### №3-2018

Full versions of the articles

https://elibrary.ru/contents.asp?id=36173225

**Merijs-Meri R.V., Plume E.Z., Zhigun V.I., Zicans J.J.**

## Evaluation of the effect of structural parameters on elastic characteristics of carbon-carbon materials

This work is devoted to exploring of few structural parameters effect on elastic characteristics of carbon-carbon composites. Two such material types have been investigated. They have different systems of reinforcing: with orthogonal fibers arrangement in three directions, and with fibers arrangement in the xy plane with angles 0º( y ), ±60º and in transversal direction. The material of the second type had three varieties which differ with the character of fibers allocation of z direction in x , y axis, of y direction in z , x axis, and with the diameter of reinforced bundles. The experimental research of these composites under four different types of loadings (tension, compression, bending and shearing) is made. The received experimental data are compared with the calculated data. The effect of diameter of reinforcement and its arrangement density on elastic characteristics of researching materials has been evaluated. It’s shown that composites with the smaller diameter of reinforcing bundles have higher values of elasticity modulus and shear modulus. The reduction of reinforcing bundles diameter contributes to their layup density with the same volumetric content and has a beneficial effect on their elastic properties. The effect of changing of volume of transversal reinforcing on the studied composites characteristics has been reviewed. It’s shown that even insignificant content of reinforcement in the direction z (to 3%) leads to significant properties increasing of the material in that direction without important reduction of elastic characteristics in another direction. The effect of reinforcement design on formation of carbon-carbon composites properties has been investigated. It’s shown that three-dimensional reinforced carbon-carbon composites with fibers arrangement in the directions 0º, ±60º help to manage effectively of elastic characteristics in specified direction because of fibers volume relocation in reinforcing direction.

Pages: 319-334 Elibrary

**Belov P.A., Lurie S.A., Lykosova E.D.**

## To the rheology of suspensions: a scale effect in fine-filled suspensions for the friction account between particles and liquid

In the paper, we investigate viscosity models of suspensions based on the “classical” Reiss averaging of the viscosity and taking into account the dynamic friction of the liquid on the particle. It was found that the allowance of friction determines the scale effect, which manifests itself in the fact that viscosities of both liquid and “dry” particles in the composition have larger values than outside of the composition. For finely dispersed suspensions with a non-zero coefficient of dynamic friction, the characteristic thickness of the boundary layer is commensurable with the distance between the particles, and the scale effect makes a significant correction to the viscosity of the suspension in comparison with the “classical” Reiss averaging. By analogy with the hypotheses of averaging the elastic moduli in the theory of composites, in the hydrodynamics of suspensions, the corresponding hypotheses for averaging dynamic viscosities are proposed. The hypotheses of effective inclusion, effective liquid, effective volume fraction and hypothesis of three phases, allowing to take into account scale effects of the first order in the Navier-Stokes hydrodynamics associated with friction of a liquid and particles, are formulated. The statements of all the above hypotheses are different forms of the same solution for the Navier-Stokes equations with boundary conditions that take into account the friction of the liquid on the particles. The appearance of a turbulent flow specific for suspensions associated with friction at the boundary of a particle with a liquid (nonideal slip of a liquid) is established. The size of the vortices of this flow is directly related to the characteristic thickness of the boundary layer and, correspondingly, to the coefficient of dynamic friction. This fact allows us to propose a new method for determining the coefficient of dynamic friction based on the known viscosity of the suspension and the characteristic thickness of the boundary layer.

