## No 4 (2017)

**Year:**2017**Articles:**15**URL:**https://ered.pstu.ru/index.php/mechanics/issue/view/15**DOI:**https://doi.org/10.15593/perm.mech/2017.4

Longitudinal resonance vibrations of the viscoelastic rod with a variable length

#### Abstract

Fluctuations of the core which is burning at one end are investigated in the paper. The research object belongs to a wide range of fluctuating one-dimensional objects with moving boundaries and loadings. The classical mathematical model considering viscoelasticity on the basis of the structural model of Foigt is used for the description of fluctuations. By introducing the dimensionless variables it became possible to reduce the number of parameters (which affect the process of fluctuations) to two. The parameters characterize the speed of the border’s motion and viscoelastic properties of the core. The method of Kantorovich-Galerkin is applied for the solution. Eigen functions of the boundary problem with a motionless border are taken as dynamic modes. The solution when the speed of the border’s motion is equal to zero is precise. The error of the solution increases, when the speed of the border’s motion increases. By neglecting the small values, it became possible to obtain quite a simple expression for the amplitude of resonant fluctuations. The expression obtained for the amplitude of fluctuations contains the integrals having no analytical solution, therefore they were obtained numerically. The solution has a mode structure that allows analyzing the resonant properties of the core. The obtained solution made it possible to analyze the phenomena of the established resonance and the process of passing through the resonance. The analytical expression describing the increase in amplitude of fluctuations is obtained for the established resonance. The passing through the resonance is analyzed quantitatively. The graphs of the amplitude of fluctuations changing in the resonant area for the first dynamic mode at various values of the parameter characterizing the viscoelasticity are presented. Also the graphs of the maximum amplitude of fluctuations are provided when passing through the resonance at the first dynamic mode, depending on the parameters which characterize the viscoelasticity and speed of the border’s motion. The results of numerical calculations are processed by means of the least squares method. The expressions for the maximum deformations of the core when passing through the resonance at the first and second dynamic mode are obtained. The gained expression allows depending on the speed of the border’s motion, the coefficient of viscoelasticity and strength of the core’s material which make it possible to estimate the core durability.

**PNRPU Mechanics Bulletin**. 2017;(4):5-15

Experimental study of deformation properties of a bulk layer from plumbum balls under dynamic and quasistatic loading

#### Abstract

Bulk layers of metal balls are a promising damping element that protects structures from impulsive actions. To assess their damping properties, the knowledge of the deformation and strength characteristics for static and dynamic compression is required. Three stages can be distinguished in the deformation of bulk porous media: re-stacking to close packing without appreciable deformation of the particles (elimination of free porosity); deformation to almost complete layout (absence of pores); deformation as a continuous material without pores. To obtain the dynamic characteristics of the layer, we used the Kolsky method with the Hopkinson bar system. The results of investigations at various deformation rates determined by the initial speed of the striker's departure are presented. Due to the small resistance of the porous sample to the deformation at the stages of repackaging and the initial stage of the deformation of the balls, most of the compression pulse formed when the striker projectile the loading bar is returned to the first measuring bar. This leads to a repeated cyclic loading of the sample with an increasingly decreasing load amplitude. In addition to the active loading curves, the technique allows obtaining unloading curves in the course of one experiment. It is demonstrated that a developed plastic flow and fragmentation of the balls are observed as a result of deformation of the samples with increasing load. Static compression of the bulk was carried out on a Zwick test machine. The tests carried out in a wide range of loads showed great differences between the static and dynamic deformation curves characteristic of highly porous media. The obtained results can be used for predictive mathematical modeling of elements of structures containing bulk layers.

**PNRPU Mechanics Bulletin**. 2017;(4):16-27

Models of molecular dynamics: a review of EAM potentials. Part 1: Potentials for single-component systems

