### №1-2021

Full versions of the articles

https://elibrary.ru/contents.asp?id=45583919

**Belov Pt.A., Golovina N.Ya.**

## Criticism of the law of deformation ugent for elastoplastic materials and an alternative to it

An empirical UGent model for an elastoplastic material with hardening is analyzed. The deformation curve, according to this model, is composite. The definition area is divided into three segments. On the first and third, the Ramberg-Osgood law with different exponents is postulated, and on the second, a curve is postulated that smoothly connects the curves on the first and third segments so that, on the whole, the curve is differentiable. It is shown here that this brilliant idea of the authors of the UGent law was implemented by them with several logical flaws. To correct these shortcomings, the spline is modified so that the deformation law on the second segment contains a linear combination of two linear polynomials and two power functions with different exponents included in the structures of the deformation laws on the first and third segments. Also, new conditions have been formulated that must be satisfied by the law of deformation on the third segment at its initial and final points. It is shown that there are just enough new arbitrariness to satisfy all the identified requirements for a differentiable three-link spline of the deformation curve. From a sample of 582 experimental points for steel 35, ten significant physical parameters of the proposed model were determined. Two of them, the coordinates of the strength point on the true curve in the strain-stress coordinates, are selected as normalizing factors. The remaining eight are determined from the minimum requirement of the objective function by the gradient descent method. The total quadratic deviation is selected as the objective function. The corresponding theoretical deformation curve with the established parameters is constructed. It is shown that the accuracy of the proposed model is quite high. The standard deviation of the constructed curve from the sample of experimental points is 1.8%.

Pages: 3-16 doi.org/10.33113/mkmk.ras.2021.27.01.003_016.01

**Nestsiarovich A.V., Starovoitov E.I.**

## Non-axisymmetric deformation of a freely supported three-layer plate in its own plane

Three-layer structural elements are widely used in aerospace and transport engineering, construction, production and transportation of hydrocarbons. The theory of bending of circular three-layer plates those are not symmetrical in thickness under external various kinds of force actions is currently developed quite fully. Here we present a statement of the boundary value problem of non-axisymmetric deformation of an elastic three-layer circular plate in its plane. The center of the plate is fixed, its contour is freely supported. The physical equations of state relate stresses and deformations to the relations of the linear theory of elasticity. The equilibrium equations are obtained by the Lagrange variational method. Force boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding two desired functions – radial and tangential displacements in the layers of the plate. These functions satisfy an inhomogeneous system of linear partial differential equations. To solve this problem, the method of decomposition into trigonometric Fourier series is applied. To determine the desired amplitude functions of each of the series members, a system of four ordinary linear differential equations is obtained. The analytical solution of this system is written out in the final form in the case of a non-axisymmetric cosine radial load with constant amplitude. The load is applied in the middle plane of the filler. The numerical approbation of the solution is carried out for a freely open plate contour. The dependence of radial and tangential displacements on polar coordinates is numerically investigated. Graphs of changes in displacements along the radius of the plate for different values of the angular coordinate are given. The dependence of the displacements on the thickness of the bearing layers and the aggregate is illustrated.

Pages: 17-30 doi.org/10.33113/mkmk.ras.2021.27.01.017_030.02

**Lokteva N.A., Nguyen D.P.**

## Unstationary interaction of a three-layer plate with a dying plane wave in elastic transient stress-strain

A study of the interaction of a three-layer plate with a damped plane wave in the soil has been carried out. As a model of an obstacle in the soil, a three-layer plate is considered, described by the system of equations of V.N. Paimushin, placed in the soil and dividing it into two parts. A flat formulation of the problem is considered. The boundary conditions correspond to the hinge fixation of the obstacle, and the initial conditions are zero. A damped plane wave induced in one of the semi-media is considered as an external influence. To describe the motion of the soil, the equations of the theory of elasticity, the Cauchy relations and the physical law, or their equivalent displacements in potentials and the Lame equations, are used. The problem is solved in a related setting, where the movement of platinum and its surroundings is considered together. All components of the equations of motion of the plate and media are expanded into trigonometric series that satisfy the boundary conditions, and the Laplace transform is applied to them. To specify a plane damped incident wave, the scalar potentials of the displacement field are considered, to which the Laplace transform in time and the expansion into a trigonometric series in coordinate are also applied. The equality of normal displacements and stresses at the interface between the medium and the plate is taken as the conditions for contact between the plate and the soil. It is also considered that the pressure amplitudes and normal stresses coincide. After determining the constants from the contact conditions, the displacement values and the values of normal and tangential stresses are found, after which their originals are found. Since the analytical determination of the originals of functions is impossible, the Durbin method is applied.

