PNRPU Mechanics Bulletin

Frequency: Quarterly

Publisher: Perm National Research Polytechnic University, Perm, Russian Federation

DOI:  10.15593/perm.mech

Languages: Russian, English

Editor-in-Chief: Professor, Dr. Sci. Anatoly A. Tashkinov

Executive Editor: C.Sci. Pavel S. Volegov

Editorial Contact:

Address: Editorial Board "PNRPU Mechanics Bulletin", Russian Federation, Perm, 614990, Komsomolsky ave., 29
Phone: +7 (342) 2-198-444

PNRPU Mechanics Bulletin  is an open-access periodical published scientific peer-reviewed journal.

PNRPU Mechanics Bulletin has no article processing and/or article submission charges.

All Journal's Content, including articles,  is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). Editorial of the Journal allows readers to read, download, copy, distribute, print, search, or link to the full texts of its articles and allows readers to use them for any other lawful purpose in accordance with Budapest Open Access Initiative's definition of Open Access.

Journal intended for researchers specializing in the field of continuum mechanics, mathematical modeling of physical and mechanical processes, micro and mesomechanics, senior students of natural areas.

Until 2012, Journal was named «Bulletin of Perm State Technical University. Mechanics» (ISSN 2078-6603).


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Current Issue

No 2 (2020)

Investigation of the mechanical behavior of a two-phase Mg-Ni compound during fast compression
Aptukov V.N., Skryabina N.E., Fruchart D.


The paper presents the results of using a pressing process well known in metallurgy as forging. However, here the Fast Forging process is used as a new way to synthesize materials having e.g. a high hydrogen sorption capacity. Firstly, the method allows delivering a final consolidated product from the initial Mg and Ni powdered elements during pressing. Secondly, it enables a phase transformation, i.e. synthesizing the Mg2Ni binary phase from the initial Mg-Ni mixture of powders, which is a unique result. It is established that the key point in the formation of the binary phase in a sample is the temperature of the Fast Forging. The temperature increase (below the level of the eutectic transition) leads to an almost full consumption of free Ni and the resulting presence of the Mg2Ni phase in the material without melting the work piece. In order to determine the optimal deformation conditions with the goal to obtain a maximum Mg2Ni phase content in the sample, a 2D calculation model of the cell is built with an arbitrary distribution of different-sized Ni particles among the Mg ones. The numerical simulation of the adiabatic compression of the cell up to an average strain of 80-90 % shows that except for preheating the sample chamber, it is necessary to account for the rise of еру sample’s temperature created with the hammer impact during the Fast Forging. It is found that when increasing the initial temperature from 20 °C to 400 °C, the contribution of the additional heat during the intense plastic deformation decreases progressively. The numerical simulation results of the temperature increase within the contact zone of the two types Mg and Ni particles are presented. The deformation fields are determined in the used 2D model cell, which makes it possible to evaluate the system’s mechanical behavior during the Fast Forging process. The calculation results are found in an excellent agreement with the experimental ones.
PNRPU Mechanics Bulletin. 2020;(2):5-15
Modeling the processes of atom structure formation of a superconducting spin valve
Vakhrushev A.V., Fedotov A.Y., Savva Y.B., Sidorenko A.S.


The paper considers the modeling of a multilayer nanocomposite, the combination of elements of which gives rise to a spin valve effect. The relevance and importance of effects in the field of spintronics and related materials and devices are described. We study the composition and atomic structure of individual layers of a multilayer nanocomposite, as well as the composition and morphology of the interface of nanocomposite layers. We analyzed a sample with a periodic superconductor-ferromagnet structure consisting of more than 20 alternating layers of niobium and cobalt. The deposition process took place in a deep vacuum. The simulation was carried out by the molecular dynamics method using the potential of the modified immersed atom method. The formation of layers was carried out in a stationary mode. The temperature was adjusted using the Nose-Hoover thermostat. The deposition of each nanofilm ended with a relaxation stage for the necessary stabilization and restructuring of the formed nanocomposite. Three deposition temperature regimes were considered: 300 K, 500 K, and 800 K. For these modes, we analysed the atomic structure of nanofilms and transition regions (interface) formed between the layers. A study of the atomic structure of nanofilms showed that niobium is formed by crystalline regions of different orientations. A cobalt nanofilm is characterized by a structure close to amorphous. The structural features of the interface between the superconductor-ferromagnet layers largely depend on a relief of the surface onto which the deposition is made. The smallest variation in atomic composition is observed in the first niobium-cobalt contact zone, since the formation of the first nanofilm occurs on an even plane of the substrate. An analysis of the influence of the temperature regime during the formation of the nanosystem shows the dependence of the processes of formation of multilayer nanofilm formation, the interface of nanolayers, as well as the composition and morphology of heterostructures on the temperature at which a nanocomposite is manufactured. An increased temperature leads to the formation of a more rarefied structure of nanolayers and an increase in the zones of the interface of nanolayers due to the diffusion of atoms of the sprayed materials.
PNRPU Mechanics Bulletin. 2020;(2):16-27
A variant of the thermoplasticity theory for monotonic and cyclic processes of nonisothermal loads
Bondar V.S., Abashev D.R., Fomin D.Y.


