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a linear theory of coupled thermoelastic behaviourand heat conduction in thin shells

ODEN, J. T.I

SUMMARY:

A linear theory of quasi-static coupled thermoelastic behaviour and heat conduc-tion of thin shells is developed by expanding the free energy and entropy densityabout the middle surface.

RESU~:

Vne theorie lincaire de thermoelasticite quasi-statiquc ct de conductivitc thermiquedes voiles minces est developpcc bascc sur I'expansion de la densitc d'energie Iibrect d'entropie par rapport a la surface mediane.

RESUMEN:

Una teoria lineal del comportamiento acoplado termoelastico y de la transmisioncalorifica casi-estatica de cascaras es desarrollada por medio de una expansion,alrededor de la super/icie media, de las densidades de energia Iibre y entropia.

1 Professor of Engineering Mechanics. Research Institute. University of Alabama in Hunts·ville, Huntsville, Alabama, U.S.A.

3~

A LINEAR THEOIY OF COUPLED THERMOELASTIC BEHAVIOUR AND HEAT CONDUCTION IN IHI>I SHELLS

1. INTRODUCTION

This paper is concerned with the development of a linear theoryof coupled thermoelastic behaviour and an approximate theory of heat con-duction in thin shells. The theory generalizes several previously availableand includes as special cases a first-approximation in the generaluncoupled theory of thin shells, the general theory of plane coupled thermo-elasticity, and a theory of heat conduction in solids. The theory of heatconduction is an approximate one involving a membrane theory, in whichtemperature variations through the thickness are omitted, ·together with aninextensional theory which assumes a linear variation in temperaturethrough the thickness. A rigorous analysis of the order-of-~agnitude ofterms in heat flux expressions is not attempted in the present work.

A LINEAl THEOIY OF COUPLED THERMOELASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN SHELLS

The theory is based on the usual Love-Kirchhoff hypotheses of thinshells: (a) points on a normal in the undeformed body lie on the samenormal in the deformed body; (b) the effect of the normal stress on surfacesparallel to the middle surface are negligible; (c) the normal displacementsare approximately equal for all points on the same normal. We first laydown the purely kinematic relations drawn from considerations of thedifferential geometry of the deformed and undeformed shell. These do notcontain thermal variables. In the case of thermoelastic behaviour, we nextexamine the restrictions that the Love-Kirchoff assumptions impose on theform of the free energy and the entropy rather than the strain energy,as is done in the uncoupled case. Following a scheme similar to thatKoiter rlJ used in the uncoupled theory, we expand the free energy andentropy about the middle surface and retain those terms consistent with theLove hypothesis. This leads to an expression for the free energy which isa quadratic form in the membrane strains, the changes in curvature, thetemperature at the middle surface, and the change in temperature over thethickness of the shell. By then examining energy balances, we obtainequations for the moments, membrane forces, and shears in terms of thestrains, changes in curvatures and temperatures, and equations for theproduction of entropy in the shell. The heat flux is assumed to obeyFourier's law in that it is given linearly in terms of the temperaturegradients. By introducing these results into the mechanical equations ofmotion and the law of entropy production and dropping nonlinear terms, weobtain a final system of couple differential equations governing theelastic behavior and heat conduction of thin shells

2. KINEMATICS

We shall confine our attention to linear theory. The location of apoint in the undeformed shell is given bX the intrinsic surface coordinatesxi(i = 1,2,3) or by the position vector Y

- 3r = r + x a 3 (2. I)

Here £ is the ~osition vector of a point on the undeformed middle surfaceand ~ = ~ (x ,x2) is a unit normal to the middle surface.

Likewise, in the deformed shell the same particle is located bythe position vector

(2.2)

where R is the position vector of a point on the deformed middle surfaceand ~3-is a normal to the deformed middle surface. By taking ~3 to be aunit vector we would neglect transverse extensional strains. we shall,however, aim toward the classical plane stress approximation.

The displacement of an arbitrary point in the shell is

u = (2.3)

A LINEAR THEOIY OF COUPLED THERMOELASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN SHELLS

where u = R-r is the displacement of a point on the middle surface.--The strain tensor y .. (i,j=1,2,3) is defined by

1.J- -2- = R • R . - r . r

Yij _,i -,J -,i _,j

where the comma denotes partial differentiation with respect to the

We now introduce the following notation [2]:

u = ua8 + wa3 = u A~ + wa- ......(T. - C1~ -3

a a = a . a o~q a.. aR= r a = a-a -,0. a 8 -J:'I. "'8 -

(2.4)

ix .

