### №2-2021

Full versions of the articles

https://elibrary.ru/contents.asp?id=46288275

**Goloveshkin V.A., Myagkov N.N.**

## Model of momentum transfer in hypervelocity impact

An analytical mechanical model of the ejection arising from a high-velocity impact of a rigid projectile on a semi-infinite target is constructed, and an estimate is given of the ejection mass and the effect of momentum amplification transmitted to the target upon impact. The effect of the momentum amplification is caused by the ejection of target fragments in the 148 direction opposite to the direction of flight of the projectie. At present, there is a steady interest in the study of this effect. This is due, in particular, to the possible use of the effect for deflecting a potentially dangerous object (asteroid) approaching the Earth by means of an impact spacecraft using the effect of momentum amplification. The model presented in this work is constructed in approximation of plane deformation using the minimum number of parameters of the projectiler and target materials. Equations for the mass of the ejection and the increment of the target momentum are obtained, depending on the depth of penetration of the projectile. The model takes into account the dependence of the emission angle of the ejection fragments on the penetration depth of the projectile. It is shown that the model adequately describes the ejection momentum, the rate of change in the ejection momentum, and the ejection mass depending on the penetration depth of the projectile. The possibility of representing the momentum and mass of the ejection by scaling ratios is checked both for the ratio of the densities of the projectile and the target ptρρ, and for the dynamic parameter 20ttVYγρ= (0V – impact velocity, tY – yield stress of the target), in which the proportionality coefficient depends only on the shape of the projectile. It was found that scaling with respect to the dynamic parameter γ takes place at сγγ>, where 330сγ≈ that, e.g., for aluminum gives the value 02.5 km/scV =.

Pages: 147-168 doi.org/10.33113/mkmk.ras.2021.27.02.147_168.01

**Ekster N.M., Movchan A.A.**

## Actuator with a series connection of a shape memory alloy’s rod and an elastic bias element

One of the most promising applications of shape memory alloys (SMA) is their use for creating working bodies of power exciters (actuators). The working body of the actuator must perform certain movements due to shape memory phenomena (heating, working stroke of the actuator) and the accumulation of direct transformation deformations (cooling, idling of the actuator). It is known that the process of deformation of SMA during cooling occurs only in the presence of mechanical action, while the return to the original form during heating occurs in the absence of a corresponding mechanical action and even in the presence of a sufficiently large counteraction. To ensure the possibility of not only working, but also idling of the actuator, the working body of the SMA is connected to the elastic bias element so that both of these elements are deformed together. In this case, when the SMA element is heated, it is deformed due to the shape memory phenomenon, which leads to the deformation of the bias element associated with the working body and the appearance, both in this element and in the working body, of mechanical stresses that increase with the temperature of the SMA element. It is these stresses, which continue to act during the cooling of the working body, that ensure its deformation in the opposite direction at the stage of cooling of the working body and the corresponding direct transformation into SMA. In this paper, the behavior of an actuator consisting of a SMA rod and an elastic rod (an bias element) connected in series, the total length of which is assumed to be unchanged, is analytically investigated. The influence of the system parameters on the stresses in the SMA rod and the value of the working stroke of the actuator is investigated. The conditions for the implementation of the closed two way shape memory effect in this system are determined.

Pages: 169-190 doi.org/10.33113/mkmk.ras.2021.27.02.169_190.02

**Du Yikun, Sheshenin S.V.**

## Homogenization of thin rubber-cord layers at moderate large deformations

Breker layers in a pneumatic tire are an important part in the tire construction. These layers have a metal cord resulting in substantial bending stiffness. When homogenizing such layers, a “shave” method is applied to the breaker layer. This results in a thinner layer having adequate stiffness in both tension and bending. In this work, a phenomenological approach is used to obtain the effective properties of a homogeneous anisotropic hyper elastic material of an equivalent layer. Two models utilize transverse isotropic or orthotropic potential used to describe the homogenized properties. Comparison is made between these models for the “shaved” rubber-cord layer based on numerical experiments. In both cases, the potentials are built on the basis of the Treloar or Mooney potentials. Note that in the case of an inhomogeneous thin layer, the traditional definition of homogenization needs to be modified. In previous works of the authors, it was proposed to determine 3D averaged elastic properties of a layer by surrounding it with a homogeneous material. This makes it possible to correctly take into account the fact that the boundary effect from the upper to lower surfaces that penetrates through the whole periodicity cell. A set of local problems formulated for the periodicity cell is proposed. This set is sufficient for determining elastic potential material parameters. Nonlinear local problems on a periodic cell are solved and the material constants of the elastic potential are determined. The applicability of the orthotropic potential (second model) is determined for the “shaved” layer. It was found that orthotropic properties are manifested relative to longitudinal shears. The results show the suitability of the proposed potential and the scheme for determining the material parameters.

