AD-A080 712 RUTGERS - THE STATE UNIV PISCATAWAY NJ DEPT OF M HAer-ECT F/ 20/13A FINITE ELEMENT SOLUTION OF THE COUPLED DYNAMICAL THEIRMOELASTI-ETC U)JAN 80 Y CHEN. H GHONEIN. J DAVIS DAA29-7-4-0052

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Two test problems are presented to show the micro- and the macro-time behaviorsof the thermoelastic slab. The responses to a realistic set of pressure andtemperature pulse inputs simulating the interior ballistic type of action arealso presented.

I-I

:, t 10

rodeS

A FINITE ELEMN SOLUTION OF THE COUPLED DYNAMICAL THERMOELASTICITY

EQUATIONS IN A SLAB

by

Y. Chen and H. GhoneimDepartment of Mechanics and Materials Science

RUTGERS, THE STATE UNIVERSITY OF NEW JERSEYP.O. Box 909

Piscataway, NJ 08854

J. DavisARMAMENT RESEARCH AND DEVELOPMENT COMMAND

Dover, NJ 07801

" ,- , 4 .* - *1 -

whr 2is the talca, and are che lne's con-stants, F is the b-:. force vectr : is -the lens icy

- :~cee sc..- .. -odcl og- h as been presented to and vis a constant given b..e ne-'ioenj. -' ~ !" lnamical cherm0n

eacc.t-? tactal '.-arzazle is3 discreticed by the =C3 ss :2)

4aeracr. scnzeso :central clccerence ec-eme.-ecue;ecce c 5 ~- th A'~k'nan o where Y. is thea cce

90' ; -n- -arma-I vx~ns~on. -he

- ~n7 - : ee itas fcunl ch-at the two last cer,, nntele-ft-hanA d )f -s o. U) is the -orca-_Va- ie cltse com,,oariaco per uinit volurre due cc tne cemperacure cra!dtent in tne

Wtton w's-to, tne cancear: ned _4--,. 7n an uncoe cclnt ereneratu;re il:.e toe ' urc -: e navicr 4-3 civen.

5- --- - 2ens_47e out coerfe L5no The second acuaticn cc toe calt of cercial dildi-, To imuace tne. racro-t:ne ='e- c-erential ecuations oc toe cacel is the energyc' alance

n--enc:-a - e-a-vco wouild disamrear. a ie vil-1ot-s ;c'-s are resntsl! to show the micro- I'

ar4 -sTar-S:e ~ avI;r D- toe toerrnoeLnst:c stah. 72 - - -7iv i-

-Ie7so~s- 2ce ct pressure antc ten-zzls -. -r s-rnul- t the interior ballistic i h if."*t! ste su~ n

ct--e of sct on' ae also Ceqanced. whr <itediffsivc,~ o etsuc n= '7/0< is tne ar!:cint temoeracure.

Assume coat toe bod 4frce Fant the h-eat source

can be dropped and tha th esl=ment vec:tor u is-he :op3 o' erroeasticicv orcblsn has b-een or cne form

create ':y many :vsatut7ators -n cns F:ast. A mart:Iallisetingof c h r=' vant caters are alven In the refer- u s grad: + curl a(4)ence -. mos 7oese cacers lea], with the oroblenoc a semin iz scece an-: a caw contain a ccnprehsn- Substituting (4) into ()and :3) y4ell s a pair ofcute n--er4:cat treecrent. tn many acolied poblems the couple PDt in and T2 and an u-ncouclad Pat in des-coup:4na z--n t -e enerrv balance eaquation is ignor- cribing the shear wave propagattlon. Chess ecuccnable tue to cne snallnesa cF its magnitude. in other are:crob!-aa su-ch- as tha cc:.. toe er-lal stresses 4-n a coint artal. 4-= ..ucn -4; rate cc loadinz the cocca721 - 1 7

