This article was downloaded by: [vinod varghese]On: 15 April 2013, At: 03:28Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UKJournal of Thermal StressesPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/uths20TRANSIENT THERMAL STRESS PROBLEM IN ATRANSVERSELY ISOTROPIC FINITE CIRCULAR CYLINDERUNDER THREE-DIMENSIONAL TEMPERATURE FIELDNaotake Noda aa Department of Mechanical Engineering, Shizuoka University, Hamamatsu, 432, JapanVersion of record first published: 27 Mar 2007.To cite this article: Naotake Noda (1983): TRANSIENT THERMAL STRESS PROBLEM IN A TRANSVERSELY ISOTROPIC FINITECIRCULAR CYLINDER UNDER THREE-DIMENSIONAL TEMPERATURE FIELD, Journal of Thermal Stresses, 6:1, 57-71To link to this article:http://dx.doi.org/10.1080/01495738308942166PLEASE SCROLL DOWN FOR ARTICLEFull terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditionsThis article may be used for research, teaching, and private study purposes. 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The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.TRANSIENT THERMAL STRESSPROBLEM IN A TRANSVERSELYISOTROPICFINITE CIRCULAR CYLINDER UNDERTHREE-DIMENSIONAL TEMPERATURE FIELDNaotakeNodaDepartment of Mechanical Engineering, ShizuokaUniversity This paper is concerned with atransient thermal stress problem inafinitecircular transversely isotropic solid cylinder subjected to an asymmetricaltemperature distribution on a cylindrical surface. Thestressfields arefoundby useofpotential functions method. As anexample, the temperature andstress fieldsfora specificcase are computed for thesake of comparisontotheisotropicbody.INTRODUCTIONIn recent years, owingto the increased use of anisotropic construction materials instructures, considerable interest has been devoted to thermal stress problems inanisotropic bodies. Anumber of papers have been published on steady-state trans-verselyisotropic thermoelasticproblems [1-6]. However, fewinvestigationsof transi-ent thermal stress problems in transversely isotropic bodies have beencarried out.Takeuti and Noda proposed a general solution method for the three-dimensionaltransient thermoelastic problems in transverselyisotropic bodies [7]. Sugano solvedtransient thermal stresses and deformation in a transverselyisotropic finite circularcylinder [8], and Sugano and Takeuti solved the transient thermal stresses in atransverselyisotropicthick plate witha cylindrical hole [9] using Singh's method[3].Thesepapers wereconcernedwith the symmetrical temperaturefields, andthereis noinvestigation for the transient thermal stress problemintransverselyisotropic bodiesunder three-dimensional temperaturefield.In this paper, we treat a transient thermal stress problem in a finite circularcylinder with the transverse isotropysubjectedtoanasymmetricallyarbitrarytempera-ture distribution on the cylindrical surface byuseof the potential functions methoddescribedinour previous paper [7]. Topresent theeffectsof thetransverse isotropy,numerical results for temperature and the stress fields are shown in figures andarecompared withthosefortheisotropic body.Journal of Thermal Stresses, 6:57-71,1983Copyright 1983 by Hemisphere Publishing Corporation0149-5739/83/010057-15$2.7557Downloaded by [vinod varghese] at 03:28 15 April 2013 58ANALYSISN.NODATemperatureFieldConsider a transversely isotropic finite solid circular cylinder, as shown in Fig. I,subjected to an asymmetrical temperature distribution expressed in the form ofTnf(O)g(z) on the cylindrical surface with a heat transfer into the surroundingmedium. The transient heat conductionequationof thetransverselyisotropiccylinderis expressedby(I)where Tis the temperature change, A2=Az/Aris the ratio of thermal conductivitycoefficientsintheaxial andradialdirections,K =Ar/er is thethermal diffusivity, ris thedensity, cis the specific heat, t istime, andthecommadenotespartialdifferentiationwithrespect to a variable.The solutionof Eq. (I) must satisfythefollowingboundaryandinitial conditionsT'r +hr{T-Taf(B)glz)) =a on r =a}T,z hzT = a on z = T a at t a(2)(3)where h, and h, are the relative heat transfer coefficients at r =a and z =I,respectively, a is the radius of the circular cylinder, Iis the length of the circularcylinder, Tn istheconstant temperature.To simplify, weassume that functions f(O) andg(z)satisfythefollowingrelationsf(B) =fl-B) r g(z) = g(-z) (4)For convenience, weintroducethefollowingdimensionlessquantities:"r ahr,Hz ah z T TIT",/; t l/a t' o::t/azP .. ria. 1;." z/a .hrTafl819(Z)chzL LzFig. I Finitecircular cylinder.Downloaded by [vinod varghese] at 03:28 15 April 2013 TRANSIENT THERMAL STRESS PROBLEMUsingthe Laplace transform, weobtainthe temperaturefield:00 00T I. L[TnmIn(ASmP)+ L n=Om=l k=lx coscosnewhere59(5)Tnmkf(6) - I fncos nen=O - I gmc os m=land k are the positive rootsof theequationsmarethe positiverootsofthe equationS sin 1 - HzcOS 1 = 0and In(p) isaBessel functionof the nth order, and In(p)is amodifiedBessel functionof nthorder.11hern1al StressesConsider an asymmetrical thermoelastic problem in a transversely isotropic solidcircularcylinder. The stress-strainrelationsare(6)where Cj jand are the material constants ofthe transverse isotropy. The displace-ment-strainrelationsareDownloaded by [vinod varghese] at 03:28 15 April 2013 60 N.NODA-I -1ZZ UZ'Zcre= rUr'S+uS'r- r Ue-1(7)eur'z+uZ'reze=us'z+ r Uz'e rzwhere u" uo, andUz arethedisplacements inther, (), andz directions, respectively.Intheabsenceofbodyforces, theequibriumequations are-1 -10rr'r+ xer'e+ azr'z+r (Orr-Oee)+-1+z8'z+ 2r-lo0 (8)rB'rree'e re+-1 -10rz'rr 0ez'e+azz'z+ r arzSubstituting Eqs.(6) and (7) into Eq. (8), thecorrespondingdisplacement equationsare-1 -2 -2C, l (Ur'rr+r Ur'r-r Ur)+{Cll-C,Zlr Ur'ee/2+C" ur'zz-I - 2 _+ (CII+C,Z)r ue're/2-{3Cl1-C1Z)r Ue'e/2+{C'3+C )Uz'rz=B,T'r(9)TosolvetheequilibriumEqs. (9), we introducethe displacement potentials, t/J,n, andXdefinedby