## No 4 (2018)

**Year:**2018**Articles:**25**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/38**DOI:**https://doi.org/10.15593/perm.mech/2018.4

Features of Deformation Behavior of Structural Steel during Forging

#### Abstract

The paper presents the studies of the structural steel subjected to forging including severe plastic deformations. The purpose of this work is to evaluate the role of severe plastic deformations during the forging of steel blanks. The computer simulation of the operations of forging steels of grades 20, 45 and 40H in the software complex DEFORM 3D was performed. A detailed analysis of the stress-strain state of the blanks during deformation is carried out, as well as the load schedule. As a result of the analysis it was revealed that the nature of the stress-strain state under equal conditions is determined by the material of the billet. An increase in stress and strain effectiveness is shown depending on carbon increasing in steel under identical conditions. The choice of the most optimal process parameters for different technological conditions using simulation in the DEFORM 3D software complex is completely justified, and can yield concrete results, even for complex deformation processes. The deformation behavior of 20, 45, and 40H steels was studied during forging with a degree of deformation at a draft of 30 % and with severe plastic deformation by the forging method in trapezoidal dies of steel 45 in industrial conditions. As a result of the forging, a uniform microstructure with a grain score 7 is formed. The use of severe plastic deformation promotes the intensification of the rotational deformation mechanism. The implementation of shear deformations provides grain refinement to 8 points. In this case, the hardness of steel 45 after the realization of severe plastic deformation increased by more than 13 % compared to the metal obtained by the conventional sediment and amounted to 164.8 HB. The influence of technological features of forging on the microstructure and properties of structural steel, especially on the prevailing deformation mechanism is revealed.

**PNRPU Mechanics Bulletin**. 2018;(4):7-19

THE STRESS-STRAIN STATE AND DURATION UNTIL FRACTURE OF ROTATING DISKS IN CREEP

#### Abstract

The stress-strain state and duration until fracture are calculated for a two-stage behavior of a rotating disk with a hyperbolic surface shape under creep. The first stage is the damage accumulation and beginning of fracture in a certain body’s area where the accumulated damage reaches a critical value. The second stage is the fracture front spreading and a complete destruction of the body. It is assumed that the fracture front propagates axisymmetrically, the fracture is brittle. The calculation method consists in the fact that the unsteady creep problem is reduced to a similar problem within the assumption of a steady material creep. In order to obtain a valid solution, it is necessary to multiply the known solution of the steady creep by some functions of the coordinates and time. To find these functions, we obtain the corresponding system of equations. We investigate the duration of stages depending on the creep kinetic theory version in the statement of Yu.N. Rabotnov and L.M. Kachanov. According to the calculations, the dependence of the second stage duration to the first stage duration can be from tenth to several tens of percent depending on the applied load, the surface shape and the size of the inner hole of the disc. With a decrease in the radius of the disc’s inner hole, the second stage duration can be comparable with the first one. In all the studied cases the first stage duration in the Rabotnov’s model is higher than on the Kachanov’s one; the dependence of the second and the first stage duration in the Rabotnov’s model is smaller than in the first stage of the Kachanov’s model. The analysis of the fracture front movement showed that the main time of the second stage (about 75-85 % according to the Kachanov’s model and 85-90 % according to the Rabotnov’s model) is 20 % of the working part of the disk radius.

**PNRPU Mechanics Bulletin**. 2018;(4):20-32

Identification of the inhomogeneous cylindrical waveguide properties

#### Abstract

The inverse coefficient problem of properties identification for the radially inhomogeneous (including layered and coated) elastic cylindrical isotropic waveguide is studied. To restore three functions - the Lame and density coefficients characterizing the variable properties of an isotropic waveguide, two modes of action on an object that excite normal and torsional oscillations are considered. The identification procedure is carried out according to the acoustic sounding of the outer surface of the cylinder. The problem by the integral Fourier transformation at the coordinate coinciding with the axis of the waveguide is reduced to one-dimensional problems concerning the averaged characteristics. The obtained problems are divided by the unknown functions which allow to realize the serial identification. The linearization of the divided inverse coefficient problems is carried out. Two iterative processes are formulated, which allow to restore the required functions sequentially. At each step of the iterative schemes, the corresponding boundary value problems are solved by the method of adjustment and the system of Fredholm’s integral equations of the first kind with smooth kernels by using of regularization methods. A computational experiment simulating normal and torsional oscillations of the waveguide is conducted. The corresponding wave fields obtained from the solution of the direct problem by the known laws of inhomogeneity are used as additional information. We study examples of identifying the laws of waveguide characteristics change which model the presence of inhomogeneous coating on the outer surface, which characteristics can significantly differ from those of the waveguide material, which are considered in these experiments to be known. We perform a representative set of computational experiments to identify the laws of changing the required mechanical characteristics - the modules of elasticity and density - for monotonic, non-monotonic and piecewise continuous functions.

