A 1D thermoelastic response of skin tissue due to ramp-type heating via a fractional-order Lord–Shulman model

Document Type : Research Paper

Authors
Ashraf M. Zenkour 1 , 2
Tariq Saeed 2 , 3
Ahlam A. Al-Raezah 1 , 4
1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia.
2 Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt.
3 Financial Mathematics and Actuarial Science (FMAS)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia.
4 Department of Mathematics, Faculty of Science, King Khalid University, Abha 9004, Saudi Arabia.
10.22059/jcamech.2023.364796.871
Abstract
This paper presents a new mathematical perfect of fractional order to deal with the response of skin tissue subjected to ramp-type heating based on the refined Lord–Shulman generalized thermoelasticity model. Three different models; the classical, the simple Lord–Shulman as well as the refined Lord–Shulman will be discussed. The governing equivalences of the present three models are attained. The general solution for the initial and boundary condition problem is found by employing the Laplace transform approach and its inverse. Numerical results are represented in figures with a comparison to the different theories with different values of fractional order to discuss the impact of the fractional order on temperature, displacement, and dilatation distributions. The effect of ramp-type heat is studied numerically and graphically on distributions of temperature, displacement, and dilatation according to different theories.
Keywords
Refined Lord– Shulman theory
fractional calculus
Laplace transforms
bio-heat response
skin tissue
ramp-type
Subjects
  1. Fox, Generalised thermoelasticity, International Journal of Engineering and Science, Vol. 7, pp. 437-445, 1969.‏
  2. M. Zenkour, T. Saeed and A.M. Aati, Refined Dual-Phase-Lag Theory for the 1D Behavior of Skin Tissue under Ramp-Type Heating, Materials, Vol. 16, pp. 2421, 2023.‏
  3. A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics, Vol. 27, pp. 240-253, 1956.‏
  4. W. Lord and Y.A. Shulman, Generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids, Vol. 15, pp. 299-309, 1967.‏
  5. E. Green and K.A. Lindsay, Thermoelasticity, Journal of Elasticity, Vol. 2, pp. 1-7, 1972.‏
  6. S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Applied Mechanics Reviews, Vol. 51, pp. 705-729,‏ 1998.
  7. B. Hetnarski, and J. Ignaczak. Generalized thermoelasticity, Journal of Thermal Stresses, Vol. 22, pp. 451-476, 1999.‏
  8. Ignaczak and O-S. Martin, Thermoelasticity with Finite Wave Speeds. OUP Oxford, 2009.‏
  9. H. Sherief, Fundamental solution of the generalized thermoelastic problem for short times. Journal of Thermal Stresses, Vol. 9, pp. 151-164, 1986.‏
  10. A. Ezzat and A.S. El-Karamany. The uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with two relaxation times, International Journal of Engineering Science, Vol. 40, pp. 1275-1284, 2002.‏
  11. Sharma, K. Sharma and R.R. Bhargava, Effect of viscosity on wave propagation in anisotropic thermoelastic with Green–Naghdi theory type-II and type-III, Materials Physics and Mechanics, Vol. 16, pp. 144-158, 2013.‏
  12. I. Othman and E. M. Abd-Elaziz, Influence of gravity and microtemperatures on the thermoelastic porous medium under three theories, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29, pp. 3242-3262, 2019.
  13. Sharma and P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses, Vol. 36, pp. 94-111, 2013.‏
  14. Lata, R. Kumar and N. Sharma. Plane waves in an anisotropic thermoelastic, Steel and Composite Structures, Vol. 22, pp. 567-587, 2016.‏
  15. D. Hobiny and I.A. Abbas, Analytical solutions of the temperature increment in skin tissues caused by moving heating sources, Steel and Composite Structures, Vol. 40, pp. 511-516, 2021.
