.,AFFDL-TR-68-150

ANALYSIS OF GENERAL COUPLED THERMOELASTICITY PROBLEMS

BY THE FINITE ELEMENT METHOD

J. T. Oden* and D. A. Kross**

Research Institute, Univ. of Alabama in Huntsville

This paper treats the formulation of discrete models of thegeneral thermoelasticity problem. It is concerned with the develop-ment of consistent finite element formulations of general dynamicproblems of coupled thermoelasticity and to application of the finiteelement models to the analysis of representative problems. Numericalexamples are included.

*Professor of Engineering Mechanics.**Senior Research Assistant.

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AFFDL-TR-68-150

SECTION I

INTRODUCTION

The theory of thermoelasticity is concerned with predicting the thermomechanical be-havior of elastic solids. As such, it represents a generalization of both the theory of elastic-ity and the theory of heat conduction in solids. The development of the modern theory ofthermoelasticity began with Duhamel's celebrated memoir of 1837 Reference 1, and has beenthe subject of intense research in recent years. Extensive references to previous work canbe found in the survey articles in References 2, 3, and 4, and in the books of Boley andWeiner, Reference 5, Nowacki, Reference 6, and Parkus, Reference 7.

Despite over a century of research on thermoelasticity, many problems of currentinterest are intractable when solutions are attempted by classical methods, This fact has ledsome investigators to consider numerical procedures for solving thermoelasticity problems,and some attempts at finite element formulations of various problems of thermal deformationhave been made. Early applications of the technique considered the temperature distributionof a certain body as a knownfunction of time and the material properties of the body as beinggiven by empirical functions of temperature. Then incremental loading procedures wereused to predict a step by step response of the body due to successive temperature and loadincrements. All coupling between thermal and mechanical effects was ignored in theseanalyses. Finite element formulations of the heat conduction problem, without mechanicalcoupling, are given in References 8 and 9 and finite-element solutions of a class of thermo-elastic problems are considered by Visser, Reference 10, who developed equations based onGurtin's variational principles for linear initial value problems, Reference 11. Visserconfined his attention to uncoupled thermoelasticity. The first finite element formulation ofthe coupled thermoelastic problem is apparently that given recently by Nickell and Sackman,References 12 and 13, who, following Gurtin's work, Reference 11, developed appropriatevariational principles for a class of linear thermoelastic materials. Nickell and Sackmanconsidered the one-dimensional initial boundary-value problem of the thermoelastic half-space with boundary layer thermal conductance and the problem of a half-space subjectedto ramp heating at the surface boundary. We also consider this problem in Section IX of thispaper.

The present paper considers the development of general finite-element models for theanalysis of coupled thermoelasticity problems. The general dynamical theory is considered,and the resulting equations are sufficiently general to include those previously presented

1092

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AFFDL-TR-68-150

as special cases, A significant feature of the analysis is that the finite element equations arederived from general energy balances, thereby freeing the development of finite elementmodels from a dependency on the availability of suitable variational principles.

Following this introduction, the general field equations for thermoelasticity continuaare reviewed, and local and global forms ofthe first law of thermodynamics and the Clausius-Duhem inequality are presented. It is assumed, as is customary, that the Helmholtz freeenergy is a differentiable function of the instantaneous strain and absolute temperature;oot specific forms of this function are not introduced until later. Attention is then given to thedevelopment of general finite-element representations of the displacement, velocity, andtemperature fields which are used in energy balances for typical finite elements. This resultsin coupled equations of motion and heat conduction for such elements. It is then shown thatthe notions of generalized nodal heat fluxes and entropy fluxes come about naturally in theanalysis. Several types of boundary conditions, including specified boundary temperatures,heat fluxes, and those for convective heat transfer, are examined. Finite element equationsfor general forms of the free energy are presented, but special emphasis is given to theclassical quadratic form for linear anisotropic solids. Numerical examples are included.