**PNRPU Mechanics Bulletin**. 2018;(4):33-46

Numerical simulation for development of methodology of stress-strain state control of composite bulkhead for aviation application with the usage of FBG sensors

#### Abstract

Fiber-optic sensors are used to monitor the state of structures in various industries. The best solution for monitoring deformations in the products made of polymer composite materials are fiber Bragg gratings (FBGs). In this paper, we consider the possibility of studying the stress-strain state (SSS), tracking structural changes in the construction with the help of FBGs, using the example of a "U" -shaped composite frame of an aircraft engine. Based on the revealed regularities and peculiarities of the FBGs operation, the main control points have been determined. In the course of the research, we created a three-dimensional computer model of an aircraft propulsion system composite frame which was used for the analysis of the stress-strain state under various loading conditions. The developed model makes it possible to perform a layer-by-layer analysis of the composite frame structure to estimate the normal and shear interlayer stresses, which determine the destruction of the structure. Numerical modeling of this problem was carried out by the finite element method (FEM) in the software package ANSYS Workbench. Numerical results were compared with the experimental data obtained in laboratory tests of a composite frame segment equipped with fiber-optic sensors. The tests were carried out on the equipment of the unique scientific installation "Unique scientific and technological complex of automated computation" using the Zwick servo-hydraulic machine and the Astro X327 interrogator. The results obtained will be used to determine the size of the effective "sensitive zone" for embedding fiber-optic sensors (FOS) based on Bragg gratings into the construction of a composite frame. The main goal of this work is the development of a methodology for detecting and identifying various types of damage and the maximum load during the working lifespan of the structure.

**PNRPU Mechanics Bulletin**. 2018;(4):47-57

The methodical Aspects of Strength, Deformation and Energy characteristic Determination of Salt Rocks under Direct Tensile Loading of Rock Specimens in Laboratory Conditions

#### Abstract

Mineral deposit development is impossible without suitable data about mineral and parting properties. Underground facilities and construction stability depend on how rocks can resist increased loads appearing due-to overburden load redistribution during mineral development. Tensile stresses usually cause stability losses. One of the main information sources for mining design is a laboratory investigation of rock specimens behavior under various loadings. The work aims to create the approaches of strength, deformation and energy determination of salt rocks under direct tensile load in laboratory conditions. The studies are carried in the salt rocks of the Verchnekamskoe potash and magnesium deposit. The main direct tensile problems of rock specimens have been analyzed at laboratory conditions. The comparative direct tensile result analysis of salt rocks has been performed depending on the specimen geometry. Based on the laboratory and numerical experiments, the specimen geometry has been suggested considering the stress state mode arising under loading process, grain size, bracing influence. A material has been suggested to enable a good specimen bracing in caps. It has good adhesion both with a salt specimen and with steel surface of a cup. The center device is developed to provide a good coaxial specimen bracing in cups. Also the device is developed to write the axial specimen deformation directly at test section by the three console type transducers set by the scheme of the equilateral triangle; and increases the reliability of the experiment. The obtained results determine the mechanical salt rock characteristics of the Verchnekamskoe potash and magnesium deposit under direct tensile loading.

**PNRPU Mechanics Bulletin**. 2018;(4):58-68

The Experimental and Theoretical Study of Large Deformations of Cylindrical Samples from Steel 09G2S with Stress Concentrators under Tension-Torsion Loading to Failure

#### Abstract

The paper presents the experimental and numerical results of elastoplastic deformation of specimens with a cylindrical working part and specimens with a stress concentrator made of 09G2C steel under monotonic kinematic tension-torsion loadings prior to failure, taking into account large deformations and inhomogeneity of the stress-strain state (SSS). A complete system of equations describing generalized axisymmetric torsion problems is written in a cylindrical coordinate system. Kinematic relations are formulated at speeds in the metric of the current state, which makes it possible to describe large deformations. The equation of motion of a continuous medium follows from the equation of the balance of virtual capacities. Since we consider active loading processes close to proportional ones, we describe the elastoplastic properties of materials by the flow theory with nonlinear isotropic hardening. The system of equations, supplemented by kinematic boundary and initial conditions, is solved by the finite element method in a combination with an explicit integration scheme of the "cross" type. On the basis of the experimental-calculation method, true deformation diagrams for tension and torsion were constructed to supplement the plasticity model with material functions. These diagrams differ significantly with deformations of more than 15 %, which is caused by the sensitivity of the plastic properties of the test material to the form of the stressed state. To describe the tension-torsion, the dependence of the deformation diagram on the form of the stressed state is introduced. The refined deformation diagram is a linear combination of tension and torsion diagrams, which coefficients depend on the parameter of the form of the stressed state, the parameter of the stress triaxiality or the Nadai-Lode stress parameter. There is a good correspondence of the numerical results with the experiment on the integral characteristics (axial force from axial displacement and torque from the twist angle). The analysis of the SSS parameters in the neck and circular concentrator is performed. The estimated the peculiarities of the mutual influence of tension and torsion on the localization of deformations and destruction of cylindrical samples and samples with a stress concentrator.