#### Abstract

This article is the first part of the review devoted to the description of modern approaches to the construction of interatomic potentials based on the embedded atom method (EAM) for various one-component materials. We consider the general ideology of the molecular (atomic) dynamics method and outline the shortcomings of the approach to modeling atomic systems using the classical pair-interaction potentials. The basic ideas and relations of the embedded atom method are presented, as well as various modifications of the method, including MEAM. The algorithm for numerical implementation of EAM potentials in calculations of molecular dynamic systems is considered, as well as the ways to improve the efficiency of the computational algorithms, including those related to the introduction of the cut-off radius and the justification of its choice for various materials. The review focuses on the methods of construction and types of embedding functions for a variety of materials, as well as the physical and mechanical properties of materials, which are well described in some kind of potential. A brief review of the methods for obtaining potential parameters is given, e.g. by using the quantum mechanical calculations (often referred to as "first principles" or "ab initio") or using experimentally determined properties (diffraction, X-ray methods or electron microscopy), as well as a list of physical and mechanical properties that can be determined for a material using molecular dynamics methods with the application of modern potentials constructed by EAM. The authors have considered in detail the works which offer those potentials that (at present) most accurately describe the properties of such "difficult" (for forward modeling) materials, such as beryllium, iron, tungsten, niobium, titanium, uranium and others. The relevance of studying the structure and properties of these materials is largely due to the prospects of its application in materials for nuclear power engineering, aerospace industry, and biomedicine. Also, this review is devoted to the construction of EAM potentials capable for describing the structure and properties of a number of metals (lithium, nickel, copper, aluminum, etc.), which are considered both in the melt state and after crystallization.

**PNRPU Mechanics Bulletin**. 2017;(4):214-237

CRITICAL DYNAMICS OF LOCALIZED INSTABILITIES OF PLASTIC FLOW IN THE AL-MG ALLOY

#### Abstract

The critical dynamics of the spatial-temporal fluctuations related to the stress of the plastic flow (Portevin-Le Chatelier effect) of inclined cylindrical samples made of Al-Mg (AMG6) alloy was investigated at strain rates 0.4-1.7·10-1 s. Such samples’ shape allows achieving large deformations (up to 80 %) without destruction. The plastic flow of the investigated alloy exhibits multiscale features of strain localization along the entire length of the plastic flow curve. The first critical value is determined by the appearance of the multiple regions of localized plasticity, with signs of autosoliton dynamics exhibiting the correlated behavior at the macroscopic size of the sample. The two values of "critical" strains corresponding to the range of stochastic dynamics were determined which is characteristic for non-equilibrium critical systems. The transition through the second critical value is associated with qualitative changes of collective modes in ensembles of defects: a transition from autosoliton modes providing the plastic strain localization to collective “blow-up” modes responsible for damage-failure transition. The localization of plastic flow was studied by the structural analysis of the morphology of the surface using the optical interferometer-profilometer NewView-5010 for the subsequent calculation of the scale invariant (Hurst index) and the spatial scale of the region on which there is a correlated behavior of microshears. It was shown that the plastic flow instability and the fully developed turbulent flow in liquids reveal universality features of momentum transfer, that could be linked to the same “universality classes”.

**PNRPU Mechanics Bulletin**. 2017;(4):28-39

Numerical solution for an inverse problem about determination of volumetric heat capacity of rock mass during artificial freezing

#### Abstract

The paper is devoted to development and implementation of algorithms for a numerical solution of a coefficient inverse Stefan problem. This problem arises in a modeling for a process of ice wall formation around a projected horizontal section of a mine shaft. The ice wall is formed by an artificial ground freezing to convert soil pore water into ice. The temperature field modeling is based on an enthalpy form of a two-dimensional Stefan problem. The aim of the study is to determine the volumetric heat capacity for the rock layer on the base of additional information about temperature evolution in thermal wells. The problem of coefficients’ identification is stated as a variation form of the coefficient inverse Stefan problem. As a result, two algorithms for a numerical solution of the stated inverse Stefan problem have been developed. The first algorithm is based on the conjugate gradient iterative optimization method. The second algorithm is based on the steepest descent iterative optimization method. Under the first algorithm the calculation of the discrepancy functional gradient and determination of parameters for the optimization method are performed by solving a sensitivity problem and an adjoint problem. Forms of these problems have been obtained for the stated direct Stefan problem. For the second algorithm the discrepancy functional gradient and parameters for the descent step method are determined by calculating sensitivity coefficients. Solutions of the direct problem, the sensitivity problem and the adjoint problem are performed by the finite element method. The special feature of the used optimization methods is that these methods have regularizing properties. In order to verify effectiveness of the proposed algorithms, the computational experiments have been performed. The first and second experiments are related to determining only one unknown volumetric heat capacity for an ice domain or a cooling domain. The third experiment is devoted to determining the volumetric heat capacity for the both domains. The results of the experiments show that both algorithms allow to determine the volumetric heat capacity with a sufficiently good accuracy. However, a convergence rate of the second algorithm is higher than the rate of the first algorithm. The presented approach to modeling the process of ice wall formation as the coefficient inverse Stefan problem and the developed algorithms can be used for designing and improving the initial data for building mine shafts with a technology of artificial ground freezing.

