THERMO-VISCOPLASTIC ANALYSIS OF HYPERSONIC STRUCTURES SUBJECTED TO SEVERE AERODYNAMIC HEATING by Earl A. Thornton l and J. Tinsley Oden 2 Texas Institute for Computational Mechanics The University of Texas at Austin Austin, TX 78712-1085 and W. Woytek Tworzydlo 3 and Sung-Kie Youn 4 The Computational Mechanics Co., Inc. 3701 North Lamar, Suite 201 Austin, TX 78705 AIAA Paper No. 89-1226 AIAA/ASMEiASCE/ AHS/ASC 30th Structures, Structural Dynamics and Materials Conference April 3-5, 1989 Mobile, Alabama 1 Visiting Scholar, Associate Fellow AIAA 2 Director 3,4 Senior Research Engineer
Transcript
THERMO-VISCOPLASTIC ANALYSIS OF HYPERSONIC STRUCTURESSUBJECTED TO SEVERE AERODYNAMIC HEATING
by
Earl A. Thorntonl and J. Tinsley Oden2
Texas Institute for Computational MechanicsThe University of Texas at Austin
Austin, TX 78712-1085
and
W. Woytek Tworzydlo3 and Sung-Kie Youn4
The Computational Mechanics Co., Inc.3701 North Lamar, Suite 201
Austin, TX 78705
AIAA Paper No. 89-1226
AIAA/ASMEiASCE/ AHS/ASC30th Structures, Structural Dynamics and Materials Conference
April 3-5, 1989
Mobile, Alabama
1 Visiting Scholar, Associate Fellow AIAA2 Director3,4 Senior Research Engineer
TlllmrvIO-VISCOPLASTIC ANALYSIS OF IIYI'EHSONIC STIUJCTUHESSUUJECTED TO SEVERE AEIWDYNAMIC HEATING
Earl A. Tliorntonl and J. Tinslcy Odcn2
Texas Institutc of Compntational I'\'fechanicsThc Univcrsity of Texils at Anstin
Anstill, Tcxas 78712-1085
W. Woytek Tworzy(l\o~ and Snng-Kie Yonn·1
Tlie Computatiolli\l Mechallic.~ Co., Illc.:J701 North Lamar, Snite 2UI
Austin, Texas 78705
Abstract
A thermo-vi~coplastic computational meth-od for hypersouic structures is presented. Themethod employs a unified viscol'lastic constitu-tive nlodel implemented iu a finite element ap-proach for qnasi-static thermal-struclural anal-ysis. Applications of the approach to con vec-tively cooled hypersonic structures illustrate theeffectiveness of the approach and provide insightinto the transient inelastic structural [",haviorat elevilted tenlflcratures.
INTIlODUCTION
Thc commitmcnt to develop tlic National AcrospacePlanc lias gcncratcd rcsnrgcnt interest ill tlie teclillol-ogy reqllired to design structures for hypersollic /light.Such structurcs will be cxposed to aerodynamic heatingof unpreccdcntcdmagnitndcs. As tlie vehicle acceleratesor decelerates at hypersonic speeds in the at nosphere,shocks will sweep across the vehiclc and intc:act withlocal shocks and boundary layers. These interactions in-troduce severe local pressures and heating ratcs. A recentexperimental study (ref. I) of interacting shock waves ona cylindrical leading edge sllows hcating rates ten timesundisturbed levcls.
Leading edges of cngine structures (Fig. I) present asignificanl design problem becausc of intensc local Ileal.-illg and pressures. Analysis of thc Oow, thcrmal andstructural behavior prcscllt serious cOlllputaliollal chal-lenges to analysts because of lhe inhcrent 1I0rdineariticsin all aspects of the multidisciplinary proLlcllls. SOllieof the critical computational issucs arc identilied in ref-erC'nce 2. Critical issues illcludc lhe diffiwllics involvedin (I) analY7.illg the viscous, cOlllprcssible (low and pre-dicting the high local aerodyn;l1nic hcatillg, (2) modd-ing and analyzing l1Iultimode nnsteady heatlransfer in il
high lelllperature convectively-cooled structure, illld (:1)simulating the trausi,:nl, nOlllinear therlllal-structural rc-
IVisiting Scholar. Associate fellow AIAA2 Director
3'~Senior llcsearch Engineer
sponse for rapid tcmperature changes. Preliminary struc-luralanalysis of all iml'ing(!Jllent cooled leading ..dge (ref.3) showed high local pbsticity thal seriously degradcdthc structnre's load carrying capacity at elevated tem-peraturC's. A recent thermostructural analysis with ex-perimental verification (ref. 4) of cowl lip designs con-fiulled that inelastic effects occur and can be significant.In the experilllcntal study, two specilllens failcd duc toburn-through bccause of intense local heating or bccauseof loss of cooling.
The pnrpose of this paper is to prcsent a thermo-vis.coplll ..~lic computationt," lIIethod for Ilypersollic struc-turcs suhjected to severe local unsteady heating. Theallalysis cmploys a unified viscoplastic constitutive modelimplemented in a finite clement approacl, capable of pre-dictillg rnte-dependent plasticity cfTects for telllperaturesup Lo about 75 percent of the mclting point. Hate-dependcnt plasticity cfTects are known to be illlportantat devatco tClllperatures.
Unified viscol'lastic constitutive 1II0dels have evolvedover the last twenly years to provide a means for ana-lytically representing a materials response frolll the elas-tic through the plastic range includiug strain-rate dc-pClldellt plastic nOIV, creep and stress relaxation. Thetlleol ies nrc guidcd by physical considerations includingdislocation dynillllics and arc based ou the principles ofcontinullm mechanics. Tile lirst 1n1iitidilncnsioIHII f"l'IIlII-latiolls of c1astic-visc:ol'lasl,ic constitutivc equations wasdue to Bodner and Partolll. Since lhcn a Illllllber ofconstitutive models have appeared; many of these the-ories are sUllllllarizcd in revicw articlcs that appcar inreferencc 5. A NASA-Lewis sponsored rcscarch progralll(1I0ST) condncted by L1leSouthwest flesearch InslitntercccnLly concluded iI four year research dfort (ref. fi ..7) Lofurt.her develop unified constitutive models for isotropicmateri;lls and t.o delllonstrat.e their usefulness for 11Iialysisof high I.emperature gas turbinc cngines. One result ofthis study is mat.erial propert.y data for high tempcrat.urenickcl-bascd alloys over a wide temperature range. Theunifcd models employed were those of Dodner-Partolnand \Vlllkcr.
