## No 3 (2022)

**Year:**2022**Articles:**17**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/302**DOI:**https://doi.org/10.15593/perm.mech/2022.3

EFFECT OF SYNCHRONIZATION OF CONTINUOUS ACOUSTIC EMISSION STATISTICAL PROPERTIES DURING STRUCTURALLY HETEROGENEOUS MATERIALS DEFORMATION

#### Abstract

In this paper, a correlation analysis of the statistical properties of continuous acoustic emission recorded in various parts of marble and glass fiber laminate samples during their quasi-static deformation is carried out. The spectral measure of coherence, which is a generalization of the square modulus of the coherence spectrum to the case of multidimensional series, is chosen as a correlation measure. The measure of coherence was estimated for the width of the multifractal spectrum and the spectrum carrier realizing its maximum, calculated in a sliding time window for acoustic emission signals. It is shown that the preparation of a macrofracture site is accompanied by synchronization of the statistical properties of acoustic emission in the selected frequency intervals. Based on the analysis of changes in the frequency-averaged measure of coherence for both types of materials, four characteristic stages are distinguished, the boundaries of which are individual for each of the materials. The onset of the fourth stage, which is characterized by a monotonic increase in the average measure of the coherence of the statistical properties of the AE, can be chosen as a possible criterion for the transition of the material to the limiting state.

**PNRPU Mechanics Bulletin**. 2022;(3):5-13

Cycle life prediction under low cycle fatigue using nonlinear Marco - Starkey model

#### Abstract

The process of deformation and fracture of structural alloys under low-cycle fatigue in conditions of uniaxial loading with axial strain control under complex cycle shape and block loading has been studied. The obtained results of experimental studies of structural alloys were used to assess the possibility of using the nonlinear Marco - Starkey damage accumulation model. The processing of the nickel alloy cyclic tests results with a simple and complex form of the cycle has been carried out. A combination of exponents included in the non-linear Marco - Starkey model was selected which for the case of low-cycle fatigue with a complex M-shaped cycle made it possible to predict durability which is in good agreement with experimental test data. The main problem of using the nonlinear Marco - Starkey model for predicting cyclic durability with an M-shaped cycle is the presence of many combinations of exponents that allow predicting cyclic durability with the same accuracy. To determine the uniqueness of the solution it is proposed to carry out a set of tests under simple and block loading. Experimental results have been obtained on the processes of deformation, fracture of D16T aluminum alloy under low-cycle fatigue conditions under simple cycle forms with constant parameters, and block loading with variable cycle parameters in tests for uniaxial loading with axial strain control. Based on the obtained experimental results according to the new method combinations of the m degrees coefficients were selected, which was carried out in the range of exponents from 0.2 to 10 with a step of 0.2. Comparison of the forecasting results for blocks consisting of three groups made it possible to find several combinations of general exponents m for two groups that are present in all blocks. The final choice of a pair of coefficients was carried out on the condition that the predicted damage is close to unity in the first (test) block. The selected values of the coefficients made it possible to predict the durability under block low-cycle loading using the nonlinear Marco - Starkey model.