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SECTION II

THERMOMECHANICAL PRELIMINARIES

Consider a continuous body B under the action of a general system of external iorcesand prescribed temperatures. To trace the motion of the body and the time variation of itstemperature, we introduce the time parameter T which is assigned the value zero in somereference configuration C of the body. Preferably, we select some natural, unstrained stateounder a uniform temperature T to correspond with the reference configuration. In theo .reference configuration, we establish an intrinsic (embedded) coordinate system Xl (i = 1,2,3)which is etched onto the body and which, for simplicity, is assumed to be rectangular carte':'sian at T = O. At some later time T = t, the body occupies another configuration C and therectangular cartesian coordinates of a particle in C which was initially xi in C ,relative to aofixed rectangular coordinate system in C , are denoted z.. The set of three functions,o 1

z. (xl, x2, x3, t) = z. (x, t) defines the motion of the body relative to the reference configuration.1 1_

We postulate that the behavior of a volume v of the body of surface area A is governed bythe following fundamental physical laws;

1. Balance of Linear Momentum

Jp U. dv010

Vo

2. Balance of Angular Momentum

= J p, F. dv010Vo

+ J S.dAI 0A

o

(I)

Jp E.. kZ.U.dV = JA E·'kz,F.dV + JE. ·kz.S.dAo IJ I J 0 0 IJ I J 0 IJ I J 0

~ ~ ~

3. Conservation of Mass

(2 )

(3)

4, Conservation of Energy

fp'~·~·dV +fp Edvo I I 0 0 0

Yo Yo

-=Jp F.u.dYo I I 0

Yo

+JS.U.dAI I 0

Ao

1094

(4)

AFFDL- TR-68-150

5. Clausius-Duhem Inequality

f h f q.- P. -d v - ~ n. dA ~ 0o e 0 e I 0

v Ao 0

( 5)

=ui

Fi =

S. =1

The quantiti.es appearing in these equations are defined as follows:

p , p = mass densities in configurations C and C, respectivelyo 0

v ,v = volume of the body in C and Co 0

A ,A = surface areas of the body in C and Co 0

z. (x,t) - x. = components of displacement relative to x. in Cl~ 1 1 0

components of body force per unit mass in C referred to x.o 1

components of surface traction per unit "undeformed area" (area in C )oreferred to x.1

n. = components of a unit vector normal to the surface A1 0

E = internal energy per unit mass in the "undeformed configuration" C0

h = heat supplied per unit mass in C from internal sources0

qi = components of heat flux per unit area in C referred to x.o 1

7] = entropy per unit mass in C0

e = absolute temperature

The superposed dots (.) in these equations indicate time rates and the entropy flux and

entrOPY source supply are assumed to be given by ~/e and hie respectively. If crij are the

components of the stress tensor per unit undeformed area referred to the convected coordinate

lines xi in C, then

S. = crm j z· . n mI I, J

(6)

where here and in the following the comma denotes partial differentiation with respect to thei / j_x (i.e oz. ox - z. j)'

1 1,

Under suitable smoothness assumptions. the following local forms of Equations 1 through 5

can be obtained.. _ mjp u. - (cr z. .) + P F.o I I tl ,m 0 I

1095

(7)

( 8)

AFFDL-TR-68-150

Here

is the strain tensor and

Po = p./G• _ i j .

P E - cr y.. + q. . + P ho I J I, I 0

p 8';' ~ q. . + P h - 81q. 8, .o I, I 0 I I

( 9)

(10)

( II)

(12)

( 13)

We now introduce the specific free energy cp and the internal dissipation C'" defined by

cr = C'" i j r. - p, (cj, + T}8 )I j 0

so that we have the dissipation inequality

cr--Lq 8 >08 i 'i -

and Equations 10.and 11 become

P,4> = crUy .. - P T} 8 - cro I J 0

p 8T, = q. . + ph + cro I, I

1096

(14)

(15 )

(16 )

( 17)

(18 )

AFFDL-TR-68-150

SECTION III

EQUATIONS OF THERMOELASTICITY

Fundamental assumptions of the theory of thermoelasticity are that the internal dissi-pation is zero and that the specific free-energy is a differentiable function of only the instan-taneous strain and absolute temperature:

.,., =

0" = 0

4> = $(y .. , 81IJ

Thus

o ~ . 0$ .4> = ~y .. + ~ 8vy.. I J 00

I J

and from Equations 17 and 18 we have

1\

04>08

p8T] = q. , + phI I I

( 19)

