in Table 2 and Table S2 in the Supplementary Material. In Table S2 in the Supplementary Material for the values of Young's modulus E 4 , E 5 , E 6 and E 7 , the values of the angles at which they are achieved are also given. In these tables, the global maximum and minimum values of Young's modulus are shown in bold. An analysis of the variability of Young's modulus showed that the value E 7 is the inflection point for all rhombic crystals from [4].
In Figure 1, the classification scheme for three stationary values of Young's modulus E 1 , E 2 and E 3 depending on two dimensionless parameters is presented, α = (∆ 2 − ∆ 1 )/s 11 and β = (∆ 4 − ∆ 3 )/s 11 . The points indicate the values of dimensionless parameters α and β for 142 rhombic crystals from [4]. Most crystals fall into the area −1 < α < 1 and −1 < β < 1. There are six zones on the classification scheme, in which various inequalities between the stationary values of Young's modulus E 1 , E 2 , E 3 are satisfied. For each of these zones, the surface of Young's moduli for some rhombic crystals are shown in Figure 2.

Young's Moduli of Tetragonal, Hexagonal and Cubic Crystals
Above, the stationary values of Young's modulus for rhombic crystals were shown. Below we present the stationary values of Young's modulus for tetragonal, hexagonal and cubic crystals as special cases of rhombic crystals. Rhombic crystals are characterized by nine independent compliance coefficients s 11 , s 22 , s 33 , s 44 , s 55 , s 66 , s 12 , s 13 and s 23 , and six anisotropy coefficients (see Formulas (3)).

Tetragonal Crystals
Tetragonal crystals have six independent compliance coefficients, which are obtained under three conditions, s 11 = s 22 , s 44 = s 55 and s 13 = s 23 , for nine compliance coefficients that were given previously.
The dependence of Young's modulus for six-constant tetragonal crystals is a periodic function ϕ, θ with periods T ϕ = π/2 and T θ = π. Such crystals will already have three anisotropy coefficients,  4. At ϕ = 0, ϕ = π/2 and limitation the fourth stationary value has the form This value corresponds to stretching in the (100) (at ϕ = 0) and (010) (at ϕ = π/2) planes. Young's moduli E 1 and E 2 also lie in the (100) and (010) planes. 5. In this case, the system of Equations (5) is greatly simplified, and it is possible to obtain a simple form for the fifth stationary value: which is achieved at ϕ = π/4, ϕ = 3π/4 and limitation Young's moduli E 2 , E 3 and E 5 lie in the same plane. A detailed analysis of the extreme values of Young's modulus for six-constant and seven-constant tetragonal crystals was carried out in [108].
Young's modulus of hexagonal crystals depends on only one Euler's angle θ. The dependence of Young's modulus is a periodic function θ with a period T θ = π. Hexagonal crystals already have two anisotropy coefficients: Young's moduli E 1 , E 2 and E 3 lie in the same plane. A detailed analysis of the extreme values of Young's modulus and Poisson's ratio for hexagonal crystals was carried out in [12]. In this article, a classification scheme for the extreme values of Young's modulus E 1 , E 2 and E 3 , depending on two dimensionless parameters, is also given. The largest differences between the maximum and minimum values of Young's modulus were found in graphite (E max /E min = 71.8), which has the greatest ratio among rhombic, tetragonal, hexagonal and cubic crystals. A large difference (E max /E min > 5) was also revealed in RbNiCl 3 (E max /E min = 5.52) and CsNiF 3 (E max /E min = 5.72 for one experimental set of compliance coefficients and 10.6 for the second set of compliance coefficients) [12]. Maximum Young's modulus with E max > 500 GPa were detected in graphite (E max = 1020 GPa), WC (E max = 827 GPa), SiC (E max = 556 GPa), Re (E max = 588 GPa) and Ru (E max = 550 GPa). Graphite with hexagonal anisotropy and diamond with cubic anisotropy have the highest Young's moduli (E max > 1 TPa) among the rhombic, tetragonal, hexagonal and cubic crystals from [4].
For a subclass of cubic crystals with ∆ < 0 from (18)- (20), opposite inequalities follow: For example, V, Cr, Mo and Nb have negative anisotropy coefficients (∆). The maximum Young's moduli with E max > 500 GPa were detected in diamond (E max = 1207 GPa), Ir (E max = 649 GPa; for the second set of elastic constants E max = 620 GPa), ReO 3 (E max = 571 GPa; for the second set of elastic constants E max = 478 GPa), NbC 0.865 (E max = 526 GPa), SiC (E max = 511 GPa; for the second set of elastic constants E max = 547 GPa) and CeB 6 (E max = 508 GPa; for the second set of elastic constants E max = 472 GPa). The largest differences between the maximum and minimum values of Young's modulus were found in InTl (25at%Tl) (E max /E min = 32.5), InTl (28.13at%Tl (4.17at%Si). In the case of cubic crystals, a relationship between the maximum ratio E max /E min and the negativity of Poisson's ratio can also be observed. All these crystals with negative Poisson's ratios have positive anisotropy ratios (∆).

Conclusions
In the article, the variability of Young's moduli of rhombic crystals was analyzed. Analytical expressions of seven stationary values were obtained. Three stationary values always exist. Four other values occur when the additional conditions are met. In the case of rhombic crystals, the six stationary values of Young's modulus were revealed upon tension in the (100), (010) and (001) planes. Three of these values have a simple form and correspond to stretching in the [100], [010] and [001] directions. In addition, these six stationary values of Young's modulus can be extremes under certain conditions. The seventh stationary value is the inflection point for all 142 rhombic crystals indicated in [4].
Analytical stationary values of Young's modulus for tetragonal, hexagonal and cubic crystals were written out as special cases of rhombic crystals. Tetragonal crystals already have five stationary values of Young's modulus, whereas hexagonal and cubic crystals have three. In the case of tetragonal and hexagonal crystals, all stationary values can be global extrema under certain conditions. For cubic crystals, only two stationary values are global extrema (E [100] or E [111] ).
In the article, a numerical analysis of the stationary and extreme values of Young's modulus of rhombic crystals was also carried out, and the angles at which these values were revealed were determined. For three stationary values of Young's moduli of rhombic crystals corresponding to tension in the [100], [010] and [001] directions, a classification scheme based on two dimensionless parameters was presented. Rhombic crystals with strong anisotropy (E max /E min ) were detected.