### №4-2021

Full versions of the articles

https://elibrary.ru/contents.asp?id=47460294

**Belov P.A., Kriven G.I., Lurie S.A., Shramko K.K.**

## On the correctness of the mathematical statement of boundary problems in gradient elasticity

Gradient elasticity theories contain, by definition, scale parameters and therefore, naturally. that they are very attractive for modeling scale effects in the mechanics of materials with a micro-nano structure, for studying phase transformations with the formation of interphase layers that change the microstructure of materials, modified composites with nanostructures on fibers, as well as for studying connected problems of thermo-mechanics and hydrodynamics, and etc. The appearance of scale parameters in gradient models is due to the fact that not only deformations, but also their gradients are considered as arguments in the variational description of such models. As a result, the governing equations in first-order gradient models are determined not only by the tensor of elastic properties of the fourth rank, but also in the general case by elastic tensors of the fifth and sixth rank, which differ in dimension from the classical elastic moduli. The paper discusses the symmetry of the tensors of the moduli of elasticity of the sixth rank under the permutation of the indices of differentiation in the gradient elasticity, which is a consequence of the fact that the second derivatives of the displacement vector do not depend on the order of differentiation. It is noted that there are cases when, for correct formulations of applied boundary value problems, it is necessary to use in the boundary conditions the tensors of the elastic moduli of the sixth rank, symmetric when rearranging the differentiation indices (moment stresses symmetric in the last indices), even if the formally constructed versions of applied gradient theories lack this symmetry feature. It is shown that ignoring the symmetry property of the tensor of moduli of the sixth rank when rearranging the differentiation indices can lead to significant errors in comparison with correct solutions that take this feature into account.

Pages: 447-458 doi.org/10.33113/mkmk.ras.2021.27.04.447_458.01

**Baigonakova G.A., Garin A.S., Gunter S.V., Marchenko E.S., Vetrova A.A., Yasenchuk Yu.F.**

## Softening effect during cyclic stretching of titanium nickelide knitwear

Samples of knitwear made of TiNi wire with a thickness of 40 microns, 60 microns and 90 microns were studied by the method of soft zero cyclic loading under the action of uniaxial tensile stress and stretching to rupture. It has been found that metal friction, when stretched, behaves like a hyperelastic material. The effect of superelasticity was found in TiNi wire, but it did not manifest itself in knitwear made of it. The effect of softening and lagging of elastic unloading was detected during cyclic stretching of the metal mesh. The calculation of the cyclic stretching of knitted material was carried out using the calculation models of Gent, Neo-Hooke, Mooney-Rivlin, Bergstrom-Boyce, using experimental data on the cyclic stretching of knitted tapes made of titanium nickelide. The discovered similarity of the hyperelastic behavior of metallotricotage and computational models will allow us to develop a method for comparative evaluation of knitted materials made of titanium nickelide wire of different thicknesses and criteria for choosing a knitted material for the plastic of hyperelastic biological tissues. The main criteria for the rheological similarity of metallotricotage and soft tissues can be considered: the value of the tensile strength; elastic modulus and the range of low-modulus and high-modulus elastic deformation under load and unloading; the amount of residual deformation under cyclic tension. It has been found that the metal mesh made of superelastic TiNi wire under soft zero cyclic loading under the action of uniaxial tensile stress exhibits a rubber-like behavior characteristic of hyperelastic materials. At the same time, in the most loaded contact sections of the superelastic TiNi wire, the martensitic phase transition did not affect the stretching diagram of the hyperelastic knitwear. The residual macro-deformation during the first two stretching cycles is caused by the slipping of the loops on the contact sections during loading and friction, which counteracts the restoration of elastic deformation during unloading. The effect of softening and lagging of elastic unloading was detected during the cyclic loading of TiNi metal drainage. This effect is due to the heterogeneity of the elastic load distribution in the knitwear loops and the friction in the contact sections of the loops, which resists the elastic deformation of the loops. It is established that the Bergstrom-Beuys model is closest in the stress-strain diagram to the knitted ribbon diagram.