Unifi('d viscoplllsLic llleorics hav .. bccn imJllcmclIl.cdby II IIIl1ubcr of fiuite elclllelit researchers. Detailed stud-
ics of several rate-lh~pcnuenL plasticity models and thcirnnlllcrical illlPlc1ncntatioll using adaptive finiLe c1cmcnLmethods were conducted by Bass 18} and Bass and Oden[9, 101. Under the NASA 1I05T prognllll, the Walkermodel was im(llemenLed in the MAfiC finite e1emcnLprogram (ref. II) and useu to IIl1aIxze the t.hermo-viscoplastic response of II tnrbine blaue under simulaLednight conuitions. In another recent finite element appli-cation (rd. 12), Lhc Bodner-Parlom and Walker thcorieswere compared for a thin circular plate subject to highlylocalized, transient heating.
In this paper the Ilodncr-Partom constitntive modelis employed, and the finite clement approach dnvelopediu references 8- to for thc isothermal case is cxtended toinclnde thermal effects. The papcr begins by dcscrib-ing the finite clement approach used for thermal anal-ysis of convectively cooled hypersonic sLructures. Thcnthe thcrl1lo-viscoplastic sLructural analysis is presenLed.The iniLial value problcm is formulaLed, the constitutivemodel is described and the finite ell:1l1ent viscoplastic so-luLion mcthod is presenLed. NexL, IIpplieatiolls of theapproach are made Lo convectively cooled structnrcs. Anappendix prcsenLs thermal and strnctural data for thccOllvecLivcly cooled hypersonic structure used iu the 1Il1-
lIIerical compuLaLions.
TIIEIlMAL ANALYSIS OF IIYPERSONICSTltUCTUllI~S
Convl'cLivcly cool cd strudur<:s arc sLl'llllg candidaLesfor usc in hypersonic night vehicles. For hypcrsonic night,some leading edges amI panels require aCLive cooliug sys-tems Lo keep strnctural temperatures wit.hin accepLableranges. The interual now in the coolanL passage has apredominant role in the thermal rcsponse of a hypersonicsLrur.Lure subject to exLernal heating. A cross-section ofa typical convccl.ively cooled strucLnre is shown in Fig. 2.An aerodynamic skin and a coolanL passage with internalheat exchanger protcct the pri mary structure from theaerodynamic heating. The thin, typically metallic, aero-dynamic skin transfers Lhc <:nergy of tile aerodynalllicheating Lo a low temperaLure coo\;lIIt /low Lhrough theheat exdlanger fins that connect the aerodynamic skillLo the primary sLructure. Iu a Lypical engine slrnctllle,the coolant is cold hydrogen that laLer is used as thepropulsion system fun/.
1I<:at Lransfer in the aerodynamic skin consists of con-duction comhined with surfacc radiation. IIeat transferbclween the aerodynamic skin, tile ileaL excllanger sur-faces and the primary structure is hy conduction at tilesolid -/lnid inLerface. The finite element represenLation ofconduction heat trausfer with mdiation boundary con-ditions folluws the sLandard proccdures described in rd.13.
The dominant mode of lIeat transfer in the coolant/low is rorced convection. Tile reprcsenLation of the lIelatLransfcr in the coolant passage is the criLical step in theheat transfer analysis. There are two hasic representa-tions tllat can he employed. The first, denoLed lIere asLhe engineering model, is hased upon a numl)(:r uf as-sumptions Lhat greaLly simpliry the problem inlo a singleenergy equation wiLh a specified mass Jlow rate. De-tailed compuLation of the nuid velociLy components and
Lcmperatures ill uoL required. The second reprcsenta-Lion idealizes the coolant /low as a continunm model, andthe partial differential equations describing conservationof mass, momentum and encrgy arc solved simultane-ously to ohtain lIuid velocity anu Lmperature distrihu-tions. This latter model is the most accnrate, buL it isalso considerably more expensive tllan thc enginccriugIllode\. In this papcr the engineering heat transfer modelis employed.
Engineering Model for Cooh1l1t Passnges
The hasic features of the <'lIgineering model arc dcvel-ope~d frolll the idealization shown in Fig. 3. A segment ofthe coolanl passage of width IV is showu; for simpliciLy,only the upper onc-half of the coolant passage with theaerodynamic skin is sllown. The engineering formulation(ref. 14) is based on the following assumptions:
I. The thermal energy state of the /lui<l is character-ized by the /luid bnlk temperilLnre 'IF which variesonly in the /low direction, i.e., 'IF(X, t).
2. Tlw flow is represenLed by the mass /low ral.e Iii
in the coolant passage specificd by lil = {IF AFVF
wlH:rc {Iv is the coolant densiLy, AI" is 1.I1ecross-sectional area of the coolant passage, and VI" is LheCOOlallt mean /low velocity.
:t. A conved inn cocllicient /, is defined such t hat theheal. /lux cit,· transferred bdwccn the sLflldure andLhe coolant may he exprcssed as
IiI" = I.('I's - 'If')
where: 'l's(;r, t) denol.es the stnH:tural ICIIII)(:ratureat the !luid-solid interface.
,I. The convection coefficicnL /, may be exprcssed <IS
a function of the /luid bulk Lemperature alolle hyusing analytical/empirical equaLions for the Nuss(,ltIII IIllb er ,
Nit = I.DkF
wllere D is I.hc hydraulic diameter of till) coolantpassage, aud k[.' is the tI.ernwl conductivity of lIwl1uid coohllL. The convection coelJ1cienL as \I'd I itS
tile solid and fllIi.1 therllial parametcrs may, in gcn-cral, be tcmperaturc depclldellt.
Witll Lhese assumptions, energy balances 011 the aerody-namic skin i1l1d coolant give the governing conservatiollequaLiolls.