Pages: 31-46 doi.org/10.33113/mkmk.ras.2021.27.01.031_046.03

**Guseva M.A., Petrova A.P.**

## Methods of testing amd research of thermosetting binders for pcm

The polymer binder (matrix) plays an important role in the composition of the polymer composite material (PCM), ensuring the integrity and shape of the product, the mutual arrangement of the reinforcing fibers, the distribution of the acting stresses over the volume of the material, distributing a uniform load on the reinforcing fibers and at the same time preventing the growth of cracks in the PCM. The level of providing increased mechanical properties of fibers depends on such properties of the polymer matrix as strength, stiffness, ductility, fracture toughness, impact strength. Properties such as heat resistance, fire resistance, heat resistance, impact strength, water and weather resistance, chemical resistance of the finished product are determined by the characteristics of the polymer binder and PCM based on it. When developing polymer binders, a large number of research and testing methods are currently involved, while the main properties of binders are monitored both at the stage of obtaining a polymer matrix and in the process of processing it into products – at the stage of obtaining a composite material, the binder is described by parameters that ensure manufacturability processing of the composite and allow you to select the mode of this process. In the presented work, the main methods of testing polymer binders, which are currently used to control their quality, as well as their physical and mechanical characteristics, are considered. A description of the main methods for determining the glass transition temperature, viscosity, pot life and gelation time of a polymer binder is given. An overview shows a wide range of methods that make it possible to obtain complete information on the properties of thermosetting binders both at the stage of their preparation and as part of PCM – thermal stability, fire safety, the influence of climatic factors, and others.

Pages: 47-64 doi.org/10.33113/mkmk.ras.2021.27.01.047_064.04

**Badina E.S., Sidorov V.N.**

## Finite element modelling of composite beams vibration taking into account damping nonlocal in time

The paper is devoted to finite element modelling of the beam vibration process, taking into account the complex internal structure of the material. In the finite element model of the beam external damping (air damping) and internal damping are taken into account. The external part of the damping forces is considered local, i.e. depending on the velocity of element nodes only at the current moment. Internal damping is considered nonlocal in time, i.e. depending on the velocities on the vibration time history. Unlike the nonlocal in space damping model nonlocal in time model can be easily integrated to the finite element analysis algorithm. The central differences method is used to solve the dynamic equilibrium equation and the continuous internal damping kernel function is replaced by its discrete equivalent. The beam vibration model considering nonlocal damping is implemented in MATLAB. The damping of glass-fiber reinforced plastic beam element vibrations is considered in this paper as the numerical example. The parameters of nonlocal model are determined with the least squares method using numerical simulation data obtained in SIMULIA Abaqus CAE. The advantage of flexible nonlocal damping model over the local one (Kelvin-Voight model) is shown for the orthotropic beam vibration simulation in cases when the one-dimensional models are preferable.

Pages: 65-72 doi.org/10.33113/mkmk.ras.2021.27.01.065_072.05

**Erasov V.S., Oreshko E.I., Utkin D.A., Yakovlev N.O.**

## Determination of shift durability of polymeric composite materials at an indentation

Fiber-matrix adhesive strength determination approach using method of pushing out the multifilament cylinder from the thin carbon fiber polymer composite sample by the cylindrical indenter with the flat basis is offered. The equipment which had a groove for sample fixing at test and an opening which provides expression of a part of a sample under an indenter was developed for tests of polymeric composite materials for shift by a method of an indentation. Also the indenter of a cylindrical form which allows to squeeze out a cylindrical part of a sample in an opening on equipment was developed and made. The method is economic as the thickness of samples does not exceed 2 mm that favourably distinguishes it from methods of determination of durability at shift by modern methods: method of a short beam and Iosipesku’s method which demand samples of the big sizes. Types of deformation and destruction of samples after tests, types of charts of deformation are considered at tests and their characteristic sites. Results of the carried-out tests show that charts of shift have three characteristic sites: – a quasilinear site of elastic loading on which the effort of pushing out linearly grows with increase in depth of introduction of an indenter; – the step site of emergence and destruction development with loss of energy of deformation on development of new free surfaces of destruction, is presented in the form of sharp falling of effort of pushing out; – a flat site of sliding with a friction of the tiny multifilament cylinder at its pushing out from a sample-section. Comparison of the received results of tests with values of durability is carried out at shift by polymer composite materials received by a method of a nanoindentation.

