Поперечное растяжение тонких двухслойных пластин из одинаково ориентированных гексагональных кристаллов Текст научной статьи по специальности «Физика»

УДК 539.31, 539.32

Поперечное растяжение тонких двухслойных пластин из одинаково ориентированных гексагональных кристаллов

В.А. Городцов, Д.С. Лисовенко

Институт проблем механики им. А.Ю. Ишлинского РАН, Москва, 119526, Россия

В рамках теории анизотропной упругости проведен теоретический анализ поперечного растяжения тонких двухслойных пластин из гексагональных кристаллов. Предполагается, что кристаллические оси шестого порядка всех кристаллов перпендикулярны плоскости пластины. Получены формулы для эффективного модуля Юнга и эффективного коэффициента Пуассона. Показано, что эффективный модуль Юнга в большинстве случаев превосходит среднее по Рейссу, так что правило смесей нарушается. Равенство этих характеристик возможно только при равенстве отношений поперечных модулей Юнга и отношений поперечных коэффициентов Пуассона исходных кристаллов, заполняющих слои пластины. Что касается эффективного коэффициента Пуассона, то он может быть как больше, так и меньше соответствующего среднего по Рейссу. Кроме того, было обнаружено, что эффективный модуль Юнга может превосходить модули Юнга обоих кристаллов, образующих двухслойную пластину. Эффективный коэффициент Пуассона может быть как больше, так и меньше, чем коэффициенты Пуассона исходных кристаллов. Общие теоретические заключения подтверждены численными оценками, использующими экспериментальные значения упругих постоянных известных гексагональных кристаллов.

Ключевые слова: гексагональные кристаллы, ауксетики, двухслойные пластины, модуль Юнга, коэффициент Пуассона

DOI 10.24411/1683-805X-2020-15003

Out-of-plane tension of thin two-layered plates of identically oriented hexagonal crystals

V.A. Gorodtsov and D.S. Lisovenko

Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526, Russia

In the framework of the theory of anisotropic elasticity, a theoretical analysis of the out-of-plane extension of thin two-layered plates of hexagonal crystals is carried out. The six-fold axes of all pairs of crystals are assumed to be perpendicular to the plane of the plates. Formulae for effective Young's modulus and effective Poisson's ratio are obtained. It is shown that in most cases effective Young's moduli exceed Reuss's average, and thus the rule of mixtures is violated. Equality of these characteristics is possible if the ratios of Young's moduli and the ratios of Poisson's ratios of crystal pairs are the same. Effective Poisson's ratio may be greater or less than the corresponding Reuss's average. In addition, it was found that effective Young's modulus can surpass Young's moduli of both crystals forming a two-layered plate. Effective Poisson's ratio can be both larger and smaller than Poisson's ratios of the initial pair of crystals. The general theoretical conclusions are illustrated by numerical estimates using the experimental values of the elastic constants of the known hexagonal crystals.

Keywords: hexagonal crystals, auxetics, two-layered plates, Young's modulus, Poisson's ratio

1. Introduction

One of the important anomalous mechanical properties of crystals is negative values of Poisson's ratio. Materials that exhibit negative Poisson's ratios are called auxetics. The type of crystal and its orientation

play a decisive role in the manifestation of this property. A high prevalence of this property among crystals was demonstrated in [1-5]. A large number of papers on various types of auxetics and materials with other anomalous properties were published [6, 7].

© Городцов В.А., Лисовенко Д.С., 2020

Today, the number of studies of various types of auxe-tics and structures from them is so large that it is difficult to review them all in an ordinary paper. We give a general list of reviews on this subject [8-32], which is long enough.

Our paper is limited to describing the macroscopic behavior of anisotropic layered structures from crystalline auxetics. The diversity in the behavior of samples from auxetics and nonauxetics is primarily due to crystal anisotropy. However, the anisotropy of the structure of layered composites can also affect their mechanical behavior. The role of isotropic auxetics in such composites was investigated in [33-39]. In these papers, it was shown that the effective properties of layered composites cannot be generally calculated according to Voigt's or Reuss's rules of mixtures. Violations of these rules become especially large when auxetics are combined with nonauxetics. The effective modulus can be larger than the moduli of both phases [39]. In [34, 35], the difference in the effective elastic properties of two-layered plates stretched in the longitudinal and transverse directions was demonstrated. The effective mechanical properties of three-layered plate composites with two stiff isotropic layers and inhomogeneous two-phase cores were studied in [40, 41]. It was shown that composites with useful effective properties can be obtained using auxetics and structure optimization.