**PNRPU Mechanics Bulletin**. 2022;(3):14-22

Postbuckling Behavior of Compressed Bars with Nonlinearly Elastic Supports

#### Abstract

Supercritical bending of a rod compressed by a longitudinal force and having a non-linearly elastic resistance to rotation of its axis in hinged supports is considered. Hooke's law is adopted for the rod material. The deflection is presented as a trigonometric series, in which only the first three terms are left. Through this row, the angle of inclination of the axis of the rod is expressed. It is chosen as the desired function, with respect to which a resolving nonlinear integro-differential equation is obtained. To simplify it, the case is considered when the bent axis of the rod is a gentle curve. It is revealed that the usual approach, in which the compressive force is given, leads to a trivial solution with zero values for the coefficients of this series. Therefore, the first coefficient of the series, through which the deflection is represented, was chosen as the initial parameter. The equation containing the remaining desired coefficients and load was solved by the simple collocation method, since the use of the predetermined collocation method is impossible due to the nonlinearity of the resulting system of equations. The calculation results were analyzed depending on the choice of collocation points, which showed a small difference between the obtained solutions (the quadratic discrepancy between the calculated values did not exceed one percent, which indicates sufficient accuracy of the deflection approximation in the form proposed in the work). In contrast to the usual approach used in numerical methods, here it is not necessary to enter any parameter of the loading process (such as the length of the arc in the arc method) to pass the limit points. An approach is presented for an approximate calculation of the compliance of the considered rods in compression, taking into account their supercritical bending. The results of numerical studies of the supercritical behavior of rods for various geometric and mechanical characteristics of rods and supports are presented, and dependences between various parameters of the stress-strain state are obtained. Their analysis made it possible to reveal a number of features in the behavior of the considered rods depending on the properties of the nonlinear elastic resistance of the supports. In particular, it was found that under the considered parameters of the system under consideration, the compression force and the angle of inclination of the rod on the support are related by a relationship close to linear, the compliance of the compressible rod can increase by tens or more times depending on the stiffness of the springs, the rod can lose stability by snap at some values parameters of nonlinear elastic supports.

**PNRPU Mechanics Bulletin**. 2022;(3):23-31

Low-cycle fatigue of V-notched cylindrical samples of bronze alloy

#### Abstract

The article presents the results of experiments on cyclic loading of samples with an annular V-shaped notch and their mathematical modeling. The tests were carried out on specimens of the BrKh08-Sh alloy with a cyclic change in the tensile load from 0 to a given value. Under such loading, the material in the groove area is subjected to cyclic elastoplastic deformation, which leads to failure due to low-cycle fatigue. When testing samples using the digital image correlation method, the deformation of the material on the surface of the undercut was measured, which made it possible to determine the nature of the change in its range from cycle to cycle. Mathematical modeling of experiments is carried out according to the finite element method. For this purpose, models of elastic-plastic deformation and accumulation of material damages are introduced into the SIMULIA Abaqus software package. The plasticity model is based on the theory of flow under combined hardening. The damage accumulation model is based on the energy strength criterion. The article presents the basic equations used to calculate the models. The material parameters are determined and verified based on the results of a basic experiment on cyclic tension-compression of a smooth cylindrical sample under asymmetric rigid loading. Based on the results of the calculation, cartograms of stresses, deformations, and accumulated damages were constructed. It is shown that when the sample is unloaded, in the area of stress concentration close to the annular groove, compressive stresses arise, the values of which are close to the values of tensile stresses at the moment of application of the maximum load. The results of mathematical modeling are compared with the experiment. A comparison of the results was carried out in terms of the range of axial deformation of the material on the surface of the undercut and the number of cycles before failure.

**PNRPU Mechanics Bulletin**. 2022;(3):32-41

Memory surfaces separating the processes of monotonous and cyclic loads

#### Abstract

The processes of elastoplastic deformation of structural materials can consist of a sequence of monotonous and cyclic loading regimes, in which peculiar effects and features arise. Mathematical modeling of such processes, as well as resource assessment and forecasting, is a very difficult task. In addition, the analysis of transient processes from cyclic to monotonous and from monotonous to cyclic shows the need to separate these processes. Based on the analysis of the results of experimental studies of samples of stainless steel 12Х18Н10T under a rigid (controlled deformation) deformation process, which is a sequence of monotonic and cyclic loading modes, under conditions of uniaxial tension-compression at normal temperature, the features and differences in the processes of monotonic and cyclic loading are revealed. To describe these features and separate the processes of monotonic and cyclic loading modes in the theories of plastic flow with combined hardening, various variants of memory surfaces are introduced. An analysis of the results of experimental studies of stainless steel showed that in the space of the plastic strain tensor, the size of the memory surface is determined by the range of plastic strains, and the position of the center is determined by the values of average plastic strains under cyclic loading. Various variants of the memory surface are considered, their capabilities and disadvantages are identified, and the most adequate variant of the memory surface is determined. To confirm the operability of this version of the memory surface, together with the equations of the Bondar plasticity model, the calculated and experimental results were compared and a reliable agreement was obtained between these results both in terms of the kinetics of the stress-strain state and in terms of the number of cycles to failure.