Pages: 459-481 doi.org/10.33113/mkmk.ras.2021.27.04.459_481.02

**Fedorov L.V.**

## Elastic stiffness coefficients of regular grid composite structutes

Composite lattice plates and shells consisting of system of unidirectional ribs and fabricated by filament winding are characterized with high weight efficiency and are used in aerospace structures. The paper concerns with a possible approach to obtain the efficient stiffness characteristics of lattice structures with a dense and periodic system of ribs. In case the lattice structure is formed by ribs forming periodic system of elementary cells translating which we can describe the whole structure, the real structure can be simulated with a smooth structure characterized by efficient stiffness characteristics. This simulation is of a special interest for the problems of design of lattice structures. The approach proposed in the paper is based on the Helmholtz theorem which allows us to link the displacements of the adjacent points in a solid. Applying this approach, the strain energy can be expressed first in terms of the cell energy and then through the displacements of the end cross-sections of the ribs in the cell and finally through the displacements and strains in the solid element simulating the cell. The strain energy of the lattice structure can be then obtained summing up the energies of the cells. Constitutive equations including efficient stiffness coefficients are derived with the aid of the Castigliano theorem. These equations allow us to determine the forces and the moments acting in the ribs. Lattice structures formed by several systems of ribs with different orientations are considered and analyzed.

Pages: 482-490 doi.org/10.33113/mkmk.ras.2021.27.04.482_490.03

**Vlasov D.A., Zertsalov M.G.**

## Analytical and numerical solutions to the problem of pile-soil and rock interaction

The paper presents an exact analytical solution to the plane stress elastic boundary value problem of a half-plane with a periodic system of linear semi-infinite one-dimensional inclusions (stiffeners) orthogonal to the surface. The solution is given in the form of Papkovich-Fadl eigenfunction series whose coefficients are found exactly using functions biorthogonal to the eigenfunctions, and the series themselves are equivalent to trigonometric ones. The solution is used to assess the stress-strain state of the soil and rock mass interacting with a vertically loaded pile. At the same time, in the analytical solution, the distance between the stiffeners (piles) is chosen in such a way that they do not affect one another. In addition, the perfect interface between the pile and the soil and rock is assumed. The analytical solution to the 492 problem of pile-soil and rock interaction is compared with the numerical simulation, which was obtained in a 3D setting using the finite element method (FEM) implemented in the ZSoil software. The comparison indicates that the use of exact solutions to elastic two-dimensional problems can be quite effective in assessing the stress-strain state of a soil and rock mass interacting with a loaded pile.

Pages: 491-499 doi.org/10.33113/mkmk.ras.2021.27.04.491_499.04

**Dyakova H.N., Kren A.P., Pratasenia T.A.**

## Estimation of the influence of the regimes of heat treatment of products from photocured polymers, manufactured by laser stereolitography technology, on their physical and mechanical characteristics

The influence of heat treatment of products obtained by laser stereolithography on their physical and mechanical characteristics is considered. It has been established that annealing of specimens from the cured Formlabs Standard White V04 photopolymer resin at 60oС for an hour leads to a significant increase in the modulus of elasticity and ultimate strength, which is confirmed by the results of standard tests. As an alternative to destructive tests to control the elastic and strength properties of articles made of hardened photopolymer resins, it is proposed to use the dynamic indentation method. Analytical expressions are given for calculating the hardness and elastic modulus of the materials under study by the main parameters of the recorded shock loading diagram.

Pages: 500-510 doi.org/10.33113/mkmk.ras.2021.27.04.500_510.05

**Saganov E.B.**

## Numerical simulation of direct martensite transformation in shape memory alloys with tension-compression assymetry

The work is devoted to the numerical simulation of direct martensite transformation (DMT) in shape memory alloys (SMA) taking into account tension-compression asymmetry. Uunder tension-compression asymmetry is meant the dependence of the stress-strain state of these alloys on the type of stress state. The parameter associated with the third invariant of the stress deviator is used as a parameter of the type of stress state. Numerical simulation of the DMT is performed using the finite element method. A model of nonlinear deformation of the SMA during phase and structural transformations was used as a material model. In this work, a velocity stiffness matrix corresponding to the process of cooling a sample from an SMA through the temperature range of the DMT was obtained. The velocity stiffness matrix obtained in this work takes into account the variability of the elastic modules of the SMA during the phase transition and the dependence of the accumulated phase strain during cooling on the magnitude of the acting stress. The cooling process is considered in a coupled formulation, taking into account the effect of the acting stress on the values of the direct transformation temperatures. The verification of the user material model is carried out based on an analytical solution to the problem of a beam made of an SMA under a constant tension stress and undergoing cooling through the temperature range of the DMT. In the framework of this work, the calculation of the stress-strain state of a spherical thick-walled SMA shell under the action of constant internal or external pressure during its cooling through the temperatures of the DMT was performed. It was found that during the cooling of the shell under the action of constant internal or external pressure, the stress state parameter does not depend on the radial coordinate of the shell. In the case of the action of internal pressure, the parameter of the type of the stress state corresponds to the case of pure compression, with the action of external pressure, to pure tension.

