## No 3 (2020)

**Year:**2020**Articles:**13**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/82**DOI:**https://doi.org/10.15593/perm.mech/2020.3

Solving the problems of strength and destruction of materials and structural elements using a complex experimental-theoretical approach

#### Abstract

The paper considers a complex experimental and theoretical approach to studying high-speed deformation of structural materials including a system of basic dynamic experiments aimed at determining the strength and deformation properties of materials under various types of stresses, a program for direct parametric identification of models of deformation and fracture, as well as a system of special verification experiments in the natural and numerical realizations, which makes it possible to assess the adequacy of the model obtained and its performance in conditions other than those in which it was received. To obtain a set of mechanical strength and deformation properties of materials under compression, tension or shear, a series of basic experiments was carried out on an installation that implements the Kolsky method using sets of split Hopkinson bar of various configurations. According to the results of these experiments, together with the data of static deformation, the parameters of the Johnson-Cook model of plasticity with various versions of the model factor responsible for the influence of the strain rate were identified. To test the adequacy of the model (verification), special test experiments have been developed that are simple enough, on the one hand, and allow an unambiguous interpretation of the results and numerical reproduction without simplifications, whilst, on the other hand, the stress state in these types of tests, as well as the history of changes in the loading parameters, differs from that in basic experiments. The chain of obtaining an adequate (verified) model of deformation and fracture used in software complexes for calculating the stress-strain state and the strength of critical structures under shock loading conditions is considered in the example of steel 3.

**PNRPU Mechanics Bulletin**. 2020;(3):5-11

About mixed forced, parametric and self-oscillations by limited excitation and delayed elasticity

#### Abstract

Mixed forced, parametric, and self-oscillations are considered if there is a delay in the elastic force in the system. A dynamic model is a friction self-oscillation system describing the frictional self-oscillations that occur in many technical systems for various purposes (metal-cutting machines, textile equipment, brakes and a number of other engineering objects). The operation of the system is supported by the energy source of limited power. For the analysis we used the method of straight linearization which is easier than the known methods of analysis of nonlinear systems, has no time-consuming and complex approximations of different orders, provides an opportunity to obtain the final design ratios regardless of the specific type and degree of nonlinearity, thus reducing labor costs and time by several orders of magnitude. By using this method, we obtained solutions of a nonlinear system of differential equations describing the system's motion. The equations of non-stationary and stationary movements are derived. To analyze the stability of stationary movements, the stability conditions based on the Routh-Hurwitz criteria are compiled. Calculations were performed to obtain information about the effect of delay on the oscillation modes. It is shown that the delay affects both the magnitude of the amplitude and the location of the amplitude-frequency curve in the frequency range depending on the magnitude of the delay, the amplitude curve is shifted to the region of lower frequencies. The stability of stationary oscillations depends both on the energy source characteristics and lag value. The interaction of the oscillating system and the energy source leads to a number of effects, both in the presence and absence of the lag. However, their course may be different depending on the lag value.

**PNRPU Mechanics Bulletin**. 2020;(3):12-19

Evolution of temperature stresses in the Gadolin problem of assembling a two-layer elastoplastic pipe

#### Abstract

The work aims at solving the problem of the theory of unsteady thermal stresses simulating the assembling of the two-layer elastoplastic pipe using the shrink fit (Gadolin problem). The plastic flow condition is taken in the form of a piecewise linear condition of maximum reduced stresses (the Ishlinsky - Ivlev condition) with a parabolic yield point depending on temperature. It is shown that when solving the mechanical part of a disconnected problem of the theory of temperature stresses, the calculations of reversible and irreversible deformations and stresses can be carried out numerically, i.e. analytically without resorting to approximate calculation procedures and, therefore, without discretizing the computational domains. We present a diagram of the emergence and disappearance of plastic flow regions under the assembly conditions and its subsequent cooling. With a different choice of problem parameters, some plastic regions may not appear. However, it is impossible to obtain other areas of plastic flow by changing the geometry of the problem, properties of assembly materials, and the level of heating. This is the adequacy of the calculations. Only those plastic areas that are shown in the diagram appear and disappear. In contrast to the classical case of uniform heating of the outer pipe, this article deals with a widely used case of an uneven heating of the outer pipe from the inner surface. In this case, irreversible deformations are calculated, and then taken into account, which originated in the pipe material before the moment of landing. A comparison of the distribution of residual stresses obtained during the uniform and non-uniform heating of the outer pipe is given. As a result, the interference with the uniform heating exceeds the interference formed with the non-uniform heating of the pipe.

