### №3-2021

Full versions of the articles

https://elibrary.ru/contents.asp?id=46626084

**Gaganova N.V.**

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

In shape memory alloys, macroscopic inelastic deformations can be caused by phase transitions and reorientation of previously formed martensitic elements during structural transformation. Despite the fact that phase and structural deformation are interrelated, there are significant differences between them. As distinct from phase deformation, structural deformation is characterized by strain hardening: the elastic region is limited by the loading surface in the stress deviator space, inelastic deformation affects the size and position of the loading surface. In contrast to structural deformation, the phenomenon of hardening is not typical for phase deformation, phase transitions can occur without load, and phase deformation can be observed with decreasing stresses or without load. In this paper, a combined model of phase and structural deformation of shape-memory alloys is considered. It takes into account the mentioned properties of inelastic deformations in shape memory alloys. The model describes both phase and structural mechanisms of inelastic deformation, and influence of the first mechanism on the second. Inelastic deformation due to structural transformation in the active process obeys the associated flow rule. The differential condition for the active process and structural deformation is formulated. Tensor increment of the structural deformation is required to be codirectional with the external normal to the loading surface, hardening parameter associated with the structural transformation correspondingly is required to be positive. Most existing models considers only formation of new martensitic meso-elements, but do not take into account the development of meso-elements formed earlier. Meanwhile, experiments show that the development of martensitic elements can significantly affect deformations. This factor is necessary for describing the phenomenon of oriented transformation correctly, and the processes in which thermoelastic transformation from austenite to martensite occurs at stepwise or smoothly decreasing stresses. The subject of this article is the extension of the combined deformation model of shape-memory alloys to the relations that allow taking into account the development of martensitic elements during the phase and structural transition. The process of proportional loading with linearly increasing stresses is considered as a model problem. The obtained results are compared with the corresponding results obtained without considering the development of martensitic elements

Pages: 295-308 doi.org/10.33113/mkmk.ras.2021.27.03.295_308.01

**Vlasov A.N.**

## Reduction of the equation of elasticity theory with random coefficients in a domain with a periodic structure to the average equation of elasticity theory with constant coefficients. effective stiffness tensor

This approach makes it possible to reduce the original equation to the equation of the theory of elasticity with constant coefficients that determine the effective stiffness tensor, which is found from the solution of the cell problem, as in the Bakhvalov asymptotic averaging method. The only difference is that in the problem on a cell, the corresponding averaged functions of random variables, components of the stiffness tensor, depending on the fast variable, are used. It is shown that for layered media an analytical dependence is obtained that determines the effective stiffness tensor, which is similar to the dependence obtained by Bakhvalov’s asymptotic averaging method.

