### №2-2018

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https://elibrary.ru/contents.asp?id=35324252

**Astapov A.N., Nushtaev D.V.**

## Stress state and adhesion strength estimates for the system of a heat proof material and a heat protection coatnig for non-canonically shaped substrates

A new finite element – based numerical model for the stress-strain state estimation in a heat loaded system of a coating and a non-canonically shaped substrate is proposed to investigate the coating protection of gas turbine blades. An once coupled plane problem of thermoelasticity is solved using the CAE system SIMULIA Abaqus. The non-homogeneous distribution of the heat-induced stresses in the system around the blade profile periphery is obtained and the stress jumps at the points of connection of shallow profile segments to the highly curved ones are found. It is shown that the well-known analytical model neglecting bending moments is useful for straight or slightly curved segments of boundaries of relatively thick substrates, otherwise this estimate fails. The bending strains and curvatures effect strongly on the stress-strain state of laminated structures of complex shape. The new method of estimation of normal and shear contact stresses inducing the adhesion failure in the coating – substrate system is proposed. The most probable zones and temperatures of the failure initiation of the adhesion bound are obtained for the blade geometry and material properties. The required level of the adhesion strength for the coating – substrate system is estimated for the given operation conditions of the turbine blade. The prognosis of the level and the distribution of stresses and adhesion strength in various coating – substrate systems offers the basis of the design of coating architectures as well as their (chemical compositions, phase constitutions, number and thickness of layers), and the forming methods. As a result, the number and the costs of physical tests can be significantly reduced.

Pages: 151-170 doi.org/10.25590/mkmk.ras.2018.24.02.157_170.01

**Russkikh S.V.**

## The problem of nonlinear dynamics of an of elastic rod system connected to a spacecraft in a reduced quasi-static bending statement

Nonlinear dynamics of a planar rod system consisting of elastic inextensible rods, connected at the ends by elastic knot joints that allow large angles of rotation, is considered. The rod system is connected to an undeformed spacecraft that makes a turn about its center of gravity and movements along the horizontal and vertical axis as a free solid body. The motion of the system under consideration is described in a mobile coordinate system. The displacements of each rod are characterized by its final rotation as a rigid body with respect to a straight line passing through two adjacent hinges and a bend with a small transverse displacement. The equations of motion in compact form with the necessary explanations are given, which are obtained at speeds for the spacecraft and in the selected generalized coordinates for the rod system on the basis of the principle of possible displacements. Equations are also obtained in a matrix form convenient for numerical integration. The statement of the problem is presented, which is obtained by reducing the initial system of equations by quasistatic bending. From the equations of motion, “rapid” movements are excluded, which represent the bending of each rod, i.e. the first and second derivatives of the angles between the tangent to the curved axis of the rod and its undeformed axis. As a result, a new system of equations with necessary explanations is written in the matrix form. Examples of calculations with the necessary comparisons between two approaches are given: the problem of the reaction of a rod system to an arbitrary perturbing impulse; the problem of the deployment of the rod system from one position to another by the inclusion of elastic-viscous clamps integrated into the hinges, due to centrifugal and inertial forces.

Pages: 171-185 doi.org/10.25590/mkmk.ras.2018.24.02.171_185.02

**Safronov P.A.**

## Tension-compression asymmetry incorporating in solution the problem of cylindrical bend of the plate from shape memory alloy

