Abstract:

The problem of convective flow in a layer of viscous fluid, caused by local heating, is considered. The problem solution was sought within the class of exact solutions of the thermo-gravitational convection generalizing well-known class of solutions of the Navier-Stokes equations involved the Burger’s and Sullivan’s vortexes. The two families of self-similar solutions of the problem for a unit Prandtl`s number are found, it allow to describe the evolution of two different types of radial – localized vortices. In both cases, the radial component of the velocity at a distance from the axis of symmetry of the vortices is inversely proportional to the radius, while the vertical component of the velocity and the temperature in the first case decayed as the square of the distance from the axis, and the second – exponentially. The separate linear equation for the azimuthal velocity with coefficients depending on the current function of meridian flow is obtained. By virtue of the similarity equation admits particular solutions with separated variables, the composition of which allows us to describe the transfer of angular momentum (circulation, if it is different from zero) from infinity to the center of the vortex, as well as to trace the evolution of an arbitrary localized initial perturbation of azimuthal velocity. These vortex formations are decay in the time due to the vortex and thermal diffusion. The Found exact solutions are the simple, usefully models of localized convective vortices and previously were not known.

Keywords: convection, Oberbeck – Boussinesq equations, exact solutions, localized vortexes.

Authors:

Aristov Sergey Nikolaevich (Perm, Russian Federation) – Doctor of  Physical and Mathematical Sciences, general scientist, ICMM UB RAS (1, Academic Korolev st., 614013, Perm, Russian Federation, e-mail: asn@icmm.ru).

Knyazev Denis Vyacheslavovich (Perm, Russian Federation) – Ph.D. in Physical and Mathematical Sciences, scientist, ICMM UB RAS (1, Academic Korolev st., 614013, Perm, Russian Federation, e-mail: dvk@icmm.ru).

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Abstract:

Authors:

Badriev Ildar Burkhanovich (Kazan, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, Department of Calculation Mathematics, Kazan Federal University (18, Kremlevskaja st., 420008, Kazan, Russian Federation, e-mail: ildar.badriev@ksu.ru)

Banderov Victor Victorovich (Kazan, Russian Federation) – Ph.D. in Physical and Mathematical Sciences, Deputy Director for Research and Innovations of Institute of Computer Mathematics and Information Technologies, Kazan Federal University (18, Kremlevskaja st., 420008, Kazan, Russian Federation, e-mail: Victor.banderov@ksu.ru)

Zadvornov Oleg Anatol’evich (Kazan, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, Head of Department of Calculation Mathematics, Kazan Federal University (18, Kremlevskaja st., 420008, Kazan, Russian Federation, e-mail: Oleg.Zadvornov@ksu.ru)

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Abstract:

We consider a steady process of filtration of the incompressible high-viscosity fluid, following multivalued filtration law. Generalized statement of this problem is formulated in the form of mixed variational inequality with monotone operator and separable generally nondifferentiable functional in Hilbert space. To this problem the problem of finding the boundaries of limit-equilibrium unrecovered visco-plastic oil in multilayer beds can be reduced. We establish the properties of operator (inverse strong monotonicity, coerciveness) and functional (Lipschitz continuity, convexity) contained in this variational inequality. This makes it possible to apply the known results in the theory of monotone operators to prove the existence theorem. To solve the variational inequality, we suggest iterative method that does not require the inversion of the original operator. Each step of the iterative process can essentially be reduced to the solution of the boundary-value problem for the Laplace operator. The investigation of the convergence of the iterative process is performed by its reduction to the successive approximation method for finding a fixed point of some operator (the transition operator). We obtain a relationship between the solution of the original variational inequality and the components of the fixed point of this transition operator. We show that the transition operator is nonexpanding; moreover, we obtain an inequality stronger than the nonexpansion inequality. We also show that the transition operator is asymptotically regular. This permits one to prove the weak convergence of the successive approximations. This method was realized numerically. The numerical experiments made for the model problems confirmed the efficiency of the iterative method. It is must to be mentioned that the suggested methods permit one to find approximate values not only of the solution itself but also of its characteristics (for filtration problems, these are approximate values of the solution gradient and also the approximate values of filtration rate on the sets according to the points of multivalence in filtration law), which is very useful from the practical viewpoint.

