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Thermally nonlinear generalized thermoelasticity: a note on the thermal boundary conditions

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Abstract

The present paper is aimed to make a detail analysis on the correct simulation of the associated natural thermal boundary conditions and their interesting effects on the thermally nonlinear generalized thermoelastic response of a continuum medium. The paper presents some explanations regarding the literally conventional thermal boundary conditions associated with Cattaneo’s heat conduction law. The importance of modeling the natural thermal boundary conditions is investigated by applying three practical thermal loads to a homogeneous one-dimensional layer. The results of the investigations are illustrated by propagation of thermoelastic waves through the layers. These results clearly show the profound effects of the correct statement of natural thermal boundary conditions on the thermoelastic response of the layer. It is shown that these effects intensify the importance of thermally nonlinear generalized thermoelastic analyses.

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Abbreviations

\((\cdot ),_{p}\) :

Partial derivative with respect to p

\((\cdot )^\mathsf {T}\) :

Transpose of a tensor

 : :

Double product symbol

\(\bar{(\cdot )}\) :

Prescribed quantities on the boundary surfaces of continuum media

\(\cdot \) :

Inner product symbol

\(\dot{()}\) :

Material time derivative

\(\hat{(\cdot )}\) :

Non-dimensional parameters

\(\varTheta \) :

Difference between current and reference temperature

\(\alpha \) :

Coefficient of thermal expansion

\(\beta \) :

Thermoelastic coupling constant of isotropic materials

\(\beta _{ij}\) :

Component of thermoelastic coupling tensor

\(\pmb {\beta }\) :

Thermoelastic coupling tensor

\(\lambda \) :

Normal Lamé constant

\(\mu \) :

Shear Lamé constant

\(\nabla \) :

Left nabla operator which produces left gradient and left divergence

\(\mathrm{T_{\infty }}\) :

Absolute temperature of fluid interacting with thermoelastic solid

\(\mathrm{T}\) :

Absolute temperature

\(\mathrm t_{r}\) :

Rise time of thermal loads

\(\mathsf {Tr}\) :

Trace of second-order tensors

\(\rho \) :

Mass density

\(\pmb {\sigma }\) :

Cauchy stress tensor

\(\tau \) :

Relaxation time of heat flux vector

\(\pmb {\varepsilon }\) :

Infinitesimal strain tensor

\(\mathbf{b}\) :

Body force vector

\(\mathrm{B_i}\) :

Coefficients of finite element equations

\(\mathrm{C}_{\pmb {\varepsilon }}\) :

Specific heat

\(\exp (\cdot )\) :

Natural exponential function

\(\mathsf {F}(\cdot )\) :

Thermal loading function

\(\mathsf {H}(\mathrm t)\) :

Heaviside function of time

\(\mathrm{h}\) :

Convective heat transfer coefficient

\(\mathbf{I}\) :

Identity tensor

\(\mathrm {K}_{ij}\) :

Component of thermal conductivity tensor

\(\mathbf{K}\) :

Thermal conductivity tensor

\({\mathrm {K}}\) :

Thermal conductivity of isotropic continua

\(\mathbf n\) :

Outward unit normal vector on the boundary surface of continuous medium

\(\mathbf{N}^p\) :

Vector of shape functions associated with the variable p

\(\mathscr {Q}\) :

Thermal power or total heat input

\(\mathbf{q}\) :

Heat flux vector

\(\mathrm{R}\) :

Volumetric heat source

\(\mathrm{s}\) :

Specific entropy

\(\mathrm{t}\) :

time parameter

\(\mathbf{u}\) :

Displacement vector

\(\mathbf{X}\) :

Global vector of nodal unknowns

\(\mathbf{x}\) :

Position vector of the material points

\(\mathbf{X}^p\) :

Vector of nodal unknowns associated with the variable p

References

  1. Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (1967)

    Article  MATH  Google Scholar 

  2. Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2(1), 1–7 (1972)