  16. M. Zenkour and I.A. Abbas, A generalized thermoelasticity problem of an annular cylinder with temperature-dependent density and material properties. International Journal of Mechanics and Sciences, Vol. 84, pp. 54-60, 2014.‏
  17. E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Royal Society, Vol. 432, pp. 171-194, 1991.
  18. E. Green, and P.M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses, Vol. 15, pp. 253-264, 1992.
  19. E. Green, and P.M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity, Vol. 31, pp. 189-208, 1993
  20. S.A. Chandrasekharaiah, Uniqueness theorem in the theory of thermoelasticity without energy dissipation, Journal of Thermal Stresses, Vol. 19, pp. 267-272, 1996.‏
  21. Kumar, A.K. Vashishth and S. Ghangas, Phase-lag effects in skin tissue during transient heating, International Journal of Applied Mechanics and Engineering, Vol. 24, pp. 603-623, 2019.
  22. K. Sharma and D. Kumar, A study on non-linear DPL model for describing heat transfer in skin tissue during hyperthermia treatment, Entropy, Vol. 22, 481, 2020.
  23. Chiriţă and M. Ciarletta, Reciprocal and variational principles in linear thermoelasticity without energy dissipation, Mechanics Research Communications, Vol. 37, pp. 271-275, 2010.
  24. Ghazanfarian, Z. Shomali and A. Abbassi, Macro-to nanoscale heat and mass transfer: the lagging behavior, International Journal of Thermophysics, Vol. 36, pp. 1416-1467, 2015.‏
  25. I.A. Othman and I.A. Abbas, Generalized thermoelasticity of thermal-shock problem in a non-homogeneous isotropic hollow cylinder with energy dissipation, International Journal of Thermophysics, Vol. 33, pp. 913-923, 2012.‏
  26. H. Sherief and W.E. Raslan, A 2D problem of thermoelasticity without energy dissipation for a sphere subjected to axisymmetric temperature distribution, Journal of Thermal Stresses, Vol. 40, pp. 1461-1470, 2017.‏
  27. M. Zenkour, T. Saeed and K.M. Alnefaie, Refined Green–Lindsay model for the response of skin tissue under a ramp-type heating, Mathematics, Vol. 11, 1437, 2023.‏
  28. L. Bagley and P.J. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology, Vol. 30, pp. 133-155, 1999.
  29. A. Ezzat and A.S. El-Karamany, Fractional-order theory of a perfect conducting thermoelastic medium, Canadian Journal of Physics, Vol. 89, pp. 311-318, 2011.‏
  30. H. Hendy, M.M. Amin and M.A. Ezzat, Two-dimensional problem for thermoviscoelastic materials with fractional-order heat transfer, Journal of Thermal Stresses, Vol. 42, pp. 1298-1315, 2019.‏
  31. A. Abbas, Eigenvalue approach to fractional-order generalized magneto-thermoelastic medium subjected to moving heat source, Journal of Magnetism and Magnetic Materials, Vol. 377, pp. 452-459, 2015.‏
  32. Sharma, R. Kumar and P. Lata, Disturbance due to inclined load in transversely isotropic thermoelastic medium with two temperatures and without energy dissipation, Materials Physics and Mechanics, Vol. 22, pp. 107-117, 2015.‏
  33. A. Abbas, Eigenvalue Approach for an unbounded medium with a spherical cavity based upon two-temperature generalized thermoelastic theory, Journal of Mechanical Science and Technology, Vol. 28, pp. 4193-4198, 2014.‏
  34. A. Ezzat and M.Z. Abd-Elaal, Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium, Journal of the Franklin Institute, Vol. 334, pp. 685-706, 1997.
