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Nuclear Engineering and Design 40 (1977) 383-391 North-Holland Publishing CompanyTHERMAL STRESS 1N A BIMODULUS THICK CYLINDERN. KAMIYADepartment of mechanical Engineering, Mie University, Kamihamacho. Tsu 514, JapanReceived 15 April 1976Certain materials are known to behave differently under tension and compression. This reports an investigation of a simplo thermal stress problem based on the bimodulus elasticity in the framework of the classical-type uncoupled thermoelasticity.Because of an unavoidable complexity of three-dimensional analysis in bimodulus elasticity, as a fundamental illustration weconsider herein an axisymmetric plane strain state of a thick orthotropic circular cylinder under axisymmetric heat conditions.A system of transcendental simultaneous algebraic equations with respect to eight unknowns are derived. Numerical analysesare carried out for a steady-state temperature field and for a quasi-stationary state with internal heat generation. Results ofcalculations show that the states of a cylinder are affected significantly by the difference between tensile and compressivemoduli of elasticity.formed body subjected to an arbitrary load.Thermal stress or deformation analysis is also important from the above-mentioned engineering pointsof view, but so far as we know there is no such researchon bimodulus material. This paper reports an investigation of a simple thermal stress problem based on thebimodulus elasticity in the framework of the classicaltype uncoupled thermoelasticity. Because of an unavoidable complexity of three-dimensional analysis inbimodulus elasticity, as a fundamental illustration weconsider an axisymmetric plane strain state of a thick1. IntroductionLittle attention has been paid to materials with different deformation responses in tension and compression, except for certain types, such as concrete or soil.Yet, recently developed artificial materials, e.g. ceramics, certain composites, high-polymers and porous materials, are said to display more or less the above-described behavior. Their uniaxial stress-strain relationsunder static loads are often approximated effectivelyby two respective straight lines with different slopesemitting from the origin, provided that nonlinearity isunimportant. Such an approximately idealized materialis called a 'bimodulus' or 'bilinear' elastic one. For example, some artificial graphites which exhibit such differences are known to have important applications inpressure vessels, nuclear reactors and aerospace engineering.Although some reports have dealt with static deformations, stability analyses of bimodulus structural elements and general theories of bimodulus elasticity,study as a whole is still in its initial stages [ 1 - 1 6 ] . Onereason why such analyses encounter difficulties is that,since the elastic constants of bimodulus material aredirectly connected to the sign of stress, their propervalues may be specified after deformation and stressstates in a body are known. It is generally not easy topredict the distribution of the sign of stress in a de-JbwSTRAIN 1ET: TENSILE ,' ODULUS[CCOMPRESSIVEMODULUSFig. 1. Bimodulus idealization.383
384N. Kamiya / Thermal stress in a thick cylindercircular cylinder under axisymmetric heat conditions.Supposing the orthotropic nature of mechanical andthermal properties of material, we first derive governingequations. Since three nonvanishing principal stresscomponents exist in a cylinder, it is supposedly dividedinto finite numbers of domain (region) by combinations of their signs. Inside each domain the elastic coefficients take respective constant values; in other words,they are 'piecewise constant' in a body. Obtained equations for each domain are arranged to satisfy the prescribed boundary conditions at the inner and outersurfaces of a cylinder and those at domain boundaries.Consequently we obtain a system of transcendentalsimultaneous equations with respect to eight unknowns.Numerical analyses are carried out by an iterationmethod for (a) a cylinder under steady-state heat flow,in which the inner surface temperature is higher thanthe reference temperature at the outer surface; and (b)a quasi-stationary cylinder with internal heat generation. For several values of a parameter representing thebimodulus property, stress and displacement distributions are determined and discussed. The