Pages: 335-348 Elibrary DOI: 10.33113/mkmk.ras.2018.24.03.335_338.02

**Tutunnikov N.P.**

## About choice of holes sizes and layouts for fastener assembly of rotor composite blades

A question of rotor composite blades intelligent design is considered in this article. Usually the layout and size ratio of aperture for fastener assembly bolts into rotor longeron blade are chosen on the basis of the constructive and technological reasons. Incidentally of course the strength and fatigue resources requirements are took into account. However, as a rule, the preference gives up to easiest decision that is simplest from the technological point of view. Meanwhile it is the stress-strain state features into this region that who determine largely strength and fatigues properties of blade on the whole. This is check out of bench test results as well as the vehicle exploitation into flight conditions. Series of finite element calculus are performed for estimation of optimal sizes and layouts of fastener assembly holes into rotor blade butt part. Nonlinear contact interaction between fastener assembly bolts and blade composite material are took into account for finite-element model. Hill’s criterion is used for estimation of stress-strain state of multilayer composite material. Incidentally the necessity are appeared of determination of preliminary estimations of strength limits for arbitrary layup multilayer material that having known monolayer characteristics. Deformation curve building algorithm for multilayer composite material is developed. Diagram data are used for estimation multilayer layup stiffness characteristics and limit linear load. Composite material stress state estimation calculus is implemented for fastener assembly regions at determined loading conditions. 45 layout and size ratio variants of aperture are considered. On the basis of the calculus results conclusion are made that actual variant of constructive solutions not optimal.

Pages: 349-361 Elibrary DOI: 10.33113/mkmk.ras.2018.24.03.349_361.03

**Kazarina S.A., Movchan A.A., Sil’chenko A.L.**

## The influence of inelastic deformations on the reverse thermoelastic transformation start temperature in shape memory alloys

The process of thermoelastic phase transformations in shape memory alloys (SMA) can be caused not only by temperature changes, but also by isothermal loading (direct transformation) or by unloading (reverse transformation). Usually this effect is described by expressing the phase composition parameter – the martensite volume fraction through the characteristic temperatures of the beginning and the completion of phase transitions, which, in turn, depend on the acting stress. These latter dependences can be obtained as sufficient conditions for the fulfillment of the dissipative inequality and are in good agreement with the experimental data. According to the experimental data, the temperature of the onset of the inverse thermoelastic martensitic transformation in the SMA depends not only on the acting stresses, but also on the accumulated phase-structural deformations, and this temperature increases if the convolution of stress deviators and phase-structural deformations is positive and decreases if this convolution is negative. According to experimental data and known thermodynamic models of SMA behavior, based on the hypothesis of the additivity of the Gibbs potential of these materials, the increment in the temperature of the onset of the inverse transformation, which depends on the phase-structural deformation of the shape change, is proportional to this convolution. However, following experimental data the temperature of the onset of the reverse transformation in titanium nickelide increases very substantially with the growth of phase-structural deformations even at zero stresses. This effect can not be obtained using the above-mentioned temperature dependence of the onset of the reverse transformation through the convolution of stress deviators and phase-structural deformations, since at zero stresses this convolution is equal to zero. The first part of the work is devoted to the analysis of relevant experimental data for SMA based on iron and titanium nickelide. The influence of the nature of the initial deformations and their values, different ways of their setting and type of the strain state on the character of the temperature change of the beginning of the reverse thermoelastic phase transformation in the absence of stresses is studied.

Pages: 362-376 Elibrary DOI: 10.33113/mkmk.ras.2018.24.03.362_376.04

**Basistov Yu.A., Karnet Yu.N., Vlasov A.N.**

## Dynamic neural network as a model of viscoelastic media

The paper utilizes a nonlinear viscoelastic model of the medium with associative and hereditary memory in the form of a system of integro-differential equations. The hereditary memory is contained (for a long time) in the Volterra integral operator and the associative (short-term) memory is determined by the differential operator. Identification of the model is solved using neural networks in the version of the finite-dimensional approximation with discrete time for a composite material based on a matrix of natural rubber (polyisoprene), filled by 20% with N-330 carbon black. The study is carried out both in the small strain mode and in the finite strain mode. The issues studied in the paper are the accuracy of reproduction by the model of the real nonlinearity function and its ability to generalize the experimental material based on the training sample. The identification of this model showed a good reproduction of the actual function of nonlinearity of a real viscoelastic material in the finite strain mode.