**PNRPU Mechanics Bulletin**. 2018;(4):69-81

Method for calculating acoustic stresses in six-beam diffraction in layered media

#### Abstract

The stresses arising in a layered medium as a result of an acoustic wave are investigated theoretically. In the general case, under the action of an incident elastic wave in an anisotropic layer, six waves are formed, three of which are directed to the reflection region and three of them are directed to the region of transmission. The stress-strain state of the layer is caused by the combined effect of these waves and is described by the equations of motion of a continuous medium and the generalized Hooke's law. This system of differential equations is solved with respect to the components of the displacement vector and the stress tensor in the Cartesian coordinate system in the matrix form. The components of the displacement vector and the stress tensor at two opposite boundaries of a layer of thickness di are expressed through each other by means of a sixth-order transfer matrix Ti = exp(Wi di). The calculation of this exponential is carried out using polynomials of the principal minors of the matrix Wi and does not require finding the eigenvalues of the matrix Wi. This method provides a more accurate and reliable calculation of the transfer matrix of the N-layer medium T = TNTN-1…T1 in comparison with other known algorithms. The amplitudes of the waves scattered by the anisotropic layer are expressed in terms of the elements of the transfer matrix. The distribution of acoustic stresses along the thickness of an anisotropic layer is determined by the amplitudes of the scattered waves and the elements of the corresponding transfer matrices. This method of calculating acoustic stresses is demonstrated for the incident SH-, SV- and P-type waves on the three-layer model: isotropic layer-crystal layer-isotropic layer. We present the comparison of the scattering spectra of elastic waves and the dependence of the stresses on the scattering angles for the crystalline layers of silicon and lead molybdate. The interpretation of the resonances of acoustic stresses arising in the crystalline layer due to the action of shear waves is given.

**PNRPU Mechanics Bulletin**. 2018;(4):82-92

A Body Failure Model with a Notch Based on the Scalable Linear Parameter

#### Abstract

Based on the concept of the interactive layer (IL), the paper considers the deformation of a body with a thin deep notch in a linearly elastic formulation. The stress state in the interactive layer is determined on the basis of the constitutive relations of the Prandtl bonds type. The proposed formulation of the task explicitly includes a linear parameter (LP). Based on the analytical solution of the task in the beam approximation, the dependence of the wedge force on the thickness (IL) is obtained. The use of the classical strength criteria in this dependence leads to a zero critical force at zero thickness IL, which contradicts the Griffith-Irvin criterion. We use it as a universal criterion for the destruction of the energy product (EP), as a product of a linear size and an increase in the specific free energy of the layer is shown. A relationship is established between the dimensions of the sample and the critical force, which ensures independence with a given degree of accuracy of the critical force from the thickness IL. By comparing the solution obtained and the classical solution for the notch in the form of a mathematical cut, the assumptions under which EP criterion and the Griffith's criterion (GC) coincide are determined. By using the Neuber-Novozhilov approach, the structural volume of the material with averaged characteristics of the stress-strain state (SSS) is identified. This volume is considered as a representative volume for determining the increment of еру average free energy that controls the state of pre-destruction. The product of the increment of the average free energy and LP determines the average EP. The expression for the average EP is determined on the basis of the obtained analytical solution. It is shown that the transition to the averaged value of the EP on a square Neuber cell does not lead to an increase in the margin of error in determining the critical force. It has been established that starting with a certain value of LP, which depends on the geometric characteristics of the damaged body, the convergence of the average EP in the pre-destruction zone to the GC takes place with a specified degree of accuracy.

**PNRPU Mechanics Bulletin**. 2018;(4):93-97

Experimental Study of Short-Term Transient Creep of the Al/SiC Metal-Matrix Composite under Uniaxial Compression

#### Abstract

The article presents the study results of the deformation behavior of a metal matrix composite (MMC) based on an aluminum alloy with 10 % SiC at temperatures ranging between 470 and 570 °C under a pressure of 4.8 MPa on the sample at the initial time (before deformation). The tests were performed in a pit furnace with the use of a specially designed and manufactured device. In the experiments, the samples are heated to a fixed temperature (470, 500, 530 and 570 °C) and held for a certain time, with a simultaneous exposure to a constant compressive axial load. The heating modes are given. The dependences of the degree of shear strain and the average strain rate on the holding time and temperature are obtained. A diagram of the pressure on the workpiece at the initial and final times for the temperatures under study is plotted. During the heating, starting from 530 °C, due to the deformation of the sample, the pressure decreases significantly and continues to decrease during holding; for example, after heating to 570 °C, it is about 40 % of the initial pressure value. It has been calculated that, during the heating period, the average strain rate does not exceed 0.00031/s. It has been found that, in the studied temperature range, the mechanisms of superplastic flow are likely to appear at the obtained values of strain rate and pressure on the workpiece. The results show that it is possible to create conditions of plastic deformation SiC of the Al/SiC MMC without fracture, with the degree of shear strain sufficient to manufacture blanks and complex shaped products in one manufacturing operation.

