Skip to main content

Moving heat source response in micropolar half-space with two-temperature theory

Abstract

The present paper deals with the moving heat source response in a homogeneous, isotropic, micropolar semi-infinite medium in the presence of a finite rotation about its axis. In this context, two-temperature generalized thermoelasticity theory has been considered. In order to obtain the physical aspects of displacement, microrotation, stress distribution and temperature changes, a complex quartic equation has been solved by employing Descartes’ algorithm with the help of an irreducible Cardan’s method. To illustrate the analytical developments, the numerical solutions have been carried out for aluminum–epoxy composite, and the variations in displacement, microrotation, stress distribution and temperature changes have been shown graphically. This work may find applications in geophysics.

This is a preview of subscription content, access via your institution.

Abbreviations

λμ :

Lame constants

\({\alpha , \beta , \gamma , \kappa}\) :

Micropolar elastic constants

υ :

\({= (3\lambda + 2\mu + \kappa) \alpha_t}\)

α t :

Coefficient of linear thermal expansion

ρ :

Density of the material per unit mass

\({\overrightarrow{\varphi}}\) :

Microrotation vector

φ*:

Conductive temperature

τ :

Thermal relaxation time

σ ij :

Stress tensor

δ ij :

Kronecker delta

ε ijk :

Permutation tensor

C E :

Specific heat at constant strain

j :

Microinertia

K :

Thermal conductivity

m ij :

Couple stress tensor

Q :

Heat source term

T :

Thermodynamic temperature above the reference temperature T 0

\({\overrightarrow{u}}\) :

Displacement vector

References

  1. Eringen, A.C.: Mechanics of micromorphic materials. In: Gortler, M. (ed.) Proceedings of Eleventh International Congress of Applied Mechanics Munich (1964). Springer, Berlin, p. 131 (1966)

  2. Eringen, A.C.: Mechanics of micromorphic continua. In: Kroner, E. (ed.) Mechanics of Generalized Continua, IUTAM Symposium Frendensstadt-Stuttgart (1967). Springer, Berlin, p. 18 (1968)

  3. Eringen A.C., Suhubi E.S.: Nonlinear theory of simple microelastic solids. Int. J. Eng. Sci. 2, 189–203 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  4. Eringen A.C., Kafadar C.B.: Polar field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. 4, Academic Press, New York (1976)

    Google Scholar 

  5. Eringen A.C.: Micropolar elastic solids with stretch. In: Anisina M.I. (ed.) Ari Kitabevi Matbaasi. Istanbul. p. 1 (1971)

  6. Eringen A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)

    MathSciNet  MATH  Google Scholar 

  7. Eringen A.C: Microcontinuum Field Theories I, Foundations and Solids. Spinger, New York (1999)

    MATH  Book  Google Scholar 

  8. Boschi E., Iesan D.: A generalized theory of linear micropolar thermoelasticity. Meccanica 3, 154–157 (1973)

    Article  Google Scholar 

  9. Dost S., Tabarook B.: Generalized micropolar thermoelasticity. Int. J. Eng. Sci. 16, 173–183 (1978)

    MATH  Article  Google Scholar 

  10. Amar M., Andreucci D., Bisegna P., Gianni R.: On a hierarchy of models for electrical conduction in biological tissues. Math. Methods Appl. Sci. 29, 767–787 (2006)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  11. Amar M., Andreucci D., Bisegna P., Gianni R.: Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues. Math. Model. Methods Appl. Sci. 14(9), 1261–1295 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  12. Bouchitte G., Bisegna M.: Homogenization of a soft elastic material reinforced by fibers. Asympt. Ann. 32, 153–183 (2002)

    MATH  Google Scholar 

  13. Chen P.J., Gurtin M.E.: On a theory of heat conduction involving two temperatures. J. Appl. Math. Phys. (ZAMP) 19, 614–627 (1968)

    MATH  Article  Google Scholar 

  14. Chen P.J., Gurtin M.E., Williams W.O.: A note on non-simple heat conduction. J. Appl. Math. Phys. (ZAMP) 19, 969–970 (1968)

    Article  Google Scholar 

  15. Chen P.J., Gurtin M.E., Williams W.O.: On the thermodynamics of non-simple elastic materials with two temperatures. J. Appl. Math. Phys. (ZAMP) 20, 107–112 (1969)