Special attention was drawn to the structural anisotropy of layered tubular composites of isotropic materials. The anomalous behavior of three-layered tubular composites from pairs of materials was analyzed using the finite element method [42, 43]. It was shown that additional auxetic cell phases have a great influence on the effective mechanical properties of stretched coaxial cylindrical composites. The analytical solution to the problems of longitudinal tension of layered composites from different crystals and numerical estimates for many crystals were given in [4448]. Experimental data on the elastic constants of these crystals are collected in the Landolt-Bornstein encyclopedic handbook [49]. In [44], the tension of two-layered tubular composites of identically oriented cubic crystals was considered. It was shown that negative effective Poisson's ratio of composites can be obtained for many pairs of initial nonauxetics. In [46], the solution of the problem for the longitudinal tension of two-layered plates with different orientations of cubic crystals filling the layers was given. The obtained formulae for effective Young's modulus and effective Poisson's ratios are used for the numerical analysis of the elasticity of plates formed of 10,000 pairs

of crystals whose elastic characteristics are indicated in [49]. In [47], the effective elastic properties of three-layered plates of identically oriented cubic crystals (auxetic and nonauxetic) under tension and bending were studied. In many cases, effective Young's moduli for the auxetic-nonauxetic-auxetic type of plates were larger than Young's moduli of all three crystals. In [48], a theoretical analysis of the in-plane tension of a thin two-layered plate formed of hexagonal crystals of the same orientation was given. It was shown that changing the thickness of the auxetic layer MoS2 makes it possible to control negative effective in-plane Poisson's ratio of layered composites. The most serious violation of the rule of mixtures for both effective elastic characteristics of two-layered hexagonal plates occurs when auxetics and nonauxetics are combined. The development of bi-stretch auxetic woven fabrics based on foldable geometrical structures was described [50]. Differential shrinkage was used to convert a parallel in-phase zigzag foldable structure into a two-layered woven fabric and an out-of-phase zigzag foldable structure into a single-layered woven fabric. The influence of the location of the weft threads was also studied. All discussed fabrics exhibit the effect of negative Poisson's ratio in a wide range of large strains.

A number of published papers touch on questions about the auxetic behavior of layered composites with nanolayers. The elastic properties of two-dimensional systems of layered plates were studied using computer simulation [51]. Alternating layers of thin plates were filled with hard disks and hard cyclic hexamers. Poisson's ratio of the considered systems depends on the orientation of the layers. The general conclusion is made that nonauxetic, partially auxetic and auxetic structures can be obtained by changing the orientation of layers, the concentration of hexamers, and the diameter of disks. The composite studied in [52] was a structure of a close-packed Yukawa crystal and a nanoslit filled by hard spheres. The introduction of a monoatomic layer of hard spheres into the Yukawa crystal can significantly change Poisson's ratio and create the strongly auxetic regime. The Monte Carlo computer simulation of the elastic properties of layered nanocomposites of solid spheres was performed [53]. Composites were composed of packets of identical solid spheres and monolayers of solid spheres of different diameters. This study showed that by changing the particle size of the nanoinclusions, it is possible to change the magnitude and sign of Poisson's ratio of the composites at high pressures. It is also shown that the difference in the elastic properties of

the hard sphere and Yukawa systems with monolayers and geometry has an ambiguous effect on the magnitude and sign of Poisson's ratio.

The elastic (in particular, auxetic) properties of chiral metal tubes made of an iron bcc crystal were investigated by the numerical method of atomic statics [54]. The nanotubes have approximately the following dimensions: the inner radius 20 nm, wall thickness 2 nm, and length 100 nm. We found that in a certain range of chiral angles the nanotubes have both negative Poisson's ratio and experiences twisting due to axial deformation. A comparison was also made with the results obtained earlier by the method of the theory of anisotropic elasticity in the framework of continuum mechanics [55, 56]. The continuum approximation implies the superiority of the considered length scales L over the atomic scales a (L >> a), resulting in a small parameter s = a/L <<1. With the atomic lattice constant 0.287 nm for bcc Fe, the characteristic sizes are L > h = 2 nm and a = 0.287 nm. The small parameter s = 0.14 indicates the accuracy of the continuum approximation of about 15 percent. The comparison of the results found by the two methods reveals significant quantitative differences but good qualitative agreement. The atomic modeling method was also applied to chiral tubes of the same sizes made of fcc crystals of Al, Cu, and Co [57]. The lattice constants a for these metals are 0.405 nm, 0.361 nm, and 0.251 nm, respectively, so that the small parameter s = a/h differs by less than two times from the small parameter for the iron tube with the wall thickness h = 2 nm. A comparison of the results for Poisson's ratios (and Poynting coefficients) found by the atomic statics method and the continuum method showed good qualitative agreement with a significant quantitative difference.