(20)

(2 I )

(22)

(23)

(24)

Equations 22 and 23 are constitutive equations for the stress and entropy. To these must be

added an independent constitutive equation for the heat flux ~. It is customary to take thelinear Fourier law of heat conduction as a first-order approximation:

ql = k. ,8 •.I J J

where the k., are coefficients of thermal conductivity.IJ

(25)

The basic equations of motion and heat conduction of coupled thermoelasticity areobtained by introducing (22) into (7) and (23) into (24):

(26)

( 27)

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AFFDL-TR-68-150

T( x,t ),..

Figure 1

of node, N, TN is the temperature (temperature change) at node N, and the interpolating

functions "'N( ~) = Ij!N(e) (~) for element e are defined as follows:

"'N(e)('~P=O if ~ = xi does not belong to element e

if node M of element e with coordinatesaM = x~ is ident ica' Iy node N of the

element (31 )

,0 if otherwise

These properties, of course. are recognized as those of generalized lagrange interpolation

functions (generally taken to be polynomials) defined locally over each finite element. Later,

we shal~ affix labels (e) to the local fields ui and T of Equations 29 and 30 to distinguish thoseassopiated with one finite element from those associated with another.

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AFFDL-TR-68-150

The absolute temperature over an element is then

8(x It) = T + o/,N( l( ) TN0--

(32)

where dependence of TN on time is understood. Moreover, from Equation 12, the componentsof strain in the element are

Yj j = ~["'N,iUNj + "'N,jUNi + "'N,k"'M,kUNjUMj] (33 )

where i, j, k = 1, 2, 3 and M, N = 1, 2, ... , N . The last term on the right side of Equation 33eis surpressed in the case of small strain gradients.

The formal assembly of local displacement and temperature fields is accomplished bymeans of the Transformations 14 through 17.

( 34)

(35 )

where Ul:i i and Tl:i are global values of displacement components and temperatures at node l:iof the assembled system of elements and

i f node N of element e co inc id eswith node fj. of the assembly of elements

if otherwise

( 36)

In (34), the repeated global node index l:i is summed from 1 to G, G being the total number ofnodes in the asaembled system of elements. Equations 34 and 35 of course, represent amathematically formal procedure for assembling the local fields. In applications to linearproblems, simpler devices can be used.

The complete finite-element representations of the fields u. (x ,t) and T(x,t) over the1 -

body can now be written in the forms

(37)

(38)

1100

AFFDL-TR-68-150

SECTION V

EQUATIONS OF MOTION AND HEAT CONDUCTIONOF A THERMOELASTIC FINITE ELEMENT

To develop finite element models of various physical problems, it is necessary that

!.Xleans be available to translate a set of relations that hold at a point (local equations) into

relations that hold for a finite subregion (global equations). Customarily variational principles

are devised for this purpose, but any means of converting a local relation into a global one,

such as the local and global forms of the energy balances, can also be used. Indeed, a point

relation is to a continuum as the equations governing a finite element are to the discrete

model of the continuum.

With these observations in mind, we temporarily confine our attention to a typical finite

element of the body with local displacement and temperature fields given by Equations 29 and

30, and we return to the global form of the conservation of energy given in Equation 4.

Introducing Equations 14 and 29 into Equation 4

mNtfMiuN i + J Po (~ + 1}8) d\lo

\Ie

where

= PN1·~N' + J (q .. + ph - pB.ryldvI I, I 0

\Ie

PN, =Jp FjIj!N(l(ld\l +fS,'ltN(Il)dAI 0 ,... 0 I"" 0

ve Ae

(39)

(40)

(41 )

Here v is the volume of the finite element A is its surface area, mNM (N, M = 1, 2, ... , N )e e eis the consistent mass matrix for the element, and PNi are the components of "consistent"

generalized force at node N of the element. The repeated index N in Equation 39 is to be

sl\mmed from 1 to N .e

1101

AFFDL-TR-68-150 '.