results o1"calculations show that the stress states of a cylinder areaffected significantly by the difference between tensileand compressive moduli of elasticity.2. Governing equationsSince general three-dimensional analysis of bimodulus material becomes significantly complicated, we consideraxisymmetric problems of a thick circular cylinder with respect to the cylindrical coordinate system r, 0 and z. Themechanical and thermal properties of a cylinder are assumed to be orthotropic. According to the bimodulus elasticity in the context of the uncoupled classical thermoelasticity, relations between stress and strain components in acylinder under an axisymmetric steady-state or quasi-stationary temperature field T are expressed as follows, e.g.[17 211:e r - C r T a l l O r a l 2 o o al3Oz ,e o - a o T a 2 1 o r a 2 2 o o a23oz ,e z - tzT a31or a32o o a33o z ,(1)where elastic constants all, a22 and a33 are specified, in relation to the signs of the corresponding principal stresses,as(a 2 1/Eg(oo 0)a22:[a 2 : alG(oo O)': [atn 1/Etr(O r 0)auta l : l l E C ( O r O ) (43 a33( 'z o)(a 3: 1/L: (% (2)o)For the expression of off-diagonal elastic compliance aq(i :/:j) appearing in eqs. (1), we assume that the followinggeneralized reciprocal relations hold for bimodulus material identically,al 2 a21 .40.Ftr.q0.Ec4rGE E (similarly for al3 and a23 )(3)where Poisson's ration is defined as usual,Vii - - e j / e i .(4)And we further suppose that the off-diagonal compliances are independent of the sign of stresses and that they maybe written only by aaz.In what follows, we consider a circular cylinder long enough to ignore the longitudinal strain,cz o.(5)
N. Kamiya / Thermal stress in a thick cylinder385Substituting the expression of oz obtained from the last of eqs. (1) into the first two of the same equations, weobtain6r ( all --; 323( r -- a12 30tz) T, ) -Or (a12-a 2100 \Ola3-a33/a 2a2eo (a12 33)Or (a22-- 121Oo (Oto]\ -a12a3 30tz T.(6)From eqs. (6) stress components are solved, to give r--D/(a22a 2' -a33)erO0 1{ (a12(a12 a 2 ea33] 0I(a22a 2 (a12 a 2 -a12Ya22) a3- (7)a212' e [ (a12 -- a 2t r (all - a122 o (a12 - all)T33%ja12 "IT},a22' (all -a33]-a33Jeroa33!a33] 0where(a 2/a33)(all a22 a33) (2a 2/a33).D alla22 -Moreover, the longitudinal stress is expressed as1 za33 D {a12(a22-a12)er a12(a11-a12)eo - [a12(a22-a12)C r a12(all-a12)C o (a 2-alla22)az ]T)(8)There are three nonvanishing principal stresses in the present problem and so eight combinations of their signoccur in principle. Consequently, if we divide a body into 'domains' (regions) in relation to the combination ofthe signs, eight kinds of domain are considered. But is is natural that we may not necessarily find all of them. Ineach domain elastic coefficients are constant; i.e. they are piecewise constant in a-body.Inside an arbitrary domain the equilibrium equations and the kinematic relations are formulated, for axisymmetric deformations, asd r1 "t-(Or- oo) O;er du/dr,e o u/r.(9; 1O)Substituting eqs. (7) into (9) and employing eqs. (10), we obtain2d2u du alla33 - a12 u [ a12(a33 - a12)a12(a12 a22)oL']! dT . . . . . LOtr0 0 "1"-dr 2 rdr a22a33-a122 ra22a33 a 2a22a33-a122 zj dr T[(a12a33 a22a33 2a 2) tr (alla33 a12a33 2a 2) s0 al2(al 1 a22) Otz] "r(a22a33 - a22)(11)Now, we define the following dimensionless variables R and s by using the linear dimension b, asr Rb,R es ,(12)and rewrite eq. (1 1), to yieldd2ualia33 - a22ds 2a22a33u eSf(s),a22(13)
N. Kamiya/ Thermalstressin a thick cylinder386wheref(s) b{[ 0 r a12(a33-a12)a22a33 a 2 ct0 a12(a12 a22) ]dTa2z ]asa22a33 12----, [(a12a33 a22a33 2a )a r (alla33 a12a33-2a 2)O o a12(all-a22)Otz ] .a22a33 - a 2Denoting, in consideration of the inequality (alla33 - a22)/(a22a33- a212) 0,X1} [(alia33 a 2)/(a22a33 a 2)] 1/2X2we obtain the general solution of eq. (13) as follows:(15)su cleMs c2eX2S (14)s[eXlSf eX2 f( )at eXZSf eM f( )d ]12 [(alia33 - a212)/(a22a33- a122)]1/2where c l and c 2 are arbitrary constants.With the aid of eqs. (7), (8) and (16), the stress components are expressed as1 [e -s [((a2 '12 ,- (a12 a 2t cleXlS ((a22 a 2 t (al 2 aff2))c2ekxS r b W t \\a22-a33] 1a33Ha33] 2--a3 12[(alia33 a 2)/(a22a33 a 2)]1/2 [((a22 a 2 Xa33] 1 (a12 a 2--(at2 a 21;fSex2S k' f( )d l?- I(a22 a 2 a33]/-s(a12 a 2' a33] o- - -a!2a33!] % 21 {eb s I( (a12 a 2])t (all a 2 \c eXls (all 0 Da33] 1a33 ff 1 ( (a12 a 2])ta331 2 2[(alia33 a 2)/(a22a33-a 2)]1/2[( (a,2 Laa33 ] 1 a12 -] T}a33 C z ] '-a 2]\233)/ , ex2s 12 f eX,S X2 f( )d 2s33a 2 t-- a;3/a 2 f s e 2s hl f( )d l -I--(al2 - a 21Otr (all ---a 21o (a12 all)Z -O z 33 -].1 T } , (17).a12 (all - 1 21 "33//a331a33] oO2a33D12((a22-a12) l (all-a12))cleXaS a12((a22 1Ia2[(alia33 - a 2)/(a22a33 - a 2)l1/212((a22- al2)X 1 (alla12((a22 a12)X2 (all al2))fSex2S Xl f( )d ]l-[a12(a22 al2)Otr al2(alla12)O o (a 2 --alla22)O z] T} .al2)X2 (all- a12))c2 ex2satz) fseXlS X2 f( )d
