\{0

UARI Research Report No. 72

Project Themis - AFOSR 69-2816TR

ON THE NUMERICAL SOLUTION OF A

CLASS OF PROBLEMS IN DYNAMIC

COUPLED THERMOELASTICITY

By

J. T. Oden

and

J. Poe

Research Sponsored by Air Force of Scientific Research, Office of AerospaceResearch, United States Air Force - Contract No. F4462-69-C-0124

Research Inst ituteUniversity of Alabama in Huntsville

Huntsvi lie, Alabama

October, 1969

This document has been approved for public release and sale; its distribution is unlimited.

i

ON THE NUMERICAL SOLUTION OF A CLASS OF

NONLINEAR PROBLEMS IN DYNAMIC

COUPLED THERHOELASTICITY

J. T. Oden and ~. Poe

Research Institute, University of

Alabama in Huntsville

ABSTRACT

This paper concerns the application of the finite-element

method to the solution of certain nonlinear problems in thermoelas-

ticity. Numerical solutions of transient, coupled thermoelasticity

problems involving bodies which exhibit material nonlinearities and

temperature-dependent thermal conductivity and specific heat are

presented. General equations of motion and heat conduction of an

arbitrary finite element are reduced so as to apply to the problem of

transient response of a nonlinear thermoelastic half space subjected

to a time-dependent temperature over its boundary.

ii

NOTATION

ao' aI' .," - Material constants

c - Specific heat at constant deformation~

Co - Reference configuration

Fm - Components of body force per unit mass in Co

G - Green deformation tensor

h - Internal heat supplied per" unit undeformed volume

I, II, III - Strain invariants

K1J - Temperature dependent thermal conductivity tensor

Ko - Conventional thermal conductivity

~- Nondimensional length

mNM - Consistent mass matrix for the element

PNk - Components of nodal generalized forces

ql - Components of heat flux

qN - Generalized nodal heat flux at node N.

sm - Components of surface traction

T - Temperature

To - Temperature of the reference configuration

urn - Cartesian components of displacement relative to Co

u - Nondimensio~al displacement

x1

- Coordinates describing the motion of the body

Xi- Rectangular cartesian coordinates in the undeformed body

a - Linear coefficient of thermal expansion

Y1J - Green-Saint Venant strain tensor

6 - Thermomechanical coupling parameter

€ - Term governing variation of thermal conductivity with

temperature

iii

~ - Nondimensional time

Tl - Entropy per unit undefonncd volume

e - Absolute temperature

e - Nondimensional temperature

A, µ - Lame' constants

~ - Internal energy per unit undeformed volume

p, Po - Mass densities in the c?nfigurations Co and C respectively

a - Internal dissipation

a1J - Stress per unit initial area

T - Time parameter

~ - Free energy per unit undeformed volume

*N(~) - Finite element interpolation function

1

INTRODUCTION

It is widely known that many physical properties of common

Innterials are dependent on temperature. A casual glance at any good.' I

handbook on physical properties of materials, for example, will show

that quanti ties such as specific heat, therma I conductivity, thermal

l'xpansion coefficients, etc., which are treated as constants in classi-

cal theories, may change significantly with a change in temperature.

The description of the behavior of such materials becomes further com-

plicated if it is also recognized that their ability to conduct heat

may be dependent on deformation. Indeed, the interconvertibility of

mechanical work and heat was recognized long ago by Joule; hence, it

is reasonable to take into account the effects of temperature in the

equations of motion of a body and to include the effects of motion in

the equations of heat conduction.

Experimental observations indicate that many thermomechanical

phenomena associated with fairly common materials are decidedly non-

linear in nature. As such, their analytical description falls well

outside the scope of the classical theory of thermoelasticity. Reiner

[1] proposed a nonlinear stress-strain law for a limited class of

thermoelastic solids and Jindra [2] used a perturbation procedure to

solve a problem in static, uncoupled thermoelasticity wherein certain

material nonlinearities were assumed. Recognizing that many materials

exhibit deviations from standard linear constitutive laws even for

infinitesimal strains, Dillon [3] developed a theory which included

mild material nonlinearities by retaining certain higher-order terms

in a series expansion of the free energy density. Dillon studied the

influence of nonlinear terms in the deviatoric strains by calculating

the temperature generated in a circular bar subjected to prescribed

time-dependent twisting .