2, 5

rutb ~ -' h eulso h nalvsis shows

cant aot ce -n*-. analysis for su-ch - Z-6

crobrleis---h-------:uar naroematcical :codel to be I

del- -:. ns -ae c3a I tounoart value problem. for a72)- = ()cair 3-;rci teff2titat esit~cns :.ove'rin theC

stres an- : e -- z-----------Is. the boundar' con-c1cions oro"cn to to e time d4ecendent stress and C2 .d

termperac on cone sujrcace and the free traction whrIi=yX2( = 0ad2 .c .aand! art:--------r on toe other surface. and -: are the dilatational and the shear wave speed

A finte elemcent approach is used in discretizing -

the szatia. as weIl as the ti'me variable. in each respectively. Since the' temnperature fields has nointeraction with the shear wave, the subs.equent

discretizse'on a ---oice of two different approximation anlss'il ietit tetont s)n (6).sc cro'id :ed to funiha c-omparison Tolhug h specialize the cair ofeqain aslbw

crotdle. analyzed is one-dine"sional in space the method-T qain t lbwolr!evelcped, c3n be transferred to the problem of an ca wrt (6 inhe on

annoucs_: witocut t o ciuco co.ance In :ts principles and I mt-=ics .20th next section the mathematical model :42 t'4.1 b---fen in a general form first and then special- I

-ze,!-T A la::.and (5) as follows

xx c x ilet the tarprst-r 4n -he medium bea 7and its has-

placement he !enoted dv -., then the equation of motion Since it is more convenient to use the stress insteadcc toe Me

4±X- sn LAivfen tn a vectorial form as follows, the pooential :as the vtariable in *one-iimensional

Lu.'. r~d -problem, we introduce the constitutive equation for!Lv, F - vradT 311I a theinsolastic medium,

2$-..VWt

xx

where 7is the- _nisxisl stress. To discretoze the spetial vlariable in (16) an:2am,.; I) t Lrminste trijm S) yi'4elds (17) we 'used two ~rar~s a.. the Salerki. troce-

dure and the t.entratl 4ot-rsnce acneme.

xx tt The Galerkin Procedure

And In this crocelurs we sna'.1' assume that bDoth

7 ~~Approximate solutiorns -f(1CS and (17) 1 and T --an z-e- tw :2 extpanded in terms the same spat al anproxima'- r =

.1x) in the following day.

The zgunlsarv iz~~:n re ;-ien as L:i's es

ix 1' 1 0c (t-C '0(i (20)7x-t 3 7 t ' - (i * t 70= (13

T x) ~ . T. (t)

T'Z:1 3~T tC = (4) where FNnr+1 n is th-e numbner of internal nodes and

P. x) is oetinec aswhere i. s tn.e t.nI:ness of the slab, 35 ="1/

2 anid h, are the hneat transfer =,oefficient-= ''c

an-c :-.e :czct :tr the material. . -

rlN-:)t.EN:l:- .o ' ?l-AO NS 0 h xx> i=0,1,2.... (21)

Ti f acilitate c:omputation the fo;llowing non-di-7nens::ral n-ant:tzes are :nt-rodu::oA where h isa dimenai:nless length if the interval. A

-racnica' '"lst-"-ii)n or tne run-ction in (22- is given--Tp I in Fioure 1.

-0 x -, - - - 15 For convenience we introduce the syrtola L and~~=7 0 \2u 2Isuch tht thDOE in (16) and (17) ran be abbreviated

After overting tne equations into the non-dimensions1 asI

form o~ne '-ars3 are drtrped for ease of writing.Thus, after some algebra, we have I.a),T) 0,

I2Z)

7x -'. T-= 0 (16) 0 ,T) =0t2

Then the reite .. and R, tan be defined to neand

R, (a,?) and (3- -- '' N \; 0(17) R ' -

Oxx -1 2'* 1 t

C-)+ u For the boundar*, conditions we abbreviate (13) and C13)3C o2 To.2~ as

The boundary conditions reduce to 1 I1c , BI()-0 ad 4

x . :.03') - ft) -0 5 2C() - 0, 5 C T) - 0 .25)

T- 3 C - -qIS)1In the Galerking procedure we require that th-e

weighted residue to vanish with the weighting fir=ro::n

x - (rl,t) - taken as v.lx' , i.e.

where 3, 5 ~ , f~t)- f*/( ',.2 2, 0 .1 .'0 j ''2.....1

For i - 1, (26) le*ads to after integration by parts andsom algebra.

