A comparison of the results obtained recently for power and power-logarithmic singular asymptotics of solution associated with a class of singular integral equations of the two-dimensional elasticity is performed. It is noted that leading parts of the integral equation contain similar terms for these singular solutions. In this connection, transcendental equations in regard to singularity exponents for additive form (superposition) of power and power-logarithmic solution asymptotics were constructed. It was established that superposition of the mentioned singular solutions has the singularity exponent which is known for the classical power asymptotics of elastic stress.

The general nature of the obtained results is discussed that is related to the description of numerous boundary value problems of the two-dimensional elasticity by means of systems of singular integral equations belonging to the class under consideration. Based on theory of the Kolosov-Muskhelishvili complex potentials power-logarithmic singular solution of a boundary value problem is constructed. This solution represents the obtained results from point of view of direct asymptotic analysis of the boundary value problems.

The parametric approach for equations on real singularity exponent is suggested to extend domain where non-oscillatory asymptotic is implemented. Numerical results on leading power-logarithmic singularity exponent for the two-dimensional problem of the elasticity theory for the crack terminating an interface are presented. The efficiency of the developed parametric approach for examined crack problem is demonstrated.

Keywords: singular integral equation, power and power-logarithmic asymptotics, elastic stress concentration, complex and real singularity exponent.

Andrey V. Andreev (Moscow, Russian Federation) – Ph.D. in Physical and Mathematical Sciences, Senior Scientist of the Joint Institute for High Temperatures of the Russian Academy of Sciences (13, Bd. 2, Izhorskaya str., 125412, Moscow, Russian Federation, e-mail:
andrey.andreev@inbox.ru).

1. Sinclair G.B. Stress singularities in classical elasticity-I: Removal, interpretation, and analysis. Appl. Mech. Rev., 2004, vol. 57, no. 4, pp. 251-297. doi: 10.1115/1.1762503

2. Sinclair G.B. Stress singularities in classical elasticity-II: Asymptotic identification. Appl. Mech. Rev., 2004, vol. 57, no. 5, pp. 385-439. doi: 10.1115/1.1767846

3. Paggi M., Carpintery A. On the stress singularities at multimaterial interfaces and related analogies with fluid dynamics and diffusion. Appl. Mech. Rev., 2008, vol. 61, pp. 020801-1-22. doi: 10.1115/1.2885134

4. Andreev A.V. Development of direct numerical integration methods for one-dimensional integro-differential equations in mechanics. Mechanics of Solids, 2009, vol. 42, iss. 2, pp. 209-222. doi: 10.3103/S0025654407020069

5. Andreev A.V. Stepenno-logarifmicheskie osobennosti resheniya odnogo klassa singuliarnykh integralnykh uravnenij ploskoj teorii uprugosti [The power-logarithmic singularities of solution for a class of singular integral equations arising in two-dimensional elasticity]. Computational continuum mechanics, 2014, vol. 7, no. 1, pp. 30-39. doi: 10.7242/1999-6691/2014.7.1.4

6. Erdogan F.E., Gupta G.D., Cook T.S. The numerical solutions of singular integral equations. Mechanics of fracture. Vol. 1. Methods of analysis and solutions of crack problems. Ed. G.C. Sih. Noordhoff Intern. Publ., 1973, pp. 368-425.

7. Muskhelishvili N.I. Singuliarnye integralnye uravneniya [Singular integral equations]. Moscow: Nauka, 1968. 511 s.

8. Duduchava R.V. Integralnye uravneniya svertki s razryvnymi predsimvolami, singuliarnye integralnye uravneniya s nepodvizhnymi osobennostiami i ikh prilozheniya k zadacham mekhaniki [Convolution type integral equations with discontinuous presymbols, singular integral equations with fixed singularities, and their applications to problems in mechanics]. Tbilisi: Metsniereba, 1979. 135 s.

9. Savruk M.P., Madenci E., Shkarayev S. Singular integral equations of the second kind with generalized Cauchy-type kernels and variable coefficients. Int. J. Numer. Meth. Eng., 1999, vol. 45, no. 10, pp. 1457-1470. doi: 10.1002/(SICI)1097-0207(19990810)45:10<1457::AID-NME639>3.0.CO;2-P

10. Williams M.L. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. App. Mech., 1952, vol. 19, no. 4, pp. 526-528.

11. Kalandia A.I. Zamechaniya ob osobennosti uprugikh reshenij vblizi uglov [Remarks on the singularity of elastic solutions near corners]. PMM, 1969, vol. 33, iss. 1, pp. 132-135.

12. Muskhelishvili N.I. Nekotorye osnovnye zadachi matematicheskoj teorii uprugosti [Some basic problems of the mathematical theory of elasticity]. Moscow: Nauka, 1966, 707 s.

13. Theocaris P.S. The order of singularity at a multiwedge corner of a composite Plate. Int. J. Eng. Sci., 1974, vol. 12, no. 2, pp. 107-120. doi: 10.1016/0020-7225(74)90011-1

14. Dempsey J.P. Power-logarithmic stress singularities at bi-material corners and interface cracks. J. Adhes. Sci. Technol., 1995, vol. 9, no. 2, pp. 253-265. doi: 10.1163/156856195X01157

15. Savruk M.P. Dvumernye zadachi uprugosti dlia tel s treshchinami [Two-dimensional elasticity problems for bodies with cracks]. Kiev: Naukova dumka, 1981, 324 s.

16. Linkov A.M. Kompleksnyj metod granichnykh integralnykh uravnenij teorii uprugosti [Complex variable boundary integral equations of elasticity theory]. St. Petersburg: Nauka, 1999, 382 s.

17. Mikhailov S.E. Singuliarnost napriazhenij v okrestnosti rebra v sostavnom neodnorodnom anizotropnom tele i nekotorye prilozheniya k kompozitam [Stress singularity near edge in the compound inhomo­ge­neous anisotropic body and some applications for composites]. Izvestiya Aka­demii nauk SSSR. MTT, 1979, no. 5, pp. 103-110.

18. Korepanova T.O., Matveenko V.P., Sevodina N.V. Chislennyj analiz singuliarnosti napriazhenij v vershine konusa s negladkoj bokovoj poverkhnostju [Numerical analysis of stress singularities at the apex of a cone with unsmooth lateral surface]. Computational continuum mechanics, 2010, vol. 3, no. 3, pp. 68-76. doi: 10.7242/1999-6691/2010.3.3.28

19. Korepanova T.O., Matveenko V.P., Sevodina N.V. Chislennyj analiz singuliarnosti napriazhenij v vershine prostranstvennykh peresekayushchikhsya treshchin [Numerical analysis of stress singularities at the tip оf intersecting 3D wedge-shaped cracks]. Computational continuum mechanics, 2011, vol. 4, no. 3, pp. 68-73. doi: 10.7242/1999-6691/2011.4.3.28

20. Fenner D.N. Stress singularities in composite materials with an arbitrarily oriented crack meeting an interface. Int. J. Fract., 1976, vol. 12, no. 5, pp. 705-721. doi: 10.1007/BF00037917

21. Yong-Li W. Crack tip stress singularities in a bimaterial with an inclined interface. Int. J. Fract., 1992, vol. 54, no. 4, pp. R65-R72. doi: 10.1007/BF00035114

The research relates to studying a mixture of grained oilseeds saturated with oil and deformed with plunger molding. Mathematical simulations aim was determination of oil extraction velocity under the set loading conditions. Current statement treats processed material as biphasic mixture. Oilseed cake presented the first phase, which also was material’s porous skeleton. Vegetable oil filling the porous skeleton was the second phase of the mixture. Multiphase dynamics approach was applied in current research for material behavior description. Balance equations were set up for each mixture components. Interfacial volumetric force introduction modelled filtration of liquid. According to former researches, viscous liquid model described properties of porous skeleton as well as properties of vegetable oil. Porous skeleton viscosity assumed to be pressure dependent. Numerical solution of problem was carried out in two-dimensional statement for expression chamber middlesection using finite element approach. The primary variables were constituent’s velocity and pressure fields. Current study used cake pressure dependent porosity model that is common in porous media mechanics. Computational domain discretization was carried out using nine-node rectangular finite elements with linear and quadratic approximation for pressure and velocity fields respectively. Oil saturation distribution along expression chamber height obtained in numerical experiments demonstrates nonlinearity under high external loads. Moreover, the study investigated porosity changes influence on vegetable oil flow during expression.

Keywords: vegetable oil extraction, porous media, filtration, biphasic mixture, viscous liquid, finite element method.

Sergey D. Anferov (Perm, Russian Federation) – Research Engineer of Laboratory of Thermoplastics Mechanics, Institute of Continuous Media Mechanics Ural Branch of the Russian Academy of Sciences (1, Akademik Korolev str., 614013, Perm, Russian Federation, e-mail: anferov@icmm.ru).

Oleg I. Skul’skiy (Perm, Russian Federation) – Doctor of Technical Sciences, Leading Researcher of Laboratory of Thermoplastics Mechanics, Institute of Continuous Media Mechanics Ural Branch of the Russian Aca­demy of Sciences (1, Akademik Korolev str., 614013, Perm, Russian Fe­de­ration, e-mail: skul@icmm.ru).

