NUCLEAR ENGINEERING AND DESIGN 10 (1969) 465-475. NORTH-HOLLAND PUBLISHING COMPANY, AMSTERDAM

FINITE ELEMENT ANALYSIS OF NONLINEAR PROBLEMSIN THE DYNAMICAL THEORY OF COUPLED THERMOELASTICITY

J.T.ODENResearch Institute. University of A/abama.

HuntfIJil/e, A/abama. USA

Received 9 May 1969

This paper is concerned with the fonnulation of imite clement models for the analysis of nonlinear problems inthe dynamical theory of coupled thennoelasticity. Displacement and temperature fields are approximated over aimite element, and energy balances are fonned which lead to general equations of heat conduction and motion ofthermoelastic fmite elements. A number of special cases are considered. Numerical results obtained from simplerepresentative problems are also presented.

I. Introduction

When the external forces acting on an elastic bodyperform work, common experience tells us that thebody becomes hotter. Likewise, when heat is suppliedto a body, mechanical work is performed. The de-scription and prediction of such phenomena is theprovence of the theory of thermoelasticity, a fieldwhich dates back over a century to the pioneeringwork of Duhamel [I] .

Typically, the thermoelastic problem is consideredto be two disjoint problems: (1) the problem ofsteady-state heat conduction in a perfectly rigid body;and (2) the problem of static deformation of an elas-tic solid subjected to steady, prescribed temperaturedistributions. Solutions for the temperature fieldobtained from the fust problem are used as input forthe second problem. We refer to such problems asstatic, uncoupled thermoelastic problems. Alternative-ly, a more general theory of quasi-static, uncoupledthermoelasticity may be used, wherein results of thetransient heat conduction problem are used as inputto a series of problems of static behavior of elasticsolids in which the time t plays the role of a param-eter. If inertia terms are retained in the equations ofmotion, the theory becomes a dynamical theory ofuncoupled thermoelasticity. Finally, if the deforma-

tion of the body is accounted for in the heat conduc-tion equation, and the influence of changes in tem-perature appear in the equations of motion, theproblem becomes one of coupled thermoelasticity.According to Parkus [2] , approximately 850 papersand books on thermoelasticity have been published inthe last two decades. The vast majority of these havedealt with the uncoupled theory; with few exceptions,solutions to linear problems in transient coupledthermoelasticity have been limited to one-dimensionalcases. Extensive references to previous work on thelinear theory can be found in books and survey articleson the subject [2- 7] .

A nonlinear theory of thermoelasticity has beenpresented by Dillon [8] , who attempted to consider"mild nonlinearities" by accounting for couplingbetween tlle thermal field and the deviatoric compo-nents of strain. This is accomplished by retainingterms of higher-order than second in expansions ofthe free energy. A similar procedure is adopted in thispaper. Reiner [9] has also suggested a nonlinearstress-strain law for thermoelastic solids, and Jindra[10] , using a perturbation technique, solved a non-linear problem in static, uncoupled thermoelasticity.

Many practical problems of current interest in thearea of thermoelasticity present considerable mathe-matical difficulties when attacked by classical methods

(6)

(5)

(4)

(2)

(3)

(I)

2"1" = u .. + u .. + U . U '.II 1,1 1,1 m.1 m,}

F ..=z .. =li .. +u ...I} I,} I} 1,1

"1ij= t(Gij-liij),

which can be written in terms of the displacementswith the aid of eqs. (I) and (2):

or the Green-Saint Vcnant strain tensor

and the deformation gradient Fjj is given by

where the comma denotes partial differentiation withrespect to Xj : Ui,j = auj/axj' A measure of the defor-mation is provided by the Green deformation tensor

Since "1ij is a symmetric second-order tensor, wemay evaluate the following three principal invariants:

2.1. Kinematic relationsThe cartesian components of displacement relative

to the reference configuration are defIned by

Alternatively. we can compute the principal invari-ants ofGi{

coordinates xi which are assumed to be initially rec-tangular cartesian. At some later time T = t, the bodyoccupies a new configuration C. The rectangular car-tesian coordinates of a particle in C which was initiallyat xi in Co' rela tive to a fIxed reference frame in Co'are denoted zi' The functions zi(xI, x2, x3, T) =zi(x, i) define the motion of tile body relative to thereference configuration. Moreover, the temperature ata particle in configuration C is To + T(x, t), whereT(x, t) is the change in temperature from Co to C.