(2.5)b = -a . a = a • a

er.S ..:.:.0. _3 S ,::J -8,0

bQ = aQAb b~8 = a~AbR8 AA A

Here a and aa are the covariant and contravariant base vectors of the sur-face ~oordi~ates xa, ua and u are the corresponding components of displace-ment of the middle surface, wais the transverse component of displacementof the middle surface, a Rand aa8 are the covariant and contravariantc0l!lPonents of the first a'fundamental form of the surface; b ,bcr, andbas are various components of the second fundamental form ofo'~he fiiiddlesurface. In these expressions, the Greek indices range from 1 to 2:~,S'A = 1,2. Similar expressions can be derived for the deformed shell.

Introducing (2.1), (2.2), (2.3) and (2.5)that normals ~ remain straight and normal to thewe obtain, after lengthy algebra,

v = + )(3o.R V~R XaR

V = 0n3

into (2.4) and requiringdeformed middle surface,

(2.6)

V ~(R 3 . R 3 -l)33 -, ~,

where v and yare the membrane strain and change-of-curvature tensorsrespectq~ely. '§pecifically,

where

2y = u + u.(tR ('f; R R ; er

2X = ~a' A + ~q. +0.8 .,. , ,0.

- 2wb0.8

bA IV +R cr.A

(2.7)

~ =a [II + u bA;a A a (2.8)

2 (f) = u - uaA 8;a aiR

(2.10)

'" LINE"'I THEon OF COUPLED THUMOELASTIC B!HAVIOUI "'ND HEII.T CONDUCTION IN THIN SHELLS

Here the semi-colon ( )'n denotes convariant differentiation with respectto the surface coordinates:

u = u - rU un;8 0.,8 '(l.A U (2.9.)

un = uCi. + r U uU;A '8 Au

in which ~ is the Christoffel sumbol of the second kind for the middlesurface. Su The quantities~ represent a rotation of the normal at themiddle surface and W describe the rotation in the middle surface aboutthe normal. a.S

We also comment here that the change-in-temperature T(xl,x2,x3)in the shell is assumed to be of the form

T(xl,x2,x3

) = T(x1,x2) + x3e(xl,x2)

where T(xl,x2) is th~ temperature of the middle surface and x3e(xl,x2) is alinear variation of T through the shell thickness.

3 . FREE ENERGY

We assume that the shell is essentially in a state of plane stress.Then the free energy ~ per unit of undeformed volume can be taken to be ofthe form

p - c- 1. ED,8)'U. -y y + C cr, y T + _~ - '2 R. ~I..l CI.R 2Tcr..

T 2 (3.l)

(3.3)

where E"RAU and CO~ are arrays of material functions, and c is the specificheat at constant deformation.

Koiter [1] reduced the strain energy per unit volume to a manageabledensity per unit surface area by expanding the strain energy in a seriesin x3 about x3 = O. We shall now follow the same procedure using insteadthe free energy. Let I denote the free energy per unit middle surfacearea. Then, if Hand K denote the mean and Gaussian curvature of theundeformed middle ~rface and h is the shell thickness,

t =s [1-2 Hx3 + K(x3)2Jidx3 (3.2)-\h

3Expanding ~ about x = z = 0, we find

- a. - (T - a. 2 a.t (x ,z) = t (x ,0) + z~ (x ,0) 3 + \z ~(x ,0) 33+ ..., ,Observing that EcYR)µ = CcY8 3 = 0 and introducing (3.1) into (3.3), we get

;3 ;

~ (x~,z) = ~Ea8Au (y -zX + ~z2X - ... ) (v(Y.S /"t'R o.B, 3 AU

A LINEAR THEOIY OF COUPLED THERMOELASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN ~HELLS

E nRAU CnR and c evaluated, ,into (3.2) and simplifying, we

+z 'l + ... ) +:... (T2 + 2z8r + Z2A2 + ... )A 2To

where En AU, C~A: and c are the values ofat the middle surface. Introducing (3.4)obtain

(3.4)

~ = ~ (EnRAUy V + 2eaS y T + ~ T2)2 a.p AU nA T

h3 0

+ - (E0BAU X X -2 ccYBy e + -=-- ~2) (3.5)24 nR AU . '0./3 To

In arriving at (3.5), we have followed a plan similar to that used byKeiter [I] in his development of a first-approximation theory of thinshells. We have effectively assumed a state of plane stress wherein termsof the following type

Kh3 cetS T- v A12 a.