N. Kamiya / Thermal stress in a thick cylinderAgain we mention that eqs. (15)-(17) hold only insideeach domain.Stress distrilgutions in the conventional isotropicelastic cylinder (E t E c) for the example temperaturedistributions considered below are as follows: while theradial stress is alwayg compressive, the circumferentialand longitudinal stresses vary from compressive in theinside region to tensile in the outside region.We follow analogous domain partitions of a cylinderand the following four concentric ring domains are considered:t z(ii)- C z(iII?go O, --a(ii)IV[IIIV(i)r : dFig. 2. Outline of stress distributions for conventional elasticmaterials.o0 0,Oz O,When a cylinder is isotropic, they becomeDomain Ill"Or 0,4III)Oz O,Domain II:Or 0,-I --Domain I:Or O,387(18)ao O,oz 0 ,O0 0 ,az 0 .XII/X II-- 1,-Domain IV:Or O,(i -vc2 vt )1/2However, domains II and III are not realized simultaneously. It is usually not known beforehand which onewill occur and depends on a temperature distribution,geometrical parameters and a ratio of elastic moduli.Denoting r d and e where o 0 and o z change theirsigns, respectively, we have (i) domains I, II and IVwhen d e; or (ii) domains I, III and IV when d e.Coefficients X1 and ;k2 for each domain are expressed as)tI1 ] : (caila 3 a 2)1/2)kI \aha;3 a 2(1ic-c)1/2- - V.rcz V zr[50x l)pt c 1 , VJ-1vtpc ptPt2 vcJ.,II/- 2 I- tvCoz vCzo Ecrrz zr c"a22a33 -- a22 !01 0atr a,ptOzpczO o IV 0at r b,,C/r(19)XIII,. ct1/2pc vtc\1/21 [alla33-a212 )(1-E O) III/ -\ c t rz zr,2/a22a33 a 2vCozvtzO Ecr)c'ic ? ( a l l a 3 3 a 2 1/2xP - - - - ( - 7 - \a22a33 -- a 2"If we suppose that the inner and outer surfaces ofa cylinder are free from mechanical loading, the boundary conditions thereupon and conditions at domainboundaries are formulated as follows:when d e,rX1 ( a l l a a 3 - a 2 2 ](19a), )v c p t E t 1/2rz zr.1 -- Vt0zUz0.t EcIOr o I,UI u II,iI 0 ! 0 or o 0II oIVratr d,ull u IV,'atr--e ozll 0 or o V 0,(20)
N. Kamiya / Thermal stress in a thiek cylinder388when d e,r/b,6,5o 1r 0,at ro Iv 0 0 IIIrrul,8.91,a,-,02at r b,ro1 ,7 u III,-,04'atr e,o! 0or2-,06Zo II o VroJ I l O ,rulll ulV-,08'atr d.O II 0or'o 0IV - 0 , "IqlYgqPIf.-.10At the domain boundaries the radial stress and displacement must be continuous, but stress components inthe circumferential and longitudinal directions maysuffer some discontinuity. Thus, the positions whereo 0 and o z vanish may belong to either I or If, and II orIV, respectively, in the case (i) d e.For each domain, corresponding equations similarto eqs. (16) and (17) are established, including the respective two unknowns c 1 and c2, and consequently, asystem of governing equations derived from eqs. (20)containes the following eight unknowns:(a),6 2 ,2.5,6,8.ncll, c , c I (or CllI'), c 1(or cIII), c W, c V, d and e.-,2The governing equations are formulated as transcendental simultaneous algebraic equations with respect to theabove unknowns. Their expression is so complicatedthat we shall not cover them here in detail.-.q-c,l S,.,TRQP[-.6(b)3. Illustrations,2Two numerical illustrations are treated: (a) whensteady-state temperature distribution is expressed, soas to satisfy the Laplace equation, as in(21)T T(r) T a log(b/r)/log(b/a),where the inner surface temperature is elevated toT a ( O) when compared to the reference temperatureat the outer surface; and (b) when uniform heat generation occurs in a cylinder, in which the temperature isexpressed asT T ( r ) 4 [ b 2 - r 2 - 2a 2 l o g ( b / r ) ],(22),2-,q/////o. (c)LqOTPOPlC1,