(42)

t43)

According to the thermoelastic law of entropy production Equation 24, the last integralin Equation 39 vanishes. Moreover, in view of Equation 23, the integral on the left side ofthe equality sign can be reduced as follows:

1\

J Ocf>.; ----- dvPo oy., Yj j 0

ve I J

; J Po dYO~ IJIN k(Sik + I/IM kUMi)UN,dvv IJ' , J 0e

1\ . ,The term P dcf>loy.. is recognized as the stress tensor (1'"IJ of Equation 22. Thus, the

o I J .

general equation of energy balance for a finite element of a thermoelastic continuum is

1\

[rnNMuMi + J Po o;.~IJIN,k ISj k + IJIM.ku~\'q)dvo - PNi ]uNi = 0v I Je

Since this equation must hold for arbitrary nodalvelocities uNi' we have the general equationsof motion for a thermoelastic finite element

(44)

1\Notice that no restrictions have as yet been placed on the form of the free energy function cpor on the order of magnitude of the displacements or displacement gradients.

Turning now to the total entropy product of the element, we observe that if Equation 24is multiplied through by the temperature increment T and use is made of the identity

we arrive at the local law

(Tq. I,. = T •. Q. + TQ ..I I I I I. I

pTe.;,==- (Tq.I,. - T"qi + pThI I I

1102

(45 )

(46 )

AFFDL-TR-68-150

To obtain the comparable global form of this law, we integrate both sides over the volume of

the element and use the Green-Gauss theorem to convert the integral of the first term on the

right side of the equation into a surface integral. Recalling Equations 19 and 28, we get

(47)

1\ 1\(For simplicity, we assume that ITI« To. Noting also that 0 cp/08 =acp aT, we introduce

Equation 30 into Equation 46 and obtain

1\

[- JPo "'N(~ITo ~t (~ dvo + f IJIN j(~lq i dvo - qN]TN = 0•Ve ve

where

qN = fp. h IJIN(xldV +fIJlN(Xlq, n. dAo "" 0 "" I I 0v Ae e

(48)

(49)

Since (47) must hold for arbitrary TN'

fIJI ,( x lq. dvN I..... I 0

V •e

,..

J d dip -- PoIJIN(~ITodt(dTldvo -qN

ve

(50)

This result is a general equation of heat conduction for a finite element of a thermoelastic

continuum.

1103

AFFDL-TR-68-150

SECTION VI

LINEAR COUPLED THERMOELAS'l'ICITY

Equations 44 and 50 represent the general coupled equations of motion and heat con-duction for any type of thermoelastic media. No restrictions have as yet been imposed onthe form of the free energy function, the order of magnitude of the strains or displacements,or the constitutive equation for heat flux. Thus, these equations apply to the general, nonlinear,coupled thermoelasticity problem. However, we shall postpone an exploration of variousnonlinear thermoelasticity problems to future investigations. In the present study, we shallhenceforth confine our attention to linear thermoelastic theory, which is characterized by afree energy function which is a quadratic form in the strains and the temperature increments:

1\ , ijkl ij I a 2p. cp = -2E y, . YkI + By .. T + -2 -T To IJ I J 0

(51)

where Eijkl and Bij are arrays of material parameters which may be functions of T but whichare usually assumed to be constants for homogeneous bodies, and which have the symmetries

ijkl jikl ijlk klijE' = E = E =E

ij j i8 = B

(52 )

..(53 )

In Equation 50, a is a thermoelastic constant (see Section 9) and lyo.1 = 0 (€) ,€z«l.IJ

As an additional assumption, we suppose that Yo 0 are gi ven by the linearized strain-IJ

displacement relations

y" =!- (u. 0 + u, .)I J I I J J I I

(54)

According to Equations 22 and 23, the stress tensor and the specific entropy are given by

(55)

(56)

1104

AFFDL-TR-68-150

Substituting Equation 55 into 44 and noting that because of the restriction to small

displacement gradients implied in Equation 54 the term (8jk + IJIM,kuMj) in Equation 44

reduces to simply 8jk, we arrive at general equations of motion of a linear thermoelastic

Unite element

where

aim = J Eikmn~ "" dvNM N,k M,n 0ve

( 57)

