.'

FINITE ELEMENT FORMULA nON OF PROBLEMS

OF FINITE DEFORMATION AND IRREVERSIBLE

THERMODYNAMICS OF NONLINEAR CONTINUA

- A SURVEY AND EXTENSION OF RECENT

DEVELOPMENTS

J. T. Oden

Professor of Engineering Mechanics, ResearchInstitute, University of Alabama in Huntsville

Huntsville, Alabama U.S.A.

JAPAN - U.S. SEMINAR ON MATRIX METHODS

OF STRUCTURAL ANALYSIS AND DESIGN

TOKYO, JAPAN

August 25-30, 1969

3b

FINITE ELEMENT FORMULA nON OF PROBLEMS OF

FINITE DEFORMATION AND IRREVERSIBLE THERMO-

DYNAMICS OF NONLINEAR CONTINUA - A SURVEY

AND EXTENSION OF RECENT DEVELOPMENTS

J. T. Oden*

University of Alabama in Huntsville

Abstract. This paper presents general finite element models of nonlinear thermomechanical

behavior of continuous media. Both spatial and material forms of the finite element equa-

." tions of motion and heat conduction are derived on the basis of local approximations of the

displacement, velocity, density, and temperature fields. Finite element formulations of a

variety of problems are considered as special cases. These include problems in finite elas-

ticity, nonlinear thermoviscoelasticity,heat conduction, and flow of incompressible and

compressible viscous fluids, among others. Incremental forms for the case of large deforma-

tions of elastic solids are also examined.

1. INTRODUCTION

Mechanical action of a continuous body does not manifest itself in mechanical effects

alone. When the external forces acting on a body perform work, the body gets hotter. Con-

versely, supplying heat to a body results in motion. If the process is reversed, not all of the

mechanical energy suppl ied is general Iy recoverable. The process is then an irreversible one,

and, as such, its description may fall well outside the realm of both classical mechanics and

classical thermodynamics. Indeed, al though the interconvertibil ity of mechanical work and

heat was recognized in the nineteenth century by Joule, works on thermodynamics have, unti Irecent years, treated the bulk of thermodynamic processes both as reversible and as completel y

uncoupled with the mechanical behavior. Likewise, classical solid and fluid mechanics, as a

whole, ignore the fact that viscous behavior, coupled heat conduction, and plasticity belong

to the realm of irreversible thermodynamics; and many phenomena associated with continuous

bodies that have traditionally been treated with reasonable accuracy as reversible, cease to be

so when finite deformations are taken into account (e. g., finite deformations of metals).

*Professor of Engineering Mechanics

- 1 -

Although the development of a rational theory of thermodynamics of continuous

media has lagged the development of the more familiar, special theories by over a

century, a significant step toward a general thermodynamics of continuous media was

made in 1964 by Coleman [1], who presented a consistent and rational theory applicable

to simple materials with memory. More recently, additional refinements of Coleman1s

theory have been discussed by Laws [2], Truesdell and Noll [3J, Truesdell [4,5], and

Muller [6J, among others. Although much experimental work remains to be done, it may

be said that sufficient basis of the theory is now available to apply it to a wide class of

problems involving nonl inear behavior of continuous media.

Applications of modern theories of irreversible thermodynamics to realistic physical

problems very quickly exceed the limits of classical methods of analysis. The mathematical

analysis of the thermomechanical behavior of continuous bodies of arbitrary shape and general

boundary and initial conditions, compl icated by the presence of finite deformations and

time-dependent material properties, constitutes one of the most difficult classes of problems

in applied physics. It appears that only through the use of modern numerical techniques is

there hope of obtaining quantitative solutions to problems of this type.

Toward this end, several applications of the finite element concept to the development

of discrete models of non' inear behavior of continua have been made. Following the pio-

neering paper of Turner, Clough, Martin, and Topp [7), applications of the method to simple,

geometrically nonlinear problems were discussed by Turner, Dill, Martin, and Melosh (8),

among others. Summary accounts of appl ications to geometrical Iy nonlinear problems invol ving

infinitesimal strains were given by Martin [9,10J, Argyris [11, 12J, Zienkiewicz [13J,

Przemieniecki [14J, and Marcal [15J, among others, and several solutions to problems in

classical thermoelasticity and elasto-plasticity have been presented (e.g., by Gallagher,

Padlog, and Bijlaard [16], Akyuz and Merwin [17], Wilson [18], Felippa [19], Pope [20],

Marcal and King [21], Zienkiewicz, Valliappan, and King [22], and Yamada, Yoshimura,

and Sakurai [23 J). Finite element formulations of problems of finite deformation of elastic and

thermoelastic solids have been given by Oden et al [24-30] and Becker [31]. Applications to

nonl inear viscoelasticity [32-34], heat conduction and coupled thermoelasticity [35-37], and

potential flow [38-40J have also been presented. There does not appear to be available attempts

at developing discrete models of general, thermomechanical behavior of both solids and fluids.

- 2 -

'.

.~

.'

In the present paper, we consider the problem of constructing general finite element

models of finite deformation and irreversible thermodynamics of nonlinear continua. A

major objective of the developments presented here is general ity, in the sense that the

resulting formulations represent discrete models of a wide range of problems in both

solid and fluid mechanics. To achieve this, both spatial and material descriptions of the

local displacement, velocity, temperature, and density fields are introduced: the former

depicting the finite element as a subregion of three-dimensional Euclidean space through

which the continuum flows, and the latter depicting the element as a material collection

of particles specified in some reference configuration. Since most of the primary dependent

variables are kinematic in nature, it is shown that the spatial forms of the finite element

equations are complicated by the presence of nonlinear convective terms. Conversely, the

material description is complicated by the fact that forces and heat fluxes are applied on

a material surface in the deformed element.

Once the basic kinematical equations for finite elements have been developed, we

examine the essential aspects of continuum thermodynamics. By considering balances of

energy for typical finite elements, we obtain spc:1tiClIand material forms of the general

equations of motion and heat conduction of finite elements. In the case of compressible

fluids, we show that the equations of motion must be supplemented by finite-element ana-

logues of the spatial form of the equation of continuity, the mass density p being an ad-

ditional unknown in the problem.

We then examine finite element formulations of a number of special cases that can be

obtained from the general finite element equations. These include thermomechanically

simple materials with memory, finite coupled thermoelasticity, finite elasticity, dynamic

coupled thermoelasticity, classical elasticity, transient heat conduction, coupled thermo-

viscoelasticity, and compressible and incompressible Stokesian fluids. The latter equations

represent finite-element models of the Navier-Stokes equations of fluid dynamics.

We conclude the investi gation by presenting incremental forms of the finite-element

equations for the special case of finite displacements of elastic sol ids. We show that these

exhibit the well-known initial stress and displacement matrices used in the stability and non-

linear analysis of elastic and elasto-plastic structures.

- 3 -

2. KINEMATIC PRELIMINARIES

We consider the motion of a continuous body. To identify the configuration of the body

at a given time t we assign to its particles the labels Xi (i = 1,2,3). The quantities Xi

are referred to as intrinsic or convected coordinates, and we interpret them as being etched

onto the body and to move with it as the body deforms. The motion of the body is a con-

tinuous one-parameter family of configurations C t, and the parameter T is associated with

time. We shall refer to the configuration Co corresponding to T = 0 as the reference

configuration. The confi guration at T = t is denoted C t and is termed the current confi g-

uration.

For simpl icity, we shall assume that when the body occupies its reference configura-

tion Co, the intrinsic particle labels Xi are rectangular cartesian. However, at all other

times T > 0 the coordinate lines Xi will be, in general, curvilinear. When the body is in

its reference configuration, we also establish a spatial reference frame Xi which is rectangu-

lar cartesian and which, again for simpl icity, coincides with X i at T = 0 . We regard the

frame Xi as inertial; that is, it is absolutely fixed in space (alternately, we may give the

Xi any time-dependent rigid translation relative to Co), It is important to realize that the

numbers X i identify a particle, whereas the numbers Xi identify oJ place in the three-climen-

sional space through which the body moves.

Mathematically, we can describe the motion of a particle X 1 by equations of the form

(1)

Equation (1) defines the position of a particle at time t in terms of its position in the refer-

ence configuration. The components of displacement Ui of a particle originally at Xi are

given by

Ut=Xi-Xi

The components of deformation gradient, denoted Fi j' are defined by

i _ OXi _F j - oX j - U 1 , j + ()i j

where the comma denotes partial differentiation with respect to Xj (i.e., uilj= Oui/

oXj) and 01j is the Kronecker delta.

- 4 -

(2)

(3)

A measure of the degree of deformation at a point in the body is provided by the

square of the material element of arc ds2 in the deformed body;

(4)

where GI J is Green's deformation tensor:

(5)

For our purposes, it is often more convenient to use the Green - Saint Venant strain tensor

(6)

." Since the same label Xl identifies a given particle at all times, forms of measures of

the rates of deformation are particularly simple when the present description of motion is

used. Components of velocity are

and rates-of-strain are given by

• _OY1J_lV1J -~-2(vllJ +vJI1 +Uk,lVk'J +Vk,lUk'J)

(7)

(8)

(11)

(10)

Higher-order strain rates are obtained by repeated partial differentiations with respect to

time.

Eulerian Forms. In the study of the motion of fluids it is usually meaningless to trace the

motion of a specific particle. Motion is then described by establishing a fixed reference

frame (the Xl) and viewing the motion of the continuum through points in three-dimensional

space. Then we use the Almansi<auchy strain tensor

1(au 1 Ou J Ou k Cluk )e 1 = - - + - - - - (9)J 2 ClxJ ClXl Ox1 ClXJ

Since we now write all quantities as functions of x 1 rather than X 1, time rates be-

come more complicated:

Clu1Vl=Vllx)=-~ Clt

etc.

- 5 -

3. MOTION OF A FINITE ELEMENT

We shall now consider a discrete model of the continuum which consists of a col-

lection of a finite number of finite elements connected appropriately together at various

nodal points. Since the process of connecting elements together to form such a model is

well-documented and is based on purely topological properties of the model, we shall not

elaborate on it here. It suffices to point out that two distinct types of finite element models

can be-developed for continuous media: 1) material models and 2) spatial models. The

material model is viewed as an actual collection of physical bodies connected continuously

together at nodal points. The nodal points themselves are materia' particles Xi. In the

spatial model, the space through which the media moves is regarded as a connected set of

subregions, each representing a finite element. Nodal points then represent places Xi

in the three-dimensional Euclidean space. In either case, a finite element is regarded as

a subdomain of a field quantity: in the material description, the domain is a specified

collection of particles, and, in the spatial case, the domain is a subregion of £3. It fol-

lows that the material model is used for material descriptions of motion and the spatial

model is used for spatial descriptions. Moreover, in either case, typical finite elements

can be isolated from the collection and appropriate fields can be described local! y over an

element, independent of the ultimate location of the element in the connected model. This

latter property has been referred to as the fundamental property of finite element models

[41].

Material Model. We consider a typical finite element e of a continuum, which, for our

present purposes, we regard as a subdomain of the displacement field Ui (and later, the

temperature field T). A finite number Nil of material particles are identified in the element

while in its reference configuration. These are called nodes and are further labeled X~,

where N = 1,2, ... , Ne• The values of the displacement field at the nodes are then

'.

(12)

Following the usual procedure, we now approximate the displacement field locally over the

element by functions of the form

(13)

- 6 -

where here (and henceforth) the repeated nodal index is summed from 1 to Ne• The inter-

polation functions ~N(~9have the property that

(14)

Moreover, 'fN(~)~ 0 for all particles ~belonging to the element, and (13) is designed so

that the local fields ulare uniquelydefined over the element in tenns of the nodal values

UN1' It is understood in (13) that UNI are functions of time.

With the displacement field approximated, it is now a simple matter to calculate all

other quantities needed to define the motion of the element:

(15)

and so forth.

Spatial Model. For the spatial model, we consider a bounded subdomain of e}' in which

we identify a number Ne of places called nodes and labeled xiN' Although it is a simple

matter to construct spatial approximations of the displacement field, it is, for our purposes,

more natural to begin with approximations of the velocity field VI:

(16)

where VN1 = vd~.N)are the values of the local velocity field at the nodal places x~ and the

interpolation functions .N(~ have the same properties described in (14) except that they are

cast in terms of the spatial coordinates Xl .

(17)

- 7 -

etc., where at is the ith component of the acceleration. Notice that the convective terms

(e.g. VIIOV1/~XIll' etc.) complicate the Eulerian description of the motion of a finite element.

Temperature Fields. In the study of thermomechanical phenomena, finite element models of

the temperature field are also needed. Let e denote the absolute temperature and To denote

a uniform reference temperature that is independent of time. Then e is written as the sum

of To and T, where T is the change in temperature.

For the local approximation of T over a finite element, we have for a material model

( 18)

or, for a spatial model,

(19)

where the functions ,N are identical to those described earlier and TN = T (X N)or T(xN)are

the nodal temperatures. Then

e = To + ~NTN

Time rates of change of temperature are then, with TH = dT N/dt,

or

for the material and spatial finite element models respectively.

4. THERMODYNAMIC RELATIONS

In the following developments, we shall largely confine our attention to material

descriptions of the deformation of a continuous body. Spatial forms will then be stated

without detai led deri votions.

(20)

(21a)

(21 b)

The thermomechanical behavior of an arbitrary continuum is governed by five physical

laws: 1) the conservation mass, 2) balanee of linear momentum C1nd, 3) ~ngullllr momentum, 4)the

conservation of energy, and 5) the Clausius-Duhem inequality. For material descriptions,

mass and angular momentum are usually (but not always [42J) assumed to be satisfied a priori

for a finite element. Linear momentum is balanced globally for afinite element,but is balanced

- 8 -

only in an average sense throughout an element. Local and global forms of the law of con-

servation of energy provide a natural means for developing the equations governing the motion

of a typical element, while the Clausius-Duhem inequal ity provides bounds on the entropy

production in an element as well as restrictions on the nature of the constitutive equations

describing the material of which the element is composed.

Consider a materia' volume V' of mass density p bounded by a surface of area A. The

global form of the law of conservation of energy for this volume is, considering only thermo-

mechanical behavior,.K+ U =O+Q

where K is the kinetic energy, U the internal energy, 0 is the mechanical power and G

is the heat energy:

(22)

(23)

Q =1'A

Here we have referred all quantities to the reference configuration Co: p is the m.,:!ssper

unit volume V in the reference configuration, ~ is the internal energy density, F1 are the

body forces per unit mass in the "undeformed" body, and h is the heat supply perunit mass

in Co' The quantities S 1 and qt are components of surface traction and heat flux per unit

initial area referred to convected coordinates Xl in the current configuration. These are un-

avoidably available only after some motion of the body takes place and are, therefore,

functions of the displacement gradients u t, j' We explore this problem more thoroughly later.

If mass is conserved and the principle of balance of linear momentum is assumed to

hold, then, provided certain continuity requirements are satisfied, it can be shown that the

local form of the law of conservation of energy is

(24)

- 9 -

•

where 0'1 j are the contravariant components of stress per unit deformed area referred to the

intrinsic coordinates Xt, and the semi-colon denotes covariant differentiation with respect

to the Xl.

Upon introducing (24) into (22) and making use of the Green-Gauss theorem, we

obtain the alternate global form

fu ,u,d'" + f' 'V"dV = (1 (25)

Entropy, Free Energy, Dissipation. TheV"entropycontent of a body is defined by introducing

the specific entropy Tj. The total entropy production r is then

r = f~d1Y 1;.!!.dA - fp~d1Y (26)

V 'A 'V

where e is the absolute temperature. According to the Clausius-Duhem inequality, r~O,or locall y,

It is convenient to introduce the Helmholtz free energy density cp and the internal

dissipation 0':

(27)

cp = e - Tje (28)

Then1 1O'+eq er1~O

and (24) can be written in the aiteroote forms

• t· • ~pcp = " •Y 1 j - pTje - 0'

pe~ = q ~t + ph + 0'

(29)

(30)

(31 )

Another useful form of the energy balance is obtained from (31) after multiplying both

sides by the temperature increment T:

(32)

- 10 -

(33)

5. EQUATIONS OF MOTION AND HEATCONDUCTION OF A FINITE ELEMENT

Material Forms. We now consider a typical finite element of the continuum on which the

displacement, velocity and temperature fields are given by (12), (15)3' and (20).

Introducing (12) into (25) and simpl ifying, we obtain for the general balance of mechanical

energy for the element (33,41]

O -' [N"''' j1J N (0 M )....... N ]- UNk mUM k + er .. 11 k J + tUM k OV - P, J k

I'

where mNM is the consistent mass matrix and pNare the consistent generalized forces fork

the el ement:

(34)

(35)

Here "'oe and Aoe are the volume and the surface area of the element in the reference

configuration Co. The quantities .N are functions of the materia' coordinates X1. Further,

we are again reminded that (35) is, in one sense, only symbolic since the surface tractions

Sk are themselves functions of the nodal displacements. We examine this aspect of the

equations in the following section.

