Abstract
This article presents new constructive formulas for three-dimensional thermal stresses Green’s functions (TSGFs) for a generalized boundary value problem (BVP) of thermoelasticity for a semi-bounded parallelepiped. These results are formulated in a theorem, which is proved using the developed harmonic integral representations method. On the basis of derived constructive formulas, it is possible to obtain many analytical expressions for TSGFs to 32 BVPs within a semi-bounded parallelepiped. An example for a particular spatial BVP for a semi-bounded parallelepiped, TSGFs of which are presented in the form of a sum of elementary functions and double infinite series, containing products of exponential and trigonometric functions is included. These results are formulated in another theorem, which is proved by using the derived general constructive formulas for TSGFs. The integration formula of Green’s type, which permits to determine thermal stresses within a semi-bounded parallelepiped, caused by a distributed inner heat source and by heat fluxes given on the surface also is derived. A numerical example for a particular BVP within a semi-bounded parallelepiped, acted by a constant heat flux, given on a boundary half-strip, is presented.
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References
Boley, B.A., Weiner, J.F.: Theory of Thermal Stresses. Wiley, New York (1960)
Kovalenko, A.D.: Fundamentals of Thermoelasticity. Naukova Dumka, Kiev (1970). (Russian)
Mayzel, V.M.: The Temperature Problem of the Theory of Elasticity. AN SSSR, Kiev (1951)
Melan, E., Parkus, H.: Thermoelastic Stresses Caused by the Stationary Heat Fields. Fizmatgiz, Moscow (1958)
Nowacki, W.: The Theory of Elasticity. Mir, Moscow (1975)
Nowacki, W.: Thermoelasticity. Pergamon Press, Oxford (1962)
Nowinski, J.L.: Theory of Thermoelasticity with Applications. Sijthoff & Noordhoff International Publishers, Alphen Aan Den Rijn (1978)
Hetnarski, R.B., Eslami, M.R.: Thermal Stresses—Advanced Theory and Applications, vol. XXXII. Springer, Dordrecht (2009)
Şeremet, V.: A new approach to constructing Green’s functions and integral solutions in thermoelasticity. Acta Mech. 225(3), 737–755 (2014)
Şeremet, 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)
Seremet, 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)
Şeremet, V.: Static equilibrium of a thermoelastic half-plane: Green’s functions and solutions in integrals. Arch. Appl. Mech. 84(4), 553–570 (2014)
Seremet, V., Wang, H.: Thermoelastic equilibrium of some semi-infinite domains subjected to the action of a heat source. J. Thermal Stresses 38(5), 509–525 (2015)
Seremet, V., Carrera, E.: Solution in elementary functions to a BVP of thermoelasticity: Green’s functions and Green’s-type. J. Thermal Stresses 37(8), 947–968 (2014)
Şeremet, V., Wang, H.: Two-dimensional Green’s function for thermal stresses in a semi-layer under a point heat source. J. Thermal Stresses 38(7), 756–774 (2015)
Şeremet, V.: Steady-state Green’s functions for thermal stresses within rectangular region under point heat source. J. Thermal Stresses 39(8), 906–927 (2016)
Şeremet, V.: A method to derive thermoelastic Green’s functions for semi-bounded domains (on examples of two-dimensional problems for parallelepipeds). J. Acta Mech. 227, 1–18 (2016)
Hou, P.-F., Li, Q.-H., Jiang, H.-Y.: Three-dimensional steady-state general solution for isotropic thermoelastic materials with applications II: Green’s Functions for two-phase infinite body. J. Thermal Stresses 36(8), 851–867 (2015)
Qin, Q.-H.: Thermoelectroelastic Green’s function for a piezoelectric plate containing an elliptic hole. Mech. Mater. 30(1), 21–29 (1998)
Pan, E., Chen, W.: Static Green’s Functions in Anisotropic Media. Cambridge University Press, Cambridge (2015)
Şeremet, V.D.: Handbook on Green’s Functions and Matrices. WIT Press, Southampton (2003)
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Şeremet, V., Şeremet, D. Solution in integrals of a 3D BVP of thermoelasticity: Green’s functions and integration formula for thermal stresses within a semi-bounded parallelepiped. Acta Mech 228, 4471–4490 (2017). https://doi.org/10.1007/s00707-017-1923-3
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DOI: https://doi.org/10.1007/s00707-017-1923-3