( 58)

( 59)

To obtain the accompanying coupled heat conduction equations for the finite element. we

assume that the linear Fourier law Equation 25 holds with k.j 8" = k ..T ,. and introduce1 J IJ J

Equations 25, 30, and 56 into Equation 50

(60)

where

(61)

(62)

If we denote by P~i and Q~ the global values of generalized force and heat flux corre-

sponding to (Equations 34 and 35), then

(e)P~i = ~ .o;N~PNi (e) (63)

(e)Q~ = ~nN~qN(e) (64)

We therefore have for the general coupled equations of motion and heat conduction for an

assembly of linear thermoelastic finite elements,

(65)

(66)

1105

AFFDL-TR-68-150

in which

and

M : L ntel mlel ntel~f e N~ NM Mf

Aim =L:.ntelaimteldel~r e N~ NM Mr

Si _ L .0.\ e Ibite InIe l6r - e N6 MN Mr

K : L nlelklel ntelLlr e N~ NM Mr

1106

(67)

(68)

( 69)

(70)

( 71 )

AFFDL-TR-68-150

SECTION VII

BOUNDARY CONDITIONS

Following the plan of Becker and Parr (Reference 9), we point out three possible boundaryconditions involving the thermal variables qN and TN. Mechanical boundary conditions involvesimply prescribing Ufj. i or P fj. i at various boundary nodes.

1. Convective Heat Transfer

If T* is the temperature of the media surrounding the body, and T* is the temperatureoacross a surface boundary layer, then the convective heat flux is given by

q . = f. (T* - TIE-)I I 0

where f. are fUm coefficients.1

2. Specified Heat Flux

The heat flux may be specified on the boundary

q. = q.I I

(72)

(73)

An important special case of Equation 73 is that of adiabatic behavior for which ~ = O.

3. Specified Temperature

Temperatures are prescribed on the boundary by simply prescribing the nodal values

TN at appropriate boundary nodes.

Combining Equations 72 and 73, we have

q '1 = q. + t, (T - T *)I 1 0

so that Equation 49 becomes

1107

(74 )

(75)

AFFDL-TR-68-150

where

and

Thus, instead of Equation 60, we may use

i . . *"(kNM - fNM ITM - TobMNuMi - CNM TM = qN

Likewise Equation 66 becomes

where

1108

(76)

(77)

( 7 8,)

(79)

(80l

(81 )

AFFDL-TR-68-150

SECTION VIII

SOME SPECIAL CASES

The above finite element equations are valid for any type of finite-element represen-tation for which the general, local interpolation Equations 29 and 30 are used. The equationsare considerably simplified, however, if the familiar simplex models for the local fields areus~. Then tetrahedral, triangular, or lineal elements with four, three, or two nodes are usedin three-, two-, or one-dimensional problems, respectively. Then the interpolating functions'itN(X)of Equation 31 are given by the linear forms

,/, (l() = a +" i (J'f'N N fJiN

(82)

where aN and {3 iN are constants which depend only on the geometry of the finite element.Physically, Equation 82 depicts the deformation of each element as being homogeneous andthe local temperature field of each element as being homogeneous (Reference 17),

In this case, the elemental mass, stiffness, coupling, and conductivity matrices ofEq,uations40, 58, 59, 61, and 62 reduce to

im (J (J ikmnQNM =ve fJkNfJNME

kNM=v k··I3.Nf3·Me I J I J

i -.1-. ik (J (J jbMN - P B fJkNlaM" + fJ· MSo e J

(83)

(84)

(85)

(86)

where me and v are the mass and volume of the element, sj are the mass moments withe .respect to the system i, and i.. is the mass-moment-of-inertia tensor of the element with

i IJrespect to the x .

1109

AFFDL-TR-68-150

Equations for the coefficients a N and 13 iN for the one-, two-, and three-dimensional

cases can be found in, for example, (Reference 17). In the two-dimensional case, for example,

x~ x;- x~ x~

I 2 I I 2aN = 2A x I x3-xl )(3

0

xix 2_x2 x'I 2 I 2

-I

2 2 2 2 2 2x2 - x3 x - x x, - x23 I~ =~

iN 2Ao I I I x I - x I I I1.3-1.

