## No 4 (2022)

**Year:**2022**Articles:**16**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/332**DOI:**https://doi.org/10.15593/perm.mech/2022.4

INFLUENCE OF HIGH TEMPERATURE TREATMENT ON THE MECHANICAL CHARACTERISTICS OF CARBON-CARBON COMPOSITE MATERIALS WITHPYROCARBON MATRIX

#### Abstract

Pyrolytic compaction of porous substrates is one of the methods for obtaining carbon-carbon composite materials. When using such materials at high temperatures, for example, as elements of heaters, it is necessary to take into account the effect of high temperatures on their mechanical characteristics. In this work, the influence of high-temperature treatment on me-chanical characteristics was studied and the mechanism of destruction of samples of the mate-rial "Argolon GR" produced by JSC "Composite" was considered. It is shown that with an in-crease in the processing temperature from 1800 to 2400 °C, the open porosity of the samples increases in proportion to the increase in the number and size of cracks in the samples. The compressive failure stress depends very weakly on temperature, however, the graph of the corresponding dependence clearly shows its slight decrease with an increase in the processing temperature from 2000 to 2400 °C, which corresponds to the accumulation of damage in the material matrix that reduces its strength. Attention is drawn to a significant increase in the ten-sile strength of the material after high-temperature treatment, which is not typical for carbon-carbon composite materials. The dependence of the breaking stress in tension with an increase in the processing temperature for material samples has a pronounced maximum at 2000 °C. The analysis of the change in impact strength, the roughness parameter of the fracture surface of the samples, and the breaking stress in tension showed that the dominant mechanism during fracture is the mechanism of strength increase associated with the violation of the compatibility condition of deformations in the components of the material, while the effect of thermal stress relaxation is very small.

**PNRPU Mechanics Bulletin**. 2022;(4):5-12

INFLUENCE OF GEOMETRIC, KINEMATIC, GAS-DYNAMIC PARAMETERS ON ROTOR DYNAMIC STATE TAKING INTO ACCOUNT GAS DYNAMIC FLOW IN LABYRINTH SEALS CLEARANCES

#### Abstract

The present work details a new approach to the study of GTU rotor vibrations, based on the solution of a related dynamic problem for the «gas – dynamic flow – deformable structure» sys-tem. The modern tendency to increase an aggregates power with a simultaneous decrease stiffness results in new phenomenons that affected a rotor vibration state. The compressor rotor model with a labyrinth seal is considered. ANSYS software product is used. The calculations were carried out on a high-performance computer complex PNRPU. The performed calculations showed a qualitative and quantitative effect of a gas-dynamic gap on the rotor dynamics. A 2FSI calculations series was performed to study the influence of geometric, kinematic and gas-dynamic parameters on the rotor dynamic state. A pressure fluctuations spectral analysis in the gas-dynamic gap and displacements has been carried out. The obtained spectrograms pro-cessing it possible to plot amplitudes and frequencies dependences of resonant pressure oscil-lations over an initial pressure in the gas-dynamic gap. It was found that the initial pressure in a gas-dynamic gap has the greatest influence. A rotor and gas oscillations resonant frequency was found, which corresponds to a change in the shaft axis spatial position. The «gas – struc-ture» system resonant frequencies were obtained for models differing in mass and stiffness. A decrease in an elasticity modulus of the structure led to a decrease in the maximum pressure fluctuations amplitude, while a decrease in mass led to its increase. For the base model and the model with lower rigidity, the resonant pressure oscillations frequency depends on the initial pressure value according to a law close to linear, while for the model with a lower mass, the dependence has a pronounced non-linear character.