Keywords: mathematical simulation, steady filtration, variational inequality, nondifferentiable functional, inverse strongly monotone operator, iterative method.

Authors:

Badriev Ildar Burkhanovich (Kazan, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, Department of Calculation Mathematics, Kazan Federal University (18, Kremlevskaja st., 420008, Kazan, Russian Federation, e-mail: ildar.badriev@ksu.ru).

Nechaeva Ludmila Anatol’evna (Kazan, Russian Federation) – research associate, Institute of Informatics of the Tatarstan Republic Academy of Sciences (36A, Levo-Bulachnaja st., 420111, Kazan, Russian Federation, e-mail: nechaeva.ludmila 64@mail.ru).

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Abstract:

Review of matrix methods, which describe the propagation of waves in layered media, is given. The method of representation transfer matrix (or characteristic matrix) as a solution of system of first order differential equations is developed. This system of equations is called the defining. The method of obtaining the defining system of equations is shown by the example of thermoelastic waves. Traditional methods of finding the matrix exponential are considered: Taylor series expansion, Chebyshev and Padé rational approximations, scaling and squaring, methods based on numerical integration of ordinary differential equation, matrix decomposition methods (eigenvectors method, QR algorithm, Jordan canonical form, Schur decomposition, block diagonal method), Lagrange-Sylvester formula, Baker formula, Newton formula, Laplace transforms. The method of symmetric polynomials is presented. Symmetric polynomial of n-th order, introduced by the author, used to express the entire functions of matrices, including matrix exponential. This method does not require the computation (or estimation) of matrix eigenvalues. Algorithm to compute integer powers of matrices, based on the use of symmetric polynomials, is the least expensive in the number of elementary multiplications and, therefore, the most accurate in comparison with other known methods. The formulas, which analytically express transfer matrix of elastic deformation of 2-4-order through elementary symmetric polynomials, are presented. Analytical estimation of the modulus of symmetric polynomials is given. Symmetric polynomial method allows to control roundoff and truncation errors in the calculation of the transfer matrix. Evaluation of the scaling factor, which provides a reliable calculation of the matrix exponential with admissible error is made. The calculation of the transfer matrix of elastic waves in layered media by symmetric polynomials have advantages compared with other approaches on a combination of parameters such as generality, reliability, stability, accuracy, simplicity, ease of use and efficiency of the numerical algorithm.

Keywords: elastic waves, layered media, matrix, exponential, polynomials, roundoff error, scaling.

Authors:

Belyayev Yuriy Nikolaevich (Syktyvkar, Russian Federation) – Ph. D. in Physical and Mathematical Sciences, Ass. Professor, Department of Mathematical Modelling and Cybernetics, Syktyvkarskiy gosudarsrvenniy universitet (55, Oktyabrskiy av., 167001, Syktyvkar, Russian Federation, e-mail: ybelyayev@mail.ru).

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Abstract:

The present paper is devoted to the fatigue durability of structurally inhomogeneous disk of constant thickness under low-cycle (LCF) and very high-cycle (VHCF) loading. For this purpose, two modelling problems of elasticity theory for circular disk loading were solved. In the first problem the centrifugal loading is applied to the disk, also the variable and periodical in the angle radial stresses are applied on the external contour, modelling the centrifugal loading from the blades (LCF). In the second problem the disc bending equation is solved under periodical in the angle torques on the external contour. The cyclic torques model the high-frequency blades vibrations and correspond to the very high-cycle fatigue (VHCF). The structure and depth of near-surface layer with the increasing fatigue characteristics are determined for which the maximum durability is achieved for each regime of cyclical loading. Total radial and tangential stresses at the rim of the disc under the action of centrifugal forces are determined. The distribution of durability in the vicinity of the radial rim of the titanium disk (LCF mode) for a homogeneous structure and inhomogeneous structure is found. The dependence of the logarithm of the durability on the fatigue limit in the surface layer is cflculated (LCF mode). We calculated also the total radial and tangential shear stresses in the rim of the disc, corresponding to a maximum torsion of the blades in clockwise and counter-clockwise directions (VHCF). The distribution of durability along the radius of the rim titanium disc and the dependence of logarithm of the durability on the fatigue limit in the surface layer (VHCF mode) are also defined.