Composites from pairs of materials in the form of thin two-layered plates are characterized by high structural anisotropy. Therefore, in-plane and out-of-plane tension of a two-layered plate has a large difference in effective elastic characteristics, as shown for isotropic materials [34, 35]. The difference between these types of deformation will be more significant for anisotropic substances. In-plane tension of thin two-layered plates of hexagonal crystals with the same orientation was analyzed in [49]. In the present paper, we will consider out-of-plane tension of thin two-layered plates of hexagonal crystals with the same orientation.

2. Analytical description of out-of-plane tension of a thin two-layered hexagonal plate

We will consider elastic out-of-plane tension of a thin two-layered rectangular plate with layers of thickness h1 and h2. These layers are filled with identically oriented hexagonal crystals. The direction of tension coincides with the hexagonal axes of both crystals (Fig. 1). The linear elasticity of such crystals is characterized by matrix compliance coefficients snk, s\2k, s\3k, S33k, S44k, which connect the six-dimensional vectors of stresses a/ and strains Smk using Hooke's matrix law [58, 59]. Superscript k is the layer number. In the case of out-of-plane tension of a thin plate under the action of specific force P, tangential stresses can be neglected and the simplified equations of Hooke's law for the matrix components of normal stresses a1k, a2k, a3k and strains S1k, S2k, S3k in layers with numbers k are as follows:

s k _ k a k , k a k , k a k

s k _ k a k , k a k , k a k

C* - J k £o — Si i(Jo

-a 2).

-3 _k311^3 ^12(a1

With the equilibrium equations in all three directions

h1al1 + h2aj2 = 0, h1a12 + h2a2 = 0, a3 = a2 = P and subject to adhesion conditions of thin layers

81 = £1, £2 = ^2,

they allow obtaining expressions for the individual components of stresses and strains and the simplest connections between them:

a1 =a2 = P—

s13 s

13

S11 + s12 ( S121 + S122)

, ^ = T1,

Fig. 1. Tension of the rectangular two-layered plate consisting of two layers 1 and 2 with thicknesses h1 and h2 under the out-of-plane force P. X and Y are the side lengths

2 2 л 1 1 2 о

CTj = ^2 = —ACTi, CT3 = CT3 = CT3 = р,

(s11 + s12)(^13 - ^13)

S1 = S1 = s2 = S 2 = P

s3 = P

s2 = p

s33 +-

s13 + ' 1 1 2 2

s11 + s12 + A(s11 + s12).

2s13(s13 s13)

11 + S12 + A( Sn + s12)

2s13(s13 — s13)

s323 A" 1,1,^2,2

S11 + s12 + A( s11 + S12). The total displacement in transverse direction 3 is composed of displacements in the layers. This gives complete out-of-plane strain S3 of a thin plate

h

h1 + h

- + s3

Ah

h1 + h2

and, as a result, for effective Young's modulus of the two-layered plate E = P/S3, we obtain 1 1

E 1 + A

S33 + As 33

2A(s13 — s13)2

33

(1)

— S1 — s1Q +

(S1! + s12)(s13- s^ . S11 + s12 + A( s121 + s122)

Ï-T 1 1С л x / / x

(2)

The separate formula for effective Poisson's ratio is more cumbersome:

S13 + (s11 + S12 )(s123 _ S13)S

—v = (1 + A)

1

s33 + As13 — 2A(s12 —s13)2 S

(3)

s11 + s

S

42 + A( sn + s12)'

The following thermodynamic constraints are imposed on the compliance coefficients sj of hexagonal crystals:

skn > 0, sk3 > 0, s44 > 0, ski + s* > 0,

ski - sik2 > 0, (ski + s*^* > 2(sk)2. The above formulae for the effective coefficients are simplified in the particular case by the same compliance coefficients si3 in both layers of the plate. Namely, when si31 = si32, we have

1 s33 + As33 л s s1

-= 33, A 33 ' — v = (1 + A) 2 ; 1 1 + A s33 + As3

4

E

—- = s1 E ~ s13'

33 /u,33

Sometimes it is convenient to use engineering elastic coefficients (several Young's moduli Etk and Pois-

son's ratios v,/) instead of compliance coefficients [58]. Such interconnects for hexagonal crystals have the form