Evgeny V. Slavnov (Perm, Russian Federation) – Doctor of Technical Sciences, Head of Laboratory of Thermoplastics Mechanics, Institute of Continuous Media Mechanics Ural Branch of the Russian Academy of Sciences (1, Akademik Korolev str., 614013, Perm, Russian Federation, e-mail: slavnov@icmm.ru).

1. Slavnov E.V., Petrov I.A. Izmenenie viazkosti ekstrudata rapsa v protsesse otzhima masla [Rape cake viscosity variation during oilseeds extraction]. Agrarnyj vestnik Urala, 2011, no. 6, pp. 42-44.

2. Slavnov E.V., Petrov I.A., Anferov S.D. Izmenenie viazkosti ekstrudata rapsa v protsesse otzhima masla (vliianie davleniia) [Rape cake viscosity variation during oilseeds extraction (pressure influence)]. Agrarnyj vestnik Urala, 2011, no. 10, pp. 16-18.

3. Slavnov E.V. Izmenenie pronitsaemosti maslichnykh kul'tur v protsesse otzhima masla na primere ekstrudata rapsa [Oilseeds permeability variation during rape oil extraction]. Doklady Rossiiskoj akademii selsko­khoziaistvennykh nauk, 2013, no. 3, pp. 58-60.

4. Iakovlev D.A. Teoreticheskie issledovaniya protsessa otzhima soka shnekovym rabochim organom s dopolnitelnym dreniruiushchim konturom [Theoretical investigation of Juice extrusive extraction process with an additional drain circuit]. Vestnik Donskogo gosudarstvennogo tekhnicheskogo universiteta, 2011, vol. 11, no. 7, pp. 997-1004.

5. Iakovlev D.A. Ratsionalizatsiya shnekovogo rabochego organa dlia otzhima soka iz zelenykh rastenij [Rationalization of the screw-working body for extraction of juice from green plants]. Vestnik Donskogo gosudarstvennogo tekhnicheskogo universiteta, 2010, vol. 10, no. 4, pp. 556-559.

6. Petrov I.A., Slavnov E.V. Modelirovanie shnek-pressovogo otzhima kak sovokupnosti protsessov techeniya viazkoi neszhimaemoj smesi i filtratsii zhidkosti skvoz poristuyu sredu [Simulation of screw-press oil extraction as a set of two processes: incompressible viscous mixture flow and fluid filtration in porous medium]. Computational continuum mechanics, 2013, vol. 6, no. 3, pp. 277-285. doi: 10.7242/1999-6691/2013.6.3.31

7. Meretukov Z.A., Kosachev V.S., Koshevoi E.P. Reshenie zadachi nelineinoj naporoprovodnosti pri otzhime. Izvestiia vysshikh uchebnykh zavedenij. Pishchevaya tekhnologiya, 2011, vol. 323-324, no. 5-6, pp. 62-64.

8. Meretukov Z.A., Koshevoi E.P., Kosachev V.S. Reshenie differen­tsialnogo uravneniya otzhima [Extraction differential equation solution]. Novye tekhnologii, 2011, no. 4, pp. 54-57.

9. Asgari A., Bagheripour M.H., Mollazadeh M. A generalized analytical solution for a nonlinear infiltration equation using the exp-function method. Scientia Iranica, 2011, vol. 18, iss. 1, pp. 28-35. doi: 10.1016/j.scient.2011.03.004

10. Sanavia L., Schrefler B.A., Steinmann P. A formulation for an unsaturated porous medium undergoing large inelastic strains. Computational Mechanics, 2002, vol. 28, pp. 137-151.

11. Aptukov V.N. Model' uprugo-viazkoplasticheskogo poristogo tela [The model of the elasto-viscoplastic porous body]. Vestnik Permskogo universiteta. Matematika. Mekhanika. Informatika, 2008, no. 4, pp. 77-81.

12. Wang S.-J., Hsu K.-C. Dynamic interactions of groundwater flow and soil deformation in randomly heterogeneous porous media. Journal of Hydrology, 2013, vol. 499, no. 30, pp. 50-60. doi: 10.1016/j.jhydrol.2013.06.047.

13. Helmig R., Flemisch B., Wolff M., Ebigbo A., Class H. Model coupling for multiphase flow in porous media. Advances in Water Resources, 2013, vol. 51, pp. 52-66. doi: 10.1016/j.advwatres.2012.07.003.

14. Kondaurov V.I. A non-equilibrium model of a porous medium saturated with immiscible fluids. Journal of Applied Mathematics and Mechanics, 2009, vol. 73, iss. 1, pp. 88-102. doi: 10.1016/j.jappmathmech.2009.03.004.

15. Khoei A.R., Mohammadnejad T. Numerical modeling of multiphase fluid flow in deforming porous media: A comparison between two- and three-phase models for seismic analysis of earth and rockfill dams. Computers and Geotechnics, 2011, vol. 38, iss. 2, pp. 142-166. doi: 10.1016/j.compgeo.2010.10.010.

16. Amaziane B., Jurak M., Keko A.Ž. Numerical simulations of water-gas flow in heterogeneous porous media with discontinuous capillary pressures by the concept of global pressure. Journal of Computational and Applied Mathematics, 2012, vol. 236, iss. 17, pp. 4227-4244. doi: 10.1016/j.cam.2012.05.013.

17. Sun S., Salama A., El-Amin M.F. An Equation-Type Approach for the Numerical Solution of the Partial Differential Equations Governing Transport Phenomena in Porous Media. Procedia Computer Science, 2012, vol. 9, pp. 661-669. doi: 10.1016/j.procs.2012.04.071.

18. Fučík R., Mikyška J. Discontinous Galerkin and Mixed-Hybrid Finite Element Approach to Two-Phase Flow in Heterogeneous Porous Media with Different Capillary Pressures. Procedia Computer Science, 2011, vol. 4, pp. 908-917. doi: 10.1016/j.procs.2011.04.096.

19. Rohan E., Shaw S., Wheeler M.F., Whiteman J.R. Mixed and Galerkin finite element approximation of flow in a linear viscoelastic porous medium. Computer Methods in Applied Mechanics and Engineering, 2013, vol. 260, pp. 78-91 doi: 10.1016/j.cma.2013.03.003.

20. El-Amin M.F., Salama A., Sun S. A Conditionally Stable Scheme for a Transient Flow of a Non-Newtonian Fluid Saturating a Porous Medium. Procedia Computer Science, 2012, vol. 9, pp. 651-660. doi: 10.1016/j.procs.2012.04.070.

21. Liu J., Mu L., Ye X. A Comparative Study of Locally Conservative Numerical Methods for Darcy's Flows. Procedia Computer Science, 2011, vol. 4, pp. 974-983. doi: 10.1016/j.procs.2011.04.103.

22. Choquet C. On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media. Journal of Mathematical Analysis and Applications, 2008, vol. 339, Iss. 2, pp. 1112-1133. doi: 10.1016/j.jmaa.2007.07.037.

23. Nikolaevskij V.N., Basnieva K.S., Gorbunov A.T., Zotov G.A. Mekhanika nasyshchenykh poristykh sred [Saturated porous media mechanics]. Moscow: Nedra, 1970. 339 p.

24. Nigmatulin R.I. Dinamica mnogofaznykh sred [Multyphase dynamics]. Part. 1. Moscow: Nauka, 1987. 464 p.

25. Albets-Chico X., Kassinos S. A consistent velocity approximation for variable-density flow and transport in porous media. Journal of Hydrology, 2013, vol. 507, no. 12, pp. 33-51. doi: 10.1016/j.jhydrol.2013.10.009

26. Torner R.V. Teoreticheskie osnovy pererabotki polimerov [Theoretical principles of polymer processing]. Moscow: Chemistry, 1977. 464 p.

27. Skulskiy O.I., Aristov S.N. Mekhanika anomalno vyazkikh zhidkostey [Anomalously viscous liquids mechanics].Yekaterinburg: Uralskoe otdelenie Rossiyskoy akademii nauk, 2004. 156 p.

28. Gershuni G.Z., Zhukhovitskii E.M. Konvektivnaya ustoichivost' neszhimaemoj zhidkosti [Convective stability of incompressible fluid]. Moscow: Nauka, 1972, 392 p.

29. Konovalov A.B. Imitatsionnoe modelirovanie rabochego protsessa v pressakh s prodolnoj filtratsiej. Tekhniko-tekhnologicheskie problemy servisa, 2012, vol. 20, no. 2, pp. 40-47.

30. Reddy J.M. An introduction to nonlinear finite element analysis. Oxford, 2004. 482 p.

31. Segal Ir.A. Finite element methods for the incompressible Navier-Stokes equations. Delft: University of Technology, 2012. 80 p.

32. Reddy J.M., Brezzi F., Fortin M. Mixed and Hybrid Finite Element Methods, Springrer-Verlag, 1991.

33. Vavilin V.A., et al. Ob al'ternativnom sposobe opredeleniya predela uprugosti gornykh porod v usloviyakh, adekvatnykh plastovym[ About alternative elastic limit estimation method for rocks in reservoir-like conditions]. Georesursy, 2008, no. 5, pp. 44-48.