J.T.ODEN

of analysis. This has led some investigators to con-sider numerical proccdures, and a number ofattempts at finite elcment solutions of variousproblems of thermal deformation have been pre-sented. Finite element formulations of the classi-cal heat conduction without mechanical couplinghave been presented by Wilson and Nickell [II],and Becker and Parr [12] , and solutions to staticuncoupled thermoelasticity were presented byVisser [13] and Fujino and Ohsaka [14]. Finiteelement solutions of linear dynamic problems incoupled thermoelasticity were given by Nickelland Sackman [15] and aden and Kross [16] . Nosolutions to coupled nonlinear problems, static ordynamic, appear to bc available.

This paper is concerned with the formulationof finite element models for the analysis of a classof nonlinear problems in dynamic, coupled ther-moelasticity. Following this introduction, the fun-damental equations of thermoelasticity are rc-viewed, and various forms of the free energy func-tion for isotropic materials are examined. Attentionis then confined to a typical finite element of athermoelastic media, throughout which the dis-placement and temperature fields are approxi-matcd. By considering the conscrvation of energyof an element, general equations of motion andheat conduction are obtained. These equationshold for finite deformations of an element, andinclude a number of important problems as specialcases. Consideration is then givcn to specific formsof the free energy for small deformations of mate-rially nonlinear thermoelastic solids. The generalequations are reduced so as to apply to some sim-ple cases wherein tile influencc of certain nonlinearterms can be evaluated and the results comparedwith the linear theory.

We shall measure the deformation of a ther-moelastic solid by rcferring its motion to somenatural, unstrained reference configuration Cowherein the body is under a uniform temperatureTo. We introduce a time parameter T which is as-signed a value of zero when the body is in Co. AtT = 0, a particle x is located by a set of intrinsic

2. Equations of finite thermoelasticity

466

FINITE ELEMENT ANALYSIS OF NONLINEAR PROBLEMS 467

II =GU=3+2JI

12 = 2!(G..G .. - G ..G ..) = 3 + 4J1 +4J2II II II 'I (7)

where oij is the stress per unit undeformed areareferred to the convected coordinate lines xi in thedeformed body.

The specific free energy <p and internal dissipationa are defined by

2.2. Thennomechanics of continuaEqs. (1)-(8) describe purely kinematical relations

for the body. To describe thermal and mechanicalphenomena, we postulate that linear and angularmomentum and mass are conserved during the mo-tion. In addition, we postulate a global law of con-servation of energy of the fonn

Also, introducing (12) into (9), we obtain an alternateglobal form

(13)

(14)

(16)

(15)

. iI' . (J'<p=o r..-1/ -0,'I

(Jil = V . q + h + a .

where 1/ is the entropy per unit undeformed volumeand (J = TO + T(x, T) is the absolute temperature.Introducing (13) and (14) into (12) and rearrangingterms, we arrive at the pair of relations

f Po" 'vdv + J oij'Y;jdv = n .vo vo

(8)

aJI-=liarij ij

We also note that

where Po is the mass dcnsity in Co' v is the velocityfield, Uo the volume of the body while in Co' € is theinternal energy per unit undeformed volume, n is themechanical power, and Q is the heat supply:

J pov.vdv+ f edv=n+Q,VO VO

n= fF'Vdv+ f S'vdA,vo AO

(9)

(10)

2.3. Thennoe/asticityThe theory of thermoelasticity is based on the

assumption that 0 is zero and <p is a differentiablefunction of the instantancous (current) strain andtemperature; <p = ,p(rij, (J). Then

and (15a) yields

(17)

Q = f h duO + f q' I1dA .VQ AO

(11) aij = a,par .. 'II(18a, b)

Here F is the body force and h the heat supply perunit undeformed volume, S the surface traction andq the heat flux per unit undeformed area referred tothe convected coordinate lines xi.

