Fractional order thermo-viscoelastic problem with temperature dependent modulus of elasticity

References

• H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, vol. 15, pp. 299–309, 1967.
   Web of Science ®Google Scholar
• A.E. Green and K.A. Lindsay, Thermoelasticity, J. Elast., vol. 2, pp. 1–7, 1972.
   Google Scholar
• R. Dhaliwal and H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math., vol. 33, pp. 1–8, 1980.
   Web of Science ®Google Scholar
• R.B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Therm. Stress., vol. 22, pp. 451–476, 1999.
   Web of Science ®Google Scholar
• H. Sherief, State space formulation for generalized thermoelasticity with one relaxation time including heat sources, J. Therm. Stress., vol. 16, pp. 163–180, 1993.
   Web of Science ®Google Scholar
• H. Sherief and F. Hamza, Generalized thermoelastic problem of a thick plate under axisymmetric temperature distribution, J. Therm. Stress., vol. 17, pp. 435–453, 1994.
   Web of Science ®Google Scholar
• N.M. El-Maghraby and H. Youssef, State space approach to generalized thermoelastic problem with thermomechanical shock, J. Appl. Math. Comput., vol. 156, pp. 577–586, 2004.
   Web of Science ®Google Scholar
• A.A. Ilioushin and B.E. Pobedria, Fundamentals of the Mathematical Theories of Thermal Viscoelasticity, Nauka, Moscow, 1970.
   Google Scholar
• R.I. Tanner, Engineering Rheology, Oxford University Press, New York, 1988.
   Google Scholar
• R. Huilgol and N. Phan-Thien, Fluid Mechanics of Viscoelasticity, Elsevier, Amsterdam, 1997.
   Google Scholar
• M.I.A. Othman, M.A. Ezzat, S.A. Zaki, and A.S. El-Karamany, Generalized thermo-viscoelasticity plane waves with two relaxation times, Int. J. Eng. Sci., vol. 40, pp. 1329–1347, 2002.
   Web of Science ®Google Scholar
• M.A. Ezzat, M.I.A. Othman, and A.S. El-Karamany, State space approach to generalized thermo-viscoelasticity with two relaxation times, Int. J. Eng. Sci., vol. 40, pp. 283–302, 2002.
   Web of Science ®Google Scholar
• M.A. Ezzat and A.S. El-Karamany, The uniqueness and reciprocity theorem for generalized thermo-viscoelasticity plane wave with two relaxation times, J. Therm. Stress., vol. 25, pp. 507–522, 2002.
   Web of Science ®Google Scholar
• M.I.A. Othman, Generalized electro-magneto-thermo-viscoelastic in case of 2-D thermal shock problem in a finite conducting medium with one relaxation time, Acta Mech., vol. 169, pp. 37–51, 2004.
   Web of Science ®Google Scholar
• J.N. Sharma and R. Chand, Transient thermoviscoelastic waves in a half-space due to thermal loads, J. Therm. Stress., vol. 28, pp. 233–252, 2005.
   Web of Science ®Google Scholar
• M.A. Ezzat, The relaxation effect on volume properties of electrically conducting viscoelastic material, Mater. Sci. Eng., vol. 130, pp. 11–23, 2006.
   Google Scholar
• M.A. Ezzat and A.S. El-Karamany, State space approach of two-temperature magneto-viscoelasticity theory with thermal relaxation in a medium of perfect conductivity, J. Therm. Stress., vol. 32, pp. 819–838, 2009.
   Web of Science ®Google Scholar
• M.A. Ezzat, M. Zakaria, and A. Abdel-Barry, Generalized thermoelasticity with temperature dependent modulus of elasticity under three theories, J. Appl. Math. Comput., vol. 14, pp. 193–212, 2004.
   Google Scholar
• M.I.A. Othman, State space approach to generalized thermoelasticity plane waves with two relaxation times under dependence of modulus of elasticity on the reference temperature, Can. J. Phys., vol. 81, pp. 1403–1418, 2003.
   Web of Science ®Google Scholar
• M.I.A. Othman and Kh. Lotfy, Two dimensional problem of generalized magneto thermoelasticity under the effect of temperature dependent properties for different theories, Multidiscip. Model. Mater. Struct., vol. 5, pp. 235–242, 2009.
   Google Scholar
• K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
   Google Scholar
• K.S. Miller and B. Ross, An Introduction to the Fractional Integrals and Derivatives—Theory and Applications, Wiley, New York, 1993.
   Google Scholar
• Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Therm. Stress., vol. 28, pp. 83–102, 2005.
   Web of Science ®Google Scholar
• H. Sherief, A.M.A. El-Sayed, and A.M. Abd El-Latief, Fractional order theory of thermoelasticity, Int. J. Solids Struct., vol. 47, pp. 269–275, 2010.
   Web of Science ®Google Scholar
• H. Youssef, Theory of fractional order generalized thermoelasticity, J. Heat Transfer, vol. 132, pp. 1–7, 2010.
   PubMed Web of Science ®Google Scholar
• M.A. Ezzat, Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Physica B, vol. 405, pp. 4188–4194, 2010.
   Web of Science ®Google Scholar
• M.A. Ezzat, Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Physica B, vol. 406, pp. 30–35, 2011.
   Web of Science ®Google Scholar
• G. Jumarie, Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time: Application to Merton's optimal portfolio, Comput. Math. Appl., vol. 59, pp. 1142–1164, 2010.
   Web of Science ®Google Scholar
• H. Youssef, State-space approach to fractional order two-temperature generalized thermoelastic medium subjected to moving heat source, Mech. Adv. Mater Structer, vol. 20, pp. 47–60, 2013.
   Web of Science ®Google Scholar
• H. Sherief and A.M. Abd El-Latief, Application of fractional order theory of thermoelasticity to a 1D problem for a half space, Z. Angew. Math. Mech., vol. 94, pp. 509–515, 2014.
   Web of Science ®Google Scholar
• L.Y. Bahar and R.B. Hetnarski, State space approach to thermoelasticity, J. Therm. Stress., vol. 1, pp. 135–145, 1978.
   Web of Science ®Google Scholar
• G. Honig and U. Hirdes, A method for the numerical inversion of Laplace transforms, J. Comput. Appl. Math., vol. 10, pp. 113–132, 1984.
   Web of Science ®Google Scholar