**PNRPU Mechanics Bulletin**. 2020;(3):20-31

Modeling the deformation process of a plate with a stress concentrator taking into account the postcritical stage of material deformation

#### Abstract

Postcritical deformation of a material is a process which is characterized by a decrease of stress during growing deformations as a result of accumulation of structural damage. The design becomes unable to withstand the external load only when zones with weakened connections are developed enough. Evolution of postcritical deformation zones can occur with an increase of the external load applied to the construction. It means that taking into account the softening of the material allows determining the strength and deformation reserves of constructions more accurately. The mathematical formulation of the boundary value problem of supercritical deformation mechanics is given in the paper. The features of the experimental study of the postcritical stage of material deformation are listed. Strain curves of various steels with a long section of softening are obtained. Numerical solutions for the problems of deforming a thin plate with stress concentrators of different geometries under kinematic loading are obtained. Piecewise linear approximations and real strain curves of steel 20 and steel 40Cr4 obtained experimentally are considered. The evolution of zones of postcritical deformation in the material is considered. The correspondence between the value of the decline modulus and the nature of the evolution of the softening zones is determined. A stress plot is constructed that reflects how the complete material deformation diagram is realized near the concentrator. The calculated loading diagrams are constructed. It is noted that even after the appearance of softening zones, an increase in external load is possible. The strength and deformation resources of structures are determined, and the influence of the geometry of the stress concentrator on their values is considered. It is noted that the consideration of softening in modeling the behavior of structures with stress concentrators is appropriate.

**PNRPU Mechanics Bulletin**. 2020;(3):32-40

About the nonlinear acoustic parameter during deformation of AMG61 alloy

#### Abstract

Various acoustic effects are used in the study of deformation processes. Acoustic emission is most often mentioned in such studies, and the effects due to nonlinear properties of a deformable metal are the subject of the present work. These properties of real solids lead to nonlinear acoustic effects of the interaction elastic waves forbidden by the theory of elasticity of a homogeneous isotropic body. The work solves the problem of using the principles of nonlinear acoustics in studying the deformation of AMg61 alloy samples. A surface elastic wave is used to control the alloy condition. The process of the elastic wave propagation in the deformed AMg61 alloy due to nonlinear effects is accompanied by generating the double frequency, both of the longitudinal component of the wave and the shear one, the latter is forbidden by equations of the classical elasticity theory. Excitation and reception in the samples was carried out by piezoelectric converters (PES). A wedge converter with a resonance frequency of 1MHz was used to excite the surfactant. The passing surfactant was recorded by a wedge converter with a resonance frequency of 2 MHz. We justified the control technique of the nonlinear acoustic parameter with respect to amplitudes of the first and second harmonics measured during the whole deformation process. An experimental device has been developed to control the nonlinear acoustic parameter in the process of changing the structural state of the sample metal. The pilot study results of the nonlinear acoustic parameter under Amg61 alloy deformation are given. It is shown that the nonlinear acoustic parameter, as well as the acoustic emission activity, is sensitive to changing mechanisms of the defective structure evolution. The non-linearity jump formation during deformation of alloy AMg61 is recorded, which may indicate adjustment of the metal structure. The presented data demonstrate the increase of acoustic nonlinearity in metal at various deformation stages, both at early stages of elastoplastic deformation and at pre-destruction stage, which can be used as the prognostic criterion.

**PNRPU Mechanics Bulletin**. 2020;(3):41-47

Structural failure criterion for spatially-reinforced carbon-carbon composite materials