2x - x

I 3 2 I

(87)

(88)

where Ao is the area of the element and x~ (N = 1, 2, 3; a = 1,2) are the coordinates of

node N. For one-dimensional elements,

a = - x /L2 I

/3 = -{3 = - I/LII 21

( 89)

where xl and x2 are the x-coordinates of nodes I and 2 and L = x2-xl.

It is interesting to note that if Equation 57 is ignored, the term -Tob~N UMi is deleted,

and the values in Equations 87 and 88 are introduced in Equations 83 and 85 for the purpose

of calculating kNM and cNM' then Equation 78 reduces to the heat conduction model of Becker

and Parr (Reference 9). If the two-dimensional simplex values of Equations 87 and 88 are

used. in Equations 83 through 86, and if the inertia term mNMuMi and the coupling term

Tob~N uMi are deleted from Equations 57 and 60 or 78, then uncoupled quasi-static equations

of the type considered by Visser (Reference 10) are obtained. Finally, if the one-dimensional

model defined by Equation 89 is introduced into Equations 83 through 86, 57, and 60, the finite-

element formulation of Nickell and Sackman (Reference 12) is obtained.

1110

AFFDL-TR-68-150

SECTION IX

NUMERICAL RESULTS

In order to dem.:nb~rate the merits of the finite element representation developed in this

paper. several numerical examples are presented in this section. Solutions of the finite

~~ement differential equations were obtained using a standard Rung-Kutta-Gill integration

seheme. The first examples consider a linear thermoelastic half-space subjected to ramp

heating on its stress-free bounding surface. This problem was first investigated by Sternberg

and Chakravorty (Reference 19) for the thermoelastically uncoupled case. The special case

of sudden step heating of the bounding plane had been studied earlier by Danilovskaya (Ref-

erence 20) for the thermoelastically coupled case and is included in this paper. Nickell and

Sackman (References 12 and 13) also investigated this problem using a one-dimensional finite

element model obtained through special variational principles.

It is customary to use the following nondimensional parameters:

where

8::.LTo

U :: u (90)

KK :: pC

v

f3 :: at 3>.. + 2µ) ( 91)

In the above equations XI is a characteristic length, t the real time, K the thermal conductivity,

C the specific heat at constant volume, A and µ are the isothermal Lame constants. andva is the linear coefficient of thermal expansion. The quantity 8 in (Equation 91) is a thermo-

mechanical coupling parameter.

Now consider a linear elastic half-space (\ > 0) with bounding surface at XI = 0 assumed

to be stress free. Let the bounding plane be subjected to a ramp surface heating of the form

TFT = - t

I to(92 )

t <o

1111

AFFDL-TR-68-150

where T1 is the surface temperature, TF is the final surface temperature, and to is the

boundary temperature rise time.

Define the nonLu',,,,·,,,,.mal rise time T byo

(93 >' ' ..'I iI,. ,

The dimensionless boundary condition is

(94)

8 = I T < TI 0 -T

The quantity TF has been set equal to unity in Equation (93) for convenience.o

The numerical results for the half-space are presented in Figures 2 through 6. Figures 2and 3 contain the dimensionless temperature Band displacementu at ~ = 1.0 with coupliugparameters 8 of 0.0 and 1.0 as a function of dimensionless time T, for the case whenT = 1.0, whereas Figures 4 and 5 are for the case when T = 0.25. In these figures, theo 0number of elements between the free surface and ~ = 1,0 is denoted by NE, whereas the totalnumber of elements used in the analysis is denoted as TNE. The "exact" solutions presentedin these figures were obtained by Nickell and Sackman (Reference 12), using a numericalLaplace Transform inversion procedure.

Excellent agreement between the "exact" solution and the finite element solution for thetemperature is seen in Figure 2. It is noted in Figure 3 that the displacement solution di-verged when NE = 2 and TNE = 10. Increasing the number of elements, however, gaveimproved results. The finite element solution for the temperature in the case of the highersloped ramp heating in Figure 4 also yielded good agreement with the "exact" solution. Anillustration of the rates of convergence characteristics of the finite element solution is .illustrated in Figure 5 for the uncoupled case.