    Article  MATH  Google Scholar 

  3. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 432(1885), 171–194 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cattaneo, C.: Sur une forme de lequation de la chaleur eliminant le paradoxe dune propagation instantanee. C. R. Hebd. Seances Acad. Sci. 247(4), 431–433 (1958)

    MATH  Google Scholar 

  5. Fox, N.: Generalised thermoelasticity. Int. J. Eng. Sci. 7(4), 437–445 (1969)

    Article  MATH  Google Scholar 

  6. Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stress. 15(2), 253–264 (1992)

    Article  MathSciNet  Google Scholar 

  7. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31(3), 189–208 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ignaczak, J.: Soliton-like solutions in a nonlinear dynamic coupled thermoelasticity. J. Therm. Stress. 13(1), 73–98 (1990)

    Article  MathSciNet  Google Scholar 

  9. Hetnarski, R.B., Ignaczak, J.: Soliton-like waves in a low temperature nonlinear thermoelastic solid. Int. J. Eng. Sci. 34(15), 1767–1787 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Abo-El-Nour, N., Ghaleb, A.F., Maugin, G.A.: Harmonic wave generation in nonlinear thermoelasticity. Int. J. Eng. Sci. 32(7), 1103–1116 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rawy, E.K., Iskandar, L., Ghaleb, A.F.: Numerical solution for a nonlinear, one-dimensional problem of thermoelasticity. J. Comput. Appl. Math. 100(1), 53–76 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Strunin, D.V., Melnik, R.V.N., Roberts, A.J.: Coupled thermomechanical waves in hyperbolic thermoelasticity. J. Therm. Stress. 24(2), 121–140 (2001)

    Article  Google Scholar 

  13. Kosiński, W., Frischmuth, K.: Thermomechanical coupled waves in a nonlinear medium. Wave Motion 34(1), 131–141 (2001)

    Article  MATH  Google Scholar 

  14. Bargmann, S., Steinmann, P.: Theoretical and computational aspects of non-classical thermoelasticity. Comput. Methods Appl. Mech. Eng. 196(1), 516–527 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bargmann, S., Steinmann, P.: Modeling and simulation of first and second sound in solids. Int. J. Solids Struct. 45(24), 6067–6073 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bargmann, S., Steinmann, P., Jordan, P.M.: On the propagation of second-sound in linear and nonlinear media: Results from GreenNaghdi theory. Phys. Lett. A 372(24), 4418–4424 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bargmann, S., Denzer, R., Steinmann, P.: Material forces in non-classical thermo-hyperelasticity. J. Therm. Stress. 32(4), 361–393 (2009)

    Article  Google Scholar 

  18. Bateni, M., Eslami, M.R.: Thermally nonlinear generalized thermoelasticity of a layer. To appear in J. Therm. Stress. (2017)

  19. Hetnarski, R.B., Eslami, M.R.: Thermal Stresses, Advanced Theory and Applications, 1st edn. Springer, New York (2009)

    MATH  Google Scholar 

  20. Ignaczak, J., Ostoja-Starzewski, M.: Thermoelasticity with Finite Wave Speeds, 1st edn. Oxford University Press, Oxford (2010)

    MATH  Google Scholar 

  21. Eslami, M.R., Hetnarski, R.B., Ignaczak, J., Noda, N., Sume, N., Tanigawa, T.: Theory of Elasticity and Thermal Stresses, 1st edn. Springer, New York (2013)

    Book  MATH  Google Scholar 

  22. Hetnarski, R.B. (ed.): Encyclopedia of Thermal Stresses, 1st edn. Springer, New York (2013)

    Google Scholar 

  23. Eslami, M.R.: Finite Elements Methods in Mechanics, 1st edn. Springer, New York (2014)

    MATH  Google Scholar 

  24. Tamma, K.K., Zhou, X.: Macroscale and microscale thermal transport and thermo-mechanical interactions: some noteworthy perspectives. J. Therm. Stress. 21(3–4), 405–449 (1998)