Kh5 cnR e- X 890 ct.

3cHh TA12To

Hh3 cCf.'\ A + T)~ VnS Xn8

Kh3

~ T2

12 To

Kh5 2--.£ 890To

(3.6)

etc., have bSen neglected in comparison with terms in (3.5). The termsinvolving Kh are clearly negligible for thin shells. Moreover, in afirst-approximation theory, we do not wish to retain terms involving coup-lings of membrane effects (y . and T) with curvat~re effects 1Xa.R ande/h). The remaining terms iRP(3:6) involve Kh3Cn~v T and Kh c T2/To,whiZh are easily shown to be negligible in comparisBH with hCCY8y . T andhcT ITo for thin shells. as

4. ENERGY CONSIDERATIONS

For simplicity, we confine our attention to quasi-static deforma-tions of thin shells; that is, we neglect inertia effects in the equationsof motion. This is not a serious omission; the heat conduction equationsare not affected by it and the equations of motion must be simply amendedby adding the well-known acceleration terms from shell dynamics.

The law of conservation of energy for quasi-static thermoelasticbehaviour of thin shells is of the form

U =(')+Q (4.11)

where U is the internal energy, n is the mechanical power of the externalforces, and Q is the heat energy. The superimposed dot (.) indicates atime rate. These energy terms may be written in the forms

U = So€' dvv

A LINEAR THEORY OF COUPLED THERMOELASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN SHELLS

(4.2)u dv + r s 'u dA, -Ai

Q = rq I,dvv· 1.

o = r F

wherein € is the internal energy density per unit undeformed volume v, Fis the body force per unit volume, S the surface tractions, ql the heat~flux vector, and the vertical stroke' denotes covariant differentialwith respect to the coordinates xl. I The possibility of internal heatsources can be accounted for by simply adding an integral! h dv, h beingthe heat supply per unit volume, to Q; but we omit this here for convenience.

The free energy is given, by definition, in terms of € by therelation

(4.3)~ = e: - T'fT'

where ~ is the entropy per unit undeformed volume and T = To + T is theabsolute temperature. Thus

€ = i + ~ ~ + T~ (4.4)

For reversible thermodynamic processes, we also have

rr ii dv = ,!cit, dv = Q (4.5)v v' I

which states that the total dissipation is zero. Hence

u = r(t + r;~)dv + Q = ,r~dv + ,Ii=i ; dv + Qv S v

where S is the middle surface of the shell and ~ is given by (3.5).

(4.6)

We now examine an expansion of the entropy-temperature term~T about the middle surfa~e to reduce the second integral in (4.6) to anintegral over S:

~hJ ~ ; dv = J J 2 2

(I-2Hz + Kz ) (n + z~ + ~z S'3+ ...)v S -~h

X (T + ZA ) dzdS

where ~ = ~(xn,u) and S = ~ (x2,O)'3' Upon simplifying and invokingarguments concerning magnitudes of various terms, we find, approximately,

.r Ti ; dv = J (hnT + h3 SA) dS

v S 12(4.7)

Here we have assumed that C'3~O, and we have neglected, in comparisonwith hnT and h3SA, terms of the type

Hh3 s?;'6

Kh6 1;890

A LINEAR THEORY OF COUPLED THERMOELASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN SHELLS

Hh3 •61]8

primarily to keep membrane terms connected with terms of the order of thethickness h and terms representing variations throughout the thickness,aswell as curvatures, of the order h3•

Introducing (4.6) and (4.7) in (4.1), we find. h3

•r (f + h~T + - ~e ) dS = ng 12'

The mechanical power can be reduced as follows:

(4.8)

..-v(4.9)

where~h

p = r F (I-2Hz + Kz2)dz

-~h(4. lOa)

z F (I-2Hz + Kz2)dz (4. lOb)

Likewise, using the Green-Gauss theorem,

(4. 11)