In this paper the problems cantilever and pure cylindrical bending of plate with solid rectangular cross-section from shape memory alloy (SMA) are solved. Tension-compression asymmetry (TCA) of the SMA stress-strain curves and different elastic modules is incorporated. This paper is database on model of SMA non-linear straining in phase and structure transitions. In this paper proposition of active processes proportional loading are used. So, numerical solution of the problem is obtaining in non-coupled formulation and a slow enough processes are considered. In this paper hypothesis of Kirchhoff-Love in cylindrical bending problem and plane strain hypothesis to second component of the full strain tensor are accept. Dependence of dimensionless neutral surface coordinate, dimensionless normal stress and the compliance of plate from the TCA of stress-strain curves and the difference in elastic modules in case of martensite non-elasticity (MN) and direct transition (DT) are shown. Dimensionless neutral surface coordinate are only depend on ratio of tension and compression elastic modules of SMA in case of small values of dimensionless bending moment (MN) and in case of small values of dimensionless martensite volume fraction (DT). Dependence of the second component of the stress tensor from the first component in implicit form are obtained. Distribution of the elastic and the phase-structure axial strains in plate cross-section are found. Difference in solution of the pure bend of the beam and the cylindrical bend of the similar plate are shown. The low expression nonlinear dependence of the dimensionless plate curvature from the martensite volume fraction is found.

Pages: 186-206 doi.org/10.25590/mkmk.ras.2018.24.02.186_206.03

**Kabaldin Yu.G., Zavodinsky V.G.**

## Shear module and adhesion energy of nanocoatings used for cutting tools

Thin coatings are widely applied to protection of cutting tools against corrosion and wear. First of all, it is refractory coatings on the basis of such compunds as TiC, TiN and TiAlN which many times over increase the work life of tools. However still there is no clear understanding of details of interaction of these coatings with material of a cutter, mechanisms of their influence on efficiency of the cutting process and on durability of the tool. This understanding can be reached fully only on the basis of quantum-mechanical modeling which allows to study physical and mechanical processes at the atomic level. One of the most important characteristics of any coatings is the adhesion energy that is the work for separation a coating from a substrate. In this article results of a quantum-mechanical study by methods of the density functional theory and the pseudo-potential, taking into account spin polarization, the adhesion durability of nanodimensional coatings of TiC, TiN and TiAlN on the surface of iron (the main component of steel) and cobalt (binding material of hard-alloy cutters like WC-Co) are described. Calculations have shown that the adhesion energy of these coatings on iron and cobalt is enough for ensuring their reliable coupling of these compounds with the cutting tool when cutting and lies in the range of 3,5-4,1 J/m that is coordinated with the available literary data. The main physicomechanical characteristic which is responsible for wear resistance of the “coating-substrate” system in the course of its use as a cutter can be considered the shear module. In this work the calculations modeling shear deformations were carried out and the following values of the shear module have been obtained: G (GPa): G (TiC-Fe)=63; G (TiN-Fe)=68; G (TiAlN-Fe)=120; G (TiC-Co)=52; G (TiN-Co)=57; G (TiAlN-Co)=115. Calculated values of the shear module for coatings of TiC, TiN and TiAlN on cobalt are slightly lower than the values obtained for the same coatings on iron. Apparently, it is connected with a fact that contribution in elastic properties of the “coating-substrate” system makes not only a covering, but also a substrate; and the iron shear module (82 GPa) is bigger than the cobalt shear module (75 GPa). Both on an iron substrate, and on cobalt the following ratios are carried out: G (TiAlN-Fe/Co)> G (TiN-F/Co)> G (TiC-Fe/Co).

