https://elibrary.ru/contents.asp?id=48174307

**Bobok D.I.**

## Analytical solution to the problem of deformation of transverse bellows and disc spring made of shape memory alloy

The paper is devoted to solving the problem of a cross-seam bellows made of a shape memory alloy (SMA) under the an axial load during a direct thermoelastic martensitic phase transformation. As special cases, two different approaches to accounting for the operator associated with the Poisson’s coefficient during the Laplace transform are considered. As a result, a hypothesis is put forward that it is possible not to take into account the operator associated with the Poisson’s coefficient during the Laplace transformation in the case of considering incompressible materials, but to consider this coefficient as a material parameter. which was verified numerically in the framework of this work. The bellows behavior was described in the framework of the model of linear deformation of the SMA during phase transformations and was modeled as the behavior of a system of ring plates. The problem was solved within the framework of an unrelated formulation, the distribution of the phase composition and temperature parameter over the bellows material at each moment of time was assumed to be uniform. Similarly, the possibility of structural transformation in the bellows material, the variability of elastic modules during the phase transition, and the property of the SMA’s resistance to diversity were neglected. To obtain an analytical solution of 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. When 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: 3-18 doi.org/10.33113/mkmk.ras.2022.28.01.003_018.01

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

## Comparative analysis of vibration absorbing properties of three-layer and homogeneous barriers under the impact of a transient load

The paper demonstrates a general approach that allows one to solve coupled problems of the interaction of an elastic medium in which non-stationary waves of various types are excited and a vibration-absorbing barrier. For this, the motion of an elastic medium and plates of various types are considered separately. All problems are solved in dimensionless form. To construct solutions, all functions were expanded into trigonometric Fourier series, and the direct Laplace transform in time was applied to them. The problem of determining the kinematic and dynamic parameters of a medium in which waves of various types were induced was solved, the damped plane wave and the cylindrical wave. The solution for the auxiliary problem of determining the surface transient functions for an elastic half-space when a displacement field appears on the boundary of this half-space is obtained. The initial-boundary value problems for transient interaction of elastic media and obstacles are solved. In this case, various approaches were used: for a homogeneous Kirchhoff-Love plate, the results announced in paragraph 3 are used, and for the plate model of Paimushin V.N. the conditions of contact between the medium and the barrier are introduced. Thus, in the image space and in the coefficients of the series, displacements in the soil after the wave passed through the barrier, as well as the stresses and strains, were found. When performing the inverse Laplace transform, it turned out to be impossible to perform the inversion in an analytical way, then the numerical-analytical modified method of F. Durbin was applied. As a result, specific examples of the interaction of barriers and waves in an elastic medium were considered, for which a homogeneous plate equivalent to a three-layer barrier was found. Based on the reduction coefficients found, a conclusion was made about the more effective absorbing properties of a three-layer plate.