WiS iPaz IS MET gVUam nWWS40

.4j Ow .5.'. 1)W

d. . r } x ) -V .. . t-x +eli ) S_ X g t d

- x : x . "'7' - f x) (x)T '(t) (x)dx

SN i N

""-'; :2; ' itx)(jxxTi'

T ... ) +xl i-(

- I Performinq :he necessaryI integration in (27; and (321

i' (x)'u. (x)T (t)'3x! (27) Yields -he x-discretized equation of (U1).

To reat the term in the bracket in (27) the ther- (7+h) -T oma- ;cun-arv zonditions in '241 Cnd 25) wil2. be used.

-et 2 -1

R ) -B , (T I - I 2 - iT

Ch 2:N N h2

I=- I I - -1 2 -i T

and -1 (1+82 h) TN

3,

N N= )x) r (t)+8 2 i %(x)r(t) (29) 2

i-C 1 i=0 5

We demand that the weighted residue of R(O) and 1 4 1 TRVl) varish. This statement, when (28) and (29) are 6 6 6ised, :an re zasted int -- e olowing forr by intro-iucinq a delta function in each term. _ 4 T2

6 6 6

, 'X) . *x)T. ct)i(x)ix +( A - - -2 -

J1 21 4 1 "

-n

I N

- V! r , ix) (x) T (t) xldx 1 4 Ni

11103

. ' x) '(X -I T( )dx 0 (31) _ 1 " T

; 0 0

6 6 1

66S f(xT (t) d ( 6 6

[i- i(xlI' fx)T. (tI) ! 0

(33)

4i.

tr

For i 2, f R2 P (X)i -I -0 (A) + [B I + X [HITi-f (35)0 n

where (4., [A 2 , etc. are the moefficient natrices in

3y a similar calculation as above we obtain the x-dis- (33) and (34).cretized equation of (12)

Central Difference Scheme

2 1 1 In this scheme the approximating functions P( X)

-1 2 -1 0j are quadratic instead of linear as in the case of theGalerkin scheme. The weighting function is taken to be

-1 2 -1 3.5 (x-x.)(3,

ILet

h2 - 1 2 - l) I i+12(x,t) " -P4 (x)O (t) (36)

i+1

4T(Xt) - i (x)Tj(t) (36)

6 6

1 4 where

6 6T (x-.) (x-xj l

1 4 1 i Xj-1 (x -X ) -- x+)6 6 6

(-j- j -1j+l

-1.(x) - (x -x )(x-x (37)

4 1! j(X-X. 1 ) (X-1 j+ 1 )

6 6(x-x. )x-x

1 '( (x j+lXj- ) x

6 j+ iIU+

graphical sketch of (31) is given in Figure 2

The residues are defined as:

6 4 1 F Ta 1 f "1. R-LI(0, T) andhi2 L2 (aT) (38)

6 66I hK and we require that

6 6 6 114 f R (x-x )dx - .0, j- 1,2. (39)

I ! Performing the operation specified in (39) yields

I I I T 0 (__ 2 T T )- I X A ) i x 06 6 6I Ch1 i- + l +1 2) 1l1

1 0 ) (40 )

6 6 ( 2a -c ) + i + I

0

h2 1- 1.~l2

(34)The boundary conditions will be discretized as

follows. Applying (40) to the boundaries at x0, weEquations (33) and (34) can be abbreviated as have

follows

Ch2 -T.2To-T I ) -0 (41)

/2 [A1)T + "l+N1 X2 ) )EDI I + I 1H f (35)

Ch Calculating the residue R(O),

R(0) a 91 (T)

t}

. + +, .. . ,..5

and settinq F0 1 2.

