An analytical analysis of the problem of the longitudinal tension of two-layered tubes with walls made of tetragonal crystals was carried out together with numerical calculations of the effective Young's moduli and Poisson's ratios of the tubes, using the known experimental data on the elastic characteristics of such crystals. The study of the effective elastic properties of two-layered tubes was carried out in the cases of layers of the same thickness, equal volumes, and greater arbitrariness. The effective Young's modulus often exceeds the largest Young's modulus of a pair of layers, and the effective Poisson's ratio can be negative, even if the Poisson's ratios are positive in both layers. In other words, an auxetic of the two-layered tube may correspond to a pair of non-auxetics in two layers of this tube.

The results of calculations of the effective Young’s modulus of longitudinally stretched two-layered plates made of identically oriented cubic crystals are presented on the basis of analytical analysis and the numerical finite element method. Analytical dependences of effective Young’s modulus on Young’s moduli and Poisson’s ratios of crystals in layers are presented. Combinations of pairs of crystals with a significant deviation of the effective characteristics from ones found by the rule of mixtures are determined. The dependences of the effective Young’s moduli on extreme values of the Young’s moduli and Poisson’s ratios of crystals in layers are established. They are presented graphically, and in some cases are reflected in the form of a table.

The problem of longitudinal tension for a two-layered plate of hexagonal and cubic crystals with different orientations of crystallophysic coordinate systems is discussed. Analytical dependencies of effective Young's modulus and Poisson's ratios on thicknesses ratio are obtained. Variability of effective properties is analyzed for all possible combinations of hexagonal and cubic crystals. Significant difference between effective Young's modulus and predictions by rule of mixtures takes place in the case when one of the layers is auxetic. Effective Young's modulus can exceed Young's modulus of the stiffer layer. The ratio of Young's moduli of crystals has significant influence on effective Poisson's ratio. A FEM (finite element method) analysis of this problem was performed and its results were compared with analytical ones. Between the analytical and numerical results qualitative and quantitative correspondence takes place.

The study analyzes the elastic properties of chiral metallic nanotubes formedby rolling up thin crystal plates with the [011] and [111] orientations withintwo frameworks of anisotropic elasticity and molecular statics. Iron, copper andaluminum nanotubes are discussed. It is shown that the tubes have a positivePoisson’s ratio in the entire range of chiral angles, unlike nanotubes obtainedby rolling up crystal plates with the [010] orientation. Poynting’s coefficientof nanotubes rolled up from plates with the [011] orientation becomes negativeat certain chiral angles, which corresponds to a change in the twistingdirection of the nanotubes under tension. The description of the dependence ofthe elastic properties of nanotubes on the chiral angle and thickness withinanisotropic elasticity theory agrees qualitatively with the results of molecularstatics simulations. The results for some chiral metallic nanotubes arequantitatively different.

The extreme values of Young’s modulus for six- and seven-constant tetragonal crystals are found using the necessary and sufficient conditions for the extrema of the functions of two variables. Theoretical and numerical analyzes of stationary and extreme values are performed based on experimental data on elastic constants collected in the Landolt–Börnstein reference book. Five such Young’s moduli are formed in the case of six-constant and seven-constant tetragonal crystals. It is found how the stationary and extremal values of Young’s modulus depend on three anisotropy coefficients that disappear in the limit of an isotropic material. Simple analytical dependences of some stationary and extreme values of Young’s moduli of six-constant crystals are obtained. In the case of stationary values of Young’s modulus of seven-constant tetragonal crystals, the coefficients included in the sufficient conditions for the extremum of the function of two variables are estimated numerically. Tetragonal crystals (Hg2I2, Hg2Br2, Hg2Cl2, TeO2, (NH2)2CO, LiY0.5Tb0.5 F4 and C(CH2OH)4) with a large difference between the maximum and minimum values of Young’s modulus are revealed. It is shown that six-constant tetragonal crystals may have a greater difference between global extrema than seven-constant tetragonal crystals. It is found that six-constant tetragonal crystals with a large difference between the global extrema of Young’s modulus have a negative Poisson’s ratio. In the case of seven-constant tetragonal crystals, such relationship has not been identified. A classification scheme based on the dependence of three stationary values of Young’s modulus on two dimensionless parameters is proposed.

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 ((CH3)3NCH2COO·(CH)2(COOH)2, I, SC(NH2)2, (CH3)3NCH2COO·H3BO3, Cu-14 wt%Al, 3.0wt%Ni, NH4B5O8·4H2O, NH4HC2O4·1/2H2O, C6N2O3H6 and CaSO4) 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 BeAl2O4 crystal. More rigid materials were revealed among tetragonal (PdPb2; 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.

Theoretical and numerical analyses of engineering elastic characteristics (Young's modulus and Poisson's ratios) of cylindrically anisotropic tubes formed by rolling-up plates of cubic crystallographic structures (110) and (111) are given. Present results are compared with the results for tubes with crystal structural orientation (001), studied previously. It is found that tubes with (110) crystal structure exhibit auxetic properties (negative Poisson's ratio) more often. Poisson's ratios averaged over the cross-section of some tubes have negative values.