FllIid:
Solid:
-fx( ksAstg;;) + 1ll1t('J's - Td + od'\(2)
DTs. .+ {lscsAsut = q
wlll're [/V(x)] arc the c~lell1<)ntinLerpolation fnncLions,Following nsnid !inite element procedures (ref. 13), thec1iscretizc~d eqnations for an eleillent of lengLh [, may hederived ill the form
Finite Elelllent Forlllulation for Coolant Passages
A typical /iuite element representing Eqs. (1) an.1 (2)is characterized by fluid and fluid-solid interface nodes.The element shown in Fig. <I has two lIuid nodes (Iand J), and two lInid-solid interface nodes (I( and L).Within the clement, the lInid and solid temperatures arcexpressed as
where the suhscripLs Jo' and S denote L1wfluid and solid,rcspecLively. In thesc equations, c denotl'.5 specific heat,q is the SLefan-BolLzlllalln conslant, ami E is the surfaceemissiviLy. Because of the Lempcrature dependence ofLhe thermal paralllCters and Lhc radiation term, Eqs. (I)and (2) constituLe a lion linear scL of partial differentialcqnations.
The heat exchanger fins arc not included explicitly inLhis model. However, the Ilcattransfer heLwcen the heaLexcliangcr fins alld coolant C1l.TI approximaLely I)e Lakeninto consideration through the nse of an effectivc widthw which represents the area over which Llle convectiveIleat exchange occurs.
Initial Value Viscopillsticity Problem
Consider a viscoplastic structure occupyiug a regionn with houudary Dn. The hehavior of Lhc structurc is dc-scribed by Llle follow ing sysLem of dirrercntial elJuiltiolls:
1. Equilibrium in rate form,
TIIEI1MO-VISCOPLASTIC STRUCTURALANALYSIS
The lwhavior of a Lhermo-viscop\;\stic sLrllctnre suh;jecLed to acrodYllamic heaLing is analyzcd assullling thaL:(I) lIu:nno-mcc:llallical coupling in the conservation ofellergy eqnaLion can be negl.~cLcd, (2) the sLructnral rc-sponse is qnasi-sLatic:, alld (3) deformations arc infillites-imal. Wit.h these assumpLions, an unsLeady Lhermalallalysis may be pcrfofllled first lo detcrmine the tem·peratures. 'I'llI'll, using Lllese temperatures, tile struc-ture's viscoplastic respollsc is deLcflllilled. The soluLionis tllus obtained by separately solving initial boundaryvalue problems for firsL the thermal and thcu tile struc-tural response.
The elmnenL eqnations given in I·:qs. (,I) show that thecoolant IHlssage model can he regarded liS an assl'mblyof clements where eadt demellt represenLs a sillgle heatLransfer mode. Thns we can r<~present thc coolanL pas-sage in Lerms of two convective elements: (I) a mass-Lransport e1(~lnent (Fig. 5a), and (2) a surface convcctionelement (Fig. 5b). The mass transport elemcnL repre-sents the downsLream convective hcaL lransfer due tomass lIOWi it lias a capacitance malrix [Cd as well ascondnctance matrices !Ii"I and [1\, ..). The surface con-vection element represents the convection heat exchangebetween the coolant fluid and the solidi it docs not have acapacitance maLrix, but iL has a conductance matrix II(h)linking lhe solid aud fluid nodes. These coolant passageelements are assembled into the finite c1emenL thermalmodel for the compleLe cOllvectivcly cooled stfllcture tllatlias conduction elements, radiatiou elements, aud surfacec1emcnts for convection to a specified convective exchangeLemperature.
The unsteady thermal analysis is nOlllinear because oftemperature depelldent Lhermal properties and surfaceradiation. The equaLions arc solved by timc marcllingwith the Crank-Nicolson algoriLhm; at cach timc step,tllc lion linear algcbrilic eC)uaticlIIs are solved by Newton-Raphson iteration.
(3)
(5)
- 1(1. ] J 7i-') J 0 )f(s + f(n lTs = lQ
7's = [N(x)J {'l's)
7'p = [N(x)J ('I'd
[CF] = le. p/-,cl.·Ap{N}(N]dx
(Cs) = [, p,scsAs{N}(NJdx
I( u + Iih + I( p
- f(h
and the clement conductance matrices arc
wher( till' e1emellt capacitance matrices arc given by
[[(hI = [, w'h(N)[N]dx
11. {dN} [dN][I(pj =. kP/IF"J; dx d:t
1', {dN} [dN][lis] =. ksAs (1:z: J; .Ix
where l1ij denotc componenLs of Lhc sLw.5S Lensor,hi are Llle body force componenLs per unit volnme,and tile sumlllation cOllv<lI!t.ionis employed.
wherc (ij dcnotes the lolal strain componenLs andsupcrscripLs R alltl P denoLe clastic and inelasticslrain componclIl s, rcsrwclivcly. The compollentsof the displilcelllcnt raLes are il;.
(Ii)
11. (INlilcv{N}(-1 ](lx
o (X[1(,,)
3. Coustitutive refatious straiu rate aud stu,ss tensors, respectively. 1'(,1' ex-alllple, lhe devialoric strcss invarianl is
(no SUIII) (!l) (1·1 )
(uo sum)
wherc Ejjk/ reprcscnls Hookc's tcnsor of elasticitypara mders, nkl arc comoncnl.s of a Lensor of tiler-mal cxpansion paralneLers, and /i'i' represenLs tll(~rate of the change in temperatnre. Both E;jk/ andnu are temperature depcndent. The constiLutivefunctions are lij and go where Zk represents inter-nal state varial,lcs. Thesc fuuctions and stat.e vari-aiJlcs characterize the viscoplastic responsc of Lhematcrial.