Keywords: very-high-cycle fatigue, low-cycle fatigue, structural inhomogeneity, near-surface layer, durability, stress concentration, vibrations of blades, flight loading cycle

Authors:

Burago Nikolay Georgievich (Moscow, Russian Federation) – Doctor of Physical and Mathematical Sciences, Leading Researcher, Laboratory of Modelling in Institute for problems in mechanics of RAS (101, Vernadskiy av., 119526, Moscow, Russian Federation, e-mail: buragong@yandex.ru).

Nikitin Ilya Stepanovich (Moscow, Russian Federation) – Doctor of Physical and Mathematical Sciences, Leading Researcher, Department of Math. Modelling Institute for computes aided RAS (19/18, 2-nd Brestskaya st., 123056, Moscow, Russian Federation, e-mail: i_nikitin@list.ru).

References:

1. Polkin I.S. Perspektivy razvitiya granulnoy metallurgii titanovih splavov [Development prospective of granular metallurgy of titanium alloys]. Tekhnologiya legkikh splavov, 2011, no. 4, pp. 5-11.

2. Garibov G.S., Grits N.M. [et al.] [Study of possibility of gas-turbine engine production with variable structure and functional-gradient characteristics from granules of different fractions]. Tekhnologia leghikh splavov, 2011, no. 4, pp. 41-50.

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5. Burago N.G., Zhuravlev A.B., Nikitin I.S. Models of Multiaxial Fatigue Fracture and Service Life Estimation of Structural Elements. Mechanics of Solids, 2011, vol. 46, no. 6, pp. 828-838.

6. Dem’yanushko I.V., Birger I.A. Raschet na prochnost’ vrashchayushchikhsya diskov [Strength calculation of rotating disks]. Moscow: Mashinostroenie, 1978, 247 p.

Abstract:

The article is devoted to the mathematical modeling of phase transformations in steels under thermomechanical loading. To construct the model, a multilevel approach based on the use of internal variables in its structure – the parameters characterizing the state and evolution of meso- and microstructure of the material – has been applied. The proposed model can be used to describe diffusionless (martensitic) and diffusion phase transformations. When modeling the diffusion phase transformations 0take into account the fact that, along with the restructuring of the crystal lattice can be redistribution of carbon and alloying elements. Statement of the general problem of individual facilitated the release of subtasks – namely, determining the stress-strain state, temperature, and describe the redistribution of alloying elements, which can be formulated in relation to independent productions. For the considered sub-tasks include different types of models. So much for the problem of determining the stress-strain state uses a direct model of the second type, and for the problems of heat conduction and diffusion – a direct model of the first type. In this case, the problem of describing the redistribution of atoms of carbon and alloying elements is considered only at the meso level, since it is at this scale diffusion processes are significant. The paper presents the general structure of the two-level model. For the problem of determining the stress-strain state, problems of heat conduction and diffusion are all considered statement on the scales. Special attention is given to obtaining a kinetic equation for the change of the volume fractions of the coexisting phases.

Keywords: phase transformation, steel, multi-level model, the internal variables.

Authors:

Isupova Irina Leonidovna (Perm, Russian Federation) – postgraduate student of Department of Mathematical Modeling of Systems and Processes, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, e-mail: enotyforever@yandex.ru).

Trusov Peter Valentinovich (Perm, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, Head of Department of Mathematical Modeling of Systems and Processes, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Fede­ration, e-mail: tpv@matmod.pstu.ac.ru).

References:

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2. Isupova I.L., Trusov P.V. Obzor matematicheskikh modeley dlya opisaniya fazovykh prevrashcheniy v stalyakh [Review of mathematical models on phase transformations in steels]. Vestnik Permskogo natsionalnogo issledovatelskogo polytekhnicheskogo universiteta. Mekhanika, 2013, no. 3, pp.