Pages: 191-204 doi.org/10.33113/mkmk.ras.2021.27.02.191_204.03

**Goncharova T.P., Khitrova N.V., Ponomarev M.V., Ponomareva G.P., Popova I.M., Sladkov O.M.**

## Strengthening of the inside layer of the multilayered composite by bazaltoplastik in the form of cellular structure

In our studies we present a multilayer structure consisting of outer layers of two-ply basalt composed of basalt fabric impregnated with an epoxy compound and an inner layer of polyurethane reinforced with basalt, having the same composition, formed in conjugate hexagonal prisms, and forming a cellular framework. The main part describes the technological sequence of obtaining basalt face layers, basalt hexagonal cellular frame, and filling it with a liquid reaction mass of rigid polyurethane system. The formation of a single multilayer composite structure is shown. We also present the results of the experimental research of strength of a laminated composite and its inner layer without facing layers when tested for static bending with concentrated load increasing at a constant rate and compression testing up to 10% of relative deformation. It describes the effect of the size of hexagonal prismatic cells of the frame on the physical and mechanical characteristics of the middle layer. The reduction of the cell size results in the increase of the composite strength. The dependence of the composite density on the size of the cells of the basalt frame has been studied.

Pages: 205-216 doi.org/10.33113/mkmk.ras.2021.27.02.205_216.04

**Belashova I.S., Bibikov S.P.**

## Influence of additional heat treatment of gas-phase inhomogeneous surface layers on residual stresses and adhesion strength of steel

The article presents the results of a study of the surface layers obtained by the deposition of organometallic chromium compounds on the base metal, and the structural changes occurring in them when exposed to additional heat treatment in the form of subsequent annealing. It is noted that the type of coating and its structure primarily depend on the deposition temperature; therefore, there are three main types – kinetic, transitional and diffusional. Each such type of surface layer is determined by its own mechanism of formation, and as the temperature changes, one transforms into another. The metastable state of such surfaces, which are supersaturated solid solutions, upon additional heat treatment leads to significant structural changes and the formation of a nanostructured composite layer. The possibility of obtaining nano-modified layers by forming the structure of a matrix solution with dispersed-dissolved secondary nano-sized particles of chromium carbides is shown. The modes at which crystallization of the amorphous phase begins with the release of nano-sized particles of chromium and chromium carbides have been determined. The modes with the maximum and minimum levels of residual stresses, which directly affect the performance of products, have been established, and the effect of additional annealing on the distribution of residual stresses has been studied. It is shown that the process of crystallization of the amorphous phase during annealing and the parallel process of coagulation of carbides causes the appearance and growth of residual stresses. The influence of residual stresses on the adhesion strength of surface layers, which largely depends on the main parameters of heat treatment – temperature and time, has been investigated. Comparison of adhesion strength before and after annealing showed that the overall level of adhesion strength during heat treatment of coatings decreases insignificantly, by 7-10%.