We revealed some features and differences in isotropic and anisotropic hardening under monotonic and cyclic loads by analyzing the experimental results of the samples made of 12X18H10T stainless steel under a rigid (controlled) deformation process, which includes a sequence of monotonic and cyclic loading modes under uniaxial tension-compression and different temperature levels. To describe these features with the theory of thermoplasticity, which belongs to the class of flow theories for combined hardening, a memory surface is introduced in the space of the plastic strain tensor components that separates the processes of monotonic and cyclic deformations. The main assumptions and equations of the thermoplasticity theory are formulated. To describe the transition from the monotonic to the cyclic and from the cyclic to the monotonic deformations, the evolutionary equations are formulated for the parameters of isotropic and anisotropic hardening. The basic experiment, which determined the material functions, consists of three stages, such as cyclic loading, monotonic loading and the subsequent cyclic up to destruction. The method of identifying the material functions based on the results of the basic experiment is considered. The material functions that close the thermoplasticity theory at different temperature levels are determined for 12X18H10T stainless steel due to the basic experiment and identification method. We considered the results of the computational and experimental studies of the rigid cyclic deformation under isothermal and non-isothermal loadings up to destruction of 12X18H10T stainless steel. The kinetics of the stress range and the average stress during isothermal and non-isothermal cyclic loadings are analyzed. A reliable compliance of the computational and experimental results is obtained.
PNRPU Mechanics Bulletin. 2020;(2):28-36
The Numerical study of the influence of a two-scale pore structure on the dynamic strength of water-saturated concrete
Konovalenko I.S., Shilko E.V., Konovalenko I.S.


Many infrastructural concrete facilities, such as dams, bridge footings, foundations of port facilities and offshore drilling platforms, operate in a permanent contact with water. The permeable fractured-porous structure of concrete determines the water-saturated state of the surface layers of such concrete elements. Under dynamic contact loading, the pore fluid is capable of exerting a significant mechanical influence on the local stress-strain state and strength characteristics of the surface layers of concrete. This has to be taken into account when assessing the wear intensity of surface layers and predicting a concrete element’s service life. The aforesaid determines the relevance of the study aimed at identifying the influence of the pore fluid and characteristics of the concrete pore structure on the strength and fracture pattern under quasistatic and dynamic compressive loading. The present work is devoted to the theoretical study and generalization of the laws of mechanical influence of the pore fluid on the dynamic strength of high-strength concrete with a two-scale pore structure. The emphasis in the study is on analyzing the contributions of each of the pore subsystems to the integral mechanical effect of the fluid. To carry out such an analysis, a coupled hydromechanical model is developed. It takes into account the compositional structure of concrete, the presence of a pore space in a cement stone of two different scales, the interaction of a pore liquid and a solid-phase skeleton based on the Bio poroelasticity model, as well as fluid filtration in a pore space. By using the developed model were performed the numerical studies of the dependence between the compressive strength of the representative concrete volumes of the mesoscopic scale on the strain rate, the sample size, the pore fluid viscosity, and pore structure parameters. The simulation results showed the possibility of combining the obtained dependencies into a generalized (master) curve in terms of a combined dimensionless parameter, which has the meaning similar to the Darcy number. We identified two key factors that control the type and parameters of the concrete master curve of the dynamic strength. The first factor is the mobility of the pore fluid in the network of the capillary pores. It determines the rate of stress equalization in the porous skeleton due to fluid flow. The second factor is the interconnection of large micropores with the network of the small capillary pore channels. It determines the magnitude of the decrease in stress concentration in micropores by filtering the excess pore fluid into the capillary pore network. It is shown that the contributions of these two factors to the amplitude of variation of the dynamic strength of the water-saturated concrete are additive, and their total contribution reaches 25 %.
PNRPU Mechanics Bulletin. 2020;(2):37-51
The effect of replacing non-analytic trajectories with break points on smooth paths to the complexity of deformation and loading processes of materials
Zubchaninov V.G., Alekseev A.A., Gultiaev V.I.


