A NOVEL APPROACH TO THERMAL AND MECHANICAL STRESSES IN A FGM CYLINDER WITH EXPONENTIALLY-VARYING PROPERTIES

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 1, pp , Wasaw 2017 DOI: /jtam-pl A NOVEL APPROACH TO THERMAL AND MECHANICAL STRESSES IN A FGM CYLINDER WITH EXPONENTIALLY-VARYING PROPERTIES Keimcan Celebi Adana Science and Technology Univesity, Depatment of Mechanical Engineeing, Adana, Tukey Dumus Yaımpabuc Osmaniye Kokut Ata Univesity, Depatment of Mathematics, Osmaniye, Tukey dumusyaimpabuc@osmaniye.edu.t Ibahim Keles Depatment of Mechanical Engineeing, Ondokuz Mayis Univesity, Samsun, Tukey A novel appoach is employed to a geneal solution fo one-dimensional steady-state themal and mechanical stesses in a hollow thick cylinde made of a functionally gaded mateial (FGM). The tempeatue distibution is assumed to be a function of adius, with geneal themal and mechanical bounday conditions on the inside and outside sufaces of the cylinde. The mateial popeties, except Poisson s atio, ae assumed to be exponentially-vaying though the thickness. Focing functions applied to the inne bounday ae intenal pessues which may be in fom of steps. These conditions esult in govening diffeential equations with vaiable coefficients. Analytical solutions to such equations cannot be obtained except fo cetain simple gading functions and pessues. Numeical appoaches must be adopted tosolvethepobleminhand.thenoveltyofthepesentstudyliesinthefactthatthe Complementay Functions Method(CFM) is employed in the analysis. The Complementay Functions method(cfm) will be infused into the analysis to convet the poblem into an initial-value poblem which can be solved accuately. Benchmak solutions available in the liteatue ae used to validate the esults and to obseve the convegence of the numeical solutions. The solution pocedue is well-stuctued, simple and efficient and it can be eadily applied to cylindes. It is also well suited fo poblems in which mechanical popeties ae gaded. Keywods: themal stesses, functionally-gaded mateials, thick cylinde, Complementay Functions Method 1. Intoduction Pessue vessel stuctual membes such as cylindes, disks and sphees find boad application fields in the industy, and thei vibation analyses ae deemed necessay fo safe design and opeation. Hollow cylindes and thick-walled cylindical shells ae common components in stuctual applications and device systems involving aeospace and submaine stuctues, civil engineeing stuctues, machines, pipes, sensos and actuatos, etc. These stuctues ae often exposed to tempeatue envionment and themal stesses ae then induced. In many cases, themal stesses will significantly depess stength and also affect functionality of stuctues. Thus, the exact analysis of themal stesses is eally impotant(ying and Wang, 2010). Thee have been many studies, such as Timoshenko and Woinowsky-Kiege(1959), Boley and Weine(1960), Das and Navaatna(1962), Das and Rath(1972), Stavsky(1963) and Thangaatnam et al.(1988), which focused on themal stesses in isotopic homogeneous ectangula plates. Yee and Moon(2002) have been obtained a closed-fom analytical solution fo the plane themal stess analysis of a homogeneously othotopic hollow cylinde subjected to an abitay, tansient, asymmetic