Pages: 217-226 doi.org/10.33113/mkmk.ras.2021.27.02.217_226.05

**Zhavoronok S.I.**

## Wave dispersion in heterogeneous waveguides: methods of solution (a review). part I

A brief review of the modern state-of-the art and tendencies of further development of various methods of solution of wave dispersion problems in heterogeneous functionally graded elastic waveguides is presented. Main types of functionally graded materials and structures, including gradient thon-walled structures, and their main engineering applications is discussed. The main difficulties of modelling of the stress-strain state of functionally graded shells and plates are pointed, as well as the possible ways to overcome such difficulties. The main theoretical bases of definition of effective constitutive constants of functionally graded materials and their possible estimates used in the practice are considered. Main dependencies of the effective constitutive constants of a functionally graded material on coordinates used for the mathematical modelling of the dynamics are also shown. The statement of the dynamics problem for a functionally graded waveguide and the appropriate statement of the normal wave dispersion problem are pointed. The presented Part I of the review consider some analytical methods of solution of dispersion problems, mainly the matrix ones based on the formulation of the steady dynamics problem in the image space as a first-order ordinary differential equations system. The state vectors corresponding to the useful Cauchy and Stroh formalisms are introduced, and the appropriate governing equations and the boundary conditions on waveguide’s faces are presented. Classical methods for solving the steady dynamics problem for a laminated waveguide are briefly described, which could be a basis for the further approximation of a functionally graded material by a system of layers with constant properties, i.e. the transfer matrix method, its main modifications developed to ensure the stability of calculations, and the global matrix method. Then, the intensively developed last 15 years reverberation matrix method, stiffness matrix method, and the Peano series method are discussed. Some key solutions of the wave dispersion problems for heterogeneous layers are presented; such solutions improve the efficiency of approximation of a functionally graded structure by a laminated one. The implicit solution of the general problem of steady dynamics for a waveguide with arbitrary gradation law is shown. The key features of the discussed matrix methods are pointed briefly as well as their main drawbacks. In the Part II, the main attention will be paid to methods of semi-analytical solution of dispersion problems based on the approximation of a waveguide by an equivalent system with a finite number of degrees of freedom: power series, generalized Fourier series, semi-analytical finite elements. spectral elements, as well as methods based on various theories of plates and shells.

Pages: 227-260 doi.org/10.33113/mkmk.ras.2021.27.02.227_260.06

**Golubkov A.K., Ivanova A.N., Kulakov V.V., Shmelev D.S.**

## Design features of carbon-carbon friction composites made by the aerodynamic method based on discrete fibers

The designing of carbon-carbon composites (СCС) for friction use is an important problem, since the existing approaches are applicable only to technologies for obtaining materials using continuous fiber in the form of tapes and fabrics and are not suitable for products with chaotic reinforcement by short fibers, the use of which improves the mechanical characteristics of ССС both under static and dynamic influences. Also, this technology has economic advantages due to the use of cheaper raw materials and a significant reduction in the time of the molding stage in the presence of comparable physical, mechanical and frictional properties with CCC based on needle-punching and needle-piercing. To ensure the strength of discretely reinforced friction products, it is necessary to ensure the required effective length of the carbon fiber in the bundles distributed in the volume of the material, due to their evaluation by the degree of separation of the filaments. This article shows the dependence of the critical length of the bundle on the number of its fibers, from which it is found out that the critical length of the bundle sets the minimum threshold value at which the material is able to realize high strength characteristics. These ratios allow us to estimate the minimum necessary rate of separation of the bundle for a given fiber length. this will allow you to realize the maximum strength of the material, taking into account the technological limitations in the rate of separation. Models are also proposed that allow us to evaluate the technological prospects for the manufacture of discretely reinforced carbon-carbon materials for friction purposes and to predict their properties. The effect of additional reinforcement of the inter-joint spaces of the composite material on its wear resistance at high specific friction energies is shown

Pages: 261-271 doi.org/10.33113/mkmk.ras.2021.27.02.261_271.07

**Feldstein V.A., Onuchin E.S., Orlov D.A., Tovarnova N.A., Vasilchenko L.B.**

## Fabric energy absorption volume under shock loading conditions

Space orbital stations operations support consists an adoption of meaningful measures to protect space station against impacts of space debris and meteoroids. This goal can be reached by using multilayered protection shields that are made with the fabric material layers. Shields designing and modeling requires specific characteristics that define energy absorbed volume by the fabric destruction under impact. The paper describes the methodology and experimental determination method for absorbed energy volume results by using multilayer fabrics of orbital manned stations shielding constructions under distributed impulse loading caused by the space debris impacts. The energy absorbed volume by the multilayer fabrics is obtained from the experiments by analysis of specimen and flat metal projectile impact. Projectile was accelerated by the air gas gun. The obtained experimental determination results of energy absorbed volume in pressure range up to 1,5 GPa are given. Using the model of fabric as a porous material its energy absorption volume dependence in pressure range up to 10 GPa and compared with experimental data. It is shown that for materials with high porosity absorbed energy volume against pressure dependence is close to linear. Corresponding asymptotic dependence for materials with high porosity under the high pressure is obtained.

Pages: 272-287 doi.org/10.33113/mkmk.ras.2021.27.02.272_287.08