  35. H. Sherief, A.M.A. El-Sayed and A.M. Abd El-Latief, Fractional Order Theory of Thermoelasticity, International Journal of Solids and Structures, Vol. 47, pp. 269-275, 2010.‏
  36. H. Sherief and W.E. Raslan, 2D Problem for a long cylinder in the fractional theory of thermoelasticity, Journal of Solids and Structures, Vol. 13, pp. 1596-1613, 2016.‏
  37. H. Hendy, S.I. El-Attar and M.A. Ezzat, Two-temperature fractional Green–Naghdi of type iii in magneto-thermo-viscoelasticity theory subjected to a moving heat source, Indian Journal of Physics, Vol. 95, pp. 657-671, 2021.‏
  38. J. Chen and M.E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift Für Angewandte Mathematik Und Physik (ZAMP), Vol. 19, pp. 614-627, 1968.‏
  39. A. Ezzat and E.S. Awad, Constitutive relations, uniqueness of solution, and thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures, Journal of Thermal Stresses, Vol. 33, pp. 226-250, 2010.‏
  40. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, 198, 1998.
  41. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition using Differential Operators of Caputo Type, Springer, 2010.
  42. A. Abbas and A.M. Zenkour, Semi-analytical and numerical solution of fractional-order generalized thermoelastic in a semi-infinite medium, Journal of Computational and Theoretical Nanoscience, Vol. 11, pp. 1592-1596, 2014.‏
  43. A. Kilbas, M. Rivero, L. Rodriguez-Germa and J.J. Trujillo, Caputo linear fractional differential equations. IFAC Proceedings, Vol. 39, pp. 52-57, 2006.‏
  44. A. Ezzat, A. S. El-Karamany, A. A. El-Bary and M. A. Fayik, Fractional calculus in one-dimensional isotropic thermo-viscoelasticity, Computes Rendus Mecanique, Vol. 341, pp. 553-566, 2013.
  45. Zhang, Y. Sunan and J. Yang, Bio-heat response of skin tissue based on three-phase-lag model, Scientific Reports, Vol. 10, pp. 16421, 2020.
  46. Sobhy and A.M. Zenkour, Refined Lord–Shulman theory for 1D response of skin tissue under ramp-type heat, Materials, Vol. 15, pp. 6292, 2022.‏
  47. Honig and U. Hirdes, A method for the numerical inversion of the Laplace transform, Journal of Computational and Applied Mathematics, Vol. 10, pp. 113-132, 1984.
  48. Fox, Generalised thermoelasticity, International Journal of Engineering and Science, Vol. 7, pp. 437-445, 1969.‏
  49. M. Zenkour, T. Saeed and A.M. Aati, Refined Dual-Phase-Lag Theory for the 1D Behavior of Skin Tissue under Ramp-Type Heating, Materials, Vol. 16, pp. 2421, 2023.‏
  50. A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics, Vol. 27, pp. 240-253, 1956.‏
  51. W. Lord and Y.A. Shulman, Generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids, Vol. 15, pp. 299-309, 1967.‏
  52. E. Green and K.A. Lindsay, Thermoelasticity, Journal of Elasticity, Vol. 2, pp. 1-7, 1972.‏
  53. S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Applied Mechanics Reviews, Vol. 51, pp. 705-729,‏ 1998.
  54. B. Hetnarski, and J. Ignaczak. Generalized thermoelasticity, Journal of Thermal Stresses, Vol. 22, pp. 451-476, 1999.‏
  55. Ignaczak and O-S. Martin, Thermoelasticity with Finite Wave Speeds. OUP Oxford, 2009.‏
  56. H. Sherief, Fundamental solution of the generalized thermoelastic problem for short times. Journal of Thermal Stresses, Vol. 9, pp. 151-164, 1986.‏
  57. A. Ezzat and A.S. El-Karamany. The uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with two relaxation times, International Journal of Engineering Science, Vol. 40, pp. 1275-1284, 2002.‏
  58. Sharma, K. Sharma and R.R. Bhargava, Effect of viscosity on wave propagation in anisotropic thermoelastic with Green–Naghdi theory type-II and type-III, Materials Physics and Mechanics, Vol. 16, pp. 144-158, 2013.‏
  59. I. Othman and E. M. Abd-Elaziz, Influence of gravity and microtemperatures on the thermoelastic porous medium under three theories, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29, pp. 3242-3262, 2019.
