# 2017 • no. 2

# TABLE OF CONTENTS

**Babeshko V.A., Evdokimova O.V., Babeshko O.M., Telyatnikov I.S., Eletskiy Y.B., Uafa S.B.**

Factorization methods in the problem of the bases and coverings of polygonal forms

**Babeshko O.M., Babeshko V.A., Evdokimova O.V.**

About starting earthquakes for full contact of the lithospheres’ plates and base

**Ganeeva M.S., Moiseeva V.E., Skvortsova Z.V.**

Nonlinear bending and stability of ellipsoidal reverse buckling disks being under the liquid pressure and temperature

**Gusev A.A., Voloshin A.E., Yakovenko N.A.**

Long-term forecasting of the dynamics of mean solar magnetic eld with the fuzzy inductive reasoning model

**Evdokimova O.V., Babeshko O.M., Babeshko V.A.**

Composition of the packed block elements into the block structure and their homeomorphisms

**Zaretskaya M.V., Babeshko O.M., Zaretskiy A.G., Lozovoy V.V.**

On blocks of various types in problems of geoecology

**Kochergin V.S., Kochergin S.V.**

The use of adjoint problems solution in the identi cation of input parameters for transport models and planning of experiments

**Lavrov I.V., Bardushkin V.V., Sychev A.P., Yakovlev V.B., Kirillov D.A.**

On calculation of the e ective thermal conductivity of textured tribocomposites

**Osipyan V.O., Leyman A.V., Chesebiev A.A., Zhuk A.S., Harutyunyan A.H., Karpenko Y.A.**

Mathematical modeling of non-standard multiplicative knapsack cryptosystems

**Rubtsov S.E., Pavlova A.V., Telyatnikov I.S.**

To the problem of vibrations of a limited volume of a liquid on an elastic foundation

**Sobol B.V., Rashidova E.V.**

Equilibrium state of the internal crack in in nite elastic wedge with the thin coating

**Stogny G.A., Stogny V.V.**

Seismicity of greater caucasus on the position of the earth’s crust block divisibility

## ABSTRACTS

*Subjects*: mechanics

**Babeshko V.A.**^{1}, **Evdokimova O.V.**^{1}, **Babeshko O.M.**^{2}, **Telyatnikov I.S.**^{1}, **Eletskiy Y.B.**^{1}, **Uafa S.B.**^{1}

**Factorization methods in the problem of the bases and coverings of polygonal forms**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 5-12.

The block element method is considered in mechanics mixed boundary problems for continuous bodies The approach was induced by Wiener-Hopf method, and its extension to space case is called integral factorization method and mostly used in applications with the smooth-boundary areas. In this work the method is used for sectionally smooth boundary areas with corner points, what requested its development for two complex-variable functions. Concerned boundary problems have numerous applications for the tasks of mechanics, theoretical and technical physics. Created method was tested on a vector contact problem for wedge-shaped stamp, taking place in the first quadrant. Means of achieving of different solution’s characteristics are described in details. They are based on convertion of one-dimensional linear integral equation system, typical to dynamic and static contact problems for stripe-stamps.

*Keywords*: vector contact problem, system of integral equations, wedge-shaped area, block element, factorization, approximate values, singular characteristics.

^{1}Southern Scientific Center RAS, Rostov-on-Don, 344006, Russia

^{2}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: babeshko41@mail.ru

- Wiener N., Hopf E.
*Über eine Klasse singulärer Integralgleichungen*. S.B. Preuss. Acad. Wiss, 1932, pp. 696-706. - Arutyunyan N.Kh., Manzhirov A.V.
*Kontaktnye zadachi teorii polzuchesti*[Contact problems of the theory of creep]. Erevan, Izdatel’stvo AN Armyanskoy SSR, 1990, 320 p. (In Russian) - Manzhirov A.V. Kontaktnye zadachi dlya neodnorodnykh stareyushchikh vyazkouprugikh tel [Contact problems for inhomogeneous aging viscoelastic bodies]. In Vorovich I.I., Aleksandrov V.M. (eds.)
*Mekhanika kontaktnogo vzaimodeystviya*[Mechanics of contact interaction], Moscow, Fizmatlit Pub., 2001, pp. 549-565. (In Russian) - Manzhirov A.V. Kontaktnye zadachi o vzaimodeystvii vyazkouprugikh osnovaniy, podverzhennykh stareniyu, s sistemami neodnovremenno prikladyvaemykh shtampov [Contact problems on the interaction of viscoelastic bases subject to aging with systems of non-simultaneously applied dies].
*Prikladnaya matematika i mekhanika*[Applied Mathematics and Mechanics], 1987, vol. 51, iss. 4, pp. 670-685. (In Russian) - Nobl B.
*Metod Vinera-Khopfa*[Method Wiener-Hopf]. Moscow, Izdatel’stvo Inostrannaya literatura Pub., 1962, 280 p. (In Russian) - Vorovich I.I., Babeshko V.A.
*Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskikh oblastey*[Dynamic mixed problems of elasticity theory for nonclassical domains]. Moscow, Nauka Pub., 1979, 320 p. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Ob odnoy smeshannoy zadache dlya uravneniya teploprovodnosti v poluogranichennoy oblasti [On a mixed problem for the heat equation in a semibounded region].
*Zhurnal prikladnoy mekhaniki i tekhnicheskoy fiziki*[J. of Applied Mechanics and Technical Physics], 2015, no. 6, pp. 31-37. (In Russian) - Babeshko V.A.
*Obobshchennyy metod faktorizatsii v prostranstvennykh dinamicheskikh smeshannykh zadachakh teorii uprugosti*[Generalized factorization method in spatial dynamic mixed problems of the theory of elasticity]. Moscow, Nauka Pub., 1984, 256 p. (In Russian) - Babeshko V.A., Glushkov E.V., Zinchenko Zh.F.
*Dinamika neodnorodnykh lineyno-uprugikh sred*[Dynamics of inhomogeneous linearly elastic media]. Moscow, Nauka Pub., 1989, 344 p. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Metod blochnogo elementa dlya integral’nykh uravneniy kontaktnykh zadach v klinovidnoy oblasti [The block element method for integral equations of contact problems in the wedge-shaped domain].
*Zhurnal prikladnoy mekhaniki i tekhnicheskoy fiziki*[J. of Applied Mechanics and Technical Physics], 2017, vol. 58, no. 2, pp. 133-140. (In Russian) - Babeshko V.A., Babeshko O.M., Evdokimova O.V. Ob integral’nom i differentsial’nom metodakh faktorizatsii [On integral and differential methods of factorization].
*Doklady Akademii nauk*[Rep. of the Academy of Sciences], 2006, vol. 410, no. 2, pp. 168-172. (In Russian) - Glushkov E.V., Glushkova N.V. Ob osobennostyakh polya uprugikh napryazheniy v okrestnosti vershiny klinovidnoy prostranstvennoy treshchiny [On the singularities of the field of elastic stresses in the vicinity of the vertex of a wedge-shaped spatial crack].
*Mekhanika tverdogo tela*[Mechanics of a rigid body], 1992, no. 4, pp. 82-88. (In Russian)

*Subjects*: mechanics

**Babeshko O.M.**^{1}, **Babeshko V.A.**^{2}, **Evdokimova O.V.**^{2}

**About starting earthquakes for full contact of the lithospheres’ plates and base**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 13-20.

The boundary problem of rigid coupling of lithospheric plates modeled by Kirchhoff plates with three-dimensional deformable layered medium as a base is considered. The possibility of occurrence of a starting earthquake in such a block structure is investigated. Thus, two states of the block structure are considered in the static mode. In the first case, the semi-infinite lithospheric plates in the form of half-planes are remote from each other, so that the distance between the ends is different from zero. In the second case, lithospheric plates are brought together to a zero distance between them. It is proved that in this case the earthquake can occur, capable of breaking the surface of the base, forming new fractures in the Earth’s crust, but the defect of the contact problems of elasticity have been fined for this case. In the previous articles of the authors, scalar and vector cases of impact on lithospheric plates have been studied. In the scalar case of vertical impact on lithospheric plates it was assumed that tangent contact stresses are absent in the domain of contact of the lithospheric plates with the base. In the vector case of horizontal impacts on lithospheric plates it was assumed that there are no vertical components in the contact domain in the presence of two components of contact tangent stresses. In the case of rigid coupling of lithospheric plates with the base, both vertical and horizontal components of contact stresses are present in the contact domain, and they are determined as a result of solving the complete three-dimensional boundary value problem. The result is not a sum of solutions to previous problems and has new properties.

*Keywords*: block element, topology, integral and differential factorization methods, exterior forms, block structures, boundary problems, starting earthquakes.