2

(hying to the extreme mathematical difficulties usually involved

in treating nonlinear ('quations, no analytical solutions to nonlinear

bOllndary-and initial value problems in dynamic-coupled thermoelasticity

"I'pear to he avai lab Ie. It is natura I, therefore, to consider numeri-

cal methods for solving such problems. This is the viewpoint adopted

in the present investigation.

This paper is concerned with the application of the finite-element

concept to the analysis of class of nonlinear problems in dynamic-

coupled thermoelasticity. After a brief review of certain fundamental

equa tions governing the behavior of thermoelastic so lids, we present

rather general coupled equations of motion and heat conduction for a

typical elwnent of a discrete model of the continuum. The form of

the free energy function and the constitutive equation for heat flux

is not specified in these equations; nor are restrictions imposed on

magnitudes of the displacement gradients. We then obtain special

forms of these equations by assuming infinitesimal strains. expanding

the free energy in a power series, and retaining terms of higher order

than the second, in the manner described by Dillon [3J. We also account

for temperature dependent specific heat and thermal conductivity,the

latter being incorporated in a nonlinear version of Fourier's law where-

in the thermal conductivity i8 assumed to be a linear function of temp-

erature. By linearizing these equations, we show that the finite-

element models of Nickell and Sackman [4] and Oden and Kross [5] are

obtained.

To demonstrate the influence of various nonlinearities, we then

consider applications of the theory to selected problems for which

solutions to the linearized problem are known. In particular. we

solve several nonlinear versions of the coupled Danilovskaya problem

[6J, which involves the transient response of a thermoelastic half-space

3

nubjected to a time-dependent temperature field applied uniformly over

its plane boundary. This problem was solved using linear theory by

St~rnberg and Chakravorty [7J; numerical solutions of the linearized

rrohl~m have also been presented [4, 5J. Our results indicate that

material nonlinearities, as manifested by nonlinear dilatational terms

ill the constitutive equation for stress, temperature-dependent thermal

conductivity, and temperature-dependent specific heat, may lead to

significant differences from the linear theories. Parametric studies

are performed in those cases in which insufficient experimental data

is available to estimate the relative magnitudes of higher-order ma-

terial constants. The present finite-element formulation of the prob-

lem leads to several hundred simultaneous nonlinear first- and second-

order differential equations in the nodal values of displacements and

temperatures. These are solved by a Runga-Kutta-Gill integration scheme.

NONLINEAR T1IERMOELASTICITY

Consider a deformable, thermoelastic continuum under the action

of a general system of external forces and prescribed temperatures.

The reference configuration Co is ideally selected to correspond to a

natural unstrained state and to be at a uniform temperature To. To

trace the motion of the continuum and its variations in temperature,

we introduce a system of intrinsic coordinates Xl which are rectangular

cartesian at T = O. T being a time parameter. At T = t > 0, cartesian

coordinates of a particle Xi are denoted Xl and the temperature at Xl

is To + T(Xi, t), where T(Xi, t) is the temperature change. The rela-

tions Xi = Xi(X1• X2, X3, t) describe the motion of the body.

The thermomechanical behavior of the continuum must be such that

the following physical laws are satisfied locally at every particle t:

4

il (al JX• ) I + PoF = pou a1 J = a1 J (1).. , J , II '" ,

p.JG = Pn (2)

-: ~ alJYIJ + q~l+ h (3)

0'h 1ql + h - - qlT (4); Ie, I

lo'n· ,,'; is tIll:stress per unit initial area Ao referred to the con-

l:clcd coordinate lines Xi; Fillare the components of body force per

Init mass in Co; u= are the cartesian components of displacement

relative to Co; S, ~, and h denote the internal energy, entropy, and

internal heat supplied per unit undcformed volume; Y1J is the Green-

Saint Venant strain tensor; ql are components of heat flux; and e =To + T is the absolute temperature. Superposed dots indicate time

rates, commas partial differentiation with respect to Xl, and semi-

colons denote covariant differentiation with respect to Xl. We also

ha vc

(5)

where

(6)

Equations (1) represent local forms of the laws of balance of linear

and angular momentum; (2) insures that mass is conserved during the

motion; (3) is a local form of the law of conservation of energy, and

(4) is the Clasius-Diehem inequality.