Pages: 309-322 doi.org/10.33113/mkmk.ras.2021.27.03.309_323.02

**Beklemishev S.A., Golovshkin V.A., Myagkov N.N., Vybornov A.N., Yushmanova O.O.**

## A model of a hard axsymmetric projectile’s high-speed impact with a deformable semi-infinite barrier

An analytical mechanical model of penetration at a high-speed impact of a rigid cylindrical projectile into a semi-infinite barrier has been constructed. The projectile is characterized by three parameters – linear transverse size, mass and initial speed. Regarding the mechanical properties of the barrier material, the hypothesis of a perfectly hard-plastic body with a condition of incompressibility has been adopted. The material of the barrier is characterized by two parameters – density and the yield strength. The problem is considered in the ax symmetrical dynamic setting. The aim of the work is to obtain acceptable engineering estimates of the following parameters – the depth of the projectile ‘s introduction, the mass of material thrown out of the barrier (in its evaluation, it was assumed that the material risen above the initial level of the undeformed barrier, leaves it and does not participate in further consideration of the movement), the effect of strengthening the momentum caused by the release (ejection) of the barrier fragments in the direction opposite to the projectile’s flight direction. The following method of research is proposed. An ax symmetric field of velocities is being built in three zones. The first is the area of the barrier material “sticking” to the projectile and moving with it, like a solid undeformable body. It is modeled as a segment of the ball. The second velocity field is a fragment of a ball layer adjacent to the first zone. The velocity field in this zone is based on the assumption that the speed ;UR, in a specially selected local spherical coordinate system, depends only on .That is ;U R g. The speed ;RUR is determined from the incompressibility condition. The third zone is a cylindrical zone moving like a solid in the direction opposite to the projectile’s movement. At the boundary of the zones a condition of continuity of normal speed component is supposed. The zone parameters are determined from the minimum power of the internal forces. The equation of motion is replaced by an equation of energy balance – the change of kinetic energy equals the power of internal forces. The assumptions made allowed to define the parameters of zones as a function of the depth of the projectile’s penetration. This allowed building a relatively simple analytical engineering model, which permits to determine the depth of the projectile’s introduction, the mass of the ejection, the strengthening of the pulse. In fact, the result is determined by two non-dimensional parameters.

Pages: 323-342 doi.org/10.33113/mkmk.ras.2021.27.03.323_342.03

**Movchan A.A.**

## Constitutive relations of the model of nonlinear deformation of shape memory alloys, resolved with respect to stress increments

A well-known variant of the constitutive relations of the model of nonlinear deformation of shape memory alloys (SMA) expresses the increments of strains through increments of stresses, the parameter of phase composition and temperature, as well as through the values of stress, strains, the parameter of phase composition and temperature themselves. However, for the development of numerical algorithms for analyzing the thermomechanical behavior of elements from the SMA in the form of the finite element method (displacement method), it is necessary to have constitutive relations resolved with respect to stress increments. It is this form of constitutive relations that allows us to obtain expressions for the incremental stiffness matrix of the finite element method for SMA. In a number of works, such an inversion was obtained numerically. Such a numerical procedure significantly slows down the solution process. In this paper, we propose an analytical inversion algorithm that uses specific features of the structure of the original system of constitutive relations, namely, the fact that the linear operator by which the increments of the components of the stress deviator enter the right part of the original system of constitutive relations is degenerate (the case of direct transformation or inverse, but without a structural transition), or the increments of the components of the stress deviator enter the right part of the original systems in the form of two terms, defined in terms of two different degenerate operators (the case of an inverse transformation together with a structural transition). The inversion of the constitutive relations is obtained for a coupled statement of boundary value problems, in which the increments of the components of the stress deviator are included in the right part of the original system of constitutive relations not only through the equations for the increments of the components of the deformation deviator due to elastic deformation, phase and structural transitions, but also due to the differential relations used for the phase composition parameter. When considering, the phase change, the volume effect of the phase transition reaction, as well as the influence of the variability of elastic modules (both volume and shear) on the increment of the phase composition parameter are taken into account.

Pages: 343-359 doi.org/10.33113/mkmk.ras.2021.27.03.343_359.04

**Yankovskii A.P.**

## Modeling the elastic-viscoplastic dynamic behavior of flexible cylindrical reinforced shells within the framework of a refined deformation theory