**PNRPU Mechanics Bulletin**. 2022;(4):13-21

SECONDARY BUCKLING OF A HEATED CIRCULAR PLATE

#### Abstract

Upon heating, a circular plate with immovable edge exhibits buckling, which results in ax-isymmetric postcritical bending. Progressive axisymmetric bending due to increasing thermal load in the postbuckling range leads to substantial redistribution of the stresses in the plate and occurrence of high compressive circumferential stresses in a narrow zone adjacent to the plate edge. In this case, secondary buckling can occur resulting in unsymmetric stress-strain state. The aim of the present work is to study stability behavior of the postbuckling axisymmetric equilibrium configurations of a circular plate subjected to uniform temperature rise. The plate edge is assumed to be immovable in the radial and transverse directions and elastically re-strained against bending rotation, which allows one to model the boundary conditions between two extremes of clamped and simply supported. The stability analysis is performed by the semi-analytical finite element method, where unknown displacements are approximated by truncated Fourier series in the circumferential coordinate. The geometrical nonlinearity is taken into ac-count in a quadratic approximation using the Fӧppl – von Karman nonlinear plate theory. To find equilibrium states of the plate, an iterative algorithm is proposed which requires determination of the coefficients of the first and second variations of the total potential energy. Stability of the equilibrium states is examined using the criterion of positive defineteness of the Hess matrix of the plate finite-element model. The critical temperature rise at which secondary buckling occurs is determined. The post-buckling nonlinear deformation characterized by local winkling near the plate edge is studied. The effect of boundary conditions and temperature-dependent material properties on the critical thermal load and secondary buckling modes is discussed.

**PNRPU Mechanics Bulletin**. 2022;(4):22-29

SIMULATION OF VIBRATORY PLATE INTERACTION WITH THE GROUND SURFACE

#### Abstract

The paper presents a three-mass rheological model of the system "soil – vibration plate base – vibration plate frame". The rheological model makes it possible to reproduce different modes of interaction between the vibratory plate base and soil: with different types of plate decoupling and without decoupling. We verify this model by comparing the experimental values of the vertical oscillation span of the base and frame of the Zitrek CNP 20 vibrating plate with the previously calculated values. As a whole, the calculated values of the span of vertical oscillations of the base and frame of the Zitrek CNP 20 vibrating plate correlate with the experimental data in the range of the dynamic modulus of soil deformation of 13…30 MPa. During the experiment we used the rheological model and obtained results are as follows: the mass of the vibrating plate (50; 150; 250; 350; 450; 550; 650; 750 kg), the coefficient of the elastic resistance of soil (30; 60; 90; 120 MN / m), and the coefficient of viscous resistance of soil (100; 200; 300 kN • s / m). The total number of combinations of parameters was 96. The processed results of the computational experiment provide the regression dependences for calculating the maximum soil reaction force, the time of soil loading (increasing the values of the reaction force of soil) t1, and the time of soil unloading (decrease the values of the reaction force of soil) t2. The simulation results show that, within one exposure cycle, the soil loading time t1 is less than the soil unloading time t2. The ratio t1/t2 is influenced by the weight of the vibratory plate, as well as the factors of elastic and viscous resistance of soil. This feature (t1/ t2 1) is typical for both vibratory rollers and rammers, which is confirmed by the results of the relevant experimental studies. The obtained regression dependences of parameters Fs, t1, and t2 on the vibratory plate mass and the factors of elastic and viscous resistance of soil are important for calculating the distribution of stresses and strains on the depth of the compacted soil.

**PNRPU Mechanics Bulletin**. 2022;(4):30-41

EVALUATION OF THE THERMAL AND MOISTURE AGING INFLUENCE IN AGGRESSIVE ENVIRONMENTS ON THE CHANGE IN THE OF THE MECHANICAL BEHAVIOR OF FIBERGLASS BY A SHORT BEAM BENDING TESTS BASED ON ACOUSTIC EMISSION TECHNIQUE