**PNRPU Mechanics Bulletin**. 2018;(4):98-105

Construction and analysis of finite-element models of inhomogeneous deformable solids based on scanning

#### Abstract

The paper presents a set of mathematical methods and its computer implementation aimed at studying the real change in mechanical characteristics (elastic modulus) and the geometry of deformable solids based on scanning. Later these data will be used in constructing the finite element (FE) models and analyzing the stress-strain state (SSS). The presented research is important for deformable solids of complex (individual) geometric shapes and pronounced properties of inhomogeneity of the material mechanical characteristics. The scanning of the deformable solids is carried out by a computer tomography (CT). As a result of this work we prepared a package of raster images of sections of the investigated body. The further stage of the investigation is determined by analyzing the pixel color characteristics of the resulting raster images for the construction of an individual geometry of deformable solids and the distribution of mechanical characteristics in it. The contour of the external geometry and geometry of the internal structure of deformable solids in the cross-section is constructed based on two stages. The first stage is the preliminary one; and is necessary for the determination in the sections of the domain, both of the body itself and of the regions in it with pronounced changes in the mechanical characteristics. At the second stage, the geometry of the contours is refined, formed based on the method of studying the gradients of the change in the pixel color indices. The determination of the field of the change in mechanical characteristics is carried out by calculating the weight coefficients obtained on the basis of two parameters: the mathematical expectation of a change in the pixel color index in the package of sections of deformable solids; as well as the results of a full-scale test on the tensile (compression) of standard samples, that is, the averaged data on the mechanical properties of the material of deformable solids. We studied the deformable solids from bone tissue, in the form of a fragment of the femur, a human tooth in the jaw and a tooth with a composite seal in the jaw. The presented choice of deformable solids is not principled, but is caused by the following circumstances: a high degree of heterogeneity of bone material and its geometry, as well as a high level of development of CT scanning technology in medicine and engineering. The analysis results of the SSS of the obtained FE models of real deformable solids, obtained taking into account the inhomogeneity of the mechanical characteristics of the material and the individuality of the geometry, allow us to reach a higher level realization of the mathematical finite element model with respect to the real object, and also prove efficiency and accuracy of the presented technology in real working conditions (design and manufacturing) of structures.

**PNRPU Mechanics Bulletin**. 2018;(4):106-118

Dynamic deformation interaction of elements of the “drummer-gasket-reinforced concrete beam” system

#### Abstract

The use of automated monitoring systems ensures the deformation safety of structures. Deformation control systems are supplemented with the tools which allow evaluating the criticality of the structure state on the basis of vibration measurements. Vibration-based damage detection of the reinforced concrete structures allows detecting the presence of defects in the structure, determining their localization, as well as promptly tracking their development. The results obtained in this study are directly related to the shock-wave vibrodiagnostics of reinforced concrete structures. Special attention is paid to the diagnostics in a “sparing mode”, which does not cause inelastic deformations in the elements of the structure under study. The vibration process in a concrete beam, caused by a local impulse action, is considered. The purpose of this study is to find the parameters of the impulse that forms an elastic wave with the desired frequency spectrum or required wavefront characteristics. One of the main parameters determining these characteristics is the impulse duration. A mathematical model of the dynamic elastic interaction of the elements of the “drummer - gasket - reinforced concrete beam” system was constructed and a series on numerical experiments on the bases of this model was carried out. According to the results of these experiments the dependence of the impulse duration on a number of factors is obtained. These factors are the velocity, mass and dimensions of the striker and the thickness and elasticity of the gasket. These parameters are chosen because they can be varied in a real experiment. It is shown that the impulse duration is most sensitive to the changes in striker velocity and gasket elasticity. Within the selected range of control parameters, it is possible to change the impulse duration from 0.25 to 2.8 ms. Since the impulse duration determines the dominant vibration frequency, one may conclude that in this case frequencies will vary in the range from 200Hz to 1500Hz.

**PNRPU Mechanics Bulletin**. 2018;(4):119-126

Determination of elastic and dissipative properties of concrete under dynamic deformation

#### Abstract

Vibration diagnostics is one of the most promising approaches for the diagnostic of reinforced concrete structures integrity, which analyzes natural vibrations caused by impact loads. It concentrates on the propagation of the shock wave through the elements of the structure and the effect of defects on this process. An important element of the shock wave control system is a mathematical model of the structure. The mathematical modeling accuracy is ensured by a precise determination of the material elastic and dissipative characteristics. This aspect is essentially important for concrete since its characteristics may vary much. The paper presents a theoretical and experimental approach to determining elastic and dissipative characteristics of concrete. In the framework of the viscoelastic model, the deformation response of a concrete specimen to a localized shock impulse load is analyzed. A numerical solution of the initial boundary value problem is obtained by the finite-element method using the ANSYS software. Based on this solution, the structural scheme of experiments has been obtained. In the experiments, free vibrations of the specimen are excited using a striker equipped with a vibrometer. The deformation response at reference points of the sample is recorded with a laser vibrometer. A special iterative procedure ensuring the agreement between the numerical and experimental results is developed. The agreement is achieved by the correction of the mechanical characteristics of the material. The reliability and effectiveness of the proposed approach are demonstrated by considering the vibration processes in a concrete specimen. Model parameters are obtained for description of elastic and dissipative characteristics of concrete at frequency range of 5.6±0.5 kHz. The proposed algorithm can be used to identify the properties of concrete in any frequency range. It represents the possibilities for determining the frequency dependence of the elastic and dissipative properties of materials. The described approach can be used to identify the elastic and dissipative characteristics of other materials.