The paper is devoted to the analysis of panel flutter of functionally graded cylindrical shells in a supersonic gas flow. The aerodynamic pressure is calculated based on the quasi-static aerodynamic theory. The inner surface of the structure is made of aluminum and the outer surface is made of zirconium dioxide. The effective properties of the material continuously changes through the shell thickness with radial coordinate according to the power law. The geometric and physical relations and the equations of motion written in the framework of the classical shell theory are reduced to the system of eight ordinary differential equations for new unknown quantities. A solution of the problem is found by integrating the obtained system of equations by the Godunov’s orthogonal marching method at each step of the iterative procedure generally used in Muller’s method to evaluate complex eigenvalues. The reliability of the method was assessed by comparing the obtained results with the available experimental and theoretical data. The paper presents the results of numerical experiments carried out to estimate the effect of the properties of functionally graded materials on the stability boundary of circular cylindrical shells for different combinations of boundary conditions and linear dimensions. It has been found that the type of loss of stability is defined not only by geometrical characteristics of the structure and boundary conditions but also by given composition of the functionally graded material. It has been shown that an effective control of critical aerodynamic loading can be executed only for shells with certain geometrical dimensions.

Keywords: classical shell theory, functionally graded material, Godunov’s orthogonal marching method, stability, flutter.

Sergey V. Lekomtsev (Perm, Russian Federation) – Ph.D. in Physical and Mathematical Sciences, Junior Researcher, Department of Complex Problems of Mechanics of Deformable Bodies, Institute of Continuous Media Mechanics, Ural Branch of the Russian Academy of Sciences (1, Akademik Korolev str., 614013, Perm, Russian Federation, e-mail: lekomtsev@icmm.ru).

1. Reddy J.N., Chin C.D. Thermomechanical analysis of functionally graded cylinders and plates. J. Therm. Stresses, 1998, 21(6), pp. 593-626. doi: 10.1080/01495739808956165

2. Sheng G.G., Wang X. Thermomechanical vibration analysis of a functionally graded shell with flowing fluid. Eur. J. Mech. A-Solid., 2008, vol. 27, no. 6, pp. 1075-1087. doi: 10.1016/j.euromechsol.2008.02.003

3. Iqbal Z., Naeem M.N., Sultana N. Vibration characteristics of FGM circular cylindrical shells using wave propagation approach. Acta Mech., 2009, vol. 208, no. 3-4, pp. 237-248. doi: 10.1007/s00707-009-0141-z

4. Naeem M.N., Arshad S.H., Sharma C.B. The Ritz formulation applied to the study of the vibration frequency characteristics of functionally graded circular cylindrical shells. Proc. Inst. Mech. Engng., Part C: J. Mech. Engng. Sci., 2010, vol. 224, no. 1, pp. 43-54. doi: 10.1243/09544062JMES1548

5. Huang H., Han Q., Feng N., Fan X. Buckling of functionally graded cylindrical shells under combined loads. Mech. Adv. Mater. Struct., 2011, vol. 18, no. 5, pp. 337-346. doi: 10.1080/15376494.2010.516882

6. Iqbal Z., Naeem M.N., Sultana N., Arshad S.H., Shah A.G. Vibration characteristics of FGM circular cylindrical shells filled with fluid using wave propagation approach. Appl. Math. Mech., 2009, vol. 30, no. 11, pp. 1393-1404. doi: 10.1007/s10483-009-1105-x

7. Khazaeinejad P., Najafizadeh M.M. Mechanical buckling of cylindrical shells with varying material properties. Proc. Inst. Mech. Engng., Part C: J. Mech. Engng. Sci., 2010, vol. 224, no. 8, pp. 1551-1557. doi: 10.1243/09544062JMES1978

8. Matsunaga H. Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory. Compos. Struct., 2009, vol. 88, no. 4, pp. 519-531. doi: 10.1016/j.compstruct.2008.05.019

9. Najafizadeh M.M., Hasani A., Khazaeinejad P. Mechanical stability of functionally graded stiffened cylindrical shells. Appl. Math. Model., 2009, 33(2) , pp. 1151-1157. doi: 10.1016/j.apm.2008.01.009

10. Khazaeinejad P., Najafizadeh M.M., Jenabi J., Isvandzibaei M.R. On the buckling of functionally graded cylindrical shells under combined external pressure and axial compression. J. Press. Ves. Technol., 2010, vol. 132, no. 6, 064501 (6 p.). doi: 10.1115/1.4001659

11. Bagherizadeh E., Kiani Y., Eslami M.R. Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation. Compos. Struct., 2011, vol. 93, no. 11, pp. 3063-3071. doi: 10.1016/j.compstruct.2011.04.022

12. Sheng G.G., Wang X. Dynamic characteristics of fluid-conveying functionally graded cylindrical shells under mechanical and thermal loads. Compos. Struct., 2010, vol. 93, no. 1, pp. 162-170. doi: 10.1016/j.compstruct.2010.06.004

13. Heydarpour Y., Malekzadeh P., Golbahar Haghighi M.R., Vaghefi M. Thermoelastic analysis of rotating laminated functionally graded cylindrical shells using layerwise differential quadrature method. Acta Mech., 2012, vol. 223, no. 1, pp. 81-93. doi: 10.1007/s00707-011-0551-6

14. Malekzadeh P., Heydarpour Y. Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment. Compos. Struct., 2012, vol. 94, no. 9, pp. 2971-2981. doi: 10.1016/j.compstruct.2012.04.011

15. Hosseini-Hashemi Sh., Ilkhani M.R., Fadaee M. Accurate natural frequencies and critical speeds of a rotating functionally graded moderately thick cylindrical shell. Int. J. Mech. Sci., 2013, vol. 76, pp. 9-20. doi: 10.1016/j.ijmecsci.2013.08.005

16. Qu Y., Long X., Yuan G., Meng G. A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions. Compos. Part B-Eng., 2013, vol. 50, pp. 381-402. doi: 10.1016/j.compositesb.2013.02.028

17. Haddadpour H., Mahmoudkhani S., Navazi H.M. Supersonic flutter prediction of functionally graded cylindrical shells. Compos. Struct., 2008, vol. 83, no. 4, pp. 391-398. doi: 10.1016/j.compstruct.2007.05.011

18. Sabri F., Lakis A.A. Aerothermoelastic stability of functionally graded circular cylindrical shells. ASME International Symposium on Fluid-Structure Interactions, Flow-Sound interactions, and Flow Induced Vibration & Noise, Montreal, Canada, August 2010. Montreal, 2010, pp. 939-945. doi: 10.1115/FEDSM-ICNMM2010-30384

19. Voss H.M. The effect of an external supersonic flow on the vibration characteristics of thin cylindrical shells. J. Aerospase Sci., 1961, vol. 3, pp. 945-956.

20. Bochkarev S.A., Matveynko V.P. Reshenie zadachi o panelnom flattere obolochechnykh konstruktsij metodom konechnykh elementov [Finite-element solution of panel flutter of shell structures]. Matem. Mod., 2002, vol. 14, no. 12, pp. 55-71.

21. Karmishin A.V., Lyaskovets V.A., Myachenkov V.I., Frolov A.N. Statika i dinamika tonkostennykh obolochechnykh konstruktsij [The Statics and Dynamics of Thin-walled Shell Structures]. Moscow: Mashino­stroyeniye, 1975, 376 p.

22. Godunov S.K. O chislennom reshenii kraevykh zadach dlya sistem linejnykh obyknovennykh differentsialnykh uravnenij [Numerical solution of boundary-value problems for systems of linear ordinary differential equations]. Uspekhi Mat. Nauk, 1961, vol. 16, no. 3, pp. 171-174.

23. Bochkarev S.A., Matveynko V.P. Ob odnom metode issledovaniya aerouprugoj ustojchivosti obolochek vrashcheniya [A method of aeroelastic stability analysis of shells of revolution]. Vestnik Samarskogo gosudarstvennogo universiteta. Estestvennonauchnaya seriya, 2007, no. 4(54), pp. 387-399.

24. Olson M.D., Fung Y.C. Supersonic flutter of circular cylindrical shells subjected to internal pressure and axial compression. AIAA J., 1966, vol. 4, no. 5, pp. 858-864. doi:10.2514/3.3558

25. Olson M.D., Fung Y.C. Comparing theory and experiment for the supersonic flutter of circular cylindrical shells. AIAA J., 1967, vol. 5, no. 10, pp. 1849-1856. doi: 10.2514/3.4315

26. Carter L.L., Stearman R.O. Some aspects of cylindrical shell panel flutter. AIAA J., 1968, vol. 6, no. 1, pp. 37-43. doi: 10.2514/3.4438

27. Bismarck-Nasr M.N. Finite element method applied to the supersonic flutter of circular cylindrical shells. Int. J. Numer. Meth. Engng., 1976, vol. 10, no. 2, pp. 423-435. doi: 10.1002/nme.1620100212

28. Ganapathi M., Varadan T.K., Jijen J. Field-consistent element applied to flutter analysis of circular cylindrical shells. J. Sound Vib., 1994, vol. 171, no. 4, pp. 509-527. doi: 0.1006/jsvi.1994.1137

29. Sabri F., Lakis A.A. Finite element method applied to supersonic flutter of circular cylindrical shells. AIAA J., 2010, vol. 48, no. 1, pp. 73-81. doi: 10.2514/1.39580

This paper considers the developed mathematical model of inelastic deformation in structural steels, describing thermoviscoplastic deformation taking into account mutual influence of plasticity and creep effects. An integration algorithm for constitutive relations of thermoviscoplasticity has been developed. It consists in the formulation of constitutive relations in increments, depending on the selected time step. In difficult areas of deformation paths, time step can be adjusted throughout the whole estimation time in case of stability calculations. Stresses, plastic deformations and creep deformations are determined by integrating the defining relations of thermal creep by Runge-Kutta method with the correction of stress deviator and subsequent determination of stress according to thermal plasticity equations with regard to the average creep strain at the next sampling time. Experimental studies of influence between creep processes and plasticity under high temperature using 12H18N9 steel have been conducted. By numerical computer simulation of stress-strain state (SSS) kinetics in laboratory samples and by comparing the obtained results with field experiments, the authors carried out certification of the developed thermoviscoplastic model and integration algorithm of constitutive relations. All of these led to the conclusion about the reliability of model concepts and methods for determining material parameters under joint actions of fatigue and creep mechanisms. The authors have compared computer and physical tensile experiments of laboratory 12H18N9 steel samples with different histories of changes in temperature and mechanical deformation. It is shown that the developed thermoviscoplastic model qualitatively and quantitatively describes main effects of inelastic deformation in structural steels with different histories of mechanical deformation and changes in temperature. It is concluded that the defining relations of thermoviscoplasticity are reliable, and the above methods of integration are accurate.