**PNRPU Mechanics Bulletin**. 2022;(3):42-50

EVALUATION OF THE DYNAMICS OF ELASTIC PLATE AND LIQUID SECTION DYNAMIC ABSORBER

#### Abstract

In this paper, the problem of studying the dynamics of transverse vibrations of plate, taking into account the elastic dissipative characteristics of the hysteresis type in conjunction with a liquid section dynamic absorber under the influence of kinematic excitations. In the differential equations of motion, the elastic dissipative characteristics of the plate material of the hysteresis type are taken into account by means of harmonic linearization coefficients based on the Pisarenko - Boginich hypothesis. The amplitude-frequency characteristic of the vibrating plate and the analytical expressions of the transfer function were determined using a differential operator from a system of differential equations of motion depending on the system parameters. In order to perform numerical calculations, the coefficients of the first three terms of the logarithmic decrement expression were found. In the amplitude interval, the function representing the vertical deviation of the amplitude-frequency characteristic decreases and the function representing the energy dissipation in the plate material increases. It has been shown that the efficiency of the liquid section dynamic absorber in quenching harmful plate vibrations at low frequencies can be evaluated based on the results of numerical calculations to ensure that the displacements of the plate point reach minimum values. Amplitude-frequency characteristics for plate points at different parameters were constructed for the distributed parametric system using the developed model and method. Recommendations for the selection of parameters of the system depending on the elastic dissipative and inertial properties are given.

**PNRPU Mechanics Bulletin**. 2022;(3):51-59

Effect of mechanical load on resultant composite piezomodules

#### Abstract

A mathematical model has been developed and, on its basis, analytical solutions have been found for the resulting "reduced" electroelastic characteristics, in particular, for deformation (shear) piezomodules and dielectric permeabilities of a transversal-isotropic piezoelectric material (composite), taking into account corrections due to the presence of mechanical axial stresses and electrodeformational reorientation of the axis of symmetry of the material properties under the action of an electric field. In the initial configuration, axial tensile and/or compressive stresses do not result in shear strains of the material, the shifts are initiated by the application of an electric field and are "amplified" by axial stresses of the initial configuration. The effect of increasing the resulting values of deformation piezomodules of the material from the action of axial stresses was revealed. The values of the applied axial stresses do not exceed the values of the loss of stability of the material (elements of the structure and, in general, the composite), which causes the preservation of the effect of increasing piezomodules also under the action of an alternating electric field. The results of numerical modeling are obtained for the transversal-isotropic polymer composite "silicone/PZT-4" with unidirectional piezoelectric fibers with a circular cross section as a partial (limit) case of the structure with oriented ellipsoidal inclusions using the known "generalized singular approximation" based on the method of Green functions of statistical mechanics of composites for calculating tensors of effective properties composite in the current configuration. Deformation anisotropy corrections are calculated through the current coordinates of the guide vector for the symmetry axis of the properties - the orientation direction of the fibers of the composite. It has been found that the most significant effect of increasing shear piezomodules is manifested for a unidirectional fibrous composite with a low-modulus polymer matrix "silicone/PZT-4" at small values of the volume fraction of rigid piezoelectric fibers from the action of compressive longitudinal axial stress.

**PNRPU Mechanics Bulletin**. 2022;(3):60-69

Vibrations of a strip with delamination in the framework of the one-parameter Aifantis model of gradient elasticity theory

#### Abstract

The problems on in-plane and anti-planar steady-state vibrations of an isotropic elastic strip with delamination at the lower boundary has been investigated. The goal of the study is to analyze the stress-strain state in the crack tips areas and to construct a crack opening function being the main mechanical characteristics in the crack theory problems. The problems under study have been solved in the framework of the nonclassical gradient elasticity theory (GET) on the basis of the one-parameter model proposed by Aifantis. The boundary integral equations (BIE) are obtained with respect to crack opening functions or their derivatives. The analysis of BIEs is carried out, regular and irregular parts are distinguished, the obtained BIEs with singular (e.g., with hypersingular, with cubic singularity) integrals are solved via collocation methods, approximating Chebyshev polynomials, quadrature formulas for singular integrals. For the in-plane problem solution, the simplified Ru-Aifantis method has been applied. The Ru-Aifantis method allows to divide the initial boundary value problem into two sub-problems - the classical linear elasticity theory (LTE) problem and the simplified boundary value problem for finding the gradient solution which includes the solution found via the classical theory. For each of the problems, semi-analytical expressions for the functions of crack opening have been constructed, and the analysis of the stress-strain state in the area of crack tips has been carried out. The problems have also been solved in the case of a crack with small relative length, the analysis of BIE depending on small parameters ratio has been carried out, and explicit expressions for the crack opening functions have been obtained. Numerical calculations have been performed; the applicability conditions for the asymptotic method are determined, and a comparative analysis of the results obtained on the basis of GET and LTE models, depending on the values of the gradient parameter and the delamination length, is realized.

