INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING. VOL. 3. 161-179 (1971)

FINITE ELEMENT FORMULATION OF GENERALELECTROTHERMOELASTICITY PROBLEMS

J. T. ODIN'r AND B. E. KELLlVt

Ikl'l'lI/'(·fr Ills/illite. Ulliv('/'si/,I' of Alliba/1/a ill HIIIIIS/lille. HIIIIIsrille. Alaba/1/a

SUMMARY

This paper is concerned with the devclopmcnt of consistent discrctc models. via thc concept of tiniteelements. of linear and non-linear electrothcrmomechanical behaviour of continuous bodies. In thedevelopment. general energy balances are utilizcd to derive equations govcrning electromagnetic fieldsover an element and coupled equations of motion and heat conduction of a typical clcmcnt of thecontinuum. These cquations make it possiblc to study a general class of field problcms involvingarbitrary geometries and boundary condit ions. Sample problems arc includcd.

INTRODUCTION

Classical physics has long recognized thc fact that mechanical, electrical and thermal fieldquantities appear to be coupled in most natural phenomena. When current is applied to a con-tinuous body it becomes hotter. and when it becomcs hottcr it undcrgocs a certain amount ofdeformation. Converscly. heating a body or subjccting it to prescribed dcformations is ac-companied by the development of electromagnetic fields. Although such phenomena are discussedlibcrally in most textbooks on elemcntary physics. relativcly few solutions of the coupled fieldcquations appear \0 be available in thc litcrature owing to their inhercnt complcxity. Indeed.solutions of the couplcd tleld equations. the gencral forms of which themsclves havc been thesubjcct of currcnt investigations. arc virtually intractablc by classical analytical methods in thecase of irregular geomctries and complicated boundary and initial conditions. For quantitativesolutions to such problems it is generally necessary to resort to numerical mcthods whicheflectively represent the continuum by an appropriate discrcte model,

[n this paper. we investigate the application of the increasingly popular concept of finiteelements to the formulation of rather gcncral discrete models of dynamic clectrotherrnoelasticphenomena. Brielly. the finite elcmcnt concept involves the rcprcscntation of a continuum as acollection of a tlnite number of componcnt parts. each of relatively simple geometric shape. whicharc connected together at prescribcd nodal points. Various field quantities are represented locallyover each element and are assumed to be defined uniquely in terms of their values at the element'snodes. An important aspect of this proccdurc is that each c1emcnt can be considered to be disjointfor the purposc or describing its behaviour locally. Thus. oncc the behaviour of a typical elementhas been defincd. a discrete model of virtually any shape with arbitrary boundary and initialconditions can be obtained by connecting thc clements appropriately together. The method.introduced in 1956 by Turner and co-workers.l has found wide application in solid and fluidmechanics.~ 5 Reccnt applications to certain clectrical field problems have also been presented.6. 7

7 Professor of Engineering Mechanics.~ Graduate Studenl.

RecC'illed 28 Jal/llary 1970R"l'i.l'ed 20 March 1970

© 1971 by John Wiley & St'ns. lid.

161

/ '

[62 J. T. ODEN AND fl. E. KELLEY

In applications of the finitc clement mcthod. it is customary to attempt to formulate an ap-propriate variational principle governing the problem at hand so as to obtain a consistent meansfor generating analogues of the field equations which hold in an average sense over an element.Wc do not follow this procedure in this paper: firstly. because no well-deflned functional appears10 be readily available which is appropriate for the general class of problems we wish to consider.and secondly, becausc Ihe use of variational principles is not necessary if we develop appropriateforms of the law of conservation of energy for an clement. We feci that the laueI' procedure is notonly more general than the conventional one but that it is more primitive in the sense that itmerely involves the application of a minimum number of fundamental physical laws to anisolated portion ofa continuum over which various Ilelds are prescribed. In thc prcsent investiga-tion. however. this approach requires that we develop a theory of electrothermoe[asticily whichdiflers in form from those we have found in the literature. We outline the development of thesefield equations in the following section. Next. we represent the continuum as a collection of finiteelements and approximate over a typical element the displacement. temperature. scalar potentialand vector potential in terms of their nodal values. Upon introducing these approximations intoglobal forms of the laws of conservation of energy. charge and magnctic llux density. wc obtaina system of equations govcrning the clectrical. thermal and mechanical behaviour of a typicalclement. Thcsc equations can be modified so as to apply to a number of intercsting special cases.including formulalions of discrete models for coupled thermoelasticity, electroelasticity. electro-static problems. etc. In the Ilnal section of the paper we present a sample problem which involvestime-dependent heating of an anisotropic clectl'Othcrmoelastic half-space.

No/arionIn this paper we adopt Einstein's index notation and thc summalion convention. Lower casc

Latin indices represent components of vectors and tensors and range from I to 3. Upper caseLatin indices indicate local nodal points of an element and range from I to Nc' where Nt. is thenumber of nodes belonging to clemcnt e. Whcn a distinction between local and global nodes isrequired. upper case Greek indices are used to represent global nodes. Other letter symbols aredetlned where they first appcar in the paper.