Keywords: plasticity, creep, basic experiment, stress-strain state, the trajectory of loading, type of stress state, complex loading, material parameters.

Ivan A. Volkov (Nizhny Novgorod, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, Head of the De­partment of Applied mathematics and lifting transport machines, Volga State Academy of Water Transport (5a, Nesterova str., 603950, Nizhny Novgorod, Russian Federation, e-mail: pmptmvgavt@yandex.ru).

Andrey I. Volkov (Nizhny Novgorod, Russian Federation) – Master Degree Student of the Department of Applied mathematics and lifting transport machines, Volga State Academy of Water Transport (5a, Nesterova str., 603950, Nizhny Novgorod, Russian Federation, e-mail: pmptmvgavt@yandex.ru).

Dmitry A. Kazakov (Nizhny Novgorod, Russian Federation) – Doctor of Technical Sciences, Senior Research Fellow, Head of the Laboratory at Research Institute of Mechanics, Lobachevsky State University of Nizhny Novgorod (23, Gagarin av., Building 6, 603022, Nizhny Novgorod, Russian Federation, e-mail: pmptmvgavt@yandex.ru).

Yuri G. Korotkikh (Nizhny Novgorod, Russian Fede­ration) – Doctor of Physical and Mathematical Sciences, Professor, Honored Russian Scientist, Department of Applied mathematics and lifting transport machines, Volga State Aca­demy of Water Transport (5a, Nesterova str., 603950, Nizhny Novgorod, Russian Federation, e-mail: pmptmvgavt@yandex.ru).

Ivan S. Tarasov (Nizhny Novgorod, Russian Federation) – Associate Professor, Head of Laboratory of the Department of Applied mathematics and lifting transport machines, Volga State Academy of Water Transport (5a, Nesterova str., 603950, Nizhny Novgorod, Russian Federation, e-mail: pmptmvgavt@yandex.ru).

1. Mitenkov F.M., Kaidalov V.B., Korotkikh Yu.G. Metody obosnovaniya resursa yadernykh ehnergeticheskikh ustanovok [Methods of study resource nuclear power plants]. Moscow: Mashinostroenie, 2007. 448 p.

2. Volkov I.A., Korotkikh Yu.G. Uravneniya sostoyaniya vyazkouprugoplasticheskikh sred s povrezhdeniyami [Equations of state viscoelastoplastic media with injuries]. Moscow: Fizmatlit, 2008. 424 p.

3. Collins J. Povrezhdenie materialov v konstruktsiyakh. Аnaliz, predskazanie, predotvrashhenie [Damage to materials in the construction . Analysis, prediction, prevention]. New York: Wiley, 1984. 624 p.

4. Gomyuk, Beau-Quoc T. Raschet dolgovechnosti korrozionnostojkoj stali 304 v usloviyakh vzaimodejstviya ustalosti i polzuchesti s ispolzovaniem teorii nepreryvnogo povrezhdeniya [Calculation durability stainless steel 304 in interaction of fatigue and creep by using the theory of continuous damage]. Proceedings of the Amer. Islands of engineer – fur. Ser. D. Teor. Ing basis. calculations, 1986, vol. 108, no. 3, pp. 111-136.

5. Gomyuk, Beau-Quoc T., Biron A. Izuchenie povedeniya stali 316 pri nagruzheniyakh po skhemam ustalosti, polzuchesti i sovmestnogo dejstviya ustalosti i polzuchesti [The behavior of 316 during loading schemes fatigue , creep and fatigue and joint action of creep]. Modern machines, 1991, no. 1, pp. 14-23.

6. Kazantsev A.G. Issledovanie vzaimodejstviya malotsiklovoj ustalosti i polzuchesti pri neizotermicheskom nagruzhenii [Investigation of the interaction of low-cycle fatigue and creep under nonisothermal loading]. Problems of strength, 1985, no. 5, pp. 25-31.

7. LeMay. Razvitie parametricheskikh metodov obrabotki rezultatov ispytanij na polzuchest i dlitelnuyu prochnost [Development of parametric methods of processing the results of creep tests and long-term strength]. Proceedings of Amer. Islands of engineer fur. Ser. D. Teor. Ing basis. calculations, 1979, vol. 101, no. 4, pp. 19-24.

8. Manson Ensign. Uspekhi za poslednyuyu chetvert veka v razvitii metodov korrelyatsii i ehkstrapolyatsii rezultatov ispytanij na dlitelnuyu prochnost [Successes over the past quarter century in the development of methods of correlation and extrapolation of the test results on the long-term strength]. Proceedings of the Amer. Islands of engineer fur. Ser. D. Teor. Ing basis. calculations, 1979, vol. 101, no. 4, pp. 9-18.

9. Volkov I.A., Korotkikh Yu.G., Tarasov I.S., Shishulin D.N. Numerical modeling of elastoplastic deformation and damage accumulation in metals under low-cycle fatigue conditions. J. Strength of Materials, 2011, vol. 43, no. 4, pp. 471-485.

10. Volkov I.A., Volkov A.I., Korotkikh Yu.G., Tarasov I.S Model povrezhdyonnoj sredy dlya otsenki resursnykh kharakteristik konstruktsionnykh stalej pri mekhanizmakh ischerpaniya, sochetayushhikh ustalost i polzuchest materiala [Damaged environment model for the evaluation of resource characteristics of structural steels in the mechanisms of exhaustion , fatigue and creep combining material]. Calc. fur. the solid. Matter., 2013, vol. 6, no. 2, pp. 232-245.

11. Volkov I.A., Kazakov D.A., Korotkikh Yu.G. EHksperimentalno-teoreticheskie metodiki opredeleniya parametrov uravnenij mekhaniki povrezhdyonnoj sredy pri ustalosti i polzuchesti [Experimental and theoretical methods for determining the parameters of the equations of mechanics damaged environment with fatigue and creep]. Vestnik Permskogo natsionalnogo issledovatelskogo politekhnicheskogo universiteta. Mekhanika, 2012, no. 2, pp. 50-78.

12. Kasakov D.A. EHksperimentalno teoreticheskoe issledovanie vyazkoplasticheskogo deformirovaniya stalej v oblasti povyshennykh temperatur i skorostej deformatsij do 10^–2 s^–1 [Еxperimental and theoretical investigations of visco-plastic deformation structural steels in high temperature and strain rate to 10^–2 s^–1]. Applied problems of strength and ductility. Algorithmic and automation solutions of problems of elasticity and plasticity. Proc. Intercollege. Sat Gorky University, 1985, pp. 89-97.

13. Likhachev V.A., Maligin G.A. Polzuchest tsinka pri teplosme­nakh [Creep and other zinc thermal cycles]. Fiz., 1963, vol. 16, iss. 6.

14. Oding I.A., Ivanova V.S., Burduksky V.V., Geminov V.N. Teoriya polzuchesti i dlitelnoj prochnosti metallov [Theory of creep and creep rupture of metals]. Moscow: Metallurgiya, 1959. 488 p.

15. Mozharovskaya T.N. Programma i metodika issledovaniya polzuchesti i dlitelnoj prochnosti materialov s uchyotom vida deviatora napryazhenij i istorii nagruzheniya [Program and method of study of creep and creep rupture of materials with regard to the form of the stress deviator and loading history]. Problems of Strength, 1984, no. 11, pp. 83-88.

16. Rabotnov Yu.N. Polzuchest' ehlementov konstruktsij [Creep of structural elements]. Moscow: Nauka, 1966, 752 p.

17. Chaboche J.L. Constitutive equation for cyclic plasticity and cyclic viscoplasticity. Inter. J. of Plasticity, 1989, vol. 5, no. 3, pp. 247-302.

18. Benallal A., Marquis D. Constitutive Equations for Nonproportional Cyclic Elasto-Viscoplasticity. Journal of Engineering Materials and Technology, 1987, vol. 109, pp. 326-337.

19. Lemaitre J. Damage modelling for prediction of plastic or creep fatigue failure in structures. Trans. 5th Int. Conf. SMRiT, North Holland, 1979, no. L5/1b.

20. Murakami S., Imaizumi T. Mechanical description of creep damage and its experimental verification. J. Mec. Theor. Appl., no 1, 1982, pp. 743-761.

