'.

II

FINITE ELEMENT APPROXIMATIONS IN NONLINEAR TH ERMOVISCOELASTICITY

J. T. Oden

1

FINITE ELEMENT APPROXIMATIONS IN NONLINEAR THERMOVlSCOELASTICITY

J. T. Oden*

Summary. It is the purpose of this lecture to describe the importantfeatures of general theories of thermomechanical behavior of materials, todemonstrate the construction of general finite-element models of thesetheories, to investigate special forms of these models which are appr~priate for the numerical analysis of nonlinear problems in thermovisco-elasticity, thermoelasticity, and heat conduction in solids, and to citenumerical results obtained by applying the finite element equations torepresentative nonlinear problems. The bulk of this lecture deals withso-called simple materials, particularly thermoviscoelastic materials,with consideration of thermomechanical dissipation, heat conduction, andcertain wave phenomena.

1. INTRODUCTION

It is a common subject of elementary physics that various types 9f

energy can be converted from one form to another. Every child is aware of

the fact that if two objects are rubbed together fast enough and long

enough, they get hotter, indicating a conversion of purely mechanical

energy into thermal energy or heat. Sometimes such conversions of energy

are manifested internally in materials; that is, the physical character of

the material is such that a great deal of heat can be generated by mechani-

cal working. Likewise, we may produce changes in mechanical energy (work)

by heating bodies; but if the same amount of heat generated in working a

body is applied to the body, it may not produce the same amount of mechad-

cal work. Then the process is an irreversible one, and energy is dissi-

pated in the system.

In the physical world, all of these phenomena take place over a finite

period of time. The "instant" recovery of an elastic body upon unloading

is a mathematical abstraction; the response of different materials to the

*Professor and Chairman, Department of Engineering Mechanics, The Universityof Alabama in Huntsville

2

same conditions generally takes place over different periods of time. The

physical properties of many materials are time-dependent, as is illustrated

by the continuing deformation of, say, concrete, certain plastics, metals,

and biological tissues under constant load. Moreover, the rate at which

such materials respond to given loads may be very sensitive to changes in

temperature.

While such thermomechanical phenomena are experienced frequently in

everyday life, their complete mathematical description has elluded scien-

tists for centuries. Only in recent times have rather general continuum

theories of thermoviscoelasticity emerged, and, in order to achieve suffi-

, cient generality to account for most of the phenomena mentioned above, these

theories are necessarily nonlinear and highly complicated. While general

theories are valuable in their own right for providing general and quali-

tative information on the character of certain classes of materials, they

are of little use in controlling, predicting, or studying specific thermo-

mechanical phenomena or in the design of specific machines and structures if

quantitative information cannot be extracted from them. It is toward this

goal, that is, the quantitative description of the nonlinear thermomechani-

cal behavior of materials, that the powerful concept of finite elements and

modern high-speed computers may find their most interesting and useful appli-

cations.

It is the purpose of this lecture to describe the important features of

general theories of thermomechanical behavior of materials, to demonstrate

the construction of general finite-element models of these theories, to in-

vestigate special forms of these models which are appropriate for the numeri-

cal analysis of nonlinear problems in thermoviscoelasticity, thermoelast~ty,

and heat conduction in solids, and to cite numerical results obtained by

,.3

applying the finite element equations to representative nonlinear problems.

Portions of this lecture are based on earlier articles and papers; the dis-

cussion of simple materials and the corresponding finite element models

follows closely that in Finite Elements of Nonlinear Continua [1], and some

of the numerical results presented are taken from pertinent papers [2,3].

The bulk of this lecture deals with so-called simple materials, particularly

thermoviscoelastic materials, with ample consideration of thermomechanical

dissipation, heat conduction, and certain wave phenomena.

2. THERMOMECHANICAL PRELIMINARIES

Kinematics. Consider a body B, the elements of which are material particles

X. We associate with each particle an ordered triple of real intrinsic par-

ticle labels Xi, i = 1,2,3, called intrinsic coordinates of X. Since this

association is one-to-one and onto, we use interchangeably X and! = (Xl,~,

~). Let C denote a subregion of three-dimensional Euclidean space, theo

points ~ = (X1,X2,X3) or which are in one-to-one correspondence with the

particles X of B. We regard C as the portion of E3 occupied by B at timeo

t = to (or T = t - to = 0) and we choose a correspondence ~:B ~ Co such that

the numbers Xl coincide with the coordinates xi. The motion of the body is

traced relative to the fixed reference configuration Co by

(2.1)

where ~ is the mapping that carries X in the reference configuration onto

the place ~ at time t. The deformation function X has the property ~(!,to)=

X (through a change of variables we may also write x = X(X,T) such that- - --X(!,O) =~. For each T'X(!,T) may define a different configuration of the

body. The configuration Ct corresponding to ~(!,t) is referred to as the

current configuration. In addition to (2.1), we introduce as the deforma-

tion gradient F of the motion ~ the second-order tensor

• •

F = ~X(X,t) = F(X,t)- ----. - --1 e

where ~ is the material gradient (i.e., ~ = £ ext).

4

(2.2)

If !t denotes a

system of basis vectors tangent to the material lines Xl in Co and ~ de-

notes a system of natural basis vectors tangent to the material lines Xl

of 6 while in C, then it is clear that

(2.3)

where F~I are the components of F. The tensor

G = F'F (2.4)

is called the Green-Saint Venant deformation tensor. Clearly, its com _.

ponents are

G = G • G = g F- Fn = x· X1J _l ~ •n • I •J ,1. ,J (2.5)

wherein commas denote differentiation with respect to the material co-

ordinates (x:1 = Ox-/exJ). The Green-Saint Venant deformation tensor is

positive definite and nonsingular; it relates the differentials dX1 to the

3square of a material line element ds according to

The Green-Saint Venant strain tensor 1is defined by

(2.6)

1Y = -(FTF - I) =2- __12(~ - I) (2.7)

where I is the unit tensor (i.e., ~ has components!t • ~ = g1J' glJ, 6~,

glJ being the material metric coefficients in Co), Thus, if ds is ao

material line element in C and ds is the same line element in C,o

· ". 5

ds~(2.8)

where the superimposed dot indicates B time-rate-of change (i.e. V\ J a

oytJ(!,t)/ot).

The vector

is the displacement vector of the particle X. The deformation gradient

F is given in terms of gradients of ~ by

H(X,t) - I--(2.10)

Consequently,

(2.11)

and

(2.12)

or, in component form,

1YtJ ="2 (ul;J+ uJ;l+ u~lul<;J)

-.!(' . k' k')Yl j - 2 u\;J+ uj,\+ U;lUk;j+ U;tUk;J(2. l3)

Here H denotes the (material) velocity gradients and the semi-colon denotes

covariant differentiation in Co'

Let !(~,t) denote a continuous function of X and a piecewise contin-

uous function of t. The functLon

• •

ft(s) == f(X,t-s)

6

(2.14 )

is called the total history of f at time t. It is often convenient to

represent the total history by the pair (ft,ft(O», where ft, called the.....r _ _

past history, is the restriction of f(!,t-s) to the open interval s€(O,~)

and ft(O) = f(~,t) is the current value of f at time t. The families of

tensors and vectors

(2.15 )

are called the total strain histories and displacement histories at X.