2 344 K. Celebi et al. tempeatue distibution. They used a stess function appoach fo obtaining hoop, adial, and shea stesses in a hollow cylinde. Shao(2005) pesented, by using a multi-layeed appoach based on the theoy of laminated composites, the solutions of tempeatue, displacements, and themal-mechanical stesses in a functionally gaded cicula hollow cylinde. Shao et al.(2008) used complex Fouie seies and Laplace tansfom techniques to investigate tansient heat conduction and themo-mechanical stesses in an FGM hollow cylinde. Jabbai et al.(2002, 2003, 2009) deived the exact solution fo one-dimensional and two dimensional steady-state themoelastic poblems of functionally gaded hollow cylindes whee mateial popeties vaied with the powe poduct fom of the adial coodinate vaiable. Recently Ruhi et al.(2005) studied themoelastic analysis of thick walled finite length cylindes of functionally gaded mateials and achieved esults fo stess, stain and displacement components though the thickness and along the length. The esults wee pesented fo unifom intenal pessue and themal loading. Ootao and Tanigawa(2006) analyzed exactly a one-dimensional tansient themoelastic poblem of a functionally gaded hollow cylinde whose themal and themoelastic constants wee assumed to vay with the powe poduct fom of the adial coodinate vaiable. The esulting govening diffeential equation then possessed vaiable coefficients. Geneal closed-fom solutions to such equations ae not available. Noda et al.(2012) studied the tansient themoelastic analysis fo an FGM solid cicula disk whose mateial popeties wee expessed by a piecewise powe law. As it was done in the woks cited above, in such situations the solution methods included integal tansfomations, development of finite element models, and, in some special cases, seies solutions wee attempted. Assuming that the membe was composed of many homogeneous layes of diffeent popeties emulating the FGM behavio, thee was anothe way of tackling the poblem on hand. All of these appoaches equied heavy mathematical manipulations and, in the case of having to discetize the domain into many elements, a high amount of computational time. In the pesent pape, the govening diffeential equation is non-homogeneous with vaiable coefficients which include mateial popeties. A novel appoach is attempted to obtain displacements, stains and stesses in a simple and efficient manne. The complementay functions method(cfm), theoetically explained in the liteatue by Aktaş(1972), Agawal(1982) and Robets and Shipman(1979) is infused into analysis to convet the poblem to an initial-value poblem which can be then easily solved by, fo example, the fifth-ode Runge-Kutta method (RK5) with geat accuacy(chapa and Canale, 1998). Shell theoies o dividing the mateial into homogeneous subelements of diffeent popeties emulating the gaded behavio contains the customay appoach of modeling FGM stuctual elements. Finite element analysis, seies expansion methods and diect methods ae pimay solution methods used in the liteatue. The pesent pape uses a novel and efficient method which employes CFM. A thick hollow cylinde of FGM unde one-dimensional steady-state tempeatue distibution with geneal types of themal and mechanical bounday conditions is analysed. Two mateial models ae used:(a) with a simple powe law with constant Poisson s atio(jabbai et al., 2002) fo which analytical benchmak solutions ae available,(b) with exponentially-vaying popeties. It should be emphasized once again that the solution pocedue is not confined to any paticula choice of the mateial model; it is equally suitable fo abitay functions defining the gadient vaiation of mateial popeties. 2. Solutions by the Complementay Functions Method The CFM tansfoms two-point bounday-value poblems to a system of initial-value poblems. It educes to a paticulaly simple solution scheme when applied to a given class of poblems, e.g.foanannuladiskofinneadius i andouteadius o.asitisshowninthepoceeding Sections, unde axisymmetic conditions, the govening diffeential equation of the dependent vaiableu()initsmostgenealfomis