Herein, the theoretical analysis of out‐of‐plane tension of a thin rectangular two‐layered plate of cubic crystals with the same orientation is provided. The dependences of effective Young's modulus and effective Poisson's ratio of the plate on Poisson's ratios of layers, the ratio of Young's moduli, and the thickness ratio are obtained. It is shown that when the ratios between Young's moduli and ratios between Poisson's ratios of the layers are equal, effective Young's modulus and effective Poisson's ratio coincide with Reuss'es means. The rules of mixtures are not fulfilled in the absence of equality of these ratios. In this case, effective Young's modulus can exceed not only Reuss'es mean but also Voigt's mean. It is demonstrated that effective Young's modulus can be larger than Young's moduli of both layers in two‐layered plates of nonauxetic−nonauxetic and auxetic−auxetic type. It is established that the behavior of effective Poisson's ratio of nonauxetic−nonauxetic plates can vary depending on the ratio of the layer thicknesses. At the same time, effective Poisson's ratio of plates from pairs of auxetics cannot be less than Poisson's ratios of both initial auxetic layers.

In the framework of the theory of anisotropic elasticity, a theoreticalanalysis of the out-of-plane extension of thin two-layered plates of hexagonalcrystals is carried out. The six-fold axes of all pairs of crystals are assumedto be perpendicular to the plane of the plates. Formulae for effective Young’smodulus and effective Poisson’s ratio are obtained. It is shown that in mostcases effective Young’s moduli exceed Reuss’s average, and thus the rule ofmixtures is violated. Equality of these characteristics is possible if theratios of Young’s moduli and the ratios of Poisson’s ratios of crystal pairs arethe same. Effective Poisson’s ratio may be greater or less than thecorresponding Reuss’s average. In addition, it was found that effective Young’smodulus 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 ratiosof the initial pair of crystals. The general theoretical conclusions areillustrated by numerical estimates using the experimental values of the elasticconstants of the known hexagonal crystals.

The stability, elastic moduli and deformation behavior of graphene-based diamond-like phases are examined by molecular dynamics simulations. Three important criteria are considered to study stability of the structure within applied methods. Molecular dynamics simulations are performed to derive stiffness and compliance coefficients, and stress-strain curves under hydrostatic compression of diamond-like phases. The analysis of elastic properties (Poisson’s ratio, Young’s modulus and shear modulus) is carried out. Analytical calculations are used. From the obtained stability criteria, it is found that only two graphene-based diamond-like phases, LA3 and LA6, instead of eight known, can be considered stable in molecular dynamics approach. Young’s modulus, measured in linear elastic regime, of stable phases orientationally dependent: values in different directions can differ two times. Analysis of the variability of shear modulus showed that for LA6 values are much higher than for LA3. Again, shear modulus are orientation-dependent for some directions. In non-elastic regime, stable diamond-like phases can be stretched until 0.012 and compressed until 0.1. The main mechanisms of non-elastic deformation is stretching of the covalent bonds and valent angles.

The article provides a brief overview of studies on crystalline materials with negative Poisson's ratio (crystalline auxetics) with cubic anisotropy. It has been demonstrated that 1/4 part of all cubic crystals has auxetic properties. Even more auxetics are found among chiral nano/microtubes with cubic cylindrical anisotropy. It has been established that chiral nano/microtubes made of cubic crystals exhibit linear Poynting's effect under torsion, in contrast to the known nonlinear effect for isotropic materials. In the case of longitudinal tension of two-layered composites, it is shown that Voigt's rule of mixtures is violated in the presence of a layer of auxetic materials.

The work studies the mechanicalproperties of chiral metallic nanotubes by the molecular staticsmethod. The atomic structure of nanotubes was obtained byrolling up thin nanoplates from cubic crystals of copper, iron,aluminum, and cobalt with the (010) orientation at variouschiral angles. It is shown that such nanotubes can experiencetorsion under tension and their Poisson’s ratio decreases withincreasing chiral angle within the range from 0° to 45°.Poisson’s ratio of stretched copper and cobalt nanotubes becomesnegative at certain chiral angles. A relationship is determinedbetween the uniaxial deformation of nanotubes and their torsionat different chiral angles (reverse Poynting’s effect). As thechiral angle increases, Young’s modulus of nanotubes alsoincreases. Atomistic modeling results are shown to agreequalitatively well with theoretical estimates obtained in theframework of anisotropic elasticity, but with significantquantitative differences for various crystalline materials.