1

1

42

v21 = v12, v23 = v13, v31 = v32 = v

23

13 31

32

13

k E3_

Ek

(4)

13

Independent coefficients for a hexagonal crystal layer are the pair of Young's moduli (in-plane Young's modulus E1 and out-of-plane Young's modulus E3) and the pair of Poisson's ratios (in-plane Poisson's ratio vu and out-of-plane Poisson's ratio v^). The dependences of effective Young's modulus and effective Poisson's ratio on the engineering coefficients can be represented as follows:

1 1 B

E Er

(5)

J11 + S12 + A(s11 + s12) _ Tension of a two-layered plate in the direction of force P is accompanied by elastic strain in the plate plane. The relative measure of this strain will be effective Poisson's ratio v = S11/S3. The ratio of effective Poisson's ratio to effective Young's modulus is reduced to the above formula for strains S11 = S12 = S21 = S22 in the layers 1

B =

v = (A +1)

2A Mv13 — M3v23)2 1+ A 1 — v12 + A|j,1(1 — v22) '

v13N — (1 — v22)( v13 — ^3v23)

( A + |3) N — 2Ai13(v13 —|3 v23)2

N = 1 — v12 + A|1(1 — v22).

(6)

Here, Er is Reuss's average for effective Young's modulus of the two-layered plate

1

1

h1 + h2

r Al+Aл E3

-3 J

A + |3 1 A +1 EC'

and |1, |3, |13, П13, A are the abbreviations:

E1

11 ■ E?

I3 ■

■ E

I13 ■

■ E

n13 ■

_ 13

43

A ■ h Ah

Then, Reuss's average for effective Poisson's ratio vr takes the form

1

1

(

h+h2

A.+A

1 2 v13 v

Л

A + n13 1 A +1 "

V v13 v13 J 1 * v13

Indices 3 and 13 characterizing the considered out-of-plane tension of a two-layered plate are omitted in the notation for effective Young's modulus and effective Poisson's ratio. The limits A ^ x> and A ^ 0 correspond to single-layered plates 1 and 2. The obtained formulae for effective Young's modulus and effective Poisson's ratio are correctly given E = E31, v = V131 in the first limit and E = E32, v = V132 in the second limit.

Important limitations of engineering coefficients correspond to the thermodynamic limitations of the compliance coefficients indicated above:

Ek > 0, Ek > 0, | > 0, | > 0, ^ > 0, (7)

1 - ak > vf2 > -1, ak = 2Ek (vk3)2/E3k > 0. The obtained formulae for the effective coefficients show that these coefficients are reduced to Reuss's averages in the case of the same ratios E31 =

E32 and V131 = V132:

E = ER, V = VR.

Moreover, there is an equality of the three ratios

er = =EL

VR v13 vl23

Since Young's moduli are positive, such special case is possible only for the positive ratio Vi31/Vi32. In any other situation, the combination of the engineering coefficients B is positive, and effective Young's modulus will be larger than Reuss's average (see (5)). Thus, the rule of mixtures will not be satisfied in most cases.

The formula for the ratio of effective coefficients

= _1_ V13^Ml(1 - v22) + ^3v23(1 - V12) (8)

E E 1 -vl2 + -V22)

reveals the positivity of effective Poisson's ratio at V13k > 0 and constraints (7). Differentiating equality (8) with respect to the parameter A, we obtain

= 1 ^1(1 -v22)(1 -v12)(v13 -^3v23)

SA E E1 [1 -v12 + (1 -v22)]2 . This derivative is also positive for V131/E31 > V132/E32 > 0. Since v/E = V132/E32 at A = 0 and v/E = V131/E31 at A ^ <», then v/E will monotonically increase from V132/E32 to V131/E31 with increasing A.

3. Results and discussion in respect to various hexagonal crystals, auxetics, and nonauxetics

Numerical estimates for effective Young's moduli and effective Poisson's ratios of two-layered hexagonal plates in out-of-plane tension were based on experimental elastic constants. These constants were taken from [49]. The transversal isotropy of equally oriented hexagonal crystals in the layers allows us to confine ourselves to the pairs of characteristics E1, v12 under in-plane tension and E3, V13 under out-of-plane tension. The numerical analysis of elasticity of 170 hexagonal crystals with the considered orientation [49] showed that the coefficient v13 is positive for all of these crystals, and the coefficient v12 is negative only for Zn and MoS2 crystals.