3 Novel appoach to themal and mechanical stesses u +P()u +Q()u=R() (2.1) subjecttoboundayconditionsontheinne(= i )andoute(= o )sufaces.hee( ) denotes the deivative with espect to. A geneal closed-fom solution of the above equation cannot be obtained. The complete solution to Eq.(2.1) is u=b j u j +u p j=1,2 (2.2) wheeu j andu p ae,espectively,homogenousandpaticulasolutions.thecoefficientsb j ae deteminedviatheboundayconditions.cfmbeginsbyassumingu i =Y (i) 1 andu i =Y(i) 2, which means (Y (i) 1 ) =Y (i) 2 (2.3) Hee,theindexi=1,2efestohomogeneoussolutionsandi=pmeansthepaticulasolution. To detemine the homogeneous solutions, the ight-hand side of Eq.(2.1) is set equal to zeo, and the following is obtained (Y (i) 2 ) = P()Y (i) 2 Q()Y (i) 1 (2.4) The system of Eqs.(2.3) and(2.4) can be solved numeically fo each homogeneous solution. The Konecke delta initial conditions given below ae used to assue linea independence of the solutions(robets and Shipman, 1979) Y (i) j =δ ji j,i=1,2 (2.5) To obtain the paticula solution, Eq.(2.4) is modified as (Y (p) 2 ) = P()Y (p) 2 Q()Y (p) 1 +R() (2.6) A paticula solution needs only to satisfy the diffeential equation and homogeneous initial conditions Y (p) j =0 j=1,2 (2.7) be imposed. Equations(2.3),(2.6),(2.7) constitute a system of equations fo the paticula solution along with the initial conditions. The fifth-ode Runge-Kutta method(rk5) is used fo allcasesconsideed.notethatbythispoceduenotonlythesolutionu()itselfbutalsoitsfist deivative ae eadily calculated. Applying the bounday conditions pescibed fo the paticula pobleminhandesultsinthefollowingsystemofalgebaicequationsfothecoefficientsb 1 andb 2 [ ][ ] A11 A 12 b1 = A 21 A 22 b 2 [ ] RHS1 RHS2 Hee,A ij includesthevaluesofthehomogeneoussolutionsattheboundaypoints.rhs1and RHS2 contain values of the paticula solutions. If the cylinde is subjected to intenal and extenal pessues, they will also be included in the ight hand-side tems. On the othe hand, implementing CFM in the heat conduction poblem yields RHS1 and RHS2 as pescibed tempeatues along the boundaies. These points will be illustated in the following Sections. (2.8)

4 346 K. Celebi et al. 3. Heat conduction in the adial diection The heat conduction equation in the steady-state condition fo a one-dimensional poblem in pola coodinates and themal bounday conditions fo a FGM hollow cylinde ae given, espectively, as 1( =0 k()t ()) i o C 11 T( i )+C 12 T ( i )=f (3.1) 1 C 11 T( o )+C 12 T ( o )=f 2 wheek=k()isthethemalconductioncoefficient, i and o aetheinneandouteadii ofthehollowcylinde.c ij aetheconstantthemalpaameteselatedtotheconductionand convectioncoefficients.theconstantsf 1 andf 2 aeknownconstantsontheinsideandoutside adii. It is assumed that the nonhomogeneous themal conduction coefficient k() is an exponential functionofas k()=k o e β (3.2) wheek o isamateialconstantandβistheinhomogeneitypaamete.usingeq.(3.2),theheat conduction equation becomes 1( =0 e β T ()) (3.3) Steady-state axisymmetic heat conduction without heat geneation is consideed. The heat balance equation in the adial diection fo a nonunifom disk yields T +B()T =0 (3.4) wheeb()=(1/)+βanditisvayingasafunctionoftheadialcoodinate.thebounday conductions ae tempeatues pescibed on the inne and oute sufaces as T( i )=T i and T( o )=T o (3.5) The complete solution is the homogeneous solution with T=b j T j j=1,2 (3.6) T =b j T j j=1,2 (3.7) Following the steps outlined in Section 2, the tempeatue distibution is obtained at the collocationpoints.theconstantsb j cannowbefoundbyimposingtheboundayconditions.this pocessesultsinasystemgivenbyeq.(2.8)whee A 11 =T 1 ( i ) A 12 =T 2 ( i ) A 21 =T 1 ( o ) A 22 =T 2 ( o ) (3.8) RHS1=T i RHS2=T o