^{1}Kuban State University, Krasnodar, 350040, Russia

^{2}Southern Scientific Center RAS, Rostov-on-Don, 344006, Russia

Corresponding author’s e-mail: babeshko41@mail.ru

- Babeshko V.A., Evdokimova O.V., Babeshko O.M. K probleme fiziko-mekhanicheskogo predvestnika startovogo zemletryaseniya: mesto, vremya, intensivnost’ [To the problem of the physicomechanical precursor of the initial earthquake: place, time, intensity].
*Doklady Akademii nauk*[Reports of the Academy of Sciences], 2016, vol. 466, no. 6, pp. 664-669. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. O startovykh zemletryaseniyakh pri gorizontal’nykh vozdeystviyakh [About starting earthquakes with horizontal influences].
*Doklady Akademii nauk*[Reports of the Academy of Sciences]. (In Press) (In Russian) - Vorovich I.I., Babeshko V.A.
*Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskikh oblastey*[Dynamic mixed problems of elasticity theory for nonclassical domains]. Moscow, Nauka Pub., 1979, 320 p. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Vneshniy analiz v probleme skrytykh defektov i prognoze zemletryaseniy [External analysis in the problem of hidden defects and the forecast of earthquakes].
*Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological bulletin of scientific centers of the Black Sea Economic Cooperation]. 2016, no. 2, pp. 19-28. (In Russian) - Nikonov A.A.
*Sovremennye dvizheniya Zemnoy kory*[Modern movements of the Earth’s crust]. Moscow, Nauka Pub., 1979, 184 p. (In Russian) - Morozov N.F.
*Matematicheskie voprosy teorii treshchin*[Mathematical problems in the theory of cracks]. Moscow, Nauka Pub., 1984, 256 p. (In Russian) - Chernov Yu.K.
*Sil’nye dvizheniya grunta i kolichestvennaya otsenka seysmicheskoy opasnosti territorii*[Strong ground motion and quantitative assessment of the seismic hazard of the territory]. Tashkent, FAN Pub., 1989, 296 p. (In Russian) - Rays Dzh.
*Mekhanika ochaga zemletryaseniya*[Mechanics of the earthquake focus]. Moscow, Mir Pub., 1982, 217 p. (In Russian) - Gamburtsev G.A. Perspektivnyy plan issledovaniy po probleme “Izyskanie i razvitie prognoza zemletryaseniy” [Perspective plan of research on the problem “Search and development of the earthquake forecast”]. In
*Razvitie idey G.A. Gamburtseva v geofizike*[Development of ideas Gamburtsev in geophysics]. Moscow, Nauka Pub., 1982, pp. 304-311. (In Russian) - Sadovskiy M.A., Bolkhovitinov L.G., Pisarenko V.F.
*Deformirovanie geofizicheskoy sredy i seysmicheskiy protsess*[Deformation of the geophysical environment and seismic process]. Moscow, Nauka Pub., 1987, 104 p. (In Russian) - Alekseev A.S. et al.
*Aktivnaya seysmologiya s moshchnymi vibratsionnymi istochnikami. Kollektivnaya monografiya*[Active seismology with powerful vibrational sources. Collective monograph]. Moscow, Izd-vo SO RAN, 2004, 388 p. (In Russian) - Sobolev G.A.
*Osnovy prognoza zemletryaseniy*[Basics of earthquake prediction]. Moscow, Nauka Pub., 1993, 313 p. (In Russian) - Keylis-Borok V.A. Dinamika litosfery i prognoz zemletryaseniy [Dynamics of the lithosphere and the forecast of earthquakes].
*Priroda*[Nature], 1989, no. 12, pp. 10-18. (In Russian)

*Subjects*: mechanics

**Ganeeva M.S.**^{1}, **Moiseeva V.E.**^{1}, **Skvortsova Z.V.**^{1}

**Nonlinear bending and stability of ellipsoidal reverse buckling disks being under the liquid pressure and temperature**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 21-27.

The nonlinear bending and stability of reverse buckling disks have been studied. The reverse buckling disks represent the oblate ellipsoidal shells being under the pressure of the heated or cooled compressed working environment on the convex side of the shells. We have obtained the numerical results depending on the temperature of the working environment. Comparison of the studied ellipsoidal shells with the spherical segments having the same base and depth of pole under the base was also presented. It was found that the biggest critical loads are observed at low temperatures with a significant reduction of their values at elevated temperatures for oblate ellipsoidal shell under the joint action of pressure of the compressed gas and its temperature. Herein, the loads of wave formation in a parallel for all the considered values of temperature are not observed. It is shown that oblate hemi-ellipsoidal shell provides an axisymmetric form of stability loss with the opening of their central part at low temperature and at the same temperature of the loading environment. This is consistent with the operating conditions of the safety reverse buckling disks. It is shown that small changes in geometry lead to a significant change in the stress-strain state and in stability of the considered shells.

*Keywords*: reverse buckling disks, ellipsoidal shell, nonlinear bending,
temperature, pressure.

^{1}Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences, Kazan, 420111, Russia

Corresponding author’s e-mail: ganeeva@kfti.knc.ru

- Olhovskij N.E.
*Predokhranitelnye membrany*[Protective Membranes]. Moscow: Chemistry Publ., 1976, 149 p. (In Russian) - Pavlov V.V., Belikov N.V., Iudin A.S., Kakurin A.M., Zanimonets U.M. Ustrojstvo dlya izgotovleniya hlopayushchih predohranitel’nyh membran [Device for the manufacture of reverse buckling disks]. Patent RF na izobretenie no. 2353456. Opublikovano 27.04.2009. Byul. No. 12. [RF patent for the invention no. 2353456. Published on 27.04.2009. Bull. No. 12.] (In Russian)
- Kovalenko A.D.
*Osnovy termouprugosti*[Foundations of Thermoelasticity]. Kiev, Naukova Dumka Publ., 1970, 307 p. (In Russian) - Thornton E. A. Thermal buckling of plates and shells.
*Appl. Mech. Rev.*, 1993, vol. 46, no. 10, pp. 485-506. - Ganeeva M.C.
*Termosilovaya zadacha v geometricheski i fizicheski nelineinoi teorii netonkikh i tonkikh obolochek*[Temperature and Force Problem in Geometrically and Physically Nonlinear Theory of Thin and Non-Thin Shells]. Kazan, Kazan Physicotechn. Inst. Kazan Branch USSR Acad. Sci., 1985, Available from VINITI, 1985, no. 4459-85. (In Russian) - Ganeeva M.S., Moiseeva V.E., Skvortsova Z.V. Reverse Buckling Disks under
Liquid Pressure and Temperature.
*J. of Machinery Manufacture and Reliability*, 2014, vol. 43, no. 6, pp. 482-489. DOI: 10.3103/S1052618814050069 - Ilgamov M.A.
*Staticheskie zadachi gidrouprugosti*[Static Problems of Hydroelasticity]. Kazan, Ime KazSc RAS, 1994, 208 p. (In Russian) - Ganeeva M.S., Kosolapova L.A. O sootnosheniiakh zakona Guka v temperaturnoi zadache uprugogo tverdogo tela [About Hooke’s Law Relations in the Temperature Problem of Elastic Solid].
*Trudy XVII Mezhd. konf. po teorii obolochek i plastin*[Proceedings of the XVII Int. Conf. on the Theory of Shells and Plates]. Kazan, Kazan State University Publ., 1996, vol. 1, pp. 33-37. (In Russian) - Iliushin A.A.
*Plastichnost. Ch. I. Uprugoplasticheskie deformatcii*[Plasticity. Part I. Elastic-plastic deformations]. Moscow, Leningrad, Gostehteorizdat Publ., 1948, 376 p. (In Russian) - Ganeeva M.S., Ilgamov M.A., Moiseeva V.E. Ustoichivost’ sfericheskogo
segmenta, nagruzhennogo davleniem szhimaemoi zhidkosti [Stability of spherical segment loaded by the force of compressed liquid],
*Problemy prochnosti i plastichnosti. Mezhvuzovskii sbornik*(Strength and Plasticity Problems. Interuniversity Collection of Papers), Nizhniy Novgorod, Lobachevsky State University of Nizhniy Novgorod, 2009, iss. 71, pp. 71-76. (In Russian)

*Subjects*: mechanics

**Gusev A.A.**^{1}, **Voloshin A.E.**^{2}, **Yakovenko N.A.**^{1}

**Long-term forecasting of the dynamics of mean solar magnetic eld with the fuzzy inductive reasoning model**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 28-31.

The accumulation of large amounts of data in a variety of domains creates a demand for the development of new tools for data processing and forecasting for decision support. The article is devoted to the long-term forecast of the mean solar magnetic field obtained with the fuzzy inductive reasoning (FIR) model. The model is generated automatically with the AimDSS computer program developed by the authors earlier. AimDSS is a cross-platform, standalone program which implements the FIR methodology for model creation. Authors generated the model for long-term mean solar magnetic field prediction until 2030. The model was evaluated against the control sample with the RMSE 0.0964. The forecast also matches the long-term solar activity forecast created by the authors with the Wolf number datasets earlier. Further study will be focused on elaboration of the constructed model with the final aim of obtaining a set of inductive models for application in different domains.

*Keywords*: decision support, long-term forecasting, data mining, fuzzy inductive reasoning, solar activity, mean solar magnetic field, magnetism.