There are pairs of hexagonal crystals in which the ratio of Young's moduli E31/E32 coincides with the ratio of Poisson's ratios vзз3/vззl with reasonable ac-

curacy. Due to relations (4), this corresponds to the equalities of the compliance coefficients s^Vs^2. Examples of two-layered plates with such crystals in layers are collected in the table. In these cases, in accordance with Eq. (5), effective Young's moduli E and Poisson's ratios v under out-of-plane tension of two- layered composites completely coincide with Reuss's averages Er and vr.

Numerical estimates show that the rules of mixtures can be satisfied with good accuracy even at a significant difference in the ratios of = E37E32 to «13 = v3:/v32. In particular, a difference of less than two percent occurs in 100 two-layered plates with zinc crystals (auxetic) in one of the layers. The Zn-Pr plate is an example of a small difference between effective Young's modulus and Reuss's average at the almost two-fold difference between «13 and ^13 (Fig. 2). The difference between v and vr is small for a relatively thick zinc layer (Fig. 2, b). In another example, the two-percent accuracy of the deviation of E/Er from unity can be achieved for a hundred of different hexagonal crystals and a BN3 crystal in one of the layers of two-layered plates.

The two-layered Zn-MoS2 plate is peculiar in that both layers are filled with auxetics. In this case, the ratio of v131/v132 to E31/E32 is 2.54. There is a significant difference between the effective elastic coefficients E, v and Reuss's averages Er, vr (Fig. 3). Thus,

Values of S13k and E31/E32 = vзз3/vззl for two-layered plates

Two-layered plates Compliance coefficients S131 = S132, TPa-1 Equal ratios of Young's moduli and Poisson's ratios Eз1IEзl = V131/V132

Zn-CdMg (3.33 at % Mg) -7.00 1.21

CdS-MgLi (10 at % Li) -5.30 1.30

Mg-InN -5.00 0.61

Pr-Bi2Ge3O9 -3.80 0.87

Ho-ZrO (8 at % O) -2.90 0.59

Er-GdY (40 at % Y) -2.60 1.13

a-Zr-ZrO (24 at % O) -2.40 0.76

GaS-Sc -2.20 0.40

a-ZnS-SmCo5 -2.10 0.70

LaF3-NbF3 -1.10 0.94

Co-Be3AkSi6C18 -0.94 1.23

WC-C (graphite) -0.33 22.70

Fig. 2. Dependences of effective Young's modulus E, Reuss's average Er (a) and effective Poisson's ratio v, Reuss's average vr (b) on the ratio of the layer thicknesses A = A1/A2 for a two-layered Zn-Pr plate. The constants of out-of-plane Young's moduli E31, E32 (a) and out-of-plane Poisson's ratios vn1, v132 (b) for the initial hexagonal Zn and Pr crystals are also given (color online)

Fig. 3. Dependences of effective Young's modulus E, Re-uss's average Er (a) and effective Poisson's ratio v, Reuss's average vr (b) on the ratio of the layer thicknesses A = A1/A2 for a two-layered Zn-MoS2 plate. The constants of out-of-plane Young's moduli E31, E32 (a) and out-of-plane Poisson's ratios vo1, v132 (b) for the initial hexagonal Zn and MoS2 crystals are also given (color online)

the rule of mixtures is not satisfied, unlike in the previous examples. Figure 3 also shows that there are the following bounds for the effective and average values

E3 < Er < E < E32, vj^ <vr <v< v}3.

The bound of effective Poisson's ratio v > V131, V132 arise for 50 two-layered plates with auxetic Zn and a nonauxetic at certain layer thicknesses. In Fig. 4, this situation is shown by the example of the two-layered Zn-MnAs plate. The bound for effective Poisson's ratio v < V131, V132 was established for 20 two-layered plates with zinc crystals in one of the layers. This possibility for the Zn-WC plate is proved by Fig. 5. Figures 4, b and 5, b for the Zn-MnAs and Zn-WC plates also demonstrate a large difference between v and vr.

Effective Young's modulus can also exceed Young's moduli of both crystals forming a two-lay -ered plate E > E31, E32. For example, this is the case for two-layered plates Zn-C (graphite), Zn-GaSe, Zn-GaS, BN3-Ti, MoS2-CuCl, C (graphite)-CuCl, C (gra-phite)-In2Bi, C (graphite)-Cd, and MoS2-RuNiCl3 at

certain thicknesses of layers. The dependence of effective Young's modulus for the C (graphite)-Cd plate is shown in Fig. 6. However, the numerical estimates showed that for most plates there are upper and lower bounds of both effective Young's moduli and effective Poisson's ratios, as well as of Reuss's average values

E3 < Er, E < E32, v}3 < vR, v < Vj23.