**PNRPU Mechanics Bulletin**. 2017;(4):56-75

Aramid fabric surface treatment and its impact on the mechanics of yarn’s frictional interaction

#### Abstract

Aramid textiles are widely used in protective armor structures (vests, helmets, etc.). The most important problem of the development of such structures is to reduce the fraction of the bullet's kinetic energy transmitted to the object located behind the armor panel. The reduction of the kinetic energy results to decrease a dynamic deflection of the back surface of the armor panel. Researches show that a significant part of the energy absorbed by the armor panel is linked to frictional forces of pulled yarns. In this paper, we present an effective method for controlling of dry friction between yarns - surface treatment by PVA suspension, rosin or silicone grease with a slight overweighting of the fabric. In the experimental part of the paper, the results of quasi-static yarn pull-out tests from an aramid fabric SVM P110 with a plane structure (with different types of surface treatment) are presented. The relationships between force and displacement are also obtained. It can be noticed that the surface treatment of fabrics with addition of only 6 wt. % leads to increase frictional interaction between the yarns by 4 times. Elastic and strength properties of aramid yarns are obtained from quasi-static tensile tests. The dry friction coefficient between yarn and neat fabric is determined by experiments. The numerical part of the paper is devoted to development of low-parametric FE model of a yarn pull-out test for P110 fabric performed in explicit FE code LS-DYNA. It is shown that different surface treatment can be effectively substitute during calculations by variation of dry friction coefficient (and increase/decrease yarn pull-out force). The calculated “force -displacement” curves are obtained by pulling out a yarn from a fabric with and without surface treatments, which imposed within the corresponding scatter band of the experimental data.

**PNRPU Mechanics Bulletin**. 2017;(4):121-137

Modeling the dynamic penetration processes of dimensional bodies in a compressible elastoplastic medium

#### Abstract

The paper considers the problem of the normal impact and penetration of rigid spatial bodies into the half-space of the elastoplastic medium. For the medium, a model of a linearly compressible elastoplastic medium is adopted with a linear dependence of the yield stress on the pressure (the Mises-Schleicher-Botkin plasticity condition). The solution of the problem is carried out numerically in a three-dimensional formulation using the software package LS-Dyna. The elastic-plastic penetration medium is considered using a fixed Euler grid with the allocation of empty cells into which the material flows during deformation. The strikers are modeled by a rigid undeformed body in the Lagrangian coordinate system. For comparison, the parameters of the penetration process, i.e. the resistance forces to penetration and the penetration depth of the strikers, were also obtained within the framework of the local interaction model on the basis of the analytical solution of the problem of expanding the spherical cavity. Earlier the applicability of the local interaction model to the determination of the force and kinematic characteristics of sharp conical bodies is shown based on the data of the inverted experiments and numerical calculations in an axisymmetric formulation. Verification of the model applicability aimed at describing the motion of three-dimensional bodies in the full three-dimensional formulation has not been carried out before. The investigated bodies, i.e. a circular cone, three- and tetrahedral pyramids, and a body with a star-shaped cross-section, have the same area of the base, the normal to the lateral surface of the bodies makes up a constant angle of 600 with the direction of motion. The peculiarity of constructing the shape of the three-dimensional bodies in question is the fact that within the framework of the local interaction model these bodies should have the same resistance to the introduction, which coincides with the resistance to the introduction of the circular cone. All these bodies have a height less than the height of the cone. The results of the three-dimensional numerical calculations of the bodies’ penetration along the normal into an elastoplastic medium with subsonic and supersonic velocities are presented, which demonstrate an increase in resistance to penetration proportional to a decrease in body height. For the bodies of the same height, the changes in the form of the cross section do not lead to significant differences in the strength of the resistance and penetration depth.

