### №3-2020

Full versions of the articles

https://elibrary.ru/contents.asp?id=44100159

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

## Influence of viscoelastic properties of the matrix on the behavior of a unidirectional composite with shape memory alloy fibers

Composites with fibers or layers of shape memory alloys (SMA) and elastic or viscoelastic matrix are promising materials for creating multiple-action actuators and power exciters, “artificial muscles”, deployable systems, surfaces of variable geometry, etc. Due to the elastic properties of the matrics, such composites can have the property of repeatedly reversible shape memory, i.e. they can have a two-way shape memory effect, which is not typical for either the binder or the SMA filler without a special thermomechanical processing. With proper design of such a composite, it is possible to achieve the closed property of the two-way shape memory effect, that is, its shape can be controlled in a certain range of changes only by changing the temperature of the filler. Modeling the thermomechanical behavior of composites with elements from SMA is complicated by the complexity of the constitutive relations for SMA, which are of a differential nature, should be considered in the coupled formulation, taking into account the variability of elastic modules of SMA. These constitutive relationships should take into account both the phase and structural mechanism of inelastic deformation of SMA, the fundamental difference between these mechanisms and their mutual influence. This paper uses a variant of the SMA nonlinear deformation model that takes into account all these effects. Polymer binders of composites have not only elastic, but also viscoelastic properties. This circumstance is not taken into account in most works devoted to describing the behavior of composites with elements from SMA. In this paper, the simplest model of linear hereditary viscoelasticity (Kelvin body) is used to evaluate the effect of viscoelastic properties of the matrix on the behavior of a unidirectional composite with SMA fibers. A significant influence of viscoelastic properties on the behavior of the composite at low rates of temperature change of SMA fibers has been established.

Pages: 287-312 Elibrary

**Krasnov L.L., Muiznieks K.I., Plume E.Z., Zhigun V.I.**

## Universal methods for determining the shear modules of composite materials

Some of the reasons limiting of the use of composite materials in structural parts have been noted. Among them we can distinguish their relatively low stiffness during interlayer shear and the lack of reliable methods for its establishment. The analysis of the methods used for estimating the modulus of interlayer shear of composite materials has been presented. It has been shown that they are mainly based on the experiments of testing samples for transverse bending with the measurement of deflection under the point of application of the load. The methods are quite labor-consuming, inconvenient in practical use, and do not allow to obtain stable and reliable values of the determined characteristics of composite materials. Modification of some of them by taking into account certain factors affecting the deflection of the sample, such us the curvature of the supports, significantly complicates the process of estimating the interlayer shear modulus. However, the reliability of the determined values is not proven. Three methods have been proposed for determining the shear modules in the three main planes of elastic symmetry of orthotropic composites. For two of them the necessary devices to ensure the simplicity of their implementation and the approach to the strain gauges glued to the sample have been developed. The nature of the distribution of tangential stresses in the sample near the points of application of concentrated forces to it has been considered. Based on this factor, the optimal parameters of samples and their loading schemes have been established. The acceptability of each method has been evaluated by comparing the determined value of the characteristic with its value obtained using another, reliable method. The test has been performed on various types of composite materials. It has been established that one of the proposed methods also allows determining also the interlayer shear modulus of composites. It has been shown that the values of the interlayer shear modulus for orthogonally reinforced composite materials have significantly lower values than the values of the shear module in the plate plane. The advantages of the considered methods have been noted: their universality, convenience and ease of implementation, as well as stability and reproducibility of the obtained values.

Pages: 313-326 Elibrary

**Fedotenkov Grigory V., Kireenkov Alexei A.**

## Motion of a composite spherical shell on a solid surface under conditions of combined dry friction

The implementation of the theory of multicomponent dry friction in some engineering problems of the contact interaction of composite shells and rough rigid support planes is proposed. The main attention is paid to the construction of analytical models of combined dry friction, taking into account the anisotropy of the dry friction coefficients and the real distribution of normal and tangential contact stresses. These models can be used for a more detailed study of transient rolling modes of pneumatics, characterized by simultaneous sliding and rotation. The quasi-static distribution of the contact pressure is based on the solution of the contact problem for the spherical composite shell ant the absolutely rigid plane. The governing equation of S.A.Ambartsumyan for a transversally isotropic elastic shell is used as a background, the superposition principle, and the transient function for a shell. Such a function is a normal translation as a solution of the problem for a shell loaded by the unit concentrated force. This problem is solved by expanding the unknowns in series with respect to Legendre polynomials and their derivatives. After construction of the transient function the contact problem is reduced to the integral equation for the contact pressure on the basis of the superposition principle. The integral equation is then reduced to the algebraic one for the expansion factors of the contact pressure using the orthogonality of Legendre polynomials. Finally, after the truncation of the series and by discretization along the meridian coordinate the obtained problem is reduced to the algebraic system for the expansion factors of the contact pressure.