**PNRPU Mechanics Bulletin**. 2018;(4):127-135

Optimization of mechanical characteristics of models of laminate composites using embedded optical fiber strain sensors

#### Abstract

The problem of analysis and prediction of the mechanical behavior of modern composite materials and structures at the stage of their design, production and in-service conditions is of great importance. One of the most promising solutions in the field of real-time monitoring of the mechanical state of composite structures is associated with smart materials and smart systems based on the sensor elements. The data obtained during operation on the state of the structure can be used both for monitoring the mechanical state of structures and for refining mathematical models to predict the failure processes. This paper is devoted to the approach according to which indications of the embedded fiber-optical strain sensors (FOSS) with Bragg gratings are used to refine the mechanical characteristics of a laminate composite material. The essence of the approach is to estimate the difference between the deformation response predicted using the model with the data obtained in real time with the help of the FOSS. To refine the model parameters in accordance with the information received from the FOSS, an algorithm is proposed, according to which the inverse problems are solved in order to ensure that the numerical and experimental results having the specified accuracy. The optimization parameters are the elastic material constants, which, in the final analysis, should ensure that the simulation results and the FOSS measurements are consistent in the control points. To optimize the parameters for the regression model, various minimization algorithms are used. The algorithm implementation is demonstrated on the example of the test problems of two types of composite samples with a concentrator (notches): with quasi-isotropic and transversal-isotropic plies.

**PNRPU Mechanics Bulletin**. 2018;(4):136-144

Applied theory of inelasticity

#### Abstract

We consider the main features and equations of the applied theory of inelasticity relating to the class of flow theories with combined hardening. The applied theory of inelasticity is the simplest engineering version of the theory of inelasticity; and can be used for calculations of the worked out and residual resource of high-performance structural materials under repeated and long-term thermomechanical loads. The strain rate tensor is represented as the sum of the elastic and inelastic strain tensors, i.e. here there is no conditional separation of inelastic deformation by deformation of plasticity and creep. The elastic deformation follows the generalized Hooke’s law. A loading surface is introduced that isotropically expands or reduces and displaces during loading. For the radius of the loading surface (isotropic hardening), an evolutional equation is generalized to nonisothermal loading and restoration of mechanical properties during annealing. The displacement of the loading surface (anisotropic hardening) is described through the evolution equation with a three-member structure, generalized to non-isothermal loading and the back stresses removal (displacement) during firing. To determine the rate tensor of inelastic deformation, the associated (gradient) flow law is used. For rigid (given deformations) and soft (given stresses) loading regimes, expressions are obtained to determine the rate of the accumulated inelastic deformation. Conditions of elastic and inelastic states are formulated. To describe the nonlinear processes of damage accumulation, the kinetic equation of damage accumulation is introduced, where the energy equal to the work of back stresses on the field of inelastic deformations is assumed as the energy spent on creation of damages in the material. Here this kinetic equation is generalized to nonisothermal loading and processes of embrittlement and healing of damages. The material functions closing the applied theory of inelasticity are singled out; the basic experiment and the material functions identification method are formulated. An example of determining the material functions from the basic experiment results is considered and material functions for 12Х18H9 stainless steel in the temperature range from 20 °C to 650 °C are given. Further we give a list of experiments and structural steels and alloys on which the applied inelasticity theory was verified under plastic and inelastic (viscoplastic) deformations, isothermal and nonisothermal, simple and complex loadings. In conclusion, we discuss the application of the theory of inelasticity.

**PNRPU Mechanics Bulletin**. 2018;(4):145-160

RELIABILITY ASSESSMENT OF MULTILAYER PIPES FROM POLYMER MATERIALS

#### Abstract

Reliability is an important problem when it comes to products made from polymer materials, as polymer materials are characterized by a significant spread of deformation and strength at the manufacturing stage, as well as the strength decrease polymers over time. In addition, products made of polymer materials lose their strength properties, when temperature is increased dramatically. The paper uses the method assessing the probability of failure taking into account these features of polymer materials in relation to single-layer and multi-layer pipes. We collect the initial information in the form of statistical data about deformation strength and limiting properties of polymer materials in the laboratory using standard test methods. Materials specimens were tested immediately after their production, as well as after their exposure for two years in different climatic zones in the open space. Further, this information is used to estimate the parameters of the stress-strain state (SSS) and the quantitative values of the fracture criteria. Since the experimental data have some variation, the parameters of the SSS product and the criteria of destruction are also probable. The proposed and implemented concept of determining the reliability of polymer materials allows obtaining the product reliability estimates when the external load parameters are also random and have a known scattering. Moreover, by taking into account the degradation of polymer matrices over time (aging), we can assess the reliability of structures over time, i.e. their resource.