Similarly, yt(s),ut(s) are the past histories of y(X,t) and u(X,t).-:..r -..:r _ --- - -

Physical Laws. In addition to the kinematical relations just described,

we may postulate certain physical laws which govern the behavior of all

cuntinuous media. Let ~(!,t), S(X,t) denote the body force density (i.~,

the body force per unit mass) and S(X,t) the contact force per unit area- -of a material surface 06 in C. Let a = alJG ~ G denote the Piola-

- -\-J

Kirchhoff stress tensor and p~,t) denote the mass density of 6 in C.

Then, under sufficient smoothness conditions, the principles of conserva-

tion of mass, balance of linear momentum, and angular momentum can be

shown to hold at a particle! it and only if

Div (~) + p~ = p~ (2.16)

wherein G = det G and a is the acceleration field. In component form,- -(2.l6)~ appears as

. ,

(1l J + pb J = pa J: J

7

(2.l7a)

in which : denotes covariant differentiation in C. However, if we intro-

duce the stress tensor tlJC (11JJG, and interpret i<!,t) as tre body force

per unit mass in C , then (2.l7a) becomeso

(2. 17b)

which is often more convenient to use in applications. If ~ is a unit

normal to an element of area on the material surface of ~ while in C , ando

n is a unit normal to the same area while in C, we also have

(2.18)

where SJ and SJ are contravariant components of the surface tractiono

per unit initial and current surface area, respectively.

In addition to (2.16), we introduce the principle of conservation of

energy and the Clausius-Duhem inequality. Glooally, these take the forms

~+U=O+Q(2.19)

r 2: 0

where K, U, 0, and Q are the kinetic energy, the internal energy, the

mechanical power, and the heat energy respectively, and r is the total

rate of entropy production:

K =

0=

• vdu

::.du +If!. .A

vdA

•

Q " J Phdv + !L·n dA

u A

ndA

8

(2.20)

Here v is the velocity field, h is the heat supply density from internal

sources, 1is the heat flux vector, ~ is the internal entropy density and

a is the absolute temperature. If (2.16) holds, we may obtain, under

sufficient smoothness conditions, the following local forms of (2.19):

pe = w + ~ . 1+ ph

• 1pa~ - ~ • q - ph + - ~a • q ~ 0- - a--

Here w is the stress power,

Alternately, we may introduce the free energy density

w = € - ~a

and the internal dissipation,

and obtain the alternative equations

.pw z W - ~e - a

.pa~ = ~ • 1+ ph + a

(2.21)

(2.22 )

(2.23)

(2.24 )

(2.25)

In subsequent calculations, two alternate global forms are convenient

to use. Observing that Q = J (~ . 1+ ph)du, we introduce (2.21), into

u

• • 9

(2.19), and obtain

,; + f wdu • 0

u

(2.26)

Likewise, let T be a uniform reference temperature of B while in itso

reference configuration C and denote T(X,t) - 9(X,t) - T. Multiplyingo ~ _ 0

(2.25)~ by T and using some algebraic manipulations, we obtain

f (p9T~ + Y:.r

u

ydu = f (ph

u

+ oldu + f T'l.

A

ndA (2.27)

All of the relations established thusfar are insufficient in number

to solve specific problems in the thermomechanics of continua. We must

supplement these equations with a collection of constitutive equations

that characterize the material under consideration. We shall examine such

constitutive laws appropriate for simple materials in Section 4.

3. THERMOMECHANICS OF A FINITE ELEMENT

We now direct our attention to the problem of constructing general

finite-element models of continuous media. We follow here the gene~al

procedures outlined in previous work [1,4.5,6J.

Following the usual finite-element philosophy, we represent B by a

discrete model B consisting of a finite number E of pieces B , calledII

finite elements, which are connected continuous together at G preselectedE

material points called nodes. Then 6 = U B. Each element may be con-e-I II

sidered to be disjoint for the purpose or describing its local behavior.

Locally, we identify N particles in element a and label them oonsecu-II II

tively 1,2 •..••N ; the particle at node N is then XN, and if X6 is theII -1l

\ . 10

same particle in the connected model B, then the decomposition of the model

is accomplished by the simple Boolean mapping

(3.1)

tel _ leIwhere 06 = 1 if node N of Be is incident on node 6 of a and ~ a 0 if

otherwise. Likewise, for fixed e,

(3.2)

tf1l6 (elAN being the transpose of ~, and we say that (3.2) establishes the con-

nectivity of the model a. Further properties of these mappings can be

found elsewhere [1,7]. What is important in regard to our present study

is that individual elements can be isolated from B and their behavior can

be described independent ot their ultimate location or mode of connection

in B; indeed, once the behavior of a typical element is described, the

discrete model of the behavior of any global collection of such elements

is obtained by using the transformations (3.2) and (3.2).

With this fundamental property in mind, it suffices to confine our

attention to the local behavivr of a typical element, and to take for

granted that equations governing the global behavior of the collection of

elements can be generated by applying the transformations (3.1) and (3.2).

Consider, then, a typical finite element Be' and let ~ and 1~denote the

restrictions of u(X,t) and T(X,t) to B. Let__ - e

~N(t)(3.3)

denote the values of ~_, and T at node XN of B. Then we approximate_ leI - e

UtA and T) locally over B by functions of the form_) (e e

. - 11

(3.4)

where ~~(!)and ~(!)are local interpolation functions wnich have the

properties

1

0, X t Be

(3.5)

with similar properties holding for ~(~. Note that the repeated index

N in (3.4) is summed from 1 to N. Instead of (3.4), we could also repre-e

sent u~land T as linear combinations of the values of various derivatives- ~at the nodes; this is discussed elsewhere [1,7J, and since the procedure

is straightforward, we limit the present discussion to first-order repre-

sentations.

With the local displacement approximation and the local change in

temperature given by (3.4), we can now proceed to compute all of the other

kinematic variables and the absolute temperature:

2~J = u~(f=Jl + f:1J + f~Jrf~;ru~)

f~ J1

cl1j1N= aXJ 5~ - wNfj 1

.2y.el = (fll + fll )tiN + P fll'r (uNuM)

1 J NJ1 N1J II Njr M1 III n

EJeI (3.6)

12

etc., where the element identification label (e) is omitted in some places

for simplicity and the dependence of ~ '~N on X and uN, TN on t 18N,1 __

understood. In (3.6)a' r~1 is the Christoffel symbol of the seconu kind

corresponding to the undeformed material coordinates.

The stress power developed in element 6 ise

'tel = tr[~}

(3.7a)

or, if the Xl are cartesian in C ,o

(3.7b)

The velocity field is

(3.8)

Thus, introducing (3.4), (3.6), (3.7), and (3.8) into (2.26) and (2.27),

we obtain

(3.9)

Ue

and

[f[P{TO + 'P,T')'P.~ +.'!.. Y.'P.1du - q. - o,}T' = 0

U"

(3.10)

where mNM is the consistent mass matrix for the element, ~ is the gen-

eralized force at node N, qN is the generalized normal heat flux at node

N, ON is the generalized dissipation at node N, and u. is the volume of

6:e

13

m" • f pv, V,du, !'" • f p£.V,du + f ~V,dA

u u AII II II

q, " f ph<p,du + J -'l.' ~<p,dA (3.11)

u AII •

",· f "'l',dA

uII

If uN and TN are each linearly independent sets, (3.9) and (3.10)

must hold for arbitrary nodal velocities and temperatures. Consequently,

the quantities in braces in these equations must vanish. This leads to

the general equat~ons of motion of a finite element

and the general equations of heat conduction of a finite element

(3.12)

f [P(To + <PM T" )<p,~ + -'l. • ~<p,JdU

uII

(3.13)

To apply these models to specific problems, we must furnish appropriate

constitutive equations for a, ], and i (and a).