#### Abstract

The work is aimed at experimental research and description of the mechanical behavior and degradation of the fibrous structural composite material strength properties during thermal and moisture aging in aggressive (operational) environments of different duration and temperatures. The object of the study was STEF structural fiberglass. STEF is a laminated fiberglass reinforced plastic obtained by hot pressing of glass fabric impregnated with a thermosetting binder based on combined epoxy and phenol-formaldehyde resins. After preliminary aging under various temperature-time conditions, fiberglass samples were tested at normal temperature for interlayer shear. To study the defect initiation and propagation during the deformation of fiberglass after preliminary aging under various temperature-time conditions and environments, the acoustic emission method is used in this work, which makes it possible to study the deformation stage structure and track the processes associated with the formation of defects in the structure of the fibrous composite. Influence of various media, such as industrial water, sea water and machine oil, at different durations (15, 30, 45 days) and temperatures (22°, 60° и 90 °С) on the processes of composite destruction and the implementation of various mechanisms damage accumulation during quasi-static interlaminar shear tests were obtained and analyzed. The paper presents the test results obtained by the acoustic emission signals processing. The data illustrating the stages of damage accumulation are presented and described, and the main mechanisms of damage to the composite structure under loading are analyzed. The results of a study of the microstructure of samples obtained using a stereomicroscope before and after thermal and moisture aging in aggressive media are described.

**PNRPU Mechanics Bulletin**. 2022;(4):42-53

ANALYSIS OF MESOSTRUCTURE AND FRACTURE KINETICS OF ELEMENTS OF LATTICE COMPOSITE STRUCTURES UNDER TRANSVERSAL COMPRESSION USING STOCHASTIC FEA MICROMECHANICS

#### Abstract

The paper analyses the mesostructure of the structural elements of lattice aircraft shells – ribs consisting of alternating layers of equal thickness and made from unidi-rectional CFRP and pure matrix material. In experimental studies, the elastic characteristics of unidirectional CFRP were obtained under three-point bending and transversal compression. As a result, the longitudinal modulus of elasticity of the layered composite turned out to be 101 GPa, and the shear modulus was 2.95 GPa. Numerical modeling of the meso- and micromechanics of the interaction of the noticed layers under transversal compression has been performed up to failure. The ANSYS FEA software (explicit and implicit formulations) was used. The regular and stochastic stacking of fibres in the cross section under compression is considered. The fiber diameters in the composite element were measured on thin sections using a Zeiss Axio Observer D1m digital microscope and were equal to 5.1 ± 0.8 µm. Layers with a fiber volume fraction of about 60 % alternate with layers of pure epoxy. It is proposed to use only the first principal stress in the matrix as a micromechanical criterion for failure under com-pression and tension. At the first stage of calculations, the problem of transversal compression of a cell with a regular laying of fibres was solved (the error in the value of the transversal modulus of elasticity was less than 2 %). At the second stage, an assessment was made of the strength and accumulation of microdamages under compression in a model of a layered structure with stochastic fibre stacking. The analysis of stress-strain state of a layered mesostructure under compression made it possible to explain the reason that the rib has a trans-verse strength twice lower than that of a homogeneous CFRP. The calculated values of the ultimate strength in transversal compression of a layered rib are in good agreement with the experimental ones.

**PNRPU Mechanics Bulletin**. 2022;(4):54-66

A NUMERICAL ANALYSIS OF FORCED STEADY-STATE VIBRATIONS OF AN ELECTRO-VISCOELASTIC SYSTEM IN CASE OF A JOINT IMPACT OF ELECTRICAL AND MECHANICAL LOADS

#### Abstract

In the current paper an estimation of a mechanical response of a structure on joint mechanical (force and kinematic) and electrical (voltage and current) impacts taking into account different options of their combinations was carried out. A mechanical response was esteemed basing on solution to the problem of natural vibrations for piecewise-homogeneous electro-viscoelastic bodies. A cantilever plate with piezoelectric element attached to its surface was chosen as an object under study. The plate was made of viscoelastic material the mechanical properties of which are described using complex dynamic moduli. A piezoelectric ceramic material was chosen as the one for piezoelectric elements. An amplitude of forced steady-state vibrations of a component of displacement vector that is perpendicular to the surface of the plate was chosen as a quantity that characterize a mechanical response. Variational equation of motion of the object under consideration is formulated based on relations of linear elasticity, linear viscoelasticity and Maxwell’s equations for a conducting media in a quasistatic approximation. Ranges of changing of values of excitation impacts for vibration modes taken into account were chosen on the basis of values of natural vibrations frequencies obtained from the solution to natural vibration problem for electro-viscoelastic bodies. Numerical implementation of the stated problem was performed in finite element software package ANSYS. Some characteristic peculiarities of a mechanical response depending on a type and magnitude of external impact were obtained as the result of the carried out numerical experiments. There was demonstrated that in case of joint impact of two loading factors there reveals an additional capability of control a mechanical response of an electro-viscoelastic system.