The reiaLiou governiug inelastic deformations is Lhe"kinetic equalion", and the form taken by Bodner-l'arLom is
where D" is the liluiLing straiu rate in shear, n is atellipera.lnre-dcpendent material parameter, and Zis illterpreted as a load history depelldent param-cLer (1Icn'in called the illtemal staLe variaiJle) thatreprcsellts the hardness of tile IllaLerial wiLh respcctto resistance Lo plasti<.; now. Combining Eqs.(II),(12), and (13) gives
The evolul.ion (:qnaLion proposcd ror the isoLropichardening comp<lIIenl (ref. 5) is
3. EvoluLion I~quatiolls of InLernal Stale Variable,
The inLerlial slate variable Z consisls of isotropicand dircclional componenLs,
The description of the problel1l is complelcd hy pre-scribiug the boundary and initial conditions,
Ii = iii 011 Dnl
(10)
wherc iii arc prescriiJed surface displaccment raLes, IIj
arc the componcnts of a unit normal vector, and i;j arcprescribed surface traction rates. The initial conditionsinclude specifying the displaccn1<:nLs, strcsscs 'and inter-nal statc valiablcs, i.e.,
iT'. _ S;j [ 1') - ,fhDoexp -2(Z2/3J2r]
z = ZI + ZD
(16)
(I i)
Ui( :r.,O),Uij(:C,O), Z,(:r., 0) :I: E fl.
Conslilulive Model
TIl<: Iloducr-I'artom constiLuLive modd is of the inLer-nal sLate variable typc that is based on phenomcnologicalobsen'alions and supported by physical concepts reiaLedto dislocation dynamics. Thc model has gonc lhronghseveral modificalions and was extendcd for anisoLropicwork hardening materials. The cuneut model (ref. 5-G)aJso includes tcmperature effects.
1. Flow Law,
For the inelasLic strain rale component, the isotro-pic form of the Praudtl- Heuss law is assumed
i~ = >'S;j
wiLh the initial condiLion, Z/(O) = Zoo In the lirstterm, Z, is the' limiting (saLuraLiou) value or ZI,IlII is the hanlening raLe, and the plaslic work rateIS
. pWJ' = Uij!;j (19)
which is takcn as the measure of hardening. Z2 isthe minimum valne of Z/ at a giv,:n Lemperature,and At and TI arc Lemperature dependenlmalerialconstants.
Thc evolution fOrlll of the directional hardeningcomponent (ref. ;1) is defincd as
(11 )
= ° (20)
where Dr and 12 are the second invarianLs of Lhe(22)
The evoluLion eqnilLioll for /3;j(t) has Lite same gen-eral form as tltat ror isotropic hardening !'ut liastensorial charadeI',
where lIij arc lite direction cosincs of LIICcnrrenlstress state,
( 12)
( 13)
IS·· = U·· - -fo·akkIJ 'J 3 I)
and irk = ° denotes plastic incompressibilily.
where Sij are lhe devialoric sLress coml)()ncnLsgiveu hy
2. Kinematic Equations,
SquiUiug Eq.(II) leads to
>.1 = Df /11
and
where(2:1)
(2,1 )
dcpeud on tempera ture, (2) nodal loads (I\) de:pend ont.lle locall.elnperilture rates, and (3) several paramel.c·rs inthe l3odnC'r- Partom constitutivc model arc temperaturedepcndent.
As in Eq.(<1.12), 1lI2 is the hardening rat~. A2 and"2 arc tempcraturc dependcnt matcrial constants.
where [111 is the straiu-displaccmeut Inatrix. FollowillgUSl1al fiuite clement procedures, the !inite clement eqna-tions for a typicnl c1eIlleut are form as,
where [J( (7')] is the clement stiffnp-ss matrix. alld theterms on the right-hand sidc arc elemcnt load vectorsdue to thc rate of plastic strains, temperature, surfacetractions and body forces, respcctively. These matriccsare deli ned hy
I. At time I, initialize C1ij, Z; for cach clement;
2. Calculate iG = /;j(Uij, Zd for each e1emelltj
3. Assemble and solve !TIllS} = {F} ;
4. Calculate iij for each c1emcnt, { i) = [DI(6);
5. Calculatei:Jij for cadi demcllt. {a l= [E] {i -i"}-(EIQ~i';
G. Calculal.c Z; for eilch c1emcnl., Zj = 9j(C1ij, Zdi
7. Illtegrilt.e i:J;j, Z forward for cacli clement lo gel Ujj
and Z; at I + ~l,j
Viscoplnstic Solution Method
Since the present approach uses an unconpled (somc-times called one-way coupled) formulation, the thermalproblem is solved lirst followed by the viscoplastic allal-ysis. The trallsiellt thermal proillem is solved by timemarching wil.h a timc step ~tT, and the nodal tcmper-atures at successive timcs 1,,12,' .. are ohtailled. Thesetempemture vectors arc used as input to the struduralaualysis.
Thc first computation ill the structural analysis is tosO!\'e an initial statics proLlem if the illitiallemperaturedistributioll 1'(x, 1I) is not cqual to a ulliform referencetemperaLure or if allY initial static loads arc presellt. Theresllits of l.lds analysis arc tIle initial conditions (displace-ments and strcss<~~) ror tllC transient viscoplastic allalysis.
The viscopla.~tic analysis time-marches with a timestep ~t,. Expcrience has shown the time step requiredfor the structnral analysis is usually smaller than for thethennal ilnalysis, i.e. ~t, < ~IT' At intermediate timesin the structural analysis, llie temperat.ures are lillcarlyinterpolated from thc tempcratures kllown at the begill-nillg and elld of the larger thermal lime illtervals.
The strat.egy employed in the viscoplastic algorithmis as follows: with til<: illitial distribution of strcss, telll-peratllre and intcmal variables specilied use the equilib-rium condit.ion (Eq. 27) 1.0 ohtaillthe nodal displaccmentratcs. Then illt.cgrate lhe cOllstitlltive equations forwardin time at thc demellt Gauss int<:gratioll poillts. Withupdated values of t.he stress, temperature alld intemalvariables at the new tiIlH:, the equilibriulll equation issolved again. This se'llll'nCe of determining the nodaldisplacenlCnt rates, then advancing thc: constitutive equa-tions ill timc is continued until the desired history of theinitial bonlldary-value problem has been ohtaillcd.
Thus, the algorithm procccds through the followingst.eps:
(2G)Ii} = [/Jj(6}
[/((1')) = 1 [nr[E(1')][ll]dn (28)0,
(Fp) = 1 [nf'[E('/')]{il')<In (29)0,
(i:r) = 1(lW[E('/')]{n(T)}~'i'dn (30)0.
{ii» = h [N]r{if}ds (31 )DO.