**PNRPU Mechanics Bulletin**. 2018;(4):161-168

Influence of the biaxial loading regimes on fatigue life of 2024 aluminum alloy and 40CrMnMo steel

#### Abstract

The work studies the fatigue life of metallic materials under various schemes of multiaxial non-proportional loading which lead to the occurrence of a complex stress-strain state. We present the results of the experimental study of the fatigue life of 40CrMnMo structural alloyed steel and 2024 aluminum alloy under biaxial cyclic loading. Cyclic tests were carried out on the Instron ElectroPuls E10000 biaxial electrodynamic test system under joint tension-compression and torsion of solid cylindrical corset-type samples. The methodological issues of conducting cyclic tests with a mixed modes loading are considered and the corresponding new experimental results are obtained. The experimental data are presented in the form of points on the graphs and corresponding approximating lines which reflect the dependence of the number of cycles to failure on the relative values of the constant components of the tangential and normal stresses. In all the tests, the specified values of the additional parts of the stress components did not exceed the values of the corresponding conditional yield strengths, which were previously determined in the quasistatic tensile and torsion tests for each material. Based on the test results, the influence of the constant component of tangential stresses on the fatigue life of the materials under cyclic tension-compression was evaluated, and the effect of the constant component of normal stresses on the fatigue life under cyclic torsion was considered. It is shown that because of the constant stress components, both under cyclic tension-compression and cyclic torsion, there is a decrease in the number of cycles before the specimens break. The obtained data demonstrate the necessity to estimate the allowable limits of the constant parts of the stress components, which will not lead to a significant reduction in the fatigue life of structures operating under cyclic loading conditions.

**PNRPU Mechanics Bulletin**. 2018;(4):169-177

Mechanics of collisions of solids: influence of friction and adhesion. II Numerical modeling

#### Abstract

This paper is the second part of the review on the physics of two-particle collisions of solids. The first one describes theoretical and experimental works on collisions of elastic and elastic-plastic solid bodies in the case when the energy dissipation is caused by the inner or interface friction, plasticity, adhesion, or some other damping mechanisms. In this second part, we concentrate on the case of collisions of elastic particles. Results of the analytical and numerical modelling within the dimensionality reduction (MDR) method are presented. MDR allows describing a three-dimensional contact by a reduction to an equivalent problem in a one-dimensional space. We consider three cases: a collision between bodies without slipping (equivalent to an infinite coefficient of friction), with a finite coefficient of friction in the interface, and with the presence of adhesion, which is described in the JKR limit. In all the considered cases, proper dimensionless variables have been identified which fully characterize the collision process, and the functions determining the transformation of the state before the impact into the state after the impact has been determined numerically. By using these dimensionless variables, it is possible to calculate the velocity components and the angular velocity, and hence the trajectory of the ball after the collision, in case if analogous parameters are known before the collision. Theoretical results are compared with the available results, the experimental data show agree well with the experiments.

**PNRPU Mechanics Bulletin**. 2018;(4):178-199

Contact problems for an elastic inhomogeneous body with a cylindrical cavity

#### Abstract

An axially symmetric elastic equilibrium problem is investigated for a continuously inhomogeneous space with a cylindrical cavity when Poisson’s ratio is being an arbitrary fairly smooth function with respect to radial coordinate while shear modulus is constant. For this case Young’s modulus is also variable with respect to the radial coordinate. A general solution is suggested which leads us to a vector Laplace equation and a scalar Poisson equation whose right-hand side depends on Poisson’s ratio. As a result, exact general solutions of the Laplace and Poisson equations are constructed in integral forms with the help of Fourier transformations. Then integral equations of two axially symmetric contact problems are derived on the interaction between the cavity surface and a rigid cylindrical insert fitted with interference. In the first problem the contact is supposed to be absolutely ideal, a singular asymptotical method is used here to solve the integral equation of the first kind with respect to the contact pressure, which is effective for fairly long inserts. In the second problem the mine surface is supposed to be rough simulated by an extra Winkler type, a collocation method is used for solving the integral equation of the second kind, which is effective for fairly short inserts. The contact pressure has typical square root singularities at end-points in the first problem while it takes finite values at those points in the second problem. For a homogeneous material, the integral characteristics of the contact pressures in both problems are close for small coefficients of roughness for some values of the insert relative length. It is shown that the rough surface distributes contact pressures more uniformly removing effect of nonhomogeneity. The calculations are made for the cases when Poisson’s ratio and Young’s modulus increase or decrease from the surface of the cavity.

**PNRPU Mechanics Bulletin**. 2018;(4):200-208

Shaped and branched analogs of triangle multi-leaf spring

#### Abstract

The effectiveness of elastic elements like leaf springs is determined by their possibility to store the maximum elastic energy for unit mass of springs. For example, in the sense of elastic energy stored in 1 kg of mass, the unidirectional GFRP is one of the best among all structure materials due to its high strength and critical strain, and low Young’s modulus and density. In this paper we discuss the possibility and effectiveness of composites application for equistrong shaped and branched elastic beams with a constant sum area of cross sections (“constarea”). These beams are the analogs of steel multi-leaf springs, and at a fixed compliance and bear capacity these beams may provide great mass advantages: approximately threefold mass reduction for the end force action and a five-time reduction for a uniform disturbed loading with no account of connection problems. It’s pointed that traditional maximum deflection computation methods lead us to infinity integrals for “sharp” shaped beams, but the stored elastic energy remains correct and finite for formally infinite deflections. It is shown that the branched structures may give approximately the same effectiveness as shaped ones but they have some additional advantages connected with: the absence of cut fibers (Leonardo’s rule); the exclusion of fiber disorientation; the possibility of a leaf spring size limitation during a “branch” connection with the bundle. In future branching and shaping composite elastic elements may become efficient for space-based constructions without size limitations due to their low mass and energy of production, and these factors allow them to be produced directly in an orbit laboratory.