^{1}Kuban State University, Krasnodar, 350040, Russia

^{2}Construction Bureau “Selena”, Krasnodar, 350072, Russia

Corresponding author’s e-mail: gusev@ftf.kubsu.ru

- Gusev A.A., Shvetsova N.A., Voloshin A.E., Yakovenko N.A. Izuchenie primenimosti instrumentalnogo sredstva nechetkogo induktivnogo rassuzhdeniya k zadache prognozirovaniya solnechnoj aktivnosti [The study of the applicability of the tool for fuzzy inductive reasoning to the problem of solar activity prediction].
*Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of The Black Sea Economic Cooperaition], 2016, no. 4, pp. 35-38. (In Russian) - Gusev A.A., Shvetsova N.A.
*Program for management decision-making “AimDSS”. Certificate of State Registration for Computer Program no. 2016610899 Russian Federation*. (In Russian) - Strizhak U.V., Poniavin D.I. Istochniki magnitnogo polya Solnca kak zvezdy [The sources of the magnetic field of the Sun as a star].
*Physics of Auroral Phenomena*, 2016, vol. 39, no. 1(40), pp. 84-86. (In Russian) - WSO - The Wilcox Solar Observatory. URL: http://wso.stanford.edu. (access date 31.05.2017)

*Subjects*: mechanics

**Evdokimova O.V.**^{1}, **Babeshko O.M.**^{2}, **Babeshko V.A.**^{1}

**Composition of the packed block elements into the block structure and their homeomorphisms**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 32-35.

Currently, the problem of building block elements of boundary problems for systems of differential equations with constant coefficients has been solved for a fairly wide range of domains. For the formation of block structures consisting of block elements with individual physical and mechanical properties, the mechanism of their conjugation is used, which is, in terms of topology, the construction of quotient topologies of topological spaces of conjugated block elements. This mechanism is described using the example of packed block elements generated by the boundary value problem for systems of linear differential equations in partial derivatives viewed as topological objects. They can be considered as manifolds with boundaries in certain spaces representing Cartesian products of topological spaces. Thus, the packed block elements are conjugated to form block structures of varying complexity. This approach is based on the methods of exterior analysis, the section of the theory of block elements, which makes it possible to build solutions of boundary problems on given carriers. The paper discusses other approaches to the application of topological methods in boundary value problems. Clear goals, approaches and opportunities of different methods are described.

*Keywords*: block element, topology, boundary problems methods, exterior forms, block structures, coverings.

^{1}Southern Scientific Center RAS, Rostov-on-Don, 344006, Russia

^{2}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: babeshko41@mail.ru

- Babeshko V.A., Evdokimova O.V., Babeshko O.M. O stadiyakh preobrazovaniya blochnykh elementov [On the stages of transformation of block elements].
*Doklady Akademii nauk*[Rep. of the Academy of Sciences], 2016, vol. 468, no. 2, pp. 154-158. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Vneshniy analiz v probleme skrytykh defektov i prognoze zemletryaseniy [External analysis in the problem of hidden defects and the forecast of earthquakes].
*Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological bulletin of scientific centers of the Black Sea Economic Cooperation], 2016, no. 2, pp. 19-28. (In Russian) - Zorich V.A.
*Matematicheskiy analiz. Chast’ 2*[Mathematical analysis. Pt. 2]. Moscow, MTsNMO Pub., 2002, 788 p. (In Russian) - Kelli D.
*Obshchaya topologiya*[General Topology]. Moscow, Nauka Pub., 1968, 384 p. (In Russian) - Mishchenko A.S., Fomenko A.T.
*Kratkiy kurs differentsial’noy geometrii i topologii*[A short course of differential geometry and topology]. Moscow, Fizmatlit Pub., 2004, 302 p. (In Russian) - Golovanov N.N., Il’yutko D.P., Nosovskiy G.V., Fomenko A.T. Komp’yuternaya geometriya [Computer geometry]. Moscow, Akademiya Pub., 2006, 512 p. (In Russian)
- Hassani B., Hinton E. A review of homogenization and topology optimization
I - homogenization theory for media with periodic structure.
*Computers and Structure*, 1998, vol. 69, pp. 707-717. - Hassani B., Hinton E. A review of homogenization and topology optimization
II - homogenization theory for media with periodic structure.
*Computers and Structure*, 1998, vol. 69, pp. 719-738. - Hassani B., Hinton E. A review of homogenization and topology optimization
III - homogenization theory for media with periodic structure.
*Computers and Structure*, 1998, vol. 69, pp. 739-756. - Xin Z.Q., Wu C.J. Topology Optimization of the Caudal Fin of Thtree-Dimantional Self-Propelled Swimming Fish.
*Adv. Appl. Math. Mech.*, 2014, vol. 6, iss. 6, pp. 732-763. - Bendsoe M.P., Sigmund O.
*Topology Optimization - Theory, Methods and Applications*. Berlin, Springer, 2003. - Bonvall T., Petersson J. Topology optimization of fluids in stokes flow.
*Int. J. for Numerical Methods in Fluids*, vol. 42, pp. 77-107. - Wachspress E.L.
*A Rational Finite Element Basis*, New York, Academic Press, 1975. - El-Sabbage A., Baz A. Topology optimization of unconstrained damping treatments for plates. Engineering Optimization, 2013. Vol. 49. P. 1153-1168.
- Zheng W., Lei Y., Li S. et. al. Topology optimization of passive constrained layer damping with partial coverage on plate.
*J. Shock and Vibration*, 2013, vol. 20, pp. 199-211. - Van der Veen G., Langelaar M., van Keulen F. Integrated topology and controller optimization of motion systems in the frequency domain.
*Structural and Multidisciplinary Optimization*, 2014, vol. 51, pp. 673-685. - Dahl J.J., Jensen S., Sigmund O. Topology optimization for transient wave propagation problems in one dimension.
*Structural and Multidisciplinary Optimization*, 2008, vol. 36, pp. 585-595.

*Subjects*: mechanics

**Zaretskaya M.V.**^{1}, **Babeshko O.M.**^{1}, **Zaretskiy A.G.**^{1}, **Lozovoy V.V.**^{2}

**On blocks of various types in problems of geoecology**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 36-41.

Developers should consider the model of the environment as close to natural as possible, apply a mathematical apparatus that adequately and reliably describes the processes and phenomena occurring in the environment under study, while constructing mathematical models of modern geophysical environment monitoring systems, environmental quality management and environmental management.

In this paper, we proposed a method based on a topological approach that allows one to set and solve boundary problems on the basis of equations of motion for media characterized by essentially different mechanical, chemical, rheological characteristics in different coordinate systems.

An algorithm of the differential factorization method for investigating processes in a block-structured medium has been developed, the individual blocks of which are formed by spherical boundaries (in spherical coordinates) and cylindrical boundaries (in cylindrical coordinates).

The developed methods make it possible to study a wide class of convective currents that arise in the atmosphere (modeling tornadoes), seas and oceans (cyclonic currents of various scales), geophysics; promptly assess the level of technogenic seismicity, which will reduce the seismogenic impact of modern industrial production and minimize the level of induced seismicity.

*Keywords*: medium, complex internal structure, topological approach,
cylindrical block element, ball block element.

^{1}Kuban State University, Krasnodar, 350040, Russia

^{2}Southern Scientific Center RAS, Rostov-on-Don, 344006, Russia

Corresponding author’s e-mail: zarmv@mail.ru

- Babeshko V.A., Evdokimova O.V., Babeshko O.M., Zaretskaya M.V., Pavlova A.V. The differential factorization method for a block structure.
*Doklady Physics*, 2009, vol. 54, iss. 1, pp. 25-28. - Babeshko V.A., Zareckaya M.V., Ryadchikov I.V. K voprosu modelirovaniya processov perenosa v ehkologii, sejsmologii i ih prilozheniya [To the problem of modeling transport processes in ecology, seismology and their applications].
*Jekologicheskij vestnik nauchnyh centrov Chernomorskogo jekonomicheskogo sotrudnichestva*[Ecological bulletin of scientific centers of the Black Sea Economic Cooperation], 2008, no. 3, pp. 20-25. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Ob osobennostyah metoda blochnogo ehlementa v nestacionarnyh zadachah [On the features of the block element method in nonstationary problems].
*Doklady Akademii nauk*[Rep. of the Academy of Sciences], 2011, vol. 438, no. 4, pp. 470-474. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. O topologicheskih strukturah granichnyh zadach v blochnyh ehlementah [On topological structures of boundary value problems in block elements].
*Doklady Akademii nauk*[Rep. of the Academy of Sciences], 2016, vol. 470, no. 6, pp. 650-654. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Ob avtomorfizme i psevdodifferencial’nyh uravneniyah v metode blochnogo elementa [On automorphism and pseudodifferential equations in the block method Element].
*Doklady akademii nauk*[Rep. of the Academy of Sciences], 2011, vol. 438, no. 5, pp. 623-625. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Topologicheskij metod resheniya granichnyh zadach i blochnye ehlementy [Topological method for solving boundary value problems and block elements].
*Doklady akademii nauk*[Rep. of the Academy of Sciences], 2013, vol. 449, no. 6, pp. 657-660. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M., Gorshkova E.M., Zareckaya M.V., Muhin A.S., Pavlova A.V. O konvergentnyh svojstvah blochnyh ehlementov [Convergence properties of block elements].
*Doklady Akademii nauk*[Rep. of the Academy of Sciences], 2015, vol. 465, no. 3, pp. 298-301. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Blochnye ehlementy s cilindricheskoj granicej v makro- i nanostrukturah [Block elements with a cylindrical boundary in macro- and nanostructures].
*Doklady akademii nauk*[Rep. of the Academy of Sciences], 2011, vol. 440, no. 6, pp. 756-759. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. The theory of the starting earthquake.
*Jekologicheskij vestnik nauchnyh centrov Chernomorskogo jekonomicheskogo sotrudnichestva*[Ecological bulletin of scientific centers of the Black Sea Economic Cooperation], 2016, no. 1, iss. 2, pp. 37-80. (In Russian)

*Subjects*: geophysics

**Kochergin V.S.**^{1}, **Kochergin S.V.**^{1}

**The use of adjoint problems solution in the identi cation of input parameters for transport models and planning of experiments**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 42-47.