    Article  Google Scholar 

  25. Joseph, D.D., Preziosi, L.: Heat waves. Rev. Mod. Phys. 61(1), 41 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51(12), 705–729 (1998)

    Article  Google Scholar 

  27. Chandrasekharaiah, D.S.: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39(3), 355–376 (1986)

    Article  MATH  Google Scholar 

  28. Hetnarski, R.B., Ignaczak, J.: Generalized thermoelasticity. J. Therm. Stresses 22(4–5), 451–476 (1999)

    MathSciNet  MATH  Google Scholar 

  29. Hetnarski, R.B., Ignaczak, J.: Nonclassical dynamical thermoelasticity. Int. J. Solids Struct. 37(1), 215–224 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Taheri, H., Fariborz, S., Eslami, M.R.: Thermoelasticity solution of a layer using the GreenNaghdi model. J. Therm. Stress. 27(9), 795–809 (2004)

    Article  Google Scholar 

  31. Bagri, A., Taheri, H., Eslami, M.R., Fariborz, S.: Generalized coupled thermoelasticity of a layer. J. Therm. Stress. 29(4), 359–370 (2006)

    Article  Google Scholar 

  32. Sanderson, T., Ume, C., Jarzynski, J.: Hyperbolic heat equations in laser generated ultrasound models. Ultrasonics 33(6), 415–421 (1995)

    Article  Google Scholar 

  33. Bagri, A., Eslami, M.R.: Generalized coupled thermoelasticity of functionally graded annular disk considering the LordShulman theory. Compos. Struct. 83(2), 168–179 (2008)

    Article  Google Scholar 

  34. Zad, S.H., Komeili, A., Eslami, M.R., Fariborz, S.J.: Classical and generalized coupled thermoelasticity analysis in one-dimensional layered media. Arch. Appl. Mech. 82(2), 267–282 (2012)

    Article  MATH  Google Scholar 

  35. Tehrani, P.H., Eslami, M.R.: Boundary element analysis of coupled thermoelasticity with relaxation times in finite domain. AIAA J. 38(3), 534–541 (2000)

    Article  Google Scholar 

  36. Tehrani, P.H., Eslami, M.R.: Boundary element analysis of finite domains under thermal and mechanical shock with the Lord-Shulman theory. J. Strain Anal. Eng. Des. 38(1), 53–64 (2003)

    Article  Google Scholar 

  37. Wang, Y.Z., Zhang, X.B., Song, X.N.: A unified generalized thermoelasticity solution for the transient thermal shock problem. Acta Mech. 223(4), 735–743 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang, Y.Z., Liu, D., Wang, Q., Zhou, J.Z.: Thermoelastic behavior of elastic media with temperature-dependent properties under transient thermal shock. J. Therm. Stress. 39(4), 460–473 (2016)

    Article  Google Scholar 

  39. Eslami, M.R., Vahedi, H.: Galerkin finite element displacement formulation of coupled thermoelasticity spherical problems. J. Press. Vessel Technol. 114(3), 380–384 (1992)

    Article  Google Scholar 

  40. Othman, M.I.A., Singh, B.: The effect of rotation on generalized micropolar thermoelasticity for a half-space under five theories. Int. J. Solids Struct. 44(9), 2748–2762 (2007)

    Article  MATH  Google Scholar 

  41. Sherief, H.H., Allam, M.N., El-Hagary, M.A.: Generalized theory of thermoviscoelasticity and a half-space problem. Int. J. Thermophys. 32(6), 1271–1295 (2011)

    Article  Google Scholar 

  42. Abbas, I.A.: Analytical solutions of thermoelastic interactions in a hollow cylinder with one relaxation time. Math. Mech. Solids 1081286515579308 (2015)

  43. Abo-Dahab, S.M., Abbas, I.A.: LS model on thermal shock problem of generalized magneto-thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity. Appl. Math. Model. 35(8), 3759–3768 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sharma, J.N., Othman, M.I.: Effect of rotation on generalized thermo-viscoelastic RayleighLamb waves. Int. J. Solids Struct. 44(13), 4243–4255 (2007)