5 Novel appoach to themal and mechanical stesses Goveningequation Consideathickwalledcylindeoftheinsideadius i andtheoutsideadius o madeoffgm. The mateial is gaded though the diection. Let u be the displacement component in the adial diection. Then the stain-displacement elations ae ε = du d ε θθ = u The stess-stain elations ae σ =(λ+2µ)ε +λε θθ (3λ+2µ)αT() σ θθ =(λ+2µ)ε θθ +λε (3λ+2µ)αT() wheeσ ij andε ij (i,j=,θ)aestessandstaintensos,t()istempeatuedistibution detemined fom the heat conduction equation, α is the coefficient of themal expansion, and λ andµaethelamecoefficientselatedtothemodulusofelasticityeandpoisson satioνas λ= νe() (1+ν)(1 2ν) µ= E() 2(1+ν) The equilibium equation in the adial diection, disegading the body foce and inetia tems, is σ +σ σ θθ (4.1) (4.2) (4.3) =0 (4.4) To obtain the equilibium equation in tems of the displacement component fo the FGM cylinde, the functional elationship of the mateial popeties must be known. To ascetain the effect of the inhomogeneity, the popeties ae consideed to vay exponentially acoss the thickness E()=E o e β α=α o e β (4.5) wheee o andα o aethemateialconstantsandβistheinhomogeneitypaamete.poisson s atio vaies vey little though the thickness in FGM mateials. Futhemoe, its effects on themal and mechanical stesses ae insignificant. Fo simplicity, Poisson s atio is assumed to be constant(akbai Alashti et al., 2013; Jabbai et al., 2015). Using elations(4.1)-(4.5), the Navie equation in tem of the displacement is whee u +P()u +Q()u=R() (4.6) P()=(β+1) 1 Q()= R()= eβ α o (1+ν) (2βT+T ) (1 ν) ( νβ 1 ν 1 ) 1 2 Following the steps outlined in Section 2, the complete displacement is obtained at the collocation points as (4.7) u=b 1 u 1 +b 2 u 2 +u p (4.8) with u =b 1 u 1 +b 2u 2 +u p (4.9) Thecoefficientsb 1,b 2 willbedeteminedusingthestessfeeconditionsoninne(σ ( i )= P i ) andoute(σ ( o )= P o )boundaies.thisstepispaticulalysimplesinceduingthesolution pocess the fist deivative of the adial displacement has aleady been calculated.

6 348 K. Celebi et al. 5. Results and discussions Asanexample,consideathickhollowcylindeoftheinneadiusa i =1mandtheoute adius o =1.2m.Poisson satioistakentobe0.3,andthemodulusofelasticityandthe themalcoefficientofexpansionattheinneadiusaee o =200GPaandα o = / C, espectively. The popeties ae consideed to vay exponentially acoss the thickness. The boundayconditionsfotempeatueaetakenast( i )=10 CandT( o )=0 C.Thehollow cylinde has pessue on its inne suface, so the bounday conditions fo stesses ae assumed asσ ( i )= 50MPaandσ ( o )=0MPa. The numeical solution in the pesent study is checked with the esults obtained by Jabbai et al.(2002) fo the validation pupose. Compaison is illustated in Tables 1-3. It can be obseved thattheesultsaeingoodageementwiththesameesultsbyjabbaietal.(2002).the numeical esults have been obtained to six-digit accuacy by picking only 11 collocation points. Table1.CompaisonofCFMwithJabbaietal.(2002)foahomogenouscylinde(m=0and β=0) i T/T( i ) u/ i σ /P i σ θθ /P i CFM Jabbai CFM Jabbai CFM Jabbai CFM Jabbai Table 2. Compaison of CFM with Jabbai et al.(2002) fo FGM cylindes with constant Poisson satioandtheelasticmodulusobeyingasimplepowelaw(m= 2andβ= 2) i T/T( i ) u/ i σ /P i σ θθ /P i CFM Jabbai CFM Jabbai CFM Jabbai CFM Jabbai Table 3. Compaison of CFM with Jabbai et al.(2002) fo FGM cylindes with constant Poisson satioandtheelasticmodulusobeyingasimplepowelaw(m=2andβ=2) i T/T( i ) u/ i σ /P i σ θθ /P i CFM Jabbai CFM Jabbai CFM Jabbai CFM Jabbai