Pages: 207-220 doi.org/10.25590/mkmk.ras.2018.24.02.187_220.04

**Chernov M.I., Movchan A.A.**

## Non-linear analysis of thermomechanical bechaviour of a unidirectional composite with shape memory alloy fibers

In the framework of the modified model of nonlinear thermomechanical behavior of the shape memory alloy during phase and structural transformations, the processes of changing the stress-strain state of a unidirectional composite with an elastic matrix and fibers of titanium nickelide are described. Preliminary, with the help of the phenomena of direct strain accumulation or martensitic inelasticity, the fibers are given the initial tension phase-structural deformation. After this, the fibers in the martensitic phase state are combined with the matrix in order to ensure joint deformation of both components. With the subsequent heating of the fibers, in them an inverse thermoelastic martensitic phase transformation occurs, accompanied by a shape memory phenomenon. As a result, the entire composite experiences longitudinal compressive deformations, internal stresses appear in its elements (compressing in the matrix and stretching in the fibers). Upon subsequent cooling of the fibers, a direct thermoelastic martensitic phase transformation will occur in them under the action of internal tensile stresses. The fibers will elongate in the longitudinal direction, providing the process of deformation by stretching the entire composite. Using the approaches of micromechanics of composite materials, in a coupled statement, taking into account the variability of the Young’s modulus of titanium nickelide, the processes of changing the stress-strain state of such a composite as a whole and its components, changes in the phase composition of fibers were studied. Special attention is paid to the investigation of the possibility of implementing in this system a closed two-way shape memory effect (the phenomenon of repeatedly reversible shape memory) in which the internal stresses in the system disappear completely at the point of the direct thermoelastic phase transformation termination in the fibers. It is shown that this phenomenon can be described only in the framework of the shape memory alloys behavior model, taking into account, not only the process of martensitic meso-elements nucleation, but the process of their development under direct transformation, i.e. in the case of using systems of constitutive equations correctly describing the phenomenon of oriented transformation. It is shown that a closed two-way shape memory effect takes place at a certain value of the filament volume fraction, which is a decreasing function of a given initial deformation of filament.

Pages: 221-241 doi.org/10.25590/mkmk.ras.2018.24.02.221_241.05

**Buketov A.V., Smetankin S.A.**

## Modifier 4,4-sulfonilbis(4,1-phenylene)bis(n,n-diethyldithiocarbamate) influence on adhesive properties of epoxy matrix

On the first stage of the work the influence of selected technological modes of formation on the properties of composite materials based on epoxy-diane resin ED-20 and modifier 4,4-sulfonilbis(4,1-phenylene)bis(N,N-diethyldithiocarbamate) was investigated. As a result of analysis of the received data the optimal input mode of modifier into epoxy resin was selected and the forming technology of modified epoxy matrix was developed. On the second stage of the work the influence of amount of modifier 4,4-sulfonilbis(4,1-phenylene)bis (N,N-diethyldithiocarbamate) on adhesive properties of the modified matrix, which interacts with the aluminum base D16 and AMg5 grades and also steel base St 3 was investigated. The optimal content of the modifier to form a matrix, having an improved adhesive strength relative to the aluminum base, was set. It is shown that the input of modifier in amount of q =1,50…1,75 wt % into the epoxy binder ensures the formation of a material that differs with the following adhesive strength under tearing: σ =51,3…55,5 MPa (base made of aluminum D16 grade), σ =52,7…54,1 MPa (base made of aluminum AMg5 grade). To form the matrix with optimal adhesive properties relative to the steel base St 3 grade the modifier in amount of q =0,25 wt % must be input into the binder. This material is characterized by the following properties: adhesive strength under tearing σ =41,9 MPa, adhesive strength under shear τ =8,6 MPa, residual stresses σ =2,2 МPа.

Pages: 242-265 doi.org/10.25590/mkmk.ras.2018.24.02.242_265.06

**Borovskaya Ja.S., Glebova M.A., Grishin V.I, Guseva N.V.**

## Strength estimation of multi-bolted metal composite joints under combined operation of bearing and bypass stresses

It is necessary to know the load distribution in the joints elements and predict the failure ways associated with the different types of loads to forecast the damage and failure in composite joints. In the case when, the relation between the bearing stresses to the normal bypass stresses of the hole is high as a rule the failure occurs due to bearing. If this relation is low, the joint is broken down by tension on net-section. The ultimate stresses values of bearing-bypass stresses can be obtained from the experimental diagram of the bearing-bypass destructive stresses. The similar diagrams for carbon fibre-reinforced plastic (CFRP) were obtained by American researchers Grevs and Nike at the first time and have rather complex polygonal view. To get such destruction diagrams, it is required to fulfill a large quantity of experiments on unique experimental facility with two independent loading channels to change independently the bearing and bypass loads simultaneously. Similar experiments require significant financial investments and aren’t always available to researches. The well-known joints strength calculation method, using the finite element method in combination with experimentally obtained diagrams of the bearing-bypass load capacity is modified in the work. A quadratic approximation of intermediate values is proposed to describe bearing-bypass diagram by isolated experimental or calculated data on the bearing and bypass destructive stresses values. The results of the similar approximation describe the Grevs and Nike polygonal relation quite well as almost all points of the obtained diagrams keep within the spread of the experimental results of these authors. The obtained technique allows to refuse from bearing-bypass diagram construction with application complex multichannel experimental facilities, and thereby save resources significantly at the designing of the new equipment units.