A dynamic problem of elastic-viscoplastic deformation of flexible cylindrical closed circular shells is formulated. Traditional criss-cross reinforcement structures along equidistant surfaces and spatial structures are considered. The inelastic deformation of the materials of the composition is described by the constitutive relations of the theory of flow with isotropic hardening, taking into account the dependence of the plastic properties on the rate of deformation. The two-dimensional kinematic and dynamic equations used, as well as the corresponding initial-boundary conditions, make it possible to calculate the mechanical behavior of flexible cylindrical composite shells with an accuracy of different orders. These equations allow simulating the possible weak transverse shear resistance of such structures. In the simplest version, two-dimensional equations, boundary and initial conditions are reduced to equations of Ambartsumyan’s non-classical theory. The numerical solution of the formulated nonlinear initial-boundary value problem is constructed according to an explicit “cross”-scheme. A comparative analysis of elastic-plastic and elastic-viscoplastic deformation of reinforced cylindrical shells dynamically loaded by internal pressure is carried out. The deformation of fiberglass and metal-composite constructions of different relative thicknesses has been investigated. It is shown that if the dependence of the plastic properties of the materials of the composition on the rate of their deformation is not taken into account, then this leads to an inadequate calculation of the dynamic inelastic deformation of such composite shells. It is shown that even in the case of relatively thin reinforced shells, the use of Ambartsumian’s theory can lead to a significant difference from the solution obtained by the refined theory. This can lead to qualitatively incorrect results when solving inverse problems (for example, rational reinforcement) using Ambartsumyan’s theory. Calculations based on the refined theory showed that replacing the traditional criss-cross structure of reinforcement along equidistant surfaces with a spatial structure of reinforcement in the case of long cylindrical shells of different relative thicknesses does not lead to a positive result. The positive effect of such a replacement of reinforcement structures is observed only for relatively thick short fiberglass shells.

Pages: 360-381 doi.org/10.33113/mkmk.ras.2021.27.03.360_381.05

**Zheleznov L.P.**

## Study of the effect of the monolayers lay-up angles on the composite cylindrical shell stability

Currently, polymer composite materials are widely used in the designs of advanced aircrafts, which significantly lightens the construction weight while maintaining its strength and stiffness characteristics. Despite the fact that there is a fairly large number of works on the study of such structures strength, the issues of their strength and stability under conditions of the initial nonlinear stress-strain state remain unsolved. The latter is especially necessary for aircraft fuselage structures, in which the composite skin stability loss is unacceptable. Methods of calculation for strength and stability of composite structures taking into account the nonlinearity of the initial stress-strain state are currently underdeveloped. Therefore, the development of reliable and efficient methods for analysis of shells made of composite materials is undoubtedly an urgent task. The most suitable method in this case is the finite element method. Its advantages are its versatility, physicality and unlimited applicability to complex structures under arbitrary loading. The application of the finite element method to the analysis of shells is associated with significant difficulties due to the thickness and curvature of the shell. The building of effective finite elements of shells is also an urgent problem to this day. Most of the developed finite elements are elements of circular cylindrical, conical or spherical shells. In the present work, the problem of strength and stability of cylindrical composite shells under arbitrary loading is solved by finite element methods and Newton-Kantorovich linearization. Finite elements of non-circular cylindrical composite shells and reinforcement elements of natural curvature, developed by the authors on the basis of the Timoshenko hypothesis, are used, in the approximation of displacements of which their rigid displacements (displacements of finite elements as a rigid body) are explicitly identified. Critical loads are determined in the process of a geometrically nonlinear problem solving using the matrix triangulation method and Sylvester’s criterion. The shapes of the shells in the prebuckling state and their buckling shapes are also calculated. The stability of a circular cylindrical shell made of a polymer composite material has been investigated under various types of loading: torsional and bending moments, edge compressive and transverse forces, and external pressure. The effect of the monolayers lay-up angles, deformation non-linearity on the critical loads of the shell instability has been determined.

Pages: 382-395 doi.org/10.33113/mkmk.ras.2021.27.03.382_395.06

**Kazachenko S.A., Zertsalov M.G.**

## Numerical-analytical method of engineering assessment of the impact of the development of the pit on the movement of the adjacent soil mass, taking into account the rigidity of the fencing of the pit

Currently, the analysis of changes in the stress – strain state of the soil mass during the construction of pits is carried out using numerical methods, the most widespread of which is the finite element method, implemented in a variety of calculation complexes. It is obvious that at the current level of development of computer technology, this method allows us to take completely into account all the factors that affect the final result. At the same time, even solving the problem in a two-dimensional formulation takes a lot of time and requires the necessary qualifications of the personnel performing the calculations.