  60. Sharma and P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses, Vol. 36, pp. 94-111, 2013.‏
  61. Lata, R. Kumar and N. Sharma. Plane waves in an anisotropic thermoelastic, Steel and Composite Structures, Vol. 22, pp. 567-587, 2016.‏
  62. D. Hobiny and I.A. Abbas, Analytical solutions of the temperature increment in skin tissues caused by moving heating sources, Steel and Composite Structures, Vol. 40, pp. 511-516, 2021.
  63. M. Zenkour and I.A. Abbas, A generalized thermoelasticity problem of an annular cylinder with temperature-dependent density and material properties. International Journal of Mechanics and Sciences, Vol. 84, pp. 54-60, 2014.‏
  64. E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Royal Society, Vol. 432, pp. 171-194, 1991.
  65. E. Green, and P.M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses, Vol. 15, pp. 253-264, 1992.
  66. E. Green, and P.M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity, Vol. 31, pp. 189-208, 1993
  67. S.A. Chandrasekharaiah, Uniqueness theorem in the theory of thermoelasticity without energy dissipation, Journal of Thermal Stresses, Vol. 19, pp. 267-272, 1996.‏
  68. Kumar, A.K. Vashishth and S. Ghangas, Phase-lag effects in skin tissue during transient heating, International Journal of Applied Mechanics and Engineering, Vol. 24, pp. 603-623, 2019.
  69. K. Sharma and D. Kumar, A study on non-linear DPL model for describing heat transfer in skin tissue during hyperthermia treatment, Entropy, Vol. 22, 481, 2020.
  70. Chiriţă and M. Ciarletta, Reciprocal and variational principles in linear thermoelasticity without energy dissipation, Mechanics Research Communications, Vol. 37, pp. 271-275, 2010.
  71. Ghazanfarian, Z. Shomali and A. Abbassi, Macro-to nanoscale heat and mass transfer: the lagging behavior, International Journal of Thermophysics, Vol. 36, pp. 1416-1467, 2015.‏
  72. I.A. Othman and I.A. Abbas, Generalized thermoelasticity of thermal-shock problem in a non-homogeneous isotropic hollow cylinder with energy dissipation, International Journal of Thermophysics, Vol. 33, pp. 913-923, 2012.‏
  73. H. Sherief and W.E. Raslan, A 2D problem of thermoelasticity without energy dissipation for a sphere subjected to axisymmetric temperature distribution, Journal of Thermal Stresses, Vol. 40, pp. 1461-1470, 2017.‏
  74. M. Zenkour, T. Saeed and K.M. Alnefaie, Refined Green–Lindsay model for the response of skin tissue under a ramp-type heating, Mathematics, Vol. 11, 1437, 2023.‏
  75. L. Bagley and P.J. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology, Vol. 30, pp. 133-155, 1999.
  76. A. Ezzat and A.S. El-Karamany, Fractional-order theory of a perfect conducting thermoelastic medium, Canadian Journal of Physics, Vol. 89, pp. 311-318, 2011.‏
  77. H. Hendy, M.M. Amin and M.A. Ezzat, Two-dimensional problem for thermoviscoelastic materials with fractional-order heat transfer, Journal of Thermal Stresses, Vol. 42, pp. 1298-1315, 2019.‏
  78. A. Abbas, Eigenvalue approach to fractional-order generalized magneto-thermoelastic medium subjected to moving heat source, Journal of Magnetism and Magnetic Materials, Vol. 377, pp. 452-459, 2015.‏
  79. Sharma, R. Kumar and P. Lata, Disturbance due to inclined load in transversely isotropic thermoelastic medium with two temperatures and without energy dissipation, Materials Physics and Mechanics, Vol. 22, pp. 107-117, 2015.‏
  80. A. Abbas, Eigenvalue Approach for an unbounded medium with a spherical cavity based upon two-temperature generalized thermoelastic theory, Journal of Mechanical Science and Technology, Vol. 28, pp. 4193-4198, 2014.‏
  81. A. Ezzat and M.Z. Abd-Elaal, Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium, Journal of the Franklin Institute, Vol. 334, pp. 685-706, 1997.