Pages: 327-340 Elibrary

**Artamonova N.B., Sheshenin S.V.**

## Modeling of a nonlinear coupled consolidation problem

Previously by the authors, a coupled physically and geometrically nonlinear formulation of the boundary value problem was obtained using the Lagrange approach with adaptation for the solid phase and the ALE (Arbitrary Lagrangian-Eulerian) approach for the fluid under the assumption of quasistatic deformation of the rock skeleton. The differential formulation in “velocities” includes the equilibrium equation, the filtration equation and the porosity change equation derived from the laws of conservation of continuum mechanics using spatial averaging over the representative volume element. In this paper, we propose a method for solving this problem and present the results of numerical simulation. The system of equilibrium and filtration equations is solved under the assumption of constant porosity, which is recalculated at each time step. To solve the system, a generalization of the implicit scheme with internal iterations at each time step is used according to the Uzawa method. The paper analyzes the stability of a linear problem when approximated by elements Q1-Q1 and Q2-Q1. Numerical examples are given of calculating a nonlinear coupled consolidation problem for a hyperelastic material when approximated by the Mooney, Mooney-Rivlin, Treloar, and Saint-Venant-Kirchhoff potentials, the influence of taking into account geometric nonlinearity is investigated, and the problem with varying porosity and filtration coefficient is solved. For modeling the constitutive relations for elastic-viscoplastic soil deformation under short-term loads, the Grigoryan-Rykov model generalized to large deformations is selected. In this theory, the associated flow law is considered in the five-dimensional Ilyushin space, and the relationship between the first invariants of the stress and strain tensors is determined according to the theory of viscoplasticity. Comparison of the results of calculations of effective elastic moduli by the averaging method based on three-dimensional and two-dimensional models of the real structure of pure limestones and experimental values is presented.

Pages: 341-361 Elibrary

**Fedotkin R.S., Fedotkina A.A., Kryuchkov V.A.**

## Friction and wear in tracked undercarriage systems of traction and transportation vehicles with rubber reinforced tracks

The article covers the issues of tribological interaction in tracked undercarriage systems of traction and transportation vehicles with rubber reinforced tracks on agricultural tractors example. Taking into account the interaction not only with units and mechanisms of tracked undercarriage system, but also with aggressive environmental factors, the most exposed to wear element is rubber reinforced track. Varieties of friction and wear processes are considered for it, their analytical evaluation is given. In particular, we analyze mechanical or physical friction: friction in contact with the supporting surface, its relief, abrasive particles and substances; friction in contact with drive and driven elements of the propulsor, sprockets, rollers, etc.; intermolecular or chemical friction: interlayer, hysteresis friction in the rubber mass. Destruction of rubber reinforced track systems as a result of wear and tear is the main limitation of tracked undercarriage system life. Possible methods of friction reduction in tracked undercarriage system and reliability increase of rubber reinforced tracks are explained. The main of them, in addition to improving recipes, physical and mechanical properties of rubber compositions and improving the technological process of their manufacture, is a rational design of structures. The latter can be achieved only by applying modern methods of stress-strain assessment. Recommendations to further work on development of calculation and theoretical methods for designing of rubber reinforced track structures under conditions of real operation and as a part of concrete agricultural machines, analysis of reasons and possible consequences of such destructions are given. It is necessary to form a complex of measures of design and technological character for prevention of destructions of elements of tracked undercarriage systems at various stages of life cycle and also their restoration.

Pages: 362-370 Elibrary

**Mishustin Ilya V.**

## Theoretical description and practical determination of the strain incompatibility in the model of a shape memory alloy

To take into account the strain incompatibility caused by phase-structural deformation of martensite, additional terms are added to the expression for the thermodynamic potential of a shape memory alloy (SMA). These terms are expressed through two material functions. The first of them characterizes the strain incompatibility between martensite formations and the austenite matrix. The second function is associated with the strain incompatibility between differently oriented martensite formations and between different variants of martensite orientation within these formations. The issue of the sign of additional terms and the arguments of corresponding functions is considered. If chaotic martensite without hardening is taken as the base state, then the term taking into account the strain incompatibility in martensite is less than or equal to zero and increases in modulus with an increase in the degree of orientation of martensite. The term that takes into account the strain incompatibility between martensite and austenite is zero in single-phase states and positive in two-phase state. The SMA with homogeneous hardening of the martensitic part of representative volume is considered in detail. The conditions for the nonnegativity of the mechanical part of dissipation represented as a sum of terms associated with phase transitions and structural transformation are formulated. From these conditions, the restrictions imposed on the material functions, taking into account the strain incompatibility, and the shifts of characteristic temperatures of phase transitions are derived. In particular, in the case of a reverse phase transition without loading in the SMA sample having a nonzero phase-structural strain, the transition start and end temperatures increase and the transition interval narrows in comparison with the chaotic martensite sample, which is consistent with the results of a number of experiments. A technique for determining material functions characterizing the strain incompatibility in the SMA using experimental data is demonstrated. The form of these functions is chosen and constants included in their expressions are found by fitting. At the same time, the restrictions ensuring the nonnegativity of the mechanical part of dissipation are satisfied.