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5.  Kondepudi D., Prigogine I. Modern thermodynamics: from heat engines to dissipative structures. John Wiley & Sons, 1998, 508 p.

6. Trusov P.V., Shveykin A.I. Mnogourovnevye fizicheskie modeli mono- i polikristallov. Statisticheskie modeli [Multilevel physical models of single- and polycrystals. Statistical models]. Fizicheskaya mezomekhanika, 2011, no. 4, pp. 17-28.

7. Trusov P.V., Shveykin A.I. Mnogourovnevye fizicheskie modeli mono- i polikristallov. Pryamye modeli [Multilevel physical models of single- and polycrystals. Direct models]. Fizicheskaya mezomekhanika, 2011, vol. 14, no. 4, pp. 5-30.

8. Trusov P.V., Аshikhmin V.N., Volegov P.S., Shveykin A.I. Konstitutivnye sootnosheniya i ikh primenenie dlya opisaniya evolyutsii mikrostructury [Constitutive relations and their application to the description of microstructure evolution]. Fizicheskaya mezomekhanika, 2009, vol. 12, no. 3, pp. 61-71.

9. Trusov P.V., Аshikhmin V.N., Shveykin A.I. Dvukhurovnevaya model uprugoplasticheskogo deformirovaniya polikristallicheskikh materialov [Two-level model for polycrystalline materials elastoplastic deformation]. Mekhanika kompozitsionnykh materialov i konstruktsiy, 2009, vol. 15, no. 3, pp. 327-344.

10. Trusov P.V., Shveykin A.I., Nechaeva E.S., Volegov P.S. Mnogourovnevye modeli neuprugogo deformirovaniya materialov i ikh primenenie dlya opisaniya vnutrenney struktury [Multilevel models of inelastic deformation of material and their application for description of internal structure]. Fizicheskaya mezomekhanika, 2012, vol. 15, no. 1, pp. 33-56.

11. Trusov P.V., Nechaeva E.S., Shveykin A.I. Primenenie nesimmetrichnykh mer napryazhennogo i deformirovannogo sostoyaniya pri postroenii mnogourovnevykh konstitutivnykh modeley materialov [Non-symmetric stress-strain measures using when construct multilevel constitutive material models]. Fizicheskaya mezomekhanika, 2013, vol. 16, no. 2, pp. 15-31.

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Abstract:

In steels all known solid state phase transformations are observed: polymorphic transformation with wide spectrum of morphological and kinetics features, eutectoid decomposition (pearlite transformation), decomposition of supersaturated solid solutions, short- and long – range ordering in austenite and martensite. The important feature of these systems is different diffusion mobility of metal and carbon atoms. Therefore, reorientation of the crystal lattice may occur simultaneously with the diffusion redistribution of carbon and alloying elements.

In the present article reviews the papers on mathematical modeling of diffusionless (martensitic) and diffusion phase transformations in steels under thermomechanical loading. There are two basic approaches to constructing models of the polymorphic transformations. The first method is based on an explicit introduction to the consideration of interfaces according to the conditions at the interface of a deformable material and the kinetics of the new phase. The second method involves the development of models based on the introduction of additional state parameters that characterize certain features of the material structure "on average" (for example, the concentration of a new phase), and the formulation of relations for them. Models used to describe the phase transitions are manifold. In the present review we present works that are considered and multi-level, and self-consistent, and direct model. We also consider the work that use gradient theory, which allows to take into account the impact of scale factors on the processes of phase transformations and behavior of the investigated steel.

Keywords: review, models, phase transformations, steels.

Authors:

Isupova Irina Leonidovna (Perm, Russian Federation) – postgraduate student, Department of Mathematical Modeling of Systems and Processes, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, e-mail: enotyforever@yandex.ru).

Trusov Peter Valentinovich (Perm, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, Head of Department of Mathematical Modeling of Systems and Processes, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Fede­ration, e-mail: tpv@matmod.pstu.ac.ru).