Diamond‐like structures, that include sp2 and sp3 hybridized carbon atoms, are of considerable interest nowadays. In the present work, various carbon auxetic structures are studied by the combination of molecular dynamics (MD) and analytical approach. Two fullerites based on the fullerene C60 and fullerene‐like molecule C48 are investigated as well as diamond‐like structures based on other fullerene‐like molecules (called fulleranes), carbon nanotubes (called tubulanes) and graphene sheets. MD is used to find the equilibrium states of the structures and calculate compliance and stiffness coefficients for stable configurations. Analytical methods are used to calculate the engineering elastic coefficients (Young's modulus, Poisson's ratio, shear modulus and bulk modulus), and to study their transformation under rotation of the coordinate system. All the considered structures are partial auxetics with the negative value of Poisson's ratio for properly chosen tensile directions. It is shown that some of these structures, in a particular tension direction, have a very high Young's modulus, that is, 1852 GPa for tubulane TA6.

Using atomistic calculations, we study the features of uniaxial deformation of nanotubes made of rolled-up thin [0 1 0] plates of Fe cubic crystals. We find that within a certain range of chiral angles these nanotubes have both negative Poisson's ratio and axial strain-induced torsion (reverse Poynting's effect) during tension and compression. The maximum torsion and the minimum value of Poisson's ratio are observed at chiral angles of [Formula: see text] and [Formula: see text], respectively. We show that Young's modulus of the chiral Fe nanotubes increases with a chiral angle. We demonstrate that in the discussed range of nanotube sizes there is a satisfactory correspondence between the results obtained by molecular statics and anisotropic theory of elasticity.

The variability of Young's modulus and Poisson's ratio for hexagonal crystals was studied. Expressions for three stationary values of Young's modulus and eight stationary values of Poisson's ratio were obtained. Numerical analysis of the extrema of the elastic characteristics for crystals was given based on experimental data from the Landolt-Bornstein reference book. Classification schemes for extreme values of Young's modulus and Poisson's ratio are proposed. Data on hexagonal materials with negative Poissons ratio are obtained. It was shown that global extremes of Poisson's ratio can exceed the value 0.5, which is the upper limit for isotropic materials. On the other hand, hexagonal crystals were not found for which the global minimum value was less than - 1 (lower limit for isotropic materials).

Previously, the authors proposed a model of surface elasticity, in which the internal boundary was considered as a thin structured layer endowed with its own elasticity. The transition to the limit of an infinitely thin boundary was carried out in two stages. For a structured boundary of an interface, the governing equations of surface elasticity are formulated, generalizing the well-known Shuttleworth equations. In the present work, such a model is supplemented by boundary conditions on the interface and with its help the problem of spherically symmetric deformation of an infinite body with a spherical inclusion is considered.

Based on the known solutions of the problems of extension and torsion of cylindrically anisotropic chiral tubes, the analysis of variability of the Young’s modulus, Poisson’s ratios and torsional stiffness of monoclinic crystalline tubes was carried out using known experimental values of the compliance coefficients of the crystals. The extreme values of Young’s modulus and torsional stiffness were determined and their variability was studied. It was shown that chirality had a significant effect on the values of the elastic characteristics of tubes made from monoclinic crystals. Thin-walled tubes (the ratio of the external and internal radii of about one), which had a negative Poisson’s ratio, were identified. The largest negative values of Poisson’s ratio among all the analysed tubes were achieved for tubes from LaNbO4 and CsH2PO4 crystals. Due to the change in the ratio of the radii and the chirality, the Young’s modulus and the torsional stiffness of such tubes were changed in several times, and their Poisson’s ratios could change by several units. Angular Poisson’s ratio could reach the value of –6.5 on the inner surface of a tube from CsH2PO4 crystal. The LaNbO4 and CsH2PO4 crystals also had large negative values of Poisson’s ratios in the case of rectilinear anisotropy, which were significantly different from the Poisson’s ratio for isotropic materials.

Eight diamond-like structures (tubulanes) of different morphology based on carbon nanotubes are studied by the combination of molecular dynamics simulation and analytical calculations. Molecular dynamics is used for the structure relaxation, stability analysis, and calculation of the stiffness and compliance coefficients of stable tubulanes. Six stable tubulanes which can be elastically deformed are distinguished among the considered structures. Engineering elastic constants such as Young's modulus, Poisson's ratio, shear modulus, and bulk modulus are found by analytical methods. Two of the studied structures, TA6 and TB, are found to be partial auxetics having negative Poisson's ratio with the minimum values of −0.01 and −0.8, respectively. The maximum Young's modulus for tubulanes TA6, TA8, and TB is found to be >1 TPa. In combination with hardness close to that of diamond, considered phases with their outstanding mechanical properties can be potentially used as protective coatings.

The paper provides a theoretical analysis of axially stretched rectangular thin plates composed of two differently oriented layers of cubic crystals. In one layer, the crystal has its principal crystallographic orientations parallel to the plate edges, and in the other, the crystal is oriented at a certain angle to the plate plane. The analysis shows that more than fifty cubic crystals with positive anisotropy can form two-layered plates whose effective Young’s modulus is greater than the moduli of both layers, which violates the well-known Voigt’s rule of mixtures. The anomalous behavior of effective Young’s modulus and Poisson’s ratios in the plates depends on the properties of their constituent crystals: the sign and value of anisotropy coefficients and Poisson’s ratio, ratio of Young’s moduli, relative orientation angle, and layer thickness ratio.