7 Novel appoach to themal and mechanical stesses Figue 1a shows vaiations of tempeatue along the adial diection fo diffeent values of the inhomogeneity paamete(β). The figue shows that as the inhomogeneity paamete β inceases, the tempeatue deceases. Figue 1b shows the plot of the adial displacement along the adius. The magnitude of the adial displacement is deceased as the inhomogeneity paamete β is inceased. The adial and cicumfeential stesses ae plotted along the adial diection and ae showninfigs.1cand1d.themagnitudeoftheadialstessisinceasedasβisinceased.it isseenthatfoβ<1thehoopstessdeceasesalongtheadialdiection.foβ>1,thehoop stess inceases as the adius inceases, since the modulus of elasticity is an inceasing function of theadius,seeeq.(2.6).physically,thismeansthattheoutelayesofthecylindeaebiasedto maintainthestessduetotheihighestiffness.theeisalimitingvaluefoβ,wheethehoop stess emains almost constant along the adius. The cuve associated with β = 1 shows that the vaiation of hoop stess along the adial diection is mino, and is almost unifom acoss the adius. To investigate the patten of stess distibution along the cylinde adius, the effective stessσ = 2 σ σ θ isplottedalongtheadialdiectionfodiffeentvaluesof o / i andthe inhomogeneitypaameteβ.figue2isplottedfo o / i =1.2.Itisinteestingtonotefom Fig.2thatfoβ=1theeffectivestessisalmostunifomalongtheadiusofthecylinde. Fig. 1. Radial distibution of:(a) tempeatue,(b) adial displacement,(c) adial stess and(d) hoop stess fo cylinde Itshouldbepointedoutonceagainthatthepuposeofthepesentwokistheintoduction ofcfmtothesolutionpocedueoftheclassofpoblemsinhand.convetingthetwo-pointbounday value poblem to a system of an initial-value poblem gives a way to the implementation of well-established numeical schemes. The Runge-Kutta method of fifth-ode(rk5) is used to

8 350 K. Celebi et al. Fig.2.Effectivestessdistibutionfo o / i =1.2 solve the system of equations. The pocedue is simple and efficiently implemented. The numeical esults have been obtained exact up to six-digit accuacy by picking only 11 collocation points in RK5. 6. Conclusion This pape pesents a numeical solution fo calculation of axisymmetic themal and mechanical stesses in a thick hollow cylinde made of FGM. The mateial popeties though the gaded diection ae assumed to be nonlinea with a powe law distibution and exponentially-vaying popeties. The mechanical and themal stesses ae obtained though the CFM of the solution of the Navie equation. The compaisons of tempeatue distibutions and stess distibutions aepesentedinfomoftables.thenumeicalesultsfoallcasesaeshowntoexactlymatch those epoted by Jabbai et al.(2002). Finally, we can conclude that: Withtheunifiedappoachpesentedinthepesentstudy,onewouldnothavetocompomise on the functional continuity of the mateial popeties. Analysis of any mateial model in fom of an abitay function subject to intenal pessue has been analyzed efficiently and accuately by employing CFM. The unified method used is accuate and moe efficient than the conventional methods. The method employed in this study allows one to find solutions of continuous functions. The CFM of solving the diffeential equation povides a complete solution, yielding