Structural Formulas and Explicit Green’s Functions for a Generalized BVP for Half-Strip under a Point-Heat Source
Publication: Journal of Engineering Mechanics
Volume 143, Issue 9
Abstract
This study presents new structural formulas for steady-state thermoelastic Green’s functions (TGFs) to a plane-generalized boundary value problem (BVP) of thermoelasticity for a generalized half-strip. The structural formulas for TGFs are expressed in terms of Green’s functions for Poisson’s equation (GFPE). These results are formulated in a special theorem, which is proved using a developed harmonic integral representations method. The development of this method consists in calculating a boundary integral of the product between two GFPEs. This integral was calculated due the proved before statement that the TGFs have to satisfy must have the following two conditions: (1) thermal boundary conditions with respect to points of application of the heat source, and (2) mechanical boundary conditions with respect to the points of finding the displacements. On the basis of derived structural formulas, it is possible to obtain many explicit Green’s functions for termoelastic displacements, deformations, and stresses. An example is presented for a concrete-plane BVP for a half-strip; TGFs are obtained in terms of elementary functions. New analytical expressions for thermal deformations and thermal stresses to a particular plane problem for a thermoelastic half-strip under a boundary constant temperature gradient were obtained. The graphical computer evaluation of thermal deformations and thermal stresses, created by a unit point heat source and by a temperature gradient, also is included.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors would like to thank the editor and the referees for their valuable comments and suggestions, which improved the quality of the paper.
References
Boley, B. A., and Weiner, J. F. (1960). Theory of thermal stresses, Wiley, New York.
Duffy, D. G. (2001). Green’s functions with applications, Chapman and Hall/CRC Press, Boca Raton, FL.
Greenberg, M. D. (1971). Application of Green’s functions in science and engineering, Prentice-Hall, Upper Saddle River, NJ.
Hetnarski, R. B., and Eslami, M. R. (2009). Thermal stresses—Advanced theory and applications, Springer, Dordrecht, Netherlands, 559.
Hou, P. F., Leung, A. Y., and He, Y. J. (2008). “Three-dimensional Green’s functions for transversaly isotropic thermoelastic biomaterials.” Int. J. Solids Struct., 45(24), 6100–6113.
Hou, P. F., Li, Q. H., and Jiang, H. Y. (2013). “Three-dimensional stady-state general solution for isotropic thermoelastic material with applications. II: Green’s functions for two-phase infinite body.” J. Therm. Stresses, 36(8), 851–867.
Kovalenko, A. D. (1970). Fundamentals of thermoelasticity, Naukova Dumka, Kiev, Ukraine (in Russian).
Kushnir, R., and Protsiuk, B. (2009). “A method of the Green’s functions for quasistatic thermoelasticity problems in layered thermosensitive bodies under complex heat exchange.” Operator theory: Advances and applications, Vol. 191, Springer, Basel, Switzerland, 143–154.
Maple 15 [Computer software]. Maplesoft, Waterloo, ON, Canada.
Mayzel, V. M. (1951). The temperature problem of the theory of elasticity, AN SSSR, Kiev, Ukraine (in Russian).
Melan, E., and Parkus, H. (1958). Thermoelastic stresses created by the stationary heat fields, Fizmatgiz, Moscow (in Russian).
Nordgreen, N. (1963). “On the method of Green’s function in the theory of shallow shells.” Int. J. Eng. Sci., 1(2), 279–308.
Nowacki, W. (1962). Thermoelasticity, Pergamon Press/Polish Scientific Publishers, Oxford/Warszawa.
Nowacki, W. (1975). The theory of elasticity, Mir Publishers, Moscow (in Russian).
Nowinski, J. L. (1978). Theory of thermoelasticity with applications, Sijthoff and Noordhoff International Publishers, Alphen Aan Den Rijn.
Qin, Q. H. (1998). “Thermoelectroelastic Green’s function for a piezoelectric plate containing an elliptic hole.” J. Mech. Mater., 30(1), 21–29.
Roach, G. F. (1982). Green’s functions, Cambridge University Press, New York.
Seremet, V. (1995). “The integral equations and Green’s matrices of the influence elements method in the, mechanics of solids.” Ph.D. thesis, Technical Univ. of Moldova, Chisinau (in Romanian).
Seremet, V. (2001). “Some new influence functions and integral solutions in theory of thermal stresses.” Proc., 4th Int. Congress on Thermal Stresses, Osaka Institute of Technology, Osaka, Japan, 423.
Seremet, V. (2003). Handbook on Green’s functions and matrices, WIT Press, Southampton, U.K.
Seremet, V. (2014a). Thermoelastic Green’s function (Steady-state BVPs for some semi-infinite domains), Publisher “Print-Caro,” Chişinău, Moldova, 236.
Seremet, V. (2014b). “A new approach to constructing Green’s functions and integral solutions in thermoelasticity.” J. Acta Mech., 225(3), 737–755.
Seremet, V. (2014c). “A new efficient unified method to derive new constructive formulas and explicit expressions for plane and spatial thermoelastic Green’s functions.” J. Acta Mech., 226(1), 211–230.
Seremet, V. (2014d). “Recent integral representations for thermoelastic Green’s functions and many examples of their exact analytical expressions.” J. Therm. Stresses, 37(5), 561–584.
Seremet, V. (2014e). “Static equilibrium of a thermoelastic half-plane: Green’s functions and solutions in integrals.” J. Arch. Appl. Mech., 84(4), 553–570.
Seremet, V., and Bonnet, G. (2008). “Encyclopedia of Domain Green’s functions (Thermomagneto-electrostatics of solids in rectangular and polar coordinates).” Editorial Center, Agrarian State Univ. of Moldova, Chişinau, Moldova.
Seremet, V., and Carrera, E. (2014). “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.
Seremet, V., and Wang, H. (2015). “Two-dimensional Green’s function for thermal stresses in a semi-layer under a point heat source.” J. Therm. Stresses, 38(7), 756–774.
Stakgold, I., and Holst, M. (2011). Green’s functions and boundary value problems, 3rd Ed., John Wiley, New York.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 143 • Issue 9 • September 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Aug 16, 2015
Accepted: Feb 22, 2017
Published online: May 29, 2017
Published in print: Sep 1, 2017
Discussion open until: Oct 29, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.