Pages: 511-522 doi.org/10.33113/mkmk.ras.2021.27.04.511_522.06

**Dumanskiy S.A.**

## Analysis of a stability of the shanley column on shape memory alloys rods during the reverse phase transition within the framework of the combined model of phase-structural deformation

This work is devoted to the analytical study of the stability of the Shanley column on SMAs rods during reverse phase transformation under the action of a constant load. For the first time, a combined model of inelastic deformation of SMAs is used for such a task. Two methods of preparing rods before starting the reverse transformation are considered, namely, deformation in the mode of martensitic inelasticity and direct phase transformation under the action of a constant load. As a criterion for the loss of stability, the quasi-static Euler method is used, which allows linearizing kinematic constraints and static equilibrium equations, with respect to small angles of deviation of the Shanley column from the vertical position.

Pages: 523-542 doi.org/10.33113/mkmk.ras.2021.27.04.523_542.07

**Gaganova N.V.**

## Superelasticity description based on the combined model of shape memory alloys deformation considering translational hardeninfg and development of the martensitic elements

A combined model of phase and structural deformation for shape-memory alloys is considered. The model takes into account kinematic and isotripic hardening and can be used for describing the phenomenon of oriented transformation. According to the model, phase deformation can increase under a decreasing load or without load after preload. Concepts of the loading surface and active process are used for structural deformation description. Structural deformation in the active process is determined by the associated law by analogy with the plasticity theory. Tensor increment of the structural deformation is required to be codirectional with the external normal to the loading surface, and the hardening parameter associated with the structural transformation correspondingly is required to be positive. The majority of models consider only formation of new martensitic meso-elements, not taking into account the development of the elements formed earlier. Meanwhile, experiments show that the development of martensitic elements can significantly influence on deformations. In the considered model, a special material function determines the relationship between the processes of formation and development of martensitic elements. The temperature of the phase transition in shape memory alloys depends on the operating stress, therefore phase transition can occur at a constant temperature. The article examines the possibilities of the combined model for describing the phenomenon of superelasticity in titanium nickelide. The nonlinear dependence of deformations on stresses and the corresponding phase-structural transformation after reaching stress thresholds is modeled. The results for different material functions are compared. Phase deformations are higher for material functions that take into account the development of martensitic elements. The model correctly describes the nonlinear growth of deformations under monotonically varying stresses at a constant temperature and the phenomenon of superelasticity. At monotonically increasing stresses at a constant temperature, the influence of the development of martensitic elements is less noticeable than at decreasing stresses.

Pages: 543-557 doi.org/10.33113/mkmk.ras.2021.27.04.543_557.08

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

## A model of the force contact of a composite spherical shell with a solid surface taking into account the combined anisotropic dry friction

The motion of an elastic composite shell over a hard rough surface in the presence of combined anisotropic dry friction is considered. This model can be used to study the dynamics of pneumatics (aviation and automotive) in conditions of combined kinematics, as well as various control robotics systems. To correctly account for the influence of the anisotropy of dry friction coefficients in such systems, it is required to construct approximate analytical models of the force state inside the contact spot, taking into account the real distribution of normal and tangential contact stresses. The contact pressure distribution is constructed using the S.A. Ambartsumyan for a transversally isotropic spherical shell. This equation is modified by introducing additional relationships for the reduced contact pressure and normal displacements. The construction of the resolving integral equation for the contact pressure is based on the principle of superposition and the method of Green’s functions. For this, the corresponding Green’s function is constructed, which is the normal displacement of the shell as a solution to the problem of the effect of concentrated pressure. Green’s function as well as the contact pressure, it is sought in the form of series expansions in Legendre polynomials, taking into account additional relations for the reduced contact pressure and normal displacements. Using the Green’s function, an integral equation solving the problem is constructed. As a result, the problem is reduced to determining the expansion coefficients in a series of the reduced contact pressure. Restricting ourselves to a finite number of terms in the series of expansions, using the discretization of the contact area and the properties of Legendre polynomials, the problem is reduced to solving a system of algebraic equations for the expansion coefficients for the reduced pressure. After that, from the additional relation, the coefficients of the required expansion of the contact pressure in a series in Legendre polynomials are determined.

