Using computer algebra, the scientists at the Institute of Problems in Mechanical Engineering of the Russian Academy of Sciences derived definitions of hundreds of elasticity constants with a high degree of nonlinearity (anharmonicity) for an anisotropic material, taking into account its atomic structure. These constants describe the elasticity of mono- and polycrystals in cases where deformations are large (about 25%), but do not form defects. The results were published in the journal Acta Crystallographica, which accompanied the article with a scientific commentary from the author of monographs on nonlinear mechanics of crystals and shock waves in them.

We all remember from school the Hooke's law for springs, according to which the tensile force is linearly (directly) proportional to the stretching, i.e. the dependence of the force on the amount of deformation is described by a polynomial of the first degree, where a single coefficient characterizes the stiffness of the spring. However, the model of a solid body, in which atoms, like balls connected by Hooke springs, execute vibrations (sound or heat) according to a harmonic law, works only with small (about 1%) deformations and small amplitudes of atomic vibrations. Such a model cannot explain, for example, the thermal expansion of materials, since in it an increase in temperature would only lead to an increase in the amplitudes of vibrations, but their average equilibrium positions (i.e., the nodes of the crystal lattice) would remain unbiased. In reality, the greater the reversible deformation, the more obvious the deviation from the Hooke law and the more power-law correction terms to it need to be taken into account. The coefficients in these correction terms are called nonlinear stiffnesses of different orders.

The nonlinear theory of elasticity is required, for example, to describe deformed states under shock waves or in the immediate vicinity of defects in the crystal lattice. It is practically impossible to find numerical values of high-order stiffness from the experiment, since the crystal must be absolutely perfect (without defects) in order not to crush before the theoretical limit under heavy loads. Therefore, they try to calculate the values of high-order stiffness using computer quantum mechanical calculations. This method allows you to draw a curve of the force dependence on the deformation up to the limit of destruction of the body.

The difficulty in calculating these stiffness constants lies in the fact that in a three-dimensional body the deformation is characterized not by one variable of tension/compression, but by six: three extensions and three shear deformations. This complication leads to a significant increase in the order of the problem. If we consider power polynomials describing the dependence of force on strain from these parameters, they become functions of six variables instead of one. And as the degree of nonlinearity increases, the number of constants increases significantly with each order of nonlinearity.

Due to these computational difficulties, nonlinearity above the third degree has hardly been considered until now, with the exception of the structures like diamond and two-dimensional graphene.

Rodion Telyatnik, a researcher at IPMash RAS, was the first to deduce definitions of hundreds of constants above 4th order for elasticity with a high degree of nonlinearity (anharmonicity) for an anisotropic material, taking into account its atomic structure. These constants describe the elasticity of mono- and polycrystals in cases where deformations are large (about 25%), but do not form defects.

According to the young scientist, the study of the features of the stress-strain state of a promising new material for electronics, created by the Laboratory of Structural and Phase Transformations in Condensed Media at IPMash RAS, motivated him to expand the limits of the known theory of crystal elasticity to such large degrees of nonlinearity.

This is SiC silicon nanocarbide of high monocrystalline quality, which is synthesized directly from the surface layer of the silicon substrate by the unique Kukushkin–Osipov method based on the coordinated substitution of a part of the atoms in silicon with carbon atoms without destroying the silicon substrate. The mismatch of the crystal lattices of the new material and the silicon base leads to strong elastic stresses arising in the new material, and a theoretical description of the stress-strain state of the new nanomaterial is impossible without expanding the boundaries of the theory of elasticity.

The results obtained by Mr Telyatnik are a significant contribution to the development of deformable solid mechanics and are of great practical importance in designing and creation of new types of structures, including the field of microelectronics. Using formulas obtained by R. Telyatnik, it is possible to determine the nonlinear elasticity constants of specific materials subjected to severe deformations that can occur at the interfacial boundaries of various semiconductor heterostructures or in disturbed areas of crystals near various kinds of defects.