**PNRPU Mechanics Bulletin**. 2018;(4):209-222

Deformation and Strength Properties Prediction of Metals with Uniformly Distributed Closed Pores under Single and Cyclic Loading

#### Abstract

The paper deals with predicting deformation and strength properties of metallic materials with defects uniformly distributed over the product volume in the form of gas bubbles under single and cyclic loadings. The proposed approach is based on the use of the corresponding characteristics of a defect-free material (stress-strain diagram, plasticity resource, curve of deformation in a cycle and cyclic deformation curve, and fatigue curve) and the results of numerical experiments on samples models with different volume fractions of defects (degree of porosity of the medium). We analyze the stress-strain state of the virtual samples using the finite element method in ANSYS. The moment of reaching the limiting state-discontinuities - was recorded with a monotonous loading using the deformation criterion of V.L. Kolmogorov. Under single loading, the elastic constants and stress-strain curves are determined up to the moment of destruction of the model medium simulating a material with different porosity (a possible change in the deformation mechanism due to distortion of the gas bubbles wall shape accompanied by plastic flow of the matrix material was not taken into account). The property of the central similarity of the stress-strain curves is noted. Under cyclic loading conditions, both deformation curves in a cycle representing the trajectory of the state point in “the stress ~ strain” space during the cycle and the cyclic deformation curve, i.e. the stress amplitude versus the strain amplitude, reflecting the hardening (weakening) of the material of various porosity degrees are obtained. The description of the fatigue curves characterizing the strength properties under these conditions was carried out using the local Manson-Langer type criterion. The results can be used to normalize the allowable defect size and density and to assign the reasonable safety factor for stress, strain and durability of a real porous medium.

**PNRPU Mechanics Bulletin**. 2018;(4):223-233

Elimination of a Rigid Body Oscillations Suspended on a Variable-Length Cable with a Controlled Horizontal Suspension Movement

#### Abstract

The paper considers a passive force (dynamic) and kinematic control problem of a heavy cargo movement (an undeformed solid) suspended on an inextensible inertia-free variable length cable with a controlled horizontal displacement of the suspension point. Differential equations with variable coefficients for small translational-rotational vibrations of the body are obtained. The following problem is stated: to move the body from the initial rest position to a given final equilibrium rest position for a preset time with oscillations elimination at the stop. In this case, the law of changing the cable length is considered to be prescribed, and the law of displacement of its suspension point is unknown. The integral conditions are established for required unknown control actions (force or acceleration of the suspension point), which should be satisfied. An approximate solution of the kinematic control problem described by two differential equations with variable coefficients for the angles of rotation of the cable and body is sought in series with unknown coefficients by the Bubnov-Galerkin method with the use of the given approximating functions of time satisfying certain initial and final conditions. Acceleration of the suspension point of the cable is sought in the form of a series of sines with unknown coefficients. A coupled system of linear algebraic equations for all unknown coefficients is obtained, which includes equations of the Bubnov-Galerkin method, equations for the initial and final conditions that are not satisfied in the choice of given functions, and one equation representing the integral condition in the form of the dependence of the acceleration of the cable suspension point on its specified finite displacement. The proposed approach for solving the problem of the oscillations finite control for a system with variable parameters is new. By using the examples of a system with a cable of constant and variable length, we performed the calculations with an analysis of the convergence and accuracy of solutions for two different sets of given functions and for different numbers of them by comparing them with numerical solutions of differential equations of the direct problem by the Adams method with the control laws found.

**PNRPU Mechanics Bulletin**. 2018;(4):234-245

Load interaction effects during near-threshold fatigue crack growth under variable amplitude: theory, model, experiment

#### Abstract

In this paper, we consider the loading interaction problems that arise in fatigue life prediction. The brief overview of recently most popular models is presented. Most of them explain this phenomenona by the crack closure effect with different nature. At the same time, cycle-sequence sensitivity can be observed at high stress ratio in the absence of crack closure. This fact calls into question the adequacy of these approaches. A new physically based model which can adequately predict fatigue life in a wide range of crack growth rates (the Paris region, near-threshold) is proposed. This model is based on the suggestion of a brittle fracture nature of the crack propagation in the near-threshold region. As a result it is shown that the threshold stress intensity, ΔKth, is not a material constant, but a variable that is extremely sensitive to load history. A numerical technique is proposed to estimate the near-tip stress for an arbitrary loading sequence including random loading spectra. This method is based on the constitutive equations with the combined (isotropic-kinematic) hardening rule and linear rule for strain prediction. The combined hardening can be interpreted as a simple modification of Frederick-Armstrong law and Chaboche model. The numerical integration of constitutive equations based on the return-mapping scheme (implicit Euler method) is performed. The experimental procedure for adjustment of models and its verification is proposed. We show the comparison of the experimental and calculated data with a constant amplitude loading under a variety of overloads and underloads and under spectral loading. In all cases, a satisfactory compliance with a high correlation factor can be observed.