References:

1. Grinfeld M.A. Metody mekhaniki sploshnykh sred v teorii fazovykh prevrashcheniy [Methods of continuum mechanics in the theory of phase transitions]. Moscow: Nauka, 1990. 312 p.

2. Kashchenko М.P., Chashchina V.G. Dinamicheskaya teoriya g–a martensitnogo prevrashcheniya v splavakh zheleza i reshenie problemy kriticheskogo razmera zerna [Dynamic theory of g–a martensite phase transformation in Fe-based alloys and problem of grain critical size]. Мoscow, Izhevsk: NITS Regularnaya i khaoticheskaya dinamica, Izhevskiy institute kompyuternykh issledovaniy, 2010, 132 p.

3. Kashchenko М.P., Chashchina V.G. Formirovanie martensitnykh kristallov v predelnom sluchae sverkhzvukovoy skorosti rosta [Formation of martensitic crystals in the limiting case of supersonic growth speed]. Pisma o materialakh, 2011, vol. 1, pp. 7-14.

4. Lebedev V.G., Danilov D.А., Galenko P.K. Ob uravneniyakh modeli fazovogo polya dlya neizotermicheskoy kinetiki prevrashcheniya v mnogokomponentnykh i mnogofaznykh sistemakh [Phase-field model equations for non isothermal kinetics of transitions in a multi-component and multi-phase system]. Vestnik Udmurtskogo gosudarstvennogo universiteta. Fizika i Khimiya, 2010, iss. 1, pp. 26-33.

5. Movchan А.А., Movchan I.А. Odnomernaya mikromekhanicheskaya model nelineynogo deformirovaniya splavov s pamyatyu formy pri pryamom i obratnom termouprigikh prevrashcheniyakh [One-dimen­sional micromechanical model of the nonlinear deformation of shape memory alloys for forward and reverse thermoelastic transformations]. Mekhanika kompozitsionnykh materialov i konstruktsiy, 2007, vol. 13, no. 3, pp. 297-322.

6. Movchan А.А., Movchan I.А., Silchenko L.G. Mikromekhanicheskaya model nelineynogo deformirovaniya splavov s pamyatyu formy pri fazovykh i strukturnykh prevrashcheniyakh [Micromechanical model of the nonlinear deformation of shape memory alloys during phase and structural transformations]. Izvestiya Rossiyskoy akademii nauk. Mekhanika tverdogo tela, 2010, no. 3, pp. 118-130.

7. Trusov P.V., Shveykin A.I. Mnogourovnevye fizicheskiy modeli mono- i polikristallov. Statisticheskie modeli [Multilevel physical models of single- and polycrystals. Statistical models]. Fizicheskaya mezomekhanika, 2011, no. 4, pp. 17-28.

8. Trusov P.V., Shveykin A.I. Mnogourovnevye fizicheskiy modeli mono- i polikristallov. Pryamye modeli [Multilevel physical models of single- and polycrystals. Direct models]. Fizicheskaya mezomekhanika, 2011, vol. 14, no. 4, pp. 5-30.

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Abstract:

In this paper mathematical model is developed for the study of metal melt flows general regularities in nonuniform alternating magnetic field, in which cylindrical area filed with paramagnetic conducting melt is considered, the symmetry axis is directed verticaly.The model include the equations describing spatial magnetic field distribution for the inductor representing itself shot coil, the equations for induced currents arising in metal volume at alternating inductor magnetic field, the equation of heat energy transfer that includes movement of media and volume heat sources, the equation of melt convection in Boussinesq approximation with accounting of Lorentz force actin to the melt. On the side and bottom solid area boundaries slip conditions for velocities, top boundary of melt is free. Heat sink on side surface is determined by Newton-Richman law, on the top it is calculated Stefan-Boltzmann law, bottom is considered as insulated. The governing equations and boundary conditions are given in dimensionless form. It is shown the given problem can be reduced to the consecutive solving the magnetic and thermoconvective subproblem. In axial symmetrical external magnetic field approximation for different magnetic Reynolds numbers the magnetic field strength spatiotemporal distributions in melt, densities of electrical currents and heat source powers are calculated. The parameters given above changes regularities at the control parameter – Reynolds number variations are detected. This information will later allow modeling the melt convective flow and revealing the effects are important for understanding the processes regulating the distribution of impurities.

