Finite Element Formulation of Problems of Finite Deformation and ...oden/Dr._Oden_Reprints/1971-001.fin… · of Finite Deformation and Irreversible Thermodynamics of Nonlinear Continua

Finite Element Formulation of Problems

of Finite Deformation and Irreversible

Thermodynamics of Nonlinear Continua

A Survey and Extension of Recent Developments

J. Tinsley Oden

INTRODUCTION

Mechanical action of a continuous body does not manifest itselfin mechanical effects alone. When the external forces acting on a bodyperform work, the body gets hotter. Conversely, supplying heat to abody results in motion. If the process is reversed, not all of the mech-anical energy supplied is generally recoverable. The process is then anirreversible one; and, as such, its description may fall well outside therealm of both classical mechanics and classical thermodynamics. In-deed, although the interconvertibility of mechanical work and heat wasrecognized in the nineteenth century by Joule, works on thermodynam-ics have, until recent years, treated the bulk of thermodynamic pro-cesses both as reversible and as completely uncoupled with the mechan-ical behavior. Likewise, classical solid and fluid mechanics, as a whole,ignore the fact that viscous behavior, coupled heat conduction, andplasticity belong to the realm of irreversible thermodynamics; andmany phenomena associated with continuous bodies that have tradi-tionally been treated with reasonable accuracy as reversible, cease to beso when finite deformations are taken into account (for example, finitedeformations of metals).

Although the development of a rational theory of thermodynamicsof continuous media has lagged the development of the more familiarspecial theories by over a century, a significant step toward a generalthermodynamics of continuous media was made in 1964 by Coleman

" [1], who presented a consistent and rational theory applicable to sim-ple materials with memory. More recently, additional refinements ofColeman's theory have been discussed by Laws [2], Truesdell and Noll

693

ANAL YSIS OF FLOW AND SPECIAL PROBLEMS

[3], Truesdell [4, 5], and Muller [6], among others. Although muchexperimental work remains to be done, it may be said that sufficientbasis for the theory is now available to apply it to a wide class of prob-lems involving nonlinear behavior of continuous media.

Applications of modern theories of irreversible thermodynamics torealistic physical problems very quickly exceed the limits of classicalmethods of analysis. The mathematical analysis of the thermomech-anical behavior of continuous bodies of arbitrary shape and generalboundary and initial conditions, complicated by the presence of finitedeformations and time-dependent material properties, constitutes oneof the most difficult classes of problems in applied physics. It appearsthat only through the use of modem numerical techniques is there hopeof obtaining quantitative solutions to problems of this type.

Toward this end, several applications of the finite element concepthave been made to the development of discrete models of nonlinearbehavior of continua. Following the pioneering 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 appli-cations to geometrically nonlinear problems involving infinitesimalstrains were given by Martin [9, 10], Argyris [11, 12], Zienkiewicz[13], Przemieniecki [14], and Marcal [15], among others; and severalsolutions to problems in classical thermoelasticity and elasto-plasticityhave been presented (for example, 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], andYamada, Yoshimura, and Sakurai [23]). Finite element formulationsof problems of finite deformation of elastic and. thermoelastic solidshave been given by Oden et al [24-30] and Becker [31]. Applicationsto nonlinear viscoelasticity [32-34], heat conduction and coupledthermoelasticity [35-37], and potential flow [38-40] have also beenpresented. There does not appear to be available attempts at develop-ing discrete models of general thermomechanical behavior of bothsolids and fluids.

In the present paper, we consider the problem of constructing gen-eral finite element models of fmitedeformation and irreversible thermo-dynamics of nonlinear continua. A major objective of the developmentspresented here is generality, in the sense that the resulting formulationsrepresent discrete models of a wide range of problems in both solid andfluid mechanics. To achieve this, both spatial and material descriptionsof the local displacement, velocity, temperature, and density fields are •introduced: the.'former depicting the finite element as a subregion of

694

FINITE DEFORMATION AND IRREVERSIBLE THERMODYNAMICS

three-dimensional euclidean space through which the continuum flows,and the latter depicting the element as a material collection of particlesspecified in some reference configuration. Since most of the primarydependent variables are kinematic in nature, it is shown that the spatialforms of the finite element equations are complicated by the presenceof nonlinear convective terms. Conversely, the material description iscomplicated by the fact that forces and heat fluxes are applied on amaterial surface in the deformed element.