In the numerical identification input model transport parameters of passive admixture in the measurement data raises the question of constructing the optimal plans for performing the measurement. From a mathematical point of view, the solution of inverse problems, the optimal plan is treated as a net of measurement points distributed across space and time that gives the best conditioning of identification input parameters of numerical simulations against measurements. Computational properties of algorithms for solving inverse problems can largely be improved by optimal schemes of measurements. At the core of the algorithms for optimal planning is the creation and study of the properties of Fisher information matrix, which is based on the Jacobian matrix characterizing the admixture concentration field variation dependence of variations parameters of the problem. For example, the one-dimensional problem of the transfer of passive admixtures discusses the assessment component of this matrix by a series of adjoint tasks of a special kind. As a result of numerical experiments with the one-dimensional model of transport of passive tracer shows that the best convergence of the variational algorithm for identifying the initial field of concentration is in the selection of measurement points in the region of maximum values. This scheme of measuring net leads to improved conditioning of the problem being solved. Additional information about the location of the boundaries of the pollution are also important in the algorithm initialization the initial distribution. Similar calculations are carried out with a three-dimensional model for the Azov sea. The results can be used for solving various ecological orientation problems in the study of the sources pollution influence of anthropogenic nature in the Azov and Black seas waters.

*Keywords*: planning of the experiment, the model of transport of passive admixture, identification of the related task, minimization, Azov sea.

^{1}Marine Hydrophysical Institute, Sevastopol, 299011, Russia

Corresponding author’s e-mail: vskocher@gmail.com

- Gorskij V.G.
*Planirovanie kineticheskikh eksperimentov*[Planning kinetic experiments]. Moscow, Nauka Pub., 1984, 240 p. - Penenko V.V. Otsenka parametrov diskretnykh modeley dinamiki atmosfery i okeana [Estimation of parameters of discrete models of atmospheric and ocean dynamics].
*Meteorologiya i gidrologiya*[Meteorology and Hydrology], 1979, no. 7, pp. 77-90. - Marchuk G.I.
*Matematicheskoe modelirovanie v probleme okruzhayushchey sredy*[Mathematical modeling in the environmental problem]. Moscow, Nauka Pub., 1982, 320 p. - Kochergin V.S. Opredelenie polya kontsentratsii passivnoy primesi po nachal’nym dannym na osnove resheniya sopryazhennykh zadach [Determination of the passive admixture concentration field from the initial data on the basis of the solution of the conjugate problems].
*Ekologicheskaya bezopasnost’ pribrezhnoy i shel’fovoy zon i kompleksnoe ispol’zovanie resursov shel’fa*[Ecological safety of coastal and shelf zones and integrated use of shelf resources], MGI NANU, Sevastopol’ 2011, iss. 25, vol. 2, pp. 270-376. - Ivanov V.A., Fomin V.V.
*Matematicheskoe modelirovanie dinamicheskikh protsessov v zone more - susha*[Mathematical modeling of dynamic processes in the sea-land zone]. Sevastopol’, EHKOSI-gidrofizika, 2008, 363 p. - Eremeev V.N., Kochergin V.P., Kochergin S.V., Sklyar S.N.
*Matematicheskoe modelirovanie gidrodinamiki glubokovdnykh basseynov*[Mathematical modeling of hydrodynamics of deep basins]. Sevastopol’: EHKOSI-Gidrofizika, 2002, 238 p. - Kochergin V.S., Kochergin S.V. Realizatsiya variatsionnogo podkhoda pri identifikatsii vkhodnykh parametrov modeli perenosa passivnoy primesi v Azovskom more [Implementation of the variational approach in identifying the input parameters of the passive admixture transport model in the Azov Sea].
*Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological bulletin of scientific centers of the Black Sea Economic Cooperation], 2016, no. 3, pp. 50-58.

*Subjects*: physics

**Lavrov I.V.**^{1}, **Bardushkin V.V.**^{1}, **Sychev A.P.**^{2}, **Yakovlev V.B.**^{1}, **Kirillov D.A.**^{1}

**On calculation of the e ective thermal conductivity of textured tribocomposites**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 48-56.

The common operator expression for a tensor of effective thermal conductivity ${\rm {\bf k}}^{\ast }$ of the inhomogeneous textured material is derived. Assuming the inhomogeneous material to consist of ellipsoidal grains let us approximate the integral operator by the constant tensor ${\rm {\bf g}}$ related with the concrete inclusion and thus obtain the generalized singular approximation for ${\rm {\bf k}}^{\ast }$ on the base of the common operator expression. It is shown that in case of coincidence of axes of ellipsoidal inclusion with principal axes of a tensor of thermal conductivity of the comparison medium the components of a tensor ${\rm {\bf g}}$ may be expressed through components of a tensor of the generalized geometrical factors of the ellipsoid placed in the anisotropic external medium.

The received generalized singular approximation is applied to calculation of a tensor ${\rm {\bf k}}^{\ast }$ of the multicomponent textured matrix composite with uniformly oriented inclusions. For a special case of the generalized singular approximation - a self-consistent approximation - the system of equations for finding of the main components of a tensor ${\rm {\bf k}}^{\ast }$ of this composite is derived. On the basis of the received system of equations numerical simulation of thermal conducting characteristics of the textured tribocomposite consisting of three components is made: epoxy ED-20 system as a matrix, polytetrafluoroethylene inclusions of spherical shape as an antifriction component and the prolate spheroidal glass inclusions as the reinforcing component. Dependences of the principal components of effective thermal conductivity tensor of this tribocomposite on volume fractions of reinforcing inclusions are given. It is shown what this tribocomposite has anisotropy of thermal-conducting properties, despite isotropic material characteristics of each components. It is also shown that values of the principal components of effective thermal conductivity tensor are less than volume average value of a thermal conductivity.

*Keywords*: tensor of effective thermal conductivity, texture, composite, tribocomposite, multicomponent, generalized singular approximation, matrix, ellipsoidal inclusion, self-consistent approximation.

