V. A. Gorodtsov

Auxetic Mechanics of Crystalline Materials

In the present paper, we analyze uniaxial deformation of crystals of different systems with negative Poisson’s ratios, known as auxetics. The behavior of auxetic crystals is studied on the basis of extensive knowledge on the experimental values of elastic constants of different crystals, gathered in the well-known Landolt-Börnstein tables. The competition between the anisotropy of crystal structures and the orientation of deformable samples results in the dependence of the elastic characteristics of deformation, such as Young’s modulus and Poisson’s ratio, on the orientation angles. In the special case of a single angle, a large number of auxetics were found among crystals of cubic, hexagonal, rhombohedral, tetragonal, and orthorhombic systems and the character of variations in their response due to changes in orientation was determined.

A Compact Analytic Model of the Strain Field Induced by Through Silicon Vias

The thermoelastic strains are induced by through silicon vias due to the difference of thermal expansion coefficients between the copper ( ~ 18 ppm / °C) and silicon ( ~ 2.8 ppm /°C) when the structures are exposed to a thermal ramp in the process flow. A compact analytic model (Bessel function) of the strain field is obtained using Kane-Mindlin theory, and has a good agreement with the finite-element simulations. The elastic strains in the silicon in the radial direction and angular direction are tensile and compressive, respectively. The linear superposition of the analytic model of a single via can be used in the multi-via configuration. Due to the interaction of vias, the slightly larger errors of strain occur between the two close vias when the linear superposition is used.

Negative Poisson’s ratio for cubic crystals and nano/microtubes

The paper systemizes numerous cubic crystals which can have both positive and negative Poisson’s ratios (the so-called partial auxetics) depending on the specimen orientation in tension. Several complete cubic auxetics whose Poisson’s ratio is always negative are indicated. The partial cubic auxetics are classified with the use of two dimensionless elastic parameters. For one of the parameters, a critical value is found at which the orientation behavior of the crystals changes qualitatively. The behavior of mesotubes obtained by rolling up plates of cubic crystals (crystals with rectilinear anisotropy) is considered in detail. Such mesotubes with curvilinear cubic anisotropy can have micron and nanometer lateral dimensions. It is shown that uniform tension of nano/microtubes of cubic crystals is possible only in the particular case of zero chirality angle (the angle between the crystallographic axis and the axis of a stretched tube). It is demonstrated by the semi-inverse Saint-Venant method that solution of the axial tension problem for cylindrically anisotropic nano/microtubes of cubic crystals with a non-zero chirality angle is possible with radially inhomogeneous fields of three normal stresses and one shear stress. In the examples considered, the cylindrically anisotropic nano/microtubes of cubic crystals are auxetics even if they are initially non-auxetics with rectilinear anisotropy.

On the Modeling of Surface and Interface Elastic Effects in Case of Eigenstrains

The constitutive equations of interface elasticity in case of eigenstrains are obtained in terms of interface (surface) values defined as integrals of the excesses of the corresponding volumetric values over the normal to the interface. The equations are consistently linearized, which corresponds to the case of both elastic strains and eigenstrains being small. It is shown that the obtained equations possess more general form then Shuttleworth equations. The obtained type of equation was confirmed by considered example: an interface formed by a thin layer of constant properties. It was also shown that the type of energetic restrictions on the surface elastic constants may depend essentially on the definition of the position of the surfaces.

Young’s modulus and Poisson’s ratio for seven-constant tetragonal crystals and nano/microtubes

In the paper, the elasticity theory was applied to consider the mechanical properties of rectilinearly anisotropic seven-constant tetragonal crystals and their cylindrically anisotropic nano/microtubes with and with no chiral angle, being the angle between the crystallographic symmetry axis and elongated tube axis. Pt is found that the number of crystals with negative Poisson’s ratio is the least for rectilinear anisotropy and is much larger for curvilinear anisotropy. With a nonzero chiral angle, all nano/microtubes can have negative Poisson’s ratio. The elastic problem on axial tension of cylindrical nano/microtubes is solved for radially inhomogeneous stresses: three normal stresses and one shear stress.