both themal stesses and tempeatue distibutions. Refeences 1. Agawal R.P., 1982, On the method of complementay functions fo nonlinea bounday-value poblems, Jounal of Optimization Theoy and Applications, 36, 1, Akbai Alashti R., Khosand M., Taahhomi M.H., 2013, Thee-dimensional asymmetic themo-elastic analysis of a functionally gaded otating cylindical shell, Jounal of Theoitical and Applied Mechanics, 51, 1, Aktaş Z., 1972, Numeical Solutions of Two-Point Bounday Value Poblems, Ankaa Tukey: METU, Dept. of Compute Engineeing 4. Boley B.A., Weine J.H., 1960, Theoy of Themal Stesses, Wiley Hoboken, New Jok, 35-45

9 Novel appoach to themal and mechanical stesses Chapa S.C., Canale R.P., 1998, Numeical Methods fo Enginees, 2nd ed., McGaw-Hill, New Yok, Das Y.C., Navaatna D.R., 1962, Themal bending of ectangula plates, Jounal of the Aeospace Sciences, 29, 11, Das Y.C., Rath B.K., 1972, Themal bending of modeately thick ectangula plates, AIAA Jounal, 10, 10, Jabbai M., Bahtui A., Eslami M.R., 2009, Axisymmetic mechanical and themal stesses in thick shot length FGM cylindes, Intenational Jounal of Pessue Vessels and Piping, 86, 5, Jabbai M., Nejad M.Z., Ghannad M., 2015, Themo-elastic analysis of axially functionally gaded otating thick cylindical pessue vessels with vaiable thickness unde mechanical loading, Intenational Jounal of of Engineeing Science, 96, Jabbai M., Sohabpou S., Eslami M.R., 2002, Mechanical and themal stesses in a functionally gaded hollow cylinde due to adially symmetic loads, Intenational Jounal of Pessue Vessels and Piping, 79, 7, Jabbai M., Sohabpou S., Eslami M.R., 2003, Geneal solution fo mechanical and themal stesses in a functionally gaded hollow cylinde due to nonaxisymmetic steady-state loads, ASME Jounal of Applied Mechanics, 70, 1, Noda N., Ootao Y., Tanigawa Y., 2012, Tansient themoelastic analysis fo functionally gaded cicula disk with piecewise powe law, Jounal of Theoetical and Applied Mechanics, 50, 3, Ootao Y., Tanigawa Y., 2006, Tansient themoelastic analysis fo functionally gaded hollow cylinde, Jounal of Themal Stesses, 29, 11, Robets S.M., Shipman J.S., 1979, Fundamental matix and two-point bounday-value poblems, Jounal of Optimization Theoy and Applications, 28, 1, Ruhi M., Angoshtai A., Naghdabadi R., 2005, Themoelastic analysis of thick-walled finite- -length cylindes of functionally gaded mateials, Jounal of Themal Stesses, 28, 4, Shao Z.S., 2005, Mechanical and themal stesses of a functionally gaded cicula hollow cylinde with finite length, Intenational Jounal of Pessue Vessels and Piping, 82, 3, Shao Z.S., Ang K.K., Reddy J.N., Wang T.J., 2008, Nonaxisymmetic themomechanical analysis of functionally gaded hollow cylindes, Jounal of Themal Stesses, 31, 6, Stavsky Y., 1963, Themoelasticity of heteogeneous aeolotopic plates, Jounal of the Engineeing Mechanics Division, 89, 2, Thangaatnam R.K., Palaninathan, Ramachandan J., 1988, Themal stess analysis of laminated composite plates and shells, Computes and Stuctues, 30, 6, Timoshenko S., Woinowsky-Kiege S., 1959, Theoy of Plates and Shells, 2nd ed., McGaw- -Hill, New Yok, Yee K.C., Moon T.J., 2002, Plane themal stess analysis of an othotopic cylinde subjected to an abitay, tansient, axisymmetic tempheatue distibution, ASME Jounal of Applied Mechanics, 69, 5, Ying J., Wang H.M., 2010, Axisymmetic themoelastic analysis in a finite hollow cylinde due to nonunifom themal shock, Intenational Jounal of Pessue Vessels and Piping, 87, 12, Manuscipt eceived May 20, 2015; accepted fo pint Septembe 13, 2016