It is convenient to introduce the free energy ~ per unit unde-

formed volume:

(7)

iiI:""1'11 ('\) can he recast in the alternate forms

f11 ;p = a1jylJ- Tl~ - a

5

(8)

g'h = q~ 1+ h + a (9),

wllt'rl' on defines a, the interna I dissipation. The theory of thermo-

~)asticity is based on the assumption that a is zero and that ~ is a

differentiable function of the current values of Y1J and T. It then

follows from (8)' that

(10)

Therefore, the equations of motion and heat conduction at a point in

a thermoelastic continuum are

[a~(YIJ' '1') ]x + -" (11)ay I J II ,I ,J Po F 1ft - PC\ urn

_ e.L [orp(y I J' T) ] = ql + h (12). dt aT ; I

Specific forms of (11) and (12) can be obtained when the form of the

function ~(YiJ' T) and the constitutive equation for qi for the

material are identified.

FINITE-ELEMENT MODELS

We shall now outline the development of general discrete models

of thermoelastic behavior obased on the finite-element concept. Follow-

ing the usual procedure, we view the continuum as a collection of a

finite number of component parts called finite elements, which are

connected continuously together. Ordinarily, the elements are of

relatively simple geometric shapes (e.g., tetnahedra, prisms, quadri-

laterals, triangles, etc.) and the connectivity of the model is

f)"'.1~~1•~if5I~~~,,1

!

6

h',",ll"l!, I i~hcd by regarding the c lemt'nts to be attached to one another

;11 I'fl'Sl'lccted noda 1 points. Since the process of connecting elements

:ll'l'rnl'rinlely together i.sbased on purely topological properties of

lIlt" ,':n<l,'l, it sufficies to isolate a typical finite and to first des-

I' r ih.. i.tsbehavior independent of the rest.

III the present investigation, a typical element e is viewed as a

:;lIhdlllll.1in of the displacement and temperature fields u1 (~, t) and

TO:. t). The local displacements and temperatures over a typical ele-

IIll'llt C1re .1ssumed to be of the form

T (13)

where tI and T are the displacement components and the temperature'II N

change at node N of the clement. Here the dependence of u I and T onN N

t is understood and the repeated nodal index is summed from 1 to N ,e

N~ bei~g the total number of nodes of element e. The interpolation

functions ,I. (X) form a basis [or the N -dimensional subspace, described'N _ e

by (13), which is a projection of the space to which the continuum dis-

placement and temperature functions belong; they are assumed to have

the following properties:Ne

6~, L\h(~)N:=l

1 (14)

Following a procedure described in previous investigations [8, 9,

10], we introduce the local approximations (13) into appropriately

modified forms of the energy balances (8) and (9) and require that

the results hold for arbitrary nodal values UNI and TN' In this way

we obtain general equations of motion and heat conduction for a typical

finite element e:

m"'u", + J"0 (e)

f [t, (!'){To

Vov·/

7

(15)

(16)

lIl'rc N, N == I, 2, ... , Ne; i, j, k

J 1jr N (~) W" (~)dU- (17)

·J t,(!,:)F,Pod\l"

"'n(,.)

+ J $'(0,.+

Ao (e)

(18)

q, • J t, (!':)hd"lf

lfo (e)

+ J q'n,t,(!,:)dA

Ao (,,)

(19)

The array mNM is the consistent mass matrix for the element, PNk

is the kth component of generalized force at node N, and qN is the

normal generalized h:! at flux at node N. The surface tractions sm and

hcat flux components ql per unit undeformed area are referred to

coordinate directions Xi in the deformed body. Equation (15) is the

discrete analogue of (11) and (16) is the discrete analogue of (12).

Again, specific forms of (15) and (16) can be obtained once forms of

~(YIJ' T) and ql(Y1J' T, T, I) appropriate for the material under con-

sideration are specified.

CONSTITUTIVE EQUATIONS FOR THERMOELASTIC SOLIDS

For isotropic thermoelastic materials, the free energy can be

expressed as a function of the strain invariants and temperature:

\vhere

(I, II, III, T)

I = Y11

1II ="2 (Y11YJJ - Y1JYlj

III = det (Ylj)

8

(20)

(2l)

Following a standard approach, we assume that the free energy can be

expanded in a power series in the strain invariants and temperature

, *1ncrements;

2+ aeIT + a91 II + aloIT + allIIT + a12T3 (22)

+ a13IzT + a14I4 + alslIz + alGI II + ...

where ao, al' a2

... arc material constants. If we aSSume the material

to be stress free in the reference state and to not be dependent on

terms in the free energy of higher than fourth order, ao = al = 0 and

al?' ale' als' .. , do not appear. We may regard this form as.having

no restriction on the magnitude of the strains and temperature but as

a free energy function of a certain class of thermoelastic materials.