^{1}National Research University of Electronic Technology, Moscow, Russia

^{2}Southern Scientific Center of RAS, Rostov-on-Don, Russia

Corresponding author’s e-mail: iglavr@mail.ru

- Kolesnikov V.I.
*Teplofizicheskie protsessy v metallopolimernykh tribosistemakh*[Thermophysical processes in metal-polymeric tribosystems]. Moscow, Nauka Publ., 2003, 279 p. (In Russian) - Garnett J.C.M. Colours in metal glasses and in metallic films.
*Phil. Trans. R. Soc.*, London, 1904, vol. 203, pp. 385-420. - Bruggeman D.A.G. Berechnung verschiedener physikalisher Konstanten von heterogenen Substanzen.
*Ann. Phys.*, Lpz., 1935, b. 24, pp. 636-679. (In German) - Zarubin V.S., Kuvyrkin G.N., Savel’eva I.Yu. Effektivnaya teploprovodnost’ kompozita v sluchae otkloneniy formy vklyucheniy ot sharovoy [Effective thermal conductivity of a composite in case of inclusions shape deviations from spherical ones].
*Matematicheskoe modelirovanie i chislennye metody*[Mathematical modeling and numerical methods], 2014, no. 4, pp. 3-17. (In Russian) - Bragg W.L., Pippard A.B. The Form Birefringence of Macromolecules.
*Acta Cryst*., 1953, vol. 6, no. 11-12, pp. 865-867. - Progelhof R.C., Throne J.L., Ruetsch R.R. Methods for Predicting the Thermal Conductivity of Composite Systems: A Review.
*Polymer Engineering and Science*, 1976, vol. 76, no. 9, pp. 615-625. - Pietrak K., Wisniewski T.S. A review of models for effective thermal conductivity of composite materials.
*J. of Power Technologies*, 2015, vol. 95, no. 1, pp. 14-24. - Fokin A.G. Dielektricheskaya pronitsaemost’ smesey [Dielectric Permittivity of Mixtures].
*Zhurnal tekhnicheskoy fiziki*[Technical Physics. The Russian Journal of Applied Physics], 1971, vol. 41, no. 6, pp. 1073-1079. (In Russian) - Shermergor T.D.
*Teoriya uprugosti mikroneodnorodnykh sred*[Micromechanics of inhomogeneous medium]. Moscow, Nauka Publ., 1977, 399 p. (In Russian) - Kolesnikov V.I., Yakovlev V.B., Bardushkin V.V., Lavrov I.V., Sychev A.P., Yakovleva E.N. Association of evaluation methods of the effective permittivity of heterogeneous media on the basis of a generalized singular approximation.
*Doklady Physics*, 2013, vol. 58, no. 9, pp. 379-383. doi: 10.1134/S1028335813090012 - Kolesnikov V.I., Yakovlev V.B., Bardushkin V.V., Lavrov I.V., Sychev A.P., Yakovleva E.N. A Method of Analysis of Distributions of Local Electric Fields in Composites.
*Doklady Physics*, 2016, vol. 61, no. 3, pp. 124-128. doi: 10.1134/S1028335816030101 - Gel’fand I.M., Shilov G.E.
*Obobshchennye funktsii i deystviya nad nimi*[Generalized functions. Properties and Operations]. Moscow, GIFML Publ., 1958, 440 p. (In Russian) - Lavrov I.V. Proizvol’no orientirovannyy dielektricheskiy ellipsoid v anizotropnoy srede: metod neortogonal’nogo preobrazovaniya prostranstva [An arbitrarily oriented dielectric ellipsoid in an anisotropic medium: the non-orthogonal space transformation method].
*Fundamental’nye problemy radioelektronnogo priborostroeniya*[Fundamental problems of radioengineering and device construction], 2013, vol. 13, no. 1, pp. 44-47. (In Russian) - Grigor’ev I.S., Meilikhov E.Z. (eds.)
*Fizicheskie velichiny: Spravochnik*[Physical Quantities: A Handbook]. Moscow, Energoatomizdat Publ., 1991, 1232 p. (In Russian) - Wiener O. Die Theorie des Mischkörpers für das Feld der stationären Strömung.
*Abh.-Sachs. Geselsch.*, 1912, b. 32, ss. 509-604. (In German)

*Subjects*: mathematics

**Osipyan V.O.**^{1}, **Leyman A.V.**^{1}, **Chesebiev A.A.**^{1}, **Zhuk A.S.**^{1}, **Harutyunyan A.H.**^{1}, **Karpenko Y.A.**^{2}

**On calculation of the e ective thermal conductivity of textured tribocomposites**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 57-64.

It investigates the development of mathematical models of alphabet cryptosystems based on the tasks a non-standard multiplicative knapsacks. Mathematical models of the cryptosystems in the form of tuples. Establishes necessary and sufficient conditions under which the generalized multiplicative injective knapsack vector over $Z_p$, $p\geqslant 2$. Developed mathematical model of the cryptosystem by overlapping scales, in which the algorithm of the inverse transformation of the closed text is reduced to an algorithmically non-solvable problem for the analyst. On the basis of the analysis previously offered a different backpack models are revealed qualitative features of non-standard multiplicative knapsack systems that increase their resistance to known attacks.

We also study the problem of constructing isomorphic additive and multiplication knapsacks. Moreover, in contrast to the standard knapsack-teams, in which when determining the entrance of a knapsack or other components of the knapsack vector are either present or not, and here we consider the case when they can be repeated a specified number of times for a given array for both generic and super generic multiplicative knapsack.

*Keywords*: alphabetic cryptosystem, mathematical model of cryptosystems, symmetric and asymmetric knapsack system of information protection, non-additive (multiplicative) knapsack, generalized (generalized super) multiplicative knapsack.

^{1}Kuban State University, Krasnodar, 350040, Russia

^{2}Adyghe State University, Krasnodar, 385016, Russia

Corresponding author’s e-mail: rrwo@mail.ru

- Shannon C. Communication theory of secrecy systems.
*Bell System Techn. J.*, 1949, vol. 28, no. 4, pp. 656-715. - Merkle R., Hellman M. Hiding information and signatures in trapdoor knapsacks.
*IEEE Transactions on Information Theory*, 1978, vol. IT-24, pp. 525-530. - Rivest R.L., Chor B. A knapsack-type public key cryptosystem based on arithmetic in finite fields.
*IEEE Transactions on Information Theory*, 1988, vol. 34, no. 5, pp. 901-909. - Shamir A. A polynomial-time algorithm for breaking the basic Merkle-Hellman cryptosystem. Information Theory, IEEE Transactions. 1984. Vol. 30. No. 5. pp. 699-704.
- Koblitz N.
*A Course in Number Theory and Cryptography*. Springer-Verlag, New York, 1987. - Osipyan V.O. Ob odnom obobshchenii rukzachnoi kriptosistemi [On a generalization of a knapsack cryptosystem].
*Izv. vuzov Sev.-Kavk. reg*. [Bulletin of the Universities of the Nord Caucasus region], 2003, no. 5, pp. 18-25. - Osipyan V.O. O sisteme zashchity informatcii na osnove funktcional’nogo rukzaka [On a security system based on functional knapsack].
*Voprosy zashchity informatcii*[Question of information’s security], 2004, no. 4, pp. 16-18. - Osipyan V.O. O sisteme zashchity informatcii na osnove problemy rukzaka [On a security system based on knapsack’s problem].
*Izvestiya Tomskogo Politekhnicheskogo universiteta*[Bulletin of Tomsk politechnical University], 2006, vol. 309, no 2, pp. 209-212. - Osipyan V.O., Harutunyan A.S., Spirina C.G. Modelirovanie rantcevyh kriptosistem, sodergashchikh diophantovuyu trudnost’ [Modelling of knapsack cryptosystem Diophantine difficulty contains].
*Chebyshevskii sbornik*[Chebyshev’s digest], 2010, vol. XI, no. 1, pp. 209-217. - Osipyan V.O., Kerpenko Y.A., Zhuk A.S., Harutunyan A.H. Diofantovy trudnosti atak na nestandartnye rukzachnye sistemy zashchity informatcii [Diophantine difficulties of attacks on non-standard knapsack security systems].
*Izvestiya UFU. Tekhnicheskie nauki*[Bulletin of South Federal University. Technical sciences]. 2013, no. 12, pp. 209-215. - Osipyan V. O. Information protection systems based on universal knapsack problem.
*SIN’13: Proc. of the 6th International Conference on Security of Information and Networks, ACM, 2013*, pp. 343-346. - Lenstra, Jr. H.W. Integer Programming with a Fixed Number of Variables.
*Mathematics of Operations Research*, 1983, vol. 8, no. 4, pp. 538-548. - Vaudenay S. Cryptanalysis of the Chor-Rivest cryptosystem.
*Advances in Cryptology - CRYPTO ‘98: Proc. of 18th Annual International Cryptology Conference Santa Barbara, California, USA August 23–27, 1998*, pp. 243-256. DOI: 10.1007/BFb0055732

*Subjects*: mechanics

**Rubtsov S.E. ^{1}, **Pavlova A.V.

^{1},

**Telyatnikov I.S.**

^{1}

**To the problem of vibrations of a limited volume of a liquid on an elastic foundation**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 65-73.

The paper presents a model of a hydro engineering structure under the influence of vibration loads, taking into account its interaction with an elastic foundation modeled by an isotropic layer or a half-space. The mathematical description of this model reduces to the formulation of a boundary-value problem for a system of partial differential equations, the solution of which leads to integral relations with respect to the unknown function of stresses, distributed over the area of contact between the surface object and the underlying foundation. Contact stresses are determined from the solution of the integral equation, the feature of which is the dependence of its kernel not only on the difference of the arguments, but also on their sum.

Formulas for calculating the hydrodynamic pressure in the contact zone of the liquid and elastic media are obtained, as well as the velocity potential in the fluid and the displacements of the elastic foundation points. Analysis of the results of computational experiments made it possible to determine the dependencies of stresses in the media contact zone from the physical, geometric and frequency parameters of the problems under consideration for two models of an elastic foundation: an elastic half-space and an elastic layer with a fixed lower bound.

The novelty of the conducted studies consists in analyzing the dynamic behavior of hydroelastic media based on solving integral equations, rather than using the average characteristics of contact stresses. Such approach will allow increasing the accuracy of the description of real processes and reliability of assessments of possible consequences of vibroseismic effects, which is especially important in regions where artificial and natural reservoirs and dams are near populated areas.

*Keywords*: vibration loads, limited pool of liquid, elastic foundation, hydrodynamic pressure distribution.

