Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

FRACTIONAL ORDER THERMOELASTIC PROBLEM FOR FINITE PIEZOELECTRIC ROD SUBJECTED TO DIFFERENT TYPES OF THERMAL LOADING - DIRECT APPROACH

  • GAIKWAD, KISHOR R. (PG DEPARTMENT OF MATHEMATICS, NES, SCIENCE COLLEGE) ;
  • BHANDWALKAR, VIDHYA G. (PG DEPARTMENT OF MATHEMATICS, NES, SCIENCE COLLEGE)
  • Received : 2021.05.22
  • Accepted : 2021.09.19
  • Published : 2021.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

The problem of generalized thermoelasticity of two-temperature for finite piezoelectric rod will be modified by applying three different types of heating applications namely, thermal shock, ramp-type heating and harmonically vary heating. The solutions will be derived with direct approach by the application of Laplace transform and the Caputo-Fabrizio fractional order derivative. The inverse Laplace transforms are numerically evaluated with the help of a method formulated on Fourier series expansion. The results obtained for the conductive temperature, the dynamical temperature, the displacement, the stress and the strain distributions have represented graphically using MATLAB.

Keywords

Acknowledgement

ORCID ID: 0000-0003-3551-301X

References

  1. H. W. Lord and Y. Shulman, A Generalized Dynamical Theory of Thermoelasticity, J. Mech. Phys. Solids, 15 (1967), 299-309. https://doi.org/10.1016/0022-5096(67)90024-5
  2. A. E. Green and K. E. Lindsay, Thermoelasticity, Journal of Elasticity, 2 (1972), pp. 1-7. https://doi.org/10.1007/BF00045689
  3. H. M. Youssef, E. Bassiouny, Two-Temperature Generalized Thermopiezoelasticity for One Dimensional Problems-State Space Approach, Computational Methods in Science and Technology, , 14 (2008), 55-64. https://doi.org/10.12921/cmst.2008.14.01.55-64
  4. H. Tianhu, T. Xiaogeneg and S. Yapeng, State Space Approach to One-Dimensional Shock Problem for A Semi-Infinite Piezoelectric Rod, Int. J. Eng. Sci., 40 (2002), 1081-1097. https://doi.org/10.1016/S0020-7225(02)00005-8
  5. W. Nowacki, Some General Theorems of Thermo-Piezoelectricity, Journal of Thermal Stresses, 1 (1978), 171-182. https://doi.org/10.1080/01495737808926940
  6. A. E. Abouelregal, Fractional order generalized thermopiezoelectric semi-infinite medium with temperature dependent properties subjected to a ramp-type heating, Journal of Thermal Stresses, 34(11), (2011), 1139-1155. https://doi.org/10.1080/01495739.2011.606018
  7. D. S. Chandrasekharaiah, A Generalized Thermoelastic Wave Propagation in a Semi-Infinite Piezoelectric Rod, Acta. Mech., 71 (1983), 39-49. https://doi.org/10.1007/BF01173936
  8. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1998.
  9. K. S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
  10. Yang X.-J., H. M. Srivastava, and Jose A. T. Machado, A New Fractional Derivative Without Singular Kernel: Application to The Modelling of the Steady Heat Flow, Thermal Science, 20(2) (2016), 753-756. https://doi.org/10.2298/tsci151224222y
  11. J. Losada and Nieto, Properties of A New Fractional Derivative Without Singular Kernel, Progress in Fractional Differentiation and Applications, 1(2) (2015), 87-92.
  12. W. E. Raslan, Application of Fractional Order Theory of Thermoelasticity to A 1D Problem for A Spherical Shell, J. Theoretical and Applied Mechanics, vol. 54(1) (2016), 295-304. https://doi.org/10.15632/jtam-pl.54.1.295
  13. H. H. Sherief, and A. M. El-Latief, Application of Fractional Order Theory of Thermoelasticity to a 1D Problem for A Half-Space, ZAMM, 94 (2014), 509-515. https://doi.org/10.1002/zamm.201200173
  14. H. H. Sherief, A. M. Abd El-Latief, Application of Fractional Order Theory of Thermoelasticity to a 1D Problem for a Half-space, ZAMM, 2(2013), 1-7.
  15. H. H. Sherief, A. M. A. El-Sayed, and A. M. Abd El-Latief, Fractional Order Theory of Thermoelasticity, Int. J. Solids Struct., 47 (2010), 269-275. https://doi.org/10.1016/j.ijsolstr.2009.09.034
  16. H. M. Youssef, E. A. Al-Lehaibi, Fractional Order Generalized Thermoelastic Infinite Medium with Cylindrical Cavity Subjected to Harmonically Varying Heat, Engineering, 3 (2011), 32-37. https://doi.org/10.4236/eng.2011.31004
  17. H. M. Youssef, Fractional Order Generalized Thermoelasticity, Journal of Heat Transfer, 132(6) (2010).