Rayleigh and Love surface waves in isotropic media with negative Poisson’s ratio

The behavior of Rayleigh surface waves and the first mode of the Love waves in isotropic media with positive and negative Poisson’s ratio is compared. It is shown that the Rayleigh wave velocity increases with decreasing Poisson’s ratio, and it increases especially rapidly for negative Poisson’s ratios less than −0.75. It is demonstrated that, for positive Poisson’s ratios, the vertical component of the Rayleigh wave displacements decays with depth after some initial increase, while for negative Poisson’s ratios, there is a monotone decrease. The Rayleigh waves are characterized by elliptic trajectories of the particle motion with the change of the rotation direction at critical depths and by the linear vertical polarization at these depths. It is found that the elliptic orbits are less elongated and the critical depths are greater for negative Poisson’s ratios. It is shown that the stress distribution in the Rayleighwaves varies nonmonotonically with the dimensionless depth as (positive or negative) Poisson’s ratio varies. The stresses increase strongly only as Poisson’s ratio tends to−1. It is shown that, in the case of an incompressible thin covering layer, the velocity of the first mode of the Love waves strongly increases for negative Poisson’s ratios of the half-space material. If the thickness of the incompressible layer is large, then the wave very weakly penetrates into the halfspace for any value of its Poisson’s ratio. For negative Poisson’s ratios, the Love wave in a layer and a half-space is mainly localized in the covering layer for any values of its thickness and weakly penetrates into the half-space. For the first mode of the Love waves, it was discovered that there is a strong increase in the maximum of one of the shear stresses on the interface between the covering layer and the half-space as Poisson’s ratios of both materials decrease. For the other shear stress, there is a stress jump on the interface and a more complicated dependence of the stress on Poisson’s ratio on both sides of the interface.

Young’s moduli and Poisson’s ratios of curvilinear anisotropic hexagonal and rhombohedral nanotubes. Nanotubes-auxetics

The study of materials with unusual mechanical properties has attracts a lot of attention in view of new possibilities for their application. One of these properties is negative Poisson’s ratio which is commonly found in crystalline materials (materials with linear anisotropy). However, until now the capabilities of negative Poisson’s ratios in tubular crystals (materials with curvilinear anisotropy), e.g., in today’s popular nanotubes, have not been studied.

Relation of Poisson’s ratio on average with Young’s modulus. Auxetics on average

A linear relation between the Poisson’s ratio averaged along the transverse directions and Young’s modulus of the tensed cubic crystal is established. It is found that the coefficients of the linear relation in the dimensionless form depend on two dimensionless elastic parameters combined from three compliance coefficients. By virtue of this fact, the form of angular regions of the crystal orientation with negative Poisson’s ratio on average varies as the magnitude of one dimensionless coefficient and the sign of the other one. We find the critical value of the dimensionless parameter at which there is the topological change in the structure of the angular regions occurs is established.

Equilibrium structures of carbon diamond-like clusters and their elastic properties

Three-dimensional carbon diamond-like phases consisting of sp3-hybridized atoms, obtained by linking of carcasses of fullerene-like molecules, are studied by methods of molecular dynamics modeling. For eight cubic and one hexagonal diamond-like phases on the basis of four types of fullerene-like molecules, equilibrium configurations are found and the elastic constants are calculated. The results obtained by the method of molecular dynamics are used for analytical calculations of the elastic characteristics of the diamond- like phases with the cubic and hexagonal anisotropy. It is found that, for a certain choice of the dilatation axis, three of these phases have negative Poisson’s ratio, i.e., are partial auxetics. The variability of the engineering elasticity coefficients (Young’s modulus, Poisson’s ratio, shear modulus, and bulk modulus) is analyzed.

Auxeticity in nano/microtubes produced from orthorhombic crystals

A solution for the tension and torsion problems for the curvilinearly anisotropic nano/microtubes made of orthorhombic crystals in the framework of the Saint-Venants approach is given. We find that the number of partial auxetics among the tubes is twice as frequent among the rectilinearly anisotropic crystals, at the same time about one third of 136 orthorhombic crystals are auxetics. It is shown that the torsion causes extension of the nano/microtubes even in the absence of a longitudinal tensile force. This Poyntings effect substantially depends on the chiral angle, and in particular, it disappears when the chiral angle vanishes. We also investigate an inverse Poyntings effect when the extension of the nano/microtubes is accompanied by their twisting. It is shown that the signs of Poyntings effect and Poissons ratio are changed several times with the change of the chiral angle.