^{1}Kuban State University, Krasnodar, 350040, Russia

^{2}Southern Scientific Center of Russian Academy of Sciences, Rostov-on-Don, 344006, Russia

Corresponding author’s e-mail: kmm@fpm.kubsu.ru

- Vajnberg A.I.
*Nadezhnost’ i bezopasnost’ gidro-tekhnicheskih sooruzhenij. Izbrannye problemy*[Reliability and safety of hydraulic structures. Selected Problems], Har’kov, Tyazhpromavtomatika Pub., 2008. 304 p. (In Russian) - Bellendir E.N., Ivashincov D.A, Stefanishin D.V., Finagenov O.M., Shul’man S.G.
*Veroyatnostnye metody ocenki nadezhnosti gruntovyh gidrotekhnicheskih sooruzhenij*[Probabilistic methods for assessing the reliability of groundwater hydraulic structures], Saint-Peretsburg, Izd-vo VNIIG im. B.E. Vedeneeva. T. 1. 2003. 553 p., T. 2. 2004. 524 p. (In Russian) - Sejmov V.M., Ostroverh B.N., Ermolenko A.I.
*Dinamika i sejsmostojkost’ gidrotehnicheskih sooruzhenij*[The dynamics and earthquake resistance of hydraulic structures]. Kiev, Nauk. dumka, 1983. 320 p. (In Russian) - Ostroverh B.N.
*Chislennaya realizaciya sejsmicheskogo vozdejstviya na gidrosooruzheniya*[Numerical realization of the seismic effect on hydraulic structures]. Sejsmostojkoe stroitel’stvo [Seismic resistant construction], 1976. Ser. XIV. Is. 7. pp. 34-37. (In Russian) - Lyahtera V.M., Ykovleva Y.S. (eds.)
*Dinamika sploshnyh sred v raschetah gidro-tekhnicheskih sooruzhenij*[Dynamics of continuous media in the calculation of hydraulic structures]. Moscow, Energiya, 1976. 392 p. - Korenev B.G. Impul’snye vozdejstviya na cilindricheskie rezervuary, napolnennye zhidkost’yu. V kn.:
*Stroitel’stvo v sejsmicheskih rajonah*[Impulse effects on cylindrical tanks filled with liquid]. Moscow, Strojizdat Pub., 1957. pp. 35-44. - Korenev B.G. Dejstvie impul’sa na cilindricheskie i prizmaticheskie rezervuary, napolnennye zhidkost’ju. V kn.:
*Stroitel’naja mehanika*. [Pulse action on the cylindrical and prismatic tanks filled with liquid. In book: Structural mechanics]. Moscow, Strojizdat Pub., 1966. pp. 213-266. (In Russian) - Klimov M.A. Opredelenie prisoedinennoj massy zhidkosti v sluchae neosesimmetrichnyh kolebanij dnishh rezervuarov. V kn.:
*Dinamicheskie naprjazhenija i deformacii v jelementah jenergeticheskogo oborudovanija*[Defining the associated mass of liquid in the case of axisymmetric vibrations of tank bottoms. Proc: Dynamic stresses and strains in power equipment elements]. Moscow, Nauka Pub., 1977. pp. 76-83. (In Russian) - Trofimchuk A.N. Vliyanie zhidkosti na osnovnuyu chastotu kolebanij bassejna konechnoj zhestkosti na uprugoj poluploskosti [Influence of a liquid on the fundamental frequency of oscillations of a finite stiffness pool on an elastic half-plane].
*Matematicheskie metody mekhaniki zhidkosti i gaza*[Mathematical methods of fluid and gas mechanics]. Dnepropetrovsk, DSU Pub., 1981. pp. 119-126. - Rubtsov S.E., Pavlova A.V., K issledovaniyu dinamicheskih smeshannyh zadach dlya ogranichennogo ob”ema zhidkosti na uprugom osnovanii [To study of the mixed dynamic problems for a limited volume of fluid on an elastic foundation].
*Jekologicheskij vestnik nauchnyh centrov Chernomorskogo jekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2016, no. 4, pp. 75-81. (In Russian). - Vorovich I.I., Aleksandrov V.M., Babeshko V.A.
*Neklassicheskie smeshannye zadachi teorii uprugosti*[Non-classical mixed problem of elasticity theory]. Moscow, Nauka Pub., 1974, 455 p. (In Russian). - Vorovich I.I., Babeshko V.A.
*Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskih oblastei*[Dynamic mixed problem of elasticity theory for nonclassical domains]. Moscow, Nauka Publ., 1979, 319 p. (In Russian) - Pavlova A.V., Rubtsov S.E., Telyatnikov I.S., Zaretskaja M.V. Issledovanie naprjazhennogo sostojanija sloistoj sredy s zhidkim vkljucheniem [Investigation of the stress state of layered medium with liquid inclusion].
*Jekologicheskij vestnik nauchnyh centrov Chernomorskogo jekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2016, no. 1, pp. 71-78. (In Russian) - Rubtsov S.E. Issledovanie ustanovivshihsja kolebanij ogranichennogo ob’ema zhidkosti na uprugom sloe [The study stationary vibrations limited volume of fluid in the elastic layer].
*Izvestija vysshih uchebnyh zavedenij, Severo-Kavkazskij region, estestvennye nauki*[Proc. of the Universities. North-Caucasian region. Natural Sciences], Rostov-on-Don, 2000. no 1. pp. 49-51. (In Russian)

*Subjects*: mechanics

**Sobol B.V.**^{1}, **Rashidova E.V.**^{1}

**Equilibrium state of the internal crack in in nite elastic wedge with the thin coating**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 74-85.

The research of stress concentration in the neighborhood of internal crack’s tops, which is on a bisector of an infinite elastic wedge, is conducted. The normal efforts are applied to crack’s coast for providing her opening. Wedge’s sides are supported with a thin flexible coating, free from tension from outer side. Coating influence on an intense and deformable state of a wedge is modelled by a special boundary condition which correctness is confirmed experimentally. Mellin’s integral transformation has allowed to reduce the task to the solution of the singular integrated equation of the first kind with Cauchy’s kernel of rather derivative function of crack opening. Solutions of the integral equation constructed by the collocation method. With various combinations of geometric and physical parameters of the problem. The aim of the research was to determine the values of the influence factor – a reduced stress intensity factor in the neighborhood of the crack’s tops. Analysis of influences of task’s geometrical and physical parameters on size of the studied parameter is carried. In particular, it is established that with increasing the wedge’s angle, with unchanged other parameters, the values of the influence factor are increasing; the increasing in the thickness and hardness of the coating leads to a decrease of the influence factor; the increase of the relative crack’s length or approaching to wedge’s top implies an increase of influence factor. The known special cases of task are considered, their results are compared with available data published.

*Keywords*: crack, infinite elastic wedge, thin coating, Mellin’s integral transformation, collocation method, stress intensity factor, influence factor.