{f.·ul = 1[Nf'{b}dl1 (32)0,
Otli~r variants of the isot.ropic alld direclionalhimlellingvariables arc described in refs. (5-7). Tlicsc referenccsalso present data for the: parameters used in constitutivemodels for se:vcral materials. Tllc particular valucs usedin the prcsent study arc presented in the Appendix.
Fi n ite Element Formu !ation
The finite e1cment formulation follows the approachof Her. 9, and the dctails of the weak formulation areprllsented there. Since the approach prescuted in Her. !)
is for the isothermal casc, this formulation cxtends theprevious approach by tile inclusion of temp(~ralllre effects.
The finite clement approach approximates displacc-mcmt rates within an clement by taking thc displaccmentrates as
tic} = [N]{b} (25)
where [N] arc the interpolation functions, and {S} rep-resellt the nodal displacement rates. Using the strnin-displacement equations in rate form, Eq. 8, an clement'sstra.in ratcs cau be cOlJlputed as
wherc n, denotes the clement volnme, and an. denotesan clement surface whcre tractions arc de!iued.
The temperature affects the viscoplastic structuralanalysis directly ill three ways: (I) the elasticity matrix[E(T)I and the cocllicient.s of thermal expausion (<«T)}
8. If t + ~l, < lj;,,,,' go to 2, olherwisc stop.
The complltationalnlcLhod above has bcell presentedfor a cOllstant time sl.ep ~l,. Complltational experienceby several invest.igators (s(,e Iter. 7-1~) illdicaLes that avery Sill,'" t.illlc step can be required hecause of the "stifl"nature of Llle ordinary dilTerelltial eCJllat.ions (Itc~cribiug
APLICATIONS
iJ = I(y,t)
tions al'l~perforlll/:d first for a silliplified one-dilll<:nsion,,1model to provide insight into the computational proce-dure. Then a two-dimeusional model of a realistic con-vcctively cooled structurc is aualY7.ed in detail.
Simplil1ed One-Dimellsional fvfodel
To gain a prclillliuary understanding of the behaviorof a couVflctivcly coolcd structure subjccted to convec-tive heating, a silllple ] DbaI' mod<:1 was iuvestigated. Asegment (Fig. 7a) of the aerodynamic skin is modclcJassuming uniform tempcrature. The coolant is assumcdto have a. constant, specified temperature, and the skin isheated convecti\'e1y. A stl'OlIg time variation of tIle con-vection r.oefficient, (Fig. 7b) simulates the passage of amoving shock. The resulting transient temperature andbar compressive stress exhibit features to be expect.ell ina morc complex modcl.
The transient tcmperature history (Fig. 8a) shows arapid rise followiug the sudden increase in till, convec-tive coellicieut, and then a smooth decay arter the largeconvective heating is removed. Tile convective coolingcauses the temperaturc to return quickly to equilibrium.The initial rapid iucreasc in temp"rature causes the barto yield in compression vllry early in the response (Fig.8b). After the telllperatnre begius to decay aL t = (l.55,thc stress rilpidly chiluges to teusion hence as Ihe tcm-perature returns to equilibrium a large residual tensilestress remains iu lIle bar. Tile tensile stress is inducedbecause t.he initial high I.llnlpcratllre causes cOlupre:lsiveyielding. This yielding teuds to pcrll1ancnlly shorten thebar. lIowever, the bar's fixed elld bOllll<lary cOlI(litionsprohibit the bar from decrcasing in length as the tem-perature decren.~c:l so lhat a tensile stress <h:velops as thematcrial cools.
Duriu~ the I.hcl'llIal-structllral analyses an uncer-tainty arose conceruillg applicability of the hardrwss cvo-lution e(luation, Eq. (18), at hi~h tCll1prratnrc rales(about 5000"C/5I:C) occurring in tlwse aualyses. At suchhigh temperature riltes this equatiou may predict SOllie-what auomalous behavior of the material. This is illus-trated by the following exam pic. Consider a specimenof super alloy lJ 1000 + II I wit.h properties preseutcd inthe appeuclix. If the virgin, load-frec spccil1lell is heatedfrom 760°C to 1060°C in 0.1 second, the tell1peralure-dependent varial)le Z2 drops from 2700 At I'a to 1200AI I'a. However, the aelual hardness variable Z', whicllfollows Z2 only through the relatively slow thcrmal re-covcry term (second term in E<I' (18)) reduces iu thistime only to 2696 AI 1'/1. This means lIlat althongh thespecimen has reachcd an elevated temperature, it has notsoftened yet-a somewhat surprisiug result.
The ueed ror rll rthcr researdl iII thermal history ef-feels 011 D 1900 + II I was uolcd ill ref. 6.
In the initial viscoplastic analysis a fixed time stepof 0.001 s was used, and 1200 time stcps were requiredfor the allalysis. The variahle time stcp algorithm shownill Fig. 6 was I.liell implemented, and the I'roillem wasresolved. Fig. 8c sllOws the history of the variable timcst(~P ami illdicales that the new alia lysis rcqnired only213 steps - a substautial savings. The figure sbows l.hal.in the "/lilt" parI. of the stress response a large l.inlC~stepwas used. bllt ncal t = 0 alld again at t = 0.55 whell the
A now chart depicting the adaptive scheme is shownin Fig. 6. The /low chart shows how the t.ime step iseither rcdnced or increased depending on the crror ill<li-calor, Ell' (36). The now claart shows that thc time stepis rednced or incrcased by a factor of two. Tlais approachis effective, but an alternate sclacme also has hccu IIsedwhere the new time stcp is based on the crror E. III thisscheme
Variable Time St.ep Algorithm
In the constant time step viscoplastic algorithm theintelllal state variahles arc advanced intil1le witla the con-ditionally stal,le Enler forward dilfmence algoritlam. Thevariahle time step algorithm is a modified Euler schemensillg a trnncation error criterion (ref. 15) to adjnst thetime step.
For simplicity, cOllsider the singlc ordinary dilferential<'«nation,
As mentioned in the introduction, scramjct engiuestructures can experiellcc intense local Iaeatillg due toshock illtrractions. Two models of such structures arc all-nlyzed for heat loads representativc or shock interactions.f\laterial properties used (see the appendix) rcpresent ahigh-temperature nickel-hased super alloy. Computa-
The solntion is advanced using a predictor-correctorsdleme. Tlae predictur plaase consists of an Euler step:
Tlae error illdicator is lIext compared with a preset errorcriterion, and if the criterion is met, the lime st.ep is suf-ficiently small enongh to (Hoceed to the corrector stage.Otherwisc, the predictor plaase Eqs. (3'1-:35) is repeatedwith a smaller time step.