Pages: 371-386 Elibrary

**Glebova M.A., Grishin V.I, Guseva N.V., Iashutin A.G.**

## Investigation of multi-row bolted joints with repair cover plates

Repair of technological defects and operational damages identified during technical control in manufacture process and preventive inspection in operation is required element of the organizational and technical system for ensuring the operational robustness of aircraft structures. An analysis of various types of repairs shows that despite a rather low efficiency, mechanical (i.e. riveted and bolted) joints have the lowest labor input that promotes their application at operation of the aircraft equipment. For verification calculation of the panels with cutouts repaired mechanically, numerical methods for strength calculation using two-dimensional and three-dimensional modeling are justified. In two-dimensional modeling, a thin-walled structure is generated from flat quadrangular or triangular elements, the size of which should be approximately equal the diameter of the fastening bolts connecting the separate structural elements. Fasteners (rivets or bolts) in two-dimensional modeling of the panel were replaced by springs, and in three-dimensional modeling, they were generated from solid elements. Formulas were used that widely applied in designing by foreign aviation companies to simulate the flexibility of fasteners for single-shear joint. In three-dimensional modeling of joints, both the panel and the cover plate are generated from prisms with four, eight or more tops. Fasteners are also developed by solid elements. In this case, the contact problem of the interaction between the fastener elements and the panel parts is solved. The modified Nuizmer criterion is used previously developed to evaluate the strength of composite plates with cutouts to estimate the load bearing of the panel with repaired cutout. The calculation reliability is checked by comparison with experimental data obtained during the testing of full-scale composite panels.

Pages: 387-402 Elibrary

**Zhavoronok Sergey I.**

## New compatibility equations for shape memory alloys undergoing thermoelastic phase transitions

The new compatibility equations are derived for shape memory alloys undergoing thermoelastic phase transitions under varying temperature and stress state. The geometrically linearized problem statement is used as a background together with the once coupled model of thermoelastic behavior of shape memory alloys accounting for the effect of stress state on the temperatures of start and finish of the phase transitions. The once coupled problem formulation assumes the temperature to be a given spatial distribution at each point in time domain. The linear strain tensor is represented using an additive decomposition into the isotropic tensor of elastic and thermal strains, the elastic deviatoric strain, and the phase deviatoric strain corresponding to direct or inverse martensite transitions. On the other hand, the additive decomposition of the strain tensor into the accumulated strain and the small increment is introduced, where the summary accumulated strain is assumed to satisfy the compatibility equations. The small increment of the deviatoric phase strain is defined by the linear function of increments of the deviatoric stress and the martensite volume ratio used as phase constitution parameter. The effect of the phase dilatation is assumed to be negligible. Given the temperature field the analogous linear dependencies of martensite volume ratio on the deviatoric stress are derived. Thus, the obtained incremental formulation of the compatibility equations for shape memory alloys obeying the once coupled model of thermoelastic phase transitions uses only stress tensor as an unknown. The stress function tensor could be introduced for such a problem, and the appropriate boundary value problem could be formulated for the partial differential equation where the components of the increment of the stress function tensor are unknowns.

Pages: 403-408 Elibrary

**Bobok D.I.**

## Analytical solution of the cylindrical tank made of a shape memory alloy under internal pressure

The paper considers the problem of the mechanics of a deformable solid about a cylindrical tank made of an alloy with shape memory (SMA) under a pressure during direct thermoelastic martensitic phase transformation under constant pressure. Both the momentless shell and the influence of the edge effect with rigid and articulated fastening are considered. The problem of relaxation in a similar shell during direct phase transformation has also been solved. In the second problem, internal pressure is applied to the shell in an austenitic phase state. Next, the shell material is cooled through the temperature range of the direct thermoelastic martensitic transformation. It is required to determine the necessary decrease in the process of such a transition of the uniformly distributed load so that the deflection of the shell remains unchanged. To describe the behavior of the shell material, we used the model of linear SMA deformation during phase transformations. The solution was obtained in the framework of the theory of thin isotropic shells and the assumption that the phase composition parameter at each moment of the process under consideration is uniformly distributed over the shell material, which corresponds to an unrelated statement of the problem for the case of uniform distribution of temperature over the material. The possibility of structural transformation in the shell material is not taken into account. It neglects the variability of the elastic moduli during the phase transition and the property of the SMA diversity resistance. To obtain an analytical solution to all the equations of the boundary value problem, the Laplace transform method using the volume fraction of the martensitic phase was used. After the transformation in the space of images, an equivalent elastic problem is obtained, solving which the Laplace images of the sought quantities are obtained in the form of analytical expressions, including operators that are Laplace images of elastic constants. These expressions are fractional rational functions of the Laplace image of the phase composition parameter. Returning to the space of originals by analytically decomposing the expressions for the sought quantities in the space of images into simple factors, we obtain the desired analytical solutions.

Pages: 409-435 Elibrary