Under suitable smoothness assumptions, the localform of (9) can be obtained:

Noting that

ij' = ij(" ) .a rij a Vim + um.; Um,j'

and

(19)

e = oijr· ..+ V 'q + hlJ '

(12) V . Tq = TV . q + q .V T , (20)

(24)

468

we fmally introduce (l8a) into (16) and (20) into(15b) after multiplying through by the change intemperature T. By then using the Green-Gauss ilieo-rem to convert a volume intcgral into a surfaceintegral, we obtain the following forms of the prin-ciple of conservation of energy for a thermoelasticcontinuum:

J.T.ODEN

u~~(r) = u~e)(x~, r) ,

t~)(r) = t(e)(x~, r) .

We now approxinlate the local fields u(e) and t byfunctions designed so as to coincide with the nodalvalues 0[(24):

J Po'" vdv + J aa:.. (o;m + um.i)um,j dv = n ,uo Uo 'I (21)

- J T(To + T) :t (;~) dvuo

-J(q'VT+h)dv+Q*, (22)Uo

u~e) = \fiN (x) u~~ '

(25)

where the repeated nodal index N is to be summedfrom 1 to Ne and the dependence of the nodal valuesu~~ and t~) on time t is understood.

The interpolation functions \fIN(x) have the prop-erties

3. Finite elements of a thermoelastic media

We emphasize that no restrictions have as yet beenplaced on the magnitudes of the defornlation, on theform of,;p. or the constitutive equation for the heatflux q.

where oZ is the Kronecker delta. The simplest con-formable set of functions \fiN (X), for example, corre-spond to simplex finite clements. These are tetra-hedral elemcnts with four nodes in three-dimensionalspace, triangular elements with three nodes in two-dimensional space, and linear elements with two nodesin one-dimensional spacc. and the functions \fiN (X)are linear in xi:

where

Q* = f q' nT(x, r) dA .Au

(23)

\fIN(XAf) = oZ, \fIN(x) ~ 0,

k \fIN(x) = I ,(26)

We now considcr the motion and hcat conductionof a finite element of a thermoelastic solid. Followingthe usual procedure the continuum is approxinlatedby a collection of a finite number of discrete elementsconnected togcther at prescribed nodal points ontheir boundaries. We isolate a typical finite element,and temporarily examine its behavior independent ofthe rest.

3.1. GeneralLet u~e)(x, r) and e(e)(x, r) denote the local dis-

placement and temperature fields over a typical finiteelement e which hasNe nodal poiryts. The local coor-dinates of a node N are denoted xhr = xN. Let u~~and t'f) denote the values of the displacement com-ponents and the change in temperature at node N ofclement e:

(27)

i= 1,2, ... , k; N= 1,2, ... , k + 1. For example, in thetwo-dimensional case

(28)x2 _ x2 xl _xl

2 3 3 2

N _ I x2 _ x2 xi -x~[11.;] -2Ao 3 I

x2 _ x2 I II 2 x2-xI

FINITE ELEMENT ANALYSIS OF NONLINEAR PROBLEMS 469

where Ao is the area of the triangle; and for one-dimensional elements

qN(r) = J h(x)'l1N(x)dvuO(e)

etc.

F~~) = 0 .. + 'l1N.(x) u'f). ,II 'I ,I I

(30)G~~) =F(e).F(e~

II ml ml'

"t~~) = ~ (G~~) - 0 ..) ,II II II

where L = xi -x~.With the temperature increment t(x, t) and the

displacement field ui(x, t) defined uniquely in termsof their nodal values by (25), it is now a simplc mat-ter to compute the absolute temperature and allkinematical variables for the element. For example,

(38)PN - I peL qN - I *La - 2' l3a 13' - 2'Q ,

0'NM .. + f ~(omim uMi o"tmi

UO(e)

+ 'l1Mn(uM.)'l1Nidv = p!'(T) , (36),lit ,

- J['l1N(x)(to + 'l1M(x)tM) ;t (~~)UOe

-qi'l1N.(X)]dv=qN(T). (37),I

Eqs. (36) and (37) apply to any typc of thermoelasticmaterial for which a free energy function "'("(ii' 8)exists. No restrictions on the magnitudes of the defor-mations or temperature changcs have been imposed.