Obviously we can obtain many forms of the free energy function by

simply adding or deleting terms in (22). For example, an incompressible

thermoelastic material of the Mooney-type is described by a relatively

simple form of the free energy function;

(23)

where

(24)

and C, and Cz are the usual Mooney constants for the isothermal case.

*The term a4T is omitted because it does not influence either a1jor S.

9

constants. However, only relatively simple forms of the free

(25)

(26)

(27)

i (E1111 - h), a(3A + 2µ)

CP. = 1E I J k my y + B i J Y T + 1. iL T2

2 IJkm IJ 2T o

for \.Jhich

Anothcr example is provided by the classical thermoelastic solids,

properties of symmetry

Ek m I J and B i J

where EIJkm and B1J are arrays of material parameters which are

specific heat at constant deformation, and A and µ are Lame'

assumed constant for isotropic homogeneous bodies, and have the

where a is the linear coefficient of thermal expansion, Co is the

These material parameters may be rewritten as

.,.~<.:nergycan be utilized in the development of manageable nonlinear

theories of thennoelasticity. We shall confine our attention to

thcrmoelastic materials for which the constitutive equations of

stress are nonlinear but th<.:strains are infinitesimal:

(28)

So as to obtain quantitative solutions, we follow the example of

Dillon [3] and aSSume that the quadratic version of the free energy

function coincides with the quadratic form (25) formulated in classi-

cal thermoelasticity. Then

1~ (~+ 2µ), as = -a(3~ + 2µ) (29)

(33)

(30)

(31)

-ze(a7 + Zal~ T + 2alO

I)

10

However, we obtain a more general form of specific heat:

Conclusive data on the variation or Co with deformation is not

some metallic-type materials the relation between the dilitational

free energy is reduced to

the deviatoric components [3]. We have nevertheless retained these

linear in dilitation. It is recognized that for small strains in

retain terms of third degree in~. With these simplifications the

We shall include, however, till'effects of temperature on Co and as-

mild nonlinearities in the constitutive equations for stress, we

Equations (31) - (33) reduce to the classical equations of linear

components of stress and strain is linear to a larger exten than

readily available and as a further silllplification we set a1

('\ = O.

slime that it varies linearly witll temperature. In order to include

thermoelasticity if we delete nonlinear terms a3' a6, a9, all' ala'

and a13' If either a3 = a6 = 89 I 0, the material is mildly non-

The stress and entropy are then

11

terms in order to quantitatively assess the effects of nonlinearities

in the dilitational strain components on the behavior of the material.

Dillon [3J observed that for <l material under oscillating dilita-

tional strains the heating in compression should equal the cooling

in tension, which requires a13 = O. However, we know that when a

body is worked heat is generated which is not completely dissipated.

Recognizing that this property may not fall within the province

of thermoelastic materials, we neverthesess retain the term in

order to study its effect on the materials behavior, if indeed

sma 11.

We now turn to the problem of identifying the constitutive

equation for heat flux in a thermoe lastic solid. Fourier I slaw

seemS to work for a wide range of materials and as in the specific

heat appears to be independent of deformation, but may vary with

temperature. With this in mind we introduce a modified Fourier

Law:

(34 )

where Kl J is the temperature dependent thermal conductivity tenser .

We shall assume as a first approximation for isotropic materials

that K1J is given by the linear form.

(35)

Here KQ is the conventional thermal conductivity of the media and

€ is a material constant of dimension l/Temperature.

For materials described by (31) and (34), the equations of

motion and heat conduction for a typical finite element are

+ f ([(-2µ + 2(A + 2µ) +3a"Y,,+ agy,,+ allT + 2a,3T)y ..

"V": (e)o + ag(yrryss- YrsYrs) - a(3A + 2µ)TJ61j-

12

- ToJ f, (x) (237 T -

~ (e)

where

f I<" (I + 'T) 6 , J t, . ,T. J d"lf

~(e)

-1

Yrs = "2 [~N (x)uNr + *N (x)u" ],5 _ I' _,5

THE ELASTIC HALF SPACE PROBLEM

(36)

I (37)

At present, detailed studies of practical problems in nonlinear

thermoelasticity are handicapped by the lack of experimental data

for real materials on the magnitudes of the constants a3- alS' To

demonstrate the influence of various nonlinear terms in (31) we

have chosen to investigate nonlinear versions of the Danilouskaya

[6J problem for which comparable solutions of the linearized prob-

lem are known. The particular example considered involves a

materially nonlinear elastic half space subjected to a linear time

dependent change in temperature over the entire boundary which is

13

assumed to be initially stress free. After a specified change in

temperature has occurred the temperature is held constant over the

sur face.

