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Fractional order theory of thermo-viscoelasticity and application

  • Hany H. Sherief
  • Mohammed A. El-HagaryEmail author
Article
  • 54 Downloads

Abstract

In this work, we derive a new fractional order theory for thermo-viscoelasticity. A uniqueness theorem for these equations is proved. A reciprocity theorem is also proved. A 1D problem for a viscoelastic half space is solved by using the Laplace transform technique. The solution in the transformed domain is obtained by a direct approach. The inverse transforms are obtained by using a numerical method. The temperature, displacement and stress distributions are computed and represented graphically.

Keywords

Fractional calculus Half Space Reciprocity theorem Thermo-viscoelasticity Uniqueness theorem 

Notes

References

  1. Adolfsson, K., Enelund, M.: Fractional derivative viscoelasticity at large deformations. Nonlinear Dyn. 33, 301–321 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  2. Adolfsson, K., Enelund, M., Larsson, S.: Adaptive discretization of fractional order viscoelasticity using sparse time history. Comput. Methods Appl. Mech. Eng. 193, 4567–4590 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bagley, R.L., Torvik, P.J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983) CrossRefzbMATHGoogle Scholar
  4. Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, 133–155 (1986) CrossRefzbMATHGoogle Scholar
  5. Caputo, M.: Vibrations on an infinite viscoelastic layer with a dissipative memory. J. Acoust. Soc. Am. 56, 897–904 (1974) CrossRefzbMATHGoogle Scholar
  6. Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, 134–147 (1971a) CrossRefzbMATHGoogle Scholar
  7. Caputo, M., Mainardi, F.: Linear model of dissipation in an elastic solids. Riv. Nuovo Cimento 1, 161–198 (1971b) CrossRefGoogle Scholar
  8. Churchill, R.V.: Operational Mathematics, 3rd edn. McGraw-Hill, New York (1972) zbMATHGoogle Scholar
  9. Dhaliwal, R.S., Sherief, H.H.: Generalized thermoelasticity for anisotropic media. Q. Appl. Math. 33, 1–8 (1980) MathSciNetCrossRefzbMATHGoogle Scholar
  10. Elhagary, M.A.: A thermo-mechanical shock problem for generalized theory of thermoviscoelasticity. Int. J. Thermophys. 34, 170–188 (2013) CrossRefGoogle Scholar
  11. Ezzat, M.A., El-Bary, A.A.: Fractional order theory to an infinite thermoviscoelastic body with a cylindrical cavity in the presence of an axial uniform magnetic field. J. Electromagn. Waves 31, 495–513 (2017) CrossRefGoogle Scholar
  12. Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A., Fayik, M.A.: Fractional calculus in one-dimensional isotropic thermo-viscoelasticity. C. R., Méc. 341, 553–566 (2013) CrossRefGoogle Scholar
  13. Ezzat, M.A., El-Karamany, A.S., El-Bary, A.A.: Thermo-viscoelastic materials with fractional relaxation operators. Appl. Math. Model. 39, 7499–7512 (2015) MathSciNetCrossRefGoogle Scholar
  14. Foutsitzi, G., Kalpakidis, V.K., Massalas, C.V.: On the existence and uniqueness in linear thermoviscoelasticity. Z. Angew. Math. Mech. 77, 33–44 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  15. Fung, Y.C.: Foundations of Solid Mechanics. Prentice Hall, New Delhi (1965) Google Scholar
  16. Gross, B.: Mathematical Structures of the Theories of Viscoelasticity. Hermann, Paris (1953) zbMATHGoogle Scholar
  17. Gurtin, M., Sternberg, E.: On the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 11, 182–191 (1962) MathSciNetCrossRefzbMATHGoogle Scholar