Pages: 558-569 doi.org/10.33113/mkmk.ras.2021.27.04.558_569.09

**Tarlakovskii D.V., Zemskov A.V., Zverev N.A.**

## Unsteady elastic diffusion of an orthotropic cylinder under uniform pressure considering relaxation of diffusion fluxes

A polar-symmetric elastic diffusion problem is considered for an orthotropic solid multi-component cylinder under the action of an external uniformly distributed pressure over the surface. The applied loads initiate a mass transfer, which affects the cylinder’s stress-strain state. A coupled system of differential equations of elastic diffusion in a cylindrical coordinate system is used as a mathematical model. The model considers relaxation diffusion effects, implying finite velocities of diffusion fluxes propagation. We solved the problem by the equivalent boundary conditions method, which consists of the fact that first some auxiliary problem is considered, the solution of which is known. This problem differs from the original problem only in the boundary conditions. Then a relation connecting the right-hand sides of the boundary conditions of both problems is constructed. The specified ratio is an integral equation, the solution of which is sought using quadrature formulas. From this equation, the right-hand sides of the boundary conditions of the auxiliary problem are found. As a result, the solution to the original problem is found in convolutions of the Green’s functions of the auxiliary problem with the functions obtained by solving the above integral equation. The equivalent boundary conditions method was developed for initial-boundary value problems, the solution of which cannot be obtained by the method of separation of variables. For unsteady problems, it is semi-analytical; in stationary and static problems, it allows one to obtain a solution in an analytical form. Article considers calculus example based on a three-component material in which two components are independent. The study of the mechanical and diffusion fields interaction in a solid orthotropic cylinder is done. The limit transitions to static mechanodiffusion processes, as well as to classical models of elasticity, are investigated. The influence of relaxation effects on the kinetics of mass transfer is modeled in the continuum. The research results are presented in analytical and graphical forms.

Pages: 570-586 doi.org/10.33113/mkmk.ras.2021.27.04.570_586.10

**Zhavoronok S.I.**

## Inverse incremental constitutive relations and compatibility equations for a shape memory alloy undergoing structure transitions

New inverse incremental constitutive relations and compatibility equations are derived for a shape memory alloy undergoing the stress-induced isothermal structural transition in the entirely martensite phase state. The once coupled model of thermoelastic phase and structure transitions in shape memory alloys together with the geometrically linear statement of the solid mechanics problem is used as a background. The initial state of the alloy under vanishing stresses is assumed be entirely twinned martensite. The structural transition consists in the untwining of the crystalline structure of entirely martensite phase constitution (so-called martensitic inelasticity phenomenon). To derive the constitutive equations and the appropriate compatibility equations, first the additive decomposition of the linear strain tensor into a sum of the elastic strain tensor and the structural deviatoric strain introduced; secondly the additive decomposition of the aforementioned tensors into sums of accumulated strains and some small increments is used assuming that the summary accumulated strain satisfies the compatibility equations. The structural deviatoric strain increment is defined as linear function of the deviatoric stress increment while the summary dilatation is resulted by only elastic strain. The obtained new compatibility relations are linearized with respect to both strain and stress increments and similar to the Hookean law for anisotropic elastic media where the instantaneous anisotropy is resulted by the structural compliance tensor being a bilinear function of covariant deviatoric tensors of accumulated stress. The new compatibility equations for the stress tensor increment are obtained accordingly to the Beltrami form. Finally, the stress function tensor is introduced and the appropriate formulation of the Beltrami equations for the stress function increment are derived.

Pages: 587-593 doi.org/10.33113/mkmk.ras.2021.27.04.587_593.11