  18. M. A. Ezzat, A. S. El-Karamany, A. A. El-Bary, Magneto-Thermoelasticity with Two Fractional Order Heat Transfer, J. of the Assoc. of Arab Universities for Basic and Applied Sciences, 19 (2016), 70-79. https://doi.org/10.1016/j.jaubas.2014.06.009
  19. G. Honig, U. Hirdes, A Method for The Numerical Inversion of the Laplace Transform, Journal of Computational and Applied Mathematics, 10 (1984), 113-132. https://doi.org/10.1016/0377-0427(84)90075-X
  20. K. R. Gaikwad, K. P. Ghadle., Quasi-static thermoelastic problem of an infinitely long circular cylinder, Journal of the Korean Society for Industrial and Applied Mathematics, 14 (2010), 141-149. https://doi.org/10.12941/jksiam.2010.14.3.141
  21. K. R. Gaikwad, K. P. Ghadle, On a certain thermoelastic problem of temperature and thermal stresses in a thick circular plate, Australian Journal of Basic and Applied Sciences, 6 (2012), 34-48.
  22. K. R. Gaikwad, K. P. Ghadle, Non-homogeneous heat conduction problem and its thermal deflection due to internal heat generation in a thin hollow circular disk, Journal of Thermal stresses, 35(6) (2012), 485-498. https://doi.org/10.1080/01495739.2012.671744
  23. K. R.Gaikwad., Analysis of thermoelastic deformation of a thin hollow circular disk due to partially distributed heat supply, Journal of Thermal stresses, 36 (3) (2013), 207-224. https://doi.org/10.1080/01495739.2013.765168
  24. K. R. Gaikwad, K. P. Ghadle, Thermal stresses in a non-homogeneous semi-infinite hollow circular disk due to internal heat generation, International Journal of Advanced Research in Basic and Applied Sciences, 2(1) (2015), 130-133.
  25. K. R. Gaikwad, Mathematical Modelling of thermoelastic problem in a circular sector disk subject to heat generation, International Journal of Advances in Applied Mathematics and Mechanics, 2(3)(2015), 183-195.
  26. K. R. Gaikwad, Two-Dimensional study-state temperature distribution of a thin circular plate due to uniform internal energy generation, Cogent Mathematics, 3(1) (2016), 1-10. https://doi.org/10.1080/23311835.2015.1135720
  27. K. R. Gaikwad, Steady-state heat conduction problem in a thick circular plate and its thermal stresses, International Journal of Pure and Applied Mathematics, 115(2) (2017), 301-310. https://doi.org/10.12732/ijpam.v115i2.8
  28. K. R. Gaikwad, Axi-symmetric thermoelastic stress analysis of a thin circular plate due to heat generation, International Journal of Dynamical Systems and Differential Equations, 9 (2019), 187-202. https://doi.org/10.1504/IJDSDE.2019.100571
  29. K. R. Gaikwad, S. G. Khavale, Time Fractional Heat Conduction Problem in a Thin Hollow Circular Disk and Its Thermal Deflection, Easy Chair, 1672 (2019).
  30. K. R. Gaikwad, Y. U. Naner, Transient Thermoelastic Stress Analysis of A A Thin Circular Plate Due to Uniform Internal Heat Generation, Journal of the Korean Society for Industrial and Applied Mathematics, 24(3) (2020), 293-303. https://doi.org/10.12941/JKSIAM.2020.24.293
  31. S. G. Khavale, K. R. Gaikwad, Generalized theory of magneto-thermo-viscoelastic Spherical cavity problem under Fractional order derivative: State Space Approach, Advances in Mathematics:Scientific Journal, 9 (2020), 9769-9780, 2020. https://doi.org/10.37418/amsj.9.11.86
  32. K. R. Gaikwad, Y. U. Naner, Analysis Of Transient Thermoelastic Temperature Distribution Of A Thin Circular Plate And Its Thermal Deflection Under Uniform Heat Generation, Journal of thermal stress, 44(1) (2021), 75-85. https://doi.org/10.1080/01495739.2020.1828009
  33. K. R. Gaikwad, Y. U. Naner, Green's function approach to Thermal Deflection of a thin hollow circular disk under axisymmetric heat source, Journal of the Korean Society for Industrial and Applied Mathematics, 25(1) (2021), 1-15. https://doi.org/10.12941/JKSIAM.2021.25.001
  34. K. R. Gaikwad, Y. U. Naner, GreenS Function Approach To Transient Thermoelastic Deformation of a Thin Hollow Circular Disk Under Axisymmetric Heat Source, JP Journal of Heat and Mass Transfer, 22(2) (2021), 245-257. https://doi.org/10.17654/HM022020245
  35. E. Bassiouny and H. M. Youssef, Two-temperature Generalized Thermopiezoelasticity of Finite Rod Subjected to Different Types of Thermal Loading, Journal of Thermal Stresses, 31 (2008), 1-13. https://doi.org/10.1080/01495730701737803
  36. M. Caputo, et.al., A New Definition of Fractional Derivative Without Singular Kernel, Progress in Fractional Differentiation and Applications, 1(2) (2015), 73-85.