J-\h

where

0.13n

~o0.8(1-2Hz + Kz2)dz (4.l2a)

a =.....3

~hm~~ = I zoo.13(1-2Hz + Kz2)dz

-~h

Noting that

~ = uo.~?+ w~ + z (~ - ~)

0. 0. B 0.(u. - b w) ~ - (w + b u) a,0. 0. ,a as

(4.12b)

(4.13)

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A LINEAR THEORY OF COUPLED THERMOELASTIC BEHAVIOUR "NO HEAT CONDUCTION IN THIN SHelLS

and using the notation

n*a$ = nUS + ~bS mnAA

we introduce (4.9) - (4.12) intocalculations, that

_ \b nmSAA

(4.8) and discover, after tedious

(4.l4)

h3

J [~-nnSy - mnSy_~ + h~T + 12 saJ dS=OS US "o.p

only if the following conditions hold:

n nS -vnbS + pS = 0*;0. nnS n

n* b + v. + p3 = 0nS 'n

where

(4.15)

(4.16)

(4.17)

8

Equations (4.16) are recognized as the equations of motion (accelerationterms neglected) of the shell and vS are the contravariant surface compo-nents of the transverse shear (cf [lJ or [2J).

Since (4.15) must hold for arbitrary S, and since the integrandis assumed to be continuous, we have

i = nnSyns + mnBXnS-h~T - ~~ S8 (4.18)

Substituting (3.5) into (4.18) and equating like coefficients of yX T, and e, we get ns,aS,

nne = h(EaSAUy + CnST)3 AU

nS = ~ (Ene~µ~ + co.Be)m 12 U

(4.l9)~ = -(Cnsy +£ T )

ne To

c~ = _(Cas~ + - e), 'ne ~

Therefore, introducing (2.7) into (4.19) and incorporating the result into(4.16) and (4.17), we arrive at the coupled equations

hEaS) U(u - (b w) ) + hCaBT,)..;w Aµ 'a. 0.

+1 h3[bSEnA~U(w + ~b6w24 AP;µ P UQ

A LINEAR THEORY OF COUPLED THERMOELASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN SHELLS

_baESAOU(~ + ~ bOw ~ be w )]) 0; U p uo U DO ;0:

+ h3

(bS caAa _ abacSA) _ v~S + pS = 024 A A;a 0.

hb EaSAµ(u -b w) + h b caSTas A;µ AU a as

A 0 0+! h3 CbS b Ea aU (co . + ~b w + ~b Wi:)24 A as ' 0 , U P uo µ 0u

-b~ ESAPU (~ . + %bOw~ + ~b~ WpO)] + h3 eb

(bS caAA as 0, µ p 24 as A

(4.20a)

o (4.20b)

~olhere

(4.21a)

and 2UJaA

(4.21b,c)

5. HEAT CONDUCTION

The temperatures in (4.20) are not assigned functions as is usuallyassumed. Rather, they must be determined as solutions of the associatedheat conduction problem. We shall construct here an approximate theory ofheat conduction which is valid for a wide class of practical isotropic shells.

We shall assume that the heat flux is given by the familiar Fourierlaw

-1 = K1J Tq ,j

where KIJ is the thermal conductivity tensor.

The following identities are useful:

g = U al3 = i a g3 = ;:3.....rr. as .......a .......S

ga = Aa as = Aas a g3 = a313- ---a

(5.1)

(5.2)

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A LINEAR THEORY OF COUPLED THERMOelASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN SHEllS

uo.= 00. - zbo. - 0 - zbaSS Uo.~- 0.13 0.13

1 A ).0.3 = aSA).o.Ao.= - eo.Pe13 U S).Up ).

u = I uo. I = I-2Hz + Kz2 (5.2)S

0.8= eo.S e =e =0 e =-e =l)wherein e is the permutation symbol (e

It is noto.S difficult to show that' 11 22 ' l2 21 •

wherein

pc-a.q = AI-' q

0.

(5.3)

(5.4)

In view of (s.l), (5.3) can be rewritten as

q1 I = Ao. [(KSAT, + zKSAe, + KS3e)1 13 A A ;0.

-b: (i(3µT 'u + ZK3 UA 'µ + ~ 38 I ] + K3 Us ,µ

where

(5.5)

(5.6)

For isotropic materials, KIJ = kglJ, where k is_toe coefficient ofthermal conductivity and g1J = gl . gJ. Then K33 = I, K3 = KUG = O. and(5.6) reduces to

(5.7)

In the following, we shall confine our attention to isotropic shells.