Pages: 377-391 Elibrary DOI: 10.33113/mkmk.ras.2018.24.03.377_391.05

**Kozel A.G., Starovoitov E.I.**

## The bending of an elastic circular sandwich plate on the pasternak foundation

Due to the wide application of composite, including three-layer, structural elements in construction and machine building, it is necessary to create appropriate mathematical models and methods for calculating their stress-strain state under different operating conditions. Here is the statement of the boundary value problem of axisymmetric deformation of an elastic three-layer circular plate on the two-parameter basis of Pasternak. This allows the influence of shear properties of the base material on the stress-strain state of the calculated design to take into account. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. The Kirchhoff hypothesis of incompressibility, straightness and perpendicular to the normal to the deformed median surface is valid in thin bearing layers. In a relatively thick incompressible thickness of the filler is performed Tymoshenko hypothesis with a linear approximation of the displacements in the thickness of the layer. The contour assumes the presence of a rigid diaphragm that prevents the relative displacement of the layers. A non-uniform system of ordinary linear differential equations of equilibrium by the variational method is obtained. Three types of boundary conditions are formulated. The solution of the boundary value problem is reduced to finding three required functions – plate deflection, shear and radial displacement in the filler. The General analytical solution of the boundary value problem in Bessel functions is obtained. The numerical analysis is carried out at evenly distributed load and rigid filling of the plate contour. The influence of shear properties of the base on the stress-strain state of the plate at different compression ratios is studied numerically. The calculated values of displacements and shear in the filler obtained using Pasternak and Winkler models are compared.

Pages: 392-406 Elibrary

**Khokhlov A.V.**

## Simulation of hydrostatic pressure influence on creep curves and poisson’s ratio of rheonomic materials under tension using the rabotnov non-linear hereditary relation

The Rabotnov physically non-linear constitutive equation for non-aging elasto-viscoplastic materials with four material functions is studied analytically in order to outline the set of basic rheological phenomena which it can simulate, to clarify the material functions governing abilities, to indicate application field of the relation and to develop identification and verification techniques. The relation is quasi-linear, it doesn’t involve the third invariants of stress and strain tensors and implies that their hydrostatic and deviatoric parts don’t depend on each other. Assuming minimal restrictions on material functions of the relation, general properties of the creep curves for shear, volumetric, axial and lateral strains generated by the model under uni-axial tension and constant hydrostatic pressure are analyzed. Conditions for creep curves monotonicity and for existence of extrema and sign changes of strains, the Poisson ratio (lateral contraction ratio in creep) evolution in time, evolution of the strain triaxiality ratio (which is equal to volumetric strain divided by deviatoric strain) and their dependences on pressure and tensile stress levels and material functions characteristics are studied. Taking into account the pressure influence and volumetric creep (governed by two material functions of the model) is proved to affect strongly the qualitative behavior and characteristic features of longitudinal creep curves and the Poisson ratio evolution and its range. In particular, it is proved that the Rabotnov relation is able to simulate non-monotone behavior and sign changes of lateral strain and Poisson’s ratio under constant tensile load (even if the pressure is zero) and the longitudinal strain may start to decrease provided the pressure level is high enough. The expressions for Poisson’s ratio via the strain triaxiality ratio and in terms of tensile stress and pressure levels and material functions of the model are derived. Assuming material functions are arbitrary, general bounds for the Poisson ratio range are obtained and the influence of pressure level is studied. Additional restrictions on material functions and stress levels are derived to provide negative values of Poisson’s ratio. Conditions for its increase or decrease and for its non-dependence on time are found. It is proved that, for any fixed tensile load, the higher the pressure level is the more creep curves for volumetric, axial and lateral strains shift down along the strain axis and the Poisson ratio decreases at any time moment. The qualitative properties of the theoretic creep curves families and Poisson’s ratio produced by the constitutive equation are compared to typical properties of test creep curves of elasto-viscoplastic materials under hydrostatic pressure in order to reveal a set of necessary phenomenological restrictions which should be imposed on material functions to provide an adequate description of typical effects. A number of specific features and quantitative characteristics of the theoretic creep curves are found that can be employed as indicators of the constitutive relation applicability (or non-applicability) for simulation of a material behavior which are convenient to check in creep tests with various levels of pressure and tensile stress. The specific properties and restrictions of the model with zero dilatational creep compliance which simulates a material exhibiting purely elastic volumetric deformation are considered.