We remark that (3.12) is the finite-element analogue of Cauchy's

first law of motion (2.16)~, while (3.13) is the analogue of (2.25)~.

Equation (3.12) insures that the principle of balance or linear momentum

is satisfied in an average sense over a and (3.13) insures that energy•is conserved in an average sense over a. Notice that (3.12) and (3.13)

II

can also be obtained by multiplying (2.16)a by WN and (2.25)2 by ~N'

14

integrating over the volume of the element, and using the Green-Gauss

theorem. Then the procedure leading to (3.12) ana (3.13) is interpreted

as Galerkin's method.

It is alsu informative to consider the case in which all quantities

are referred to initial material volumes u ) and areas A ) in C and glJo~ o~ 0

61J' Then ue' Ae' a1J, ~, !' and p in (3.12) and (3.13) are replaced,

respectively, by U ....' A I.) , tl J, b, S , and p ; (3.12) then becomesalbl o,e _ -.Jl 0

(3.14 )

and

(3.15)

4, SIMPLE MATERIALS

A detailed discussion of cunstitutive theory is to be covered in

another lecture in this series; summary accounts are also available in a

number of books and monographs [1, 8, 9]. Rather than to consider a

variety of classes of materials, we shall concentrate on one which is

sufficiently broad to include most of the classical theories as special

cases: the theory of thermomechanically simple materials [10]. A material

is said to be simple if its response at particle X at time t is determined

by the history of the local deformation at X and the history of the temp-

erature at X at time t. Mathematically, a simple material is described

by a collection of four constitutive equations which define the free

energy, stress, entropy, and heat flux as functionals of the total histcry

15

of deformation and temperature and generally the current value of the

temperature gradient ~(t) = ~e(t). If we use the strain tensor y(X,t)--as a measure of the deformation, and if we decompose the total histories

yt(s), Qt(s), into past histories yt(s), et(s) and current values vt(O),- .....:..r r ...l-

et(O), then a simple material is described by the four functionals,

co

(] = ~ (~,e; ;'y,e)~Js=O

co

1) = J('(y;, e; ;1.' e,~l8=0

(4.1)

Here the dependence of W, ~, 'Tl~i' y, e, and [ on ! and t is understood.

The functionals in (4.1) must obey the principle of objectivity and they

must be consistent with the Clausius-Duhem inequality (2.21)1' We see that

this inequality can be written in the alternate form

w - Pw- p'Tl~+ l g • q ~ 0e __ (4.2)

Thus, to test if (4.1) satisfies (4.2), we must compute the time-rate-of

change of W. Since W is a functional of y, e, we have

+ a If [- ] • g&~=O

(4.3)

where, for simplicity in notation, we have used [-] = [yt, ~t; v, e] and..;.J' l' -!.

the vertical stroke indicates that the functional is linear in the argument

16

following the stroke. The operators 0 , On' D , De' 0 are Frechet andy 11 Y ~- -

partial differential operators defined by

00

& '1' [-Ih]Ys-o -

o 00

= Urn eo. ( '1' [t + a~, e~;l' e , gJ }a-t() s=o

00

= lim ~ ('1' [yt,et + as ;y,e,gJ}ua .:.J' r r _ _a-t() s=O

00

000 00

D '1' [-] • h = lim -- ( '1' [yt,et;y + ah,e,gJ}YO - -iOeo. O-u'r- --s= a s=

00 000De '1' [-] • S = lim eo. ( '1' [y;,e~;r,e + as,~.J}

s=O ~ s=O

00 000o '1' [-] • f = lim -- ( '1' [yt,et;y,e,g + afJ} (4.4)& - -iO Oa =0 -u' r - - -~=o a s-

Introducing (4.1) and (4.3) into (4.2) and arguing that the result must

hold for arbitrary rates y, ~, gJ we find that

00

o '1' [-1 = 0~=O

00

x [-J =s;;"O

00

pD '1' [- J.Ys=0

00

J('[-Js=O

00

-De '1' [-]

s=O

00 00

cr = -p(& '1' (-Iyt] + 6 '1' [-let])Y ~ e l'~=o s=O

(4.5)

That is, the

functionals

free energy

00

functional '1' [-J is independent of g, and 'the constitutive00 s=O -~~ [-] and ~C-J for stress and entropy are determined by the

s=O s=Ofunctional. Moreover, by Frechet differentiation, we can also deter-

00

mine the internal dissipation cr from '1' [-1.s=O

Special Forms of the Constitutive Equations. In trying to obtain specific

forms of the constitutive functionals for simple materials, it is natural

17

to first consider possible expansions of the free energy functional.(Xl

Let ot(s), ot(s) be the zero histories of yand q, and suppose that ~ [-]- s=o

is Frechet differentiable to admit a Taylor expansion about ~t(8), ot(s)

of the form

(Xl

\II Cot + yt ot + ~t.0 + Y 0 + ~] =1 _~ ...:.J' r r' ,0- - -s=

()) (Xl

(Xl (Xl (Xl

+ 6 e ~ C:~*Ie:] + D Ii' [~*J .i + De ~ [~*]es=o ~=o 8=0

(Xl (Xl

1+ 2~

(4.6)

where 0* = (ot,ot;o,O). Then, by truncating this series at various points,...... r_

we obtain constitutive functions for various simple materials.

For example, Achenbach, Vogel, andHerrmann [llJ proposed the free

energy functional obtained from (4.6) by retaining cubic terms in y, e,and their histories:

(Xl

{> = A + A 1 j Y + A 1 .I Y Tel+ A 1 J Y T + A 1 .I • n y yo 1 iJ 2 1.1 3 1.1 " 1.1.n

S=O

(4.7)

(Xl

Here ~ is the free-energy functional per unit undeformed volume, T =s=O

e - T is the change in temperature from a uniform constant referenceo

temperature T , ando

A =o

ro ro ro

<P [-] +.!. 6:a <P [-ITt Tt] +.!. O~ <P [-Iyt yt]2 T r'r 2 y ~'~S=O 8=0 ~=Oro ro

+ 00 iP (-Iyt Tt] +.!. 63 <P [-ITt Tt Tt]Y r ~'r 6 T r' r' r- s=O s=O

ro ro

+ .!. 020 ~ (-Iyt yt Tt] + 1 0 O~ ~ (-Iyt Tt Tt]2 y T ~'..:.J" r 2 y T -u-' r' r- s=O - s=O

18

(4.8a)

ro ro ro

A~J = 0 (~[-] + 0 iP [-Iy;] + or ~ [-IT~]YiJS=O 16=0 s=O

ro m

+ .!. 02~ [-ITt TtJ + 0 6 iP [-Iyt TtJ'2 T r' r y T -u-' r r

s=O - S=O

ro ro ro

A~J = 0 DT( ~ [-] + 0T ~ [-IT:' + 0 ~ [-IY;JJYlJ s=O s=O 1s=o

ro

= 1D D (iP (-12 Yt J Y. n 8=0

ro

+ Or ~ [-IT~])s=O

(4.8b)

(4.8c)

(4. 8d, e)

(4.8f)

ro

1 ro m+ - OCl ~ [-Iyt yt] + 0 6 ~ [-Iyt Tt]}2 Y _r'r yT _r'r"