Pages: 266-280 doi.org/10.25590/mkmk.ras.2018.24.02.266_280.07

**Vlasov A.N., Volkov-Bogorodsky D.B.**

## Asymptotic averaging of equations of thermoviscoelasticity with rapidly oscillating coefficients

A method of asymptotic averaging is developed for the equations of thermal viscoelasticity with rapidly oscillating coefficients. In contrast to the traditional approach, an asymptotic analysis of the equations introduces an additional parameter corresponding to the dependence of the material characteristics on temperature, and the functions of fast variables are considered in the parametric space. The averaging procedure is formulated in a corresponding manner so that nonlinear dependences that have a smoothly changing character with respect to fast variables are resolved in the asymptotic analysis parametrically. To implement this scheme, a complex analogy is used for the integro-differential equations of thermal viscoelasticity. The defining relationships between stresses and deformations are integral equations with relaxation kernels of difference type, and therefore they can be described by means of the integral Laplace or Fourier transformation through complex moduli for elasticity equations with rapidly oscillating coefficients that depend on spatial coordinates and temperature. A two-level scheme for solving auxiliary problems of the considered asymptotic method is developed, based on the analytical-numerical and finite-element approaches to determining the auxiliary functions at the micro- and macro levels, which is involved in the asymptotic representation of the solution. In particular, an algorithm for calculating the effective parameters of relaxation kernels is determined taking into account the dependence of the viscoelastic properties of the material on temperature. The developed approach allows to determine the effective properties of materials with a quasiperiodic structure, for example, the functional-gradient properties of viscoelastic composite materials with dependence of these characteristics on temperature. For solving auxiliary problems for fast variable functions at the micro level, a special block analytic-numerical method is developed for approximation of the solution with inclusions of arbitrary geometrical shape, in particular, for a spherical one with intermediate interphase layer.

Pages: 281-301 doi.org/10.25590/mkmk.ras.2018.24.02.281_301.08

**Anisimova V.V., Goloveshkin V.A., Myagkov N.N., Oskin D.A., Yushmanova O.O.**

## Model of changing the ejection angle depending on the depth of the projectile penetration

Due to the ejection phenomenon, a high-velocity encounter of a meteoroid or a space debris particle with a spacecraft surface produces the ejecta particles, which may represent a danger to exterior equipments of the spacecraft (the antennae, solar batteries, etc). Nowadays, the ejecta particles are considered as one of the main sources of the near-Earth space pollution that justifies an interest to their study. The main parameters characterizing the ejecta are the angles of their ejection with respect to the surface of the target, the spatial distribution by mass, size, velocity. In the present paper we consider analytically the problem of penetrating a rigid projectile into an infinite target in order to determine the dependence of the output angles of the ejecta particles (the ejection angles) on the penetration depth of the projectile. The problem is considered under the assumption of a plane deformed state of the target material. To describe the mechanical properties of the target, the hypothesis of incompressibility and ideal plasticity is adopted. On the basis of the energy balance, the equation of motion (penetration) of the projectile is obtained. The ejection angles are determined from the condition of minimum power of internal forces. In the case of a compact-projectile penetration (the penetration depth in units of transverse dimension is h ≈2¸3), the ejection angles are about 65-67 degrees, which corresponds to the known experimental data. In the case of a rod, for large values of the penetration depth h >>1, the ejection angle varies slightly with increasing h . For example, at h ≈10-20, it is approximately 72-75 degrees.

Pages: 302-313 doi.org/10.25590/mkmk.ras.2018.24.02.302_313.09