The work is devoted to determining isotropic shell of stress-strain state for arbitrary Gaussian curvature with a circular hole, located in the center of the structure. An axial tension or an internal pressure is applied to the surface of the shell. The isotropic shallow shell theory equations were used, which coincide with the isotropic shell theory equations with a large measure of variability. The integral Fourier transformation and the theory of generalized functions were applied. As a result the problem was reduced to solving the system of boundary integral equations. One benefit of using the method of boundary integral equations for the study of shell stress-strain state weakened by a hole is the ability to define the unknown quantities directly on the contour of the hole, not evaluating them on the whole surface of the shell. To obtain the kernels of the singular integral equations, integral representations of displacements and shallow isotropic shells static equations fundamental solutions were used. As the unknown functions, a combination of displacements, rotation angles and their derivatives were used. Analytical calculations are considerably simplified if it is assumed to take into account not four unknown functions on the contour, as it is customary, but five. In this paper it is chosen to use the differential equation which relates the unknown functions as the fifth equation of the boundary integral equations system. In order to obtain numerical solution of the problem, the method of mechanical quadratures for systems of integral equations and finite difference method for the fifth differential equation were used to reduce a problem to a system of linear algebraic equations. The stress concentration factors values depending on the isotropic shell curvature are given. Also the results were compared with other researchers.

Keywords: circular hole, isotropic shell, stress concentration factor, Fourier transformation, method of boundary integral equations.

Dovbnya Ekaterina Nikolaevna (Donetsk, Ukraine) – Doctor of Physics and Mathematics Sciences, Professor, Professor of the Department of Applied Mechanics and Computer Technology Donetsk National University (24, Universitetskaya str., 83055, Donetsk, Ukraine).

Krupko Nataliia Andreevna (Donetsk, Ukraine) – Doctoral student Department of Applied Mechanics and Computer Technology Donetsk National University (24 Universitetskaya str., 83055, Donetsk, Ukraine, e-mail: nataliekrupko@gmail.com).

1. Lur'e A.I. Statics of Thin-walled Elastic Shells. State Publishing House of Technical and Theoretical Literature. Moscow, 1947; translation, AEC-tr-3798, Atomic Energy Commission, 1959.

2. Shevliakov Iu.A. Napriazhenie v sfericheskom dnishche, oslab­len­nom krugovym vyrezom [Stresses in the spherical bottom weakened by a circular cut]. Inzhenernyj sbornik, 1956, vol. 24, pp. 226-230.

3. Dyke P. Stresses about a circular hole in a cylindrical shell. ATAA Journ., 1967, vol. 5, no. 9, pp. 87-91.

4. Lekkerkerker J.G. On the stress distribution in cylindrical shells weakened by a circular hole. Delft: Vitgeverij Waltman, 1965.

5. Pirogov I.M. Vliyanie krivizny na raspredelenie napriazhenij okolo otverstiya v tsilindricheskoj obolochke [Effect of curvature on the stress distribution near the hole in the cylindrical shell]. Applied Mechanics, 1965, vol.1, no. 12, pp. 116-119.

6. Imenitov L.B. Zadacha o sfericheskoj obolochke s nepodkreplennym otverstiem [The problem of a spherical shell with an unsupported hole]. Inzhenernyi zhurnal, 1963, vol. 3, no. 1, pp. 93-100.

7. Syaskii A.A., Lun E.I. State of stress in an isotropic spherical shell with a circular hole. Prikladnaya mekhanika, 1974, vol. 10, no. 8, pp. 884-887.

8. Savin G.N. Raspredelenie napryazhenij okolo otverstij [Stress distribution around holes]. Kiev: Naukova dumka, 1968. 888 p.

9. Guz A.N., Chernyshenko I.S., Chekhov Val.N., Chekhov Vik.N., Shnerenko K.I. Metody rascheta obolochek: v 5 t. T.1. Teoriya tonkikh obolochek, oslablennykh otverstiyami [Methods of calculation of shells. In 5 vols. Vol. 1. The theory of thin shells weakened by holes]. Kiev: Naukova dumka, 1980, 636 p.

10. Zakora S.V., Chekhov V.N., Shnerenko K.I. Stress concentration around a circular hole in a transversely isotropic spherical shell. Prikladnaya mekhanika, 2004, vol. 40,  no. 12, pp. 1391-1397.

11. Loo Wen-da, Cheng Yao-shun. The effect of transverse shear deformation on stress concentration factors for shallow shells with a small circular hole. Prikladnaya matematika i mekhanika, 1991, no. 12(2), pp. 195-202.

12. Shevchenko V.P., Zakora S.V. O vliyanii sdvigovoj zhestkosti na napryazhennoe sostoyanie v transtropnoj sfericheskoj obolochke s dvumya krugovymi otverstiyami pri ikh sblizhenii [On the effect of shear stiffness on the stress state in transtropic spherical shell with two circular holes on their approach]. Dopovіdі Natsіonal'noї akademії nauk Ukraїni, 2010, no. 12, pp. 56-62.

13. Khoma I.Yu., Starygina O.A. Napryazhennoe sostoyanie pologoj sfericheskoj obolochki s krugovym otverstiem, na poverkhnosti kotorogo zadany kasatelnye napryazheniya [Stress state of shallow spherical shell with a circular hole where shear stresses are given on the surface]. Teoreticheskaya i prikladnaya mekhanika, 2010, no. 1 (47), pp. 62-68.

14. Nitin K.J. Analysis of Stress Concentration and Deflection in Isotropic and Orthotropic Rectangular Plates with Central Circular Hole under Transverse Static Loading. World Academy of Science, Engineering and Technology, 2009, vol. 36, pp. 407-413.

15. Reissner E., Wan F.Y.M. Further considerations of stress concentration problems for twisted or sheared shallow spherical shells. International Journal Solids Structures, 1994, vol. 31, no. 16, pp. 2153-2165.

16. Hsu Chin-yun, Yeh Kai-yuan. The General Solution of Bending of a Spherical Thin Shallow Shell with a Circular Hole at the Center under Arbitrary Transverse Loads. Applied Mathematics and Mechanics, 1980, vol. 1, no. 3, pp. 341-356.

17. Chernyshenko I.S., Storozhuk E.A., Kharenko S.B. Elastoplastic state of flexible conical shells with a circular hole under axial tension. International Applied Mechanics, 2011, vol. 47, no. 6, pp. 679-684.

18. Dovbnya E.N. Deformatsiya kontura krugovogo otverstiya v ortotropnykh obolochkakh pri rastyagivaiushchej nagruzke [Deformation of the contour of a circular hole in orthotropic shells under tensile load]. Visnik Donetskogo universitetu. Ser.A, 2000, no. 1, pp. 51-55.

19. Dovbnya E.N. Napryazhenno-deformirovannoe sostoyanie orto­tropnoy obolochki s krugovym otverstiem [Stress-strain state of orthotropic shell with a circular hole]. Teoreticheskaya i prikladnaya mekhanika, 1997, no. 27, pp. 154-158.

20. Dovbnya K.M. Rozvitok metodu granichnikh integral'nikh rivnian' v teoriї ortotropnikh obolonok z rozrizami ta otvorami [The development of the method of boundary integral equations in the theory of orthotropic shells with holes and cuts]. Thesis of doctors degree dissertation physics and mathematics sciences. Donetsk, 2001, 362 p.

21. Dovbnya E.N. Sistema granichnykh integral'nykh uravnenii dlia ortotropnykh obolochek nulevoi i otritsatel'noi krivizn, oslablennykh razrezami i otverstiiami [System of boundary integral equations for orthotropic shells with zero and negative curvature, weakened by cuts and holes]. Vіsnik Donets'kogo unіversitetu. Ser. A: prirodnichі nauki, 1998, vol. 2, pp. 45-52.

22. Goldenveizer A.L. Teoriya uprugikh tonkikh obolochek [Theory of elastic thin shells]. Moscow: Nauka, 1976. 512 p.

23. Vlasov V.Z. Izbrannye trudy: v 3 t., T. 1. [Selected Works, in 3 vols, vol. 1]. Moscow: Akademiya nauk SSSR, 1962. 528 p.

24. Panasiuk V.V., Savruk M.P., Datsyshin A.P. Raspredelenie na­prya­zhenii okolo treshchin v plastinakh i obolochkakh [The stress distribution around cracks in plates and shells]. Kiev: Naukova dumka, 1976. 444 p.

Currently biology and medicine become one of the most attractive areas of applied mathematics. To fix certain pathologies of children, growth modelling for living tissue and growth management are the issues of major importance. In the process of growth a growing body itself experiences deformation that proves a fundamental difference of mechanics of growing bodies from the classical mechanics of bodies of constant composition. This paper presents an analysis of publications related to various models of the mechanism of living tissues growth and a brief analysis of biological growth concept. The authors considered basic principles of growth modelling and specified major areas for developing certain models of body-growing tissue. The following classification of growth models for living tissue has been given: models based on the hypothesis about the influence of intracellular pressure on tissue growth as a stimulating factor; models of multiphase media, the so-called “mixture theory”; model based on the hypothesis about the influence of residual stresses on tissue growth as a stimulating factor; models connecting the rate of growth from the deformations known from observations and experiments. The analysis resulted in specifying factors influencing the growth of living tissue. These are the chemical composition, concentration, transport and stresses in the material body. Stress is a significant factor affecting growth. The practical importance of growth model for mechanical deformation is based on its wide application for describing normal and pathological growth of hard tissues in the human body. In this case, from mechanical point of view, it becomes possible to model and control growth.