Figure 6 shows the temperature distribution through a portion of the half-space for thecase of a suddenly applied step temperature rise on the bonding surface, Exact solutions ,..were obtained by Boley and Tolins (Reference 21). It is noted that, as the time increases,the finite element solution is lower than the exact solution for signWcant distances into thebalf-space (e > 4.0). Since the total number of elements for this analysis is 48 and the

1112

0.7~~I'%j •t:l~I

t-:l!;OI

0)

00I.....

CJ1o

2.0

!;,:l

1.5

8 =O.O(UNCOUPLEO)

" 8=lO

0.5 1.0

DIMENSIONLESS TIME T

T.= IO-HE=6.TNE=300- N E = 2. THE = 10

o o

0.1

0.6t-!"....• 0.5

ICDbJ0:04::)....«

..... 0:

..... LLI 0.3

.....c.:> Q.

~LLI.... 02

Figure 2

AFFDL-TR-68-150

I()

-

....IJJ::E-I-(J)(J)

0 IJJ C':l...J Q)

Z ~0 .....~-(J)

ZIJJ::E-Q

I()

0

oN

oo

"'"Q1&1.JA.:;)ouz:;)-.."

o 0d ...:.. II

~~I)oo

~2II ..1&11&1ZZ......• ..

CDNC! .. IIiil&ll&l~zz

I I

00I

q- N 0 CD to q- N-: - 0 Q Q 0

.n. J.N3 W3~"'dSla

1114

2.01.51.00.5o o

07 :>T.=o.25 I'%j

I'%j

o -HE-IO. THE =48 t:lt"'I

0.61 O-HE= 8. THE =40I-j~- I0)(XlI....

0 I -~ I c.nI- 0

" 0.5

~ I S:O.O(UNCOUPLED)~

ICD 04LtJ0::::)I- 0.3.... «.... 0::.....

c.n LtJQ..2 0.2LtJ....

0.1

DIMENSIONLESS TIME T

Figure 4

AFFDL-TR-68-150

a&&J•

.-..(()

oo o o o o

ooeDN-'-.. .. II&&JLLILLIZZZ1-1-1-

10 • • •N.N2d .. .. II• &&J &&J LLIL!ZZZ... I I I

Doeo ~ 0 ~C\J - --: 0o 0 0 0

.!l J.N3W3~\fldSla

1116

oo

vo.'

QN

oo

· .AFFDL-TR-68-150

/.0

0.9

0.8

0.7

~ 0.6

"t-..ICD 0,5l&.I~~t-ee 0.40::l&.IQ.2~ 0.3

0.2

0.1

0.0 o

STEP HEAT INPUT8=0.03O-NE =6. TNE ':.4 8

I 2 3 4 5

01MENSIONLESS DISTANCE (

Figure 6

1117

6

AFFDL-TR-68-150

number of elements from ~ = 1.0 to the boundingsurface (e- = 0.0) is 6, the "infinite" half-space extends only to e = 8.0. At the "infinity" point, the temperature and displacementare prescribed as zero. The finite element solution, therefore, yields results which are lowin regions neal ~he "infinity" point. An improved solution can be obtained by simply in-creasing the total number of finite elements while maintaining the same number betweene- = 1.0 and the surface.

Acknowledgment

Support of this work by the National Aeronautics and Space Adminstration through GeneralResearch Grant No. NGL 01-002-001 is greatly appreciated.

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SECTION X

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

" , / /Duhamel, J. M. L., "Second memoire sur les phenomenes thermomecaniques, "Journalde l'Ecole Polytechnique, Vol. 15, cahier 25, pp. 1-57, 1837.

Chadwick, P., "Thermoelasticity, The DYnamical Theory," Progress in Solid Mechanics,Vol. 1. Edited by I. N. Sneddon and R. Hill, North-Holland Publishing Co., Amsterdam,1960.

Boley, B. A.. "Thermal Stresses." Proceedings of the First Symposium on NavalStructural Mechanics, Edited by J. N. Goodier and N. J. Hoff, Pergamon Press, Oxford,1960.