The linearized version of this type ramp heating problem was

first investigated by Sternberg and Chakravorty [7J for the thermo-

elastically uncoupled case. Solutions for the coupled case were

obtained by Nickell and Sackman [4] and Oden and Kross [5] through

finite element techniques.

Consider a materially nonlinear elastic half space (x, ~ 0)

constrained to only uniaxial motion characterized by the displace-

ment field

(38)

The bounding surface at Xl = 0 is assumed to be stress free and is

subjected to a uniformly distributed ramp heating of the form

T = 0 -oo<t$;O1

TrT = - t o ~ t ~ to (39)

1 to

T1 = Tr to ~ t < 00

where T1

is the in~tial surface temperature Tr is the final surface

temperature, and to is the rise time of the boundary temperature.

Since the body is assumed to be initially at rest, the displacements

and stresses resulting from the temperature field T are governed1

by the initial conditions

u1

(x, 0) = 0,0u

1(x, 0)

at = 0 (0 < x < 00) (40)

These boundary conditions are supplemented by the regularity con-

ditions

14 '" I.

u1 (x, t), at (x, t) ~ 0 jli

and

T (x, t) ....0 as x ~ QO (41)

We assume that the material in the half space is characterized

by a free energy function of the form given in (31). Since the body

is constrained to only uniaxial motion the second and third strain

invariants vanish reducing (31) to

(42)

For simplicity we use simplex approximations of element displace-

ment and temperature fields so that

~ n (x)T_ n (43)

where

I, 2 (44)

With these seIl.:ctions, the equations of motion (15) and heat con-

duction (16) for a typical finite element become

(45)

15

pncf)(T)To. . K.n €

q2 = 6 (T1 + 2T;:> - L [I + 2L (T1 + Tz)](Tl - Ta)

To To (48)- - a(3t.. + 2µ) (il - il ) - - a (u - u ) (u - u )2 1;.: L \3 1 2 1 Z

where L is the length of the element. We introduce the usual

dimensionless variables as follows:

Ii

~ =~ XIt 1

Te =-To

where

KoIt =-

POcf)

aZC = - t

It

U = ~±-M(49)

KST U. 0

A + 2µP

f"j

I•• '!

S = a Ot.. + 2µ) {) =(50)

:t'i:11I1",

In the above relations, Xl' is D characteristic length, t the

real time, Ko thermal conductivity, cf)(T) temperature dependent

specific heat at constant deformation, ~ and µ are Lame' constants,

a is the linear coefficient of thermal expansion, and the quantity

6 is the thermomechanical coupling parameter.

NUMERICAL RESULTS

Numerical results showing the influence of thermOtnechanical

coupling, material nonlincarities, and temperature dependent specuic

heat and thermal conductivity in the solution of the half space

problem are presented in figures 1 - 8.

Solutions of the finite element differential equations (45) -

(48) were obtained by a Runge-Kutta-Gill integration scheme. In

16

the solutions the maSHCH and tumpcratures of each element were

lumped at the nodes in order to uncouple the elemental accelera-

tion and timc rate of temperature changc terms. This method of

approximation proved very satisfactory for a large number of ele-

ments and is illustrated in Figs. land 2 which contain numerical

results for the linearized material as well as the "exact" solu-

tions [lOJ. Figures I and 2 contain the dimensionless temperature

e and displacement U at 1= 1.0 with the thermomechanical coupling

parameter 6 = 0.0 and 6 - 1.0 as a function of dimensionless time

C for the cases Cn = 1.0 and C = 0.25 respectively. These results

werc obtained using a fifty element model having 10 elements between

the 1:0 unding sur face and 1= i. O.

A qualitative description of the effect of material nonlineari-

ties in the half space are illustrated in Figs. 3 and 4. These data

were generated using the same fifty element model for a material

having Q = 0.0, 6 = 1.0 for the caSe '0 = 1.0. It should be noted

that a coupling parameter of the magnitude used represents a high

degree of thermomechanical coupling for metallic materials. The

influence of the non-linearitics introduced by the as and a13 terms

on the heat conduction equation must be transmitted through this

coupling term. The magnitudes of these terms in (45) - (48), which

when non-dimensionalized according to (49), are denoted As and A13.