Keywords: biological growth, growth deformation, biomechanical modelling, one-dimensional model of growth, mixture theory, residual stresses, tensile forces, eigenstrain, full deformation of the system, small deformation.

Olga Yu. Dolganova (Perm, Russian Federation) – Doctoral Student of Department of Theoretical Mechanics, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, e-mail: aoy85@yandex.ru).

Valeriy A. Lokhov (Perm, Russian Federation) – Ph.D. in Physics and Mathematics Sciences, Department of Theoretical Mechanics, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, e-mail: valeriy.lokhov@yandex.ru).

1. Ambrosi D., Vitale G. The theory of mixtures for growth and remodeling compression. Mini-Workshop: The mathematics of growth and remodelling of soft biological tissues, 2008, no. 39, pp. 9-10.

2. Ateshian G.A. The role of mass balance equations in growth mechanics illustrated in surface and volume dissolutions. Journal of Biomechanics Engineering, 2011, vol. 133, no. 1, pp. 381-390.

3. Ateshian G.A., Chahine N.O., Basalo I.M., Hung C.T. The correspondence between equilibrium biphasic and triphasic material properties in mixture models of articular cartilage. Journal of Biomechanics, 2004, vol. 37, no 3, pp. 391-400.

4. Chuong C.J., Fung Y.C. On residual stresses in arteries. Journal of Biomechanical Engineering, 1986, vol. 108, pp. 189-192.

5. Delfino A., Stergiopulos N., Moore J.E., Meister J.J. Residual Strain Effects on the Stress Field in a Thick Wall Finite Element Model of the Human Carotid Bifurcation. Journal of Biomechanics, 1997, vol. 30, no. 8, pp. 777-786.

6. Goriely A., Robertson-Tessi M., Tabor M., Vandiver R. Elastic growth models. Program in Applied Mathematics. RUMMBA, University of Arizona, 2010, 45 p.

7. Greenwald S.E., Moore J.E., Rachev A., Kane T.P.C., Meister J.J. Experimental Investigation of the Distribution of Residual Strains in the Artery Wall. Transactions of the ASME. Journal of Biomechanical Engineering, 1997, vol. 119, pp. 438-444.

8. Hoger A. Residual Stress in an Elastic Body: a Theory for Small Strains and Arbitrary Rotations. Journal of Elasticity, 1993, vol. 31, pp. 1-24.

9. Hsu F. The influence of mechanical loads on the form of a growing elastic body. Journal of biomechanics, 1968, vol. 1, no. 4, p. 303-311.

10. Klarbring A., Olsson T., Stalhad J. Theory of residual stresses with application to an arterial geometry. Arch. Mech., 2007, vol. 59, no. 4, pp. 341-364.

11. Lanyon L.E., Magee Р.Т., Baggott D.G. The relationship of functional stress and strain to the processes of bone remodelling. An experimental study on the sheep radius. J. Biomech., 1979, vol. 12, no. 8, pp. 593-600.

12. Lockhart J.A. An analysis of irreversible plant cell elongation. J. Theoretical Biology, 1965, vol. 8, no. 2, pp. 264-275.

13. Lubarda A., Hoger A. On the mechanics of solids with a growing mass. International Journal of Solids Structure, 2002, vol. 39.

14. Mura T. Micromechanics of Defects in Solids. Dordrecht: Kluwer Academic Publ, 1991.

15. Rodriguez E.K., Hoger A., McCulloch A.D. Stress-dependent finite growth in soft elastic tissues. Journal Biomechanics, 1994, vol. 27, no. 4, pp. 455-467.

16. Taber L.A., Eggers D.W. Theoretical Study of Stress-Modulated Growth in the Aorta. Journal of Theoretical Biology, 1996, vol. 180, pp. 343-357.

17. Kizilova N.N., Logovenkov S.A., Stein A.A. Matematicheskoe modelirovanie transportno-rostovykh protsessov v zhivykh tkanyakh [Mathematical modeling of transport and the growth processes in multiphase biological continuum media]. Mechanica zhidkosti i gaza, 2012, no. 1, pp. 3-13.

18. Logvenkov S.A., Stein A.A. Upravlenie zhivym rostom kak zadacha mekhaniki [Management of biological growth as a problem of the mechanics]. Rossiiskij journal biomechaniki, 2006, vol. 10, no. 2, pp. 9-19.

19. Lochov V.A., Dolganova O.Y. Algoritm poiska optimalnykh usilij dlya lecheniya dvustoronnej rasshcheliny tverdogo neba [Optimum force-searching algoritm for ortopaedic treatment of two-sided cleft of the hard palate]. Rossiiskij journal biomechaniki, 2012, vol. 16, no. 3 (57), pp. 42-56.

20. Lichev S.A. Kraevye zadachi mekhaniki rastushikh tel i tonkostennykh konstruktsij [Boundary value problems in mechanics of growing solids and thin-walled structures]. Abstract of the thesis of the doctors dissertation physical and mathematics sciences. Moscow, 2012. 32 p.

21. Masich A.G. Matematicheskoe modelirovanie ortopedicheskogo lecheniya vrozhdennoj rascheliny tverdogo neba [Mathematical modeling of orthopedic treatment of congenital cleft palate in children]. Thesis of the candidate dissertation physical and mathematics sciences. Perm, 2000. 132 p.

22. Regirer S.A., Stein A.A. Metody mekhaniki sploshnoj sredy v primenenii k zadacham rosta i razvitiya biologicheskikh tkanej [Methods of continuum mechanics applied to the objectives of the growth and development of the biological tissue]. Sovremennye problemi biomechaniki, 1985, vol. 2, pp. 5-67.

23. Stein A.A., Yudina E.N. Matematicheskaya model rastuschej rastitelnoj tkani kak trekhfaznoj deformiruemoj sredy [Mathematical model of the growing plant tissue as three-phase of a deformable medium]. Rossijskij zhurnal biomekhaniki, 2011, vol. 15, no. 1, pp. 42-51.

The unbonded contact problem for bending of two-leaf spring is considered; the leaves are curved along the circular arc in their natural states. The lengths of the leaves are different; each leaf has one end clamped and the other free. The angle formed by the long leaf is less than the right one. The cross-sections of leaves are the rectangles of the same width but of the different thickness. These thicknesses are considered to vanish while the elastic lines of the leaves are analyzed geometrically. The real thicknesses affect only the bending stiffness of the leaves. The given loading is applied transversely to the leaves. There is no friction between the leaves. The bending is described by Bernoulli – Euler model. The problem is reduced to finding the density of the leaves interacting forces. This density is the sum of the piecewise-continuous part and the concentrated forces. The rigorous problem statement is formulated, the uniqueness of solution is established and the complete analytical solution of the problem is provided. This construction also proves the existence of the solution. The substantiation of the solution includes proving of the non-negativity of the contact forces and contact distances and the proving of the existence of the root of transcendental equation that gives the length of the contact segment. The proving of the non-negativity of the contact distances uses a new approach that is based on the fact that these distances can be regarded as the solutions of some variational problems. It is shown that three patterns of the leaves contact are possible: the contact along the whole short leaf; the contact at the point on the end of the short leaf; the contact along the part of the short leaf and at the point. The pattern kind depends on the given loading. The obtained results generalize the known sufficient condition for the pointwise contact of the leaves.

Keywords: two-leaf spring, curved beam, Bernoulli–Euler model, bending, contact problem, mathematical statement, contact forces, contact distances, analytical solution, contact pattern.

Michael A. Osipenko (Perm, Russian Federation) – Ph.D. in Physics and Mathematics, Associate Professor, Department of Theoretical Mechanics, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, e-mail: oma@theormech.pstu.ac.ru).

1. Rabotnov Yu.N. Mekhanika deformiruemogo tverdogo tela [Mechanics of deformable solids]. Moscow: Nauka, 1988, 711 p.

2. Osipenko M.A., Nyashin Yu.I., Rudakov R.N. A contact problem in the theory of leaf spring bending. International Journal of Solids and Structures, 2003, no. 40, pp. 3129-3136.

3. Osipenko M.A., Nyashin Y.I., Rudakov R.N. On the theory of bending of foot prosthesis containing the curved plates. Russian Journal of Biomechanics, 1999, vol. 3, no. 3, pp. 73-77.

4. Osipenko M.A., Nyashin Y.I., Rudakov R.N. The sufficient condition for the pointwise contact in the two-leaf curved elastic element of the foot prosthesis under bending. Russian Journal of Biomechanics, 2000, vol. 4, no. 3, pp. 33-41.

5. Ponomaryov S.D. [et al.] Raschety na prochnost v mashinostroenii. T. 1 [Stress calculation in mechanical engineering, vol. 1]. Мoscow: Mashgiz, 1956. 884 p.

6. Parhilovsky I.G. Avtomobilnye listovye ressory [Automotive leaf springs]. Moscow: Mashinostroenie, 1978. 232 p.

7. Talantsev N.F. Kriterii otsenki ressor [The criterions for evaluation of leaf springs]. Avtomobilnaya promyshlennost, 1988, no. 10, pp. 20-21.