This article is devoted to an experimental study of the effect of rounding off corner points of two-link strain trajectories on complex loading processes during elastoplastic deformation of materials. Replacing corner points in their vicinity with local sections of circles allows a nonanalytic trajectory to be replaced with a smooth trajectory. Experimental studies were performed on thin-walled tubular specimens of the low-carbon steel St3 on an SN-EVM automated testing system. The loading programs for tubular specimens were set in the Ilyushin's deviatoric strain space. The rounding of the corner point of a two-link strain trajectory with an angle of 90° between the branches by arcs of circles with curvatures of 200, 400, as well as the rounding of the corner point of a two-link strain trajectory with an angle of 135° between the branches by arcs with curvatures of 400, 800 are considered. The experimental data characterizing the vector and scalar properties of the material are presented. The experimental data show that the effect of complex loading on the relationship between stresses and strains in a curved section is not immediately apparent. In the curved section, the magnitude of the stress vector modulus first increases, and then decreases with the formation of stress dives. The minimum point of the stress dive is located on the next straight branch of the strain trajectory. In the curvilinear section, the angle of delay increases, and in the next straight branch it decreases, and with the increase of the strain it tends to be zero. The rate of decrease of the angle of delay depends little on the differences in the geometry of the previous history of strain trajectory. In the second straight branch, the experimental results for a smooth and original two-link strain trajectories become little distinguishable from each other. Thus, replacing the original non-analytical strain trajectory to a smooth trajectory affects the complexity of the process of deformation and loading of the materials only in the vicinity of the corner point. This circumstance can be taken into account when numerically modeling the processes of elastoplastic deformation of materials and integrating the defining relations, replacing nonanalytic trajectories with smooth ones. This can be taken into account in the numerical calculation of elastic-plastic deformation and integration of constitutive relations, replacing non-analytical strain trajectories by smooth ones.
PNRPU Mechanics Bulletin. 2020;(2):52-63
Numerical Modeling of Electroelastic Fields in the Surface Piezoelectric Luminescent Optical Fiber Sensor to Diagnose Deformation of Composite Plates
Pan’kov A.A., Pisarev P.V.


We developed a three-dimensional numerical model of a piezoelectric luminescent optical fiber sensor fixed on a composite’s plate. The computational region of the sensor is the optical fiber with two concentric (with 6 sectors) shells of electroluminescent and piezoelectric materials, two control electrodes on interface surfaces, such as optical fiber-electroluminophore and piezoelectric-cover. The external sensor’s cover is made in the form of a semi-elliptic cylindrical polymer shell, which rectangular base is fixed on the surface of the fiberglass plate. In the piezoelectric shell sectors, the polarization directions of the PVDF transversal-isotropic polymer piezoelectric are different and non-planar for any three sectors. Deformation of the plate causes deformation of the sensor fixed on its surface, as well as the occurrence of informative piezoelectric fields in it, thus the occurrence of informative glows of electroluminescent elements. As a result, we find the requested information about the combined deformed state of the composite plate along the length of the sensor based on the digital processing of the integral intensities of the polychrome light signals at the output of the optical fiber. In simple cases of electric and mechanical loads, we present new numerical results of simulating the distribution of non-uniform electroelastic fields in the sensor multiphase volume, the sensor’s external cover and inside fragment of the composite plate. Loading of the sensor-covering-plate system is performed by controlling electric voltage on the sensor’s electrodes and the plate’s mechanical deformation by stretching along the transverse and longitudinal axes, as well as by twisting around these axes and bending in transverse and longitudinal planes. Numerical values of the control and informative transfer coefficients of the piezoelectric luminescent optical fiber sensor are determined, which makes it possible to perform a reliable and high-precision diagnostics of complex deformations of the composite plates and design sensors of this type.
PNRPU Mechanics Bulletin. 2020;(2):64-77
Investigation of nonlinearity of longitudinal displacement function from mechanical and geometric characteristics of the plate
Osadchy N.V., Malyshev V.A., Shepel V.T.