THEORY OF ELECTROTHERMOELAST[C[TY

General

The theory of piezoelectricity. which is the study of quasi-linear electrothermoelasticity inanisotropic crystallinc solids. was formulated by Voigt in ]894 on the basis of the works ofBecquerel. the Curies. Duhem, Lord Kelvin. Pockels and Maxweli.8 This formulation appearedin Voigt's monumental Lehrbuch del' Krislallphysik9 in 1910. Cady's comprehensive worklO

contains a list of other refercnces in Ihis field. Recent work by Trucsdcll and Toupin,u Toupin.12.13

Eringcn.H•1'. Jordan and Eringenl6.17 and Dixon and Eringen18 on electromagnctic clastic andthermoelastic solids. in addition to the ealier works of Dcbye19 and Stratton.20 provide the basisfor a general non-linear theory of electrothermoelasticity in dielectric solids. Piezoclectricity iscontaincd as a special casc of Ihc general non-linear theory \vhich includes electrostriction andother non-linearities as well as electroelastodynamic effects. Supplementing this basic theory arethe works of Landau and Lifshitz21 and Penfield and Haus22 on thc clectrodynamics of a movingmedium and the works of Fano and others,23 Hayt24 and Ramo and others,2.'i on electromagneticfields, and Walker~6 on elastic conductors. In this section. we develop a theory of elcctrothermo-elasticity which is based on the existencc of a differentiable free energy functional which dcpendsupon the current values of the formation, temperature. elcctric fic[d and magnetic field.

GENERAL ELECTROTHERMOELASTICITY PROBLEMS 163

Consider a continuous body which. at time T = O. occupies a configuralion Co in thrce-dimensional Euclidean space. A material point Po in Co is located by intrinsic (material)co-ordinates Xi which are initially rcctangular Cartesian. At T = t. thc body occupies another con-figuration C and thc location of the particle Po relative to a tlxed reference frame in Co is givenby the Cartesian co-ordinates Xi. The relations Xi = xi(X. t) define the trajectory of Po and therebydescribe the motion of thc body. The componcnts of displacemcnt are given by

IIi = ulX, t) = xi(X, t) - Xi (I)

and thc deformation of the body is characterized by the Green-Saint- Venant deformation tensor

Gij = Sif+2Yif

where S;f is the Kronecker delta and Yij is the strain tensor

(2)

(3)

Hcre commas dcnotc partial dilrcrcntiation with respect to Xi. (i.e. lIiJ ;: ('/lJe X f).Aparl from the purely kinematical rc[ations embodied in cquations (J )-(3). thcre is associated

with each particle a measure of mass and elcctric charge. which. for a perfect continuum. areaccounted for by introducing the scalar-valued functions Po and p. the mass densities in Co andC. p~ and P:. the volume and surface charge densities in C. We postulate that the behaviour ofthe body must be such that the following physical laws are satistied:

COllsen;ation of mass

r. Podl'o = r pdv.. 1'0 J v

Balallce of linear mometllutll

~ f Polifdl'o= r poFidl'n+r Tfxf.idAot • "0 .Ir. , ..10

Balallce of allgular momentum

COllserNuion of energy

d [J' .. d /'dt2 ropolli/l;dl'o+ dt., .•Pogdl'o

= J'. PoFi1i;dl'o+ r. pohdro+ r (T;xj,ilij+qini+S;nj)dAol·o .' lO .. Ao

where fA S;nidA is defincd by

J. Sin;dAo=" (DiEi+I3"Hi+JiEi)dl'o.::lo .. ,~o

The Clallsills-Duhem inequality

r ('. . 11) ,'(// .. PoTJ- (}J' Ei- Poe drn- enidAo~O... to ' I .. ..10

COllselTafioll of charge

(4)

(5)

(6)

(8)

(9)

( 10)

164 J. T. ODEN AND II. E. KELLEY

In these equations. Fi arc the components of body force per unit mass in Cn (and are gencrallyfunctions of the electromagnctic field). "i are components of a unit normal to All' lij are compo-nents of the strcss tensor per unit undeformed area referred to thc convected lines Xi, Eilk is thepermutation symbol. II is the hcat supplied per unit mass from intcrnal sources:Ti. qi, Si, Ji andIJi are components of thc surface traction, heat flux. Poynling vector. conduction current densityand clectric !lux density. rcspectively: g and TJ are thc internal energy and entropy densities withrespect to Co' respectively: 0 is the absolute temperature: G = det Gij: and Ei and Hi arccomponents of the clectric and magnetic fields. Obscrve that Si = Ejjk Hj Ek for polarizableand magnetizable mcdia. In these equations. we ignorc the influence of rclativistic effects. It isunderstood that the body force density Fi is a function of the electric and magnetic fields. As isseen by the form of equation (6). we develop herc an apolar theory, i.e. couplc stresses. bodycouples. etc. are assumed to be zero.

Assuming that these quantities satisfy thc usual rcquirements of smoothncss. the surfaceintegrals can be transformcd to volume integrals using the Green-Gauss theorem. This leads tothc following local forms of thc abovc equations:

Pn = ,1(G)p

Ii} = ,ji

Po~ = Ii}Yij+q\+ Pol1+S.ii

Si = bj E+iJi /I.+Ji E·,I , I l

D.ii = Pc in 1'0' P~ = [Dl}lj on An

(II)

( 12)

( 13)

( 14)

( 15)

(16)

(17)

Here tU is the stress tensor (i.e. Ti = tji"j) and [Di] is the jump in Di at the boundary.We now introduce the frec encrgy dcnsity function II and the intcrnal dissipation cr*defined by

cr* = tijyu- PoO'1 +TJ8)- Dj Ej+ /lj Hj

Introducing thcsc quantities into equation ([4) and using equation ([5). we find that

Po fI = Ii}Yij-PnTJ8- Djtt+ /lj Bj-a*

Po Or, = q,ii+ Pnl1+Ji Ei- Dj :, (Eikj IIi Bk)+a*

[n addition, from equation (16) we obtain the dissipation inequality

* I i 0cr -7Jq .j~O

( 18)

( 19)

(20)

(21 )

(22)

Equations (20) and (21) provide thc ncccssary basis for thc development of a thcory of elcctro-thermoelasticity. In equations (19) and (20). Ej is the stationary-medium elcctric field.

GENERAL ELECTROTlIERMOELASTICITY I'ROIlLEMS 165

(26)

E lee (/'o (he/'/I/()(' las (ici ()'

We now considcr a medium which is non-dissipative in the sense that (]'* = 0 and which ischaracterized by a free energy n which is a differentiable function of thc current strain Yij'

absolute temperature e. electric field E1 and effective magnetic flux density Bj:

n = II(Yij. 6. E1. BJ) (23)Then

. _ i/O. cIT II ali: * af[.[[ - ~Yij+ ~O u+ ':IE*E i +1B Bi (24)

0Yu v U i (i

Substituting equation (24) into equation (22) with (]'* = O. and noting that the r.:sulting equationmust hold for arbitrary YiJ' (J, Et and Ri. we arri\'e at the following relations:

.. en e([(/)= Po~' 1] = -TO (25a. b)

uYiJ (

. iJfI . (:II'If)) = - Po ~E*' If) = Po TB. (25e. d)

v_J ( )

Thus. the forms of thc constitutive equations for (ii. 1]. f)J and IfJ are determined from the freeenergy function (I.

Additional constitutive equations for the heat flu:,,>qi and the conduction current density Jimust be introduced to complete the formulation of the theory. Technically. according to theequipresence principle. these should involve all of the depcndcnt variables Yij' 8. Ej and Bj. Ho\\'-ever. in applications we shall use as a first approximation thc classical linear Fouricr law of heatconduction

qi = KijOJ

where Kij is the thermal conductivity tensor. and the linear Ohm's law.

Ji = Zij Ej (27)

where Zii is the electrical conductivity tcnsor. However. there is no need to inlroduce thcselinearized approximations at this point.

Returning to equation (25). wc can now obtain thc equations of motion, hcat conduction.charge and magnctic l1ux dcnsity of an clectrothcTmoclastic continuum by introducing equation(25) into equations (12). (21). ([ 5) and (17):

. en(po~ xj.,,) + Po F" = Po ih

(Yij .i

(}d ('(1(1). .- Po eI( {)f} _q.tj = Poh+l' Ej

d· ·'rr-· '1'"{.. " (. C) ('..-(el)' Ei Hj)./: = --d Po ':lE* Ei+ Po ;)B. B,:+)' Ei(. U i.· ( I

. Ofl- (Po OE~) . = P~\ I' ,I

It is understood that by E" and If" in thcse equations we mcan the effective ficlds, i.e.

Ei = Ei + eij" lij B"

IIi = H~-f.ij"lij Dk

where E~ and Hi arc the stationary-mcdium fields.

(28)

(29)

(30)

(31)

(32)

(33)

166 J. r. ODEN AND B. I. KEI.I.EY

FINITE ELEMENT FORMULAT[ONGeneral

We now outline the procedure for constructing tlnite element models of continuous fields.Let F(X) denote a continuous function whose domain is a region Jf of three-dimensional Eclideanspace P. Following thc usual procedure. we replace;jf by a discrete model consisting of a finitenumber E of Iinite elements. all connected togethcr at G prescribed nodal points (shown inFigure 1 for E~). The values of FtX) (or its approximation) at a nodal point X~al'edenoted F~.

Figure t. finite element model of a region in £2

~ = 1.2 ..... C (i.e. F~ = F( X.l»' Now consider a typical finite element e. isolated from thecollection. which contains Ne prescribed nodal points. The function F(X) is approximated locallyover element e as a linear combination of its local valuesJ;-~; at the Nc nodal points:

F(X):::: ~1~)(X)f{f). N= 1.2 ..... Nc (34)

wherc the inlerpolation functions if1i~l(X). generally polynomials. satisfy

(35)

The global nodal values F.l arc mapped into the local values f~) by

f'Y' = f!\,!l;l F.l (36)

where n~~).l is unity if node N of element e coincides with node ~ of the connected model. and iszero if otherwisc. [n equation (34). the repeated index N is summed from I to N,: in equation (36)~ is summed from I to G.

Except for a sct of Lebesgue mcasure zero. the final discretc model of F( X) is given byE

F(X) ::::~ n~I.l~(~I(X) F.le=1

(37)

For further details on the topological properties of the model consult Reference 3.An important feature of this discrctc model is that thc proccss of connecting clements or of

decomposing ,tjf into elemcnts [e.g. cquation (36)] is purely a formal one dependent only ongeometric and topological properties of the model. Indeed. thc loca[ approximation (34) can be

GENERAL ELECTROTlIERMOELASTICITY PROBLEMS 167

constructcd for an isolatcd clemcnt. indepcndcnt of other elcments in thc systcm or of its ultimatclocation in the connected model. For this reason. wc omit the clement idcntification label e inthe following developments and confine our attention to loca[ approximations over a typicalelement. [t is understood that globa[ representations can then be obtained by simple Booliantransformations of the type indicated in equation (36).

Approximmion of electrical. thermal and mechanical fields

In the present investigation, we select as primary unknowns those fle[d quantities observedmost naturally in the phenomcna (or the potentials from which they can be derivcd): the dis-placement field IIi' the temperature e. the scalar potential '? (voltage) and, in the case of magneticfields. thc vector potential Ai' [n the casc of temperature. it is convenient to rcpresent the absolutetemperature e as the sum of a uniform rcfcrcnce tcmperature To and an incrcment 1'( X.t). and toapproximate l' instead of e. Thus. according to equation (34). we have for a typical finitc element

ui(X.I) = ifr"( X) u.dt) (38)

T(X.I) = ifr\'(X)T:\(t) (39)

?(X.t) = l/J,\'(X)?s(t) (40)

Ai(X, t) = ifiS(X) A.dt) (41)

Here USi• T.v.9,V and A\'i are thc components of displacement. the temperaturc, the voltage andthe components of the magnctic potential at node N of element e. It is understood that USi ..... ASi

are functions of only time t, and that the repeated nodal index N is to be summed from I to Nc'

where N" is the total number of nodes belonging to clement e.It follows that for a finite elemcnt. the strain tensor is

"Iii = HiflJ 1I.\'j+ifI} lISi+ifIJ ifi:Y u,nuJU')

the absolute temperature iso = T..1 + ifI'\' T.\,

the electric field isEi - ,IS .I,.\' A'- -'fl,i ?\'-'I' Si

and the magnetic !lux dcnsity is

(42)

(43)

(44)

Bk = eiik l/J'~'A \.. (45).J . I

Here .p} = NN(X)/OXi. ASi = dA,v;(t)/dt and the dependence of ifI.\' on X and US;, .... As. on tis understood.

ELECTROTHERMOMECHANICAL BEHAVIOUR OF A FINITE ELEMENT

The behaviour of an clcment is govcrned by four syslems of coupled cquations: motion. heatconduction. electric lield and magnetic tleld.

t:C/uations of II/otion (~ran element

Introducing equations (16)-(20) into the energy cquation (7), and incorporating equation (38)into the resulting exprcssion. we obtain for the law of conservation of energy of a electrothemlO-elastic tlnite element

(46)

]68

where

J. T. ODEN AND A. L KILLEY

II/S.l1 = I Po rP-" ¢r" dl',.. ' ",

pSi = r Po,p.\' F, dr" + r ji,pS dAe•. f'r .. 4-"

(47)

(48)

Here l'e is the initial volumc of the elcment. IIlS.1l is the mass matrix for the element and pSi arethe components of generalized force at node N. For finitc deformations. it is understood that fiacts on a material area Ae in thc dcformed clement and, consequently. will depend upon thedeformations (see Reference 4) .

Since equation (46) must hold for arbitrary nodal velocities. we have

. 'Iitll.\'.1l,·,· + I p C .1..\'( t> + ,1..11") d,' - pSi.IIi ()~ 'I'.k °kj 'I'.k .l/je-

• ,', (Yij(49)

Appropriate forms of (I() must be introduced for specilic materials. Equation (49) representsthe general equations of motion for an electrothermoelastic linite element.

Equarions of !teat conduction for Iltl elemellf

Multiplying equation (2l) by the temperature increment T( X. t) = T. and obscrving that

Y'Tq = TY'q+q-YT

r Y'Tqd/'e= r Tq·ndA, ... ", .. A",

(50)

(51)

where q is the heat flux vcctor. we arrive at the alternate form of the conservation of energy:

wherein a* has been equated to zero. Now for a linite element wc recall that 0 and T are given byequations (43) and (39) and. accordingly. Ei is given by equation (44). Moreovcr. '1/ is defined byequation (25b) for an electrothermoclastic mcdia. Thus. introducing the finite elemcnt approxi-mations into cquation (52) and noting thai the result mllst hold for arbitrary nodal temperatures.we arrive at the coupled equation of heat conduction for an electrothcrmoelastic finite element:

.• [ d "iii-I P (T. +,pJlT ),p'\·_(~)_qi,p.\'-J.,p.\'(Iµ.1.1'P• I', () lJ .11 dl '.(IT .1 I ,1.11

.t..I/· ) I X .1,.11 (\ fI d..1.11 .1.11' 1 ,.+'1' A.1/i +IIJ 'I' oEjdi(II.11k('I'.j AW"-'I'.kARj»Jdl:r=cr

where qS is the gcncralized heat tlux at node N:

q-Y = r. ph,ps dr, + r ,p.\' (/ "i dAr.. r", .·,·'r

(53)

(54)

Again .. forms of equation (53) for spccific matcrials can be obtaincd aftcr identifying thcappropriate constitutive cquations for (I. qi and Ji. We obscrvc that the contributions of thccurrent and potentials 9 and Ai appcar non-linearly in equation (53).

GENERAL. ELECTROTlIER~IOF.LASTlCITY PROBLEMS 169

Electromagnetic field equations .lcw a finite i!lell/eflt

To dcve[op the equations governing the electromagnetic field in an elemcnt. we utilize threepotential functions: Ihe free energy n. the scalar potenlial ? and the vector pOlential A. Thesepotentials have thc properties:

(55a. b, c)

(56)

(60)

Additionally we must usc the tlnite elemcnt representation for the conscrvalion of charge,equation (17). Thcsc equations are formulated in the following development.

Multiplying equation (17) by 'P and using the identity

V'(1'D) = 1'(V·D)+(V1')·D

we obtain the following global form of the conservation of charge:

- r Di-:p.ide, = r p,.-:pdv,.- r 1'DinidA,. (57).. ,', .. rf' .. J" r

Introducing equations (25e) and (40) into equation (57) and observing that the resulting equationmust hold for arbitrary nodal voltages. we obtain the equations of charge for an electrothermo-elastic finite element

.. eli .I ( - P ;.-) 1/1\ dl' = eX (58)IIcE"f ,I ,

.. r, I

wherc cX is the gencralized e[cctric charge at node N:

c"\' = - r t/r\'p"dl',+ r t/1NDil/;dA, (59).' t', .. A ..

Using equations (8) and (55a. b. c) and cxpressing S,il in terms of E; and Hi wc sec that theelectromagnetic conscrvation principle can be written in the form

_,'I ijk (1 ([" A I . _ j" d ( (! fI ") r • I ."JpoE iJBl(t'.I.+ k\id1, - • ,.,\PlldtcEf (t'.I.+Ak), dIe

, c'l'l 'I (' "" C ')d 1 Jk . d- . Pllf')' ('BlAk.1 1',- . (1'.k+llk) e,.. i.e> , ' .. t·,

Using equations (40) and (41). the equations for a typical finite elemcnt are givcn by:

( ..d ( (Ii .1.\.) (" c tI ') .I.X" Id' _ /' 'd (. CIi') kI[,t, \' .l.\·· ] ..1/') I POcBl'l')·,./'x+ PO(iBi.;'I' . Xkl t,- ••.,ldt PnOEj. -J 1'I'},?x+'11 AXk dl,

(61 )

Sincc equation (61) must hold for arbitrary nodal values of 1's and A.\·i' we conclude that

and

'/ . ''fl' d' "fl' Ij f·;lk(p ~)t/1.\'_-(p ~\t/1.v dl' =j.\'/;.1,.,1 °cBl,.1 dt IIEfl I 'whcrc JXk is the generalized current density at node N:

J XI.' = -I' .try jk' de + /' f;lk II .try 1/. dA'I' r l'l',.,.. tOt .. At'

(62)

(63)

(64)

170 J. T. ODEr-; AND II. r. KELLEY

(65a. b)

(65c. d)

Equations (58). (62) and (63) govcrn the electromagnetic fields in a finite elcment. Additionallywe observc that equation (55a) does not uniquely definc thc vcctor potential A. i.e. a vector is notuniquely delincd by specifying its curl. However. we will defcr a complete specification of A(i.e. specify div A) until specific forms of the free energy function are considcred. sincc we desirethe simplest formulation in each specil1c case.

Finite element analogues oj Maxwell's equations

A[though a multitude of special cases can be dcrived from equations (49). (53). (58). (62) and(63). we shall cite here only one example of special interest: the finitc element analogues of thec1assica[ Maxwell field equations for slationary media (i.e. E:~ = £f, B~ = Bi, etc.)

div B = O. curl E+ DB/ot = 0

div 0 = Pe. curl H = .J +BD/itAlso. we have

divJ = _ llPe(1t

(65e)

(67a, b)

Equations (67a) and (67b) are identically satisficd ovcr an element due to thc definitions of?and A and thc fact that thc interpolation functions .fi-"(X) arc continuously differentiable. Supposethat Yil = T == O. and that the free energy is givcn by

-I I t:PoI = -2 B· B- J E· E (66)µ -

wherc f{ is thc permeability and t· is the permittivity of the mcdium. Thcn

. 8n I . . DO .H' = P - = - B' and [)' = - P - = eE'o a Bf µ II [l£i -

Thus. for a finite element.

. I ..fi"H' = - e781 • A .µ .7 .,"and (68a,b)

Hence, equations (58), (62) and (63) can be rewritten, rcspectively. in the forms

r .fi} Di dl',. = c·\~,.~

and

(69)

r [eilk H!,.fi:X]dl'e = (:."+ r .fi;'fJkdl',. (70)• ~ ,r,

r. [eilk HJ.fi.\]dl'" = :t r..fiX Dkdl'e+ r .fi.vJkdl'e (71).' t ~ .. tt .' ,...

Equation (69) is the finite element analogue of cquation (65c) and equation (71) is the finiteclement analogue of equation (65d). Equation (70) is the finite element analogue of the continuityequation (65c). The term on the left side of equation (70) reprcsents a finitc element approxi-mation of div (curl H) which. for thc continuum. is zero, but in the discrctc model is onlyapproximately zero over a tlnite element. As the finitc element network is retlned. this termvanishes. Consequently. it is meaningful to rewrite cquation (70) in the form

(,.\ = - r ifJJJk lI,',.. I',.

which is now analogous to equation (65e).

(72)

GENERAL ELECTROTHER~IOELASTICITY PROBLDIS 171

LINEAR ELECTROTHERMOELAST[CITYUp to this point. we have placed no restrictions on the form of the constitutive equ<ttions forII(Yij. 8. £i' I1j). qi or Ji or on the magnitudes of Yij' T.Ei or Bj' Henceforth. we restrict ourattention to linear electrothermoelasticity. for which Ihe heat nux and current density are givenby Fourier's law [equation (26)] and Ohm's law [equation (27)1. the strain tensor is linear in thedisplacement gradients (2Yij = Ui.j+/lj.i)' C and Co arc approximately the same. E~ = £i.H~ = Hi. etc. Further. we assume that thc material is characterized by a frec energy functionof the following quadratic form:

n - IEiJkm + Bij 1'+ Cr' 1'2 +~Xij £ EPo - ~ Y.jYklll Yil ?T. ~ i j- II

+ Ri BiT + ~ yij Bi IJj+Gi Ei T+ Il'Uk YijE,,+ [ijk'YiJ ilk (73)

Here Eiikl1l. BU. Xii. yij. Wijk and [ijk are arrays of matcrial parameters which. for homo-geneolls materials. are rcgarded as constants. and c,,, Ri and Gi arc the spccific heat. pyrornagneticconstant and pyroelectric constant. respectively. These arrays exhibit thc following symmetrics:

£iJkm = Elikm = Eijmk = Ekmil (74a)

(74b. c)

(74d. e. f)

According to equations (25)-(27). the constitutive equations for stress. entropy ..... etc. forsuch materials are

. an . .. . .If' = P - = R' 1'+ Y'J B.+LJlIlIy

II oBi J 1m

(75)

(76)

(77)

(78)

qi = Kij T.j' Ji = Zij Ej (79)

where Kij = Kji and Zij = Zii. We can now obtain explicit cquations for typical finite elemcntsby introducing these relations in those derived in the prcviolls section.

Eq/lations of //Iotiol/ .//I.\'.l1jj .+k·V.1lmu +b-Y.lIT +/V.Y.1lm +lv·VMmA +cS.1lmA =p.\'i.111 I .Itl/l I .11 I r.11 I .1Inl i .1/1/1

Eq/lations of heat cOl/duction

0.\'.1/ T." - kS.lt 7:" + To(lljUt liJIi +g.UI 'PJI +g·tlf A.I/i + r;Y-lf A.1/i) = q.\'

Eq/lations of electromagnetic field

d.\'Jt 'P.1t+ei""1 A.lH - I V ru/I.l1i - g.\'Jt T.ll = c.\'

h;~'olf T." +n;~·.Il 1'." - x'~' SI/I '{11m +q-".\' A.l1/1 +s·~··l/mli.\11/1 +f·]/.\'I/I /I.llm = J.\'II

Z~JSj A .Ilj = )'~t.vCP.1/

(80)

(81)

(82a)

(82b)

(82c)

172 J. T. ODEN AND 8. E. KELLEY

In these equations. 11/.,,:11 is the mass matrix defined in equation (47). k·rllm is the stilrnessmatrix, a·\'.I1 is the specific heat matrix and p'JI is the thermal conductivity matrix:

a·\'.11 = r c, . .ps .p.1I dr,.. rl"

(83)

kSM = f' K'i .I..y .~M de (84)"",1 'I'.J f..' ft

Also p-''''' ttY. c-\' and J s" are the generalized nodal forces. heat flux. charge and current densityat node N. Other matrices for the discrete model are defined in Table I.

Table I. Element matrices

b~·.11 = r B ii rp'\' rp;~l dr,• I',

H/;Y.lIr. = /' Wimk rp;i rp.1l dl:,.. I'..

g.\·.ll = r Gi rp-" rp;'f dl',e t',

r;\·.ll = r R'rp'" rp.Il dr,· "

e;\'.Il = r X i} if,:; if,.ll dr',· "

t1~·.ll = r EiJA if':~ rp.ll Prlk Rj dl',O· "[0

q.\'.Il = r rpx rp.Il dl',.. I',

!;/.\·m = r EliA' rp} 'µ;'/ P ok L""i dr,• l't'

".l/X _ , •• I..V .1..1l P vAi dl'.'.n - 'P 'P.I ",1. .fl'l'

.1',

In Table I we have used the formulae

~VlY.ll = /' Wiik rp} if':'l dr',• I',

CSM ... = I LijA E"nA .~X .1..11 de, 'f'.J 'fl.' ~.' I',

KiY•1J = r Gi rpx rp.Il d,',• t',

c/NM = r Xii rp} rp:~' dr',· "

,,~'.I[= r rp-" rp.ll P ok GA dv,• t'"

X~/N ... = I rp-" rp.ll Pnk XA," dr,.l't'

S~·.I1m = r if,.y r/r.~1PnA W·mk dr',.I'r

Z·II,\' i = I' .1..11 .IS dl',. tp,n 'P,l ,• t'"

(85)

In the case of homogeneous bodies for which the matcrial coellicients in equation (73) areconstants. some simplification in the definitions of the e1cment matrices is possible. In this case

aSJl = c" mS.l1 = c •.q'\·.Il = C,. Ri r;-'M = c" G, gi\'.l1Po RlII Rm GmGIII

(86)

(87)

etc. Further reduction is possible if isotropy is assumcd.

GI:NERAL EI.ECTROTHI:RMOELASTICITY PROBLEMS

SAMPLE PROBLEMS

173

In this section. we considcr a sample problem which involves time-dependent heating of ananisotropic e[ectrothermoelastic half-space. A simplitled linear model utilizing onc-dimensionalspatial and material forms of the finite elemcnt equations is used. This formulation representsthe finitc element equivalcnt of Voigt's9 classical linear equations for piezoelectricity. Althoughthis problem is. in some respects. of a simplc nature due to the absence of magnetic effects. itillustratcs the merits of the finite element representation as we have developed it in this papcr.

Using a standard simplex model. the non-dimensionalized. coupled finite element equationsof motion. heat conduction and electromagnetic field for a typical interior node N are givcn by

Equations of 11/ot;on

(88)

Equat;ot/s of heat cot/duc/iot/

Equations of electric field

$.\ = 7/0\ Ox-gABswhere wc have utilized the non-dimcnsiona[ equation parameters listed in Table [I.

Table II. Non-dimensional equation parameters

(89)

(90)

~ = ~'\'IK

CI'.=-/

K

l7 _ aE*- KP* 1;, U

8- T- To

ii, _ (/8- K*lll III

"OJ = To 8P*

pc,. E*

8% = 8. ~.\ (K* q,~'8To (/,)

1\*K=-pCe

a~ _ E*p

.\ = G8Topc" XIII"

_ IVTJ = XID o

In thc abovc table .. \\ is a characteristic length. t the real time. $0 a reference voltage. c,. thespecific heat at constant volumc. p the mass density. B. G. E. X. K material parameters. To areference temperature. 7/ an electromechanical coupling constant, ,\ an clectrothermal couplingconstant and 8\. O2 are elcctrothermomechanical coupling constants.

We consider a linear elastic half-space (Xt >0) subject to a ramp surface heating of the form

(91)

(92)

174 J. T. ODEN AND n. E. KELLEY

where Tl is thc surface tempcrature. Tf' is the final surface tempcraturc and II) is thc boundarytemperature rise time. Additionally we define the following non-dimensional parameters

.,a-(93)TO = -/0

K

T81 =-. O~T~TII (94)TO

81 = [. T~T (95)

Our numerical results for the half-space are presented in Figures 2-9. This system of overtwo hundred ordinary diflcrcntial cquations was solved using standard Runge--Kutta numericalintegration. Figures 2-5 contain the dimensionless temperature B. displacement D and voltage if>as a function of dimensionlcss timc T. whereas Figurcs 6-9 contain thc same dimcnsionlessvariablcs as a function of dimensionless distance ~. In all ligures thc valucs of~. 8t and TO are 1·0.