    Article  MATH  Google Scholar 

  45. Sharma, J.N., Chand, R., Othman, M.I.: On the propagation of Lamb waves in viscothermoelastic plates under fluid loadings. Int. J. Eng. Sci. 47(3), 391–404 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sharma, J.N., Thakur, M.D.: Effect of rotation on RayleighLamb waves in magneto-thermoelastic media. J. Sound Vib. 296(4), 871–887 (2006)

    Article  Google Scholar 

  47. Abo-el-nour, N., Yahia, A.A., Abo-Dahab, S.M.: On the reflection of the generalized magneto-thermo-viscoelastic plane waves. Chaos Solitons Fractals 16(2), 211–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  48. Brock, L.M.: Basic problems of coupled thermoelasticity with thermal relaxation and pre-stress: aspects observed in exact and asymptotic solutions. J. Therm. Stress. 32(6–7), 593–622 (2009)

    Article  Google Scholar 

  49. Brock, L.M.: Transient plane wave solutions to homogeneous equations of orthotropic thermoelasticity with thermal relaxation: Illustration. J. Appl. Mech. 79(6), 061006 (2012)

    Article  Google Scholar 

  50. Abbas, I.A., Kumar, R.: Deformation in three dimensional thermoelastic medium with one relaxation time. J. Comput. Theor. Nanosci. 12(10), 3104–3109 (2015)

    Article  Google Scholar 

  51. Bagri, A., Eslami, M.R.: A unified generalized thermoelasticity; solution for cylinders and spheres. Int. J. Mech. Sci. 49(12), 1325–1335 (2007)

    Article  Google Scholar 

  52. Bagri, A., Eslami, M.R.: A unified generalized thermoelasticity formulation; application to thick functionally graded cylinders. J. Therm. Stress. 30(9–10), 911–930 (2007)

    Article  Google Scholar 

  53. Bagri, A., Eslami, M.R.: Generalized coupled thermoelasticity of disks based on the LordShulman model. J. Therm. Stress. 27(8), 691–704 (2004)

    Article  Google Scholar 

  54. Sharma, J.N., Chand, R.: Transient thermoviscoelastic waves in a half-space due to thermal loads. J. Therm. Stress. 28(2), 233–252 (2005)

    Article  MathSciNet  Google Scholar 

  55. Zhou, F.X., Li, S.R., Lai, Y.M.: Three-dimensional analysis for transient coupled thermoelastic response of a functionally graded rectangular plate. J. Sound Vib. 330(16), 3990–4001 (2011)

    Article  Google Scholar 

  56. Eslami, M.R., Shakeri, M., Ohadi, A.R., Shiari, B.: Coupled thermoelasticity of shells of revolution: effect of normal stress and coupling. AIAA J. 37(4), 496–504 (1999)

    Article  Google Scholar 

  57. Bahtui, A., Eslami, M.R.: Coupled thermoelasticity of functionally graded cylindrical shells. Mech. Res. Commun. 34(1), 1–18 (2007)

    Article  MATH  Google Scholar 

  58. Bahtui, A., Eslami, M.R.: Generalized coupled thermoelasticity of functionally graded cylindrical shells. Int. J. Numer. Meth. Eng. 69(4), 676–697 (2007)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran

    M. Bateni

  2. Iran Academy of Sciences, Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran

    M. R. Eslami

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  1. M. Bateni
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Correspondence to M. R. Eslami.

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This paper is dedicated to the memory of Franz Ziegler

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Bateni, M., Eslami, M.R. Thermally nonlinear generalized thermoelasticity: a note on the thermal boundary conditions. Acta Mech 229, 807–826 (2018). https://doi.org/10.1007/s00707-017-2001-6

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  • DOI: https://doi.org/10.1007/s00707-017-2001-6

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