Figure 3 shows variations in temperature at i = 1.0 as a function

of C for the four cases: Au = 0.05, A13 = 0.0; A6= 0.25, A13= 0.0;

As = 0.0, A13 = 0.05; A6 = 0.0, A13 = 0.05. Figure 4 depicts the

variation in displacement at t = 1.0 versus time for the same four

cases. Very little variation in the temperature is observed for

17

A_ = A 3 = 0.05. For the case A = 0.25 no significant departureo 1 ,;

from the linear theory occurs until , ~ 1.2. This is due to the

fact that this non-linearity manifests itsclf only in terms of

second order in the displacements and apparently requires a rather

large value for As in order to influencc the temperature equations.

The effect of the A13 is much more pronounced in the temperature

variations due to the fact that it appears in the elemental heat

conduction equations as well as in the equations of motion. The

temperature variations become apparent at , ~ 0.8 for A13

~ 0.25.

Noticeable deviations in the displacements occur at , = 1.0 for both

the At; and A13 terms and increase significantly with time. For

I••·

,I;;:,

is the term governing the rate of change in specific heat with

nonlinear. We note again that the effects of shear do not appear

(51)co

the purpose of illustrating quantitatively the effect of a tempera-

corresponding to '0 = 1.0. For simplicity) we assume the specific

ture-dependent specific heat on the response of the material.

Figures 5 and 6 display the effects of temperature dependent

temperature. Although this form of the specific heat was not used

material with 6 = 1.0 and As = A13 = a subjected to a ramp heating

in assessing the magnitude of the incremental temperature as des-

and is of the form

heat varies only with temperature changes at each individual node

where Co is the conventionally llsed constant specific heat and a

values of Af; amI A13 greater than 0.25 the material becomes highly

cribed in the derivation of Eqs. (45) - (48) , it is included for

coefficients of specific heat and thcrmal conductivity for a

in this examplc problem.

18

According to [ll] , the relation (51) accurately describes

the variation in specific heat with temperature obtained experi-

menta lly for both iron and a luminum between the range of 00 to

4000

centigrade. Using the value of Co for iron at 00 centigrade

[11] the corresponding value calculated in (51) for e had an in-

significant effect on the temperatures or displacements at Jl= l.O.

The results displaying the influence of S shown in Figs. 5 and 6

were obtained using the specific heat of iron at 00 centigrade

with 8 = 0.25 and B = O. I for E: = 0.0. For this case 13 = 0.1 ap-

proxi~ately doubles tIlemagnitude c and B = 0.25 increases c byn n

approximately 250 percent for a change in temperature of T = 1.0.

Similarly the value e calculated from experimental data (l2J for

iron has essentially no effect on the temperatures or displacements

at ~ = 1.0 for the case for the linearized material with B = O. The

cases shown in Figs. 5 and 6 correspond to € = - 0.1 and -0.25 which

represent a ten and twenty-five percent decrease in thermal conduc-

tivity respectively for T = 1.0. It should be noted that negative

values of E: are used because experimental data indicates that

thermal conductivity decreases with increases in temperature for

certain metallic materials such as iron [12J. The results indicate

that effective values of E: and B decrease the temperatures in the

material. These temperative effects are transmitted into the

equations of motion through the thermomechanical coupling para-

meter and tend to dampen the displacements at ~ = 1.0.

Figures 7-8 display the combined quantitative effects

of the nonlinear terms As and A13' and the temperature dependent

thermal conductivity and specific heat on the response of the

material to ramp heating with ~ = 1.0. The cases shown are:

·:1:

0.25; S = E:

19

0; An = A13 = 0; 3 = 0.25, € = -0.25;

A13 = 8 = 0.25, € -0.25. W~ note that experimental data

indicate that the values of lc; and A used in this study are very

unrcalistil: for metallic l1lat~rials. Also, the values used for

the An, and A ~ t~rms an' thought to indicate a much larger in-, 1 .:<

fluence on the response of a metallic material than actually

exists in nature.

Acknowledgement. The research reported in this paper was supported

through Contract F44620-69-C-0124 under Project Themis at the

University of Alabama Research Institute.

REFERENCES

1. Reiner, M., "Rheology," Encyclopedia of Physics, Vol VI,

Springer-Verlag, Berlin, 1958, p. 507.

2. Jindra, F., "Warmespannungen bci eincm nichtlinearen

Elastizitatsgeset," Ing~nieur Archiv, Vol. 38, 1959, p. 109.

3. Dillon, O. W., Jr., "A Nonlinear Thermoelasticity Theory,"

Journal of the Mechanics and Physics of Solids, Vol. 10, 1962,

pp. 123-131.

4. Nickell, R. E. and Sackman, J. J., "Approximate Solutions in

Linear, Coupled Thermoelasticity," Journal of Applied Mechanics,

Vol. 35, Series E, No.2, 1968, pp. 255-266.