Pages: 407-436 Elibrary

**Pyrikov M.D., Shumikhin T.A.**

## The distribution of materials in a cloud of fragments at the fragmentation of a drop on a mesh bumper

As it was shown in the previous works the high-velocity perforation of thin steel meshes by aluminum projectile can give rise to cumulative jets generated by the restrained flow of the projectile material through the mesh cells. The intensity and velocity of jets and their spatial distribution depend on the geometry of meshes and on the location of an impact point relatively the symmetry centers of the mesh cell. In this work we estimate the distribution of material among the jets produced as a result of an impact of a falling water drop on a rigid mesh. We consider the process of water drop splitting is similar to the high-velocity case with an aluminum projectile. The hydrodynamic of the interaction of a liquid or plastic projectile with some barrier or surface can be characterized by Weber number. The estimate of these numbers for the case of aluminum projectile and water drop in the range of corresponding velocities reveals the similarity of their values what approve the use of the experiments with a water drop for the modeling of high-velocity experiments with an aluminum projectile. Opposite to the specific features of high-velocity experiments the experiments with falling water drop allow us to study in-situ the process of interest. For the purpose of direct observation we used high-speed photography. The distribution of the material among the jets was estimated by the area of spots left on the sheets of filter paper by the fragment of the drop colored with ink dies. The high-speed photography allowed us to estimate the velocities of the jets that revealed that they had higher velocity than the velocity of the water drop prior to impact – this result is similar to the effect of the acceleration of the jet material observed in the high-velocity experiment with aluminum projectile. Besides we found that the regime of the fragmentation of the water drop lowered the difference among the jet intensities comparatively the regime of quasistatic squeezing of the water drop through the mesh cells. We classified the spatial distribution of jets depending on the distance of the impact points from the centers of symmetry of the mesh cells.

Pages: 437-460 Elibrary DOI: 10.33113/mkmk.ras.2018.24.03.437_460.08

**Stolbova O.S.**

## Numerical modeling of the thermomechanical behavior of shape memory alloys under finite deformations

In the framework of finite deformations theory a model of the behavior of shape-memory alloys (SMA) in view of the austenite-to-martensite phase transition and plastic deformation is constructed. All the nonlinear relations that occur under finite deformations are linearized, an approach based on the kinematics of superposition of small deformations on finite ones is used. It is assumed that the rates of change in the elastic, temperature, phase, structural and plastic deformations are additive. To describe the change in phase and structural deformations a simplified version of the model of nonlinear deformation of the SMA generalized to finite deformations is used. We take into account the shift of characteristic temperatures of the phase transition in the loaded material and also the dependence of the elastic moduli on the fraction of the martensitic phase. To describe the elastic behavior of the material a simplified Signorini law is used. The statement for the boundary value problem in differential form and the variational formulation in the Lagrange form are obtained. As boundary value problems we consider problems on the cantilever beam bending and torsion of a cylindrical sample of SMA. At the initial time the samples clamped at the left end face are in the austenitic state. To the right end the forces causing bending/torsion are applied, so the elastic and after the yield point plastic deformations appear in the material. After that at the same temperature the samples are partially unloaded, then they are cooled under the load with the phase and structural strains occurred as a result of direct martensite transition. The problems are solved in three-dimensional formulation by the finite element method using the step-by-step procedure. The distributions of the intensity of plastic, phase and structural deformations and the intensity of stresses are obtained.