For simplicity in notation, we introduce the following surfacetensors:

hO$

10

A \he = J UZAo.K13A dz

-~h 13;0.

~h= r 'Z~JAo.KASdz-~h A

(5 .8a, b)

(5.8c,d)

" LINEAR THEORY OF COUPLED THERMOELASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN SHElLS

and the modified thermal conductivity" ~h~ = I 2k (H-zK)dz

-!;h(5.9)

then

(5.l0)

Following essentially the same procedure described earlier, wearrive at the approximation

. h3 •J Ar;dv~ J(hToT1 - "6 Hro~) dS (5.11)v S

were, according to (4.19),

(5.12); = -(CasXae+ ce/To)T1= _(eaSy + cT/To)as

Substitution of (5.9) and (5.10) i.nto(4.5) yields the heat conductionrelation for the shell. We consider two cases: (I) the membrane theory,in which only T is considered and A = 0 and (2) the inextensional, thermal-curvature theory, in which the middle surface is insulated and kept at auniform temperature To; T, T.~. and T'A are then zero. The governingheat condution equations for ~hese case~ are

(s.13)

Finally, introducing (s.ll) and (2.7) into (5.12). we arrive at thecoupled equations

as·-hTo (C ua;S (5.l4a)

3 as.!J--HTo (c CO. Q- 6 a, fJ

h3 Hce6

+ ~ caSbAlli + ~aSbA~ )p oA a SA

eAR + hale - ~ A,A ,a>.. (5.14b)

6. REMARKS

The relative importance of the coupling terms for various materialsis of interest. Consider in (4.20) and (5.l4) the terms representing acoupling between mechanical and thermal behavior. In (4.20a), for example),thermal coupling is represented by the term h CaST, while in (s.14a),mechanical coupling is represented by -hToCas(u a - b w). For an

n;p as

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A LINEAR THEORY OF COUPLED THERMOHASTIC BEHAVIOUR AND HEAT CONDUCTION IN THIN SHEllS

isotropic shell

(6.1)

12

Awhere a is the linear coefficient of thermal expansion, and A and µ arethe Lame constants. The influence of the term involvinu h deoends on thecurvature of the shell and is negligible on zero for shallow shells or plat~sSince, in the uncoupled case, the first term in (5.14a) does not appear, itsinfluence can be determined by comparing it with the term c h T. Followingthe same plan as Boley a~d Weiner [4J, we find that the coupling term isnegligible if ga~u . /3aT« (A + 2µ!3)/5(A + 2µ) where 6 = (3A + 2µ)ca2

To / c (A + 2µ). a,s Assuming that the strain and temperature rates are ofcomparable magnitude (a reasonable assumption for quasi-static behavior),then coupling is small if 6«1. For steel and aluminum with To = 200°F,5 = .014 and .029 respectively (Cf [4J) while for concrete 5 = .002. Thus,for most common materials coupling is small and the equations of motion andheat conduction can be treated independently. However, for certain polymersand plastics, values of 5 of around .3 may be found, in which case it may notbe safe to ignore thermoelastic coupling.

It should also be mentioned that the order-of magnitude analysisused to derive (5.11) was not used in obtaining (5.7) which, in its statedform, is exact in the sense of the basic hypothesis of linearity and validityof Fourier's law. Thus (5.13) is inconsistent in the sense that the specificforms of the tensors in (5.8) can likely be greatly simplified. However,a detailed analysis of such aspects of the theory is outside the scope ofthis expository work.

7. REFERENCES

1. KotTER, W. T. "A Consistent First Approximation in the GeneralTheory of Thin Elastic Shells," Proceedings of the Symposium on the Theoryof Thin Elastic Shells. Edited by W. T. Koiter, North-Holland PublishingCo., Amsterdam, 1960, pp. 12-33.

2. GREEN, A. E. and ZERNA, W., Theoretical Elasticity. Second Edition,Oxford at the Clarendon Press, Oxford, 1968

3. ERINGEN, A.C., Mechanics of Continua, John Wiley and Sons,New York, 1967, pp. 114-131.

4. BOLEY, B,A. and WEINER, J,H., "Theory of Thermal Stresses".John Wiley and Sons, pp. 42-44.