Th~ corrector plaivse is tlae modified Newton schcme,
An error indicator E (Her. 15) is then computed from
tla(' internal state variithles. To gain improved dliciencyand reliahility a variahle time step algorithm has heenilliplementcd.
stress is challgiug rapidly, slllall lillie steps arc u<:cd"dto capturp. the respouse accurately. This examplc sho\Vsthat the adaptivc tilllc step algorithm is au importautstep towards produciug a reliahlc, accurate soilltiou.
Convectively Cooled Structu"e
A 1II0rc realistic 2D model of a convcdi vely cooledhypersonic structure is shown iu Fig. 9. The modelr('prescllts a segment of a convectively cooled structuresuch as a wall of a scramjet engine fuel illjcction st.ruL..The fiuite elcment thcrlllalmodcl (Fig. 9a) iucludes: (I)conduction heat lransfer ill the aerodynamic skin, healexchanger fius and aud pi imary structure, (2) couvec-tive heat lransfer between thc walls of the coolant pas-sage and coolanl, (3) lIIasS trausport convection in lhecoolant which has an uuknowu bulk temperature, alHI(.() surface radiation on the aerodyualnic skin. The acro-dynamic skin is uniformly convectivcly heal.ed over itslength, I1nd supcrimposed ou the uniform heatiug is alocal, intense heatiug simulating iI transient shock. Thetransieul heating is induced by the Lime-dependent con-vcction coefficicnl shown in Fig. !Jb.
In the plane strain strucl.ural finite c1elllent lIIodel,the primary structure and aerodynlllllic skin havc unitthickness, butlhe heat cxchanger fins in the coolant pas-sage are approximately represented by a single fin withthe total thickness of O.OGOin. The wall segmcnt hasfixcd displacemcnl boundary cOIHlilions al the left amIright cnds. The lIeroclynamic skill. cxchaugcr fins a IIIIprimary structnre are also loaded by internal pressurc(7.0 M I'a = 1000 psi) iu the coolaut passage which is lel-alively large compared to extefllal aerodynamic pressure(1l.35 I'>.ll'a = 50 psi).
The 1II0del waS analyzed first for steady aerodynamiclleatiug and inte\'l1al pressure operating conditions. Thenthe suddell localized acrodynamic healing was applied.Thus, the steady tcmpcratllfes alld stresses serve as ini-tial condiliolls for the transient thermo-viscoplastic aual-ysis.
The viscoplasLic solutiou was computed with the vari-ahle time stcp algorithm, and the time slep was variedusing Eq. (39). A total of 265 steps \Verc rcquired tocompute the response for a total time duration of \.2 s.
The thermal response of thc wall segment is shown inFig. 10. Figure lOa shows the time history of the tcmper-ature at a poiut on I.he aerodynalnic skin directly underthe transient heating; Fig. lOb shows contours of temper-atures at I = 0.5s whcn tcmperat1l1es arc maximum. Thetempera t.ure history is qualitatively similar to thc rcsultsohtained in the I)) model showittg the rapid skiu temper-atnre rise aud fall with the ~illllllat('d sllock heating. Thetemperature contours show relatively steep thermal gra-dients induced hy the local heal.iug, and t.hat the coolautsubstautially limits the extent of the il\lluced high lem-peratures. Thns thc high temperatures are confiued Lothe aerodynamic skin, and the prilliary strllcture experi-CIICes only small temperature changcs. The lemperaturegradients iu the skin particularly at the coolant-skin iu-lerface arc not predictcd with high acc1l1acy hecause ofthe engineering model of the coolant heat transfer. \low-ever, local tempNature levels ill the skin arc reasollahlyaccnrate siuce n<:l energy transfer to the ·coolant is mod-eled satisfactorily.
lIistorics ~f the horizontal stress componellt ar at t.wopoints through the skill thickness arc shown in Fig. II.Thc strcss histories follow the telnperature, and underthe intense local heating stresses arc very similar to thcresults oht;tined from the one-dilllcnsionalmodei. At thislocaliou the skin yields through most of its thickness. Af-tel' the heating ceases there is a rapid decay of stress, andullder the intensc local heating there arc residualtcnsilestresscs. Figme 12 shows the time history of the verticalcomponcnt of stress ay in the heat exchanger fins. II ightensile sLresses arc illduced with significant local yieldingwhich are follow(~d by residual comprcssive stress. Thccrude fillite clement model of the fins only approximatesthese stresses, bllt the high tensile stresses can poten-tially canse it. bond fail1l1e at the heat exchangcr/skinjoint. The conse<luences of such a failurc are studied inthe next example.
The viscoplastic stress histories are compared withstresses predicted assuming clastic behavior in Figs.13- J.1. The clastic computations were made withtcmperatllfe-dcpendent clastic properties. The loss ofstiffncss due to the elevated temperatures for 0 < t < 0.55accollllts for the stress "rolling-over" with an appear-ance of yielding. Generally, the clastic stresses arc toohigh and, of course, since yielding does not occur residualstresses are not predicted. Maximum deformations (notshowu) predicl.ed by the elastic analysis and viscoplasticanalysis arc ahout the same, approximately 0.00 I iu.
Thc two-dimensional character of the slrcss compo-neuts (]rand ay are shown ill Figs. ]5 and IG, respec-tively. The stress distrilJlltions arc shown at three timesI = 0.1I5s, 0.5". 1.25 iu the rCllpousc. The stress('s alI = 1.2.'.1 arc the residual stresses. Figure 17 shows con-tours of the prillcipal plastic strains at t = 1I.5s. Thccontours show the relatively localized nature or lhe highstress gra<lients as well as the tensile and compressiveregions. The significant residual stresses suggest the pos-sibility of cllmulative damage under repeated load cycles.This possihility nccds furtller investigat.ion.