**PNRPU Mechanics Bulletin**. 2022;(3):70-82

Calculation of effective properties of geocomposites based on computed tomography images

#### Abstract

Biot’s parameter is included in the formula for calculating effective stresses and should be taken into account when assessing the stress-strain state of a water-saturated rock mass. A method for calculating Biot’s tensor parameter based on asymptotic averaging of the equilibrium equation for a fluid-saturated porous medium is proposed. Calculations of elastic properties and Biot’s coefficient were carried out on various types of rocks - limestone, dolomite, hyaloclastite, basalt. The calculations were carried out using 3D models of geocomposites built from X-ray computed tomography images. The results of 3D calculations of Young's modulus and Biot’s coefficient coincided with the results of experimental determinations of these properties by the ultrasonic method. This fact shows the expediency of using a computational approach that implements asymptotic averaging to estimate the effective properties using 3D models of the real rock structure. The results of 3D and 2D simulation of effective properties of geocomposites are compared. 2D models were built using photographs of rock sections. It was found that the values of Young's modulus and Biot’s parameter for 2D models differ from the corresponding experimental and 3D calculation results by 20-30 %. Therefore, 2D modeling is not suitable for evaluating the effective properties of porous geomaterials. Based on the results of calculations and experiments, the dependences of Young's modulus and Biot’s coefficient on porosity were obtained and studied. These dependencies are used in non-linear numerical simulation of the consolidation of water-saturated soils. The results of the study showed that Biot’s coefficient does not depend on Young's modulus of the rock skeleton material. The influence of the pore shape on Young's modulus and Biot’s coefficient was studied using 2D calculations. A method for predicting the shape of pores from the values of porosity and Young's modulus using neural networks is proposed. Using this method, a specific algorithm for predicting the shape of pores in hyaloclastites has been implemented and studied.

**PNRPU Mechanics Bulletin**. 2022;(3):83-94

Formulation and Numerical Solution of the Stability Loss Problem of Elastic-Plastic Shells of Revolution with an Elastic Filler under Combined Axisymmetric and Torsional Loadings

#### Abstract

A dynamic statement and a method for numerically solving the buckling problems of elastoplastic shells of revolution with filler in axisymmetric and non-axisymmetric shapes under quasi-static and dynamic loading are presented within the framework of two approaches. In the first approach, the problem of elastic-plastic deformation and buckling of shells of revolution with an elastic filler under combined axisymmetric loading with torsion is formulated in a two-dimensional (generalized axisymmetric) formulation based on the hypotheses of the shells theory of the Timoshenko type and the Winkler foundation. The constitutive relations are written in the cylindrical system of Euler coordinates. For each shell element, a local Lagrangian coordinate system is introduced. Kinematic relations are recorded in the current state metric. The distribution of the displacement velocity components over the shell thickness and strain rate tensors in the local basis is written as the sum of the momentless and moment components, which, in turn, are written as the sum of the symmetric and asymmetric parts in the local and in the general basis. The elastoplastic properties of the shell material are taken into account within the framework of the theory of flow with nonlinear isotropic hardening. To take into account non-axisymmetric forms of buckling, the desired functions (both displacements and forces, moments, contact pressure) are expanded into a Fourier series in the circumferential direction. The variational equations of shell motion are derived from the general equation of dynamics. The contact between the shell and the deformable filler is modeled based on the conditions of non-penetration along the normal and free slip along the tangent. The variational equations of shell dynamics for axisymmetric and nonaxisymmetric processes are interconnected through the physical relations of the theory of plasticity. They take into account large axisymmetric shape changes and the instantaneous stress-strain state of the shell. At the initial stage of the nonaxisymmetric buckling process, the deflections are small; therefore, the equations of nonaxisymmetric buckling are obtained as linearized with respect to nonaxisymmetric forms. To initiate nonaxisymmetric buckling modes, initial nonaxisymmetric deflections are introduced. To solve the defining system of equations, a finite-difference method and an explicit time integration scheme of the “cross” type are used. The second approach is based on continuum mechanics hypotheses and implemented in a three-dimensional setting. Both approaches make it possible to simulate the nonlinear subcritical deformation of shells of revolution with an elastic filler, to determine the ultimate (critical) loads in a wide range of loading rates, taking into account geometric shape imperfections, to study the processes of buckling in axisymmetric and non-axisymmetric shapes under dynamic and quasi-static complex loading by tension, compression, torsion, internal and external pressure. The results of numerical simulation are compared with experimental data on the torsion of steel cylindrical elastoplastic shells ( R / h = 1.45) with an elastic filler.