-43=0 - s=O

1 ro ro ro

B:a = "2 ~( ~ [-] + 0T ~ (-IT~J + 0 ~ [-Ii])s=O 8=0 1s=0

1B = - 03 ~ [-J

3 6 Ts=O

(4.8g)

(4.8h)

(4.8i)

ro ro

wherein ~ [-I .. J == ~ [ot,o;;o,ol .. ], 0T = 0e- == oS' and DT = De_T == De's=0 s=0 ~ - -T 0

More specifically, we may introduce the strain- and temperature-dependent

integral coefficients

19

A c: a + R *"lry' + R *"*T' + R *T*T' + R *'my'*T'o .:.:.11 r" _l~ ro

~2 .:..:.112 .J."_

+ R *T*T'*T' + R *'mT'*T'3a a .::.l a a .r0

'

B = H + L'*T + LtiJ*y + L' *T*T' + L'iJan*y *y1 1 a 1 J 11 ~~ 1J .1\

+ L 11 J*y *T I1~ t J

1B3 ::0 '3 F1(0)

(4.9a)

(4. 9b)

(4.9c,d)

(4.ge,f)

(4.9g)

(4. 9h, 1)

Here ~1' ~2, •••,GiJ.n, M~J,•••, L1l' L~~·n,•••, F1 are material kernals

and * denotes a generalized convolution operator which, with an appro-

priate change-of-variables, indicates integrals of the form

~l*r-it t

" JlJIRfl··<t-.,.t- ••)Y,,<.,)o 0

(4.l0a)

(4.10b)

t

M~ 1 J*T = JI dMiJ (t-s)

d(t-s)o .

T(s)ds (4.10c)

etc. The lower limit of integration is assumed to be the time at which

the deformation is initiated.

20

For the constitutive equation for heat flux, we might use the law

proposed by Christensen and Naghdi [12]

00.r aT ,(sJQl = x.1 J (t-s) , ds

as

0

where x.1J(t-s) is a thermal conductivity kernal.

use the generalized Fourier law

00

Ql = x.lJ[yt Tt.y T]T...:.r' r' _, Js=O '

00

(4.11)

Alternately, we may

(4.12 )

is a functional of the indicated arguments. If x.1J (-]s-O

by a constant x.1J, (4.12) reduces to the classical Fourier

where x.1J[-]s=O

is replaced

law of heat conduction.

Thermorheologically Simple Materials. Experimental evidence obtained

from tests on a large class of viscoelastic materials has led to the iden-

tification of an important subclass of materials with memory, commonly

referred to as thermorheologically simple materials. This classification

arose from the observation that, among certain amorphous high polymers

which approximately obey established linear and nonlinear viscoelastic

laws at uniform temperature, are a group which exhibit a simple property

with a change of temperature: namely, a translational shift of various

material properties when plotted against the logarithm of time at differ-

ent uniform temperatures. The shift phenomenon is the basic character-

istic of all thermorheologically simple materials and makes it possible

to establish an equivalence relation between temperature and tn t. Now

in the case of a thermorheologically simple material, it is possible to

write the relaxation moduli at uniform temperature e, e.g., J(t-s), as a

function of tn t, denoted Ee(tn t). Then the shift property is apparent

21

if

Ee(.tn t) = Ee (.ent + f(e»o

where f(9) is a shift function relative to e such that df/dt > 0 ando

f(e ) = O. With this property, the relaxation modulus curve will shifto

toward shorter times with an increase in e. By introducing a shift

factor,

b(e) = exp f(e)

Then

Je(t) = Ee(.tn t) = Ee (p,n(tb(A»] = Je (~)o 0

where S is a reduced time given by

s = tb(e)

(4.13)

(4.14)

If it is possible to invert (4.14) so as to obtain t = g(~), then a

constitutive equation for stress can be obtained from the form

s= S J(~-~')

o

1;

1 J EX+ 3 0lJ K(g-g') og'

o

d~' (4.15 )

wherein Y;J and yare the deviatoric and dilitational components of YlJ

and J(t-s) and K(t-s) are relaxation moduli, and we have used the trans-

formations

That is, when transformed to a reduced time, the constitutive equations

for a thermorheologically simple material assume the same form as the

constitutive law for the isothermal case.

While the above development applies to the special case of linear

response of a body at various uniform temperatures, the basic ideas can

22

be extended to finite deformations of bodies subjected to transie'nt,

non-homogeneous temperatures. For example, if it is assumed that the

relaxation properties at a particle ~ depend only on the current tempera-

ture 9(!,t) at ~, it may be postulated that an increment 6~ in reduced

time is related to an increment 6t in real time according to 6~ =

b[S(!,t)]6t. Then, instead of (4.14), we have

t

~ - f b [e (~, ~') Jd ~ '

o

(4.16)

Clearly, (4.16) reduces to (4.15) in the case of uniform, constant tempera-

tures.

As a representative example of a constitutive functional for the

free energy of a thermorheologically simple material, consider the func-

tional proposed by Cost [13J:

S

+f D (~-s')1 .!

o

0"Yqar- d~'

s- f f (~-~') ~~,d~'

o

fsf~ OYt oy+ i 0l.!0an [3K(S-~',S-S") - 2G(~-S"S-~"] O~I.! os~n dS'dS"

o 0

s s "

SSOY1.! 09

- 01.! 3aK(~-S' ,s-s") ~ os"o 0

S S

~ S S m(~-~' .~-n~~,g~"d~'d~"

o 0

d SId S"

(4.17)

23

Here all quantities have been transformed to reduced time; i.e. Y1j(~'~) =

Ylj(!,g(S», ••• etc.(t = g(S». Also ~ = f(t), as given by (3.8), ~I =

f(t') and S" = f(t") are dummy variables in reduced time, qlo is a constant,

D1j ( ), K( , },G( , ), and m( , ) are material kernals, and a is the co-

efficient of thermal expansion.

Integrating (4.19) by parts and introducing the resulting equation

into (4.5) yields the following constitutive equations for stress, entropy,

and internal dissipation:

~

+ ~ 0" f [3K(~-n

os

0" f Q'3K(~-~')

o

(4.18)

~

p~ = f(O) + 3a!(O) + m(O)em f ~~, [3a!(~')]

o~

• YII (~-~')d~' - f ~~.[m(~·)]e(H')d~'

o

(4.19)

a =

0"[a3K(s-s' ,s-~t1)l 0~~1 ~~II d~'dS·

" "[m(~-~I ,~-s")] ~~I ~:" dS'd~"

24

(4.20)

In these equations 3K(S') = 3K(O,g'), m(~') ~ m(O,s') = 3~. 3K(0,S').

The material kernels 3K( ) and 2G( ) are relaxation bulk and shear moduli.

Thermoelastic Materials. If the free energy depends only upon the current

values of y and e, then

(4.21)

and (4.5)a and (4.5)3 reduce to

(1 = PDy'f(y, e) or (1lJ = P o'f(x.,e)°Y1J

T] = -De 'f(Y, 9) or T] == - o'f (~f.'e)(W1 J

(4.22 )

Materials obeying constitutive laws such as these are called thermo-

elastic materials.