Layered composite materials are characterized by a high transverse anisotropy and low values of relations between the transverse shear modulus and the modulus of longitudinal elasticity. As a result, the behavior of longitudinal displacements and longitudinal normal stresses differs from the linear law, and the behavior of transverse tangential stresses differs from the parabolic law. The paper presents the analysis of the degree of the plate displacement nonlinearity functions depending on its elastic properties and geometric shape. The article considers a 2D square plate deformation problem. In non-dimensional terms, the authors could received a complete and demonstrative solution. A similar 3D problem is more bulky but it has no principle differences. The study of the degree of the longitudinal displacement nonlinearity functions due to elastic properties and the geometric shape of the deformable plate is based on the finite element method. The potential energy of the deformable plate is expressed through a square form of the variables with coefficients that are polynomials of dimensionless parameters such as plate size, ratio of the elasticity and shear moduli, Poisson's modulus. It is shown that the variational principle reduces the problem to the solution of the system of linear equations. As a result, the subareas of linearity and nonlinearity of the plate longitudinal displacements are constructed with an accuracy acceptable for engineering calculations of 5 %. It is necessary to consider the plate nonlinear longitudinal deformations for a length of the composite plate on order more than its thicknesses. As for steel, nonlinearity is characteristic for quite thick plates. The constructed areas of linearity and nonlinearity of the longitudinal displacements make it possible to construct the strain-stress state models with a smaller number of variables.
PNRPU Mechanics Bulletin. 2020;(2):78-84
The grain structure refinement of metals and alloys under severe plastic deformation: experimental data and analysis of mechanisms
Ostanina T.V., Shveykin A.I., Trusov P.V.


Wide opportunities of using fine-grained materials as structural and functional materials with advanced physical and mechanical properties have proved the importance of improving the existing technology and creating new processing methods and treatment conditions for such ma-terials. At the same time, a preliminary theoretical analysis using mathematical models gives an opportunity to significantly reduce the cost of such experimental studies. Thus, it is necessary to develop multilevel models of polycrystalline metals and alloys based on crystal plasticity with the description of structure, deformation mechanisms and refinement at various scale levels. To con-struct a correct model of such a class, it is necessary to analyze information and arrange a large amount of experimental data about grain structure refinement. The article presents a review of the experimental studies describing and analyzing the grain structure refinement during severe plastic deformations of various metal alloys. The refine-ment mainly occurs at low temperatures that are a priori lower than the temperatures at which re-crystallization becomes an important factor and the solid-state phase transitions may take place. We have summarized the significant physical mechanisms of the grain refinement during cold deformation based on the arranged experimental data from the review. All the considered articles pay attention to the local accumulation of lattice dislocations inside the grains (in the form of flat clusters), which leads to the lattice curvature and separation of grains into cells. As a result of a further accumulation of dislocations in the walls, there comes an increase in misorientation of the neighboring cells. The curved lattice is unstable (it seems clear that the flat clusters become a source of such curvatures) and relaxes with the formation and movement of the partial disclina-tions, which leads to the rotation of the adjacent grain regions and creation of new grain bounda-ries. In addition, the mesoscale defects located at the junctions of the grains (including the boundary intersection disclinations), flat clusters of the dislocations of the orientational mismatch at the grain boundaries and partial dislocations in the grains have a significant effect on the frag-mentation. The articles about the severe plastic deformation at high temperatures are briefly de-scribed here. It is noted that recrystallization is the main mechanism of the fine-grained structure formation under these conditions. We suggest including the description of the discussed mechanisms in the multilevel con-stitutive material models. When new experimental data appear for a specific process of the severe plastic deformation, the considered refinement mechanisms can be added.
PNRPU Mechanics Bulletin. 2020;(2):85-111
The current state and tendencies of using rheological models for self-forming wooden structures
Ponomarev V.S., Kashevarova G.G.