    MATH  Article  Google Scholar 

  16. Boley B.A., Tolins I.S.: Transient coupled thermoelastic boundary value problem in the half-space. J. Appl. Mech. 29, 637–646 (1962)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  17. Iesan D.: On the thermodynamics of non simple elastic materials with two-temperatures. J. Appl. Math. Phys. (ZAMP) 21, 583–591 (1970)

    MathSciNet  MATH  Article  Google Scholar 

  18. Quintanilla R.: On existence, structural stability, convergence and spatial behavior in thermoelasticity with two temperatures. Acta. Mech. 168, 61–73 (2004)

    MATH  Article  Google Scholar 

  19. Youssef H.M.: Theory of two temperature-generalized thermoelasticity. I.M.A. J. Appl. Math. 71, 383–390 (2006)

    MathSciNet  ADS  MATH  Google Scholar 

  20. Youssef H.M., Al-Lehaibi E.A.: State-space approach of two temperature generalized thermoelasticity of one dimensional problem. Int. J. Solids Struct. 44, 1550–1562 (2007)

    MATH  Article  Google Scholar 

  21. Mukhopadhyay S., Kumar R.: Thermoelastic interaction on two temperature generalized thermoelasticity in an infinite medium with a cylindrical cavity. J. Therm. Stress. 32, 341–360 (2009)

    Article  Google Scholar 

  22. Ezzat M., Hamza F., Awad E.: Elactro-magneto-thermoelastic plane waves in micropolar solid involving two temperatures. Acta. Mech. Solida. Sin. 23, 200–212 (2010)

    Google Scholar 

  23. Kumar R., Gogna M.L.: Steady state response to moving loads in micropolar elastic medium with stretch. Int. J. Eng. Sci. 30, 811–820 (1991)

    Article  Google Scholar 

  24. Katz R.: The dynamic response of a rotating shaft subject to an axially moving and rotating load. J. Sound. Vibr. 246, 757–775 (2001)

    ADS  Article  Google Scholar 

  25. Kumar R., Deswal S.: Steady state response of a micropolar generalized thermoelastic half-space to the moving mechanical/thermal loads. Proc. Indian Acad. Sci. 110, 449–465 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  26. Kumar R., Deswal S.: Steady state response to moving loads in a micropolar generalized thermoelastic half-space without energy dissipation. Ganita 53, 23–42 (2002)

    MATH  Google Scholar 

  27. Kumar R., Ailawalia P.: Effects of viscosity with moving load at micropolar boundary surface. Int. J. Appl. Mech. Eng. 10, 95–108 (2005)

    MATH  Google Scholar 

  28. Kumar R., Ailawalia P.: Moving load response of micropolar elastic half-space with voids. J. Sound. Vibr. 280, 837–848 (2005)

    ADS  Article  Google Scholar 

  29. Kumar R., Ailawalia P.: Moving load response in micropolar thermoelastic medium without energy dissipation possessing cubic symmetry. Int. J. Solid Struct. 44, 4068–4078 (2007)

    MATH  Article  Google Scholar 

  30. Othman M.I.A., Singh B.: The effect of rotation on generalized micropolar thermo-elasticity for a half-space under five theories. Int. J. Solid Struct. 44, 2748–2762 (2007)

    MATH  Article  Google Scholar 

  31. Kumar R.: Rupender: effect of rotation in magneto-micropolar thermoelastic medium due to mechanical and thermal sources. Chaos Solitons Fractals 41, 1619–1633 (2009)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  32. Gauthier, R.D.: Experimental investigations on micropolar media. In: Brulin, O., Hsieh, R.K.T. (eds.) Mechanics of micropolar media. World Scientific, Singapore (1982)

  33. Kumar R., Sharma N.: Propagation of waves in micropolar viscoelastic generalized thermoelastic solids having interfacial imperfections. Theor. Appl. Fract. Mech. 50, 226–234 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Soumen Shaw.

Additional information

Communicated by Francesco dell'Isola and Samuel Forest.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Shaw, S., Mukhopadhyay, B. Moving heat source response in micropolar half-space with two-temperature theory. Continuum Mech. Thermodyn. 25, 523–535 (2013). https://doi.org/10.1007/s00161-012-0284-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-012-0284-3

Keywords

  • Moving heat source
  • Micropolar
  • Two-temperature generalized thermoelasticity