Pages: 73-88 doi.org/10.33113/mkmk.ras.2021.27.01.073_088.06

**Bobok D.I.**

## Analytical solution of the problem of a toroidal, ellipsoidal and spherical tank made of shape memory alloy under internal pressure

The paper focuses on solving the problem of mechanics of a solid deformable body about a toroidal tank made of shape memory alloy (SMA) under internal pressure during a direct thermoelastic martensitic phase transformation under constant pressure. As special cases, a shell with a circular cross-section, as well as an ellipsoidal and spherical tank are considered. For a spherical tank, the problem of relaxation under direct transformation was solved, where it was necessary to determine the necessary decrease in the uniformly distributed load during cooling during direct thermoelastic phase transformation so that the deflection of the shell remained unchanged. The behavior of the shell was described in the framework of the model of linear deformation of the SPF under phase transformations and the theory of thin isotropic shells. Also, the problem was solved within the framework of an unrelated problem statement, that is, the distribution of the phase composition and temperature parameter over the shell material at each moment of time was assumed to be uniform. Similarly, the possibility of structural transformation in the shell material, the variability of the elastic modules during the phase transition, and the property of the SPF resistivity were neglected. To obtain an analytical solution to all the equations of the boundary value problem, the Laplace transform method was used in terms of the volume fraction of the martensitic phase. After the transformation in the image space, an equivalent elastic problem is obtained. In solving this problem, the Laplace images of the desired quantities are obtained in the form of analytical expressions that include operators that are Laplace images of elastic constants. These expressions are fractional-rational functions of the Laplace image of the phase composition parameter. To return to the original space, the expressions for the desired values in the image space are decomposed into simple fractions. As a result of the inversion of these fractions, the desired analytical solutions are obtained.

Pages: 89-109 doi.org/10.33113/mkmk.ras.2021.27.01.089_109.07

**Tarlakovskii D.V., Vestyak A.V., Zemskov A.V.**

## Unsteady elastic diffusion bending model for a bernully-euler beam on a winkler foundation

We consider the unsteady elastic diffusion vibrations problem of an orthotropic Bernoulli-Euler beam on an elastic foundation under the action of a distributed transverse load. The Winkler model is used as an elastic foundation model. For the mathematical formulation, we use the system of Bernoulli-Euler beam bending equations taking into account diffusion. These equations are obtained with use the d’Alembert variational principle, which is applied to the elastic diffusion model for a continuum. The resulting model consider the diffusion fluxes relaxation. The problem formulation is closed by homogeneous boundary conditions, which expressing the simple support conditions and zero initial conditions (internal disturbances absence at the initial time). The problem solution is sought using the Green’s functions method and is represented as convolutions of Green’s functions with functions defining unsteady volumetric disturbances. To find the Green’s functions, the integral Laplace transform in time and the expansion in Fourier series in the longitudinal coordinate are used. As a result, the original system of equations for elastic diffusion beam vibrations is reduced to the linear system of algebraic equations with respect to the sought functions Fourier coefficients in the Laplace transformation. The Laplace transform inversion is done analytically due to residues and operational calculus tables. Calculation examples for a beam with rectangular section are considered. The beam deflections and the diffusants concentration increments under the action of a impulsively applied distributed transverse load are found. Numerical study of the mechanical and diffusion fields interaction in a beam is performed. We used three-component continuum as an example. The solution is presented in analytical form and in the graphs form of the displacement fields and concentration increments as functions of time and coordinate. At the end of the article, the main conclusions about the coupling effect of the stress-strain state and mass transfer in the beam are represented.

Pages: 110-124 doi.org/10.33113/mkmk.ras.2021.27.01.110_124.08

**Kriven G.I., Lykosova E.D.**

## Strength analysis of fiber composites modified with various nanofibers in the case of pure shear along the fiber

The paper considers a fiber composite, the fibers of which on the surface contain special layers formed by whiskers. The presence of whiskers leads to an increase in the strength of the composite material during longitudinal shear. Traditionally, such a composite consists of three phases: a fiber, an interfacial whiskerized layer, and a matrix. There are investigated the modified composites consisting of two phases – fiber and an interfacial whiskerized layer acting as a binder. Two types of fiber composites with an epoxy binder are considered, in the first, the base fiber is IM7 carbon fiber with a whiskerized zinc oxide layer, and in the second, the base fiber is T650 carbon fiber with a whiskerized CNT layer. In both cases, the length and diameter of the whiskers are selected in such a way as to ensure the maximum volumetric content of whiskers in the whiskerized layer. For a modified composite loaded with longitudinal shear, calculations of the stress-strain state in each of the phases and calculations of effective properties are carried out. The deformed state in the whiskerized layer and the matrix is investigated, and a comparative analysis with the deformed state of a similar classical composite is carried out. The deformation fields of the modified and classical composites were determined under the conditions of the equivalence of force loading. The results of the study showed that the modified composites are able to withstand the load significantly exceeding the load that can be withstood by similar classical composites. This makes it possible to judge the increase in the strength of modified composites with whiskerized fibers in comparison with similar classical composites in the case of pure shear along the fibers. At the same time, it was shown in the work that from the point of view of strength and effective properties, the whiskering of the composite by microfibers made of zinc oxide is more preferable than the whiskering of the composite by the more rigid CNTs.

Pages: 125-142 doi.org/10.33113/mkmk.ras.2021.27.01.125_142.09