Since (33) represents a general law of balance of energy for the element, it must

hold for arbitrary nodal velocities. Thus, the term within brackets must vanish, and we

arri ve at the general equations of motion of ~ finite element:

mNMij + fer1 J,IIN (0 + ,~M u ) d'll' = pN (36)Mk ',1 kJ "J Mk k

VoeObserve that 1) no restrictions have been placed on the order-of-magnitude of the deforma-

tions - (36) holds for finite deformations of the element - 2) no restrictions have been placed

on the materia' of which the element is composed - the constitution of the material is r~f1ected

in the constitutive equation for stress which is, as yet, unspecified.

To obtain the corresponding equations of heat conduction, we turn to (18) and (20)

which we introduce into (32). Upon simplifying, making use of the Green-Gauss theorem,

- 11 -

--and requiring that the result hold for arbitrary nodal temperatures, we obtain the general

equation of heat conduction for ~ finite element of ~ continuum [30,34]:

+ J ph."dV

'Ir.

J p(T. +."T")."~dV+ J q'.:,dll' =q" +a"'V'o. Yo •

where qN and crNrepresent the generalized normal heat flux and the generalized npdal

dissipation at node N:

q" ~J."q'n,dA. 'Ao. .

crN = f O'1jrNdV'

'Yo.

(37)

(38)

(39)

Again, no restri'ction is placed on the magnitudes of various quantities appearing in (37)

or on the nature of the material. Specific forms of (37) val id for specific material can be

obtained when corresponding constitutive equations for 11, q1, and cr are introduced. We also

note that the heat flux q 1 in (38) is avai 'able to the observer in the current configuration

(on a material surface). Consequently, the quantities qN are, like the generalized forces

pN , functions of the nodal displacements UN 1 •k

Spatial Forms. To obtain spatial forms of the general finite element equations for applications

to problems in, say, fluid dynamics, it is necessary to regard the interpolation functions 'ltN as

functions .N of the spatial coordinates Xl' Convective terms must be added to

time rates, but the tractions S 1 and heat fluxes gl act now on spatial surfaces which are

independent of the motion.

In the case of compressible fluids, the density P is represented by WN(X)PN'PN being the .

value of the density field at node N. Then the equations of motion assume the highly non-

Iinear form

MQ N • +bMQ R N +!t otNd - Na PMVQl III PMVR.VQl e lj Ox

jV' -P1

whe re t 1 j = F 1 III F j n crilln is the Cauc hy stress tensor and

a"" ~j.'.'."dV b:'RM {.".' ~' •• "dV"V'" V".

e

(40)

(41)

- 12 -

It is again emphasized that the ,N are now functions of Xi and thatV"'e/ and, in computing

pN , Ao, are now spatial volumes and areas. Note also that the spatial equations of motion1

are cast in terms of nodal velocities rather than displacements, and that inclusion of inertial

effects and an unknown density leads to terms which are nonlinear in the nodal velocities

and densities. To (40) we must add the finite element analogue of the continuity equation

op/ot + O(PVk)/Xk= 0, which, by following a procedure similar to that used in deriving (40),

is found to be

"

MN' +dNMR - 0C PM k PRvMk- (42)

(43)

(44)

Here nk are the components of a unit vector normal to the bounding surface of the element.

For incompressible fluids, p is known and the equation of continuity is satisfied.

Then, instead of (40), we use [39J

mNMvM 1 + n~MRV ... III vR 1 +f(t1 j - P01j )~~ j dV" = p~ (45)1/"e

where 11 j is the deviatoric stress, p is the hydrostatic pressure, and nNMR is the convectiveIII

mass matrix

nN: R = I ptNtM ~rdV (45)'V'e III

We mustalso require that the incompressibil ity condition be satisfied in an average sense over

each element:

(46)

The spatial equations of heat conduction for the element are identical in form to

(37) except that Ve and Ae are regarded as spatial volumes and areas, and corresponding

interpolations are given to T1, q1 , and a. When specific forms of TIare introduced, the term

containing ~ will give rise to nonlinear convective terms in the nodal velocities and temp-

eratures.

- 13 -

6. GENERALIZED FORCES AND FLUXES FOR FINITE DEFORMA nON

We have remarked previously that for material descriptions of the motion of finite

elements, the tractions S1 and heat fluxes ql act on areas in the deformed body. Conse-

quentl y, the general ized nodal forces and heat fluxes must be expressed as functions of the

nodal displacements. We shall now examine certain forms of these functions.

