THERMAL SCIENCE

International Scientific Journal

Yuriy Povstenko

,

Derya Avci

,

Beyza Billur Iskender Eroğlu

,

Necati Özdemir

CONTROL OF THERMAL STRESSES IN AXISSYMMETRIC PROBLEMS OF FRACTIONAL THERMOELASTICITY FOR AN INFINITE CYLINDRICAL DOMAIN

ABSTRACT
In this paper, we study a control problem of thermal stresses in an infinite cylindrical body. The temperature distribution is defined by the time-fractional heat conduction equation with the Caputo derivative of the order 0 < α ≤ 2. The problem is formulated for axisymmetric case. The sought-for heat source function is treated as a control of stress and displacement components. For this purpose, we find the control function which guarantees the distribution of the stress component in some section of a body and at some time at a prescribed level. Integral transform technique is applied to obtain the desired control function, stresses and displacement components. Numerical results are illustrated graphically.
KEYWORDS
control of thermal stresses, time-fractional heat conduction, Caputo derivative
PAPER SUBMITTED: 2016-04-21
PAPER REVISED: 2016-05-20
PAPER ACCEPTED: 2016-06-25
PUBLISHED ONLINE: 2016-10-01
DOI REFERENCE: https://doi.org/10.2298/TSCI160421236P
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2017, VOLUME 21, ISSUE Issue 1, PAGES [19 - 28]
REFERENCES
  1. Noda, N., Hernarski, R. B., Tanigawa, Y., Thermal Stresses (2nd edn.), Taylor and Francis, New York, USA, 2003
  2. Nowacki, W., Thermoelasticity, Polish Scientific Publishers, Warszawa, 1986
  3. Parkus, H., Instationäre Wärmespannungen, Springer-Verlag, Wien, 1959
  4. Povstenko, Y., Fractional Heat Conduction Equation and Associated Thermal Stresses, J. Thermal Stresses, 28 (2005), 1, pp. 83-102
  5. Povstenko, Y., Two-Dimensional Axisymmetric Stresses Exerted by Instantaneous Pulses and Sources of Diffusion in an Infinite space in a Case of Time-Fractional Diffusion Equation. Int. J. Solids Structures, 44 (2007), 7-8, pp. 2324-2348
  6. Povstenko, Y., Thermoelasticity which Uses Fractional Heat Conduction Equation, J. Math. Sci., 162 (2009), pp. 296-305
  7. Povstenko, Y., Fractional Thermoelasticity, in: Encyclopedia of Thermal Stresses, (Ed. R.B. Hetnarski), Springer, New York, 2014, 4, pp. 1778-1787
  8. Povstenko, Y., Fractional Thermoelasticity, Springer, New York, 2015
  9. Gorenflo, R., Mainardi, F., Fractional Calculus: Integral and Differential Equations of Fractional Order, in: Fractals and Fractional Calculus in Continuum Mechanics (Eds. A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, pp. 223-276
  10. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006
  11. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, USA, 1999
  12. Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, Netherlands, 1993
  13. Carpinteri, A., Cornetti, P., A Fractional Calculus Approach to The Description of Stress and Strain Localization, Chaos, Solitons & Fractals, 13 (2002), 1, pp. 85-94
  14. Mainardi, F., Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics (Eds. A. Carpinteri, F. Mainardi), Springer-Verlag, Wien, 1997, pp. 291-348
  15. Mainardi, F., Applications of Fractional Calculus in Mechanics, Transform Methods and Special Functions (Eds. P. Rusev et. al.), Bulgarian Academy of Sciences, Sofia, 1998, pp. 309-33.
  16. Uchaikin, V.V., Fractional Derivatives for Physicists and Engineers, Springer, Berlin, 2013
  17. Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010
  18. Rabotnov, Yu. N., Creep Problems in Structural Members, North-Holland Publishing Company, Amsterdam, The Netherlands, 1969
  19. Rabotnov, Yu. N., Elements of Hereditary Solid Mechanics, Moscow, Mir, 1980
  20. Rossikhin, Yu. A., Shitikova, M.V., Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids, Applied Mechanics Reviews, 50 (1997), 1, pp. 15-67
  21. Rossikhin, Yu. A., Shitikova, M.V., Applications of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results, Applied Mechanics Reviews, 63 (2010), 1, pp. 010801
  22. Wei, S., Chen, W., Hon, Y.-C., Implicit Local Radial Basis Function Method for Solving Two-Dimensional Time Fractional Diffusion Equations, Thermal Science, 19 (2015), Suppl. 1, pp. S59-S67
  23. Pang, G., Chen, W., Fu, Z., Space-Fractional Advection-Dispersion Equation by the Kansa Method, Journal of Computational Physics, 293 (2015), pp. 280-296
  24. Chen, w., Pang, G., A New Definition of Fractional Laplacian with Application to Modeling Three-Dimensional Nonlocal Heat Conduction, Journal of Computational Physics, 309 (2016), pp. 350-367
  25. Povstenko, Y., Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, Birkhäuser, New York, 2015
  26. Vigak, V.M., Optimal Control of Nonstationary Temperature Regimes, Naukova Dumka, Kiev, 1979 (In Russian)
  27. Vigak, V.M., Control of Temperature Stresses and Displacements, Naukova Dumka, Kiev, 1988 (In Russian)
  28. Vigak, V.M., Control of Thermal Stresses and Displacements in Thermoelastic Bodies, Journal of Soviet Mathematics, 62 (1992), 1, pp. 2506-2511
  29. Vigak, V.M., Kolesov, V.S., Velichko, L.D., Optimal Control of Heating of a Thermoviscoelastic Cylinder, Mathematical Methods and Physicomechanical Fields (in Russian), 14 (1981), pp. 81-84
  30. Vigak, V.M., Lisevich, Ya. L., Optimizing Control Over the Nonstationary Temperature Regime of a Thermoelastic Orthotropic Cylinder, Mechanics of Composite Materials, 22 (1987), 6, pp. 756-760
  31. Özdemir, N., Povstenko, Y., Avcı, D., İskender, B.B., Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation, J. Thermal Stresses, 37 (2014), 8, pp. 969-980
  32. Sneddon, I.N., The Use of Integral Transforms, McGraw-Hill, New York, 1972
  33. Povstenko, Y., Axissymmetric Thermal Stresses in a Half-Space in the Framework of Fractional Thermoelasticity, Scentific Issues of Jan Dlugosz University in Czestochowa, Mathematics, 19 (2014), pp. 207-216
  34. Prudnikov, A.P., Brychkov, Yu. A., Marichev, O.I., Integrals and Series, Volume 2: Special Functions, Gordon and Breach, Amsterdam, 1986
  35. Green, A.E., Naghdi, P.M., Thermoelasticity without Energy Dissipation, Journal of Elasticity, 31 (1993), pp. 189-208
PDF VERSION [DOWNLOAD]
Control of thermal stresses in axissymmetric problems of fractional thermoelasticity for an infinite cylindrical domain

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence