0 

DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS 

COLLEGE OF ENGINEERING AND TECHNOLOGY 

OLD DOMINION UNIVERSITY 

NORFOLK. VIRGINIA 23529 

THERMO-VISCOPIASTIC ANALYSIS OF HYPERSONIC STRUCTURES 

SUBJECTED TO SEVERE AERODYNAMIC HEATING 

BY 

Earl A. Thornton, Principal Investigator 

Progress Report 

For the period December 31, 1988 to September 30, 1989 

Prepared for the 

National Aeronautics and Space Administration 

Langley Research Center 

' Hampton, Virginia 23665-5225 

Under 

NASA Research Grant NSG-1321 

A.R. Wieting, Technical Monitor 

SMD-Aerothermal Loads Branch 

Submitted by the 

Old Dominion University Research Foundation 

P.O. Box 6369 

Norfolk, Virginia 23508-0369 t 

0 

__ October 1989 *. 

(NA5A-CR-185YI 5 1 

ANALYSIS 7F HYPFRSdNIC STRUtlUAES SUBJECTEO 

TO SEV€Rk AERF3YNAMIC HEATING Proqress 

Report, 31 Oec, 1989 - 30 Sep, 1989 told 

Dominion Univ,) Id p CSCL 0AA Gj/O2 0234444 

THESMrJ-VI SCQf LASTIC N53-2 9 3 2 3 

Uncl as 
TIIEIZklO-VISCOPLAS‘TIC ANALYSIS OI: IIYI’INSONIC S‘L‘RUCTURES 

S l.1 L)J EC‘I’EL) TO SEVISIt I2 A EX0 DY N A M IC I I EAT IN G 

Earl A. Tliornton’ aid J. Tirisley 0clciiz 

Texas lnstitute of Computational Mechanics 

The University of Texas at Austin 

Austin, ‘l’exas 78712-1085 

W. Woytek Tworzydlo’ atid Sung-Kie Youn4 

The Conipiitational Mechanics Co., IIIC. 

3701 North Laniar, Suite 201 

Austin, Texas 78705 

A bst rnct 

A tliermo-viscoplaatic computational method for hypersonic structures is presented. The 

method employs a unified viscoplastic constitutive niotlel implemented in a finite elcment approach lor quasi-static tlicrinal-strrictiiral analysis. Applications ol the approach to conveclively cooled hypersonic structures illiistrate the 

eflcctiveness of the approach and provide insight 

into the transient inelastic structural behavior 

at elevated teniperatures. 

INTRODUCTION 

Tlic comrnitmcnt to develop the National Aerospace 

I’laiie lias generated resurgeiit interest in Llie teclinology reqiiired to design structures for hypersonic flight. 

Sucli striictrires will be exposed to aerodynamic heating 

of rinprccetlcntetl niagnitutles. As the veliicle accelerates 

or tlecclcrates at hypersonic speeds in tlie atiiiosplicre, 

shocks will sweep across tlie vcliicle antl interact with 

local shocks arid boundary layers. These iiiteractioiis introducc severe local pressure8 and liealirig rates. A recent 

experimental study (ref. 1) of interacting shock waves on 

a cylindrical leading edge sliows healing rates tcn times 

uiidistrirbed levels. 

Lcatliiig edges of cngiiie structures (Fig. 1) present a 

significant design problem because of intense local heating and pressures. Analysis of the flow, thermal and 

structural behavior preseiit serious computatioiial cliallcnges to analysts because of the inherent nonlinearities 

iii all aspects of tlie niullidisciplinary probleins. Soiiie 

of the critical computational issries are identified in reference 2. Critical issues inclticle tlic clificulties iiivolvctl 

in (I) analyzing the viscous, compressible flow and preclictiiig tlie high local aerodynamic heating, (2) iiiodcling and analyzing multimode unsteady lieat transfer in A 

liigli ternperatlire convcctivcly-coolctl structurc, and (3) 

simiilatirig tlic transient, lionlinear tliernial-structciral KCI Visiting Scholar, Associate Fellow AIAA 

Director 

3*4Senior Research Engineer 

sponse for rapid teinperaturc cliaiiges. I’rclirninary striicturd analysis of an impingement cooled leading edge (ref. 

3) sliowetl high local plasticity that seriously tlcgratlctl 

the structure’s load carrying capacity at clcvat.otl tcinpcratiires. A recent thermostructiiral analysis wit.li expcriiiiciital vcrification (ref. I) of cowl lip clcsigns colifirriictl I.liat iiielastic cffects occitr and can be significant. 

In the expcrimeiital study, two specimens failctl di~e to 

burn-through because of intense local heating or bccause 

of loss of cooling. 

Tlie purpose of this paper is to present a thcrnio--viscoplastic compiitationtal method for hypersonic structures siil)jected to severe local unsteady healing. ‘L’lie 

analysis einploys a unified viscoplastic constitutive iiiotlcl 

implementctl in a finite element approacl! capable of predicting rate-tlependeiit plasticity effects for tcinpcraturcs 

up to al>out 75 percent of tlie melting poiiit. Ratetlcpendcnt plasticity effects are known to he irnportaiil 

at elcvatcd temperatures. 

Iliiified viscoplastic constitiitive riioclels liavc evolved 

ovcr the lad twenty years to provide a incans lor analytically rcprcseiiting a rriaterials responsc from tlie elastic tlirougli the pliisl.ic range iiicliitliiig strain --rate ilcpeiitlciit plastic flow, crecp antl stress relaxation. The 

theories are giiitletl hy physical consiclcrations including 

tlislocatioii tlynaiiiics aid arc I)asetl on the priiiciplcs of 

coritiiiiiiciri riicclialiics. ‘llie first rmil~iciimensioiiil foriiiiilations of elastic-viscoplirstic constitutive eqiiatioiis was 

due to Bodner antl Parloin. Since tlicn a iiiiiiibcr of 

constitutive niotlels have appcarctl; many of tllcsc theories are sriiriniarizetl iii review articles that appear in 

rcfcrence 5. A NAS11-l.cwis sponsored research prograiii 

(IlOS‘l’) conclucted by tlie Southwest Ilesearcli Iiistitiite 

recenlly coiiclutlctl a four year research effdrt (ref. 6 -7) to 

frlrllier tlevclop iiiiifietl constitutive models for isotropic 

inaterials aiitl to dciiionstrate tlicir iisefuliicss for analysis 

of liigli tempcratiire gas turhiiie engines. Oiie rrsiilt of 

this sl.iitly is matcrial property (lata for high tciiipcratiirc 

nickcl-l)asecl alloys over a wide temperature raiigc. ‘Hie 

unifed models employed were those of Botlricr-Partoiri 

ant1 Walker. 