Once the basic kinematical equations for finite elements have beendeveloped, we examine the essential aspects of continuum thermody-namics. By considering balances of energy for typical finite elements,we obtain spatial and material forms of the general equations of motionand heat conduction of finite elements. In the case of compressiblefluids, we show that the equations of motion must be supplemented byfinite-element analogues of the spatial form of the equation of contin-uity, the mass density p being an additional unlrnown in the problem.

We then examine finite element formulations of a number of specialcases that can be obtained from the general finite element equations.These include thermo mechanically simple materials with memory, finitecoupled thermoelasticity, finite elasticity, dynamic coupled thermo-elasticity, classical elasticity, transient heat conduction, coupled ther-moviscoelasticity, and compressible and incompressible Stokesian fluids.The latter equations represent finite-element models of the Navier-Stokes equations of fluid dynamics.

We conclude the investigation by presenting incremental forms of thefinite element equations for the special case of finite displacements ofelastic solids. We show that these exhibit the well-lrnown initial stressand displacement matrices used in the stability and nonlinear analysisof elastic and elastoplastic structures.

KINEMATIC PRELIMINARIES

We consider the motion of a continuous body. To identify theconfiguration of the body at a given time, we assign to its particles thelabels Xi (i = 1,2,3). The quantities Xi are referred to as intrinsic orconvected coordinates, and we interpret them as being etched onto thebody and to move with it as the body deforms. The motion of thebody is a continuous one-parameter family of configurations Ct , andthe parameter T is associated with time. We shall refer to the configu-ration Co corresponding to T = 0 as the reference configuration. Theconfiguration at T = t is denoted Ct and is termed the current config-uration.

695


For simplicity, we shall assume that, when the body occupies itsreference configuration Co' the intrinsic particle labels Xl are rectangu-lar cartesian. However, at all other times T > 0 the coordinates linesX1 will be, in general, curvilinear. When the body is in its referenceconfiguration, we also establish a spatial reference frame xl which isrectangular cartesian and which, again for simplicity, coincides with Xlat T = O. We regard the frame x 1 as inertial; that is, it is absolutelyfixed in space (alternately, we may give the Xl any time-dependentrigid translation relative to Co). It is important to realize that the num-bers Xl identify a particle, whereas the numbers Xl identify a place inthe three-dimensional space through which the body moves.

Mathematically, we can describe the motion of a particle Xl byequations of the form

123Xl = Xl(X , X , X , t) (1)

Equation (1) defines the position of a particle at time t in terms of itsposition in the reference configuration. The components of displace-ment Ul of a particle originally at Xl are given by

Ul = Xl - Xl (2)

The components of deformation gradient, denoted Fl are defined by

Fj = ;~J.. ul,J + 0lJ (3)

where the comma denotes partial differentiation with respect to XJ

(Le., Ul , J == OUl/OXJ ) and 01 J is the Kronecker delta.A measure of the degree of deformation at a point in the body is pro-

vided by the square of the material element of arc ds2 in the deformedbody;

(4)

where Gl J is Green's deformation tensor:

(5)

696


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

Since the same label X1 identifies a given particle at all times, formsof measures of the rates of deformation are particularly simple whenthe present description of motion is used. Components of velocity are

Vl = Vl (x) = ~; = Ul (7)

and rates-of-strain are given by

• "dy 1Y1J = ~ = '2 (Vl,J + vJ,l + ulc,lvlc,J + VIc,lUIc,J)

(8)

Higher-order strain rates are obtained by repeated partial differenti-ations with respect to time.

EULERIAN FORMS

In the study of the motion of fluids, it is usually meaningless totrace the motion of a specific particle. Motion is then described byestablishing a fixed reference frame (the Xl ) and viewing the motionof the continuum through points in three-dimensional space. Then weuse the Almansi-Cauchy strain sensor

(9)

Since we now write all quantities as functions of Xl rather than Xl,

time rates become more complicated:

( 10)

(11)

etc.697

ANALYSIS OF FLOW AND SPECIAL PROBLEMS

MOTION OF A FINITE ELEMENT

We shall now consider a discrete model of the continuum whichconsists of a collection of a finite number of finite elements connectedappropriately together at various nodal points. Since the process ofconnecting elements together to form such a nodal is well documentedand is based on purely topological properties of the model, we shall notelaborate on it here. It suffices to point out that two distinct types offinite element model can be developed for continuous media: the ma-terial model and the spatial model. The material model is viewed as anactual collection of physical bodies connected continuously together atnodal points. The nodal points themselves are material particles Xl . Inthe spatial model, the space through which the media moves is regardedas a connected set of subregions, each representing a finite element.Nodal points then represent places Xl in the three-dimensional euclid-ean space. In either case, a finite element is regarded as a subdomainof a field quantity: in the material description, the domain is a specif-ied collection of particles; and, in the spatial case, the domain is a sub-region of e3• It follows that the material model is used for materialdescriptions of motion and the spatial model is used for spatial descrip-tions. Moreover, in either case, typical finite elements can be isolatedfrom the collection and appropriate fields can be described locally overan element, independently of the ultimate location of the element inthe connected model. This latter property has been referred to as thefundamental property of finite element models [41].

MATERIAL MODEL

We consider a typical finite element e of a continuum, which, forour present purposes, we regard as a subdomain of the displacementfield Ul (and later, the temperature field T). A finite number Ne ofmaterial particles are identified in the element while in its referenceconfiguration. These are called nodes and are further labeled X~whereN = 1, 2, ... , Ne The values of the displacement field at thenodes are then

UNl = Ul(XN) (12)

Following the usual procedure, we now approximate the displacementfield locally over the element by functions of the form

( 13)

698

(14)


where here (and henceforth) the repeated nodal index is summed from1 to Ne • The interpolation functions \IN (X) have the property that

N~H(XM) = o~ ~ ~H(X) - 1

Moreover, WN (X) <:: 0 for all particles X belonging to the element, and(13) is designed so that the local fields Ul are uniquely defined over.the element in terms of the nodal values UN 1. It is understood in (13)that UN 1 are functions of time.

With the displacement field approximated, it is now a simple matterto calculate all other quantities needed to define the motion of theelement

and so forth.

SPATIAL MODEL

For the spatial model, we consider a bounded subdomain of e.3 inwhich we identify a number N. of places called nodes and labeled x~.Although it is a simple matter to construct spatial approximations ofthe displacement field, it is, for our purposes, more natural to beginwith approximations of the velocity field V1 :

(16)

where VN 1 = v1 (XN) are the values of the local velocity field at thenodal places x~ and the interpolation functions I\tN (x) have the sameproperties described in (14) except that they are cast in terms of thespatial coordinates xl.

699


Other kinematic variables are computed as follows:

etc., where a1 is the i th component of the acceleration. Notice thatthe convective terms (e.g., Vm OVl /Oxm, etc.) complicate the euleriandescription of the motion of a finite element.

TEMPERATURE FIELDS

In the study of thermomechanical phenomena, finite element modelsof the temperature field are also needed. Let e denote the absolutetemperature and To denote a uniform reference temperature that isindependent of time. Then e is written as the sum of To and T, whereT is the change in temperature.

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

(18)

or, for a spatial model,

(19)

where the functions W (x) TN are identical to those described earlierand TN = T (XN) or T (xN) are the nodal temperatures. Then

(20)

Time rates of change of temperature are then, with TN '" dTN / d t,

• N •T = W (X)TN

or

for the material and spatial finite element models, respectively.700

(21a)

(21b)


THERMODYNAMIC RELATIONS

In the following developments, we shall largely confine our attentionto material descriptions of the deformation of a continuous body. Spa-tial forms will then be stated without detailed derivations.

The thermomechanical behavior of an arbitrary continuum is govern-ed by five physical laws: (1) the conservation mass, (2) balance oflinear momentum and (3) angular momentum, (4) the conservation ofenergy, and (5) the Clausius-Duhem inequality. For material descrip-tions, mass and angular momentum are usually (but not always [42])assumed to be satisfied a priori for a finite element. Linear momentumis balanced globally for a finite element, but is balanced only in anaverage sense throughout an element. Local and global forms of thelaw of conservation of energy provide a natural means for developingthe equations governing the motion of a typical element, while theClausius-Duhem inequality provides bounds on the entropy productionin an element as well as restrictions on the nature of the constitutiveequations describing the material of which the element is composed.

Consider a material volume u of mass density p bounded by a surfaceof area A. The global form of the law of conservation of energy forthis volume is, considering only thermomechanical behavior,

(22)

u

where ft is the kinetic energy, U the internal energy, n the mechanicalpower, and Q the heat energy.

1 f ..ft = '2 pU1 U1 du

U

u '" !PF1Ul du + fS1u1 dA

U A

Q = fqln1 dA + fPh du

A U

(23)

Here we have referred all quantities to the reference configuration Co:P is the mass per unit volume u in the reference configuration, E: is the

701


internal energy density, Flare the body forces per unit mass in the"undeformed" body, and h is the heat supply per unit mass in Co. Thequantities S 1 and q 1 are components of surface traction and heat fluxper unit initial area referred to convected coordinates Xl in the currentconfiguration. These are unavoidably available only after some motionof the body takes place and are, therefore, functions of the displace-ment gradients ul , J. We explore this problem more thoroughly later.

If mass is conserved and the principle of balance of linear momentumis assumed to hold, then, provided certain continuity requirements aresatisfied, it can be shown that the local form of the law of conservationof energy is

pe = a1JY1J + q~l + Phl

(24)

where 01 J are the contravariant components of stress per unit of

deformed area referred to the intrinsic coordinates Xl and the semi-colon denotes covariant differentiation with respect to the Xi.

Upon introducing (24) into (22) and making use of the Green-Gausstheorem, we obtain the alternate global form

(25)

ENTROPY, FREE ENERGY, DISSIPATION

The entropy content of a body is defined by introducing the specificentropyil. The total entropy production r is then

r = fpn du - f~.n dA - fp ~ du

u A u(26)

where e is the absolute temperature. According to the Clausius-Duheminequality, r;;:: 0, or locally,

. 1pe~ - q~l - Ph + e qle,J ;;::0 (27)

It is convenient to introduce the Helmholtz free energy density cP andthe internal dissipation 0':

702


cp = e: - 11e

Then

. .p(w + 118) (28)

1 1cr+eqe,1~0

and (24) can be written in the alternate forms

p~ = crlJY1J - p11e - cr

(29)

(30)

(31)

Another useful form of the energy balance is obtained from (31)after multiplying both sides by the temperature increment T:

(32)

EQUATIONS OF MOTION AND HEAT CONDUCTIONOFA

FINITE ELEMENT

MATERIAL FORMS

We now consider a typical finite element of the continuum on whichthe displacement, velocity, and temperature fields are given by (12),(15)3, and (20). Introducing (12) into (25) and simplifying, we obtainfor the general balance of mechanical energy for the element [33, 41]

o = uNk[mNMuMk + f crlJ$~1(6kJ + $~JUMk) du - p~J (33)

Uo II

where mNM is the consistent mass matrix and p~ are the consistentgeneralized forces for the element:

mNM = J P $NI\rMdu

Uo II

703

(34)


p~ = J PFk $N du + f Sk $N dA

uoe Aoe

(35)

Here Uo II and Ao II are the volume and the surface area of the elementin the reference configuration Co. The quantities $N are functions ofthe material coordinates Xl. Further, we are again reminded that (35)is, in one sense, only symbolic since the surface tractions Sk are them-selves functions of the nodal displacements. We examine this aspect ofthe equations in the following section.

Since (33) represents a general law of balance of energy for the ele-ment, it must hold for arbitrary nodal velocities. Thus, the term withinbrackets must vanish, and we arrive at the general equations of motionof a finite element:

mNMuMk + f (11JW~dOkJ + $~JUMk) du = p~

uoe

(36)

Observe that (1) no restrictions have been placed on the order-of-magnitude of the deformations - (36) holds for finite deformations ofthe element - (2) no restrictions have been placed on the material ofwhich the element is composed - the constitution of the material isreflected in the constitutive equation for stress which is, as yet, un-specified.

To obtain the corresponding equations of heat conduction, we turnto (18) and (20) which we introduce into (32). Upon simplifying,making use of the Green-Gauss theorem, and requiring that the resulthold for arbitrary nodal temperatures, we obtain the general equationof heat conduction for a finite element of a continuum [30, 34] :

f P (To + WMTM )$N~ du +Uo e

f ql1j.r~ 1 du = qN + (1N

Uo e

(37)

where qN and (1N represent the generalized normal heat flux and thegeneralized nodal dissipation at node N:

704


Nq (38)

(39)

Again, no restriction is placed on the magnitudes of various quantitiesappearing in (37) or on the nature of the material. Specific forms of(37) valid for specific material can be obtained when correspondingconstitutive equations for 11 , q1 , and crare introduced. We also notethat the heat flux q1 in (38) is available to the observer in the currentconfiguration (on a material surface). Consequently, the quantities q N

are, like the generalized forces p~, functions of the nodal displacementsuNi •

SPATIAL FORMS

To obtain spatial forms of the general finite element equations forapplications to problems in, say, fluid dynamics, it is necessary toregard the interpolation functions ~N as functions $N of the spatialcoordinates Xi' Convective terms must be added to time rates, but thetractions Si and heat fluxes ql act now on spatial surfaces which areindependent of the motion.

In the case of compressible fluids, the density P is represented by$N (x) PN, PN being the value of the density field at node N. Then theequations of motion assume the highly nonlinear form

MQN. MQRN J odrNa PMVQi + bm PMVRcVQi + t1J ~ du

OXJUe

where tl J = FiF~crZ n is the Cauchy stress tensor and

NPi (40)

aMQN = ~*M*Q*N du

ue

(41)

705


It is again emphasized that the 1\1 N are now functions of x 1 and that Ue,

and, in computingp~ , Ae are now spatial volumes and areas. Note alsothat the spatial equations of motion are cast in terms of nodal velocitiesrather than displacements, and that inclusion of inertial effects and anunknown density leads to terms which are nonlinear in the nodal veloci-ties and densities. To (40) we must add the finite element analogue ofthe continuity equation op I Ot + o(Pvld/xk = 0, which, by followinga procedure similar to that used in deriving (40), is found to be

(42)

where PM = dPM/dt,

(43)

dNMRk = (44)

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

For incompressible fluids, P is lrnown and the equation of a contin-uity is satisfied. Then, instead of (40), we use (39).

f" N+ (t1J - p61J)~,J du

ue

NPl (45)

where tl J is the deviatoric stress, P is the hydrostatic pressure, andn~MR is the convective mass matrix.

(46)

706


We must also require that the incompressibility condition be satisfiedin an average sense over each element:

VNk (47)

The spatial equations of heat co~duction for the element are identi-cal in form to (37) except that Ue and Ae are regarded as spatial vol-umes and areas, and corresponding interpretations are given to '£),q 1 t

and cr. When specific forms of '£)are introduced, the term containing'£)will give rise to nonlinear convective terms in the nodal velocities andtemperatures.

GENERALIZED FORCES AND FLUXESFOR

FINITE DEFORMATION

We have remarked previously that for material descriptions of themotion of finite elements, the tractions 81 and heat fluxes q 1 act onareas in the deformed body. Consequently, the generalized nodal forcesand heat fluxes must be expressed as functions of the nodal displace-ments. We shall now examine certain forms of these functions.,.

Let n denote a unit normal to the bounding surface <lAo e of theelement while it occupies its reference ("undeformed") configuration,and n the unit normal to the same material area ~d4' in the currentconfiguration ("after deformation"). Further, let 5 ,q and 5 ,q denotethe surface tractions and heat flux per unit undefo~d and deformedarea, respec"tively. Then the mechanical power Os and the heatQq de-veloped by 5 and q are given by

/

"'1 N·a, = S Ff $ UN r dAo

Ao

Qq = fq~ va;;m dAo

Ao

707

(48)

(49)

(50)


where S 1 and q m are the contravariant components of 5 and q referredto the convected coordinate lmes Xi , Fr are the deformation gradientsdefined in (3), and G = IG1 j I is the determinant of the deformationtensor G1J of (5). Observing that for the finite element

F r = (0 r + .I,M r )I 1 '1',1 tiM

and

(51)

and recalling (35) and (38), we find for the generalized nodal forces andnormal heat fluxes

NPic f N / "1 M NPoFlcW du + S (On + W,kUM1H dAo

Uo e Ao e

f Poh$N du + f q'l....jG ~m dAo

Uo e Ao e

(52)

(53)

Here Fit is the cartesian component of body force per unit mass in thereference configuration.

To specialize still further, consider the case in which a uniform pres-sure loading and a uniform heat flux of intensities P and q act on thedeformed areas of the element. Then, ignoring body forces and internalheat sources for simplicity, (52) and (53) become

N f r:: _~" M NPic = - P G VG n.(okr + 1/I',kUMr)1/! cIA

Aoe

qN = f qvGGl"inrnm 1/IN dAo

Ao e

708

(54)

(55)


where GrIII is the inverse of the deformation tensor Gr Ill'

Fortunately, simplified forms of these equations can be obtained forspecific shapes of finite element boundaries. For simplicial approxima-tions (i.e., cases in which the WN are linear in Xl ), the generalized forcesand heat fluxes assume significantly more manageable forms [25, 26] .We shall not elaborate on these in this paper.

MATERIALS WITH MEMORY

Finite element formulations of Coleman's [1] thermodynamics ofthermo mechanically simple materials lead to results which can beapplied to a wide range of problems. For materials whose responsedepends upon the history of the deformation and temperature, thefree energy cP is given by a functional J of the histories Gl J ( t- s ),e(t-s) where s = t - T ~O is a parameter:

(56)

Coleman has shown that for such materials the stress tensor and en-tropy are determined from cp as follows:

1 co lJ cocr J = X [G1J(t-S), e(t-s)]=P.DQ1J :1 (57)s=o s=o

(58)

(59)

(60)

where Xl J [] and g [ 1are functionals of the indicated historiesand DQ 1 J and De are Frechet differential operators. The heat flux isgiven by an independent functional }:l1 [ ]:

co 1ql = s~o [G1J(t-S), e(t-s); ~e(t)}

and the internal dissipation can be obtained directly from cp, crl J, and 11by means of (28), and is itself a functional of the same histories:

cr = s!o[Gl J (t-s), e(t-s)]

Upon introducing (57)-(60) into (36) and (37) and integrating overthe volume of the finite element, we eliminate any dependence on Xl.The equations of motion and heat conduction of a typical finite ele-ment then become

709


NM ••m uMk

~ NM lE ] N+U/<11 ilk URI'(t-s),TR(t-S) =Pk (61)s=o

f M 00Po(To + W TM)S~o[URI'(t-S), TR(t-s)] du

~e

Nq +

OON~

s=o(62)

00 N [ f N 001 JS~Ok --J = 01kW,JS~O ( } du

Uo e

SUO~Ml[ ] = f$~J1J;,\!:Jt} duUo e

00 d)1i>e [ ] = -d ~ [G1 J (t-S), S(t-S)'s=o t s=o ~

710


CXlN f N 0:>S~o = ~ S~O[G1J(t-S),

Uo e

e(t-s)] du (63)

Thus, the behavior of the finite element model depends upon thehistories of the nodal displacements and temperatures.

SPECIAL CASES

We now cite as examples a number of finite element formulationsthat are reducible as special cases of the general equations of motionand heat conduction derived previously.

FINITE COUPLED THERMOELASTICITY

In this case the free energy cp is an ordinary function of the currentdeformation and temperature:

cp = ~(Y1J,T) (64)

Then, according to (55) and (56),cr1J= O~/?N1J and 11= -o~/oTand(36) and (37) are put in terms of (64) by direct substitution.

In the case of isotropic thermoelastic solids, cpo can be expressed as afunction of T and the principal invariantsh, 12,and T3 of Gl J :

11 = 3 + 2Yl1

(65)

Then the equations of motion and heat conduction of a finite elementbecome (30)

(66)

711


(67)

where ex, i ,j , = 1,2 , 3; M,N = 1, 2 , .•. , Ne . Here the internal dis-sipation C1 is zero and x,Nr.! is the thermal conductivity matrix

l1.~1N = f x.1 J *~J ~ ~ 1 d U

Uo e

(68)

where x.1 J is the thermal conductivity tensor. It has been assumed in(66) that the heat flux is given by the Fourier Law

(69)

FINITE ELASTICITY

In the case of finite isothermal deformations of incompressible elas-tic solids, CP= W(ll , 12),where W( ) is referred to as the strain energydensity. Then the equations of motion become

NM ..m U Mk

(70)

where \1::], 2 and p is a uniform hydrostatic pressure for the element.To determine p , we must add to (70) the incompressibility condition

f (13 - 1) du = 0

Uo e

712

(71)


Solutions of static forms of the nonlinear equations (70) have beenpresented by Oden [28, 29], Oden and Sato [25, 27] and Oden andKubitza [26].

DYNAMIC COUPLED THERMO ELASTICITY

Consider the case of finite displacements but infinitesimal strains ofan anisotropic thermoelastic solid for which the free energy is a quad-ratic form in the strains and temperature increments:

~ = 1E1JklY1JYkl + B1JY1~T + ---2C

T2

To(72)

Here E1J Ie 1 and B 1J are thermoelastic constants and c is the specificheat at constant deformation. Then motion and heat conduction in anelement are governed by

mNM tiM It + J E 1 J an YIII n$~J (0 1k + $~;1 UM k) d U

Uo II

+ f B1J$R$~J(01k+ V~1UMk) du TR = p~ (73)

Uo e

-f Po (To + 1\rMTM)$N[B1J$~1(OJII + $~JuSm)URm

Uoe

where i,j,k,l,m = 1,2,3 and M,N,R,S = 1,2, ... , Ne• AgainFourier's law of heat conduction is assumed.

CLASSICAL INFINITESIMAL ELASTICITY

If displacements and strains are small and only isothermal deforma-tions are considered, (73) reduces to

NM.. + KNMmu _ pNm llMk Mm - k

713

(75)


where

f E1 J lIl.nO ,liN ,I,M dulk'f,J'f,1I

Uoe

The array K~M III is the well-known stiffness matrix for the element.

TRANSIENT HEAT CONDUCTION

(76)

(77)

Finite element formulations of the classical problem of heat conduc-tion in a rigid solid are obtained by linearizing (74) and neglecting termsinvolving the nodal displacements:

hNMTH + )1.NMTM = qN

Here hN M is the specific heat matrix:

hNM= - f PoctN,M du

Uo e

(78)

Equations similar to (77) have been derived by several authors [43, 36,37] .

TRANSIENT COUPLED THERMOVISCOELASTICITY

The possible characterizations of thermoviscoelastic solids are practi-cally unlimited, and appropriate finite element models can always begenerated by introducing the appropriate constitutive equations into(36) and (37). For completeness, we cite here just one such application.This involves a thermoviscoelastic material of the Voigt type, describedby Eringen [44], for which

(79)

(80)

714


qi = ~lJT + rlmny', J m n (81)

whereE1Jllln F1Jllln H1h B1J ~lJ andr1llln are arrays of mat-, , , , ,erial constants. The motion and heat conduction in a finite element ofsuch materials are governed by

(82)

(83)

where

D~MIII = f F1JnIII1jI~n1jl~JOlk du

Uo e

HNM = J 0 .IIN (BiJ,I,M + H1Ja,I,M ) duk lk'l',J 'I' '1',.

Uo e

= f 1jI~J(PoToB1J1jIN + rmlJ1jI~1II) du

Uo e

(84)

(85)

(86)

Na = ~ ~N(F1JlllnY1JYllln + H1JIllY1JT,III) du

o e

(87)

(88)

COMPRESSIBLE STOKESIAN FLUIDS

To develop finite element models of general form for the case ofcompressible Stokesian fluids, we note that the stress tensor tl J forsuch fluids is of the form

OVk Ovi OvJt1 J = (TT + A" ~_ ) 01 J + 2µ. ( ~ + '::L_ )

" uxk \J UXJ uXl

715


where TT = TT( p) = ocp/op-l is the thermodynamic "pressure" definedby the equation of state of the fluids, and A\I and µ'\I are the dilatationaland shear viscosities. Introducing (16) into (88), incorporating theresult into (40), and recalling that for compressible fluids (42) mustalso be satisfied, we obtain for the equations governing compressibleflow through a finite element

MQN. bMQRNa PMVQ1 + m PMVQlvR1

f~ ~+ 0 [(TT + A 0 v~'k) 01 JXJ Xk

Ue

evH ~ ~ N+ 2µ'\I OXJ Viol 1 + OXl V~, J ] du = Pl (89)

(90)

where the arrays aMQ N, ••• , etc. are defined in (41), (43), (44), andM,N,Q,R. = 1,2, ... ,Ne; i,j,k,m= 1,2,3. Equations (89) and(90) are the finite element analogues of the Navier-Stokes equations forcompressible fluids.