#### Abstract

This paper defines the structural strength criterion for 4DL-reinforced carbon-carbon materials. For this scheme, fiber reinforcement consists of four groups of reinforcing elements, three of them are located in parallel planes with the angles of 120° between them and the fourth one is normal to them. The paper addresses the first failure of the material corresponding to its yield stress, in this point, one of the material components deviates from linear elastic behavior. A composite material is considered to be non-uniform structurally and consists of a matrix and reinforcing elements, rods. Those rods, in their turn, represent a unidirectional composite. To analyze the stress-strain state of individual components of the material, a three-level elastic model is built that uses the analytic approach at the micro level, while at higher levels it uses the finite element method. For numerical calculations, a structural cell of the material is taken. The boundary conditions provide small to negligible influence of the edge effects, thus simulating the behavior of the infinite volume of the material. For the material components, local strength criteria are introduced, where the fields of the criterion quantities are averaged over the volume of the structural cell. The strength surface of the material that corresponds to its first failure is obtained, and the conclusion is made that the suggested criterion provides a reasonable agreement with the available data on the typical carbon-carbon composite characteristics. Based on the calculated dependencies of the material’s yield stress on the load direction, a procedure is suggested to identify the model parameters based on the material failure behavior analysis using standard tensile and compressive tests. Estimated discrepancies between the results calculated using the suggested criterion and those obtained using the limiting stress criterion for biaxial stress states are given. It is shown that the discrepancy may reach tens of percent and in some cases the material strength increases in comparison with that in the uniaxial stress state. The results are subject to verification tests in order to verify the model for advanced spatially reinforced carbon-carbon composite materials.

**PNRPU Mechanics Bulletin**. 2020;(3):48-59

Diagnostics of impregnation defects of reinforcing filaments of polymer composite with built-in fibre-optic sensor with distributed Bragg grating

#### Abstract

Mathematical model of unidirectional fibrous polymer composite material with optical fiber sensor built into reinforcing fiber (filament of elementary fibers) with distributed Bragg grating is developed in order to diagnoste defects of filament impregnation - finding probability of impregnation defect as relative length of local sections of filament without impregnation, i.e. without filling binder of space between its elementary fibers. The technique of digital processing of reflection spectrum according to the solution of the integral Fredholm equation of the 1st kind is used in order to find the desired informative function of density of distribution of axial strains along the length of the sensitive section of the fibre-optic sensor. The approach assumes that the optical fiber sensor is embedded in the composite material at the stage of its manufacture, wherein the low-reflective nature of the sensitive portion of the optical fiber allows linear summation of reflection coefficients from its various local portions regardless of their mutual positions. Algorithm of numerical processing of strain distribution density function is developed for finding of sought probability of presence of impregnation defects along filament length. It has been revealed that the distribution density function has pronounced informative pulses, from the location and value of which the sought-after values of probability of presence of impregnation defects along the length of the filament can be found. The results of diagnostics of different values of the sought probability of the filament impregnation defect are presented based on the results of numerical simulation of the measured reflection spectra and the sought function of strain distribution density along the length of the sensitive section of the optical fiber sensor at different values of the volume fraction of the filaments, combinations of transverse and longitudinal loads of the representative domain of the unidirectional fibrous composite material in comparison with graphs for the case without load.

**PNRPU Mechanics Bulletin**. 2020;(3):60-72

Asymptotic stress fields in the vicinity of the crack in perfectly plastic solids under mixed mode loading

#### Abstract

In the paper presents the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for a full range of the mode mixities. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities both for plane strain and plane stress conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions, all values of the mixity parameter within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained using the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the solution whereas six and five sectors define the solution for the mixity parameter higher 0.33 and less than 0.89 and higher 0.89 respectively for plane strain conditions and seven sectors determine the asymptotic solution for the mixity parameter less than 0.39, while five sectors determine the solution for other values of the mixity parameter for plane stress conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. It is interesting to note that the characteristic feature of the asymptotic solution obtained is the presence of a segment of values of the mixity parameter for which the solution does not depend on the mixity parameter (the solution does not depend on the mixity parameter for the mixity parameter from 0.89 to 1 and the solution coincides with the solution for mode I crack in perfect plastic materials for plane strain conditions). Thus, the salient point of the study is that the asymptotic solution is described by the same formulae for all values of the mixity parameter from 0.89 to 1 for plane strain. For plane stress conditions this segment can’t be observed. The solution in each sector corresponds to the certain value of the mixity parameter. The obtained solutions for plane strain and plane stress conditions can be considered as the limit solution for power law hardening materials and creeping power law materials.