  18. Honig, G., Hirdes, U.: A method for the numerical inversion of the Laplace transform. J. Comput. Appl. Math. 10, 113–132 (1984) MathSciNetCrossRefzbMATHGoogle Scholar
  19. Il’yushin, A.A.: Method of approximations for solving problems of the linear theory of thermoviscoelasticity. Mech. Compos. Mater. 4, 149–158 (1968) Google Scholar
  20. Kovalenko, A.D., Karnaukhov, V.G.: A linearized theory of thermoviscoelasticty. Mech. Compos. Mater. 2, 194–199 (1972) Google Scholar
  21. Li, N.T.: A method for solving the integrodifferential equations encountered in the dynamics of viscoelasticity. Mech. Compos. Mater. 14, 657–663 (1978) Google Scholar
  22. Lord, H., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967) CrossRefzbMATHGoogle Scholar
  23. Malyi, V.I.: An approximate method for solving viscoelasticity problems. Strength Mater. 8, 906–909 (1976) CrossRefGoogle Scholar
  24. Medri, G.: Coupled thermoviscoelasticity, a way to the stress-strain analysis of polymeric industrial components. Meccanica 23, 226–231 (1988) CrossRefGoogle Scholar
  25. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) zbMATHGoogle Scholar
  26. Oberhettinger, F., Badii, L.: Tables of Laplace Transforms. Springer, New York (1973) CrossRefzbMATHGoogle Scholar
  27. Pobedrya, B.E.: Coupled problems in thermoviscoelasticity. Mech. Compos. Mater. 5, 353–358 (1969) zbMATHGoogle Scholar
  28. Povstenko, Y.Z.: Fractional heat conduction and associated thermal stress. J. Therm. Stresses 28, 83–102 (2005) MathSciNetCrossRefGoogle Scholar
  29. Povstenko, Y.Z.: Thermoelasticity that uses fractional heat conduction equation. J. Math. Sci. 162, 296–305 (2009) MathSciNetGoogle Scholar
  30. Povstenko, Y.Z.: Fractional Cattaneo-type equations and generalized thermoelasticity. J. Therm. Stresses 34, 97–114 (2011) CrossRefGoogle Scholar
  31. Raslan, W.: Application of fractional order theory of thermoelasticity in a thick plate under axisymmetric temperature distribution. J. Therm. Stresses 38, 733–743 (2015) CrossRefGoogle Scholar
  32. Raslan, W.: Application of fractional order theory of thermoelasticity to a 1D problem for a spherical shell. J. Theor. Appl. Mech. 54, 295–304 (2016) CrossRefGoogle Scholar
  33. Sherief, H., AbdEl-Latief, A.M.: Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity. Int. J. Mech. Sci. 74, 185–189 (2013) CrossRefGoogle Scholar
  34. Sherief, H., AbdEl-Latief, A.M.: Application of fractional order theory of thermoelasticity to a 1D problem for a half-space. Z. Angew. Math. Mech. 94, 509–515 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  35. Sherief, H., El-Sayed, A.M.A., AbdEl-Latief, A.M.: Fractional order theory of thermoelasticity. Int. J. Solids Struct. 47, 269–275 (2010) CrossRefzbMATHGoogle Scholar
  36. Sherief, H., Allam, M., Elhagary, M.: Generalized theory of thermoviscoelasticity and a half-space problem. Int. J. Thermophys. 32, 1271–1295 (2011) CrossRefGoogle Scholar
  37. Sherief, H.H., Hamza, F.A., Abd El-Latief, A.M.: 2D problem for a half-space in the generalized theory of thermo-viscoelasticity. Mech. Time-Depend. Mater. 19, 557–568 (2015) CrossRefGoogle Scholar
  38. Stratonova, M.M.: A method of solving dynamic problems of viscoelasticity. Mech. Compos. Mater. 7, 646–648 (1971) Google Scholar
  39. Welch, S.W.J., Rorrer, R.A.L., Duren, R.G. Jr.: Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials. Mech. Time-Depend. Mater. 3, 279–303 (1999) CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AlexandriaAlexandriaEgypt
  2. 2.Department of MathematicsDamiatta UniversityNew DamiattaEgypt

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