N. Kamiya/ Thermal389stress in a thick cylinder,5,5ET/E c 0,05,O2,40,5 ,6,7,8,9i, 0,-,o4,32-,06,2(a)ISOTRr)PICET/ ,4,5',6',7',8,9i' ,r/b(d) ,2sFig. 3. (a) - (d) Distributions of stresses and displacement inisotropic cylinder under steady-state temperature distribution.8, .?-r/b.91,-,2ISOTROPIC(b)where the inner surface is thermally insulated and theouter surface remains at the reference temperature. Forboth cases, couplings between thermal effects and deformations are neglected, and so temperature distributions are expressible independently of the stress distributions from the heat conduction equations (b/a 2).Numerical calculations of the stresses and the radialdisplacement are performed for isotropic bimoduluscylinders in both temperature distributions (a) and (b)[figs. 3 ( a ) - 3 ( d ) and 4 ( a ) - 4 ( d ) ] , and for orthotropicbimodulus cylinders with circumferential and longitudinal elastic moduli twice as large as the radial modulusin case (a) [figs. 5 ( a ) - 5(d), the key is listed in table 1 ].Results are obtained for several values of Et/E c, andsome typical ones, especially for Et/E c 1,2, 0.5 and0.05, are shown in the figures. When Et/E c 1 andEt/E c 1, results correspond to those for tensilestrong (compressive-weak) and tensile-weak (compressive-strong) materials, respectively. In particular,Et/E c 0.05 displays extremely weak material in tension, such as concrete, and it approximates a so-called'no-tension' and/or 'tensionless' material. In the abovecalculations even for orthotropic materials, it was assumed that the thermal expansion coefficients wereidentical in every direction for simplicity's sake.Poisson's ratios were taken as v c 0.2 and v 0 0.2.As a whole, the magnitudes of stress become large,22 ohiLI-,L E T / E-,6C 0,050.5]2ISOTRDPIC(c)ET/E C 0.05.3 ",2O .5.'6,7'i8,9'I,(d)Fig. 4. (a) - (d) Distributions of stresses and displacement inisotropic cylinder under uniform heat generation.
390N. Kamiya / Thermal stress in a thick cylinderr/5,6,7,8LR3,gi,.,02\\--- . . . . . . /,04e ,00LRI(a)j,2 . ,4&LR14 CLR2.LR3CRICR2o,5, -- -,6 /.7---,8,91,r/bi -,4(b),5,'6,'7 r/b,8 -,2.47"/ % # CR3/ / " ""CRI,G5/(c),8CR3,5,4 . ----CR2JR2"-d ,3.2CRL - - - - - - L R ],10 ,5,6,7(d),r/ ,i,Fig. 5. (a) - (d) D i s t r i b u t i o n s o f stresses a n d d i s p l a c e m e n t inorthotropic cylinder under steady-state temperature distribution.Longitudinal reinforcement; . . . .circumferentialreinforcement.with an increase of the bimodulus parameter, Et/E c.While variations of o r and o 0 are pronounced withchange of EriE c, we find only slight variations of %.From the present results, we detect that when Et/E cprogressively decreases, the positions where the stresscomponents change their signs and the magnitudesundergo significant changes. In other words, the stressdistributions for Et/E c 1 and those for, say, Et//',"c 0.05 differ intrinsically from each other, and it will bedifficult and sometimes incorrect to forecast the latterresults from the former conventional elastic solutions.According to the classical elastic solution, tensileweak material such as concrete is said to be dangerouswhen tensile stress occurs in a body. Although this isnot incorrect we must, however, pay close attentionto the fact that the results with a near no-tension ortensionless material model exhibit a substantial difference from those with the E t E c model.We can recognize plain differences between the results for two kinds of temperature distributions (a)and (b), but the above qualitative nature for severalvalues of Et/E c remains unchanged. Parallel discussionsare possible for orthotropic materials reinforced in thecircumferential and longitudinal directions, respectively (larger moduli in these directions). Displacementsbecome small when E t / E c becomes large. Maxima of uare found when Et/E c is large, but distributions aremonotonous when Et/E c is extremely small.In general, for the present problems of a thick cylinder, since the radial stress component is small compared with other two components, its variations, eventhrough relatively significant, have only extrinsic effects. On the other hand we must be careful, in designing structural elements, of the changes of distributions of o 0 and o z whose values are not minute. Particularly, in the case of a material with different deformation responses in tension and compression, not onlythe magnitudes of stresses but also their 'signs' mustbe taken into account. In consequence, for the presentcylinder under steady-state and quasi-stationary temperature distributions, we must bear in mind that thecircumferential stress changes its distribution as well asthe magnitude with variations of Et/E c.This paper clarified by way of numerical illustrationsthat the difference of the elastic moduli in tension andcompression affects markedly the thermal stress stateof a cylinder.