**PNRPU Mechanics Bulletin**. 2022;(3):95-106

Study of deformation structures maps in metals under tension

#### Abstract

The paper considers the regularities of macroscopic inhomogeneity of plastic flow during uniaxial tension of flat samples of Fe-Cr-Ni 2 mm thick. Their tension axis was oriented along the rolling direction. The average grain size was 12.5 ± 3 mm. The plastic flow curves of the alloy had long stages of linear strain hardening over the entire test temperature range 180 K < T < 297 K. For the experimental study of plastic deformation, we used the method of accurately reconstructing the fields of displacement vectors and calculating the components of the plastic distortion tensor using speckle photography with increments of the total strain between exposures 0.001. The field of displacement vectors as a whole over the sample during loading is inhomogeneous both in the directions of the displacement vectors and in values; in some areas, the displacement vectors nonmonotonically change directions relative to the tension axis. It has been established that in the test temperature range 180 K < T < 297 K, plastic flow is localized at all stages of the process. The appearance of the a′-martensite phase during the deformation of the alloy under study leads to a change in the mechanical characteristics, the work hardening coefficient, and the deformation localization parameters. The maps of deformation structures are analyzed in the form of spatial distributions of the components of the plastic distortion tensor: local elongations, narrowings, shifts and rotations. The non-linear nature of the change in the coefficient of transverse deformation from the level of acting stresses is established. The general form and quantitative parameters of the evolution of the components of the plastic distortion tensor indicate the connection of this process with the self-organization of a defective subsystem in a deformable medium.

**PNRPU Mechanics Bulletin**. 2022;(3):107-115

ON THE INFLUENCE OF THE MECHANICAL CHARACTERISTICS OF A THIN ADHESION LAYER ON THE COMPOSITE STRENGTH. PART 1. ELASTIC DEFORMATION

#### Abstract

The problem of deformation of a DCB sample, which is a composition of bodies bound by an adhesive layer of finite thickness, is considered. Based on the variational equilibrium equation containing the layer thickness as a linear parameter, a finite element solution of the problem of loading the layer with a normal discontinuity in the plane strain state is constructed. The stresses averaged over the layer thickness are related to the stresses along the layer boundary by the equilibrium equations. The boundary stresses of the layer form the boundary conditions for the mating bodies. In the layer, along with shear stresses, stresses orthogonal to shear are also taken into account. The constitutive relations in the layer are represented in terms of average stresses. With a significant difference in the Young's moduli of the adhesive and mating bodies, the convergence of the value of the J-integral with a decrease in the layer thickness is shown. To find the J-integral, its representation is used as a product of the specific free energy at the end of the layer and its thickness. It has been established that the Poisson's ratio of the bodies affects the value of the J-integral, and the Poisson's ratio of the adhesive layer has almost no effect on the value of the J-integral. Using the theory of plates Mindlin - Reisner at zero Poisson's ratio of the adhesive, an analytical representation of the J-integral is obtained. The representation includes energy terms related to the pull-off stress and the axial stress in the layer. In this case, the term associated with the axial stress in the layer is proportional to the square of the ratio of the Young's moduli of the adhesive layer and the bodies mating with it. From the solution obtained, it follows that the mechanical properties of the adhesive layer with a small thickness compared to bodies do not affect the value of the J-integral if the elastic modulus of the adhesive layer is significantly less than the elastic modulus of the mating bodies. Thus, the use of replacing the adhesive layer with a layer of zero thickness is correct under these restrictions.

