We construct examples of exact solutions of the temperature problem for a square: the sides of the square are (i) free and (ii) firmly clamped. Initially, we solve the inhomogeneous problem for an infinite plane. The known exact solutions for a square, with which the boundary conditions on the sides of the square are satisfied, are added to this solution. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas.
MeleshkoVV.Thermal stresses in an elastic rectangle. J Elast2011; 105: 61–92.
2.
Al-AliAYAlmutairiKHRawyEKGhalebAFAbou-DinaMS.Deformation of a long thermoelastic rod of rectangular normal cross-section under mixed boundary conditions by boundary integrals. J Egypt Math Soc2016; 24(3): 449–457.
3.
VermaSKulkarniVS.Thermal stress analysis of rectangular plate due to convection using finite element method. Int J Eng Innov Technol2014; 3(7): 111–117.
4.
RoyHSBagadeSHKhobragadeNW.Thermal stresses of a semi infinite rectangular beam. Int J Eng Innov Technol2013; 3(1): 442–445.
5.
RahmanMASalehinN. Effects of thermal stresses and boundary conditions on the response of a rectangular elastic body made of FGM. In: Proceedings of the 7th Int. Conf. on Mechanical Engineering, Dhaka, Bangladesh, 29–31 December 2007, paper no. AM-76.
6.
MendonçaTSRibeiroGOVecciMA. Analysis of self-balanced thermal stresses in long flat rectangular isotropic plates. In: Proceedings of the 1st Pan-American Congress on Computational Mechanics, Buenos Aires, Argentina, 27–29 April 2015, pp. 1530–1541.
7.
KovalenkoMDMenshovaIVKerzhaevAPYuG.A boundary value problem in the theory of elasticity for a rectangle: exact solutions. Z Angew Math Phys2020; 71: 199.
8.
VlasovVV. Determination of thermal stresses in rectangular plates by the method of initial functions. In: ObraztsovIF (ed.) Strength and Stability of Thin-walled Aircraft Structures. Moscow: Mashinostroenie, 1971 (in Russian).
9.
VlasovVV.Method of Initial Functions in Problems of the Theory of Elasticity and Structural Mechanics. Moscow: Stroiizdat, 1975 (in Russian).
10.
MatrosovAVKovalenkoMDMenshovaIVKerzhaevAP.Method of initial functions and integral Fourier transform in some problems of the theory of elasticity. Z Angew Math Phys2020; 71: 24.
11.
LevinBJ.Distribution of Zeros of Entire Functions (Translations of Mathematical Monographs, Vol. 5). Providence, RI: American Mathematical Society, 1980.
12.
TimoshenkoSPGoodierJN.Theory of Elasticity. New York: McGraw-Hill, 1951.
13.
KovalenkoMDMenshovaIVShulyakovskayaTD.Expansions in Fadle–Papkovich functions: examples of solutions in a half-strip. Mech Solids2013; 48(5): 584–602.
14.
MarkushevichA.I.Entire Functions. New York: Elsevier, 1966.
15.
PrudnikovAPBrychkovYAMarichevOI.Integrals and Series (Elementary Functions, Vol. 1). New York: Gordon and Breach Science Publishers, 1986.
16.
YuGKovalenkoMDMenshovaIVKerzhaevAP.Two problems for a strip with a transverse crack: Exact solutions. J Phys: Conf Ser2019; 1215: 012037.