Pages: 461-476 Elibrary DOI: 10.33113/mkmk.ras.2018.24.03.461_476.09

**Grishanina T.V., Russkikh N.M.**

## Aerodynamic characteristics of the deformable airfoil with quazistatic subsonic flow

The linear problem of deformation and aerodynamic loading of a straight wing thin airfoil of a large elongation is considered. The wing airfoil consists of an undeformed fore part and an elastic tail. The transverse displacement and the small angle of rotation of the fore part are considered to be given functions of time The transverse displacement of the elastic tail is represented by the Ritz method in the form of expansions in terms of given functions with unknown coefficients, which are taken as generalized coordinates. The aerodynamic load is determined by the theory of plane attached airflow of an airfoil with quasistationary subsonic flow of compressible gas. The equations of aeroelastic oscillations of a deformed airfoil for generalized coordinates are obtained on the basis of the principle of possible displacements. The calculations for two types of the power schemes of the elastic tail of the airfoil are performed. In the first case, the tail is formed by a thin elastic plate of constant thickness rigidly connected with the fore part, the aerodynamic shape of which is obtained by means of an overhead profiled foam. The filler in this case does not work for bending and shear and calculations are carried out for a profile with constant characteristics along the length without allowance for shear. In the second case, the elastic part of the airfoil consists of honeycomb filler working for shear and a thin skin of constant thickness, working for stretching-compression. In this case, the thickness of the elastic tail decreases linearly to zero on the rear edge. The distributions of the aerodynamic load along the chord of the deformed profile and the values of the quasistationary aerodynamic coefficients of the lift and pitch moment are obtained for the attack angle and pitch velocity of the fore part by quasistatic elimination of generalized coordinates.

Pages: 477-489 Elibrary DOI: 10.33113/mkmk.ras.2018.24.03.477_489.10

**Firsanov Vic.V.**

## Features of the bending of a thin rectangular plate of material with an unchanged volume

The maximum value of the coefficient of transverse compression (Poisson) for isotropic linearly elastic materials is limited to 0.5. If we add linear deformations for a material with , we obtain as a consequence of physical relationships, which is valid for small deformation. This indicates that when a body from such a material is deformed, only its shape changes, and the volume remains unchanged. The deformation of the change of volume, therefore, is zero, and the modulus of elasticity characterizing the resistance of the medium to a change of volume tends to infinity. Therefore, in the physical relationships resolved with respect to normal stresses and containing the product of this module by the deformation of the volume change θ, instead of this product, which becomes indeterminate as a result of multiplying zero by infinity, a force function with a dimension of stress is introduced. Materials that have the property of constant volume during deformation are called incompressible. These include rubber, various types of rubbers and some others. These materials, which are not very common in technology, thanks to a unique property, allow us to test some of the classical problems of mechanics of a solid deformable body, based on certain hypotheses. One of these problems is the bending of thin plates. The classical problem of plate bending is based on Kirchhoff’s hypotheses: the absence of linear deformation in the direction perpendicular to the base of the plate, transverse shear deformations and normal stress in the transverse direction. The deflection of the plate from the action of forces acting in planes perpendicular to the bases of the plate due to one of the hypotheses is a two-dimensional function, which significantly simplifies the problem, in spite of the fact that the other desired functions of displacements and stresses linearly depend on the coordinate perpendicular to the base of the plate. The transition to integral characteristics of the state of stress by integrating the differential equations of equilibrium in stresses over the plate thickness allows to get rid of the linear function and finally formulate the bending problem as two-dimensional one. Static boundary conditions are also formulated with respect to the integral characteristics of the state of stress which leads to some errors in the solution near the boundaries of the plate according to the Saint-Venant principle. Some classical hypotheses of the theory of bending plates are inconsistent with respect to the incompressibility condition, therefore, it is necessary to refuse from classical hypotheses and build the problem using other assumptions. Also, for some problems, depending on the type of load and the boundary conditions, it is possible to obtain fairly simple solutions without going over to the integral characteristics of the state of stress.

Pages: 490-498 Elibrary DOI: 10.33113/mkmk.ras.2018.24.03.490_498.11