**PNRPU Mechanics Bulletin**. 2022;(4):67-79

PREDICTION OF THE BEARING CAPACITY OF THE HIP JOINT ENDOPROSTHESIS FROM THE С/C COMPOSITES

#### Abstract

The article evaluates the bearing capacity of the hip joint endoprosthesis made of carbon/carbon (С/C) composites under anatomical load. С/C is made on the basis of carbon fiber Ural N/400-22 and phenolformaldehyde resin novolachny type RP-010. A significant complexity in modeling an endoprosthesis is the prediction of the properties of С/C. The complexity of mathematical modeling based on the mechanics of composite materials consists in the impossibility of experimentally determining the mechanical characteristics of a pyrocarbon matrix. The pyrocarbon matrix does not exist in its pure form separately from the composite material. The mechanical characteristics of the С/C are calculated, mathematical models of bone tissue and endoprosthesis are constructed, taking into account the complex composite structure. The femur is represented by a combination of cortical and spongy tissue. The femur in the study was fixed along the lateral and medial condyles of the femur. This fixation corresponds to the contact spot of the condyle with the lateral and medial menisci. The hypothesis is confirmed that the weakest component in the С/C is the pyrocarbon matrix. The position of the highest values of normal and tangential stresses are revealed. The position of concentration of tangential stresses coincide with the areas of destruction of the endoprosthesis during compression testing, which were previously visually identified. New design and technological requirements for the structure have been identified, which will contribute to increasing the reliability of the endoprosthesis.

**PNRPU Mechanics Bulletin**. 2022;(4):80-89

REFINED DISCRETE METHOD FOR CALCULATING STIFFENED ORTHOTROPIC SHELLS

#### Abstract

The author proposes a refined discrete method for taking into account stiffeners in the simulationof thin-walled shell structures. According to the method, it is necessary to add different reduction factors along different coordinate axes. For ribs directed perpendicular to the considered direction, a reduction factor is introduced equal to the ratio of the width of the ribs in this direction to the linear size of the shell in the considered direction. This method supplements the previously developed geometrically nonlinear mathematical model, which takes into account transverse shears and material orthotropy. The model is written as a functional of the total potential strain energy and can be used for different types of shells by specifying the Lame parameters and the radii of principal curvatures. The computational algorithm is based on the Ritz method and the method of continuation of the solution with respect to the best parameter. The software imple-mentation was carried out in the Maple software package. The applicability of the refined discrete method is shown by the example of orthotropic shallow shells of double curvature, simply supported along the contour and under the action of an external uniformly distributed transverse load. Material parameters were selected for T-10/UPE22-27 and 0/90 Woven Roving E-Glass/Vinyl Ester fiberglass. A comparison was made of the values of critical buckling loads for different stiffening options (a grid of ribs from 0 to 12 ribs in each direction) and a comparison of the values with the conventional discrete method, which showed that with the conventional discrete method, the values of buckling loads are significantly overestimated, especially with an increase in the number stiffening ribs. Comparison of the results of the test problem with the results of experiments obtained by other authors showed good agreement between the refined discrete method.