We remark that in computing the generalizedforces p~ and heat fluxes qN of (34) and (35), thesurface tractions Sex) and the applied heat flux q(x),though measured per unit undeformed area, are avail-able to us only in the deformed body. Thus, thcsequantities are generally functions of the deformation.For example, consider the case of plane strain of atriangular boundary element subjected to a uniformlyapplied load of intensity p and a uniform applied heatflux of intensity q*. Then the generalized nodalforces and heat fluxes are approximately (fig. I)

where epa is the two-dimensional permutation sym-bol, a, 13 = 1, 2 and

+ f q' n'l1N (x) dA . (35)AO(e)

Here mNM is the consistent mass matrix for the ele-ment, p~ are the components of generalized force atnode N, and qN is the generalized heat flux at node Nof the element. For simplicity, we have omitted theelement identification label (e) in (31) and (32).

Since eqs. (31) and (32) must hold for arbitrarynodal velocities U Ni and nodal temperature changestN' the quantities inside brackets must vanish. Thus,we obtain the general equations of motion and heatconduction for a finite element of a thermoelasticcon tinuum:

(29)

(34)

(33)mNM = f 'l1(x)'l1(x)Po(x) dv,uO(e)

p~(r) = J F;(x) 'l1N(x)dv

UO{e) + J Si(x)'l1N(x)dA,

A o(e)

a1=xiIL, a2=-xtJL,

13~= -l3i = - IlL,

where uO(e) is the initial volume of the element,

3.2. Thermomechanics of a fmite elementWe now obtain two forms of the conservation of

energy for a finite element by introducing (25) into(21) and (22):

. [NM" f 0'" ("UN; m uMi + O"tk' ukiUO(e) I

+ 'l1,f (x)uMi) 'l1N(x)) dV-P~(T)J = 0, (31)

tN [- f {'l1N

(x)(to + 'l1M

(x)tM) d~ (~~)UO(e)

-qi'l1N.(x)} dv-qN(T)] = 0, (32),I

470 J.T.ODEN

Little else can be said about the form of (36) or(37) until the constitutive equation for heat flux andthe specific form of.p( ) for the material is identified.The process of connecting elements together to formthe entire connected model of the body is the same asin the linear case, and since it is well-documented. itis not dicussed here.

(41)

(40)

.p( )=ao +a1JI +a2J2 +a3J3 +a4T

2 3 2+a5J1 +a6J1 +a7T +a8J1T

+ a9J1J2 + alOJl T2 + allJ2T

32.4+a12T +a13JIT+a14Ji

+aI5J~ +aI6J/3 + ...

where aO' a1 ... are material constants. We shall con-ftne our attention to materials which are stress free inthe undeformed state, and for which the free energyis a function of no terms of order higher than third inthe strain invariants and temperature increments.Then ao and a17' a18, al9 ... are zero. We note thatno assumptions as to magnitudes of strains or tem-perature are required in specifying such materials;rather, we merely identify a class of thermoelasticmaterials which are characterized by free energyfunctions of this form.

Special forms of (40) can be obtained by expanding.p( ) in a power series in the strain invariants andtemperature increments [8] :

(39)

,,-r--.... 2I ....

/ I~(J/1\~9\'I - \~ / \

3 I \Co I' / \, /--¥----1

4.1. Finite thermoelasticityIn the case of finite thermoelasticity, manageable

theories can be developed for only relatively simpleforms of the frcc energy. For example, a certain in-compressible thermoelastic material of the Mooney-type is described by a free energy function of theform

Fig. 1. Plane deformation of a boundary element.(42)

where

4. The free energy function (43)

We now examine forms of the free energy functionwhich, when introduced into (36) and (37) lead tononlinear equations for a finite element. For isotropicthermoelastic materials, <p may be expressed as a func-tion of the strain invariants J1,J 2,J 3 of (6) and thetemperature change T:

and C1 and C2 are the usual Mooney constants forthe isothermal case. Also, the following incompressi-bility condition must be satisfied:

(44)

FINITE ELEMENT ANALYSIS OF NONLINEAR PROBLEMS 471

According to (8) and (I8), the stress and energy den-sity are given by

-T'/=2a7T+agJ1 +2alOJ1T+allJ2

322+ al2T +a13J1· (50)

where h is an arbitrary hydrostatic pressure.