Keywords: mathematical modeling, magnetic field, magnetic field diffusion equation, magnetic Reynolds number, induction current, convection.

Authors:

Nikulin Illarion Leonidovich (Perm, Russian Federation) – Ph.D. in Technical Sciences, Ass. Professor, Department of General Physics, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, е-mail: nil@mail.ru).

Perminov Anatoliy Victorovich (Perm, Russian Federation) – Ph.D. in Physical and Mathematical Sciences, Ass. Professor, Department of General Physics, Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation, е-mail: perminov1973@mail.ru).

References:

1. Hripchenko S.Ju. [et al.] Kristallizacija cilindricheskiih alju­mi­nie­vykh slitkov pri MGD-peremeshivanii [The Crystallization of Cylindrical Aluminum Ingots under MHD-sreering]. Tezisy dokladov Rossijskoy konferentsii po magnitnoy gidrodinamike: Perm, 18–22 ijunja 2012. Perm: Institut mekhaniki sploshnykh sred. Ural’skoe otdelenie Rossiyskoy akademii nauk, 2012, pp. 101.

2. Lyubimova T.P. [et al.] Numerical Investigation of Dynamic Magnetic Field Influence on Vertical Bridgman Crystal Growth. Proc. of Int. Conf. «Advanced Problems in Thermal Convection». Perm, 2004, рp. 343-349.

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4. Demin V.A., Makarov D.V. Vlijanie vrashhajushhegosja magnitnogo polja na rasplav v cilindricheskoj zhidkoj zone [The Rotating Magnetic Field Influence on the Melt in the Cylindrical Liquid Zone]. Vestnik Permskogo universiteta, vypusk 1, Fizika, 2004, pp. 106-111.

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Abstract:

In the earlier works of authors there was proposed the structural-phenomenological model of the materials behavior under shock-wave loading, based on the statistical-thermodynamic description of the medium with the typical mesoscopic defects (microcracks and microshears). The independent structural variables: the defect density tensor associated with the strain due to the defects and the structural scaling parameter that depends on two structural scales, ratio of the characteristic defect size and the distance between them are introduced in the model. The thermodynamic state of the system is described using the thermodynamic potential (Helmholtz free energy, which depends on the introduced structural variables). In the present study there was modified the earlier proposed model of a plate plane collision on the basis of the equations describing the evolution of the bulk and shear defects with the failure criterion and up to a volume fraction of defects in a given critical value. Formulated boundary problem of planar shock wave loading was solved numerically in the application package MATLAB using the finite difference method and the multi-step time integration with the automatic step selection. Verification of the model was carried out on the experimentally obtained free-surface velocity profiles of vanadium sample at pressures of 6 GPa. Comparison of numerical results with the experimental data shows a good agreement. Model identification of the vanadium spall strength made it possible to establish the dependence between the spall strength growth and the external influence.

Keywords: shock-wave loading, spall failure, defects

Authors:

Savelieva Natalia Vladimirovna (Perm, Russian Federation) – postgraduate student, Department of Mathematical Modeling of Systems and Processes, of Perm National Research Polytechnic University (29, Komsomolsky av., 614990, Perm, Russian Federation), research engineer ICMM UB RAS (1, Akademic Korolev st., 614013, Perm, e-mail: saveleva@icmm.ru).

Bayandin Yuriy Vitalievich (Perm, Russian Federation) – Ph.D. in Physical and Mathematical Sciences, scientist ICMM UB RAS (1, Akademic Korolev st., 614013, Russian Federation, Perm, e-mail: buv@icmm.ru).