**PNRPU Mechanics Bulletin**. 2022;(3):116-124

Features of diffusion-controlled processes of regulated volumetric synthesis from powder mixtures Ti-Al-Fe-Fe2O3

#### Abstract

Experimentally and theoretically analyzed processes occurring in mixtures of powders Ti-Al-Fe-Fe2O3 during reactive sintering (sintering from chemically interacting components) under controlled heating. As one of the components wastes of steel billets were used. This makes the study relevant. The heating rate was varied. Possible physical and chemical stages, their sequence and the assumed phase composition of the synthesis products were analyzed. Although reactive sintering has attracted considerable interest for the production of composite materials, the regularities established for binary systems cannot be purely transferred to sintering powders of more than two kinds. It is shown that the multistage process results in the formation of sets of nonequilibrium phases of variable composition based on α-titanium solid solution. The structures obtained from powder compositions «Ti+Al+Fe» and «Ti+Al+metal chips» do not differ much from each other, despite the presence of an appreciable share of oxides in the latter. The examples of particular problem solutions which correspond to the possible variants of diffusion interaction in a heterogeneous system are presented. Analytical solutions are obtained in the quasi-stationary approximation. The diffusive nature of changes in the position of interfaces is the same for constant and variable diffusion coefficients. The shape of the refractory reagent particle has little effect on the dynamics of the process. An example of calculating the stresses and strains in a conditional reaction cell when the phases are separated by moving boundaries is given. In the quasi-static approximation, the time for the mechanical part of the problem is a parameter. It is shown that the driving forces of diffusion transfer may be related to local stresses, which, in turn, may be related to changes in composition.

**PNRPU Mechanics Bulletin**. 2022;(3):125-134

Numerical simulation of the nickel alloy microstructure formed in the process of hot fogging

#### Abstract

The paper presents a comprehensive analysis of the deformation and thermal states of the Waspaloy alloy billet heated to different initial temperatures of 1100°C and 1150°C and subjected to free upsetting to an average diameter of ~1060 mm at the deformation rate of 100 mm/s. The thermodynamic forces acting on the billet trigger the process of dynamic recrystallization, which is associated with the appearance and growth of low-defect nuclei of new grains instead of the deformed ones. To describe the material microstructure evolution, the phenomenological approach implemented in the DEFORM-2D/3D software package was applied. The simulation was based on the modified Johnson - Mehl - Avrami - Kolmogorov (JMAK) model, whose equations allow calculating the volume fraction of recrystallized material and describing the grain structure transformation of metal alloys. The results of solution of the non-stationary temperature problem are used to construct the temperature fields in the Waspaloy alloy billet during its transportation through air from the furnace to the deforming equipment within 45 seconds and during the subsequent upsetting process. For the latter, the force and strain characteristics, including the force required to complete this process, are determined in the framework of the plastic flow theory, and the characteristics of the grain structure of the nickel alloy, such as the average size of recrystallized grains and their volume fraction, are determined in the framework of the JMAK model. The results obtained by numerical simulation make it possible to substantiate an optimal selection of parameters of billet deformation ensuring the formation of the required material structure.

**PNRPU Mechanics Bulletin**. 2022;(3):135-146

Development of the digital image correlation (DIC) method for mechanical testing at elevated temperatures

#### Abstract

The results of experimental studies on the evaluation of the performance of the non-contact optical video system Vic-3D and the digital image correlation method during tensile tests of composites under normal and elevated temperatures are presented. The dependences of the recorded strain values are analyzed by two independent measurement methods: the digital image correlation method (DIC) and a mounted extensometer. The tests were carried out at the Center of Experimental Mechanics of the Perm National Research Polytechnic University using the Instron 5882 universal electromechanical testing system, which includes a thermal chamber with an operating temperature range of -100° to +350 °С. Samples of a composite material made of STEF fiberglass and carbon fiber were studied. Tensile tests were carried out at temperatures of 2°2, 100°, 120°, 200°, and 300°C. Recommendations for conducting temperature tests using the DIC method are proposed. Based on the test results, deformation diagrams were constructed. Fields and diagrams of longitudinal deformations are obtained at different temperature levels. A detailed evaluation of the finely dispersed coating required for the implementation of the DIC method in temperature tests has been carried out.