Pages: 396-409 doi.org/10.33113/MKMK.RAS.2021.27.03.396_409.07

**Mehtiyev R.K.**

## Interaction of the two-periodic system foreign elastic inclusions and straight line cracks at lateral shift of the composite

An elastic medium weakened by a doubly periodic system of circular holes filled with washers of foreign elastic material, the surface of which is covered with a cylindrical film, is considered. The medium (binder) is weakened by doubly periodic systems of straight through cracks. An external load xy∞τ in such an environment around the holes is formed by zones of increased stress, the location of which has a doubly periodic character. The presented stresses and their displacements are expressed in terms of an analytical function. General representations of solutions are constructed that describe the class of problems with a doubly periodic stress distribution outside circular holes and straight-line cracks. For the solution, the well-known position is used that the displacement in the case of a transverse shear is a harmonic function. A known representation of the solution in each area is applied through the corresponding complex analytical function. The three analytical functions are represented by Laurent series. Satisfying the boundary condition on the contours of holes and crack faces, the problem is reduced to two infinite algebraic systems with respect to the sought coefficients and to one singular integral equation. Then the singular integral equation is reduced to a finite algebraic system of equations by the Multopp – Kalandia method. A procedure for calculating the stress intensity factors is presented. For the numerical implementation of the described method, the cases of the location of the holes at the vertices of the triangular and square meshes were taken. The results of calculations of the critical load depending on the crack length and elastic geometric parameters of the perforated medium are presented. The relevance of such studies is due to the widespread use in engineering of structures and products made of composite materials. Research on the development of mathematical models of the theoretically described stress-strain state of the reinforced composite near the inclusion at shear and cracks is practically small. The transverse shear of a linearly reinforced medium with three mutually perpendicular planes of symmetry in a shear state in a plane perpendicular to the orientation of the fibers is considered. Due to the symmetry of the medium, its deformations along the orientation of the filler are absent, and the stress – strain state is a function of only the variables 2x and 3x; it is obvious that the shears of the medium in the plane under consideration will be independent of the shear deformations in the reinforcement plane. All of the above conditions reduce to the fulfillment of the equalities 12 13 110, 0, 0= = =γγε and 10u =. As always, the 1Ox axis is directed along the fiber orientation.

Pages: 410-426 doi.org/10.33113/mkmk.ras.2021.27.03.410_426.08

**Babaytsev A.V., Nasonov F.A., Rabinskiy L.N., Yan N.M.**

## Application of probing hole methods to determine residual stresses in composite materials

Residual stresses can appear due to technological features of the manufacturing process, due to design features (asymmetric stacking), due to long-term force or temperature influences, etc. This work investigates the application of digital image correlation in the probing hole method to determine residual stresses in composite materials. Samples cut from a panel of a polymer composite material with asymmetric stacking were investigated, in which deformations that appeared after drilling a hole in the sample were determined. Deformation fixation was performed non-contactly, using the image correlation method. To improve the accuracy of strain fixation, a pattern was applied to the specimens. Two variants of the pattern were considered in this work: large (marker size 0.05-0.8 mm) and small (marker size 0.02-0.2 mm). Since the residual deformations were small, the coarse pattern did not produce representative results. The experimental deformations obtained were compared with numerical calculations using the finite element method. The solution of the inverse problem of elasticity theory to identify residual stresses was performed in the Comsol system using numerical finite-element modeling, the Monte Carlo method, and the Nelder-Meade method. The Monte Carlo method was used to find the global minimum of the residual stress function, and the Nelder-Meade method to refine the local minimum. The function of deviations of the calculated and experimental data was calculated in the root-mean-square approximation. Small correction values were introduced to refine the mean strain field in the experiment and to ignore the unrecoverable errors associated with the inaccuracy of the experimental data. As a result of the study, the numerical and experimental results were found to sufficiently coincide. The difference of the implemented method from other methods of measuring deformations around probing holes is the obtaining of a complete picture for all strain components.

Pages: 427-440 doi.org/10.33113/mkmk.ras.2021.27.03.427_440.09