**PNRPU Mechanics Bulletin**. 2022;(4):90-102

IDENTIFICATION OF THE IMPACT POSITION IN A REINFORCED CONCRETE STRUCTURE BASED ON THE ANALYSIS OF THE RESPONSE OF VIBRATION SENSORS

#### Abstract

The article presents the results of an experiment to study the vibration response of a large-scale reinforced concrete model structure to an impulse load. The load was a series of impacts along the normal to the surface of the element and was applied to all the main structural ele-ments (columns, crossbars and floor slabs). The vibration response was recorded by a system of sensors-accelerometers distributed over the structural elements and synchronized with the accelerometer mounted on the striker. The results of measurements of acceleration vibrograms were saved as digital files. An array of vibrograms recorded by the entire complex of sensors in response to test impacts on the main structural elements made up a vibration portrait of the structure. As a result of processing this information, an array of data was obtained on the propa-gation time of the vibration signal from each signal source to each of the sensors of the registra-tion system (basic array of responses).Key words: Impact localization, accelerometer, The data obtained were used to solve the problem of determining the location of an arbi-trary impact on a structure. To do this, the vibration response recorded by the sensor system during an arbitrary impact was compared with the base array of responses. The comparison was made on the basis of the calculation of the pair correlation coefficients. The resulting spatial distribution of the correlation coefficients made it possible to identify the position of the shock load application. It corresponds to the structural element that has the maximum value of the correlation coefficient. The proposed algorithm was demonstrated on an example where one of the test shocks that participated in the FORMATION of the basic vibration portrait was used as an unknown load. In a numerical experiment using the proposed algorithm, it was found that the accuracy of the impact site identification corresponds to the characteristic step of the structural elements. It is shown that the accuracy correlates with the number of sensors of the registration system and their distribution throughout the structure. The developed algorithm for identifying the place of impact load application can be effec-tively used in the development of automated systems for deformation monitoring.

**PNRPU Mechanics Bulletin**. 2022;(4):103-115

MODELING OF CREEP PROCESSES OF SINGLE-CRYSTAL ALLOYS WITH TAKING INTO ACCOUNT OF RAFTING

#### Abstract

The research is devoted to the development and verification of a two-level micromechani-cally motivated model of viscoelastic deformation of two-phase nickel-based single-crystal al-loys, predicting behavior under high thermomechanical loading with taking into account the presence of γ and γ' phases. The model is relevant for computations of the stress-strain state of cooled single crystal blades of gas turbine units. The formulation of the constitutive equations for each of the phases considered the anisot-ropy of elastic and viscous properties, the presence of octahedral slip systems, the features of the cubic system, and the presence of viscous properties both below and above the yield stress. Model parameters for γ and γ' phases were identified based on known creep curves for each phase. The effective properties of a single-crystal alloy, considering the presence of γ and γ' phas-es, were determined both based on finite element homogenization for a representative volume and using the simplest rheological (structural) models of the material, considering the series and parallel connection of phases. Based on multivariant computational experiments and analytical estimates, the dependences of the viscoelastic properties of nickel-based single-crystal alloys on the volume fraction of the γ' phase are determined. Phenomenological creep models that take into account the change in the volume fraction and the morphology of γ' inclusions have been proposed. The simulation results using the proposed two-level microstructural model of the material demonstrate a good agreement with the experimental data for the ZhS32 single-crystal heat-resistant alloy.

**PNRPU Mechanics Bulletin**. 2022;(4):116-134

STABILITY OF POSTCRITICAL DEFORMATION DURING TORSION OF THICK-WALLED CYLINDRICAL SOLID

#### Abstract

Ensuring the strength, reliability and safety of structures requires studying the issues of ine-lastic deformation zones occurrence and development which result from equilibrium accumula-tion of damages. Material postcritical deformation characterized by a decrease in the stress level during growing deformations is one of the damages accumulation effects. Application of postcritical deformation theory basic provisions to carry out structures refined strength analysis with additional deformation and strength reserves identification is expedient. Calculation of sof-tening processes stability related to loading systems rigidity is necessary. Consideration of analytical solutions where occurrence and development of softening zones are taken into account is expedient to illustrate main theoretical positions of postcritical defor-mation mechanics. The analytical solution to the hollow cylindrical solid torsion problem where material softening stage and loading system rigidity are considered is obtained. Two-link and three-link approximations of the complete material deformation diagram are considered. Dia-grams of shear stresses distribution over the cross-section are shown; various postcritical de-formation zones development scenarios existence is noted. Loading diagrams are built; torque maximum value dependence on material parameters and rod geometry is defined. The strength and deformation reserves of the structure are revealed. Conditions for complete loading diagram implementation are determined, the loading system rigidity influence is noted. Therefore, ration-ality and necessity of taking into account material softening stage and loading system rigidity in structural design are concluded.