Unified viscoplastic tlieories have I)ccii irnpleinciitctl 

hy a number of finite clciiieiit researclicrs. I)ctailctl stud- 
ies of sevcral rate-tlcpeiitlciit plasticit-y rriotlcls aid tlicir 

niimcrical i~ii~)lciiie~itatio~i using adaptive finite elcinent 

melliocls were conducted by Bass [8] and Dass and Odcn 

[!J, IO]. Uiitlcr the NASA IIOS1' prograin, the Walker 

model was irnplcrneiitcd in the MARC finite elcmeiit 

program (ref. 11) and used to analyze the tlicrnioviscoplastic response of a turbine blade under simulatetl 

flight conditions. In another recent finite elenient application (ref. 12), the Dodiier-Partom and Walker theories 

were conipared for a thin circular plate subject to higlily 

localized, transient heating. 

111 this paper the Bodner-Partom constitiitive modcl 

is eniployed, and the finite eleniciit approach devclopctl 

in rcfercnces 8-10 for the isotliernial case is cxtendcd lo 

include tlicrinal elkcts. The paper begins by dcscribing lhe finite elemelit approach used for tlierrrial analysis of convectively cooled Iiypcrsonic structurcs. Tlicii 

the tlicrmo-viscoplastic striictiiral analysis is presented. 

The initial value problem is lorrnulated, tlie constitutive 

rnodcl is described and the finite element viscoplastic solulion nictliod is preseritcd. Next, applications of tlie 

approach are made to coiivcctively coolcd striictiires. An 

appendix presents tlierinal and structural data for the 

convectively cooled Iiypersoiiic structure used iri tlic nuiiierical coniputations. 

THERMAL ANALYSIS OF HYPERSONIC 

STRUCTURES 

Convcctively coolctl strcictcircs are strong caiitlidates 

for use in hypersonic flight vehicles. For hypersonic flight, 

some leading edges arid paiicls require active cooling systems to kccp structiiral tciiiperaturcs within acceptal,lc 

ranges. l'lie interiial flow iii the coolant passage lias a 

predominant role in the thermal rcsponsc of a hypersonic 

structure subject to exleriial Iieatiiig. A cross-section of 

a typical convectively cooled structure is sliowii in Fig. 2. 

An aerodynamic skin aiid a coolant passage with internal 

lieat exchanger protcct the priinary structure frorri llie 

aeroelynainic heating. l'lie thin, typically metallic, acrodynaiiiic skin lraiisfers the energy of the aerodynaiiiic 

heating to a low temperature coolaiit flow through tlic 

lieat cxcliaiiger fins that connect the aerodynamic skiii 

to tlic priinary structure. lii a typical eiigine striictiirc, 

the coolant is cold liydrogcn that later is used as tlic 

propulsion system fuel. 

Ileat transfer in llie aeroclyiiarnic skiii consists of conduction combined with surface radiation. lleat transrcr 

between the aerotlynarnic skin, the lieat exchanger surlaces and the primary structiire is by conduction at tlic 

solid-fluid interface. The finite clcnient representation of 

concliiction lieat transfer with radiation boundary coiiditions follows the standard proccdures dcscribecl iii rcl. 

13. 

The dominant mode of licat transfcr in tlie coolaiit 

llow is forced convectioii. 'l'lic rcprcsciitation of tlic Iimt 

transfcr in the coolant passage is the critical step iii tlic 

lieat transfer analysis. 'llicre are two basic represenlalions that can be employed. 'l'lie first, clcnoted liere as 

the engineering moclcl, is based iipo~i a niinil)er of assumptions that greatly simplify thc problem in1.o a single 

energy equation with a specifictl niass llow late. I)ctailed computation of the fluid velocity coinponeiits aiid 

tcmperatiires is not reqiiiretl. 'l'lie second rcprcscntation itlcalizcs tlie coolant flow as a contiiiiiuiii modcl, and 

the partial tlilTcrciitial equat,ions describing conservatioii 

of mass, iiiomciitiini arid ciicrgy are solvcd siiiiultancously to oblaiu fluid velocity and tmperatiire distrihtioiis. 'l'liis lattcr inotlel is the most accurate, l)ut it is 

also consit1cral)ly inore expensive tliaii tlie engineering 

moclel. In this paper the cngiiiecriiig lieat transfer model 

is eiriployctl. 

Engineering Model for Coolant Passages 

Thc Oasic leatiires of the engiiiceriiig model arc elevcloped froiii the itlealization sliown in Pig. 3. A scgiiiciit of 

tlic roolait1 passage of width w is sliown; for siinplicity, 

only the uppcr one-half of the coolant passage with tlic 

acrotlynainic skiii is sliowii. The ciiginccriiig foriiiiilation 

(id. 1.1) is based on the followiiig assumptions: 

1. Tlie thermal cncrgy state of the fluid is cliaractcrized by tlic fluid bulk temperature TF wliicli vaiics 

only in the flow direction, i.e., Tr(+, t). 

2. l'hc flow is represented by the mass flow ralc lit 

in the coolant passage specified by yn = pr; A,.-Vr; 

wlicre pr; is tlic coolant tlciisity, AF is the crosssectioiial area of the coolaiit passage, aiitl \/, is the 

coolaiit mean Ilow vclocity. 

3. A coiivertion cocficicnt /I is dcfined siicli that tlie 

lieat fliix ir; transfcrrecl bctween the striicl iiic aiid 

tlie coolant may be expressed as 

qr; = h(Ts - TF) 

wlicre Ts(z, 1) tlcnotes the structural tciiipcraturc 

at tlic fluid-solid interface. 

4. 'rhe convectioii coclficieiit h may be expressed as 

a function of lhe Illrid bulk temperature alone I)y 

using analytical/ciril)irical equations lor thc Niissrlt 

iiiiriiber, 

wlirrc U is llic Iiytlrniilic tliariicter of tlic coolant 

passage, aiitl kr; is tlic tlicriiial conductivity ol lh! 

Iliiid coolaiit. 'llic coiivection coelliciciit i~s wc.lI its 

tlic solid atid Ilui(l tlierinal pararncters iiiay, iii g(:iieral, I)(! tcinpcrahire tlcpciitlcrit. 

With tliesc assiiinpl ions, energy balances on tlic acrotlynairiic skiii and coolant give tlie govcriiiiig conscrvalioii 

cqiiatioiis. 

Fluid: 

Solid: 
wlicrc tlic su1)script.q I.' atid S denote tlic Iliiid and solid, 

rcsl)ectivcIy. 111 tlicse cclitatioiis, c ticiiotcs spcciic Iicat, 

u is tlie Stcfaii-l~oltziiiatin constaiit, and L is the surface 

ciiiissivity. llccausc of tlie teiiipcratrire tlcpentlciice of 

the I.liernial parameters aid the radiation term, Eqs. (I) 

atitl (2) coiistitiite a iioiiliiiear sct of partial tlillcrcritial 

ccliiations. 

'I'hc licat exchanger fins are not inclutled explicitly iii 

this motlcl. IIowevcr, the heat traiisfer between tlie heat 

cxcliangcr fitis and coolant can approximately be taken 

into consideration tlirougli tlie use of an ellective width 

IO which represents the area over wliicti the convectivc 

licat excliange occurs. 

Finite Element Foriiiulatioii for Coolant Passages 

A typical fiiiite elemctit representing Eqs. (1) and (3) 

is clinracterized by fluid and Ilriitl-solid interfacc nodes. 

The elctncnt shown iii Fig. 4 has two fluid nodes (I 

aiicl J), aiitl two fliii&solid interface nodes (IC and L). 

Witliiii the clenietit, the fluid and solid temperatures are 

expressctl as 

TF = [N (z ) 1 (TI.. ) 

whaIc [N(z)] are tlic elernelit intcrpolaI.ion hinctions. 

Following usual finite elctiiciit procedures (ref. 13), the 

tliscrctizcd equations lor aii elcinelit of length L may be 

tlerivcd iri the lorrn 

CI; 0 [* cs]{ 

wlierc the clciiicnt capacitaiicc matrices arc given by 

and tlie element conductaiicc iiialrices are 

'rlic clcineiit eqiiatioiis given iii Eqs. (4) show tliat the 

coolaiit passage iiiotlcl can be rcgartlcd as an assciiibly 

of elements where each eleinent rcprcsciits a single Iicirt 

transfcr mode. 'l'liiis we can represcnt tlic coolant passage in terms of two convective elements: (1) a masstransport cleinciit (Fig. 5a), and (2) a surface convection 

element (Fig. 5b). The mass transport element represents the downstream convective heat transfer due to 

niass flow; it lias a capacitance matrix [CF] as well as 

conductance matrices [IC,] and [KF]. The surface convection element represents the convection heat exchange 

betwcen the coolant fluid and the solid; it does not have a 

capacitance matrix, but it bas a conductance matrix [l(h] 

linkiiig tlie solid and fluid nodes. These coolant passagc 

elements are asseniblctl into tlie finite element thermal 

model for the complete convectively cooled structure that 

lias condiiction eleiiieiits, radiation elements, aiid surface 

elements for convection to a specified convective excliange 

temperature. 

Tlic unsteady thermal analysis is nonlinear bccaiise of 

temperature dependent thermal properties and surface 

radiation. The equatioiis are solved by time marching 

with tlie Crank-Nicolson algorithm; at each time step, 

the nonlinear algebraic equations are solved by NewtonRaplison iteration. 

TIIERMO-VISCOPLASTIC STRUCTURAL 

ANALYSIS 

Tlic bcliavior of a tliermo-viscoplastic structure suhjectetl to aerodynamic heating is analyzed assuming that: 

(1) thermo-mechanical coupling iii the conservation of 

energy equation can be neglected, (2) tlic structural response is quasi-static, and (3) dclormations arc iiifinitesirrial. Willi tlicse assumptions, an unsteady tlierrnal 

analysis may be performed first to determine the temperatures. Then, using tliese temperatures, the structure's viscoplastic response is determined. Tlie solution 

is thus obtained by separately solvirig initial houiidary 

value problems for first the thermal and then the structural respouse. 

Initial Value Viscoplasticity Problem 

Consider a viscoplastic structure occupying a rcgioii 

R with boundary DR. 'llie beliavior of the structure is dcscribed I)y the following systeni of dilleretitial equations: 

1. Equililhin in rate form, I 

where uij denote coinpoiieiits of the stress tensor, 

bi are the body force components per unit volumc, 

aid the suiriniatioti conveiitioii is einployccl. 

2. I<inciriatic rclation for velocity gratliciits, 

whcre t,) dcnotca the total strain componciits and 

siipcrscripts E aiid P denote elastic aiid iiiclaslic 

strain components, respectivcly. The components 

of the displacement rates arc it,. 
strain rate aiitl strcss tcnsors, respcctivcly. lbr cxninplc, the deviatoric stress invariant is 

(110 sum) (9) 

wliere El;kr represents Ilooke's tensor of elasticity 

paramcters, nkl arc coinoncnts of a tensor of tliernial cxpatision parameters, and AT represeiits the 

rate of the cliange iri temperature. Doth Eijk, and 

(2k1 are temperature dependent. The constitutive 

Iunctioiis arc /;j and g; wliere 21, represents interiial state variahlcs. 'I'liese fwctions and state varial)les cliaractcrize the viscolblastic response of tlie 

iiiatcrial. 

l'he dcsrription of thc prol)lem is cornpletcd by presc r i I) i ng t lie bo ti iitl ary and in i t ial cond i lio ns, 

ri = ~i, oll an, 

(10) 

whcrc iil are prescriImI sui face tlisplacernciit rates, n, 

arc the components or a unit noriiial vector, and are 

prcsci ihed surface traction rates. The initial conditions 

iiicliide specifying the displacements, stiesses and internal statc variables, i.e., 

&,,lt, = i+& on an2 

210~~"1J(",o~IzI("Io~ E 

Constitutive Model 

'l'li~ 13ocliicr-Partoin coiistitutive model is of the internal statc variable type tlial is based on phenomenological 

observations and supported by physical concepts related 

to dislocation dyiiarnics. 'I'lie modcl has gone tlirougli 

scveral inotlifications atitl was extcndctl lor anisotiopic 

rvork liarderiing materials. The current model (ref. 5-6) 

also includes tcmperature ellects. 

I. Flow Law, 

For tlic inelastic straiii rate component, the isotropic forrn of tlie Praiitlll-Reiiss law is assumed 

ilJ = AS, 

(11) 

i[k=o A>o 

wlicre SI, are tlie deviatoric shes components 

given by 

(12) 

I 

3 s,j = 0,j - -6:jokk 

ancI i[k = O denotcs plastic incompressibility. 

2. I<iiieniatic Equations, 

Squaririg Eq.( 11) leads to 

A' = U,'/J' (13) 

where /I[ and J2 are the secoiid iiivariaiits of tIw 

3. 

1 

Jz = -S,,S,j 2 

The relation governing inelastic tlcrormations is tlic 

"kinetic ecpiatioii", and tlic forin takcn by BotlncrPartoin is 

where Do is tlie limiting strain rate in shear, n is a 

teinperatiire-depeii[lent inaterial parameter, arid 2 

is intcrpretcd as a load history tlcl)cndcnt parainetcr (hewin callcd the internal state variable) that 

rcpresciits thc hardness of tlie iiiatcrial with rcspcct 

to rcsistaiice to plastic flow. Combining I;qs.( I I), 

(12), and (13) gives 

Evolution Eqitations of Iiitcrnal State Variable, 

The internal state variable Z consists of isotropic: 

and directional cornpoileiits, 

z = 2' -t ZD (17) 

The evoliition eqtiation proposed for the isotropic 

Iiardeiiitig coinpoilent (ref. 5) is 

i'(l) = 

with the initial condition, Z'(O) = Z,,. In tIic firs1 

term, Z1 is the limiting (sattiration) value or Z', 

ml is the hardening rate, and the plastic work rate 

is 

wliicli is take11 as the iiieasiire of Iiardeniiig. 21 is 

lhe minimum value of Z' at a given tcinpcraturc, 

and AI and rl arc tcinpertittirc tlcpcndent illaterial 

constants. 

'The evolution lorrn of the directional liarclcniiig 

coinponelit (ref. 5) is defined as 

iV1) = o,,i; (19) 

wliere I[,] are tlie tlircctiort cosiries of tlic ciirrciit 

stress stale, 

The evolution cqiiation for &(l) 11.75 tlic saiiie geiiera1 form as that for isotropic liardcniiig 1)ut lins 

lciisorial character, 
wllerc 

arid 

118J(t) = @lJ(t)/[~k~(t)~k,(f )I* (23) 

PIJ(’) = ’ (24) 

As in Eq.(4.12), nil is tlie hardening ratr A, and 

rl arc temperature dependelit material coiistaiits. 

Otlier variaiits ol the isotropic and directional Iiartlciiiiig 

variables are described in refs. (5-7). l’liese references 

also present data lor the parameters used in coiistitutive 

inodcls for several inaterials. The particular values used 

in tlie present study are presented in the Appeiidix. 

Finite Eleiiieiit Formulation 

‘I‘lie iiiiite element forinulatioii follows the approach 

of Ref. 9, aiid the details of the weak foriiicilalion aic 

prcisciitcd the. Sincc tlie approacli presented iii Ref. 9 

is for the isotherinal case, this forniulation extends tlic 

previous approach I)y tlie iiiclusioii of teinpcrature erects. 

I lie finite element approach approximates displaceriicril rates witliiii ail eleiiieiit by taking tlie displacement 

rates as 

,. 

{h) = I”&) (25) 

where [N] are the interpolation functions, and (6) represent the nodal displaceinent rates. Using tlie strainclisplacemeiit equations iii rate form, Eq. 8, an element’s 

straiii rates can be computed as 

where [ 111 is tlic straiii-dis~)lacemcnt matrix. Followiiig 

usual iiiiite element procedures, the finite element equatioiis for a typical clerneiit are for111 as, 

where [K(Z’)] is the clenient stillness matrix. and the 

ternis on he right-hand side are elcinelit load vectors 

the to tlie rate of plastic strains, teinpcrature, surface 

tracI.ioiis atid body forces, respectively. ‘These rriatrices 

are tlcfiiictl by 

[f<(l’)] = 

{PPI = 

(&) = 

{&) = 

(PB) = (32) 

wlicre Re denotes the element. voliime, and OR, denotcs 

an clement surface where tractions are defined. 

Tlie teiiiperature alrects the viscoplastic structural 

analysis directly in thee ways: (I) the elasticity matrix 

[E(T)] and the cocllicieiits of Llicriiial expaiision (47’)) 

tlepciitl on triiiprriitiirc, (3) nodal loads ( FT) (Irl)cti(I on 

tlic local tertipcrntuic ialcs, and (3) several paraiiictrrs iii 

the Dotliicr-l’artoni conslitutive inotlel are tcinpcratiirc 

dependent. 

Viscoplastic Solutioii Method 

Since the present approach uses an uncoupled (soinetimes called one-way coupled) formrilation, the thoriiial 

problem is solved first followed hy the viscoplastic analysis. The transient thermal problcni is solvctl by tiiiic 

marching with a time step AfT, and the nodal leiiiperaturcs at successive tiines tl, t2,. . . are obtained. ‘Tlicse 

teinperature vectors are used as input to tlie structural 

analysis. 

Tlle first computation in the structural analysis is to 

solve an initial statics problem if the initial temperaturc 

clistribution 7’(r,O) is iiot equal to a uiiiforrn rcfcrence 

temperature or if any initial static loads are present. ‘l’lie 

results of this analysis are the iiiitial conditions (displaccmenls arid stresses) for the traiisieiit viscoplastic atiidysis. 

‘flie viscoplastic analysis time-marclies with a tiiiic 

step At,. Experieiice lras sliown tlie time stcp required 

for the structural arialysis is usually snialler than for the 

tlierinal aiiillysis, i.e. At, < A~T. At intermediate tiines 

in the structural analysis, the temperatures arc liiicarly 

interpolated from tlie teriiperatures known at tlic bcgiiining and end of the larger tlierrnal tinie intervals. 

l‘lie strategy employed in the viscoplastic algoritlirii 

is as follows: with tlie initial distribution of stress, t.em 

pcrature and iiiteriial variables specified use thc eqtiilibriuin contlitioii (Eq. 27) 10 obtain the notlal tlisplacciiiciit 

rates. Then iiitcgrate tlie constitutive equatioiis forward 

in time at the eleiiiciit Gauss integration poiiits. With 

updated values of the stress, tcinpcrature aiitl iiitcrnal 

variables at the iiew time, tlic ecliiilil)rium cqiiatioii is 

solved agaiii. l’liis sequence of dcteriniiiiiig tlic iiodal 

displacemelit rates, tlieii aclvaiiciiig tlic constiliitivc cqiialions in time is contiiiuetl until the desired Iiistory of tlic 

initial boundary-value problein lias becii ol,lniiirtl. 

‘rhus, tlie algoritliiii proccetls ~hr0ugh tlic following 

sleps: 

I. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

At tiiiie 1, initialize ai,, Z, for cacli clcincnt; 

CaIciiIatc C: = J,,(u,j, z,) for cacli elcinelit; 

Assemble and solve [K](6) = (k) ; 

Calculate i,, for eacli c~ciiiciit, ( C) = [DI(~); 

Calculate u,J for each cIeiiieiit, (u) = [El(;-;‘‘} - 

[ E]aAl‘; 

Calculate Z, for eacli e~ciiiciit~, i, = g,(aiJr ~k); 

Integrate ui,, 2 forward for cacli clcinciit to gct a,, 

and 2, at t t At,; 

Iff i- At, < Ljina~ go to 2, otherwise slop. 

The computational niethod above has bccn prcs~iiilctl 

for a constanl tiine step At,. Coinpntational cxpcrimcc 

by several investigators (see Ilcl. 7-12) iiitlicalcs lliat a 

vcry stliitll titne step can he rcqiiitetl I)ccaiisc of tlic “stilr” 

tiilttitc of llic orditiary clifTcrciitial ccliialions tlc:scril)itig 
tli~ iiitoriial state variii1)lcs. 'I'n giiiii irnprnvc(1 clliciawy 

niid reIiabiliI.y, a variable tinie step algorithtii has been 

irt ipleriic~itecl. 

Variable T i m e S t e p Algoritliin 

Iii tlic constatit tinie step viscoplastic algoritlitn tlie 

interiial state variahlcs are advaticed in time witli tlie COIIditionally stahle Eider forward dilrereiice algoritlim. The 

variablc time step algoritliin is a modified Eulcr sclieme 

ctsiiig a truiicatioti error criterion (ref. 15) to adjust the 

time step. 

For simplicity, cotisider the siiigle ordinary differential 

cqiiation, 

'l'lie soliltion is advanced itsiiig a predictor-corrector 

scliciiie. 'Hie predictor phase consists of ail Eulcr step: 

Y = f(Y,1) (33) 

(34) 

(35) 

An crror indicator E (Ilcf. 15) is llicn computed from 

Tlic error indicator is next cotiipared witli a preset error 

criterion, and if the criterion is inct, the time step is sufficiently small enoirgli to proceecl to the corrector stage. 

Otlierwise, tlic predictor pliase Eqs. (34-35) is repeated 

with a siiialler time step. 

The corrcctor pliasc is the tiiodified Newton scltenic, 

(37) 

A flow chart depicting tlie adaptive sclierne is sltowti 

in Fig. 6. 'l'lie flow chart sliows how the time step is 

either reduced or iiicreased depending oii the error indicator, Eq. (36). The flow chart sliows that the time step 

is reduced or increased by a factor of two. l'liis approacli 

is effective, biit an alteriiate scliriiie also has I,ecii used 

where the uew time step is bnsetl on tlie error E. In tliis 

sclicnic 

wlicre Eo,,, = ax. Error tolerances E,,,,,, arid 

&,in arc sl)ccifictl I)y tlie user, typical values being 

E,,,,, = 0.05, E,,,i,, = 0.005. 

APLICATIONS 

As tiieiil.ioiied iii tlic ititrocliictioii, scraiiijet ciigitic 

striictrircs can expcriciice iiit.ciise local heating diie to 

sliock intrractioiis. 'I'wo iiiodels ol siicli structures are analyzed lor lieat loads rrpreseiitative of sliock interactions. 

hlaterial properties itscd (see tlic appeiidix) represent a 

high-tcrnperature nickcl-basccl siil)cI alloy. Computations arc pcrforniad first fnr a siinplificd one-ditiiciisioiial 

model to provide iiisiglit into the computational proce- 

(litre. Then a two-tlitneiisional model of a realistic convectively coolcd structure is analyzed in detail. 

Simplified One-Diiiiensional Model 

To gain a preliiiiinary understanding of tlic behavior 

of a convectively cooled structure subjected to coiivective heating, a simple 1D bar model was investigated. A 

segment (Fig. 7a) of the aerodynamic skin is riiodeled 

assuming uniform temperature. 'rhe coolant is assumed 

to have a constant, specified temperature, and tlie skin is 

heated convectively. A strong time variation of the convection coeficicnt, (Fig. 7b) simulates the passage of a 

moving shock. Tlie resulting transient temperature and 

bar compressive stress exhibit features to be cxpcctetl in 

a more complex model. 

Tlie traiisient temperature history (Fig. Sa) sliows a 

rapid rise following the sudden increase in tlic convective coeficieiit, and tlien a smooth decay after tlie large 

convective heating is removed. The convective cooling 

caiises tlie temperature to return quickly to equilibrium. 

The initial rapid increase iii teinpcratiire causes the bar 

to yield iri compression very early in the respoiisc (Fig. 

8b). After the temperature begins to decay at 1 = 0.59, 

tlie stress rapidly clianges to tension hence as tlie teniperature returns to cquilibriuni a large residual tensile 

stress rcmaitis iti the bar. The tensile stress is iiitlucetl 

because the initial liigli temperature causes coiiiprcssivc 

yielding. 'I'his yielding tends to permanently sliortcii tlic 

bar. Ilowever, the bar's fixed end botititlary coritlitiotis 

proliibit tlte bar from decreasing in lciigtli M tltc kinperatiire decreases so that a tciisile stress dcvclops as the 

nialerial cools. 

During tlie tlicrnial-striictural analyses an itncertiiiiity arose concerning applica1)ility of the Iiardtiess cvoIiil.ioii qiiation, Eq. (IS), at liigli tcnipcratitre ratcs 

(almiit 5000°C/sec) occitrring iii tliese aiialyscs. At such 

liigli ternpcirature rates this cqitatioii may prcdict somcwhat anonialous behnvior of the material. Tliis is illustratctl by the following cxample. Consider a specimen 

of siiperalloy B 1900 + 11 j with properties preseiitetl in 

tlic appendix. If tlie virgin, load-free spwinieii is licatcd 

from 760°C to 1060°C in 0.1 sccoiicl, tlic teiiipcraliirctlepcnderit variable 2, drops horn 2700 MPn to 1200 

MPo. IIowcvcr, the actrial tiardiiess variable Z', wIiicIi 

follows 22 only throiigli the relatively slow tliertiial rccovery term (secotid term iii Eq. (18)) reduces iii tliis 

time only to 2696 AIPa. This means that altlioiigii tlic 

sl)ecitiicri has rcaclictl ail elevated tcitiperature, it has not 

softened yet-a somewhat surprising result. 

Tlie nced for fiirlher research in tliermal liistory cffects on B 1900 -I- Iff was noted in ref. 6. 

In tlie initial viscoplastic analysis a fixed time stcp 

of 0.001 s was used, and 1200 time steps were reqiiirctl 

for tlie analysis. 'I'lie variable time step algorithm sliowii 

in Fig. 6 was then iinplemented, and the prohlein was 

rcsolvetl. Fig. 8c sliows the liistory of tlie variable tinic 

skp arid iiidicatcs that tlie new analysis rcqiiirctl otrly 

213 steps - a siildaiitial savings. 'I'lie figure sliows tlmt 

in tlie ''flat" part of the stress respottsc a large 1,itiii: step 

was uscd, hit iicnr 1 = 0 and agaiii at 1 = 0.59 wlicti tlw 
stress is cliaiigilig rapidly, small tittic sttys are iicctlccl 

to captrirc the response accurately. ‘rliis example shows 

tliat the atlaptive time step algorithm is an iinportalit 

step towards producing a relial)le, accurat.e solution. 

Convectively Cooled Structure 

A inore realistic 2D modcl of a coiivectively cooled 

Iiypersoiiic structure is shown iri Fig. 9. llie model 

rcprmcllts a segment of a convectively cooled structure 

sclcll as a wall of a scramjet engine furl injection strut. 

‘rIie finite element tlicrtnal model (Fig. 9a) includes: (I) 

contliiction lieat transfer in tlie aerotlynamic skin , heat 

excliaiiger fins and antl primary structure, (2) CotIveclive Iicat transfer between tlie walls of the coolant passage and coolant, (3) mass transport convection in the 

coolant wliicli has an unknown bulk temperature, and 

(4) surface radiation on tlte aerodynarnic skiii. The aeroclynarnic skin is uniformly coiivectively heated over its 

Ierigtli, aiid superimposctl on the iiniform heatitig is a 

local, intense licatitig siinrilatiiig a traiisient shock. Tlie 

transient heating is induced by tlle time-depeliderit convection coeficient sliowii in Fig. 9b. 

In the plane strain structural finite element model, 

the primary structure and acrodytiaiilic skiti liave unit 

thickness, but tlie lieat excliangcr fins in tlie coolalit passage are approxiinately represented by a single fin with 

tlic total thickness of O.OG0 in. The will1 segtriciit has 

fixed displacemcnt boundary coditions at the left atid 

right eiids. The aerodynamic skin, exclianger fins and 

primary structure are also loaded by internal pressure 

(7.0 MPa = 1000 psi) in the coolaiit passage which is relatively large coiiipared to cxtcrtial ircrodytiamic prcssurc 

(0.35 Ml’a = 50 psi). 

’l’lie inoclcl was analyzed first for steady aerodynamic 

heating aid ititeriial pressiirc operating conditions. ‘rlien 

the sucldcii localized aerodynamic lieatiiig was applied. 

TIIIIS, tlie steady teniperatures and stresses serve as iiiitial coiiditioiis for the trailsiclit tlicrmo-viscoplastic analysis. 

‘I‘lle viscoplastic soliition was cotnputed with the variable time step algoritliin, ant1 tlie time step was varied 

using Eq. (39). A total of 265 steps were required to 

cornpiite the rcsponse for a total tiirie duration of 1.2 s. 

‘I’hc Lhcrmal response of the wall segment is shown in 

Fig. IO. Figure IOa sliows the time history of tIie tcmperaturc at a poitit on 1.11~ aeroclyiiaiiiic skiii tlirectly tiiiclcr 

the traiisieiit heating; Fig. lob sliows contours of teniperatiires a1 1 = 0.5s when tcinperalures arc rnaxirniim. Tlie 

lempcralure history is qualitatively siinilar to the results 

ol)lainctl iii the ID model sliowiiig the rapid skin lempcrattire rise and fall with the siiriiilatrtl shock heating. ‘rlic 

tcrnpcrature coiitoiirs sliow relatively stecp lheriiial gratliciils iiitliicetl 1)y tlie local heating, and tliat the coolant 

siibstaiitially limits the extent of the induced high teiiipcratiires. ‘l’lius the high temperatures are confined to 

tlic acrotlynatiiic skin, and tlie priiiiary structure experieiiccs only small tcrnperatiire cliaiiges. ‘l’lie temperature 

gradiciils in the skin particularly at the coolant-skin iiitcrface are not predicted witti high accuracy because of 

tlie cnginceriiig tnotlel of the coolant heat transfer. IIowever, local tciiiperatitre levels iii the skiti are reasonably 

acciiratc since net energy traitsfer to the coolant is rnotlcled satisfactorily. 

llistorics of tlic horizontal strcss coinpoiient n, at I.wo 

points tliroiigli the skiii thickiiess are shown iti Fig. 1 I. 

The stress liistories follow tlic teniperature, atid uiitlcr 

the intense local licatiiig stresses are vcry similar to the 

results obtaiiicd from tlieoiiedirrietisiotial motlcl. At tliis 

locatioii the skin yiclds tliroitgli most of its tliickness. After the heating ceases there is a rapid decay of stress, aiitl 

under tlie intense local lieating tliere are residual tensile 

stresses. Figitre 12 sliows the time ltistory of tlic vertical 

component of stress uu in tlie lieat exchanger fiiis. lligli 

tensile stresses are induced with significant local yielding 

which are followed by residual compressive stress. The 

crude finite element model of the fins only approximates 

these stresses, but the liigli tensile stresses can poteiitially cause a bond failure at the heat excliaiiger/skiii 

joint. The consequences of such a failure are stiitlietl in 

the next example. 

The viscoplastic stress histories are compared witli 

stresses predicted assuming elastic behavior in Figs. 

13-14. Tlie elastic computations were made with 

temperature-dependent elastic properties. The loss of 

stiffness due to the elevated temperatures for 0 c 1 < 0.5s 

accounts for the stress “rolling-over” with an appcarance of yielding. Generally, the elastic stresses are too 

high antl, of course, since yielding does not occur residual 

stresses are not predicted. Maxiinuiri deformations (not 

sliown) predicted by the elastic analysis and viscoplastic 

analysis are about the same, approximately 0.001 in. 

The two-dimensional character of tlie stress components u, and nu are shown in Figs. 15 and IG, rcspcclively. The stress distributions are shown at three times 

1 = 0.053, 0.53, 1.2s iii the respotise. The strcsscs at 

t = 1.2s are the resiclual stresses. Figure 17 sliows coiltours of the priricipal plastic straiiis at L = q.53. ‘& 

coritoiirs show tlic relatively localizctl nature of tlic Iiigli 

stress gradients as well as the tensile and coniprcssive 

regioiis. Tlie significant residual stresses suggest the possibility of ciimulative damage under repeated load cyclcs. 

Tliis possibility nceds further iiivestigation. 

Coiivectively Cooled Structure With Daiiiage 

The preceding aiialysis sliowetl Iiigli tensile strcsscs 

in the heat exclianger fins and suggested tlic possibility 

of an aeroclyiiamic skin/heat exclianger bond failurc. ‘To 

siiiiulate such a possibility, the analysis was rcpcatcd hit 

witti the fins given greatly retliiccd material properties 

for a width of 0.30 in. under llie high localized Itcat iiig. 

Two results of this analysis are sliowii in Figs. 18-19. 

Since the fins can no longer support tensile stress, 

the internal coolant pressure causes substantial local tlcformation of the aerodynamic skin. Figure 18 sliows the 

clcforiiied structure at f = 0.5s. The deformalioiis, sltowii 

uiitnagnifietl, rrpresent a significaiit permanent displacci~ieiit. This result can be seen in Fig. 19 wliicli coiiiparcs 

displacement histories for tlie undamaged antl (lainaged 

structure. For tlie damaged heat exchanger case, a rcsitliial permanent deforinalion of 0.007 in. is inlrotliiccd IJY 

tlie therinal loading. 

‘The iiiaxirniim cleformation intluced hy tlic fin tlntiiage depcntls strongly on the width of the clairingctl rcgion. Increasing tlic wicltli or the tlainagod rcgioii by a 

rririall amount will sitl)staiitially iiicreasc tlic tlcforirintioti. 
‘[‘lie analysis of tliesc largc tlcforiiintioiis reqiiires that tlic 

prcscnt lorrniilation be modified to account for large dcformation elrects. 

A possible consequence of this permanent tlcformation is to cause a disturbance of the extcrnal flow. Protiiberaiices into high specd flows can cause significant local aiigmcntation of the aerodynamic heating. Thus, the 

significant permanent deformation introduced by a fin 

bond failiire is likcly lo cause a complex llow-tliermalstructural interaction. 

CONCLUDING REMARKS 

A thermo-viscoplastic computational method for hypersonic structures subjected to severe local heating is 

presented. The method employs tlie Uodner-Partoni 

unified viscoplastic constitutive rnodel implemented iii 

a finite elcnient approach for qiias-static tlicrrnalstructural analysis. A variable time step algorillini is 

iiscd to provide an clficient solution sclieine for tlie slin 

ordinary dimerential equation that characterize the evolulioii of the internal slate variables. 