J R(O) d(x-O}),x - 001

result +A'2

1 1 -n_1-h(T-,-TI) - SITo +

61g 0

0 n

from wh.Lch we can calculate T.I,

T 1 2h0IT + 2hB 1q (42) 8

ER 9 -LI

uhaituti ng (42) into (41) yields 0

I ), 28,-[2T (l+hg.)-2T I + ('+ ) I _ 0

C I (45)

1 0S-L1arly, stting f R(1)6(x-l)dx-0, we have

0 0

C-'([-2TN42(12 hT0)(44) and

S2 2h)TN+ 0 (44)Ch

From (40, (42) and (44) we finally obtain the x-dis-

cretized equations. - 2 -1 a

,, 1h2 1

~(1+0 h) -l1 T --- ]

-1 2 -1 T-1 2 -1 T 2 1 Ci

Ch 2 22

-1 2 -1 T-

-1 (1+6 2 h) TN L 0I

0 1 [T 1

2 (46)

1 2 2

IIh2132 g1(1I) "(G-

T " I

I 2qui~ons (45) and (4G) can be-labbreviated as

1 n follows.

T.3

-V a2 1--

(47)

T**J +: .... .. .. ++, , -+

where [E 1] rE ], etc. are the respective coefficientmatrices in (41) and (46). Notice the equations in M C K(45) are of the same format as those in (35), only the (-l

2 i-y +l _(-Xi1+Zi+,)+At K(yi-i4*y. i )

coefficient matrices are different.

THE TIME DOMAIN 6 (ili-i-i+4

Since Eqs. (35) and (47) are of the identical Rearranging,format, only (35) will be discussed in detail. 22Equation (35) can be further abbreviated as follows. At at 2 At At2

10 [01 4_AI 'X [nBra IiXe.)

(0] -01o t)

F 2 1Br

5

1 [+1 It -(F. _+4F. +F (2

X2HT iLJ6 -- i~+.(21 [) [IJ [i 1 01 (01

Approximating the First Step

TFor the recurrence scheme to proceed, both andch y must be known. An approximation of y is obtained

+ 1 (48) by linear interpolation of the function y(t) over the

1 'AJ interval. i.e.

To further abbrevi.ate, write (48) as Yt otz 4(~l(3

Again requiring the weighted residue to vanish, i.e.Mj + Cj + Ky - F -0 (49) AMj+Cj+~-F-OAt

R N 1 (t)dt = 0

where v - other symbols are self-iden-we have, after a lengthy calculation,

(N+n) ,

C A C + t''__[tifiable. 6- + -tK)yl (L--K)vo+ (At) f (54)

2 3 1 2 6 o 3 -.

In discretizing the time domain we shall developthe recurrence schemes via i) Galerkin and ii) back-ward difference schemes. Backward Difference

In this scheme a quadratic interpolation between

Galerkin two successive time increments is used and the weight-ing function is 5 (t-tj~)

Expanind y in an infinite series (61

The vanishing of the weighted residue yields

X NI t ) Y ., (M. 3 A t CF - ( 2 2t +C .T '1 (55 )2 C)~4. 1-(2M+ _A + C +4t-+K) y, 1 -

-_It-tlIi t COMPUTTION AND DISCUSSION

where Ni(t) 4 In the previous sections methodologies based on

e-0, It-ti t at finite element approach have been developed to treat

the boundary value problem of the coupled PDE ofdynamic thermelasticity. See computational results

Abbreviating (49) as L(y) - 0 and define will be discussed.R(y) L (v). Letting the weighted residue vanish Different combinations of the spatial and timeand performing the necessary integrations, we have discretization were implemented. In the graphs pre-

sented at the end of this paper a notation such as"CD/BK" would mean central difference in spatial die-cretization and backward difference in time discreti-zation. The notation GK stands for Galerkin.