**PNRPU Mechanics Bulletin**. 2020;(3):73-89

The effect of the angle of «meeting» of fullerite C60 with a solid substrate on the deposition process

#### Abstract

Carbon forms a large number of allotropic forms, one of which is fullerene. Fullerene is a convex closed polyhedron with carbon atoms at its vertices. The most common is fullerene, consisting of 60 carbon atoms and designated - C60. In turn, fullerenes are able to agglomerate, forming a molecular crystal - fullerite. In the interaction of fullerite C60 with a solid, it is possible to deposit on the surface of the body both whole fullerite and the fullerenes that form it. The interaction process in the C60 fullerite system - the substrate of a solid, and then on - the fullerite - substrate system - is multi-parameter. So, when modeling the interaction of fullerite with a substrate, the following were taken into account: the temperature of the system - 300, 700, 1150 K; the speed of fullerite movement is 0.005, 0.01, 0.02 Å / fs. In addition, in the study, we varied the angle between the fullerite velocity vector and the normal to the contact surface of the substrate, called the “meeting angle”. An iron crystal Fe (100) was modeled as a solid body substrate, as one of the most common structural materials. Fullerite C60 was in contact with the solid substrate with its face. Computer simulation of the process of contact of fullerite C60 with the substrate was carried out in the LAMMPS software package. The main result of this study is to determine the effect of the angle of “meeting” of C60 fullerite in contact with a solid substrate, which will significantly complement the overall picture of the process of C60 fullerite deposition. In turn, this can allow the creation of various films and wear-resistant coatings on the surface of materials.

**PNRPU Mechanics Bulletin**. 2020;(3):90-97

Fracture resistance parameters for the compressor disk imitation model

#### Abstract

This paper presents a calculation and experimental technique for determining stress intensity factors in an imitation model of a titanium alloy disk. We studied a low-pressure compressor disk of a gas turbine engine (GTE) D-36. During operations, there occur fatigue cracks initiated and developed in the slot fillet under the blade at the place of transition of the bottom to the lateral surface of the inter-groove projection, which lead to a separation of the disk’s part within its rim. The mixed-mode crack growth occured in the compressor disks. Based on the principles of imitation modeling, the geometry and loading condition of the imitation model of the compressor disk was developed. The fatigue test of the imitation model was carried out with a frequency of 5 Hz, at room temperature and with stress ratio Rc = 0.1, by means of a biaxial testing machine. The crack growth was monitored using an optical microscope. The criterion for failure was the condition for reaching a growing crack of the compensation hole. During the test, the positions and sizes of the crack fronts were fixed, which are the basis for the numerical calculation of the fracture resistance parameters. In the order of the numerical studies, six three-dimensional finite element models with different positions and sizes of the crack fronts are considered. The results of the numerical calculations based on the finite element method were used to determine the distributions of elastic and plastic stress intensity factors along each crack front. We demonstrated the advantages of the calculation and experimental methods for solving the problems of interpretation and prediction of the crack growth in the rotating disks of turbomachines using the methods of fracture mechanics.

**PNRPU Mechanics Bulletin**. 2020;(3):98-107

Numerical algorithm for searching for layouts of electroelastic bodies with external electric circuits for obtaining the best damping properties

#### Abstract

This paper presents an algorithm which allows finding such layouts of electromechanical systems that provide the best vibration damping whether for one mode or a set of vibration modes within some continuous frequency ranges. The basis for the algorithm is the problem solution about natural vibrations. An elastic structure with a piezoelectric element located on its surfaces, which electrodes are connected to a passive external electric circuit, is treated as an electromechanical system. The piezoelectric elements shunted with an electric circuit are the devices where energy dissipation occurs, thus leads to damping of vibrations. A change in damping properties of such systems can be reached by a proper choice of parameters of the electric circuits and corresponding location of the piezoelectric element, which provides the highest energy withdrawal into the electric circuit. The paper presents the mathematical formulation of the natural vibrations problem for piecewise-homogeneous electroelastic bodies shunted with passive external electric circuits. Within the proposed mathematical statement, the problem solution of natural vibrations for such objects is based on values of complex natural vibration frequencies. Real parts of the complex natural vibration frequencies are the circular frequency of vibrations, and the imaginary parts are the damping indices of vibrations. We proposed techniques aimed at determining the location of the piezoelectric element and selecting parameters of shunting the external electric circuit. These approaches are based on values of the complex natural vibration frequencies obtained as results of solving the natural vibrations problem. The proposed approach is demonstrated using a specimen of a thin-walled shell in the semi-cylinder form. The piezoelectric element has a form of a segment of a ring made of PZT-4 piezoceramics. Electrodes of the piezoelectric element are connected to the series resonant RL -circuit. The formulated problem is solved numerically using the finite element method and ANSYS commercial package.