4.2. Material nonlinearity, infinitesimal strainsIf it is assumed that the displacement gradients

IUiJl « I, the classical strain-displacement relations,

+ h {2cI)i/(1+ 4J2) + 4JI(&ii

_2cI)im[/n"( )+4&im[/n"(. (&.mn Ik Ik

4.3. Simplified nonlinear theoriesThe constitutive eqs. (49) and (50) are too compli-

cated to be manageable in the simplest stress analysisproblems. Moreover, experimental data on the relativemagnitudes of the thirteen material constants are notavailable. Therefore, it is natural to seek rational sim-plifications which will lead to models from whichquantitative information can be obtained. Towardthis end, we make the follOwing assumptions and ob-servations:

I) We assume that the lin<:arized versions of (49)and (50) coincide with the classical equations describ-ing a linear isotropic thermoelastic solid. Then

(45)

(46)

as a characterization of a materially nonlinear thermo-elastic solid. Here it is assumed that terms of orderhigher tllan fourth do not contribute substantially tothe free energy. Then

hold and we may use [8]

.p( ) = al2 + a3J3 + asJf + a6J~ + a7T2

J 2 2+a8 IT+a9JIJ2 +aIOJ1T +allJ2T

2 2 4+ al2Jl T+ a13J1 T + al4J I

(52)

(51)a2 = -2µ, as = HA + 2µ),

ag = - a(3A + 2µ) , - 2Ta7 = C ,

where A and µ are the Lame constants, a is the linearcoefficient of thermal expansion, and c is the specificheat at constant deformation. We observe that a7 =- c/2T is inconsistent with (50) which is based on theassumption that a7 is independent of temperature.However, this discrepancy is also found in the lineartheory, wherein c is introduced as a constant forwhich qu = cT for constant deformation.

2) In general, the specific heat at constant defor-mation is

(47)

(48)

2"( .. = U '1 + ul' i'I I, '

ail = &ii[a2JI + a3J2 + 2aSJ1 + 3a6Ji + agT

2 2+acl2 +acl2 +alOT + all TJI3+ 2a13J1 T + 4a14J 1 + 2a15J IJ 2

+aI6J3(1 +J1)] -"(i/a2 -a3Jl +acll

+allT+2a15J2 +aI6Ji)(49)

We shall assume that Cd is independent of tempera-ture changes and the deformation. Then a12 = a10 = O.

3) Dillon [8] has noted that for specimens sub-jected to oscillating dilational strains, if the heating ofthe material in compression equals the cooling whilein tension, it is required that an = O.

4) It is widely recognized that for small strains ofmost metallic-type materials the relation between thedilational components of stress and strain is linearto a larger extent than the deviatoric components(e.g. [8]). We can furtl1er reduce.p by dropping termsof fourth and higher order in the strains, but retaining

472 J.T.ODEN

(53)

1__0 -I

(56)

time

t.T

t

qi = kT .,,I

-G • • • -3 -2 -I 0' I 2 3 •• • GFr""f~"r~T'T'T' .(-t±··.·~T,'J···T.?··n~~ ..'~,.~ ....,.-.'7'h- . . , .,'. , ," ...+---Ch\\, ,.\"..\: :.,\",'{',;""\',.J."",'., "\"."\".""\<:"" '\.::-,~ "'\\\." .>' <.,' ,,\. >," • ,,' ..

W"'~",\1\.. :'·i -:t~~'+:,l, '''', ,\~ .J:t-.d....J.:~ .....~\,i....~ .,-ill8.,-~-...... .....~ ..\...-..::~ '.""_.-~-_. __ .~---,.~.,

I.. L -I L -I

wherein k is the coefficient of thermal conductivity.As a specific example, consider the problem of

transient, coupled thermoelastic behavior of the thickslab indicated in fig. 2. The slab is subjected to anapplied ramp temperature rise on each of its boundingsurfaces, x = ± L, so that a thermoelastic wave propo-gates symmetrically inward. Material particles near thefree ends of the slab begin to displace shortly after thetemperature impulse due to thermomechanical coup-ling. The quasi-static, linear, uncoupled version of thisproblem has been considered by Visser [13].

aI - a16 for real materials. However, some estimate ofthe influence of nonlinear terms can be obtained byconSidering cases involving infinitesimal strains ofmaterially-nonlinear thermoelastic solids described byrelations of the form in (53)-(55). To these relationsmust be added the constitutive equation for heat fluxq, and for a widc class of materials we may use thelinear Fourier law,

Thus

aij = <5ii[AJI + 3a6JT -a(3A + 2µ)T

+ a9(J2 +Ji) + allJI T]