Wood is a natural and ecological material. Exceptional mechanical properties of wood allow it to be used in building structures subject to chemically active substances. However some changes in mechanical characteristics of wood under the influence of water and temperature, as well as its anisotropy, limit the application areas of this material. Based on the results of the research by S.P. Timoshenko, foreign scientists proposed a method that makes it possible to use these shortcomings of wood to create wooden structures of unique shapes. This article considers the self-forming process of glued layered wooden structures, which is based on various rheological processes taking place in wood. In order to predict the shape of a glued layered wooden structure obtained with the self-forming method, an accurate rheological model is required, which takes into account all wood deformation mechanisms. We analyzed the available rheological models of wood proposed or described both by Russian and foreign scientists, as well as the international experimental studies and numerical modeling results of the self-formed glued layered wooden structures. Based on the analysis results, it was found that the present rheological models do not take into account a number of factors that influence the process of self-forming of glued layered wooden structures. The rheological model, which includes a complete list of factors affecting wood deformation (elastic and plastic deformation, wood shrinkage or swelling, viscous-elastic creep deformation and mechanical and sorption deformation), is the most promising for creating unique architectural objects from glued wood. But it also has a number of shortcomings, mainly due to inconsideration of some factors and few experimental studies. Such factors include the geometric parameters of layers of the glued wooden structures, the surface treatment of layers, temperature, as well as wood species. These factors influence the process of wood shrinkage and swelling, thus the final shape of a wooden structure. It is necessary to have additional experimental and theoretical studies of the mechanical behavior of glued wooden structures which will take into account these factors, as well as to study other species of wood including their combinations.
PNRPU Mechanics Bulletin. 2020;(2):112-122
The method of reconstruction of residual stresses and plastic deformations in thin-walled pipelines in the delivery state and after bilateral vibro-shock surface hardening with a shot
Radchenko V.P., Pavlov V.P., Berbasova T.I., Saushkin M.N.


We suggest the phenomenological method of reconstructing the fields of residual stresses and plastic deformations in thin-walled cylindrical tubes made of Х18N10Т steel in the delivery state and after a simultaneous bilateral surface plastic hardening by the vibration-shot blasting of the surface with beads on a special vibrating stand. A cylindrical container filled with three-millimeter beads was attached to it. The tubes were 50 % filled with one-millimeter beads, and they were placed inside the container. The axis of the tube and the container coincided. The space between the tube and the container was 80 % filled with beads. The vibrational frequency of the stand was 18.5 KHz, the hardening time was 20 minutes. The tube in the container was rotated to ensure uniform hardening. We determined the experimental values of residual stresses σθ and σ z in the surface layers using the method of rings and strips with the procedure of the layer-by-layer electrochemical picking of the hardened layers. For this purpose, the experimentally measured values of the beam-strip deflection and the angular opening of the cut ring (changing the diameter) were used. The hardening anisotropy parameter which relates the axial and circumferential components of plastic deformation was introduced into the mathematical model. In solving the stated problems the hypotheses of plastic incompressibility of the material, the absence of secondary plastic deformations of the material in the compression region of the surface layer, as well as the hypothesis of flat sections and straight radii were used. We described the method aimed at solving this type of boundary value problems of reconstructing stress-strain states, which makes it possible to determine the missing component σ r and all the components of the tensor of residual plastic deformations (off-diagonal components of the tensors of stresses and deformations were not considered). The method of reconstructing the stress-strain state is universal, because it has shown its operability both in determining the technological fields of residual stresses, as well as the irreversible strains in the samples in the delivered state after mechanical operations, and after bilateral surface plastic deformation. The adequacy of the calculated data was verified, which was obtained using the phenomenological method of reconstructing the stress and strain fields of the experimental data for the samples in the delivery state and after hardening. The correspondence of the calculated and experimental data was matched. The numerical values are given for the anisotropy parameter connecting the circumferential and axial irreversible strains, for samples, in the delivery state, its numerical value is 0.1, and, for the hardened samples, it is 4.2. This indicates a significant anisotropy of the distribution of the axial and circumferential components of the residual strain tensor. It has been established that the compressive residual stresses are observed in the delivery state in the region adjacent to the inner surface, and the tensile stresses are observed in the layer on the outer surface. Only compressive stresses are observed in both regions after hardening, which significantly exceed in module similar stresses for the samples in the delivery state. The main results are illustrated by the tabular data and the corresponding diagrams of the distribution of residual stresses along the depth of the hardened layer.
PNRPU Mechanics Bulletin. 2020;(2):123-133
The Dissipative Properties Assessment of the Oscillatory System of a Serial Sample of the Coriolis Flowmeter
Romanov V.A., Taranenko P.A.