5, FINITE ELEMENT MODELS OF SIMPLE MATERIALS

In view of (3.6), the components of the history of the strain tensor,

the temperature histories, and the temperature gradient for a typical

finite element Baree

2yt (s) = g • Ut (s) + g • U t (s) + u t (s) • Ut (s)1 J _1 _ J ~ -' 1 ~ 1 ...., J

(5.1)

25

Here rJ are the Christoffel symbols of the second kind for the system1r

Xl while in C , u~U(s) and T~~(s) are the total histories of the nodalo I

displacement components and temperatures. Thus, the arguments of the

functionals (4.5) can be written in terms of the histories ~~t)(s) and

TN(t)(s):

(Xl co(j = pD If [-] = rs [uN/.t)TN(t).u NTN]Y ....J' ' I' ' ,~=O s=O -

(Xl (Xl

11 = -De If [-J = lJ! [uN(t)TNW'uN TN] (5.2)....J' ' I' ,_ '

s""O s=Oco co

s.= Q [-] = ~ [UN~ TN~'uN TN]....J' ' r' ,

s~O s~O -

Likewise, the internal dissipation for the element is of the form

(j = (5.3)

Thus, the equations of motion and heat conduction for a finite ele-

ment of a thermomechanically simple material are

+ f ~IJ[-Js=O

u•d (Xl co

dt ~l [-] + <PN 1 131[-1}dus=O ' s=O

fcp ~ [-IuN~t)TNttl]duN ....J' ' r

s=OUe

(5.S)

where, for simplicity, we denote by [-] the arguments listed in (5.2).

In order to obtain specific finite-element models, it now becomes a simple

matter of introducing the appropriate constitutive equations into (5.2)

and (5.3) to determine the forms of the integrals in (5.4) and (5.5)

26

Thermorheologically Simple Elements. The reduction of (5.4) and (5.5)

for the case of thermorheologically simple materials requires special

considerations. To outline briefly the procedure, we consider the case

in which g1J = 61J, that is, the material coordinates X1 are assumed to

be initially cartesian. The following discussion is essentially a repro-

duction of that given by Oden and Armstrong [2].

Consider a class of thermorheologically simple materials for which

it is possible to transform all field quantities to corresponding quanti-

ties in reduced time; i.e., if f(X,t) is a function of real time t, a

function g(S) = t exists such that we can write

Then it is easily shown that

A,. ,.where f = df/d~ and b( J is the shift factor at X at reduced time ~ cor-

responding to t.

• •If 2Y1J = u1,J + uJ,1' pe~ ~ PTo~' and the constitutive equations

are of the form (4.18), (4.19), and (4.20), then it can be shown [1,2,14J

that the equations of motion and heat conduction for an element are G£ the

form~

b:.,J [3K(H')

os

c.;, J3K(~-n

o

OT d~' = P~kO~I (5.6)

27

~. J Nbl11 iN + h<,ah~N + H-3\ 2.... [m(~-~')] f, d~''" N "'!II 1 "'!II o~ S

os s S

+k,., J K(~-n ~~:d~'+cIJJ:1 JJ~~[3K(~-~"~-~")-2G(H'.H")l

o 00~ ~

oUL oUL Sl ( oUN oUL

)( os~ a~IJ,d~ld~" + d~~~ J~~[2G(~-~' ,~-~")] o~~ a~IJ,ds'd~"

o 0

JSJ~ aUN "-~)~L ~~ [3K(s-~' ,~_~")] a'~:~r,d~'d~"

o 0~ ~

-d~\" J J ~s [m(s-s' .~-~")l ~: ~~,ds'ds"00

where

~_ = jlPbOV,V.dU ~)- = J Pb~., V,duu u

II II

1r 'a~"'r = '2 b(1/I/04,jOlk + 1/I/04,lOJk] ~NJJdu

UII

1 - 1 .r- e.,. - a JlCb., .• ,dUbN/04k - 3 b~""k~N,ldu

u ue •and

h\\!, ~ Tom(O) JI "'_"',du k_. - J "',. I"'•• IdUu U

II "

'0\ - JI d &>l" =-1 J'" '" <Pt duliMN - 3OToK(0) CP",CPN,l u M NL 6 M N, 1 , j

U Ue II

(5.7)

(5.8a)

(5. 8b)

28

Also

(5.8c)

To further specialize (5.6) snd (5.7), Oden and Armatrong[2]

assumed that each material kernal was representable in the form of a de-

caying Prony series; e.g.

n

K(~) = LKl exp(-~h)t)

i=l

where Kt and vt

are experimentally-determined constants, ~ ~ 0, and

Vt

> O. Kernals of the type G(~,~') Ire assumed to have the property

G(~_~I ,l,;-F")= G(2S-l,;'_~")so that,

n

G(S-~',~-~') = LGtexp

1=1

(-2S+~'+S")/A 1f

(5.9)

etc., A1 being constants> O. Then, using Simpson's rule for approxi-

mating integration between reduced times ~ and S l' we represent certain• m+

integrals according to

SJ 2G(s-n

o

~NJ

oS' dS' (5.10a)

29

o.jN (~+h 12)J II III

4 ~S

~

f8+1 AN ~~NOUJ h OUJ

O~' exp(s' lA, )d~' ~ t 1 os (~8) exp(~m/A,) +

Sm OUN(~ +h ) !x exp [(~.+ha/2)/A,J + o~J 8 • eXP[(~m+h.)/A1~

Finally, incorporating such approximations into (5.6) and (5.7),

(5.l0b)

we obtain

for the equations of motion and heat conduction,

PA = ~u ~M + J:a) ~M + (a 1 _ b 1 )" G IN (u A)NJ NM J loiN J MNJ MNJ:L. k tic '

k

(5.11a)

+ kNMLEkJ~(T,€)k

in which

etc., and

r-l- AN ~ 1 AN AN) (!r 1\) 2 (AN AN )HG(uJ,At)" L.. (12(uJIiI+1)- UJ{m-l) exp ';)m 1\1 + 3' UJ{1lI+1f U.Ka)

m=l

(5.11b)

(5.l2a)

)(exp[ (s.+h./2) I A11 + f2 (3u~~ +1'- 4u~(.t U~III_1)exp[ (~.+h) 1 AlJ}(5 .12b)

CG(u~, \)

30

_ 1 (A AN) (rr I ) 2 (AN AN-12 UJtr+ll- UJ(r-l) exp -'=>1' ~I +"3 UJlr+1f uJll'»,

eXP[-(sr+hr/2)/\ ] + i2(3u~(r+l)-4u~\r)+u~(r_l»exP[-(~I'+hr)/A11

(5.12c)

In (5.12a) we have separated the histories HG(u~'~1) from the current

values CG(U~,Al)' that is, the value of the integral at the current time

~ + h is obtained by adding to the accumulated sum at S the contribution'=>1' r

between sand S +h.r r

uN(s -h), etc.J •

iAN AN AN(~ )By notat on u"."uJl.!I-1)etc. we mean uJ '=>. '

6. TEST CASES - NUMERICAL RESULTS

We now consider several applications of the theory presented

previously to the finite-element analysis of a number of representative

problems.

Torsional Oscillations of a Thermoviscoelastic Cylinder. As a first ex-

ample, we reproduce the results of Cost [13] on the finite-element

analysis of a solid circular cylindrical rod of a thermorheologically

simple material subjected to torsional oscillations about its longitudinal

axis. The free end of the cylinder is subjected to tangential displace-

1ments -u = a sin wt, a being a given amplitude and w a given frequency,r q 0 0

and r being the radial coordinate (x1,x2,x3) a (r,z,e). The only nonzero

strain component is then y~ = a r sin wt/L, L being the length of theOZ 0

cylinder. With the strain so specified as a function of time, we can con-

centrate on the phenomena of heat generation in the solid. The axis and

exterior surface of the rod are insulated, and the temperature at the end

of the rod is set equal to zero. In this case, the internal dissipation

31

(4.20) assumes the form

(6.1)

The kernal G(g,gl) is assumed to be of the form (5.9).