C:oJlvectively Cooled Structure 'Vith Damage
The preceding analysis showed lIigh tensile stressesin the lIeat exchanger fins alld suggested the possihilityof au aerodynamic skiu/hcal excllanger Lond failnre. Tosimulate such a possibility, the analysis was repeated butwith the fins given greatly reduwd material propertiesfor a width of n.30 in. undcr the high localized healing.Two results of til is aualysis are sllown in Figs. 18-19.
Since the fillS can no longer support tensile stress,the inlernal coolallt pressure callses suhstantiallocal <Ie-formation of the aerodyuamic skin. Figure 18 shows thedcforlned structure at I = 0.5s. The deformations, shownunlilagnifi(~d, rl'present a significant permanent displace-ment. This resllit can be seen in Fig. 19 which comparesdisplacemcnt histories for the undamaged alld damagcdstruct1ll'e. For the dalT1aged IIcat exchanger case, a resid-ual pcrmancnt dcformatioll of 0.007 in. is introdllc<~<1bytile L!1I'rmal loading.
The maximum deformation induced hy tilt; fin dam-age depends strongly on the width of the damagl~d re-gion. Increasing the width of the <lillnagl!<1region by asmall amount will substantially increase tlie deformation.
The analysis of tlll;He lilrge dclorlll<d.ions requires t.llat tilepresent formulation be modified to account for large <Ie-formation effects.
A possiblc conscqnence of this pennanent deforma-tion is lo cause a disturbance of tile exlernaillow. Pro-tnberances into lIigh speed 1I0ws can cause significant lo-cal augmenlalion of the acrodynamic heating. Thns, lhesignificant permanenl deformation int.rodllced hy a finhond fail1ll'c is likely to cansc a cOlllplex now-thermal-slruc:tural inleraction.
CONCLUDING HEMAnKS
A lhcrlno-viscoplaslic compntationalmelhod for hy-pcrsonic structures snhjected to severe local heating isprcsllnle'\. Tile mcthod employs the lJodner-l'arlolllunified viscoplasLic constitutive model implementcd ina finite clement approach for qnilsi-static tllermal-structural analysis. A variablc lime slep algorithm isused to provide an cJficient solutioll schcmc for the stiffordinary differenlial eqnation that charac:t.erize tile evo-lution of tile int.ernal slate variables.
An analysis of a simplified model of a ouc-dimension-ill convcctively coolcd slructurc shows the basic featuresof the t1lenno-viscoplastic response and demonstratesthe substanlial computer lime savings as well as the re-liability of the variable time step algorillun. An analy-sis of a two-dimensionillmodel of iI realistic convedivelycooled slructure provided morc delailcd understilnding ofthe thermal structural behavior. Tlul coolanl flow dom-inales the thcrmal response provilling a relatively shortt1lf'rmaltransienL. Under intense lIealing, significant lo-cal plasticity occurs in the ilcrodyn;\Inic: skin, hnt theprimary structure remains undamaged. Il<lal exchangerfins I:xllf:rience high lensile stresses aJllI bot II the aelody-nilluic skin and heat exchauger fius havc significant resid-ual stresses. An analysis of an aerodynamic skin/heatexchanger fin bond failure showed that the acrodynamicskin would experience significant local plastic dcforlna-Iion dne to the intcrnal coolalll passage prcss1II'e. Thelocal deformation is pronounced enongh to disturh thcexternal aerodynalllic /low and introduce a c011lplcx flow-thermal-slruclural interacLion.
The thcrlllo-viscoplastic analysis of con\'ectivcIycooblhypersonic structures umler intense local heatingshows the important role that such analyses cau makein understauding complex transient inelastic structuralhehavior al elcvated lemperatnres.
ACKNOWLEDGf,M I~NT
The aulhors are pleascd to acknowledge the supporlof LlII!Aerothermal Loads nranch at the NASA LangleyIleseal(:h Center and the support of L1leAir Force Ofllceof Scientific Hcsearch.
HEFEIlENCES
1. Wieting, A. It., and 1lolden, M. S.: "Experi11len-tal Study of Shock Wave Interference lIeating ona Cylindrical Leading Edge," AIAA 221111Thermo-physics Conference, 1I0nolulu, Hawaii, JUlie 8-10,1987, AIAA P'1I1er 87-151 \.
2. Decllau11lphai, P., Thornton, E. A., and Wieting,A. It.: "flow-ThennaISlrucLurit\ Study of Aerody-namically Ilealed Leading Edges," AIAi\/ ASMI~/ASeE/ AilS 29th Structures, Structural ])ynillnicsand I'daterials Conference, \-Villiallishurg, Virgiuia,April 18-20,1988, AIAA Paper 88-22·15.
3. Declaaulllplli.i, 1'., Wieting, A. H.., and Thoruton,E. A.: "Thcrmal-Structurall'erformanccs of an Ac-tively Cooled I.eading Edge Subjected to Typc IVShock Wave Interference lIcaling," Third NalionalAero-Space PliLne Symposium, June 2-,1, 19S7, Pa-per No. 24.
.\. t-.lelis, M. W'J and Gladden, II. J.: "Thermostruc-tural Analysis with Experimental Verification ina High lIeal Flux Facility of a Simulated CowlLip," AIAA/ ASME/ ASCE/ A lIS 29th Struclures,SLrucl. ural Dynalllics and Materials Con ference,Williamshurg, Virginia, April 18-20, 1988, AIAA88-2222.
G. Chan, 1<'5., Lindholm, 1I. S., Ilodn(!r, S. H., Ilill, J.Il., Weber, It. M., and t\l<:ycr, T. C., "Cc;ustil.ul.iveModeling for Isotropic Malerials (\lOST)," ThirdAnnual Statns Report, Southwest Hcseilrch Insti-tute, San Autouio, Texas, August, I!JSG, NASA Cll179522.
7. Chau, 1<' S., Lindholm, lJ. S., and Bodner, S.It., "Constitutive Modeling for Isotropic f\litl.erials(IIOST), Final Ileport, Southwest llesearch lusti-tute, San Antonio, Texas, June, 1!J88, NASA CIl-182132.
8. !lass, J. M., "Numerical Implemcnlation of Con-stitutive ~Iodcls iu Hate- Dependeut Plasticity,"Ph.D. DisscI'I"lioll, TIle University of Texas,Austin, 1985.