  82. H. Sherief, A.M.A. El-Sayed and A.M. Abd El-Latief, Fractional Order Theory of Thermoelasticity, International Journal of Solids and Structures, Vol. 47, pp. 269-275, 2010.‏
  83. H. Sherief and W.E. Raslan, 2D Problem for a long cylinder in the fractional theory of thermoelasticity, Journal of Solids and Structures, Vol. 13, pp. 1596-1613, 2016.‏
  84. H. Hendy, S.I. El-Attar and M.A. Ezzat, Two-temperature fractional Green–Naghdi of type iii in magneto-thermo-viscoelasticity theory subjected to a moving heat source, Indian Journal of Physics, Vol. 95, pp. 657-671, 2021.‏
  85. J. Chen and M.E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift Für Angewandte Mathematik Und Physik (ZAMP), Vol. 19, pp. 614-627, 1968.‏
  86. A. Ezzat and E.S. Awad, Constitutive relations, uniqueness of solution, and thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures, Journal of Thermal Stresses, Vol. 33, pp. 226-250, 2010.‏
  87. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, 198, 1998.
  88. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition using Differential Operators of Caputo Type, Springer, 2010.
  89. A. Abbas and A.M. Zenkour, Semi-analytical and numerical solution of fractional-order generalized thermoelastic in a semi-infinite medium, Journal of Computational and Theoretical Nanoscience, Vol. 11, pp. 1592-1596, 2014.‏
  90. A. Kilbas, M. Rivero, L. Rodriguez-Germa and J.J. Trujillo, Caputo linear fractional differential equations. IFAC Proceedings, Vol. 39, pp. 52-57, 2006.‏
  91. A. Ezzat, A. S. El-Karamany, A. A. El-Bary and M. A. Fayik, Fractional calculus in one-dimensional isotropic thermo-viscoelasticity, Computes Rendus Mecanique, Vol. 341, pp. 553-566, 2013.
  92. Zhang, Y. Sunan and J. Yang, Bio-heat response of skin tissue based on three-phase-lag model, Scientific Reports, Vol. 10, pp. 16421, 2020.
  93. Sobhy and A.M. Zenkour, Refined Lord–Shulman theory for 1D response of skin tissue under ramp-type heat, Materials, Vol. 15, pp. 6292, 2022.‏
  94. Honig and U. Hirdes, A method for the numerical inversion of the Laplace transform, Journal of Computational and Applied Mathematics, Vol. 10, pp. 113-132, 1984.
  95. Fox, Generalised thermoelasticity, International Journal of Engineering and Science, Vol. 7, pp. 437-445, 1969.‏
  96. M. Zenkour, T. Saeed and A.M. Aati, Refined Dual-Phase-Lag Theory for the 1D Behavior of Skin Tissue under Ramp-Type Heating, Materials, Vol. 16, pp. 2421, 2023.‏
  97. A. Biot, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics, Vol. 27, pp. 240-253, 1956.‏
  98. W. Lord and Y.A. Shulman, Generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids, Vol. 15, pp. 299-309, 1967.‏
