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New Green’s functions for a thermoelastic unbounded parallelepiped under a point heat source and their application

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Abstract

This paper presents new analytical expressions for displacements Green’s functions to a steady-state spatial BVP of thermoelasticity for an unbounded parallelepiped, subjected to a unit point heat source. These results are obtained on the base of special structural formulas for displacements Green’s functions, which are expressed in terms of respective Green’s functions for Poisson’s equation. An example of the application of derived new analytical expressions is presented for a particular spatial BVP for a thermoelastic unbounded parallelepiped, subjected to a constant heat source, given inside of a rectangle. Both analytical expressions for displacements Green’s functions and thermoelastic displacements in the case of a particular problem are obtained in the form of double infinite series, containing product between exponential and trigonometric functions, which satisfy basic equations, boundary conditions on the marginal strips and vanishes at infinity. The presented example of a new steady-state spatial BVP of thermoelasticity for an unbounded parallelepiped, subjected to a unit point heat source will permit readers to derive the other examples to new analytical expressions for Green’s functions. These Green’s functions can be applied as kernels in the method of the boundary integral equations to solution of many particular BVP for thermoelastic unbounded parallelepiped. All these analytical results can be used also as some test problems for different numerical methods.

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Abbreviations

2D:

Two dimensional

3D:

Three dimensional

BVP:

Boundary value problem

GFs:

Green’s functions

GFM:

Green’s function method

GFPE:

Green’s functions for Poisson equation

TVD:

Thermoelastic volume dilatation

HIR:

Harmonic integral representations

HIRM:

Harmonic integral representations method

MTGFs:

Main thermoelastic Green’s functions

TSGFs:

Thermal stresses Green’s functions

References

  1. Boley, B.A., Weiner, J.F.: Theory of thermal stresses. Wiley, New York (1960)

    MATH  Google Scholar 

  2. Kovalenko, A.D.: Fundamentals of thermoelectricity. Naukova Dumka, Kiev (1970). (Russian)

    Google Scholar 

  3. Mayzel, V.M.: The temperature problem of the theory of elasticity. Publisher AN SSSR, Kiev (1951)

    Google Scholar 

  4. Melan, E., Parkus, H.: Thermo-elastic stresses caused by the stationary heat fields. Fizmatgiz, Moscow (1958)

    Google Scholar 

  5. Nowacki, W.: The theory of elasticity. Mir, Moscow (1975)

    MATH  Google Scholar 

  6. Nowacki, W.: Thermo-elasticity. Pergamon Press and Polish Sc. Publishers, Oxford, Warszawa (1962)

    MATH  Google Scholar 

  7. Nowinski, J.L.: Theory of thermoelasticity with applications. Setoff and Noordhoff International Publishers, Alphen Aan Den Rijn (1978)

    Book  MATH  Google Scholar 

  8. Hetnarski, R.B., Eslami, M.R.: Thermal stresses - advanced theory and applications, XXXII+ 559 pages, Springer, Dordrecht (2009)

  9. Șeremet, V.: A new approach to constructing Green’s functions and integral solutions in thermoelasticity. Acta Mech. 225(3), 737–755 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Șeremet, V.: Recent integral representations for thermoelastic Green’s functions and many examples of their exact analytical expressions. J Therm. Stresses 37(5), 561–584 (2014)

    Article  Google Scholar 

  11. Șeremet, V.: A new efficient unified method to derive new constructive formulas and explicit expressions for plane and spatial thermoelastic Green’s functions. Acta Mech. 226(1), 211–230 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Șeremet, V.: Static equilibrium of a thermoelastic half-plane: Green’s functions and solutions in integrals. Arch. Appl. Mech. 84(4), 553–570 (2014)

    Article  MATH  Google Scholar 

  13. Şeremet, V.D.: Handbook on Green’s functions and matrices. WIT Press, Southampton (2003)

    MATH  Google Scholar 

  14. Seremet, V., Carrera, E.: Solution in elementary functions to a BVP of thermoelasticity: Green’s functions and Green’s-type integral formula for thermal stresses within a half strip. J. Therm. Stresses 37(8), 947–968 (2014)

    Article  Google Scholar 

  15. Seremet, V., Wang, H.: Thermoelastic equilibrium of some semi-infinite domains subjected to the action of a heat source. J. Therm. Stresses 38(5), 509–525 (2015)

    Article  Google Scholar 

  16. Şeremet, V., Wang, H.: Two-Dimensional Green’s function for thermal stresses in a semi-layer under a point heat source. J. Therm. Stresses 38(7), 756–774 (2015)

    Article  Google Scholar 

  17. Şeremet, V.: Steady-state Green’s functions for thermal stresses within rectangular region under point heat source. J. Therm. Stresses 39(8), 906–927 (2016)

    Article  Google Scholar 

  18. Şeremet, V.: A method to derive thermoelastic Green’s functions for bounded domains (on examples of two-dimensional problems for parallelepipeds). Acta Mech. 227(12), 3603–3620 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Peng-Fei, H., Qiu, H., Hai-Yang, J.: Three-dimensional steady-state general solution for isotropic thermoelastic materials with applications II Green’s Functions for two-phase infinite body. J. Therm. Stresses 36(8), 851–867 (2015)

    Google Scholar 

  20. Qing-Hua, Q.: Thermo-electro-elastic Green’s function for a piezoelectric plate containing an elliptic hole. Mech. Mater. 30(1), 21–29 (1998)

    Article  Google Scholar 

  21. Pan, E., Cheng, W.: Static Green's functions in anisotropic media, Cambridge University Press (2015)

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Authors and Affiliations

  1. Institute of Mathematics and Computer Sciences, Academy of Science of Moldova, Laboratory of Mathematical Modeling, Academiei Street 5, MD 2028, Chisinau, Moldova

    Victor Șeremet & Ion Crețu

  2. Technical University of Moldova, Department of Civil Engineering and Geodesy, Dacia Avenue 41, MD 2060, Chișinău, Moldova

    Victor Șeremet & Ion Crețu

Authors
  1. Victor Șeremet
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  2. Ion Crețu
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Correspondence to Victor Șeremet.

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Șeremet, V., Crețu, I. New Green’s functions for a thermoelastic unbounded parallelepiped under a point heat source and their application. Acta Mech 234, 6515–6528 (2023). https://doi.org/10.1007/s00707-023-03675-3

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  • DOI: https://doi.org/10.1007/s00707-023-03675-3

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