5. Oden, J. T. and Kross, D. A., "Analysis of the General Coupled

Thermoelasticity Problems by the Finite Element Method," Pro-

ceedings, Second Conference on Matrix Methods in Structural

Mechanics, Air Force Fli~ht Dynamics Laboratory, (15-17 Octo-

ber, 1968), Wright-Patterson AFB, Ohio (in press).

'.I',II

20

6. Danilovskaya, V. 1., "On a Synamical Problem of Thermoelasticity"

(in Russian), Prikladnaya Hatcmatika i Mechanika, Vol. 16, Nu

3, 1962, pp. 341-344.

7, Sternberg, E. and Chakravorty, J. G., "On Inertia Effects in

a Transient Thermoelastic Prohlem," Journal of Applied Mechanics,

Vol. 26, No.4, TraTlH<lctions ASNE, Vol. 81, Series E, 1959,

pp. 503-509.

8. Oden, J. T., "A Genera 1 Theory of Finite Elements; II. Applica-

tions," Internal Journal [or Numerical Methods in Engineering,

Vol. 1, 1969 pp. 247-259.

9. Oden, J. T., "Finite Element Formulation of Problems of Finite

Deformation and Irreversible Thermodynamics of Nonlinear Continua-

A Survey and Extension of Recent Developments", Proceedings,

Japan-U.S. Seminar on Matrix Methods in Structural Analysis and

Design, Tokyo, August 1969.

10. Oden, J. T. anti Aguirre-Ramirez, G., "Formulation of General

Discrete Models of Thermomechanical Behavior of Materials with

Memory," International Journal of Solids and Structures, Novem-

ber, 1969 (to appear).

11. Hodgman, C. D., Weast, R. C., Selby, S. M., Handbook of Chemistry

and Physics, Chemical Rubber Publishing Co., Cleveland, Ohio, 1956.

12. Jakob, M., Heat Transfer, Vol. I, John Wiley and Sons, New York,

1962.

----' ..--

2.01.5

-<:5

-o--o--o-.o-.o-.o-..o-.o--O-<:1-<:J

.-<J.(J<1"

-<5-<5.0-6

lJ"o

.5

LI N EA R COUPLED - 8• 1.0LINEAR UNCOUPLED- 8=0.0--- ·EXACTU SOLUTION-8-1.0

UEXACT" SOLUTION-8=o.o

1.0

DIMENSIONLESS TIME CFigure 1. Temperature at 1= 1.0 linear coupled and uncoupled half space with :_ = 1.0,

.6

.7

.I

t2 .5....'•ICD

~.4::>ti0:lLI .3a.~lLI.... .2

• ~ - .-.-.. ' __ '..:~. __ • _., ..... 4 •••

.04

.06

o LINEAR COUPLED- 8= l.0o LINEAR UNCOUPLED- 8=0.0

--- uEXACTII SOLUTION

8 = 1.0U EXACT" SOLUTION-

8:1 0.0

2.01.51.0.05

.12

.14

.02

.10

I=>t- .08zw~UJU<{...J0-(f)-a

DIMENSIONLESS TIME tFigure 2. Displacement at 1= 1.0 in linear coupled and uncoupled half space \,.7ith ~- = 1.0.

'. _0. ~ ..... __ .'._4. . '.--.• _"_ .. .-..

.7o A =.25 A =A=c=O 8=1.013 '6 I"" ,~ ~3=·05,A6=,,8=E:=0,8=1.0o A6=.25'~3=(i=f.. =0,8=1.0

<> A6=D5,~=,B=€aO,8=I.O

o LINEAR COUPLED- 8=1.0

.6

.5

o.kIcb Aw(t:::::>~ .30::UJQ..~w .2to-

.I

oo .5 1.0 1.5 2.0 2.5

DIMENSIONLESS TIME ~3.0 3.5 4.0

Figure 1. !emperature at J' = 1.0 in nonlinear coupled half space with ~~ = l.O for various

va~ues ,i nonlinear material constants.

--~-,~-, -,=..'-- -- ,~, 4',

.10

o

I::>..- -.10zw:el1Ju«-' -.20a.ma

-.30

-AO

1.5

o A :a 25 A cA=€1I0 8-1013 . , 6 '/'"' ,.