  99. E. Green and K.A. Lindsay, Thermoelasticity, Journal of Elasticity, Vol. 2, pp. 1-7, 1972.‏
  100. S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Applied Mechanics Reviews, Vol. 51, pp. 705-729,‏ 1998.
  101. B. Hetnarski, and J. Ignaczak. Generalized thermoelasticity, Journal of Thermal Stresses, Vol. 22, pp. 451-476, 1999.‏
  102. Ignaczak and O-S. Martin, Thermoelasticity with Finite Wave Speeds. OUP Oxford, 2009.‏
  103. H. Sherief, Fundamental solution of the generalized thermoelastic problem for short times. Journal of Thermal Stresses, Vol. 9, pp. 151-164, 1986.‏
  104. A. Ezzat and A.S. El-Karamany. The uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with two relaxation times, International Journal of Engineering Science, Vol. 40, pp. 1275-1284, 2002.‏
  105. Sharma, K. Sharma and R.R. Bhargava, Effect of viscosity on wave propagation in anisotropic thermoelastic with Green–Naghdi theory type-II and type-III, Materials Physics and Mechanics, Vol. 16, pp. 144-158, 2013.‏
  106. I. Othman and E. M. Abd-Elaziz, Influence of gravity and microtemperatures on the thermoelastic porous medium under three theories, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 29, pp. 3242-3262, 2019.
  107. Sharma and P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, Journal of Thermal Stresses, Vol. 36, pp. 94-111, 2013.‏
  108. Lata, R. Kumar and N. Sharma. Plane waves in an anisotropic thermoelastic, Steel and Composite Structures, Vol. 22, pp. 567-587, 2016.‏
  109. D. Hobiny and I.A. Abbas, Analytical solutions of the temperature increment in skin tissues caused by moving heating sources, Steel and Composite Structures, Vol. 40, pp. 511-516, 2021.
  110. M. Zenkour and I.A. Abbas, A generalized thermoelasticity problem of an annular cylinder with temperature-dependent density and material properties. International Journal of Mechanics and Sciences, Vol. 84, pp. 54-60, 2014.‏
  111. E. Green and P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Royal Society, Vol. 432, pp. 171-194, 1991.
  112. E. Green, and P.M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses, Vol. 15, pp. 253-264, 1992.
  113. E. Green, and P.M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity, Vol. 31, pp. 189-208, 1993
  114. S.A. Chandrasekharaiah, Uniqueness theorem in the theory of thermoelasticity without energy dissipation, Journal of Thermal Stresses, Vol. 19, pp. 267-272, 1996.‏
  115. Kumar, A.K. Vashishth and S. Ghangas, Phase-lag effects in skin tissue during transient heating, International Journal of Applied Mechanics and Engineering, Vol. 24, pp. 603-623, 2019.
  116. K. Sharma and D. Kumar, A study on non-linear DPL model for describing heat transfer in skin tissue during hyperthermia treatment, Entropy, Vol. 22, 481, 2020.
  117. Chiriţă and M. Ciarletta, Reciprocal and variational principles in linear thermoelasticity without energy dissipation, Mechanics Research Communications, Vol. 37, pp. 271-275, 2010.
  118. Ghazanfarian, Z. Shomali and A. Abbassi, Macro-to nanoscale heat and mass transfer: the lagging behavior, International Journal of Thermophysics, Vol. 36, pp. 1416-1467, 2015.‏
  119. I.A. Othman and I.A. Abbas, Generalized thermoelasticity of thermal-shock problem in a non-homogeneous isotropic hollow cylinder with energy dissipation, International Journal of Thermophysics, Vol. 33, pp. 913-923, 2012.‏
  120. H. Sherief and W.E. Raslan, A 2D problem of thermoelasticity without energy dissipation for a sphere subjected to axisymmetric temperature distribution, Journal of Thermal Stresses, Vol. 40, pp. 1461-1470, 2017.‏
  121. M. Zenkour, T. Saeed and K.M. Alnefaie, Refined Green–Lindsay model for the response of skin tissue under a ramp-type heating, Mathematics, Vol. 11, 1437, 2023.‏