Figures 2 and 3 illustrate the etlects of the electromechanical coupling parameters 82 ontemperature B and displacement D. The curves for 82 = 0,0 arc identical to the thermoelasticcurves of Oden and KrosS.27 Figure 4. which contains the dimensionless voltagc <1> as a functionof dimensionless time T. illustrates that thc thermoelastic curves (82 = 0·0) arc also valid for anelectrothermoclastic half-space if the parameter definitions devcloped in this paper are utilizcd.Figure 5 illustrates the effect on the non-dimensionless voltage if> of a non-zero 82, It is apparentthat these curves are significantly different in character from the 'uncoupled' curvcs of Figure 4and additionally that the sgn 82 is of major importance in determining the shape and amplitudeof thesc cun'es.

0·5~ ='-08,=1'0

20

82=0'082=1'082'-10

1505

o·t

0-4

1(2) 0.3

'":;'0~c.E 0·2~

t·ODimensionless lime T

Figure 2. Tcmperatun: variation at unit distance in clcctrothermoclastic half-space subjected to rampheating

GENERAL ELECTROTlIERMOELASTlCITY PROllLBIS 175

2·01·51·0Dimensionless time r

0,5o

01

O-oa

D 006CQ)

Eg{=10

- 0·044

~a I

°1=10-002 '

Figurc J. Displacement variation at unit distance in clectrotherrnoelastic half-space subjected !O rampheating

0·5

0·4

~ = 1081 = 108

2= 0·0

Ie 0·3Cl>C7'.E'0::> 0·2

01

oo 05

A=OO1"1= 1·0

\'- "1 = 0,0

A = -10

15 20Dimensionless time r

Figure 4. Voltage variation (uncoupled at unit distance) in electrothermoclastic half-space subjected10 ramp healing

]76 J. T. ODEN AND n. E. KELLEY

05

0-4

0·1

o

t= ,·08= ,·0

1

0,5 1-0Dimensionless time T

t'5 2·0

Figure 5. Voltagc variation (coupled at unit distancc) in electrothermoelaslic half-space subjectedto ramp heating

Figures 6 and 7 illustrate the tempcrature and displacement as a function of half-space non-dimensionless distancc (for different values of T. Only the curves for j)2 = 0,0 arc included as theyare considered representative.

ICO 06~::l

liQ)

~ 04Q)t-

10 20

Dimensionless distance {'

{' =1081= 1082=00

50

Figure 6. Tcmpe ....lure distribution for vOlrious lime parOllllctcrs (T) of an electrolhermoelaslic half-spacesubjected to ramp hcating

::eoI:l:r-m'7

iF.

c~z~;o:l;;.r-~r-;'1I"')..,;o:lo..,:I:

"";l::..(5'"r-;;.'"::::?.......,;

03

05

Dimensionless distance l'

0'4

02

10& 0·1Q)t:Jl12;§

00IJ

,=1,0

-01 I-J o{l'O82=-1'0A =-1,0

-0·2 ~ TJ =1·0

I I I I0 10 20 30 4·0 50

Figure 8. Voltage distribulion for various time parameters(T)of an electrolhermoclastic half-space subjected to ramp

heating

5·0

, =1·00,= 1·00=00

2

-10

-1,2o

-0·8

+0'2

+0·4

c'"E'"uoa.III

i:5

I=>

1·0 2·0 3,0 4·0Dimensionless dislance~

Figurc 7. Displaccmcnt distribution for various timc para-meters (T) of an electrothermoetastic half-space suhjected to

ramp heating

-..I-..I

178 J. T. ODEN AND B. E. KELLEY

The non-dimensional voltage curves of Figurcs 8 and 9. which arc plotted as a function of non-dimensional distance for various 82, arc significantly different for the dilferent cases The 'coup[ed'curves of Figures 8 and 9 are linear combinations of the tempcrature and displacement curves.In Figures 8 and 9 the sgn 82 is a major contribution to the shape and amplitude of these curves,

2-4

20~ l = 1·08, = 1082 = 1·0

l6t \ A =-1-0~ =-10

12

1080 08Ql0'E(5>

04

00

-0-4

-080 10 20 30 40 50

Dimensionless distance fFigurc 9. Vohagc distrihution for various timc paramctcrs (T) of an clcctrothcrl11oclastic half-spacc

subjected to ramp hcating

and since the sgn-amplitude combinations can be different for various materials. almost anyamplitude or shape combination of the voltage curves is possible.

The merits of the formulation are clearly demonstrated by this example. Whilc morc comp[i-cated geometries and boundary conditions involve lengthier and more cumbersome calculations,the general approach is essentially the same as that used here.

ACKNOWLEDGBIENT

The support of this research by the Air Force Office of Scientific Research through ContractF44620-69-C-OI24 is gratefully acknowledged.

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GENERAl. ELECTROTlIERMOELASTlCln' PROnLEMS 179

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