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Micropolar Thermoelasticity with Voids Using Fractional Order Strain

  • Lavinia Codarcea-Munteanu
  • Marin Marin
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 179)

Abstract

The chapter is dealing with the study of the thermoelasticity of the micropolar materials with voids that uses the fractional order strain, in order to determine some equations of this linear thermoelasticity theory, as well as of a reciprocity relation for the mentioned bodies. Finding the form of the constitutive equations and using them for analyzing the reciprocity, toghether with obtaining the equation of thermal conductivity under the terms of our theory is the main purpose, realizing a parallel between classical theory and this specific case, leading to a better understanding of the behaviour of these materials.

Keywords

Micropolar materials with voids Fractional derivative Thermoelasticity Reciprocity 

References

  1. Agrawal, O.P.: Generalized variational problems and Euler - Lagrange equations. Comput. Math. Appl. 59(5), 1852–1864 (2010).  https://doi.org/10.1016/j.camwa.2009.08.029MathSciNetCrossRefGoogle Scholar
  2. Aouadi, M.: A theory of thermoelastic materials with voids. Z. Angew. Math. Phys. 61, 357–379 (2010).  https://doi.org/10.1007/s00033-009-0016-0MathSciNetCrossRefGoogle Scholar
  3. Băleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos, vol. 5, 2nd edn. World Scientific, Singapore (2012).  https://doi.org/10.1142/10044
  4. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13(5), 529–539 (1967).  https://doi.org/10.1111/j.1365-246X.1967.tb02303.xCrossRefGoogle Scholar
  5. Cattaneo, C.: Sulla conduzione del calore. Atti. Sem. Mat. Fis. Univ. Modena 3, 83–101 (1948)MathSciNetzbMATHGoogle Scholar
  6. Chirilă, A.: Generalized micropolar thermoelasticity with fractional order strain. Bull. Transilv. Univ. Braşov, Ser. III: Math. Inf. Phys. 10(59)(1), 83–90 (2017)Google Scholar
  7. Ciarletta, M., Scalia, A.: Some results in linear theory of thermomicrostretch elastic solids. Meccanica 39, 191–206 (2004)MathSciNetCrossRefGoogle Scholar
  8. Codarcea-Munteanu, L., Marin, M: Thermoelasticity with fractional order strain for dipolar materials with voids. Bull. Transilv. Univ. Braşov, Ser. III: Math. Inf. Phys. (2017) (accepted)Google Scholar
  9. Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13, 125–147 (1983)CrossRefGoogle Scholar
  10. El-Karamany, A.S., Ezzat, M.A.: On fractional thermoelasticity. Math. Mech. Solids 16(3), 334–346 (2011).  https://doi.org/10.1177/1081286510397228MathSciNetCrossRefGoogle Scholar
  11. Eringen, A.C.: Microcontinuum Field Theories I. Foundations and Solids. Springer, New York (1999)CrossRefGoogle Scholar
  12. Goodman, M.A., Cowin, S.C.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44, 249–266 (1972)MathSciNetCrossRefGoogle Scholar
  13. Hetnarski, R.B.: Thermal Stresses IV. Elsevier, Amsterdam (1996)Google Scholar
  14. Hetnarski, R.B., Ignaczak, J.: Generalized thermoelasticity. J. Therm. Stress. 22, 451–476 (1999)MathSciNetCrossRefGoogle Scholar
  15. Ieşan, D.: Generalized mechanics of solids. Univ. Al. I. Cuza, Centrul de multiplicare, Iaşi (1980)Google Scholar
  16. Ieşan, D.: A theory of thermoelastic materials with voids. Acta Mech. 60(1–2), 67–89 (1986)CrossRefGoogle Scholar
  17. Iovane, G., Passarella, F.: Some theorems in thermoelasticity for micropolar porous media. Rev. Roum. Sci. Tech. Mech. Appl. 46(1–6), 9–18 (2002)MathSciNetGoogle Scholar
  18. Lianngenga, R.: Theory of micropolar thermoelastic materials with voids. IJPAMS 9(1), 1–8 (2016)Google Scholar
  19. Lianngenga, R., Lalawmpuia: Micropolar elasticity containing voids. IJISET 2(12), 838–844 (2015)Google Scholar
  20. Malinowska, A.B., Odzijewicz, T., Torres, D.F.M.: Advanced Methods in the Fractional Calculus of Variations. Springer Briefs in Applied Sciences and Technology. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-14756-7CrossRefGoogle Scholar
  21. Marin, M.: The lagrange identity method in thermoelasticity of bodies with microstructure. Int. J. Eng. Sci. 32(8), 1229–1240 (1994)MathSciNetCrossRefGoogle Scholar
  22. Marin, M.: On weak solutions in elasticity of dipolar bodies with voids. J. Comput. Appl. Math. 82(1–2), 291–297 (1997).  https://doi.org/10.1016/s0377-0427(97)00047-2MathSciNetCrossRefGoogle Scholar
  23. Marin, M.: A temporally evolutionary equation in elasticity of micropolar bodies with voids. UPB Sci. Bull. Ser. A Appl. Math. Phys. 60(3–4), 67–78 (1998)Google Scholar
  24. Marin, M.: Harmonic vibrations in thermoelasticity of microstretch materials. J. Vib. Acoust. 132(4), 044501,6 (2010a)CrossRefGoogle Scholar
  25. Marin, M.: Some estimates on vibrations in thermoelasticity of dipolar bodies. J. Vib. Control 16(1), 33–47 (2010b)MathSciNetCrossRefGoogle Scholar
  26. Marin, M.: An approach of a heat-flux dependent theory for micropolar porous media. Meccanica 51, 1127–1133 (2016).  https://doi.org/10.1007/s11012-015-0265-2MathSciNetCrossRefGoogle Scholar
  27. Marin, M., Codarcea, L., Chirilă, A.: Qualitative results on mixed problem of micropolar bodies with microtemperatures. Appl. Appl. Math. (2017) (accepted)Google Scholar
  28. Markov, K.Z.: On the dilatation theory of elasticity. ZAMM Z. Angew. Math. Mech. 61(8), 349–358 (1981)CrossRefGoogle Scholar
  29. Nunziato, J.W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Ration. Mech. Anal. 72, 175–201 (1979)MathSciNetCrossRefGoogle Scholar
  30. Podlubny, I.: Fractional Differential Equation: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications. Academic Press, New York (1998)zbMATHGoogle Scholar
  31. Postvenko, Y.Z.: Thermoelasticity that uses fractional heat conduction equation. J. Math. Sci. 162(2), 296–305 (2009)MathSciNetCrossRefGoogle Scholar
  32. Sheoran, S.S., Kundu, P.: Fractional order generalized thermoelasticity theories: a review. Int. J. Adv. Appl. Math. Mech. 3(4), 76–81 (2016)MathSciNetzbMATHGoogle Scholar
  33. Sherief, H.H., El-Sayed, A.M.A., Abd El-Latief, A.M.: Fractional order theory of thermoelasticity. Int. J. Solids Struct. 47, 269–275 (2010)CrossRefGoogle Scholar
  34. Youssef, H.M.: Theory of generalized thermoelasticity with fractional order strain. J. Vib. Control 22(18), 3840–3857 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceTransilvania University of BraşovBraşovRomania

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