Naimark Oleg Borisovich (Perm, Russian Federation) – Doctor of Physical and Mathematical Sciences, Professor, laboratory’s head of ICMM UB RAS (1, Akademic Korolev st., 614013, Perm, Russian Federation, e-mail: naimark@icmm.ru).

References:

1. Kanel G.I., Razorenov S.V., Utkin L.V., Fortov V.Е. Udarno-volnovye yavleniya v kondensirovannykh sredakh [Shock-wave phenomena in condensed matter]. Moscow: Yanus-K, 1996. 408 p.

2. Garkuschin G.V., Kanel G.I., Razorenov S.V. Soprotivlenie deformirovaniyu i razrusheniyu alyuminiya AD1 v usloviyakh udarno-volnovogo nagruzheniya pri temperaturax 20 i 600°C [Resistance to deformation and fracture of aluminum AD1 under shock-wave loading at 20 and 600°C]. Fizika tverdogo tela, 2010, iss. 11, vol. 52, pp. 2216-2222.

3. Mescheryakov Yu.I., Divakov А.К., Gigacheva N.I., Makarevich I.P., Muschnikova S.Yu., Kalinin G.Yu. O mexanizmakh mikro-makro e'nergoobmena pri udarnom nagruzhenii tverdykh tel [On the mechanisms of micro-macro of energy under impact loading of solids]. Pis'ma v zhurnal tekhnicheskoj fiziki, 2010, iss. 11, vol. 36, pp. 54-60.

4. Razorenov S.V., Savinych А.S., Zaretskiy Е.B., Kanel G.I., Kolobov Yu.R. Vliyanie predvaritel'nogo deformacionnogo uprochneniya na napryazhenie techeniya pri udarnom szhatii titana i titanovogo splava [The influence of strain hardening prior to the flow stress by shock compression of titanium and titanium alloy]. Fizika tverdogo tela, 2005, iss. 4, vol. 47, pp. 639-645.

5. Bo Ren, Shaofan Li, Jing Qian, Xiaowei Zeng. Meshfree simulation of spall fracture. Computer method in applied mechanics and engineering, 2011, vol. 200, pp. 797-811.

6. Danian Chen, S.T.S. Al-Hassani, M.Sarumi, Xiaogang Jin. Crack straining-based spall model. International Journal of Impact Engineering, 1997, vol. 19, no.2, pp. 107-116.

 

7. Chen Danian, Yu Yuying, Yin Zhihua, Wang Huanran, Liu Guoqing, Xie Shugang. A modified Cochran-Banner spall model. International Journal of Impact Engineering, 2005, vol. 31, pp. 1106-1118.

8. Naimark О.B. Kollektivnye svojstva ansamblej defektov i nekotorye nelinejnye problemy plastichnosti i razrusheniya [Collective properties of ensembles of defects and some nonlinear problems of plasticity and fracture] Fizicheskaya mezomekhanika, 2003, vol. 6, iss. 4, pp. 45-72.

9. Saveleva S.V., Bayandin Yu.V., Naimark O.B. Chislennoe modelirovanie deformirovaniya i razrusheniya metallov v usloviyax ploskogo udara [Numerical simulation of deformation and fracture of metals under plane shock wave loading]. Vychislitel'naya mekhanika sploshnykh sred, 2012, vol. 5, no. 3, pp. 300-307.

10. Bayandin Yu.V. Issledovanie avtomodel'nykh zakonomernostej formirovaniya plasticheskikh frontov v metallakh pri intensivnykh vozdejstviyakh [Investigation of self-similar regularities of formation of plastic fronts in metals in the intensity of the impact]: PhD thesis. Thesis of doctors degree dissertation, Perm, 2007, 119 p.

11. Bayandin Yu.V., Naimark О.B., Uvarov S.V. Strukturno-skejlingovye perekhody pri dinamicheskikh i udarno-volnovykh nagruzkakh v tverdykh telakh [Structural-scaling transitions in dynamic and shock waves in solids loadings]. Fizika e'kstremal'nykh sostoyanij veshhestva – 2008. Chernogolovka, 2008, pp. 122-124.