An analysis of a simplified modcl of a one-diinensional convectively cooled structure sliows the 1)asic features 

of the tliernio-viscoplastic response and demonstrates 

tlie siibstaiilial computer time savings as wcll as llic reliahility of tlie variable time step algoritlim. Ail aiialysis of a two-diinensional model of a realistic coiivectively 

cooled structure provided more detailed understanding of 

the tliermal structural behavior. ‘l‘lie coolant flow tiominates the thermal response providing a relativcly short 

tlicrinal transient. Under intense heating, significant local p1asticit.y occurs in tlie aerodynamic skin, but tlie 

pririiary structure reinains unt~aniagcd. [kat exchanger 

611s cxpcriciice high tensile stresscs and both the acrotlynaniic skin and heat excliariger fins have significant rcsitlita1 stresses. An analysis of an acrodynaiiiic skin/lieat 

exchanger fin bond failure showed that the aerodynamic 

skin rvoiild cxpcricncc significant local plastic tlcformation tluc to tlie intcriial coolant passage pressure. ‘I’lic 

local clcforination is pronounced enoiigli to disturb the 

extcrnal acrodynaniic flow and introduce a corriplcx Ilowtlicrnial-structural interaction. 

1 lie tliermo-viscoplastic analysis of convectivcly 

coolccl hypersonic striictiires iintlcr inlensc local licatiiig 

sliotvs t.1~ iiiiportant rolc that siicli aiiidyses cnti makc 

in understantliiig coniplex traiisient inelaslic structural 

bcliavior at clcvated temperatures. 

,* 

ACKNOWLEDGEMENT 

‘I’lie autliors are pleased to acknowledge the support 

of tlic Aerotlicrinal Loads Uraiicli at the NASA Langley 

Ilcsc~arcli Ceiitcr and tlie support of llie Air Forcc OlTice 

of Sciciitific Research. 

REFERENCES 

1. Wieting, A. It., and Iloldcii, hl. S.: “Experinicntal Study of Sliock Wave Iiiterfcrcnce Ilcating on 

a Cyliiitlrical Leading Edge,” AIAA 22nd ‘Tlierinopliysics Confcrence, Iloiiolulu, Ilawaii, June 8-10, 

1987, AIAA Paper 87-1511. 

2. Decliaumpliai, P., Tliornton, E. A., and Wieting, 

A. 12.: “Flow-TherinalStriictiiral Study of Aerodynamically Heated Leading Edges,” AIAA/ ASME/ 

ASCE/ AIIS 29th Structures, Structural Dynamics 

and Materials Conference, Williamsburg, Virginia, 

April 18-20, 1988, AIAA I’aper 88-2245. 

3. Decliaumphai, l’., Wicling, A. R., and ‘I’lioriiton, 

E. A.: “Tlierinal-Structural I’crforinanccs of ail Actively Cooled Leading Edge Subjected to Type IV 

Sliock Wavc Inlcrfercnce Ilcatiiig,” ‘l’liirtl National 

Aero-Space I’lane Symposium, June 2-4, 1987, Paper No. 24. 

4. Melis, M. W., and Gladden, 11. J.: UTliermostrrictural Analysis witti Experimental Verification iii 

a IIigli Heat Flux Facility of a Simulatcd Cowl 

Lip,” AlAA/ASME/ASCIS/AIIS 29th Striiclurcs, 

Structural Dynamics aiid Matcrials Conference, 

Williainsburg, Virginia, April 18-20, 1988, AIAA 

88-2222. 

5. Miller, A. li. (editor), Unified Conslifiifiue EpnLions /or Cvecp atid Plnsficily, Elsevier Applictl Science Publislicrs, 1987. 

6. Clian, I<. S., Lintllioliii, U. S., l3odner, S. !t., Ilill, J. 

It., WAer, It. M., ant1 Mcycr, 1’. G., “Coiistilutivc 

bfodcliiig for Isotropic Matcrials (110S‘l’),” ‘l‘liirtl 

Annual Stat us Report , Soli t liwes t Resea rcli Ins tiLute, San Antonio, Texas, t\ugust, 1986, NASA CR 

179522. 

7. Chi, I<. S., l,iiidlinliii, U. S., atid Ilodiicr, S. 

R., “Constitiitive Motleliiig for Isotropic hlalcrials 

( IlOS‘I’), Filial Itcport, Soultiwest lkscarcli Institute, San Antoiiio, Texas, Junc, 1988, NASA CR182132. 

8. Dnss, .I. hl., “Nul~~cricitl Iiiil’loinciital.ioil ol Constittilive Models in Ratc--Ucpcndcnt I’lidcity,” 

I’1i.U. DisserftiLiotr, ‘l’lie University of ‘lkxas, 

Ailstin, 1965. 

9. Bass, J. M., ant1 Odcn, J. ‘l’.: “Atlapl.ivc 12iiiitc 

Elcmcnt Methods for a Class of Evolution I’rolhnis 

in Viscoplaslicily,” hrf. J. Etrg. Sci.,. Vol. 25, NO. 

G, 1987, pp. 623453. 

IO. Uass, J. M., atid Otlen, J. ‘l’., “Niiinerical Solution 

of tlie EvoItit,ion 1”:qiiatioiis of Uamagc and IlatcDcpenrleiit I’lasticily,” Id. J. Enpig. Si., Vol. 

26, NO. 7, 1988, 1’1). 713-710. 

11. MARC Gericral I’urpose 1;iriitc Ihncril. I’rograrii, 

MARC Corporation, Palo Alto, CA. 
12. Cliaiig, 11. T., aiitl Allen, I). 11.: “Annlysis ol 

Viscoplastic Plates Subjected to Ilapid External 

Ileating,” AIAA/ASMl3/ASCE/AlIS 29th Striictures, Structural Dynamics and Materials Coiiferewe, Williainsburg, Virginia, April 18-20, 1988, 

AIAA Paper 88-242‘1. 

13. Iliiebner, I<. II., and Thornton, E. A.: 7’he Fide Eleinenf Mefliod jor Engineers, Secoild Edition, John Wilcy and Sons, 1982. 

14. Tlioriiton, E. A,, and Wieting, A. lt.: “Finite Eleinent Methodology for ‘l’raiisieiit Contluctioii/Forced Convection Tlicrmal Aiialysis,” Progress in Asttvriardies and Aeroiiaulics: Neal TransJer, Thermal c07ilt.O/ and Ileal Pipes, Vol. 70, 

Edited by Walter 11. Olslntl, AIAA, New York, pi). 

77-103. 

15. Kuiiiar, V., Morjaria, M., arid Miikherjee, S., “Nuiiicrical Integration ol Soiiie Coiislilutive Motlcls 

ol liiclastic Deformation,” J. Erigineeriitg Alalerinls 

a7ld Techdogy, Vol. 102, Jan. 1980, pp. 92-96. 