8. Pestrenin V.M., Pestrenina I.V., Talantsev N.F. Chislennyj analiz napryazhenno-deformirovannogo sostoyaniya listovykh ressor [The numerical analysis of stress-strain state of leaf springs]. Vychislitelnaya mekhanika sploshnykh sred, 2009, vol. 2, no. 2, pp. 74-84.

9. Osipenko M.A., Nyashin Y.I. Novyi iteratsionnyj metod rascheta mnogolistovoj ressory [The new iterational method for calculation of multi-leaf spring]. Vychislitelnaya mekhanika sploshnykh sred, 2012, vol. 5, no. 4, pp. 371-376.

10. Osipenko M.A. Analiticheskij raschet staticheskogo izgiba dvukh­listovoj ressory s parabolicheskim profilem korotkogo lista [Аna­lytical calculation of the static bending of the combination two-leaf spring with the parabolic profile of the short leaf]. Vestnik Izhevskogo gosudarstvennogo tekhnicheskogo universiteta, 2012, no. 3 (55), pp. 146-150.

11. Feodosyev V.I. Izbrannye zadachi i voprosy po soprotivleniyu materialov [Selected problems and questions on the strength of materials]. Moscow: Nauka, 1973. 400 p.

12. Grigoluk E.I., Tolkachuov V.M. Kontaktnye zadachi teorii plastin i obolochek [The contact problems for plates and shells]. Moscow: Mashinostroenie, 1980. 415 p.

13. Johnson K.L. Contact mechanics. Cambridge University Press; Cambridge, New York, New Rochelle, Melbourne, Sydney, 1985. 510 p.

14. Elsholtz L.E. Differentsialnye uravneniya i variatsionnoe ischislenie [Differential equations and calculus of variations]. Мoscow: Editorial URSS, 2000. 320 p.

15. Kravtchuk A.S. Variatsionnye i kvazivariatsionnye neravenstva v mekhanike [Variational and quasi-variational inequalities in mechanics]. Moscovskaya gosudarstvennaya akademiya priborostroeniya i informatiki, 1997, 340 p.

The plane-stress state in the vicinity of singular point, in a plate composed of two identical elements with interlayer such as glue is studied.

The purpose of this study is determining characteristics of this solid mechanics problem and influence of material properties and interlayer thickness on stress concentration near the edge of the combining the elements (at the singular point ).

Analytical estimation of the restrictions count on the state parameters at the line edge of junction plate element and interlayer is conducted. It is shown that the count of independent restrictions depends on the material properties of the plate elements and is usually redundant (non-standard). Standard restrictions count is only an exceptional case, when certain combinations of plate elements material parameters take place. In this particular case the solution is named as a “basic-solution”. For considered case of stress-strained plate basic-solution has uniform stress and piecewise-homogeneous strain state solution. With material parameters near the basic-solution the nonstandard problem is considered by iterative numerical-analytical method based on minimizing the residual divergence of all the boundary conditions at the singular point vicinity.

A set of stress state problems is calculated. It is shown that the solution is practically independent of the interlayer thickness, but it depends drastically on the material properties of the elements. The more rigid material has the highest value concentration ratio.

Keywords: plane-stress state, singular point, stress concentration, composite plate, iterative method.

Valery M. Pestrenin (Perm, Russian Federation) – Ph.D. in Physical and Mathematical Sciences, Associate Professor Department Continuum Mechanics and Computational Technologies, Perm State National Research University (15, Bukireva str., 614990, Perm, Russian Federation, e-mail: PestreninVM@mail.ru).

Irena V. Pestrenina (Perm, Russian Federation) – Ph. D. in Tecnical Sciences, Associate Professor Department Continuum Mechanics and Computational Technologies, Perm State National Research University (15, Bukireva str., 614990, Perm, Russian Federation, e-mail: IPestrenina@gmail.com).

Lidia V. Landik (Perm, Russian Federation) – Laboratory Head, Department Continuum Mechanics and Computational Technologies, Perm State National Research University (15, Bukireva str., 614990, Perm, Russian Federation, e-mail: LidiaLandik@gmail.com).

Ekaterina A. Polyanina (Perm, Russian Federation) – Masters Degree, Department Continuum Mechanics and Computational Technologies, Perm State National Research University (15, Bukireva str., 614990, Perm, Russian Federation, e-mail: Katenapolyanina@gmail.com).

1. Goland M., Reissner E. The Stresses in Cemented Joints, Trans. ASME. Journal of Applied Mechanics, 1944, vol. 11, A17-A27.

2. Bo Zhao, Zhen-Hua Lu, Yi-Ning Lu. Closed-form solutions for elastic stress–strain analysis in unbalanced adhesive single-lap joints considering adherend deformations and bond thickness. International Journal of Adhesion & Adhesives, 2011, vol. 31, pp. 434-445.

3. Kurennov S.S. Model dvukhparametricheskogo uprugogo osnovaniya v raschete napryazhennogo sostoyaniya kleevogo soedineniya. Trudy MAI, 2013, no. 66, pp. 1-7, available at: www.mai.ru/science/trudy/

4. Bogy D.B. Two Edge-bonded Elastic Wedges of Different Materials and Wedge Angles under Surface Tractions. Trans. ASME. Ser. E, 1971, vol. 38, no. 2, pp. 87-96.

5. Chobanyan K.S. Napryazheniya v sostavnykh uprugikh telakh. Erevan: Akademiya nauk Armyanskoj SSR, 1987. 338 p.

6. Aksentyan O.K. Osobennosti napryazhenno-deformirovannogo sostoyaniya plity v okrestnosti rebra. Prikladnaia matematika i mekhanika, 1967, no. 1, pp. 178-186.

7. Aksentyan O.K., Luschik O.N. Napryazhenno-deformirovannoe sostoyanie v okrestnosti vershiny stykovogo soedineniya. Prikladnaia mekhanika, 1982, vol. 18, no. 7, pp. 66-73.

8. Sinclear G.B. Stress singularities in classical elasticity. – I: Removal, interpretation and analysis. App. Mech. Rev., 2004, vol. 57, no. 4, pp. 251-297.

9. Crocombe A.D., Adams R.D. Influence of the spew fillet and other parameters on the stress distribution in the single lap joint. Int. J. Adhes. Adhes, 1981, pp. 141-155.

10. Xu L.R., Kuai H., Sengupta S. Dissimilar material joints with and without free-edge stress singularities: Part I. A Biologically Inspired Design. Experimental mechanics, 2004, vol. 44, no. 6, pp. 608-615.

11. Xu L.R., Kuai H., Sengupta S. Dissimilar material joints with and without free-edge stress singularities: Part II. An integrated numerical analysis. Experimental mechanics, 2004, vol. 44, no. 6, pp. 616-621.

12. Matveenko V.P., Fedorov A.Y. Optimizatsiya geometrii sostavnykh uprugikh tel kak osnova sovershenstvovaniya metodik ispytanij na prochnost kleevykh soedinenij [Optimization of the geometry of compound elastic bodies with aim to improve strength test procedures for adhesive joints]. Vychislitelnaya mekhanika sploshnykh sred, 2011, vol. 4, no. 4, pp. 63-70.

13. Barut A., Guven I., Madenci E. Analysis of singular stress fields at junctions of multiple dissimilar materials under mechanical and thermal loading. Int. J. of Solid and Structures, 2001, vol. 38, no. 50-51, pp. 9077-9109.

14. Adams R.D., Atkins R.W., Harris J.A. et al. Stress analysis and failure properties of carbon-fibre-reinforced-plastic/steel double-lap joints. J. Adhes., 1986, vol. 20, pp. 29-53.

15. Pestrenin V.M., Pestrenina I.V., Landik L.V. Napryazhennoe sostoyanie vblizi osoboj tochki sostavnoj konstruktsii v ploskoj zadache [The stress state near a singular point of the flat compound designs]. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2013, no. 4(10), pp. 78-87.

16. Pestrenin V.M., Pestrenina I.V., Landik L.V. Nestandartnye zadachi dlya odnorodnykh elementov konstruktsii s osobennostyami v vide klinjev v usloviyakh ploskoj zadachi [Non-standard problems of homogeneous structural elements with wedge shape features in the plane case]. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 2014, no. 1(27), pp. 95-109.

The work deals with development of a new approach for forecasting a high-cycle fatigue life-time of bolted connection of hydro turbines runner. Operation of hydro turbines on normal operation condition does not lead to high stresses rates in bolted connection. However the high cycle fatigue failures have been occurred. High rates stresses occur in bolted connection in transient (start/stop) regimes of hydro turbines operation. The frequency of transient regimes occurrence depends from many factors and defined in this paper as a random function of time. Long-time bolted connection operation lead to natural degradation of material (aging). The degradation process is also a random process of time. So, this work pays attention to developing stochastic mathematical model of damage accumulation that take into account stochastic nature of degradation process and frequency of transient regimes occurrence. Application of the developed models is shown on real engineering example.

Degradation of properties has been modeled as a process of the reduction of fatigue (endurance) limit in time. Kinetics of damage accumulation is introduced in the context of the effective stress concept. Mathematical expectation, correlation function and the continuum damage parameter variance have been obtained as functions of time. Analysis of the influence of natural aging process on statistical parameters of damage accumulation as well as on the life-time has been carried out. The stress-strain state of bolted connection is determined by finite element method.