A number of other forms of the frce energy whichlead to nonlinear constitutive equations for stress andentropy can be obtained by deleting or adding appro-priate tcrms in the power series cxpansion in (48).Which form is appropriate for a given material can bedetermined only through careful experimentation.Eqs. (42)-(55) are cited only as examples. Eqs.(53)-(55) reduce to the classical equations of linearthermoelasticity upon deleting nonlinear terms. Bysetting a3 = a6 = a9 = 0, the material described byDillon [8] which is nonlinear in the deviatoric strainsis obtained. If eithcr of a3' a6 and a9 =1= 0, a materialwhich is also mildly nonlinear in dilation is obtained.For purely dilational strains, 'T/ is identical to that ofthe classical theory.

+'Yi;C2µ+a3J1-a9JI-auT)

+a3'Yik'Yjk' (54)

11= - 2a7T+ a(3A + 2µ)11 - 2a1112T. (55)

the term all J 2T2, since it represents a nonlinear con-tribution to the deviatoric strain components. Thenal4 = a15 = al6 = O. Further simplification can be ob-tained by retaining terms no higher than seconddegree in the strains in the constitutive equation forstress.

With these simplifications, \p reduces to

ojJ = - 2µ.J2 + a3J3 + t (A + 2µ)JT + a6Jf

+ a7T2 -a(3A + 2µ)JI T + a9J1J2

2+ all J2T

S. Sample problem

A detailed study of practical problems in nonlinearthermoelasticity must await furthcr experimcntal dataon the relative magnitudes of the material constants

Fig. 2. Nonlinearly thermoelastic slab subjected to a ramptemperature rise.

FINITE ELEMENT ANALYSIS OF NONLINEAR PROBLEMS 473

+ (3a6/az)(uf - 2uluZ + u~) , (58b)l:!0.31 • .

ql =,ca(2t1 +tz)+(k/a)(t2-tl)

--} Toa(3A + 2µ)(ul -Uz), (58e)

'"1 . . :::02q2 =6'ca(tl +2t2)-(k/a)(tz-tl)

- t Toa(3A + 2µ)(ul - U2) , (S8d)

In the present case we assume that the material ischaracterized by a free energy function of the formgiven in (53). For simplicity, a linear simplex approxi-mation of the element displacement and temperaturefields are used, so that

u = 'l1N(X)UN ' t= 'l1N(X)tN ' (57)

where N = 1, 2 and 'l1N(x) is defined by (27) and (29).We assume that the temperature is applied suddenlyand uniformly over the boundary surfaces of the slab,and that the thickness is such that deformations areessentially uniaxial. Then the second strain invariantvanishes and only one material constant correspond-ing to a nonlinear term in ~( ) appears in the finiteelement equations (a6)' Willi tllese restrictions, thcnonlinear differential equations of motion and heatconduction for a typical fInite element become (see(36) and (37):

PI = ~ poa(2u1 + uz) + i(A + 2µ)(ul - uz)

+ ~a(3A + 2µ)(t I + t2)

- (3a6/a2)(u~ - 2u Iu2 + u~) , (58a)

P2 = ~ poa(ii I + 2u2) + ~A + 2µ)( - "l + "2)

- ra(3;\ + 2µ)(tl + tz)

where a is the length of the slab element. We intro-duce the customary nondimensional parameters, asfollows:

~ = SZ

-xK 'T=S "t,

O=!.. if= seA + 2µ)T'0 K(3To

K=~ s2 = (A + 2µ)(59)

pCv' p

(32 To13 = a(3;\ + 2µ), 0 = pc,iA + 2µ) .

In the above equations. t is the real time, k the ther-mal conductivity, Cu the specific heat at constantvolume, A and µ arc the isothermal Lame constants,and a is the linear coefficient of thermal expansion.The quantity 0 in (59) is a thermomechanical coup-ling parameter.

A qualitative description of the influence of mate-rial nonlinearities is indicated in figs. 3 and 4. Theseresults were obtained from a four element model forthe case in which a uniform step temperature impulse(tl = 0, TG = I) is applied. In the present case, acoupling parameter 0 = 1.00 was used, which, for

0.4

o LINEAR (A6' 0.01o NONLINEAR (A6 • 0.333)

0.1

Fig. 3. Slab temperatures in linear and nonlinear thermo-elastic solid.