**PNRPU Mechanics Bulletin**. 2020;(3):108-124

Limit states and safety factors under repeated loading

#### Abstract

Verification of the structures operating possibility using numerical modeling beyond the elastic limit requires standardization of safety factors and calculation methods used to get them. In the framework of the discussion on the improvement of the strength standards of the aviation and nuclear industries for structures operating under low-cycle mechanical and reversible dilatation (temperature, hydrogen) external influences, the article discusses the limiting states; the deformation properties of materials necessary for their calculation; safety factors for loads and durability; calculation methods. The article divides limit states of structures under low-cycle actions into two groups: typical, corresponding to a qualitative change in the deformation type, and individual, determined by allowable displacements and cracks for a particular structure. The following types of deformation are considered: inelastic deformation only at the running-in stage (that changes to elastic after the auspicious residual stresses develop and cyclic hardening of the material); alternating flow (that continues with the number of cycles); progressive accumulation of strains and displacements; combined deformation (when both strain span and strain increment are non-zero in a stable cycle). The types of deformation differ in possible consequences for the structure and the initial data for the calculation: mechanical properties of the material required for modeling different types of deformation should be determined by fundamentally different tests. An analysis of individual limit states without taking into account differences in the types of deformation - and thus typical limit states - may be incorrect. The main focus of the article is on typical limit states. The limit states vary depending on the stage of operation at which inelastic cyclic deformation is allowed. Inelastic deformation expands allowable load range, the expansion due to the inelastic deformation at the running-in stage only is usually more significant than additional expansion due to the continuous inelastic deformation; besides, the inelastic deformation only at the running-in stage does not demand analysis of low-cycle fatigue and accumulated strains. Further expansion of the permissible load range, as well as solution of safety problems based on risk assessments, requires a more complete study of the deformation properties of materials at the pre-fracture stage, where cyclic softening predominates.

**PNRPU Mechanics Bulletin**. 2020;(3):125-135

Models of hydrogen influence on the mechanical properties of metals and alloys

#### Abstract

The article yields a survey of the key models of mechanics that are used to describe the effects of hydrogen embrittlement, hydrogen cracking, and hydrogen-induced destruction. The main attention is paid to models which are used to calculate the stress-strain state of metal samples, parts and machine components and have the potential for specific engineering applications. From a mechanical perspective, the effect of hydrogen on the material properties is a classic problem of the influence of a small parameter, since the hydrogen concentrations critical for the strength and ductility of metals are usually small. In the vast majority of models this effect is reduced to the hydrogen redistribution within the material volume and localization of concentrations in the critical fracture zones. The authors identified four main approaches that allow one to take into account the influence of a small parameter: (i) hydrogen-enhanced decohesion (HEDE), (ii) hydrogen-enhanced localized plasticity (HELP), (iii) account for additional internal pressure due to the hydrogen dissolved in metals, and (iv) bi-continuum approach that takes into account the internal hydrogen pressure and weakening of material in the framework of a special model of a solid. The links between the main approaches are established. Systematization of publications was carried out, similarities and differences in the description of the internal transport and accumulation of hydrogen in metals are highlighted. It is indicated that the predominant number of publications is devoted to the HEDE model, but so far there is no published data on the application of this model to real problems of engineering practice; only modeling the results of mechanical tests of cylindrical and prismatic samples were considered. In fact, other less popular approaches have more practical applications. The main unresolved issue in the verification of all models is the local concentration of hydrogen, which is a source of premature destruction of metals under load. All the methods for measuring local concentrations are indirect. Even in the case of applying sophisticated physical methods, mechanical surface preparation is required, which destroys the initial natural concentration of hydrogen. The lack of reliable data on the distribution of hydrogen concentration excludes the possibility of unambiguously determination of all the model parameters. On the one hand, it allows fitting to any experimental data, and on the other hand, it reduces the predictive engineering value of all models, since a qualitative fitting is not sufficient for engineering strength analysis.

**PNRPU Mechanics Bulletin**. 2020;(3):136-160