  122. L. Bagley and P.J. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology, Vol. 30, pp. 133-155, 1999.
  123. A. Ezzat and A.S. El-Karamany, Fractional-order theory of a perfect conducting thermoelastic medium, Canadian Journal of Physics, Vol. 89, pp. 311-318, 2011.‏
  124. H. Hendy, M.M. Amin and M.A. Ezzat, Two-dimensional problem for thermoviscoelastic materials with fractional-order heat transfer, Journal of Thermal Stresses, Vol. 42, pp. 1298-1315, 2019.‏
  125. A. Abbas, Eigenvalue approach to fractional-order generalized magneto-thermoelastic medium subjected to moving heat source, Journal of Magnetism and Magnetic Materials, Vol. 377, pp. 452-459, 2015.‏
  126. Sharma, R. Kumar and P. Lata, Disturbance due to inclined load in transversely isotropic thermoelastic medium with two temperatures and without energy dissipation, Materials Physics and Mechanics, Vol. 22, pp. 107-117, 2015.‏
  127. A. Abbas, Eigenvalue Approach for an unbounded medium with a spherical cavity based upon two-temperature generalized thermoelastic theory, Journal of Mechanical Science and Technology, Vol. 28, pp. 4193-4198, 2014.‏
  128. A. Ezzat and M.Z. Abd-Elaal, Free convection effects on a viscoelastic boundary layer flow with one relaxation time through a porous medium, Journal of the Franklin Institute, Vol. 334, pp. 685-706, 1997.
  129. H. Sherief, A.M.A. El-Sayed and A.M. Abd El-Latief, Fractional Order Theory of Thermoelasticity, International Journal of Solids and Structures, Vol. 47, pp. 269-275, 2010.‏
  130. H. Sherief and W.E. Raslan, 2D Problem for a long cylinder in the fractional theory of thermoelasticity, Journal of Solids and Structures, Vol. 13, pp. 1596-1613, 2016.‏
  131. H. Hendy, S.I. El-Attar and M.A. Ezzat, Two-temperature fractional Green–Naghdi of type iii in magneto-thermo-viscoelasticity theory subjected to a moving heat source, Indian Journal of Physics, Vol. 95, pp. 657-671, 2021.‏
  132. J. Chen and M.E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift Für Angewandte Mathematik Und Physik (ZAMP), Vol. 19, pp. 614-627, 1968.‏
  133. A. Ezzat and E.S. Awad, Constitutive relations, uniqueness of solution, and thermal shock application in the linear theory of micropolar generalized thermoelasticity involving two temperatures, Journal of Thermal Stresses, Vol. 33, pp. 226-250, 2010.‏
  134. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, 198, 1998.
  135. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition using Differential Operators of Caputo Type, Springer, 2010.
  136. A. Abbas and A.M. Zenkour, Semi-analytical and numerical solution of fractional-order generalized thermoelastic in a semi-infinite medium, Journal of Computational and Theoretical Nanoscience, Vol. 11, pp. 1592-1596, 2014.‏
  137. A. Kilbas, M. Rivero, L. Rodriguez-Germa and J.J. Trujillo, Caputo linear fractional differential equations. IFAC Proceedings, Vol. 39, pp. 52-57, 2006.‏
  138. A. Ezzat, A. S. El-Karamany, A. A. El-Bary and M. A. Fayik, Fractional calculus in one-dimensional isotropic thermo-viscoelasticity, Computes Rendus Mecanique, Vol. 341, pp. 553-566, 2013.
  139. Zhang, Y. Sunan and J. Yang, Bio-heat response of skin tissue based on three-phase-lag model, Scientific Reports, Vol. 10, pp. 16421, 2020.
  140. Sobhy and A.M. Zenkour, Refined Lord–Shulman theory for 1D response of skin tissue under ramp-type heat, Materials, Vol. 15, pp. 6292, 2022.‏
  141. Honig and U. Hirdes, A method for the numerical inversion of the Laplace transform, Journal of Computational and Applied Mathematics, Vol. 10, pp. 113-132, 1984.
Journal of Computational Applied Mechanics
Volume 54, Issue 3
Summer 2023
Pages 365-377

  • Receive Date 03 September 2023
  • Revise Date 28 September 2023
  • Accept Date 29 September 2023