The quantitative estimates of the flow rate (or density) of the flowing fluid obtained by the measurements using the industrial Coriolis flowmeters are made by using the laboratory experiments previously performed with the exemplary sensor. In this case we face two limitations, such as the unavailability of the facilities because of intense laboratory schedules and little time to upgrade the sensor oscillatory system. So we suggest using the virtual prototyping approaches as an alternative to the descriptive approaches. One of the fundamental problems of creating a virtual prototype of the Coriolis flowmeter is to separate the main parameter measured by the flowmeter (the phase shift) into the parts connected to the gyroscopic and dissipative forces. To solve this problem, we need to identify the dissipative forces model of the flowmeter oscillatory system. The article discusses the experimental results determining the dissipative properties of the mechanical oscillatory system of one of the commercially available Coriolis flowmeter samples. The algorithm identifying the model of the dissipative properties of the flowmeter oscillatory system is based on studying the nonlinearity degree of the envelope of the vibrogram of free damped oscillations. The experiments were carried out at the pouring stand of the Center for Experimental Mechanics of the South Ural State University, which allows controlling the speed and phase composition of the fluid flowing through the flowmeter. The article describes the processing algorithms for vibrograms of the damped oscillations, which make it possible to isolate the contribution into the dissipated energy from the dry (Coulomb model), the linear viscous (Rayleigh model) and quadratic viscous friction. The pronounced dependence of the vibrational system dissipation of the Coriolis flowmeter on the features of the fluid flow (velocity, mode: continuous, slug) was experimentally proven, the solutions of identifying the model of the dissipative forces are presented. The identification algorithm for the model of the dissipative properties of the flowmeter oscillatory system is based on studying the nonlinearity degree of the envelope of the vibrogram of the free damped oscillations. The use of the pouring stand made it possible to control the speed and phase composition of the fluid flowing through the flowmeter. The article describes the processing algorithms for the vibrograms of the damped oscillations by isolating the contribution into the dissipated energy from the dry (Coulomb model), linear viscous (Rayleigh model) and quadratic viscous friction. The pronounced dependence of the dissipation of the vibrational system of the Coriolis flowmeter on the features of the fluid flow (velocity, mode: continuous, slug) was experimentally proved, and the results of identifying the model of the dissipative forces are presented. The experiments included water acts as a fluid medium and air acts as a dispersed phase.
PNRPU Mechanics Bulletin. 2020;(2):134-144
Nonlinear dynamics and stages of damage of Ti6Al4V and Ti45Nb titanium alloys in very high cycle fatigue
Bannikov M.V., Oborin V.A., Bilalov D.A., Naimark O.B.


The paper presents an experimental methodology aimed at evaluating a very-high cycle resource for aviation titanium alloys Vt-6 (Ti6Al4V) and Ti45Nb for medical applications with different microstructures (large-crystal and submicrocrystalline ones). The submicrocrystalline (SMC) state was obtained by an intensive plastic deformation realized in two ways: the three-dimensional forging for Ti45Nb and radial-shear rolling for Ti6Al4V. The experimental program tests high-cycle and very-high-cycle loading (number of cycles 107-109) realized using the in situ determination method of the accumulation of the irreversible fatigue damage by analyzing nonlinear forms of feedback in a closed system ultrasonic fatigue setup. This makes it possible to establish the connection of the microscopic fatigue mechanisms with the model views and consider the stages of the damage development based on the nonlinear kinetics of the defect accumulation under cyclic loading in high- and gigacycle fatigue modes. We established various relations between changes in the amplitude of the second harmonic of vibrations of the free end of the samples with different internal structures, which are associated with the mechanisms of stress relaxation and damage accumulation. The grain size reduction in Ti45Nb alloy by the three-dimensional forging improved the fatigue properties by 1.3-1.5 times, whereas for VT-6 alloy, the radial-shear rolling method could not increase the fatigue properties in the very high cycle fatigue range, which may be caused by the presence of large residual internal stresses. Based on the scale parameters obtained earlier from the fracture surface morphology and the relations established in this work, the kinetic equations for the origin and growth of fatigue cracks in the gigacycle loading range will be constructed. This equation, based on the empirical power parameters related to the structure of the material, will allow us to determine the number of cycles for the origin of an internal crack and its growth to the surface.
PNRPU Mechanics Bulletin. 2020;(2):145-153

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