**PNRPU Mechanics Bulletin**. 2017;(4):92-108

Stability of heated orthotropic geometrically irregular plate in a supersonic gas flow

#### Abstract

Thin-walled geometrically irregular objects in the form of orthotropic rectangular plates are considered on the basis of the linear thermoelasticity, they are supported by the ribs symmetric with respect to middle plane. The location of the ribs coincides with the direction of the supersonic gas flow. The continuum model of the thermoelastic system “plate- ribs” was chosen. Singular differential equations of quasi-static and dynamic state of the elastic system contain tangential efforts and transverse force. Tangential efforts occur during heating of the plate. The transverse force caused by a small deflection plates is determined in the standard way via the “forcer” theory. The tangential effort is pre-determined by the solutions of singular differential equations of thermoelasticity for a geometrically irregular plate with given boundary conditions. The solution of the singular differential equations of thermoelasticity of the plate in a supersonic gas flow in quasi-static and dynamic formulation of the objectives sought in the form of sums of double trigonometric series, respectively, with the constant and variable along the time coordinate coefficients. The coefficients - approximating the function of trough - of the ranks are determined using Galerkin method as a solution of the homogeneous algebraic systems or homogeneous systems of differential equations of the second order in the case of a dynamic formulation of the problem. The solution is given in the second approximation. The critical values of the gas flow rate are determined on the basis of the standard methods of analysis of static and dynamic stability of thin-walled structures. Quantitative results are presented in tables illustrating the influence of the geometrical parameters of the “plate-ribs” thermoelastic system, the relative height of the ribs, number of ribs, the ratio of the sides of the plate, temperature, the material anisotropy on the stability of the geometrically irregular plate over the sound of the gas flow.

**PNRPU Mechanics Bulletin**. 2017;(4):109-120

Determination of the critical plane and assessment of fatigue durability under various cyclic loading regimes

#### Abstract

Analytical solutions are obtained to determine the critical plane orientation for multiaxial stress state. This critical plane is the plane of development of fatigue damage under cyclic loading. The cases of in-phase and antiphase cyclic loadings are considered for the classical fatigue range (low-cycle and high-cycle fatigue). Generalizations of the Findley fatigue criterion are proposed taking into account the orientation of the critical plane for modes of very-high-cycle fatigue under in-phase and antiphase cyclic multiaxial loadings. These generalizations are based on the similarity of the left and right branches of the bimodal fatigue curve. The procedure aimed at determining the parameters of the generalized criterion is described according to the data of two uniaxial fatigue tests for tension-compression at various coefficients of the cycle asymmetry. The stressed state of the compressor disk of the gas turbine engine are calculated by the finite element method for a low-cycle fatigue (take off-flight-landing cycles) and for very-high-cycle fatigue (vibrations of blades). For low-cycle fatigue the effect of aerodynamic, centrifugal and contact loads for the contact system of disk, blades, fixing pins and shroud are taken into account. For very-high-cycle fatigue the additional stresses due to high-frequency torsional vibrations of blades and shroud are calculated. In both cases, a stress concentration zone was defined in the vicinity of the contact between the disc and the blades, in which the fatigue damage originates. The distribution of stresses and the proposed generalizations of the fatigue fracture criteria were used to obtain estimates of disk durability for both low-cycle fatigue and very-high-cycle fatigue modes. Numerical analysis showed that real time durability for the considered fatigue modes can be very close taking into account the characteristic cycle period. Therefore, in predictions of a safe operation life both mechanisms of fatigue must be taken

**PNRPU Mechanics Bulletin**. 2017;(4):238-252

Perturbation of the stress-strain state of an elastic half-space by the spherical inhomogeneity of elastic properties under shear in horizontal plane taking account of gravitational forces

#### Abstract

The present paper is devoted to the solution of the problem related to the perturbation of the stress-strain state of an elastic isotropic half-space by the spherical inhomogeneity of elastic properties for the case of only gravity and the case of an additional superimposed pure shear on the half-space. The analytical solution is obtained for the displacement and volume strain components, expressed in terms of the potentials and quasipotentials of the sphere, for the case of only gravity forces acting on the half-spaces and the case of superposition of gravity and pure shear. Using the example of a specific seismic event, it was shown that the contribution of gravitational forces to the perturbation of the stress-strain state of a half-space by the spherical inhomogeneity of elastic properties (the distribution with respect to the volume deformation) is not small. Using the obtained analytical solution for the case of the action of gravity only, it is shown that the volume strain distribution is axisymmetric with respect to the axis . A spherical inhomogeneity of the shear modulus in an isotropic elastic half-space forms a zone of a relative compression located along the strike of the inhomogeneity, and zones of relative stretching above and below the inhomogeneity. A distinctive feature of the zones of relative stretching is their limiting size, while the size of the zone of a relative compression increases when the volume strain decreases. In this case, between these planes, the solution in the neighboring octants differs only in sign. When adding a term associated with the gravitational forces in the solution, the symmetry is not violated but the equality in magnitude of the volume deformations in neighboring octants is violated. Dimensions and geometry of isosurfaces of volumetric deformation varies significantly when gravity is taken into account. Depending on the direction of the axis of the maximum extension of the half-space with the inhomogeneity, the size of the zones of the relative compression increases and the corresponding reduction in the size of the zones of the relative stretching changes with the shape of both.