Pages: 19-35 doi.org/10.33113/mkmk.ras.2022.28.01.019_035.02

**Zhavoronok S.I.**

## Wave dispersion in heterogeneous waveguides: methods of solution (a review). part ii

A brief review of modern methods of solution of problems of dispersion of normal waves in functionally graded and laminated elastic waveguides as well as of some ways of their improvement is presented. The early published Part I of review presented some typical functionally graded materials and appropriate constitutive relations; the methods of transfer matrix, reverberation, global matrix and the Peano series method were briefly described as well as possible approximations of functionally graded waveguides by the laminated structures with properties being constant or variable across the thickness. The main ways to improve the numerical stability of matrix methods were also mentioned. In the presented below Part II, the main attention is paid to methods of semi-analytical solution of dispersion problems based on the approximation of a waveguide by an equivalent system with a finite number of degrees of freedom, i. e. to power series, generalized Fourier series, semi-analytical finite elements, as well as methods based on higher-order theories of plates and shells. The basics of power series method are stated, the appropriate recursive relations for a plane layer and a hollow cylindrical waveguide with sectorial cross-section are presented. The main alternative to power series could be based on the expansion of the unknowns into generalized Fourier series on orthogonal polynomials of the normal coordinate; contrarily to power series the so-called “orthogonal polynomial approach” does not require the solution of transcendental equation and results in the statement of the generalized eigenvalue problem, moreover the known recursive properties of orthogonal polynomials allow one to obtain the equations coefficients analytically. The formulation of the Fourier series method in terms of state-vector formalism is presented, and its application to the study of evanescent wave modes is considered. The semi-analytical finite element method is briefly described. Finally, one variant of higher-order shell theory based on the Lagrangian formalism of the analytical dynamics of continua with constraints and biorthogonal expansion of unknown functions is discussed. It is shown that both orthogonal polynomial approach and semi-analytical finite element method follows from this kind of shell theory as particular cases generated by choice of different base functions of the normal coordinate on the background of the unified variational formalism. Accounting for constraints following from boundary conditions on shells’ faces allows one to satisfy the reflection conditions for the waveguide model based on the shell model of arbitrary order.

Pages: 36-86 doi.org/10.33113/mkmk.ras.2022.28.01.036_086.03

**Firsanov Vic.V.**

## Thermal bending of a mechanically incompressible beam

The incompressibility condition for an isotropic linearly elastic material seriously limits the application of the classical hypotheses of the beam bending theory formulated by Bernoulli for small deformations and displacements. At the same time, it is assumed that such a strong kinematic condition as the volume immutability condition, which is valid for elastic components of linear deformations, must be unconditionally fulfilled. When an incompressible beam is bent by a force load, the incompressibility condition, which characterizes the absence of deformation of the volume change, is obviously homogeneous, that is, the volume of the beam at the micro and macro levels does not change during deformation, and under the action of a bending thermal load, the deformation of the volume change is proportional to the operating temperature, and the elastic component of the total deformation of the volume change is zero. Consequently, mechanical incompressibility manifests itself when a force load acts on the beam, but in the case of thermal action, the deformation of the volume change is a function of temperature. Of course, this is a serious difference between the two conditions, however, even in the case of a temperature load, the condition of partial or mechanical incompressibility may be conflicting with respect to the classical hypotheses of beam bending, which may lead to the degeneration of the problem. It would be a mistake to completely reject the classical hypotheses of beam bending, but some should be abandoned and other hypotheses should be introduced that will not lead to a serious complication of the tasks being solved If we accept the absence of linear deformation in the direction transverse to the neutral axis of the beam, the absence of shear deformation and at the same time fulfill the condition of proportionality of the deformation of the volume change to the operating temperature load, then the two desired displacements, one transverse, the other longitudinal, can be determined from these ratios, and the solutions obtained will not correspond to the temperature problem being solved. Because of this, it is advisable to abandon the hypothesis of the absence of shear deformation, then the two kinematic desired functions will be related only by the dependence of the deformation of the volume change on the temperature load. In this case, the physical connection between shear deformations and stresses is restored. Taking into account these shear deformations is especially important for materials with low shear stiffness in transverse directions. The hypothesis about the non-compressibility of the beam fibers in the transverse direction will remain valid for physical relations, but these stresses are preserved in the equilibrium equations.

Pages: 87-97 doi.org/10.33113/mkmk.ras.2022.28.01.087_097.04

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

## Modified newmark method for the dynamic analysis of composite structural elements considering damping with memory

In this paper, we consider the problem of numerical dynamic simulation of bending structural elements made of structurally complex materials (composites, nanomaterials) with the consideration of their internal damping properties. The internal damping is considered nonlocal in time. It depends not only on the strain rate value in current moment, but also on the strain rate values on the whole history of element vibration. The nonlocality level depends on the scale factor that can be determined by the experimental data. The nonlocal in time damping model is integrated into the algorithm of the finite element method – the most widely used numerical method for mechanical systems analysis. The equilibrium equation for a structure in motion is solved numerically using an implicit scheme. In this case, the damping matrix of the calculation model is obtained from the condition of stationarity of the total deformation energy of the moving mechanical system. The previous research shows that the results obtained using such a model approximate the empirically determined dissipative properties of composite elements with sufficient reliability. The article discusses the results of a study of a one-dimensional non-local in time computational model implemented in MATLAB software. The possibility of calibrated model adjustment due to better experimental data approximation is estimated. For this purpose, the modified Newmark method is used. It is shown that it is possible to adjust the attitude and phase of the vibration process using the modified Newmark method.