These bounds are determined by the characteristics of the pairs of crystals. In addition, there is the general bound Er < E for Young's moduli, and the deviation of the ratio E/Er from unity can exceed 0.1 for many plates. For example, this happens if one layer of the two-layered plate is filled with SiC or WC crystals, and the other layer contains one of 110 hexagonal crystals.

A different situation holds for effective Poisson's ratios. The study of various two-layered plates shows that in many cases the opposite inequalities v > vr, v < vr can be satisfied. For example, if SiC (or Co) crystals

Fig. 4. Dependences of effective Young's modulus E, Reuss's average Er (a) and effective Poisson's ratio v, Reuss's average vr (b) on the ratio of the layer thicknesses X = h\lhi for a two-layered Zn-MnAs plate. The constants of out-of-plane Young's moduli E31, E32 (a) and out-of-plane Poisson's ratios V131, V132 (b) for the initial hexagonal Zn and MnAs crystals are also given (color online)

Fig. 5. Dependences of effective Young's modulus E, Reuss's average Er (a) and effective Poisson's ratio v, Reuss's average vr (b) on the ratio of the layer thicknesses X = h1lh2 for a two-layered Zn-WC plate. The constants of out-of-plane Young's moduli E31, E32 (a) and out-of-plane Poisson's ratios V131, V132 (b) for the initial hexagonal Zn and WC crystals are also given (color online)

fill one of the layers of two-layered plates, then the ratio v/vr in 120 cases was less than 0.9. In the case of the C (graphite) crystal and 130 other hexagonal crystals, this ratio is larger than 1.1. Approximate equality v » vr is satisfied with the two-percent accuracy only in the case of a few pairs of hexagonal crystals.

The above results are obtained in the framework of continuum theory, the theory of anisotropic elasticity. The concept of a thin plate in this approach means a small thickness as compared to the length and width of the plate. Approximation of the continuum implies the fulfillment of the strong inequality L >> a between the considered scales L and atomic sizes a. The atomic lattice constants of cubic crystals are a = 0.25-0.65 nm [60], and therefore plates with thickness h greater than 10 nm > 15a can be described with satisfactory accuracy in the framework of the theory of anisotropic elasticity.

4. Conclusions

The theory of out-of-plane tension of thin two-layered plates from hexagonal crystals with the same

orientation is developed in the framework of the theory of anisotropic elasticity. It was found that Reuss's rule of mixtures for Young's moduli is fulfilled exactly if the ratio of Young's moduli of the crystalline pair of a two-layered plate coincides with the ratio of

45-.-

25 T-1-1-1-1-

0 2 4 6 8 À

Fig. 6. Dependences of effective Young's modulus E, Reuss's average Er on the ratio of the layer thicknesses X = h1lh2 for a two-layered plate C (graphite)-Cd crystals. The constants of out-of-plane Young's moduli E31, E32 for the initial hexagonal C (graphite) and Cd crystals are also given (color online)

Poisson's ratios. In all other cases, effective Young's modulus exceeds Reuss's average. Numerical estimates using experimental elastic characteristics confirm large violations of the rule of mixtures. It was also established that effective Young's modulus can often exceed both Young's moduli of the initial crystalline pairs.

A slightly different situation holds for effective Poisson's ratio. This coefficient can be larger and smaller than Reuss's average. The same concerns the comparison of effective Poisson's ratio with Poisson's ratios of many pairs of hexagonal crystals. An important feature of effective Poisson's ratio in out-of-plane tension of two-layered plates is that this coefficient is always positive. This occurs even in the situation of two-layered plates from two partial auxetics MoS2 and Zn. Effective Poisson's ratio behaves quite differently under in-plane tension of two-layered plates of identically oriented hexagonal crystals [48]. In this case, when auxetics and nonauxetics are combined, effective Poisson's ratio reverses sign with changing thickness of the auxetic layer.

The reported study was supported by Russian Science Foundation (project No. 18-19-00736).

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Received 29.09.2020, revised 29.09.2020, accepted 02.10.2020

Сведения об авторах

Городцов Валентин Александрович, д.ф.-м.н., проф., внс ИПМех РАН, gorod@ipmnet.ru Лисовенко Дмитрий Сергеевич, д.ф.-м.н., зав. лаб. ИПМех РАН, lisovenk@ipmnet.ru