INCOMPRESSIBLE STOKES IAN FLUIDS

In the case of incompressible Stokesian fluids, TT becomes p , thehydrostatic pressure, the density is assumed to be lmown, and

We have then, instead of (89) and (90),

NM· NMR f~~ (~m Viol 1 + nm Viol m VR 1 + ox Lfl'\I OXJ Viol 1

Ue J

+ ~ VMJ) - p01J] du = p~OX1

d~VNk :;;;;0

716

(91)

(92)

(93)


where, according to (47), d~ -fu O~NiOXk duo Equation (92) was alsoobtained by Oden and Somogyi [39], and finite element solutions ofthe problem of incompressible potential flows were considered byMartin [38].

INCREMENTAL FORMS

In applications of the finite element method to problems of instab-ility and large deflections of elastic solids under infinitesimal strain butfinite rotations and displacements, a popular procedure that is usuallyfollowed is to derive certain "initial stress" and "initial displacement"matrices to enable solutions to be obtained by solving a sequence oflinear problems (e.g., see [10], [45], [46], [47]). We shall nowdemonstrate that all of these special stiffness matrices, including termserroneously omitted in previous work, can be obtained by consideringincremental forms of the nonlinear finite element equations for thespecial case of a linearly elastic solid.

Consider a material finite element in a state of finite deformationcharacterized by the quantities u~ 1 , (J~ J, F~ , and 85. Now consider aneighboring motion characterized by corresponding quantitites r:.l ,(J~J, Ft, and 81 where

n ** 0u~,1 = UM 1 + llt<11 (94)

(95)

Here uM1, (J1 J , F 1 and S1 represent small perturbations in the initialquantities. We assume that the stress increment is linear in the strainincrements:

(96)

Introducing (94)-(96) into (36), linearizing in the increments, and not-ing that the initial values must also satisfy the equations of motion, weobtain for incremental equations of motion for a finite element

NM" + (l(NMlII -I- KNMlll + KNMIll J. + RNMm = N (97)m uMX·~ (a)x (u)~ UMIll k UM: Pk

where K~M 1:I is the ordinary stiffness matrix of (75) corresponding tothe initial linearized problem, K~~~k is the "geometric" or "initialstress" stiffness matrix [10, 45, 47], K~~)kis the "initial displacement"

717

(98)


matrix r 471, .R~101 ID is a load-correction stiffness matrix, apparentlyomitted in previous incremental forml}lations (e.J{.[49], [191, with theexception of those which began with nonlinear forms of the finiteelement equations such as [25], [26], [29], [50], [51]):

KNM:: ::;: f 01 J ~N ~M 15M du(a)K ,1 ,J k

Uo e

R~MIIl = f 8*IDI\INI\I~k dA

Aoe

The generalized nodal forces p~ are now given by

p~ = f PoFkl\lN du + f 81 (51k + l\I~kU~1)I\IN dA

v.oc Aoe

ACKNOWLEDGMENTS

(99)

(100)

(101)

Portions of the work reported in this paper were supported by theNational Aeronautics and Space Administration through a general re-search grant GK-1261.

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718


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719


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DISCUSSION

COMMENT (H. C. Martin)

This interesting paper strongly suggests that finite elements havemuch to offer the researcher interested in problems in continuummechanics. The general theory can, through the introduction of thefinite element method, be used in obtaining solutions to actualproblems.

The question of non-conservative systems has come up previously inthis Seminar. Since the general formulation of Dr. Oden includes thenon-conservative cases, it would now be interesting to have his com-ments on this subject.

RESPONSE (J. T. Oden)

In my work, I have derived equations without neglecting nonlinearities

723


in nodal displacements. Then anyone of a number of availablemethods, including incremental procedures, can be tried.

For nonconservative forces, the location of the material surface onwhich forces act is given in terms of the displacements of boundarynodes of the model. In most cases, however, the expressions for thedeformed surface area and orientation are extremely complicated. Insuch cases, I suggest that a reasonable approximation can be obtainedby fitting a lower·order polynomial through displaced nodes on theboundary. Discussions of some simple expressions obtained usinglinear approximations are given in References 25, 26, and 49 (forexample) of the paper.

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Finite Element Formulation of Problems of Finite Deformation and ...oden/Dr._Oden_Reprints/1971-001.fin… · of Finite Deformation and Irreversible Thermodynamics of Nonlinear Continua

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