We assume that the material is a polyurethene rubber for which the

shift factor is such that

and

_ 8.89(8 - 89.6°f)- .6 + 183.5 + (B - 89~60F)

8

G ( t) = L G 1 exp ( - t / AI)

i=l

where AI = 1.0 X 10-13+1, i = 1,2, ...,7, As = 00' G = 1653 psi, 1 ' G =a

4747 psi, G3 = 5539 psi, G4 = 4701 psi, Gs = 3527 psi, Gs = 858 psi, G7=257 psi, and Ge = 143 psi. These properties correspond to the Thiokol

Chemical Co. rubber, Solithane 113 [13], for which ~ = 1.46 x 10-%F;

the coefficient of thermal conductivity is 8.56 x 10-3 BTU/in. hr. of, the

specific heat c 0.48 BTU/lb.m.oF, and p = 0.0361 1b.m./in.~.

The cylinder and a finite-element model of it consisting of composite

rectangular finite elements is shown in Fig. 1. In this case,a rectangu-

lar element is generated by connecting four triangles together. Thus,

over each triangle we use local interpolation functions of the form

(6.2)

Introducing (6.1) and (6.2) and the thermal and mechanical properties

mentioned into (3.10) [or an appropriately modified form of (5.11b)] leads

to the system of differential equations governing heat conduction in the

r

-1·5

~~0"0c:30

I" ~"0Q,)-0-~IIIC'-

insulated boundary.. I." N

Figure 1. Torsional oscillations of a thennoviscoelastic cylinder and 8

finite-element model of a quadrant of the cross-section of thecylinder.

32

element. These were integrated numerically, and the reduced time S at

r at time tn+1 was generated using the approximate quadrature formula

s(r,z,tn+1)

n

~L i (b[e(r,z,tt] + b[e(r,z,tt_1)]}(tt-tt_l)

i=l

(6.3)

\

wherein the.same time increment used in integrating the differential

equations governing the finite element model is employed.

In the case of convective heat transfer boundary conditions, the

heat flux ql at boundary is given by

qt a fl (1'* - To*) (6.4)

where T* is the temperature of the media surrounding the body, 1'* is theo

temperature across a surface boundary layer, and fl are film coefficients.

of (3.11)3 is of the form,Then the corresponding generalized heat flux

q = Jrn n f (rn TM*Pol T~ 1 t TM

Ae

- T*)dAo

(6.5)

Considering again the cylinder of Fig. 1, we prescribe convection

boundary conditions at the ends of the cylinder and the exterior of the

cylinder. Cost obtained film coefficients of f = 0.012 BTU/hr.-in.2-oF by

adjusting computed temperature distributions to fit those determined ex-

perimentally for the case in which the cylinder is rotated at 1000 cycles

per minute but under no strain. Computed and experimental results are

reproduced in Figs. 2 and 3. With the film coefficients so determined,

then torsional oscillations of 1000 cycles per minute at 5.34 percent

17'-'"

160

150

140

l.L.0 .4l

130...::7.-0...4l0-E4l 120I-

110

100

------------------ - ----- -4- ......

"'- ...time • 0 "'_..._

time • 30 min

- ...... - ...... "''''G). .. .... ........ ............. ........ .... ....

Finite Elemlnt Solution----(i)-----. Explr Imlntal

... ... ...... ......"', ...

90o 0'2 0'4 0,6 0,8 1·0 1,2 1,4

Radial Coordinate, In.

Figure 2. Temperature distribution in radial direction in a thermoviscoelasticcylinder (after Cost [13]).

Anite EIement Solution

'-0O-g0-8

.......... -.....

0'7

--- -0...

............0...

.... ........."

............. ....

0'6

--- ..... _--

....

0'5

- --- -- --

0-40-30-2

---------------------

0,'

---0---- Experimental.

time = 60 min

---time = 30 min

time =0----------- -------------

-----------

o

120

160

170

110

l1.o

Q)~::l 140-o~Q)

Q.

EQ)

~ 130

Ax ial Coordinate, in_

Figure 3. Temperature distribution in axial direction in a thermoviscoelastic cylinder (after Cost [13J.

33

strain were induced. The computed and measured temperature distributions

are shown in Fig. 4, together with the finite-element solution for the

case of temperature-independent material properties. In the latter case,

the predicted temperature generated is considerably higher than that

measured or computed using the complete thermoviscoelastic model.

Transient, Thermomechanically Coupled Phenomenon. As another test case,

we cite the recent results of Oden and Armstrong [2,14J on the transient,

nonlinear response of a hollow cylinder subjected to axisYmmetric

mechanical and thermal boundary conditions. Again, a thermorheologically

simple material with b(9) = 10~T, ~(T) -9T/IOO + T °C is considered. The

material properties are: p = 10-3in-lb/in °C sec., a = 10-6/0C, K(~) -

0.1 X 106 exp (-f)oo) psi, and, using the here.dity-type heat conduction law,

~(~) = .1 exp (-S/10-6) + 0.01 exp (-~/oo)in-lb./in.oC sec. The outer

radius of the cylinder is 11.0 in., and the inner radius is 10.0 in.

The complete nonlinear system (5.6) and (5.7) [(5.11)J is used in

this case for a system of twenty finite elements, each of 0.05 in. thic~

ness. The variation (in reduced time) of various quantities appearing

in the equations of motion and heat conduction we assumed to vary quad-

ratically in reduced time. For example, if, for an arbitrary function

the reduced time step, then quadratic interpolation in time yields the

difference approximations

(-4fl+ 3f 1+ f 1)/2h1 + \-

(6.6)

f1 - (fl+1- 2f\+ f1-1)/2h

I.L 940

4)~;:,...0~ 924)Q.E~

88

Isothermal Properties (Finite Element Solution)

_____ ~ -_ Experlml'ntal

Thermorheologlcally Simple MaterialFinite Element Solution

o 0.2 0.4 0.6 0.8 1.0

Nondimensionallzed Radial Coordinate, rIo

Figure 4. Comparison of compllted and IIlcaslJrcd tplllperatllres in a thermo-rheologically simple cylinder \llIdertorsional oscillations attime t = 30 minutes, 5.34 vercent strain (after Cost (131).

34

Thus, for each h, we encounter a system of nonlinear equations in fl+1'

The Newton-Raphson method was employed for solving these nonlinear equa-

tions, its starting values being obtained from difference approximations

of the initial conditions on uN and T~ All quantities (stresses, dis-\'

placements, temperatures, etc.) are computed as functions of reduced ti~

and then transformed to real time using fit= fi~/b[e(r,s)1; that is, the

reduced time t at step r + 1 is t + h/b[e(r,; ), where h a 6S is the\' I'

reduced time step. In this procedure, we must also account for the fact

that the stress tensor is associated with a shift factor that involves

the entire element temperature and not simply the temperature at a node.

We set the real time increment associated with nodal displacements and

temperatures at a node N in the connected model equal to h/bN, where bN

is the value of the shift factor at node N. In computing stresses, we

set the real time increment in an element e equal to h/b" where b isIe leI

an averaged shift factor for element e.