391N. Kamiya / Thermal stress in a thick cylinderTable 1Key of figs. 5 ( a ) - 5(d)KeyEt Ec to/ rEC/E c0 rt cEo/EoEt/E cz- rEC/E cz- rt entNomenclatureaaq(i, j 1, 2, 3)b i n n e r radius o f c y l i n d e r elastic c o m p l i a n c e o u t e r radius o f c y l i n d e r 1, c2d arbitrary constants radius w h e r e o 0 changes sign radius w h e r e o z changes signE, E r, E o, E z qrRsT, T auz( )t( )c( )1, ( ) n .e , act, s 0 , c ze r, e o , e z0X I, X2APij(i, j r, 0, z)o r, o o, o z Young's modutii n t e n s i t y o f h e a t sourceradial c o o r d i n a t en o n d i m e n s i o n a l radial c o o r d i n a t e ,eq. ( 1 2 )n o n d i m e n s i o n a l variable, eq. ( 1 2 )temperaturesradial d i s p l a c e m e n tlongitudinal coordinaterefer to t e n s i o nrefer to c o m p r e s s i o nrefer to d o m a i n s I, II . coefficients of thermal expansion strain c o m p o n e n t s circumferental coordinate see eq. ( 1 5 )thermal conductivityPoisson's r a t i ostress c o m p o n e n t s .References[ 1 ] S.A. Ambartsumyan, Izv. Acad. Nauk Arm-SSR, Mekh.19 (2) (1966) 3 - 1 9 (in Russian).[2] S.A. Ambartsumyan, Izv. Acad. Nauk SSSR, Mekh. (4)(1965) 77 85 (in Russian).[3] S.A. Ambartsumyan and A.A. Khachatryan, Inzh. Zhur.MTT, (2) (1966) 4 4 - 5 3 (in Russian).[4] S.A. Ambartsumyan, Inzh. Zhur. MTT, (3) (1969)5 1 - 6 1 (in Russian).[5] S.A. Ambartsumyan, in: Theory of Thin Shells, F.I.Niordson, ed. (Springer, 1969) pp. 316-327.[6] F. Tabaddor, AIAA J. 10 (1972) 516-518.[7] R.M. Jones, AIAA J. 9 (1971) 5 3 - 6 1 .[8] R.M. Jones, AIAA J. 9 (1971) 917-923.[9] M. Farshad, Int. J. Mech. Sci. 16 (1974) 559-564.[10] N. Kamiya, J. Struct. Mech. 3 (1974-1975) 317-329.[ 1 l ] N. Kamiya, J. Eng. Mater. Tech., Trans. ASME, Ser. H97 (1975) 5 2 - 5 6 .[12] N. Kamiya, Nucl. Eng. Des. 32 (1975) 351-357.[13] N. Kamiya, Z. ang. Math. Mech. 55 (1975) 375-380.[14] Z. Wesolowski, Arch. Mech. Stos. 21 (1969) 4 4 9 - 4 6 8 .[15] Z. Wesolowski, Arch. Mech. Stos. 22 (1970) 253-265.[16] Z. Wesolowski, in: Trends in Elasticity and Thermoelasticity, M. Sokolowski et al., ed. (Wolters-Noordhoff Pub.,Groningen, The Netherlands, 1971) pp. 281-292.[17] H.S. Carslaw and J.C. Jaeger, Condition of Heat inSolids, 2nd edn. (Oxford Univ. Press, London, 1959).[18] B.A. Boley and J.H. Weiner, Theory of Thermal Stresses(John Wiley and Sons Inc., New York-London Sydney,1960).[19] W. Nowacki, Thermoelasticity (Addison-Wesley Pub.,Reading, Mass., 1962).[20] H. Parkus, Thermoelasticity (Blaisdell Pub., Waltham,1968).[21 ] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity,3rd edn. (McGrow-Hill Book Co., New York, 1970).
Department of mechanical Engineering, Mie University, Kamihamacho. Tsu 514, Japan Received 15 April 1976 Certain materials are known to behave differently under tension