^{1}Don State Technical University, Rostov-on-Don, 344000, Russia

Corresponding author’s e-mail: b.sobol@mail.ru

- Melan E. Zur plastizität des räumlichen kontinuums.
*Archive of Applied Mechanics*, 1938, vol. 9, no. 2, pp. 116-126. - Rejssner En.
*Nekotorye problemy teorii obolochek. Uprugie obolochki*[On some problems in shell theory. Elastic casing]. Moscow, Inostr. lit. Publ., 1962, 263 p. (In Russian) - Koiter W.T., Warner T. On the nonlinear theory of thin elastic shells.
*Koninklijke Nederlandse Akademie van Wetenschappen*, 1966, vol. 69, no. 1, pp. 1-54. *Razvitie teorii kontaktnyh zadach v SSSR*[The development of the theory of contact problems in the USSR]. Moscow, Nauka Publ., 1976, 493 p. (In Russian)- Aleksandrov V.M., Mkhitaryan S.M.
*Kontaktnye zadachi dlya tel s tonkimi pokrytiyami i proslojkami*[Contact problems for bodies with thin coatings and interlayers]. Moscow, Nauka Publ., 1976, 493 p. (In Russian) - Akopyan V.N. Ob odnoj smeshannoj zadache dlja sostavnoj ploskosti,
oslablennoj treshhinoj [On a mixed problem for a compound plane weakened by
a crack].
*Izv. NAN Armenii, Mehanika*[Mech. Proc. National Acad. Sci Armenia], 1995, vol. 48, no. 4, pp. 57-65. (In Russian) - Arutyunyan L.A. Ploskie zadachi so smeshannymi kraevymi usloviyami dlya
sostavnoj ploskosti s treshhinami [Plane problems with mixed boundary
conditions for a composite plane with cracks].
*Izv. NAN Armenii, Mekhanika*[Mech. Proc. National Acad. Sci Armenia], 2012, vol. 65, no. 3, pp. 5-9. (In Russian) - Rizk A. Stress intensity factor for an edge crack in two bonded
dissimilar materials under convective cooling.
*Theoretical and Applied Fracture Mechanics*, 2008, vol. 49, no. 3, pp. 251-267. doi: 10.1016/j.tafmec.2008.02.006 - Shatskij I.P. Rastyazhenie plastiny, soderzhashhej pryamolinejnyj razrez
s sharnirno soedinennymi kromkami [Stretching of a plate containing a
rectilinear cut with pivotally connected edges].
*Prikl. mehanika i tehnicheskaja fizika*[J. Appl. Mechanics and Technical Physics], 1989, no. 5, pp. 163-165. (In Russian) - Antipov Y., Bardzokas D., Exadaktylos G. Partially stiffened elastic
half-plane with an edge crack.
*International journal of fracture*, 1997, vol. 85, no. 3 pp. 241-263. doi: 10.1023/A:1007428813410 - Cook T.S., Erdogan F. Stress in bounded material with a crack
perpendicular to the interface.
*Int. J. Engng Sci.*, 1972, vol. 10, pp. 677-697. doi: 10.1016/0020-7225(72)90063-8 - Panasyuk V.V., Savruk M.P., Dacyshin A.P.
*Raspredelenie napryazhenij okolo treshhin v plastinakh i obolochkakh*[Distribution of stresses near cracks in plates and shells]. Kiev, Naukova Dumka Publ., 1976, 443 p. (In Russian) - Grekov M.A., Dal’ Yu. M., Kurochkin V.A. Predel’noe sostoyanie uprugoj
polosy s vnutrennej treshhinoj [Limit state of an elastic strip with an
internal crack].
*Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela*[J. Russ. Acad. Sci., Mechanics of Solids], 1992, no. 6, pp. 148-155. (In Russian) - Krasnoshhekov A.A., Sobol’ B.V. Ravnovesnoe sostoyanie vnutrennej
poperechnoj treshhiny v polubeskonechnom uprugom tele s tonkim pokrytiem
[Equilibrium of an Internal Transverse Crack in a Semiinfinite Elastic Body
with Thin Coating].
*Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela*[J. Russ. Acad. Sci., Mechanics of Solids], 2016, no. 1, pp. 136-150. (In Russian) *Spravochnik po koehffitsientam intensivnosti napryazhenij: V 2-kh tomakh, T. 1*[Handbook on stress intensity factors: In 2 volumes, vol. 1], Moscow, Mir Publ., 1990, vol. 1, 448 p. (In Russian)- Wigglesworth L.A. Stress distribution in a notched plate.
*Mathematika*, 1957, vol. 4, no. 7, pp. 76-96. doi: 10.1112/S002557930000111X - Irwin G.R. The crack-extension force for a crack at a free surface
boundary,
*NRL Rept*, 1958, no. 5120, PB 132875. - Bowie O.L. Rectangular tensile sheet with symmetric edge cracks.
*Trans. ASME. Ser. E. J. Appl. Mech.*, 1964, vol. 31, no. 2. pp. 208-212. doi:10.1115/1.3629588 - Bowie O.L., Neal D.M. Single edge cracks in rectangular tensile sheet.
*Trans. ASME. Ser. E. J. Appl. Mech.*, 1965, vol. 32, no. 3, pp. 708-709. doi:10.1115/1.3627290 - Srivastav R.P., Narain Prem. Certain two-dimensional problems of stress
distribution in wedge-shaped elastic solids under discontinuous load.
*Proc. Cambridge Phil. Soc.*, 1965, vol. 61, no. 4, pp. 945-954. doi: 10.1017/S0305004100039347 - Srawley J.E., Gross B. Stress intensity factors for crackline-loaded
edge-crack specimens.
*Mater. Res. and Stand.*, 1967, vol. 7, no. 4, pp. 155-162. - Smetanin B.I. Nekotorye zadachi o shhelyakh v uprugom kline i sloe
[Certain Problems of Cracks in Elastic Wedge and Plate],
*Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela*[J. Acad. Sci. USSR, Mechanics of Solids], 1968, no. 2, pp. 115-122. (In Russian) - Smetanin B.I. Ob odnoj smeshannoj zadache teorii uprugosti dlya klina
[On a Mixed Problem of Elasticity Theory for a Wedge]
*Prikl. Mat. Mekh.*[J. Appl. Math. Mech], 1968, vol. 32, no 4, pp. 708-714. (In Russian) - Tamate I., Kondo O. Stress singularities around a crack in an elastic
wedge.
*Trans. Jap. Soc. Mech. Eng*., 1978, vol. 44, no. 379, pp. 756-761. doi: 10.1299/kikai1938.44.756 - Aleksandrov V.M., Smetanin B.I., Sobol’ B.V.
*Tonkie kontsentratory napryazhenij v uprugikh telakh*[Thin stress concentrators in elastic bodies]. Moscow, FIZMATLIT Publ., 1993, 224 p. (In Russian) - Bantsuri R.D. Reshenie pervoj osnovnoj zadachi teorii uprugosti dlya
klina, imeyushhego konechnyj razrez [Solution of the first fundamental
problem of the theory of elasticity for a wedge having a finite cut].
*Dokl. AN SSSR*[DAN SSSR], 1966, vol. 167, no. 6, pp. 1256-1259. - Doran H.E. The wedge with a symmetric crack at the vertex in plane
elastostatics.
*IMA J. Appl. Math.*, 1969, vol. 5, no. 4, pp. 363-372. doi: 10.1093/imamat/5.4.363 - Khrapkov A.A. Beskonechnyj treugol’nyj klin s nadrezom na bissektrise
pod dejstviem sosredotochennykh sil, prilozhennykh k beregam nadreza [An
endless triangular wedge with a notch on the bisector under the action of
concentrated forces applied to the banks of the incision
*Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela*[J. Acad. Sci USSR, Mechanics of Solids], 1972, no. 5, pp. 88-97. (In Russian) - Quchterlony F. Symmetric cracking of a wedge by concentrated load.
*Int. J. Engng Sci.*, 1977, vol. 15, no. 2, pp. 109-116. doi: 10.1016/0020-7225(77)90026-X - Quchterlony F. Some stress intensity factors for self-similar cracks,
derived from path-independent integrals.
*J. Elast.*, 1978, vol. 8, no. 3, pp. 259-271. doi: 10.1007/BF00130465 - Gregory R.D. The edge-cracked circular disc under symmetric pin-loading
*Math. Proc. Cambridge Phil. Soc.*, 1979, vol. 85, no. 3, pp. 523-538. doi: 10.1017/S030500410005595X - Stone S.F., Westmann R.A. Stress intensity factors for cracked wedges.
*Int. J. Solids and Struct.*, 1981, vol. 17, no. 3, pp. 345-358. \newline doi: 10.1016/0020-7683(81)90068-8 - Sadykhov A. Eh. Klin s treshhinoj [Cracked Wedge].
*Azerb. Gos. Ped. Inst., Baku Dep. VINITI AN SSSR 28.03.79*[Baku State Ped. University Azerbaijan], 1979, no. 1091, 32 p. (in Russian). - Kipnis L.A. Uprugoe ravnovesie klina s treshhinoj [Elastic Equilibrium
of a Wedge with a Crack].
*Prikl. Mat. Mekh.*[J. Appl. Math. Mech], 1979, vol. 43, no. 1, pp. 153-159. (In Russian) - Sadykhov A. Eh. Ob odnoj zadache teorii uprugosti dlya klina s
polubeskonechnoj treshhinoj pod dejstviem sosredotochennogo momenta
[Elasticity-Theory Problem for Wedge with Semiinfinite Crack Subjected to
Point Moment].
*Prikl. Mekh.*[Int. Appl.Mech.], 1980, vol. 16, no. 5, pp. 91-96. (In Russian) - Savruk M.P.
*Mekhanika razrusheniya i prochnost’ materialov. T. 2: Koehffitsienty intensivnosti napryazhenij v telakh s treshhinami*[Mechanics of failure and strength of materials. Vol. 2: Coefficients of stress intensity in bodies with cracks]. Kiev, Naukova Dumka Publ., 1988, 619 p. (In Russian) - Nekislykh E.M., Ostrik V.I. Zadacha ob uprugom ravnovesii klina s
treshhinami na osi simmetrii [The problem of elastic equilibrium of a wedge
with cracks on the axis of symmetry].
*Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela*[J. Russian Academy Sci., Mechanics of Solids], 2010, no. 5, pp. 111-129. (In Russian) - Pozharskij D.A., Molchanov A.A. Asimptoticheskie resheniya smeshannykh
zadach dlya uprugoj polosy i klina [Asymptotic solutions of mixed problems
for an elastic strip and a wedge].
*Vestnik DGTU*[Vestnik Don State Technical University], 2010, vol. 10, no. 4(47), pp. 447-454. (In Russian) - Elliotis M.C., Charmpis D.C., Georgiou G.C. The singular function
boundary integral method for an elastic plane stress wedge beam problem with
a point boundary singularity.
*Applied Mathematics and Computation*, 2014, vol. 248, pp. 93-100. doi: 10.1016/j.amc.2014.09.090 - Lur’e A.I.
*Teoriya uprugosti*[Theory of elasticity]. Moscow, Nauka Publ., 1970, 940 p. (In Russian) - Gradshtejn I.S., Ryzhik I.M.
*Tablitsy integralov, summ, ryadov i proizvedenij*[Tables of integrals, sums, series and products]. Moscow, Fizmatgiz, 1963, 1100 p. (In Russian) - Irwin G.R. Analysis of stresses and strains near the end of a crack traversing a plate.
*Journal of Applied Mechanics*, 1957, vol. 24, no. 3, pp. 361-364.

*Subjects*: mechanics

**Stogny G.A.**^{1}, **Stogny V.V.**^{1}

**Seismicity of greater caucasus on the position of the earth’s crust block divisibility**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2017, no. 2, pp. 84-96.