To initiate the Newton-Raphson iterations, initial conditions (e.g..uN(Sl) = ~N(Sl) = 0, TN(S1) = TN(Sl) = 0) are cast in central difference

approximations. Then, if h = fiSis the time step in reduced time, ~N

GN(;l+ h) = 2u~(;1+h)/h~; similar equationsUN(Sl+ h) = 2uN(Sl+h)/h and. ..hold for iN and TN. Taking

1800 ~N TN(Sl +h)

h[lOO + ~ fM (S1+h)]llOO + ~ fN (Sl+h) J (6.7)

where M, N, R = I, 2 and 7ff!= ae + be ( rl + na ) /2 (here we use the linear'f'M M M

approximations ~M = aN+ bNr, with ~M computed in certain integrations

using the average radius, r =(r1+ r~)/2), then we use as the shift factor

"NI "" 9 (p. T [100 + ~ T M] "" •b = 10 ~ M b = b tn'lO T (6.8)

3S

Equations (6.6), (6.7), and (6.8) are then used in calculations over the

second time step. Adequate data then exists for carrying the solution

forward tn (reduced) time. With these approximations, the mass matrices

of (S.8a) become

" 1.en (10) • T • mM "l (6.9)

Here r1

and r2 are the radial coordinates of nodes land 2 of the element

a ,b are the nodal coefficients appearing in the local interpolationN N

functions (~"l = aN + bNr; N = 1, 2). Specific forms of the remaining

matrices and the equations of motion and heat conduction for a typical

finite element are given in an Appendix.

As a first example, consider the case in which the model subjected

to a step-pressure loading of 10 pounds-per-square-inch (psi) on the

inner boundary. The outer boundary is stress-free, and the problem is

considered to be isothermal. Figure S presents the radi~l displaccments

as a function of real time. The stress components in the radial and

circumferential directions are presented in Figs. 6 and 7. It is seen

that all stresses are initially compressive in nature. This is a signifi-

cant difference from the conventional quasi-static solution which gives

compressive stress only in the radial direction. The quasi-static solu-

tion, obtained from the finite-element program by deleting inertia terms,

is compared with the exact solution of the quasi-static isothermal problem

in Figs. 8 and 9. It is also noted in Figs. 6 and 7 that the dynamic

circumferential and meridional stresses become tensile at later times. The

solution was not carried past 10-3 seconds, but it is expected that these

stress components would eventually reach values equal to the quasi-static

condition.

102 I TIME IN SECONDS

-3l= Ix 10

11.010.810.610.410.2

-.s::.uc:::--~ 164~~ t= Ix 164~u<t...J k ~ ~;r8X/05Cl.(/)

a..JS 105a<ta:

RADIUS (inch

?igure 5. Radial displacement profiles for dynamic isothermal casewith a step load of 10 psi at inner boundary.

-10

-8-II)

Q.-V)en -6UJa:.-en-oJ -41- \t«0ct

00)a:

\I-2

o10.0 10.2 10.4 10.6

RADIUS (inch)

TIME IN SECONDS

10.8 11.0

Figure 6. Radial stress profiles for dynamic isothermal case with a step load of 10 psi at innerboundary.

-8

-6.....-en~-(f) -4(/)

LLJa::I-00

...J -2~

.....zUJa::LLJu..~ 0::> I ~O.2 10.4 10.6ua:: RADIUS (inch)u

2

4

TIME 'N SECONDS

Figure 7. Circumferential stress ~rofl1e8 for dynamic isothermal case w~th a.step load of 10 psi atinner boundary.

FINITE ELEMENT SOLUTION-10

-8

-..a.-(I)(I) -6~Iii..Jctc -4<lQ:

-2

o10.0 10.2 10.4

- - - - EXACT SOLUTION

10.6

RADIUS (inch)

10.8 II. 0

Figure 8. Radial stress profiles for quasi-static isothermal case ,nth a step load of 10 psi at innerboundary.

100

IIIQ.

(I)(I)IJJQ:.....(I)

140

r " ,.." " " ~ ~ ,..

...J<{

.....zWD::

~~::>ua:u

~Ol- 0-0--0-0-0- FINITE ELEMENT SOLUTION

EXACT SCl.UTION

o10.0

I L10.2

J J f L10.4 10.6

RADIUS (inch)

, ..L10.8

1

Figur~. Circucfer·:mtial stress pro':iles ::or :r.:asi-static isother-:.al ::ase ·.:ith a ste'f. loac. 0;:" Ie l-'siat inner boundary.

36

The final example to be considered is a dynamic, coupled nonisothermal

problem with a step loading of 10 psi applied at the inner boundary while

the thermal boundary conditions correspond to convective heat transfer.

Film coefficients of 0.1 and 1.0 in. lb./in~ °C·sec. respectively are as-

sumed for the inner and outer boundaries of the cylinder. The heat supply

from internal sources is assumed to be zero and the initial temperature

of the cylinder and the surrounding medium are taken to be 300oK. Heat

is thus generated through thermomechanical coupling. Since energy is dis-

sipated in the system, the internal temperature increases during the tran-

sient part of the motion while heat is expelled at different rates at

each boundary. The displacements are presented in Fig. 10 and the radial

and circumferential stresses are shown in Figi. 11 and 12 compared with

the isothermal solution. The meridional stress is indicated in Fig. l3.

While stress and displacement profiles are obtained at different times

for the isothermal and nonisothermal cases owing to the necessity of

transforming all computed quantities to real time in the nonisothermal

case, results do indicate a lag in the nonisothermal stress waves and a

noticeable decrease in amplitude. Circumferential stresses are initially

compressive but become tensile almost everYwhere in the cylinder at 10-3

seconds. The circumferential stress wave requires approximately 2.5 x

10-s seconds to traverse the 1.0 inch thickness of the cylinder in the

isothermal case and approximately 4.0 x 10-sseconds in the nonisothermal

case. Displacement profiles are qualitatively the same for the isothermal

and nonisothermal cases. The nonisothermal displacements are higher than

those predicted by the isotlaermal analysis at the outer boundary, but

they are lower on the inner boundary. ThiH is due chiefly to the rclativl'

DYNAMIC ISOTHERMAL CASE

DYNAMIC NONISOTHERMALCASE

TIME IN SECONDS

- -- -0ta8xlO

-.cu

= -6- 10....zUJ~lJ.JU4:...Jn..(/)

a -7, 10...JctCi4:a:

\,. \

-8 t- \\ \ \ \ ~ \10 \01

U\

~I \ \ \~

'. ,m

I

-910 , , ,

10.0 10.2 10.4 10.6 10.8 11.0RADIUS ( Inch)

Figure 10. Radial displacenlfmt profiles isothcnllal and noniaothcrllisl

//.O/0.8

TIME IN SECONDS

10.6

DYNAMIC ISOTHERMAL CASE

---_ DYNAMIC NONISOTHERMAL CASE

10.410.20.0

10.0

ILla:::::>(f)

IDa::0..

...J<t~ -0.6w...~

"!-0+- h ~~~ \;- \("\ \\

~''Oel'

...J I \~, \O~ \-<t \i -0.2\

RADIUS (Inch)

Figure 11. Nondimensional radial stress profiles in a thermorheologically simple cylinder-isothermaland nonisothermal cases.

UJa:~(f)(f)UJcr:Cl.