Let .6.. denote a unit normal to the bounding surface dAo e of the element while it

occupies its reference ("undeformed") configuration, and n the unit normal to the same,..material area, dA, in the current configuration ("after deformation"). Further, let~,

q and~, q denote the surface tractions and heat flux per unit undeformed and deformed area,- -respectively. Then the mechanical power 0.t..and the heat Q, developed by ~ and q are- -given by

(47)

Ql.=.[ q \fGnlldAo (48)o

A ,..

where S1 and q. are the contravariant components of ~ and ~ referred to the convected

coordinate lines Xl, F~ are the deformation gradients defined in (3), and G = IG1J I is the

determinant of the deformation tensor G1j of (5). Observing that for the finite element

(49)

and(50)

and recall ing (35) and (38), we find for the general ized nodal forces and normal heat fluxes

P: ~J p,F•• 'dV" +f 5'(6,. + .~.UM,) .'dA, (51)oe Ao e

qN = Jpoh~Nd1f+ f qll..jGn dAo (52)A II11"0. 0 e

Here Fk is the cartesian component of body force per unit mass in the reference configuration.

To specialize still further, consider the case in which a uniform pressure loading

and a uniform heat flux of intensities p and q act on the deformed areas of the element.

- 14 -

Then, ignoring body forces and internal heat sources for simplicity,

pN = -1 pGrm {Gn (Okr + +~kuMr)'fNdAk A II

oe

q' =1 q..JGG"n,n:.'dAo'Ao e

(51) and (52) become

(53)

(54)

where Gr lIis the inverse of the deformation tensor Gr II'

Fortunately, simpl ified forms of these equations can be obtained for specific shapes

of finite element boundaries. For simpl icial approximations (i. e., cases in which the ~N

are linear in Xl), the generalized forces and heat fluxes assumes.ignificantly more manage-

able forms [25,26J. We shall not elaborate on these in this paper.

7. MATERIALS WITH MEMORY

(55)(Xl

cp =~ [G1j (t-s), S(t-s)]s=o

Finite element formulations of Coleman's [1] thermodynamics of thermomechanicall y

simple materials lead to results which can be applied to a wide range of problems. For

materials whose response depends upon the history of the deformation and temperature, the

free energy cp is given by a functional~of the histories G1j(t-s), e(t-s), where s = t- 'T ~ 0

is a parameter:

Coleman has shown that for such materials the stress tensor and entropy are determined from

cp as follows:(Xl (Xl

crib 1t' 1j [G1j(t-s), e(t-s)] = D~j ~ [ Js=o ~s-o

(56)

(57)(Xl (Xl

T1 = $ [G1j(t-s), e(t-s)] = Df1~ [ ]s=o s-o

where fj [ J and $CJ are functionals of the indicated histories and D~ and Df1are Frechet

differential operators. The heat flux is given by an independent functional.~1 ( J:(Xl

ql = J:tl[G1J(t-s), e(t-s), ~e(t-s)]s=o

(58)

- 15 -

and the internal dissipation can be obtained directly from tp, 0.1.1, and" by means of (28),

and is itsel f a functional of the same histories:

00

o =1f[G1j(t-s), e(t-s)]s=o

(59)

Upon introducing (56) - (59) into (36) and (37) and integrating over the volume of the

finite element, we el iminate any dependence on X1. The equations of motion and heat con-

duction of a typical finite element then become00 00

mN"'U"'k +L:[uRr(t-s), TR(t-s)] +u",d! :"'1[URr(t-s), TR(t-s)] =p~s=o s=o

(60)

00

fio{To +,"'TM)~ [~r(t-s), TR(t-s)]dV" +"\roe s-o

HN[URI'(t-s), TR(t-s)J =qN + £Ns~o s=o

(61 )

00 00

H N[URr(t-s), TR(t-s)] = fi~1}j,1[] dV-s=o s=o

110 e

00 00

11j(} =']t1j([Oim +'~jURm(t-S)] [6.111 +,s'.lus..(t-s)]; To + wMT",(t-s)}s=o s=o

(62)

00 d m

E [] =Tt .$ [G1.l(t-s), e(t-s)]S=0 s =0

lXlN co

£. = fiHJ:rCG1j(t-S), e (t-S)] dV'S=0 s =0Voe

Thus, the behavior of the finite element model depends upon the histories of the nodal dia-

placements and temperatures.