9. Bass, J. M., and Oden, J. '1'.: "Adaptive FiniteElement !vlcthods for a Class of Evolution I'rohlel1lsin Viscoplasticily," Jill. J. Ellg. Sci., Vol. 2:1, No.G, 1987, pp. 623-653.
Ill. Bass,.1. M., and Odcn, .I. '1'., "Nu1l1erical Solutionof the Evoilition Eqnations of Dalllage and Hale-J)<lpendcnt I'lasticity," 1111. J. E1I9119. Sci., Vol.2li, No.7, 1988, PI'. 713-7'10
II. M An.C General Purpose Finite EI<:lllellt Program,MAile Corporation, Palo Allo, CA.
12. Chang, II. '1'., and Allen, I). II.: "Analysis ofViscoplastic I'laLc>~Subjectcd to Itapid ExternalIIcating," A IAA/ ASME/ ASCE/ A liS 29th Struc-tures, Structural Dynamics and Materials Confer-ence, Williamsburg, Virginia, April 18-20, 1988,A IAA Paper 88-2·122.
13. Huebner, 1<' II., and TlIorutoll, E. A.: The Fi-nile Elcmenl Me/hod fOl' Engineers, Second Edi-tion, JulIn Wiley and Sous, 1982.
H. Thornton, E. A., and Wieting, A. It.: "fi-nite Element Methodology for Transient Conduc-tion/Porced CO\1vection Thermal Analysis," Pro-gl'ess ill As/mlllln/ics and Ael'Ollaulics: /leal Tmns-fel', Thermal COIl/rol and Ileal Pi/les, Vol. 70,Edited uy Walter B. Olstad, AIAA, New York, pp.77-103.
15. ((umar, V., Morjaria, M., ali(I Mukherjee, S., "Nu-mCI:ical Integration of Sonle Constitul.ive Modelsof Inelastic Dcformat ion," J. Bl!ginecl'illg MalerialsIllld Techl/ology, Vol. 102, Jan. 1980, pp. !J2-9G.
APPENDIXTIIEIlMAL-STIWCTUIlAL DATA
ThcrJllal Data
I. Skin, heat exclIanger nns and primary structnre
(I == 0.28:llb",/iu3
Temperature Dependent Condnctivit.y andSpecific I1eat
T J( CJI
[Oil] [1J1'U /ill - sUI [UTU/lb", - RI
0 1.23 X 10-1 0.12
500 1.8 x 10-'\ 0.135
1500 3.25 x 10-.1 0.177
300n 6.0 x 10-'1 0.26
2. Coolant
Iii == 7.IG x 10-3 lb",/sec
CJI == 3A8 lJTU / Ibm - R
to == GG5 R
It == 6.2 X 10-3 BTU/s - in2 - a
3. Surfacc convection
1'00 == 5:J.l5 J?
h == G.O X 10-5 BTU /05 - ill2 - R
Additional It at time 0 ~ / ~ 0.5 sec:
o ~ x < 0.:H5: hudd == 0
(3: - 0.15) II0.375 ~ x ~ 0.625: Irudd == 3.0 x J 0-3 cos 0.05 2
0.625 < x ~ 1.0: hadd == 0
4. Surface radiation
a = 3.3063:1 X 10-5 BTU/s-in2-R"
c: == U.8
Q == 0.8
Structural Data (superalloy U I!JUU+ lIJ)
J. Temperature-independcnt constants
Du == 1.0 x 10" sec-I
ZI == 300U MI'Il
Z3 == 1150 Mfa
"I == 2
1'2 = 2
1111 = 0.270 MPIl-1
nl2 == n
2. Temperature-dependellt parameters
'[' IOC) ~ 7GO 871 982 10!J:l
Il 1.055 1.03 0850 0.70
Z2 == Zo [Mi',,] 2700 2-100 I!JUO 120U
AI /sec-II 0 0.0055 U.02 02.5
Ih [sec-I] U 0 U 0
3. Elastic constants
E = 1.987 X W + 16.781' -0.10341'1+1.143 x IO-sr [M Pal
G = 8.650 X W -17.581' + 0.023211'2-3.464 X lO-s1'3 [Al PilI
(with temperature in "C)
4. Thermal expansion coefficient
l' a
[OCI I/"CI-\O-5
0 1.15
200 1.3·1
400 1.36
600 \.<l1
800 I.M
1000 1.60
1200 1.66
ExternallIow
~
Sidewall
Slrul bolY shock
Cot>laut now
Aerodynamic: .kin
Coolant p.unge-beatexchanger rins Dot shown
Primary structUUI
Fig. 1: Fluid-thermal-structural interactions on anaerospace plane scramjet engine leading edge. Fig. 2: Convectively cooled structure.
,
Uniform FluidVelocity Profile
Fig. 3; Enginccring model of coolant passage heat transfer. Fig. 4; Finite element model of coolant passage.
LINEAR FLUIDITMPERA1lJRE VARIATION""\.
T 1 '\TJ
\.. l't'P ICAl fLU IDfEMPERATURE
(a) mass transport element
NO
~, .1 PREDICTORfI ERROR ~ E If
KTYP ICAl SURFACE NODE
TYPICAL FLU 10 NODE
l
OJUIIIFORM flUID IVELOC I rY PROF J L£
Fig. 6; Flow chart for adaptive time step algorithm.
(b) suriace convection element
Fig. 5; Coolant passage convective elemcnts
healing
cooling (665 R)
t .015 In
(5345 "J S ~,,""""'(a) thermal-slructnral lIIadei
.00196
.00006_
o
2(Blulln S R)
.5
(b) convcction coe/liciellt history
OTIME (sec)1.
Fig. 7: I-D Mo<lel of convecLiv<:ly cooled structurc
TEMI'ERATUIlE (R)
600~o .5 1.0
TIME (sec)
600.
o
-600.
STRESS (M!',)
I
.5
J
1.0 TI~IE (secl
DT (sec)
_01
(a) Tcmperature history (I» Stress histOlY
..---
________ £IXED-.9T_ --
oo .5 1.0
FIXED - 1200 STEl'S, AUArnVE - 21J STEPS
(c) Adaptive time step history
TIME (ICC)
Fig. 8: TIH'rl11o-visr.oplastic response of I-D model