...J«zcr:UJ....Z

"(/)(f)UJcr:....(f)

...J~....zUJcr:UJ11..~~ua:o

-0.6

-0.4

0.2

DYNAMIC ISOTHERMAL CASE

-- -- DYNAMIC NONISOTHERMAL CASETI ME IN SECONDS

10.6

RADIUS (inch)

Figure 12. Nondimensional circumferential stress profiles in a thermorheologically simple cylinder-isothermalaDd nonisothermal cases

lit0--(f)(f)IJJa::~(f)

-.l~z -40Q

a::UJ~

-2

o10.0 10.2 10.4

TIME IN SECONDS

\0.6 10.8 II. 0

RADIUS (inch)

Figure 13. Meridional stress profiles for dynamic nonisothermal case with a step load of 10 psi at innerboundary.

37

magnitudes of the film coefficients at these boundaries. The results

indicated in all examples were computed using a reduced ti~e increment

of 10-6 seconds; rather rapid convergence of the Newton-Raphson scheme

was observed at each time step.

The temperature distribution is shown in Fig. 14. Note that the

mechanically loaded inner boundary is the first to experience an increase

in temperature. It is seen that the temperature change reaches maximum

values for the material parameters and boundary conditions considered, at

approximately 4 x 10-6 seconds. Values for times greater than 8 x 10-6

were not obtained.

II. 010.810.4 10.6RADIUS (inch)

10.2

0.0 -10.0

1.0

rTIME I N SECONDS

I0.8

u0-wl!)z1 0.6uw0:::J

~ 0.40::wQ..

~.....

0.2

Figure 14. Radial tewperature profiles :or dynamic nonisothermal case \.1th a step load oi 10 psi atinner boundary.

· ..38

7. REFERENCES

1. Oden, J. T., Finite Elements of Nonlinear Continua, McGraw-Htl1,New York, 1971.

2. aden, J. T. and Armstrong, W. H., "Analysis of Nonlinear Dynamic,Coupled Thermoviscoelasticity Problems by the Finite Element Method,"Journal of Computers and Structures, Vol. 1, No.1, 1971.

3. aden, J. T. and Poe, J. W., "On the Numerical Solution of a Class ofProblems in Dynamic Coupled Thermoelasticity," Developments inTheoretical and Applied Mechanics, V,(Proceedings, 5th SECTAM, April1970), Edited by D. Frederick, Pergamon Press, Oxford).

4. Oden, J. T. and Aguirre-Ramirez, G., "Formulation of General DiscreteModels of Thermomechanical Behavior of Materials with Memory,"International Journal of Solids and Structures, Vol. 5, No. 10, pp.1077-1093, 1969.

5. Oden, J. T., "A Generalization of the Finite Element Concept and itsApplication to a Class of Problems in Nonlinear Viscoelasticity,"Developments in Theoretical and Applied Mechanics, IV, Edited byD. Friederick, (Proceedings, '4th SECTAM, March 1968), Pergamon Press,Oxford, pp. 581-593, 1970.

6. aden, J. T., "Finite Element Formulation of Problems of Finite De-formation and Irreversible Thermodynamics of Nonlinear Continua,"Recent Advances in Matrix Methods of Structural Analysis and Desi~,(Proceedings, U.S.-Japan Seminar on Matrix Methods in StructuralAnalysis and Design, Tokyo, 1969), Edited by R. H. Gallagher, Y.Yamada, and J. T. aden, University of Alabama Press, Tuscaloosa, 1970.

7. aden, J. T., "A General Theory of Finite Elements.Considerations," International Journal of NumericalEngineering, Vol. 1, No.2, pp. 205-221, 1969.

I TopologicalMethods in

8. Truesdell, C. and Noll, W., "The Nonlinear Field Theories of Mechan:lc8,"Encyclopedia of Physics, Vol. 111/3, Springer-Verlag, New York, 1965.

9. Eringcn, A. C., Nonlinear Theory of Continuous Media, McGraw-Hill,New York, 1962.

lO. Coleman, B. D., "Thermodynamics of Materials with Memory," Archivesfor Rational Mechanics and Analysis, VoL. l7, pp. 1-46, 1964.

11. Achenbach, J. D., Vogel, S. M., and Herrmann, G•• "On Stress Waves inViscoelastic:Media Conducting Heat," Irreversible Aspects of ContinuumMechanics and Transfer of Physical Characteristics in Moving Fluids,Edited by H. Parkus and L. I. Sedov, Springer-Verlag, New York,pp. 1-15, 1968.

· .' .39

]2. Christensen, R. M. and Naghdi, P. M., Linear Non-IsothermalViscoelastic Solids," Acta Mechanica, Vol. III/I, pp. 1-12, 1967.

13. Cost, T. L., "Thermomechanical Coupling Phenomena in Non-IsothermalViscoelastic Solids," Ph.D. Thesis, University of Alabama, University,1969.

14. Armstrong,W. H., "Analysis of Nonlinear ThermoviscoelasticityProblems by the Finite Element Method," M. S. Thesis, University ofAlabama in Huntsville, Huntsville, 1971.

8. APPENDIX - EQUATIONS OF MOTION ANDHEAT CONDUCTION OF CYLINDRICAL ELEMENTS

We present here explicit forms of the equations of motion and heat

conduction for finite elements of a thermorheologically simple material,

obeying (4.17) - (4.20). These are special cases of (5.6) and (5.7)

for the case in which ~ and T are assumed to vary linearly in the radial

coordinate r and quadratically in time. For an element joining nodes 1

and 2 of a finite element model, let

b = -l/L1

(A. 1)

where L = r2- r1• Then the equations of motion of a finite element are

of the form

M,N = 1,2

(A.2)

,. ". .,40

where MMN is the term in brackets in (6.9a) and

)(~~+h) - a( )(~.)J

)(~.+h) - 4a( )(~.) + 8( )(~.- h)J)

(A.3a)

e -~t/81eC( )a( )Sk) = 12 [a( )(~.e+ h) - a( )(~rh)]

2e - (~t+·6h )Sk+"'3 [a( )(St+ h) - a( )(~J,)]

-(~ +h)/Be t k[ ]+ 12 3a( )(~t+h) - 4a( )(~t)+ a( )(St- h)

(A.3b)

The equations of heat conduction for an element are

S K- ~[~ <e-(~t+h)/\lk'HK<TN,\Jk) + CK('fN,vk»]J

L... \l1ck=;:l

s ~ ~ ~

L 3UN(~J,+ h) - 4uN(~t) + uN(~t- h)+ QI'T 'D2 • [ K· [o M ~ Ie 2h

k=l

S KL:t< . (e-(SJ,+h)/\lkoHK(UN,\lIe) + CK(uN,vlc»l}

k=lr(2: [E~(e-(St+h)/€P.HE<TN'€p) + CE(TN'€p»lJ

p=l

, ..' .,

+ 2e-(~t+h)/AI'HG(UN,Al)'CG(Gl,Al) + CG(UN,At'CG(Gl,At»JJ

s'\:" Kk - (II' +h)/V -,. -,.

D5MNl'(L.,[Vk

.(e a ....t k.HK(TN,vlc)·HK(Tl,vk)

k=l .

+ 2e-(St+h)/Vk.HK('fN,VIc)·CK('fl'VIc) + CK(TL,VIc)'CK(Tl,Vk»)}

S K- D6"'NL'· (~[Vk (e-~(St+h)/Vk'HK(GN'Vk)'HK<Tl ,Vic)Lk

k=l

+ e - ( St+h ) / V1<•HK (GN' Vic) • CK<TL ' Vic)

+ e - (~t+ h) / V~ •HKd N' Vic) • CK(uL ' Vic)

41

K, L, M, N ~ 1, 2 (A. 4)

where HK, CK, HG, CG, HE, and CE are defined by (A.3) and

.. ~ . .42

(A.5)