**PNRPU Mechanics Bulletin**. 2022;(4):135-147

SIMULATION OF INTERIM FORGING OF DEPOSITED PRODUCTS IN ANSYS MECHANICAL APDL (IMPLICIT ANALYSIS)

#### Abstract

The paper considers the problem of simulating the interim forging during additive manufac-turing. Wire-arc additive technologies are associated with formation of technological residual stress fields, porosity, inhomogeneous structure and anisotropy, as well as unwanted defects, such as cracks, delamination or warping of the part. Interlayer hardening via forging both com-pensates such disadvantages and improves the mechanical properties of structures. Mathemat-ical modeling is one of main methods of studying these processes. There are a lot of publica-tions on modeling the formation of fields of residual stresses and heat shrinkage deformations in products obtained using additive technologies, including the method of wire welding. The work aims at checking the adequacy of using ANSYS Mechanical APDL for numerical modeling of interim metal forming processes. In this work, the Johnson – Cook viscoplastic model from Ex-plicit Dynamics was adapted to the capabilities of ANSYS Mechanical APDL for three materials: Amg6, 12X18H10T, VT6. As a physical model in ANSYS Mechanical APDL, a multilinear iso-tropic MISO plasticity model is chosen, which, unlike the Johnson – Cook model, does not take into account the effect of strain rate on the elastic-plastic behavior of the material. The values of the material constants for the MISO model are identified. The adequacy of replacing the non-stationary statement with a quasi-static one is proved, due to a slight loss of accuracy. A three-dimensional model of bar forging for three types of materials was built and implemented, its identification and verification were carried out by comparing with the results of a full-scale exper-iment. A good agreement between the calculated data and experiment is shown. Based on the data obtained, a conclusion was made about the admissibility of using the implicit solver ANSYS Mechanical APDL for calculating the processes of interim forging of the deposited products with acceptable accuracy.

**PNRPU Mechanics Bulletin**. 2022;(4):148-162

DETERMINATION OF THE CONTACT AREA UNDER CYCLIC LOADING OF CONTACT PAIRS OF ISOTROPIC MATERIALS ON THE BASIS OF 3D SURFACE MICRORELIEF MODELLING

#### Abstract

The parameters of the surface microrelief are paramount in the problems of frictional interaction of parts, the flow of liquid and gas in channels, ensuring the required thermal conditions and the stress-strain state of the structure. The solution of the problem of ensuring the optimal thermal regime for a significant range of technical products often becomes decisive in the design of products operating under conditions of high-intensity heat flows. Transit heat fluxes flowing through the product, as well as heat fluxes from own heat sources, must be either accumulated or removed to the external space. In this case, the directions of the heat flux vectors are determined by the design features of the products, including through various contact connections. Obviously, a reliable determination of the parameters of the contact interaction of product parts is the basis for a reliable analysis of the stress-strain and thermal state of a wide range of structures operating under conditions of high-intensity heat flows. The operational characteristics of the contacting parts of the structure are directly determined by the properties of the contact of the mating surfaces. When solving many problems of thermal, mechanical and electrical contact interaction, surface roughness is a major factor. The processes of friction and wear occur precisely on the actual contact area and depend not only on the properties of the material, but also on the intercontact pressure on this area, since the magnitude of the actual pressure determines the destruction of surface films and the appearance of adhesive bonds in the contact. In the presented work, the change in the actual contact area under cyclic loading of contact pairs of materials based on digital twins of contact surfaces in a wide range of compressive pressures is considered.