Parkus. H., "Methods of Solution of Thermoelastic Boundary Value Problems," High.Temperature Structures and Materials; Proceedings of the Third Symposium on NavalStructural Mechanics, Edited by A. M. Freudenthal, B. A. Boley, and H. Liebowitz,Pergamon Press, Oxford, 1964.

Boley. B. A. and Weiner, J. H., TheorY of Thermal Stresses, John Wiley and Sons,New York, 1960.

Nowaki, W.. Thermoelasticity, Addison-Wesley Publishing Co., Reading, 1962.

Parkus, H., Thermoelasticity, Blaisdell Publishing Co.. Waltham, Mass .. 1968,

Wilson, E. L. and Nickell, R. E., "Application of the Finite Element Method to HeatConduction Analysis," Nuclear Engineering and Design, Vol. 4, No.3, pp. 276-286,Oct. 1966.

Becker, E. B. and Parr, c. H.. Application of the Finite Element Method to Heat Con-duction in Solids, Technical Report S-117, Rohm and Haas Redstone Research Labora-tories, Huntsville, 1967.

Visser, W., "A Finite-Element Method for the Determination of Non-Stationary Temper-ature Distributions and Thermal Deformations," Proceedings, Conference on MatrixMethods in Structural Mechanics, Ed it e d by J. S. Przemieniecki, R. M. Bader,W. F. Bozich, J. R. Johnson, and W, J. Mykytow, Air Force Flight Dynamics Laboratory-TR-66-80, Dayton, pp. 925-943, 1966.

Gurtin, M. E., "Variational Principles for Linear Initial-Value Problems," Quarterlyof Applied Mathematics, Vol. 22, No.3, pp. 252-256, 1964.

Nickell, R. E. and Sackman, J, J., "Approximate Solutions in Linear, Coupled Thermo-elasticity ," Journal of Applied Mechanics, Vol. 35, Series E, No.2, pp. 255-266,June 1968.

Nickell, R. E. and Sackman, J. J., The Extended Ritz Method Applied to Transient,Coupled Thermoelastic Boundary- Value Problems, Structures and Materi als ResearchReport, No. 67-3, University of California at Berkeley, Feb. 1967.

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REFERENCES (CONT)

14. Wissmann, J. W., "Nonlinear Structural Analysis; Tensor Formulation," Proceedings,Conference on Matrix Methods in Structural Analysis, Air Force Flight DynamicsLaboratory, TR-66-80, Dayton, pp. 679-696.

15. Oden, J. T.. "Numerical Formulation of Nonlinear Elasticity Problems," Journal ofthe Structural Division, ASCE, Vol. 93, No. ST3, pp. 235-255, June 1967.

~6. Oden, J. T. and Sato, T., "Finite Strains and Displacements of Elastic Membranes bythe Finite Element Method," International Journal of Solids and Structures, Vol. 1,pp. 471-488, 1967.

17. Oden, J. T., "A Generalization of the Finite Element Concept and Its Application to aClass of Problems in Nonlinear Viscoelasticity," Developments in Theoretical andApplied Mechanics, (Proceedings, 4th SECTAM), Pergamon Press, London (in press).

18. Danilovskaya. V. I., "On a Dynamical Problem of Thermoelasticity," (in Russian),Prikladnaya Matematika i Mechanika, Vol. 16, p. 391, May-June 1952,

19. Sternberg, E. and Chakravorty, J. G.. "On Inertia Effects in a Transient ThermoelasticProblem," Journal of A pp lied Mechanics, Vol. 81, Series E, No.4, p. 503-509,December 1959.

20. Danilovskaya, V. I., "Thermal Stresses in an Elastic Half-Space Arising After a SuddenHeating of Its Boundary," (in Russian), Prikladnaya Matematika i Mechanika, Vol. 14,No.3, p. 316, May-June 1950.

21. Boley, B. A. and Tolins, I. S., "Transient Coupled Thermoelastic Boundary ValueProblems iIi the Half-Space," Journal of Applied Mechanics, Vol. 29. Series E, pp. 637-646, December 1962.

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