**PNRPU Mechanics Bulletin**. 2018;(4):246-255

On Modeling of Overhead TLC Galloping and Parametric Analysis of Pendulum Dampers Efficiency

#### Abstract

The paper presents the results of a numerical simulation of the galloping of overhead transmission lines conductors (TLC) with pendulum dampers in order to analyze the efficiency of their application. Studies show that during galloping, along with the pronounced vertical movements of the conductors, there are synchronized torsional vibrations associated with them. Twisting of the conductors leads to an additional increase in the lifting aerodynamic forces and, as a result, the amplitudes of the vibrations in comparison with the "clean" vertical movement of the conductors. Therefore, the control of torsional vibrations is one of the effective ways to protect the transmission lines conductors from low-frequency oscillations. For this purpose, pendulum dampers of various structural schemes are used. Wires are considered as absolutely flexible threads, which have elasticity for tension and torsion. The tension within the span is considered constant. It is assumed that the points of the conductor axis move only in a plane that is perpendicular to the longitudinal axis. The equations of dynamics in partial derivatives describing translational and torsional vibrations of a wire are written. To account for the dampers, the conductor is divided into sections between which the dampers are placed. The equations of dampers motion under the action of the conductors are written. The numerical integration procedure is based on the finite-difference analogue of the obtained oscillation equations. As a result of the numerical experiments for single conductors, the scheme is justified for the joint installation of vertical and horizontal pendulums and their parameters, under which low-frequency oscillations of conductors of the most dangerous forms of galloping are successfully damped. The initial installation angles are determined for horizontal pendulums which are most effective for detuning torsional and transverse oscillations.

**PNRPU Mechanics Bulletin**. 2018;(4):256-265

The Problem Solution of Natural Vibrations of Electroelastic Bodies with External Electric Circuits Based on Their Electrical Analogue

#### Abstract

The optimization of dissipative properties of electroelastic bodies with external electric circuits suggests the selection of parameters for these circuits which can provide the most effective damping of vibrations at the prescribed frequency. Since the external circuits can be represented as a set of elements with concentrated parameters connected to the system with distributed parameters, solving the electroelastic problems for such systems using their full formulation requires high computational and time costs. In mechanics, there exist several approaches, which allow the representation of the mechanical systems with distributed parameters as discrete systems with concentrated parameters of the spring-mass-damper type. In this paper, an approach to the analysis of the dynamic behavior of electromechanical systems with external electric circuits is developed based on the equivalent electric circuits representing discrete electric systems with concentrated parameters. The above discrete systems are the analogues of the initial electromechanical systems in relation to frequency characteristics and electrical processes occurring in them. The solution to the problem on natural vibrations of electromechanical systems are the complex natural vibration frequencies, the real part of which defines the frequency of vibrations and imaginary part - the damping index. The present paper considers the problems of determining the magnitude of elements parameters of the equivalent circuit. The proposed approach made it possible to obtain the mathematical relations binding the limit values defining the dynamic characteristics of systems under consideration (boundaries of the frequency range for natural vibration frequencies change, maximal reachable damping index, the valued of resonant frequency of a system when the resistive external circuit is attached) with the values of natural vibration frequencies of the original structure with piezoelectric element operating in the open circuit and short circuit modes.

**PNRPU Mechanics Bulletin**. 2018;(4):266-277

Generalized dynamic theory of a solid particle grinding by pulse-force compression with two non-deformable balls

#### Abstract

Machines such as ball mill, vibrating mill, planetary mill, and stirred mill using spherical grinding media as a destruction energy transmitter are widely used among the manufacturing equipment for materials grinding. Providing a necessary product particle size at an optimal energy efficiency and productivity during grinding makes it necessity to investigate new ways of improving the mentioned types of mills, supplementing them with a system for regulating the grinding regime, taking into account the physical and mechanical properties of the feed particle. As the grinding media motion is quite complicated, it is necessary to develop a new theory of a particle grinding for an accurate and adequate evaluation of a particle stress-strain state with the grinding media using simple physical and mathematical relations. So, we obtained a new general solution of the fundamental-applied multifunctional problem of materials resistance during a direct inelastic collision of two identical isotropic weightless balls of a small diameter and high rigidity with a given form moving towards each other. The solution is based on a complex application of classical Hertz-Staerman analytic dependences on the force contact of the spherical grinding media and the energy conservation law (the Reilly-Young method). In the corresponding quasi-static physical and mathematical model we considered local deformations within the framework of Hooke's law, own weight of spherical media, the impact time and the radial dimensions of the interacting elements of the constructively nonlinear mechanical system. To calculate the bearing capacity of the grinded particle and determine the crushing force, the well-known Galilei-Leibniz strength criterion is used. The criterion is interpreted by the largest tensile stress, which adequately characterizes the limiting state when a wide class of brittle homogeneous solids is destroyed. The results of the developed theory are presented by the formulae aimed to regulate and optimize the grinding process of stone materials.

**PNRPU Mechanics Bulletin**. 2018;(4):278-289