**PNRPU Mechanics Bulletin**. 2017;(4):138-153

The Stress-Strain State and Failure of Structural Elements with Sharp Stress Concentrators under Bending

#### Abstract

The paper considers two approaches to assessing the strength of constructions’ elements made of brittle material with sharp stress concentrators under the three-point bending: using a non-local theory of strength (in the top of the sharp notch the stresses are averaged on some basis) and a linear fracture mechanics (the stress intensity factors and the degree of singularity are determined). The finite element method was widely used (ANSYS Workbench, 2D model). These two approaches are connected by the analysis of the stress state at the sharp v-shaped notches with the corresponding approximations of the stress field, which allowed to find the correct values of the stress intensity factors and degrees of singularity on the basis of the optimization procedures. There were conducted the experimental studies of the strength of prismatic samples made of polymethylmethacrylate (PMMA) with a v-shaped notch (the two-edge angles of 25, 90 and 120°) in static three-point bending. In the finite element calculations using a nonlocal strength criterion it is obtained, that a sharp notch with the two-edge angle up to 90° leads to an increase in load of destruction for 15-20 % only compared with the crack-like cut (the angle of 25°) and agrees well with the experimental data. The increase of the two-edge angle of a sharp notch from 25 to 120° leads to increased loads of destruction in 1,5...2 times. There was studied the morphology of fractured surfaces, it is shown that the shallow cuts (up to 1.5 mm) when bending have a grooved fracture surface (a high velocity, periodic picture of cracks’ moving), while the deeper cuts (2.5...5 mm) are characterized by a slow crack propagation and a smooth fracture surface. Information about step of grooves was necessary to establish the size of averaging zone in nonlocal failure criterion.

**PNRPU Mechanics Bulletin**. 2017;(4):40-55

Modeling of the product termomechanical behavior during 3D deposition of wire materials in ANSYS

#### Abstract

Additive technologies allow to produce products due to the layer-by-layer synthesis and obtain products of a complex shape. The numerical modeling of the physical process of manufacturing parts using additive technologies is complex and needs to consider a variety of thermomechanical behavior. This is connected to the extensive use of the finite element computer simulation by means of specialized software packages that implement mathematical models of the processes. The paper considers the algorithm of calculating nonstationary temperature fields and stress-strain state of the structure during the 3D deposition of wire materials developed and implemented in ANSYS. The verification of the developed numerical algorithm is carried out to solve the three-dimensional problem related to the production of metal products using arc 3D deposition of wire materials with the results of the experiment. The problem is divided into the boundary value problem of nonstationary heat conductivity and thermomechanics boundary value problem of stress-strain state that are uncoupled. For its solution the technology of “killing” and subsequent “birthing” is used (it is realized in ANSYS). Continuous building of the material is made discretely on each step of calculation corresponding to “birthing” of the next subarea from "dead" elements, the boundary problem of heat conductivity and thermomechanics is solved. And the result of solving the previous step is the entry conditions for the next step. The Anand's model is used for the description of the viscoelastoplastic behavior of the studied alloy. Identification of the Anand's model for the studied material was carried out according to the stretching experiment with the set speed at various temperatures. The data obtained from the calculations on the developed numerical model are in good agreement with the experiment.