Pages: 98-111 doi.org/10.33113/mkmk.ras.2022.28.01.098_111.05

**Khokhlov A.V., Nekrasov V.V., Shaporev A.V., Stolyarov O.N.**

## Creep and recovery of multifilament polyester yarns and their modelling using the nonlinear maxwell-type elasto-viscoplastic model

The data of tensile creep and recovery tests for a complex polyester yarn under two-step and four-step loading programs obtained by the authors are presented. Their analysis was carried out in order to study viscoelastic and viscoplastic properties of polyester yarns. The applicability of the physically nonlinear Maxwell type viscoelastoplastic constitutive equation governed by two material functions (which have been studied in detail in a series of articles) to the description of the polyester yarn behavior was examined. It was found that the basic applicability indicators of the constitutive equation found earlier are fulfilled with sufficiently small errors. It has been established that the measured values of the steady-state creep rates and plastic strain magnitudes in creep and recovery tests under different prescribed stress levels (tensile load levels) depended on stress as power functions, i.e. their dependences on stress are well approximated by linear functions in log-scale. Thus, the performance of specific applicability indicators for the model with power-law material functions was verified. The material functions for polyester yarn were determined in the class of power functions, i.e. four parameters specifying a pair of power-law material functions were found, using the family of five creep and recovery curves under different load levels. Two identification techniques were developed for the constitutive equation under consideration, realized and compared. The second technique improves the first one by the preliminary reduction (via approximation) of test creep and recovery curves to the qualitative form of theoretical creep and recovery curves generated by the constitutive relation. Verification of the calibrated model with the found power-law material functions was implemented based on the data of tensile tests of polyester yarn under four different four-steps loading test programs lasting for eight hours. It was shown that the calibrated model fits the test data for polyester yarn under rather complex loading programs sufficiently well.

Pages: 112-132 doi.org/10.33113/mkmk.ras.2022.28.01.112_132.06

**Dudchenko А.А., Maskaykina А.А.**

## The algorithm for calculating the accumulation of damage on the example of the wing joint with the center section

Currently, one of the most common microdefects are cracks in the transversal layers of composite materials. There is also a decrease in the elastic modulus and shear modulus of composite materials during cracking of transversal layers. One of the most dangerous places of fatigue damage in structures are stress concentrators: joints, areas of holes for hatches in the wing. In this paper, the accumulation of damage in the area of the junction of the wing console with the center section of the aircraft is investigated, taking into account fatigue durability. An algorithm for calculating the accumulation of damage on the example of this zone is proposed. In the course of the work, a micromechanical approach is used, which is the most effective for describing the accumulation of damage. Previously, the optimization problem is solved by determining the rational zone of the junction node. After determining the rational length of the joint zone, a butt joint is designed. Then the stress-strain state of the structure is determined. Next, the zones and types of cracks are determined. The calculations assumed that cracking occurs in the transverse direction between fibers. In the crack zone, the damaged layer is unloaded, and the adjacent layers, therefore, are reloaded. As a result of this study, an algorithm was obtained that takes into account the influence of the current stress state on the process of degradation of mechanical properties. An analytical calculation was also carried out on the example of the junction of the wing console with the center section of the aircraft. Changes in the elastic modulus and shear modulus for the composite structure are obtained depending on the number of cycles. The dependence of stiffness in the composite structure taking into account the length of the interlayer crack is also shown. The algorithm obtained in this work is planned to be used to predict changes in various structures of composite materials depending on damage under the action of cyclic loading.

Pages: 133-144 doi.org/10.33113/mkmk.ras.2022.28.01.133_144.07