12. Tonks D.L. The DataShop: а Database of Weak-Shock Constitutive Data. LosAlamos, New Mexico, 1991, 135 p.

Abstract:

Three dimensional equations and boundary conditions of non-interacted thermoelasticity of micropolar orthotropic bodies with free fields of displacements and rotations are studied. Taking into consideration qualatative aspects of asymptotic solution of boundary-value problem of three-dimensional micropolar thermoelasticity in thin domain of the shell, adequate kynematic and static hypotheses are formulated for the construction of applied two-dimensional theory of thermoelasticity of micropolar orthotropic thin shells with free fields of diaplacements and rotations. Accepted kynemathic hypotheses are Timoshnenko's kynemathic hypotheses generalized for micropolar case. Beside of the hypothesis of normal stresses, accepted in theory of thin shells, some static assumptions are formulated which are conformable to asymptotic theory. For temperature function assumption of its linear distribution by shell thickness is accepted.

On the basis of the accepted generalized hypotheses applied theory of thermoelasticity of micropolar orthotropic thin shells with free fields of displacements and rotations is constructed. Theories of thermoelasticity of micropolar orthotropic thin bars and plates with free fields of displacements and rotations are obtained as private cases.

Keywords: micropolar elasticity, orthotropic material, thin shell, temperature tension general theory.

Authors:

Sargsyan Samvel Hovhannes (Gyumri, Republic of Armenia) – Ph.D., Doctor of Sciences, Professor, Correspondent-member of NAS RA, Head for Higher Mathematics Chair, Gyumri State Pedagogical Institute (4, Paruyr Sevak st., 377526, Gyumri, Republic of Armenia, e-mail: s_sargsyan@yahoo.com).

Farmanyan Anahit Jora (Gyumri, Republic of Armenia) – Ph.D., Vice-rector on Scientific and International Affairs, Gyumri State Pedagogical Institute (4, Paruyr Sevak st., 377526, Gyumri, Republic of Armenia,
e-mail: afarmanyan@yahoo.com).

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5. Nowacki W. Theory of asymmetric elasticity. Pergamon Press. Oxford. New York. Toronto. Sydney. Paris. Frankfurt, 1986, 383 p.

6. Sargsyan S.H. Matematicheskaya model mikropolyarnix uprugikh tonkikh plastin I osobennosti ikh prochnostnikh i joskostnikh xarakteristikh [Mathematical model of micropolar elastic thin plates and features of strength and rigidity]. Applied mechanics and technical physics, 2012, vol. 53, no. 2, pp.148-155.

7. Sargsyan S.H. Obshaya teoriya mikropolyarnix uprugix tonkix obolochek [General theory of micropolar elastic thin shells]. Physical mesomechanics, 2011, vol. 14, no. 1, pp. 55-66.

8. Sargsyan S.H. Mathematical models of micropolar elastic thin shells. Advanced structured materials. Shell-like structures. Non-classical theories and applications. Springer, 2011, vol. 15, pp. 91-100.

9. Sargsyan S.H. Termoelasticity of thin shells on the basis of asymmetrical theory of elasticity. Journal of Thermal Stresses, 2009, vol. 32, no. 6, pp. 791-818.

10. Sargsyan S.H. Termouprugost mikropolyarnikh tonkikh obolochek [Thermoelasticity of micropolar thin shells]. Sbornik nauchnykh trudov mezhdunarodnoy konferencii «Aktualnye problemy mehaniki sploshnoi sredy». 8-12 October 2012. Tcaxkadzor. Armenia. Yerevan, 2012, pp. 184-189.

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13. Grigorenko Ya.M., Vasilenko A.T. Teoriya obolochek peremennoy jostkosti [Theory of shells of variable rigidity]. Kiev: Naukova dumka, 1981, 544 p.

14. Podstrigach Ya.S., Pelekh B.L. Termouprugie zadachi dlya obolochek I plastin s nizkoj sdvigavoy joskostu [Thermoelastic problems for shells and plates with low shear rigidity]. Teplovye napryazheniya v e’lementakh konstruktsiy, 1970, vol. 10, pp. 17-23.