**PNRPU Mechanics Bulletin**. 2017;(4):154-172

Theoretical and experimental study of structural elements’ influence of fiber-optic strain sensor on its readings and correction procedure for transfer function

#### Abstract

Fiber optic sensors (FOS) is a rapidly developing part of measuring equipment. They are regularly expanding their application field in various engineering branches due to high operational properties. A large number of research works are carried out in this field all over the world. Most of the published articles are devoted to the application of FOS and their operating principles in terms of optics and electronics. In the present paper a fiber optic sensor is considered as a mechanical system. A sensor forms its readings while interacting with environment, which is the carrier of the measured parameter. In this process it is possible to distinguish three different aspects that determine the measurement result: the physical (optical) laws underlying the sensor transfer function, the nature of the sensor and environment interaction, and the interaction of sensor structural elements (even in the simplest FOS structure it is possible to distinguish several components with different physical and mechanical properties). All these aspects play a decisive role in the readings formation and must be taken into account by sensor designers. The present study describes the methodology aimed at adjusting FOS transfer function by taking into account all the indicated aspects of the readings formation. A fiber optic strain sensor based on the Bragg grating has been considered as an example. The influence of its structure on the readings has been determined by means of mathematical modeling. The contribution of single structural elements (such as glue, optical fiber, substrate with a technological hole, etc.) to this value has been evaluated. General algorithm for determining the effect of the sensor structural elements in its readings has been formulated as the results’ generalization. The algorithm has been tested using the existing model of the fiber optic strain sensor. The efficiency of the method has been proved by a series of laboratory experiments.

**PNRPU Mechanics Bulletin**. 2017;(4):173-188

STOCHASTICAL MODEL OF MICROSTRUCTURAL FAILURE BASED ON RESTORATION OF DISTRIBUTIONS LAWS FOR RANDOM STRESS AND STRAIN FIELDS IN MICROHETEROGENEOUS MEDIA

#### Abstract

A common direction in the micromechanics of heterogeneous media with a random structure is related to the methods of stochastic mechanics. Microstructure of a heterogeneous material has a significant effect on the distribution of stress and strain fields. Methods and tools of statistical analysis allow to take into account interactions within a many-particle system and investigate fields distributions from an analytical point of view, which is used to create stochastic models describing the mechanical behavior of a material. This paper presents an approach to the analysis of the distribution of stress and strain fields in representative volumes of inhomogeneous media based on the restoration of their distribution laws. The techniques of finding the parameters of the distribution laws are described. In the framework of the stochastic model, the central moments of the random variable obtained for the stress and strain fields in individual phases of the material are used for this. An algorithm for calculating the fracture probability using the laws of distribution of stress and strain fields is proposed on the basis of the probability representation of the failure criteria. Based on the analysis of the probability of failure of phases of a representative volume, a stochastic model of a progressive failure is presented. Some numerical results were derived for the particular case of a non-homogeneous structure; the estimates obtained with the help of the stochastic model are compared with the results of finite element modeling. The different types of parametric distribution laws used to reflect a real distribution of stress fields in a representative volume are compared on the basis of the obtained finite element analysis and the calculation of the moments in the framework of the stochastic model. For the discussed case study, a technique for calculating the fracture probabilities at static loading of a representative volume is implemented.

**PNRPU Mechanics Bulletin**. 2017;(4):76-91

ESTIMATION OF CRACK PROPAGATION DIRECTION ANGLE UNDER MIXED-MODE LOADING (MODE I AND MODE II): GENERALIZED FRACTURE MECHANICS CRITERIA AND ATOMISTIC MODELING (MOLECULAR DYNAMICS METHOD)

#### Abstract

This study is devoted to the estimation of the crack propagation direction angle in the plate with the central crack under mixed-mode loading (Mode I and Mode II) in an iso-tropic linear elastic medium using two approaches: the generalized criteria of continuum mechanics (the generalized maximum tangential stress criterion and the generalized strain energy density criterion) and atomistic modeling of the Cu single crystal with the central crack. Molecular dynamics simulations of the central crack’s growth in a plane medium using Large-scale Molecular Massively Parallel Simulator (LAMMPS) are performed. The inter-atomic potential used in this investigation is the Embedded Atom Method (EAM) poten-tial. The specimens with the initial central crack were subjected to Mixed-Mode loadings. The crack propagation direction angles under different values of the mixed parameter in a wide range of values from pure tensile loading to pure shear loading in a wide range of temperatures are obtained and analyzed. It is shown that the crack propagation direction angles obtained by the molecular dynamics method coincide with the crack propagation direction angles given by the multi-parameter fracture criteria based on the strain energy density and the multi-parameter description of the crack-tip fields.

**PNRPU Mechanics Bulletin**. 2017;(4):189-213