Lt AI3=.05,A6

=/J=€lI:o,8=1.0

o A6-.25,A13 =p= E =0,8 =-1.0

¢ A6c·05'~3-,8c€.0,8-1.0o LINEAR COUPLED-8-LO

• ---. ---.- - - .... -.-.- - .. 4,.' .. ....... _ ...... _... •• _ .0 _~ ... , .... _.... ...... _ .....

3.0 4.0

. .....

DIMENSIONLESS TIME ~

Figure 4. Displacement at L ~ 1.0 in nonlinear coupled half space with :~ = 1,0 for various

values of nonlinear material constants.

.7

C>€ =.1 , A6=A'3=,8a 0,8= 1.0D €=.25,A6=AI3=fJ= 0,8= 1.0

013=·1 ,A6=AI3= E. =0,8=1.0

\1ft=.25,A6=Af3= € =0, 8 =!=1.0o LINEAR - COUPLED - 8=1.0

.6

.5

~P-I

I(l) .4w0:::>t-<{ .3a:=l1JQ.:!:~ .2

.I

o .5 1.0 1.5 2.0 2.5

DIMENSIONLESS TIME ~

3.0 3.5 4.0

Figure 5. Temperature at ,f = 1.0 in coupled ha If space having temperature dependent specific

heat and thermal conductivity with ~: ~. 1.0.

.... ._.~_ ••• _ •. ..... 4 •• _._ ...... _._ •••••••• , ._. .. __ •• _ •• _ ..... _ ••• ._--,---.-...

t> E:=.1 ,AsD AI3-,I..1l!E0, 8 = 1.0o (-.25,A aA -,8-0,8 -1.0

6 13o ,8-.I,A6-~3-€-0,8-1.0"V A-.25,A -A ·€-O 8-1.0

/- 6 13 •o LINEAR-COUPLED - 8-1.0

.10

o

I=>...z -.10w~l1.Io~-Jn. -:20(I)-a

-:-30

1.0 1.5 2.0 3.0 3.5

DIMENSIONLESS TIME ~

Figure 6. Displacement at J'= l.O in coupled half space having temperature dependent specific

heat and thermal conductivity with Cc = 1.0.

--. J ~

.6

.7

oo 4.03.53.01.0.5

b. A =A =.25 I1=Ez:0 8 =106 13 ,1./ , .

'r7 a = E: = 25 A =A = 0 8= I 0v /oJ •• , 6 13' .

o A6·~3 ,B=E: = .25,8 II: 1.0a LINEAR COUPLED- 8= /.0

1.5 2.0 2.5

DIMENSIONLESS TIME ~Figure 7. Temperature at I = 1.0 in coupled half spac.e having both material nonlinearities

and temperature dependent specific heat and thermal conductivity with ;~ = 1.0.

.1

.4

.3

.2

•I<DlLJQ::::>l-e:{Q:lLJQ.~111I-

1-0 .51-"

-- ---

.10

o

I:::>.-z -.101LJ~lJJUc:t-.J0.. -.20(/)

o

-:30

-.40

,.-.,.- -... ~

1.0 1.5

6. A6...~3=·25,ft= € =0,8= 1.0\l /3 =E:=.25,As =A13 = 0 ,8 = 1.0o A6=AI3=fJ=€=·25,,8z:1.0o LINEAR COUPLED-8= 1.0

DIMENSIONLESS TIME ~

Figure 8. Displacement at A'- 1.0 in coupled half space having both material nonlinearities and

temperature dependent specific heat and thermal conductivity with ,~ = 1.0.

~~ -=- -.J-=-=-~ =--x_~ .".~ ---- - .~~-_ ... - -~

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':;" u,I'.,jA riNG AC TIVITV (<':orpO'IJ'~ nuthorJ 2•• REPORT S£CUFHTV CL.ASSIFICATIOI.

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ON THE NUNERICAL SOLUTION OF A CLASS OF NONLINEAR PROBLEMS IN DYNAMIC COUPLEDnIER}10~ASTIGITYI If"ll'fll'fIVI "1."),t'5(TypttolrttllOrltlll,I'"CIIl,.}v"d"'fliIl)

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This paper concerns the application of the finite-element method to the

solution of certain nonlinear problems in thermoelasticity. Numerical solutions

of transient, coupled thermoelasticity problems involving bodies which exhibit

material nonlinearities and temperature-dependent thermal conductivity and

specific heat are presented. General equations of motion and heat conduction

of an arbitrary finite element are reduced so as to apply to the problem of

transient response of a nonlinear thermoelastic half space subjected to a time-

dependent temperature over its boundary.

I:1

DO IF.?o~M,,1473 UNCLASSIFIED .I

I