In the following article analysis of seismic-generating structures of Greater Caucasus’ megaanticlinorium based on its block divisibility is carried out. According to the issued problem on results of geological and geophysical materials’ complex interpretation there was designed composition scheme of consolidated Earth’s crust in geoblock system and its composing blocks of the higher order. Consolidated Greater Caucasus’ Earth’s crust is divided by sublongitudinal Laba-Batumsky and Derbent-Lenkoransky deep faults on three geoblocks: Eastern-Black Sea, Caspian-Black Sea and Southern-Caspian, that are characterized by the different distribution of earthquakes with $M\ge 5.5$. For Eastern-Black Sea and Southern-Caspian geoblocks earthquake magnitude is commonly less than 5.5, in Caspian-Black Sea geoblock there were registered many earthquakes with $M =5.5$ and more.

On result of investigations there was established that maximum earthquake’s magnitude of Greater Caucasus is associated with the area of consolidated crust’s seismic-generating block. Earthquake magnitude of Anapa and Sochi blocks which area is about 4 thousands sq. km is less than 6.0. Earthquakes with $M >6.0$ of Caspian-Black Sea geoblock are localized in blocks which area is about 100-300 thousands sq. km. Most of the earthquake epicenters are mainly concentrated in Svaneto-Alazansky and Bezhitinsky deep faults’ effective areas.

*Keywords*: Greater Caucasus, earthquake, magnitude, gravity field, geoblock, tectonic block, deep fault, Earth’s crust.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: stogny_ vv@mail.ru

- Krasniy L.I. Geobloki [Geoblocks].
*Geotectonika*[Geotectonics], 1967, no. 5, pp. 103-120. (In Russian) - Galetsky L.S., Shmidt A.A., Titov V.K., Kossovskaya V.A. Tektonika i metallogeniya Vostochno-Evropeyskoy platformy na osnove geoblokovogo stroeniya i aktivizatsii zemnoy kory [Tectonics and metallogeny of the East-European platform on the basis of a geoblock structure and activation of the Earth’s crust].
*Geologicheskiy zhurnal*[Geological journal], 1990, no. 2, pp. 49-56. (In Russian) - Oparin V.N., Kurlena M.V. Zonal’naya dezintegratsiya i ierarkhicheskiy ryad geoblokov [Zonal disintegration and hierarchical series of geoblocks].
*Mekhanika gornykh porod*[Mechanics of rocks], 1994, no. 2, pp. 14-26. (In Russian) - Sharov V.I. Tektonicheskiy protsess kak neravnovesniy protsess razrusheniya gorykh porod (k probleme smeny paradigmy v seysmologii) [Tectonic process as a nonequilibrium thermodynamic process of rocks destruction (to the problem of paradigm shift in seismology)].
*Fizika Zemli*[Physics of the Earth], 1992, no. 5, pp. 121-127. (In Russian) - Babeshko V.A., Evdokimova O.V., Babeshko O.M. Vneshniy analiz v probleme skrytykh defektov i prognoze zemletryaseniy [External analysis the problem of latent defects and the prediction of earthquakes].
*Ekologicheskiy vestnik nauchnykh tsentrov chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2016, no. 2, pp. 19-28. (In Russian) - Stogny G.A., Stogny V.V. Seysmotektonicheskaya model’ Verkhoyano-Kolymskoy orogennoy oblasti (Severo-vostok Rossii [Seismotectonic model of the Verkhhoyansk-Kolyma orogenic region (Northeast of Russia].
*Ekologicheskiy vestnik nauchnykh tsentrov chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2014, no. 4, pp. 79-87. (In Russian) - Soloviev Al.A., Gorshkov A.I., Soloviev Al.A. Primenenie dannykh po litosfernym magnitnym anomaliyam v zadache raspoznavaniya mest vozmozhnogo vozniknoveniya zemletryaseniy [Application of data on lithospheric magnetic field anomaly in problem of recognition of places possible occurrence of earthquake].
*Fizika Zemli*[Physics of the Earth], 2016, no. 6, pp. 21-27. (In Russian) - Vardapetyan A.N. Pozdnekaynozoyskaya tectonika plit Chernomorsko-Kaspiyskogo regiona [Late Cenozoic plate tectonics of the Black Sea-Caspian region].
*Okeanologiya*[Oceanology], 1979, vol. XIX, no. 6, pp. 1066-1075. (In Russian) - Kazmin V.G., Lobkovsky L.I., Pustovitenko B.G. Sovremennaya kinematika mikroplit v Chernomorsko-Yuzhno-Kaspiyskom regione [Modern kinematics microplates in the Black Sea-South Caspian region].
*Okeanologiya*[Oceanology], 2004, vol. 44, no. 4, pp. 600-610. (In Russian) - Nalivkin D.V. (ed.)
*Geologitcyeskaya karta Kavkaza. Masshtab 1:500000*[Geological map of the Caucasus. Scale 1:500000]. Moskva: NPO “Aerogeologiya”, 1978. - Stepanov P.P., Yakushevich M.A. (eds.)
*Gravimetricheskaya karta SSSR. Masshtab 1:2500000*[Gravimetric map of the USSR. Scale 1:2500000]. Moscow, VNIIGeofizika Pibl., 1978. - Godzikovskaya A.A.
*Katalog zemletryaseniy Kavkaza*[Catalog of earthquakes in the Caucasus]. http://zeus.wdcb.ru/wdcb/sep/caucasus/catrudat.html. (In Russian) - Buriyanov V.B., Soloviev V.D. Issledovanie gravitatsionnykh maksimumov severnogo i zapadnogo obramleniya Chernomorskoy vpadiny [Investigation of the gravity maxima of the northern and western regions of the Black Sea depression]. In
*Geofizicheskie parametry litosfery yudgnogo sektora al’piyskogo orogena*[Geophysical parameters of the lithosphere of the southern sector of the Alpine orogen]. Kiev, Naukova Dumka, 1996, pp. 111-119. (In Russian) - Malovichko A.A., Gabsatarova I.P. Seysmicheskaya opasnost’ i seysmicheskiy monitoring Prichernomor’ya Rossiyskoy Federatsii [Seismic danger and seismic monitoring of Black Sea coast of the Russia Federation].
*Ekologicheskiy vestnik nauchnykh tsentrov chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2012, no. 1, pp. 98-105. (In Russian) - Gabsatarova I.P., Nikonov A.A., Kamenskaya O.P., Freyfel L.D. Rekostruktsiya makroskopicheskikh poley zemletryaseniy Severo-Zapadnogo Kavkaza s tsel’yu utochneniya seysmicheskoy opasnosti regiona [Reconstruction of macroscopic fields of earthquakes in the North-Western Caucasus with the aim of clarifying the seismic hazards in the region]. In
*Glubinnoye stroeniye, minerageniya, sovremennaya geodinamika i seysmichnost’ Vostochno-Evropeyskoy platformy i sopredel’nykh regionov*[Deep structure, metallogeny, modern geodynamics and seismicity of the East European platform and adjacent regions]. Voronezh, Nauchnaya kniga Publ., 2016, pp. 117-122. (In Russian) - Stogny G.A., Stogny V.V. Tektonicheskie usloviya seysmichnosti Severo-Zapadnogo segmenta i yuzhnogo sklona Tsentral’nogo segmenta Bol’shogo Kavkaza [Tectonic conditions seismisity of the North-West segment and the Southern slope of the Central segment of the Greater Caucasus]. In
*Glubinnoye stroeniye, minerageniya, sovremennaya geodinamika i seysmichnost’ Vostochno-Evropeyskoy platformy i sopredel’nykh regionov*[Deep structure, metallogeny, modern geodynamics and seismicity of the East European platform and adjacent regions]. Voronezh: Nauchnaya kniga Publ., 2016, pp. 383-387. (In Russian) - Pavnenkova G.A. Stroenie zemnoy kory Kavkaza po profilyam GSZ Stepnoe-Bakuriani i Volgograd-Nakhichevan’ (rezul’taty pereinterpretatsii pervichnykh dannykh) [The structure of the Earth’s crust of the Caucasus on the profile of the DSS Stepnoye-Bakuriany and Volgograd-Nakhichevan (the results of reinterpretation of primary data)].
*Fizika Zemli*[Physics of the Earth], 2012, no. 5, pp. 21-27. (In Russian) - Pavlenko O.V. Otsenka dobrotnosti kory i verkhney mantii v okrestnostyakh Sochi i Anapy (Severniy Kavkaz) [The estimation of the quality factor of the crust and upper mantle in the vicinity of Sochi and Anapa (Northern Caucasus)].
*Fizika Zemli*[Physics of the Earth], 2016, no. 3, pp. 19-30. (In Russian) - Ulomov V.I., Danilova T.I., Medvedeva N.S., Polyakova T.P., Shumilova L.S. K otsenke seysmicheskoy opasnosti na Severnom Kavkaze [To assess the seismic hazards in the North Caucasus)].
*Fizika Zemli*[Physics of the Earth], 2007, no. 7, pp. 31-45. (In Russian)