Keywords: fatigue, life-time, bolted connection, hydro turbine, material degradation, aging, damage models, probability approach, transient regimes, finite element method.

Oleksiy O. Larin (Kharkiv, Ukraine) – Ph.D. in Technical Sciences, Assoc. Professor, Department of Dynamics and Strength of Machines, National Technical University «Kharkiv Polytechnic Institute» (21, Frunze st., 61002, Kharkiv, Ukraine, e-mail: ale­xeya.larin@gmail.com).

Oleksandr I. Trubayev (Kharkiv, Ukraine) – Ph.D. in Technical Sciences, Assoc. Professor, Department of Dynamics and Strength of Machines, National Technical University «Kharkiv Polytechnic Institute» (21, Frunze st., 61002, Kharkiv, Ukraine, e-mail: trubayev@gmail.com).

Oleksii O. Vodka (Kharkiv, Ukraine) – Ph.D. student, Department of Dynamics and Strength of Machines, National Technical University «Kharkiv Polytechnic Institute» (21, Frunze st., 61002, Kharkiv, Ukraine, e-mail: a_vodka@mail.ru).

1. Failure Analysis Case Studies: A Sourcebook of Case Studies Selected from the Pages of Engineering Failure Analysis 1994–1996. Elsevier, 1998. 433 p.

2. Failure Analysis Case Studies II: A Sourcebook of Case Studies Selected from the Pages of Engineering Failure Analysis 1997–1999. Pergamon, 2001. 444 p.

3. Failure Analysis Case Studies III: A Sourcebook of Case Studies Selected from the Pages of Engineering Failure Analysis 2000–2002. Elsevier Science & Technology, 2004. 460 p.

4. Esaklul K.A. Handbook of Case Histories in Failure Analysis. ASM International, 1993, vol. 2. 583 p.

5. Yongyao Luo, Zhengwei Wang, Jidi Zeng, Jiayang Lin, Fatigue of piston rod caused by unsteady, unbalanced, unsynchronized blade torques in a Kaplan turbine, Engineering Failure Analysis, 2010, vol. 17, iss. 1, pp. 192-199.

6. Calin-Octavian Miclosina, Constantin Viorel Campian, Doina Frunzaverde, Vasile Cojocaru. Fatigue Analysis of an Outer Bearing Bush of a Kaplan Turbine. Analele Universităţii Eftimie Murgu Reşiţa. Fascicula de Inginerie, 2011, vol. XVIII, iss. 1, pp. 155-162.

7. Viorel C. Câmpian, Doina Frunzăverde, Dorian Nedelcu, Gabriela Mărginean. Failure analysis of a kaplan turbine runner blade. 24th Symposium on Hydraulic Machinery and Systems October 27–31, foz do iguassu, 2008, pp. 1-10.

8. Diego G., Serrano M., Lancha A.M. Failure analysis of a multiplier from a Kaplan turbine, Engineering Failure Analysis, 2000, vol. 7, iss. 1, pp. 27-34.

9. Arsić M., Bošnjak S., Međo B., Burzić M., Vistać B., Savić Z. Influence of Loading Regimes and Operational Environment on Fatigue State of Components of Turbine and Hydromechanical Equipment at Hydropower, International Conference Power Plants, Zlatibor, Serbia, 2012, pp. 1-10.

10. Mackerle J. Finite element analysis of fastening and joining: A bibliography (1990–2002), International Journal of Pressure Vessels and Piping, 2003, vol. 80, iss. 4, pp. 253-271.

11. Kulak G.L., Fisher J.W., Struik J.H.A. Guide to design criteria for bolted and riveted joints. Wiley, 1987. 333 p.

12. Bickford J.H. Introduction to the Design and Behavior of Bolted Joints. Fourth Edition: Non-Gasketed Joints, Taylor & Francis, 2007. p. 568.

13. Casanova F. Failure analysis of the draft tube connecting bolts of a Francis-type hydroelectric power plant. Engineering Failure Analysis, 2009, vol. 16, iss. 7, pp. 2202-2208.

14. Cetin A., Härkegård G. Fatigue life prediction for large threaded components. Procedia Engineering, 2010, vol. 2, iss. 1, pp. 1225-1233.

15. Libin Z., Fengrui L., Jianyu Z. 3D Numerical Simulation and Fatigue life prediction of high strength threaded bolt. Key Engineering Materials, 2010, vol. 417-418, pp. 885-888.

16. Zhang L., Feng F., Fan X., Jiang P. Reliability Analysis of Francis Turbine Blade Against Fatigue Failure Under Stochastic Loading. International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering, 15–18 June 2012, pp. 987-990.

17. Sayano-Shushenskaya power station accident, available at: http:// en.wikipedia.org/wiki/2009_Sayano-Shushenskaya_power_sta­ti­on_ac­ci­dent.

18. Dunn P.F. Measurement and Data Analysis for Engineering and Science. CRC Press, 2005. 504 p.

19. Sveshnikov A.A. Applied methods of a random functions theory. Moscow: Science, 1968. 457p.

20. Gidromekhanicheskie perekhodnye protsessy v gidroenergeti­ches­kikh ustanovkakh. Ed. G.I., Krivchenko. Energiya, 1975. 367 p.

21. Vladislavlev L.A. Vibratsiya gidroagregatov gidroelektricheskikh stantsij. Energiya, 1972. 176 p.

22. Yang L., Fatemi A. Cumulative Fatigue Damage Mechanisms and Quantifying Parameters: A Literature Review. ASTM Journal of Testing and Evaluation, 1998, vol. 26, no. 2, pp. 89-100.

23. Fatemi A., Yang L. Cumulative Fatigue Damage and Life Prediction Theories: A Survey of the State of the Art for Homogeneous Materials. International Journal of Fatigue, 1998, vol. 20, no. 1, pp. 9-34.

24. Schijve J. Fatigue of structures and materials in the 20th century and the state of the art. International Journal of Fatigue, 2003, no. 25, pp. 679-702.

25. Lemaitre J., Desmorat R. Engineering damage mechanics: ductile, creep, fatigue and brittle failure. Springer, 2005. 380 p.

26. Murakami S. A Continuum Mechanics Approachto the Analysis of Damage and Fracture. Springer, 2012. 402p.

27. Kachanov L.M. Introduction to continuum damage mechanics. Nijhoff Publ., 1986. 135 p.

28. Manson S.S. Fatigue and Durability of Structural Materials. ASM International, 2006. 456 p.

29. Schijve J. Fatigue of structures and materials. Springer, 2008. 642 p.

30. Lvov G.I., Movavgar A. Theoretical and experimental study of fatigue strength of plain woven/epoxy composite. Strojniskivestnik – Journal of mechanical Engineering, 2012, no. 58(2012)3, pp. 175-182.

31. Plumtree A., Lemaitre J. Interaction of damage mechanisms during high temperature fatigue. Adv: in Fracture Research, vol. 5. Pergamon Press, 1981, pp. 2379-2384.

32. Durrett R. Probability: theory and examples. Cambridge University Press, 2010. 428 p.

33. Zhovdak V.A., Tarasova L.F. Predicting the reliability of mechanical systems. NTU «KhPI», 2007. p. 108 (in Russian).

34. Larin A.A. Prediction and analysis of reliability of engineering structures. NTU «KhPI», 2011. p. 128 (in Russian).

35. Rathod V., Yadav O.P., Rathore A., Jain R. Probabilistic Modeling of Fatigue Damage Accumulation for Reliability Prediction. International Journal of Quality, Statistics, and Reliability, 2011, vol. 2011, 10 p.

36. Baldwin J.M., Bauer D.R., Ellwood K.R. Rubber aging in tires. Part 1: Field results, Polymer Degradation and Stability, 2007, vol. 92, pp. 104-109.

37. Mott P.H., Roland C.M. Aging of natural rubber in air and seawater. Rubber Chemistry and Technology, 2007, vol. 74, pp. 79-88.

38. Botvina L.R. [et al.] Effect of long-term aging on fatigue characteristics of steel 45. Zavodskaya Laboratoria. Diagnostika Materialov, 2011, vol. 77, pp. 58-61 (in Russian).

39. Botvina L.R., Petrova I.M., Gadolina I.V. High-cycle fatigue failure of low-carbon steel after long-term aging. Inorganic Materials, 2010, vol. 46 (14), pp. 134-141.

40. Zaletelj H. [et al.] High cycle fatigue of welded joints with aging influence. Materials and Design, 2013, no. 45, pp. 190-197.

41. Singh S., Goel B. Influence of thermomechanical aging on fatigue behavior of 2014 Al-alloy. Bulletin of Materials Science, 2005, vol. 28, iss. 2, pp. 91-96.

42. Hanaki S., Yamashita M., Uchida H., Zako M. On stochastic evaluation of S-N data based on fatigue strength distribution. International Journal of Fatigue, 2010, no. 32(3), pp. 605-609.

43. Zheng X.-L., Lü B., Jiang H. Determination of probability distribution of fatigue strength and expressions of P-S-N curves. Engineering Fracture Mechanics, vol. 50, 1995, pp. 483-491.

44. Kowalewski Z.L., Szymczak T., Makowska K., Pietrzak K. Damage Indicators During Fatigue of Metal Matrix Composites. Proceedings of the Fourth International Conference «Nonlinear Dynamics-2013», 2013, Tochka, pp. 386-391.