The parameters of the mathematical model are onlysuqqestive values which would provide some idea about

lpew-

a realistic problem. Within this objective, we have A Modified Step Temperature Input. q(t) = 30(l-e'it

chosen the following set of parameters X, - 1.0,-3 -2 Uncoupled Problem. Figure 6 shows the temperature

2 = 10 ( 0-, TO 1 2 profile for various times due to an input of a modifiedThere are two scales of time for the thermo- temperature step with amplitude 30, for the uncoupled

elastic problem. One time scale corresponds to the equation (X = 2 - 0). Since the thermal problem hastime of travel through the slab of a dilatational a long time scale, the result demonstrates that usingwave. The other is related to the time of diffusion At = 20 and 500 yields no significant difference.phenomenon. The difference in magnitudes of these Figure 7 shows the time response at two differenttwo times span five orders of magnitude. The wave field points. On the same figure the instability oftravel time is unity in the non-dimensional time the Galerkin time discretization is demonstrated.defined in Section 3. The "diffusion" time is about

Coupled Problem. Figure 8 and 9 are the results105 non-dimensional units. Therefore, basically there in responses for the coupling specified by X2 = .01exist two types of responses in this type of thermo-elastic problem, which we shall call micro-time and and two values of XI = 0.1 and 10.0. It is observedmacro-time behavior respectively. Thus, the time in- that the temperature responses show significantcrements At used in the computation also differ by difference when the coupling parameter XI is 10.0,similar orders of magnitude. which is an upper bound value.

Three test problems will be discussed: (1) re-sponses to a unit-step stress f(t), (2) responses to Dual Pulses, f(t) - .1 te

- ' t g(t) = 8.1548 te

-'It

a unit temperature step g(t), (3) responses to both ta stress and a temperature pulse, simulating the gas Simultaneous inputs of gas pressure and tempera-pressure and temperature in a gun barrel. ture as approximated by the given set exponential func-

ft)=l-e'it tions yield responses at x - .2 in micro-time as shownA Modified Unit-Step Stress Input,~tin Figure 10. It is observed that the wave frontarrives at x - .2 at t = .2. The successive cycles of

Uncoupled Problem. The theoretical response wave travel are depicted by the ripples oscillating(solution to the wave equation) of a slab to a unit- about a sean curve following the general input pulsestep stress input imposed at x - 0 is given in wave form.Figure 3 by the rectangular waves for x = 0.24. This Figure 11 shows the micro-time response in thetheoretical answer is used for comparing the accura- temperature and the stress. For XI - 1.0 and X = .01cies of the responses to the modified step computed 2by the several -ombinations of At and Ix as exhibited the top figure shows that the coupling effect onin the tabulation in Figure 3. This figure shows stress is minimal, whereas the lower figure shows thatthat decreasing at and 4x improves the accuracy. the coupling has a strong effect on the wave-like tem-The amplitude of the response using 6x - .05 and perature profile.at - .002 matches the theoretical solution. Other Figure 12 and 13 show the macro-time temperaturecombinations show deterioration of accuracy after one vs. time and the temperature profile under the action

or two cycles, of a dual stress and temperature input. Notice thatFigure 4 shows the comparison of the quality in macro-time the stress remaining is low. This is

of the computational results based on different due to the fact that the slab is free to expand.discretization schemes. It is shown that CD/BK andGK/UK give quite close results. In other words, the CONCLUSIONspatial discretizations by CD and GK do not yieldsignificantly different results. The backward scheme The theory and the implementation of a finitein time introduces some artificial "damping, causing element methodology in solving the problem of theinaccuracy, wher-as the Galerkin scheme in time couple dynamic thermoelastic slab has been established.generates oscillations of the amplitude about the Test problems of a unit-step stress input as well as aexact value. These oscillations can be reduced bv de- unit-step temperature input are used to exhibit thecreasing both At and &x. Figure 4 shows that using two different types of responses, namely, the Micro-Ax - .05 and at - .02 reduces the oscillations when and the macro-time behavior of the slab. In thecompared to larger &x and At. The graphs showing the spatial discretization both the Galerkin scheme andresults of larger increments are omitted for the sake the central difference are satisfactory whereas theof space. backward difference scheme is preferred in the time

discretization. A realistic set of the stress and theCoupled Problem. Figure 5 shows the stress and temperature inputs, simulating the interior ballistics

the temperature response due to a modified =nit stress of a gun barrel, is used to generate responses.applied at x - 0 and zero temperature input at x - 0with coupling parameters A1 . .01 and 2 .1. ACXI2WLZDGKMT