**PNRPU Mechanics Bulletin**. 2022;(3):147-159

The influence exerted by active cooling on the permanent connection formed as a result of electric arc welding

#### Abstract

The problem of temperature stresses is solved in the work. The process of electric arc welding with guaranteed penetration of steel plates of various thicknesses is simulated. The possibility of reducing residual stresses formed as a result of welding by applying heat removal from the surface of the near-weld region of a steel plate is being investigated. The temperature distribution is determined by solving a non-linear heat equation, in which the specific heat and thermal conductivity depend on temperature. The heat source is modeled using the double ellipsoid method proposed by John A. Goldak. The horseshoe-shaped heat sink follows immediately after the anode spot and is set by the heat flux in such a way that the main temperature of the plate does not decrease below its initial temperature. This limitation reduces the level of the temperature gradient, thereby reducing the stresses in the metal plate. To illustrate the effect of active heat sink, the temperature distribution in the top layer at the moment of termination of welding is compared with and without active heat sink. The material is assumed to be elastoviscoplastic, the deformations are small and consist of reversible and irreversible ones. Reversible deformations are related to stresses by the Duhamel - Neumann law. Irreversible strains increase when the von Mises plastic flow condition is satisfied, in which there is a plastic strain rate component responsible for the plastic flow viscosity. The elastic moduli (Young's modulus, Poisson's ratio) and the yield strength are assumed to be temperature dependent. The solution of the mechanical part of the problem is found by the method of simple iterations. For plates with different thicknesses, diagrams of residual stresses are given, located in the center of the plate from the seam to the periphery. As a result of the work, according to the obtained distributions of residual stresses in the plate material, it is concluded that the use of an active horseshoe-shaped heat sink from the near-weld zone when welding thin steel plates reduces residual stresses, therefore it is recommended to use in the welding process. The use of an active heat sink on the reverse side of the plate leads to an increase in the stress level, therefore it is not recommended for use.

**PNRPU Mechanics Bulletin**. 2022;(3):160-169

Modal analysis of a hydrogen-damaged pipe as shell inhomogeneous on thickness

#### Abstract

Hydrogen corrosion of main gas and oil pipelines often leads to accidents. It is necessary to assess the life of hydrogen-weakened pipes. An inner layer with deteriorated mechanical characteristics is formed in them under pressure of hydrogen-containing medium. In the paper the calculation of free vibrations of such a tube taking into account the degradation of its material is performed. The tube is modeled as a bilayer cylindrical shell according to the classical theory. Influence of hydrogen-impacted layer is taken into account when calculating stiffnesses and displacement of neutral line of the shell. Three variants of averaging of parameters determining the stiffness of the shell are considered, numerical experiments are carried out and the natural frequencies of the shell are found. Comparison with calculations using the method of finite elements in the ANSYS software made it possible to estimate the degree of applicability of each version of averaging. The method of averaging Young's modulus over the thickness of the shell does not "feel" asymmetry of layers relative to the neutral line. The method of adding a correction for the neutral line radius of the shell works satisfactorily for axisymmetric and beam modes. The next more accurate approximation is to average the shell stiffnesses over its thickness with the reduced radius. This method allows us to obtain satisfactory results in a wide enough frequency range and for the modes related to the deformation of the shell cross section. A method for the principle reconstruction of the parameters characterizing the stiffness of the shell using three experimentally obtained fixed frequencies has been proposed. On the basis of these studies formulas are proposed which allow to reconstruct parameters of weakening of pipe material, as a result of hydrogen corrosion, both in thickness and in time from the frequency characteristics of the pipe.

**PNRPU Mechanics Bulletin**. 2022;(3):170-182