8. SPECIAL CASES

We now cite as examples a number of finite element formulations that are

reducible as special cases of the general equations of motion and heat conduction derived

previously.

- 16 -

"

1. Finite Coupled Thermoelasticity. In this case the free energy cp is an ordinary function

of the current deformation and temperature:

cp = ~(Y1j, T) (63)

Then, according to (54) and (55), cr1j= ~~1j and 11 = - ~: and (36) and (37) are put in

terms of (63) by direct substitution.

In the case of isotropic thermoelastic sol ids, cp can be expressed as a function of T

and the principal invariants h, la, and 13 of G1j:

11 = 3 + 2Y11

(64)

Then the equations of motion and heat conduction of a finite element become [30J

(65)

+ KMNTM =qN (66)

where Q',i,j, = 1,2,3; M,N = 1,2, ••• ,NIl• Here the internal dissipation cr is zero and

K M N is the thermal conductivity matrix

MN = /"1j ,~M ,I.N dV" (67)K K ".1,,1

Vee

where ~1.1 is the thermal conductivity tensor. It has been assumed in (66) that the heat flux

is given by the Fourier Law

(68)

2. Rnite Elasticity. In the case of finite isothermal deformations of incompressible elastic

sol ids, cp = W(I1, 12), where W ( ) is referred to as the strain energy density. Then the equations

of motion become

(69)

- 17 -

where N=l,2 and p is a uniform hydrostatic pressure for the element. To determine p,

we must add to (69) the incompressibil ity condition

[(I. -1) dV = 0 (70)

Solutions of static forms of the nonl inear equations (69) have been presented by Oden [28,29J,

Oden and Sato [25,27J and Oden and Kubitza [26J.

3. Dynamic Coupled Thermoelasticity. Consider the case of finite displacements but infini-

tesima~ strains of on anisotropic thermoelastic sol id for which the free energy is a quadratic

form in the strains and temperoture increments:

Here E1.lle1 and B1.l are thermoelastic constants and c is the specific heat at constant

deformation. Then motion and heat conduction in an element are governed by

mNMij +fE11"1nv doN (0 + doM U ) duoIe 1.n'lJ lie ',1 Hie "

~e

+f B1J .. R"~/6tlc + ..~ tUMIe)dlf TR = P:Voe

- fio(To +'f"'TM) .. N[Blj'~t(5jl +"~JUSID)URmVoe

+ TC'fRf R J dv + /(M NT M = q N

o

(71 )

(72)

(73)

where i,j ,k,l,m = 1,2,3 and M, N, R,S = 1,2, •.. , Ne• Again Fourier's low of heat conduc-

tion is assumed.

4. Classical, Infinitesimal Elasticity. If displacements and strains are small and only isothermal

deformations are considered, (72) reduces to

(74)

where(75)

- 18 -

The array KNM 111 is the well-known stiffness matrix for the element.Ie

5. Transient Heat Conduction. Finite element formulations of the classical problem of heat

conduction in a rigid sol id are obtained by linearizing (73) and neglecting terms involving

the nodal displacements:

hNMT M + KNMT M = qN

Here hNM is the specific heat matrix:

hNM = - f Poc~NtMdV'"'Ifoe

Equations similar to (76) have been derived by several authors [43,36,37].

(76)

(77)

6. Transient Coupled Thermoviscoelasticity. The possible characterizations of thermoviscoelas-

tic solids are practically unlimited, and appropriate finite element models can always be

generated by introducing the appropriate constitutive equations into (36) and (37). For com-

pleteness, we cite here just one such application. This involves a thermoviscoelastic materiol

of the Voigt type, described by Eringen [44], for which

(78)

(79)

(80)

where E1jmTl, F1jlll\ H1jlll, B1j, K1j, and rian are arrays of material constants. The motion

and heat conduction in a finite element of such materials are governed by

where

mNMij +DNMlIIu +KNMIIU +HNMT =pN. Mle Ie Ms 1: Mil Ie M Ie

aNMjUMj +CNMT,., + KNMTM = qN +aN

- 19 -

(81)

(82)

(83)

(84)

(85)

eN = f."N(F1Jmny. y' + H1J.y' T ) dV" 1J .n 1J ' mVoe

(86)