It is shown the propagation of the stress wave and The authors Y. C. and H. G. gratefully acknowledgealso the temperature wave due to the coupling effect. the support of their work by Grant DAAG29-79-G-0052,The solid lines show the wave front at various time Army Research Office.instants. The dotted lines show the temperature wavebeing synchronous with the stress wave. The magnitude REEENCESof the non-dimensional temperature is 1/10 of that ofthe stress. This ratio is clearly determined by thecoupling parameter A1 (-0.1). It is also obeerved that 1 Nowacki, W., Thermoelasticity, Addison-Wesleyat the time corresponding to that of the wave travel Publishing Co., 1962, p. 296.through the slab (micro-time), no diffusion effect is 2 Sternberg, E., and Chakravorty, J. G., "Onobserved. Inertial Effects in a Transient Thermoelastic Problem,"

Journal of Applied Mechanics, Vol. 26, 1959, pp. 502-509.

"1M

3 Dillon, Jr., 0. W., "Thermoelasticity Whenthe Material Coupling Parameter Equals Unity," Journalof Applied Mechanics, Vol. 32, 1965, pp. 378-382.

4 Dillon, Jr., 0. W., "Coupled Thermoelasticityof Bars," Journal of Applied Mechanics, Vol. 34, 1967pp. 137-145.

5 Solar, A. I., and Brull, M. A., "On the Solu-tion of Transient Coupled Thermoelastic Problems byPerturbation Techniques," Journal of Applied echanics,vol. 32, 1965, pp. 389-399.

6 Zienkiewicz, 0. E., The Finite Element Method,Third Edition, McGraw-Hill, 1977, pp. 570-574

71 k,

I

0(X) p i (X)-

0 12 2 1- i+i IN

Figure I. -Galerkin

P.(X) 'P.(X) ik. Wx

2. i+

Figure 2. -Central. Difference

CN.

-E

-E

41 >

0 0 U)

'41

P4-

PAn

CD/GK

Ax = 0. 05

1.2 - t = 0.02

2 =o0.01X, -0.1

1.0 -

.8t=..4

.6 t=1.4

.4.

C.

E- ~ .2

.2 4 6 8 1.0

Figure 5. -Stress and Temperature vs x

Unit stress at x -0 for various t

rwr4.

16

14

12

1X1 4.Ox

10

8

34J

E4 6

21. 0 4

.2 .4 .6 .8 1.3x

Figure .6. - Temperature Profiles (Xi-x - 0)

for Input g - 30(1-e it)

NAM

x

4.

0

41.

Em

* 0

12 none(2

10 / Coupled problem

8 At = 500, Ax =0.1

E

S

2

1224 6 8 10X10 4

8

6

$44

caU2 2

(2)

42 4 6 a lOx1O

Time, t.

*Figure S. -Temperature and Stress vs. Tim*

10

8

4)4

16

I..4

4134.4

2. 4 6 a 1

1.6

1.2e9 Taeaur rfl

Ve

wU)

00.U,C,

U,U,Ci

4) $.i0 4)

* '-I U2ClS Ci

".4 2

9.4 .-44)

0

U

".4

0'-4

Cl'.4

-4C'.

'.4 D '33SZ~S - -- r.Si

-

4

3

q - S.1548te* it

if = .1te i

U(2

CD/GK

At=0.02AX= 0.05

4 a, =~ = 0.4

--Uncoupled

X 1=1.0

3--- Coupled

x =1.0

2=0.01

Em 2

(1) t -2h(3

.hJ(2) t =10

ha(3) t =20

.2 .4.6 .8 1.0

Figure ii. -Temerature and Stress Profile

E-4

4.

A. 1. 280xl10 3

.3

3S. 120x10 1

1.10

. 2

.2 .4 6. .

x

?iaure 13. Teaemrature Profilej

-- ___ __ ___ ___ __ ___ __7__