474 J.T.ODEN

Acknowledgement

to begin to take place at T ~ 0.8, and for T = 1.0 tem-perature differences on the order of 100 percent wereobtained.

References

It is a pleasure to acknowledge the assistance of Dr.RJ.Polge and Mr. James W.Poe. Their assistance andencouragement during the course of this project isgreatly appreciated.

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(3) P.Chadwick, Thennoelasticity, the dynamical theory,in: Progress in Solid Mechanics, Vol. I, eds. l.N.Sneddonand R.HiIl (North-Holland, Amsterdam, 1960).

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(5) B.A.Boley and J,H.Weiner, Theory of Thermal Stresses(Wiley, New York, 1960).

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(7] H.Parkus, Thennoelasticity (Blaisdell Publishing Co.,Waltham, Mass., 1968).

(8) O.W.Dillon Jr., A nonlinear thennoelasticity theory,Journal of the Mechanics and Physics of Solids 10 (1962)123.

(9) M.Reiner, Rheology, Encyclopedia of Physics, Vol. VI(Springer-Verlag, Berlin, 1958) p. 507.

(IOJ F.Jindra, Wiirmespannungen bei einem nichtlinearenElastizitiitsgesetz,lng. Archiv 28 (1959) 109.

( II J E.L.Wilson and R.E.Nickell, Application of the finiteelement method to heat conduction analysis, Nuc1. Eng.Design 4 (1966) 276.

(12J E.B.Becker and C.H.Parr, Application of the Unite ele-ment method to heat conduction in solids, TechnicalReport S-117, Rohm and Haas Redstone Research Labo-ratories, Huntsville, 1967.

(13J W.Visser, A finit~Iement method for the detenninationof non-stationary temperature distributions and thennaldefonnations, Proceedings Conference on Matrix Methodsin Structural Mechanics, eds. J.S.Przemieniecki, R.M.Bader, W.F.Bozich, J.R.Johnson and W.J.Mykytow, AirForce Flight Dynamics Laboratory-TR-66-80, Dayton,1966, pp. 925-943.

o LINEAR THEORY (A6• 0.0)a NONLINEAR THEORY (A6' 0.3331

0.9

1.0

0.8

0.7

0.6

0.5

'"'"'"..J~ O.~

'"z'"~0

0.3

0.2

0.1

metallic materials, represents a very high degree ofmaterial thermomechanical coupling. The influenceof all material nonlinearities on the hcat conductionequation in this case must be transmitted through thiscoupling term. The relative importance of the Q6 termappearing in (58) which, when non-dimensionalized inaccordance with (59), is denoted A6' The classical,linear thermoelastic solid corresponds to A6 = 0 anda highly nonlinear thermoelastic solid would have anA6 on the order of 1.0. Since fue illustrative materialnonlinearity manifests itself in terms of second orderin the displacements, a rather large (and possibly un-realistic) value of A6 is needed to make thc influenceof these terms felt in the temperature equations. Fig.3 shows the variation of the temperature at the mid-point of the slab as a function of non-dimensionaltime T for the case in whichA6 = 0.333 andA6 =0.000. Fig. 4 illustrates the variation of the end dis-placemcnt with time for the same material. Somesignificant departures from the linear theory are seen

000.1 0.2 0.3 o.~ 0.5

OIMENSIONLESS TIME

Fig. 4. Slab displacement in linear and nonlinear thermo-elastic solid.

FINITE ELEMENT ANALYSIS OF NONLINEAR PROBLEMS 475

(14] T.Fujino and K.Ohsaka, The heat conduction and ther-mal sUess analysis by the finite element method, Pro-ceedings Second Conference on Matrix Methods inStructural Mechanics, Air Force Flight Dynamics labo-ratory, 15-17 October 1968, Wright-Patterson AFB,Ohio, in press.

[15] R.E.Nickeli and J.J.Sackman, Approximate solutions inlinear coupled thennoelasticity, J. AppL Mech. 35(1968) 255.

[16] J.T.Oden and D.A.Kross, Analysis of the generalcoupled thennoelasticity problems by the finite elementmethod, Proceedings Second Conference on MatrixMethods, Air Force Flight Dynamics Laboratory, 15-17October 1968, Wright-Patterson AFB, Ohio, in press.