T I( 

[OR] [lll*U/ii~ - sR] 

0 1.23 x lo-’’ 

APPENDIX 

TIIERMAL-STRUCTURAL DATA 

CP 

(BTUIlb, - R] 

0.12 

Tlieriiial Data 

1. Skin, lieat cxcliangcr fins ant1 primary structure 

p = 0.283 1b,,,/i113 

[“C] 5 760 

1.055 

[MPu] 2700 

[sec-l] 0 

(sec-‘1 O 

Temperature Dependent Conductivity and 

Specific IIeat 

871 

1.03 

2400 

0.0055 

O 

I 1;” 1 1.8 x 10-4 1 0.135 

3.25 x 10-4 0.177 

3000 6.0 x IO-” 0.26 

2. C;oolallt 

rii = 7.16 x lb,,,/sec 

Cp = 3.48 DTUIlb, - R 

3. Stidilcc convcclioii 

7k = 5345 R 

11 = 6.0 x lo-’ Bl’Uf.9 - ill2 - R 

Additional It at tiiric 0 5 1 5 0.5 sec: 

0 5 X < 0.375 : hDdd = 0 

(X - 0.15) I1 

0.05 2 0.375 5 x 5 0.625 : h,dd = 3.0 x 10-3~~~ 

0.625 < z 5 1.0 : hod = 0 

4. Surface radiation 

u = 3.30633 x lo-‘ BTUIs - in2 - R‘ 

E = 0.8 

a = 0.8 

Structural Data (superalloy U 1900 t Ilj) 

I. Tcniperature-iiitlopeiideiit coiistaiils 

23 = 1150 M Pa 

r1 = 2 

r2 = 2 

till = 0.270 AlPa-’ 

ni2 = 0 

2. ?kml’eratiire-tlel,eiidcnt paraincters 

11 = (5.2 x OTUIs -ina - R 
3. Elastic constants 

I'C] 

0 

200 

400 

600 

800 

1000 

1200 

E = 1.987 x lo'+ 16.783 -0.1034T2t 

1.143 x [hfPal 

>l/"C].lo1.15 

1.34 

1.36 

1.41 

1.54 

1.60 

1.66 

C = 8.650 x 10' - 17.58T + 0.023213' 

-3.464 x lO-'T' (MPa) 

(with temperature in "C) 

4. Thermal expansion coefficient 

Aarodynamlc skin 

Coolant paaaag*-b*.t 

rxchan[*r fins not shown 

Primary structurr 

Sidewall 

Coolant flow Shock 

interaction 

coo I i n g 

Fig. 1: Fluid-thermal-structural interactions on an 

aerospace plane scramjet engine leading edge. Fig. 2: Convectively cooled structure. 
Uniform Fluid 

Conv.ctltr . I 

Constant Mass Flow 

Fig. 3: Engineering model of coolant passage heat transfer. Fig. 4: Finite element model of coolant passage. 

LINfAR FLUID 

TEMPLRANRE VARlAllON \ ~ 

T 

TYPICAL FLU10 '1 I TEMPERANRE 

LINfAR FLUID 

TEMPLRANRE VARlAllON 

UN I 

VELOClM PROFILE 

(a) mass transport element 

CONVECTION SURFACE \ 

yp A r. YIP+ Ai 

I dr-Ar I2 c 

CORRECTOR Ya,. Y,:,, 

a dr = 2.Y 

I c 

Fig. 6: Flow chart for adaptive time step algorithm. 

(b) surface convection element 

Fig. 5: Coolant passage convective elements 
,00196 cooling (665 R) 

.ais In 

.00006 

b 

TIME (sec) 0 .5 1 .o 

healing (5345 R) 

600. 

0 

(a) tlicriiial-slrucliiral iiiodcl (b) convection coelliciciit 11 islory 

Fig. 7: I-D Model or coiiveclivcly cooled structure 

I 

c 

TlhlE 

.I 

0 

J 

.5 

-600. 

(a) Temperature history 

..< 

1 .o 

TIME (ICE) 

0 .5 1 .o 

TIME (see) 

FIXED - I200 STEPS. ADAI‘TIVE - 213 STEPS 

(c) Adaptive lime step history 

(b) Stress history 

Fig. 8: ‘I’licrmo-visco~~l;lic response of 1-D inoclel 
CY licitiiie (5345 R) ertcriiol pressure .I5 hlPn 

I cool;liil pressure 7.0 hlPa 

I 

I 

I I 'I' L1l"'ll"' 111 II II IL LI II I I Y 

Ir W 

WO 

a 

'.L ?.. 

c 

o 

0 

" t (B'u/in2 R' 

I 

l 

.US coolant 

t fitla 

.I primary 

slrucwic 

TIME (sec) 

.J 1 .o 

Fig. 9: 2-D model of convectively cooled structure 

T = 2254 R 

665 R 

= 0.5s. 

=o 

Fig. 10: Thermal response of 2-D model 
ql 

rinE ISCCI 

Fig. 11: Viscoplastic stress histories in aerodynamic Fig. 13: Comparison of viscoplastic and elastic stress 

skin for 2-D model. histories in aerodynamic skin. 

skin 

fins 

0 4 0 ---- 0' 

/ 

L 

/ \ 

0 

OI 

------- 

Tin€ ISECI 

z9 zs N" N-- 

0 __g_ viscoplastic 

*A 0 

.- I 

----o---. elastic TIHE ISEC) 

Fig. 12: Viscoplastic stress liistory in heat exchanger Fig. 14: Comparison of viscoplastic and elastic stress 

F fins for 2-D model. histories in lieat exchange fins. 
STRESS a (MPa) X 

7 

-650 Mpa 

- 

420 Mpa 

I - I 

(a) t = 0.053 

.. 

-580 Moa 

J 

(b) I = 0.55 

450 Mpa 

(c) t = 1.2s 

Fig. 15: Contours of 0, stress in 2-D model 
STRESS oy (MPa) 

-100 Mpa 

(a) t = 0.05s 

-400 MPa 

500 Mpa/ 

(b) 1 = 0.5s 

- -200 Mpa j 

- 

(c) 1 = 1.23 

Fig. 16: Contours of d, stress in 2-D model 
Fig. 17: Principal plaslic slrain in 2-D model at t - 0.5s. 

Aerodynamic skin with fin deboiid 7 rDamaged firis 

Fig. 18: Mnxiiiiuiti tlrloi tiintioil ol 41 IIC~.IIII- (I = 0.53) willi tlalttagcd llcnl cxcltniigcr litis. 

Ucloriiialioii is sI~o\vtt 1.0 scale. 

t u = 7.68.10-’in 

with fin damage 

n 

I;i/ f 

Pig. 19: Viscoplastic delornialion ol aerodyirainic 

skin for undamaged and damaged lieat exchange fins.