7. Compressible Stokesian Fluids. To develop finite element models of general form

for the case of compressible Stokesian fluids, we note that the stress tensor t1j for such fluids

is of the form

t = ( + A OVk ) 6 + 2 . (OV1 + OV, ) (87)1J n \)OXk 1J µ\) OxJ Ox1

where n = n(p) = Ocp/Op-l is the thermodynamic "pressure" defined by the equation of state of

the fluid, and A\) and µ\) are the dilitational and shear viscosities. Introducing (16) into (87),

incorporating the result into (40), and recalling that for compressible fluids (42) must also be

satisfied, we obtain for the equations governing compressible flow through a finite element

M Q N· bM Q R N f 0t N [( ~)a PMVQ1+ m PMVQ1VR1+ OxJ

n+A OxkVMk01JVe

+2 µ\)(~VM1 + ~: vMJ)J dv= p~ (88)

CMNpM+d~MRpRvMk =0 (89)

where the arrays aMQN, •.• , etc. are defined in (41), (43), and (44) and M,N,Q,R = 1,2,.

.• ,Ne; i,j,k,m = 1,2,3. Equations (88) and (89) are the finite element analogues of the

Navier-Stokes equations for compressible fluids.

8. Incompressible Stokesian Fluids. In the case of incompressible Stokesian fluids, n

becomes p, the hydrostatic pressure, the density is assumed to be known, and

(~1 ov.)t1J=-P01J+ µ -+~\) OxJ OX1

We then have, instead of (88) and (89),

N M • N M R f o~N [(~ ~) l:. J..h'" Nm VM1+n VMIlV/l1+ -s- µ '::> VM1 + '::> vMJ - pV1j Ulf =pm oXj \) oXJ oXt 1V"e

(90)

(91)

dNvNk = 0 (92)k

where, according to (46), d: = [O~N/OXk dv • Equation (91) was also obtained by Oden

and Somogyi [39J, and finite element solutions of the problem of incompressible potential

flows were considered by Martin [38].

- 20 -

"

9. INCREMENTAL FORMS

In applications of the finite element method to problems of instability and large de-

flections of elastic sol ids under infinitesimal strain but finite rotations and displacements, a

popular procedure that is usually followed is to derive certain "initial stress II and "initial

displacement" matrices to enable solutions to be obtained by solving a sequence of linear

problems (e.g., see [10], [45J, [46], [47]). We shall now demonstrate that all of these

special stiffness matrices, including terms erroneously omitted in previous work, can be obtained

by considering incremental forms of the nonl inear finite element equations for the special case

of a Iinearly elastic sol ids.

Consider a material finite element in a state of finite deformation characterized by

the quantities u~ 1, ao Ij, F~, and S~. Now consider a neighboring motion characterized by

corresponding quantities u: l' a/j, F~, and 51 where

(93)A #It. A

F* = FO + F S1 = S 1 + S 1 (94)1 1 1 * 0

Here uM 1, 0'1.1' F1 and S 1 represent small pertubations in the initial quantities. We assume

that the stress increment is linear in the strain increments:

al.1 = E 1.1klll Y = E 1.1lcm .~~ U (95)I< m 1', I< loll!

Introducing (93)-(95) into (36), linearizing in the increments, and noting that the initial values

must also satisfy the equations of motion, we obtain for the incremental equations of motion

for a finite element

mNMij + [KNill III +KNMm +KN".Ju +RNMmu =pN (96)loll< I< (£.h (~.h 10111 k 101m I<

where K:M• is the ordinary stiffness matrix of (75) corresponding to the initial linearized

problem, K(~):is the "geometricll or "initial stress II stiffness matrix [10,45,46J, K(~):- -

is the lIinitial displacement II matrix [47], R~MIII is a load-correction stiffness matrix,

apparently omitted in previous incremental formulations (e .g. [48J, [19], with the exception

of those which began with nonlinear forms of the finite element equation~ such as L25], [26],

[29J, (49J, [50]):

(97)

- 21 -

RNM• = r S*-,I,N,I,M dAk JA '"t',k

oe

The general ized nodal forces pN are now given byk

(98)

(99)

(100)

Acknowledgement. Portions of the work reported in this paper were supported by the

National Aeronautics and Space Administration through a general research grant, NGL-Ol-

002-001 and by the National Science Foundation through research grant GK-1261.

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I1I

.(