**PNRPU Mechanics Bulletin**. 2022;(4):163-169

ANALYSIS OF THE FORM AND EVOLUTION OF THE SOLID PHASE DURING DIRECTIONAL CRYSTALLIZATION OF NON-FERROUS METALS WITH ELECTROMAGNETIC INFLUENCE BY ULTRASONIC AND TEMPERATURE METHODS

#### Abstract

The paper is devoted to the experimental study of new mechanisms of controlling the process of directional crystallization of non-ferrous metals. The focus of the study is on the development and testing of measuring techniques applicable both to laboratory modeling and to processes under real operating conditions. The mechanism of controlling the rate and homogeneity of crystallization of a metal melt by changing the phase angles between the supply currents of a traveling magnetic field induction stirrer is proposed. This makes it possible to generate vortex flows of various topologies in the liquid metal, in particular, to change the number of large-scale vortices or to suppress the large-scale flow. It is shown that the hydrodynamic flows have an impact on the crystallization front shape, which allows one to control the homogeneity of metal solidification by changing the characteristics of the power supply of the inductor. It is important to note that a change in the phase angles of the currents while maintaining the supply amplitude does not significantly affect the crystallization rate, which opens up wide possibilities for controlling processes by changing both the current strength and the phase angles. The temperature method for determining the position of the crystallization front was successfully applied and verified by ultrasonic velocimetry measurements. It has been found that, in the presence of developed flows in a liquid medium, thermocouple measurements provide good agreement (up to a few percent) of the measured position and geometric shape of the crystallization front with the ultrasonic measurement data. In the absence of liquid phase stirring, the difference between the thermocouple and ultrasonic measurements increases slightly. Nevertheless, even in this case, the thermocouple method can be used to correctly determine the position and velocity of the crystallization front.

**PNRPU Mechanics Bulletin**. 2022;(4):170-179

ELECTROMAGNETIC COUPLING OF PIEZOELECTRIC/FERRITE COMPOSITE WITH INITIAL STRESS STATE

#### Abstract

A mathematical model of electromagnetic thermoelasticity for an initial-stressed transversal-isotropic composite with piezoelectric magnetostrictive phases has been developed. To solve the related boundary value problem of electromagnetic thermoe-lasticity, the Green function method was used as part of a generalized singular approximation of the statistical mechanics of composites, taking into account the initial stress state of the representative domain of the composite at micro- and macro-levels. The solution of the problem of "effective module" for tensors of effective elastic, piezomechanical, magnetostrictive properties, dielectric permittivity and magnetic permit-tivity, temperature, pyroelectric, pyromagnetic coefficients and (which appeared only at the macro level) electromagnetic and magnetoelectric couplings of a quasi-periodic composite with an initial electromagnetic elastic state was obtained. The solutions for the desired tensors of effective properties of the quasi-periodic composite are presented in the form of analytical formulas of simple linear decompositions by solutions for tensors of effective properties of the periodic structure and statistical mixture, decomposition coefficients are the coefficient of "periodicity" (correlation of quasi-periodic and periodic structures) p and "disordering" 1-p, respectively. Results of calculation of all independent components of tensors of effective coefficients of electromagnetic and magnetoelectric couplings of different structures (periodic, quasi-periodic and statistical mixture) are presented in unidirectional direction of fibrous composite "PZT-4/ferrite" with axisymmet-ric tensor of initial macrostrain of composite. For a quasi-periodic composite (with initial macrostrain), the significantly non-monotonic nature of the dependencies of relative (to values in the absence of an initial stress state) values of effective coefficients of electromagnetic and magnetoelectric couplings from the volume fraction of ferrite fibers was revealed. It was revealed that we have an increase in the absolute values of the electromagnetic and magnetoelectric coupling coefficients of composite at negative values of its initial axisymmetric axial macrostrains. The most significant effect is the initial comprehensive macrostrain in the transversal plane on the coefficients of the transversal magnetoelectric coupling of the composite.

**PNRPU Mechanics Bulletin**. 2022;(4):180-195