+

w

SLOW CRACK GROWTH ANALYSIS OF ADVANCED STRUCTURAL CERAMICS UNDER COMBINEDLOADING CONDITIONS - DAMAGE ASSESSMENT IN LIFE PREDICTION TESTING

Sung R. Choi"Ohio Aerospace Institute, Cleveland, Ohio 44142, USA

[Sung.R.Choi@grc.nasa.gov]

John P. GyekenyesiNASA Glenn Research Center, Cleveland, Ohio 44135, USA

[John.P.Gyekenyesi@grc.nasa.gov]

ABSTRACTSlow crack growth a,-udysis was ped'onned with tJ_ee different

loading histories including constant s_'css-mteJconstantslress-ratctesting (Case I loading), constant skess/constant stress-rate testing(Case II loading), and cyclic stress/constant stress-rate testing (Case ITIloading_ Strength degradation due to slow crack growth and/ordamage accumulation was determined numerically as a function ofpercentage of interruption time between the two loading sequences fora given loading history. The numerical solutions were examined withthe experimental data determined at elevated temperatures using fourdifferent advanced ceramic materials, two silicon nitrides, one siliconcarbide and one alumina for the Case I loading history, and alumina forthe Case H loading history. The numerical solutions were in reasonableagreement with the experimental data, indicafin 8 that notwithstandin 8some degree of creep deformation presented for some test materialsslow crack growth was a governing mechanism associated with failurefor all the test materials.

INTRODUCTION

Advanced ceramics m¢ candidate materiedsforhigh-temperaturestructural applications in heat engines and heat recovery systems. Oneof the mayor limitations of thesematerialsin high temperatureapplications is delayed failure, where slow crack growth (also called

mechanics specimens in which the crack velocity measurements aremade. Constant s_xess-rate testing detcrmmes the strensth for a givenapplied s_-ess; whereas, constant stress and cyclic stress testingmeasures time to failure for given constant stress and cyclic stresses,respectively. Of these test methods, constant stress-rate testing hasbeen widely utilized for decades to chamcter_ SCG behavior ofceramic materials at both ambient and elevated temperatures. Theadvantage of constant stress-rate testing over other methods lies in itssimplicity: Strengths are measured in a routine manner at four to fiveapplied stress rates by applying either displvzement-control mode orload-controlmode. The SCG paran_tm'sfor l_e-prediction/desisn aresimply calculated from a rehtionship between strength and appliedstress rate. Because of its advantages, conatant stress rate testing hasbeen devdoped as an ASTM test standard (C1368-97) to determineSCG parameters of advanced ceramics at ambient temperature [1]. Theadvantages of constant stress-rate testing have also promoted an effortto develop a companionteststandardto evaluate SCG parametersatelevated temperatures, which is under consi_on within ASTM C28Advanced Ccramics Committee [2].

One ofthedifficulties possibly encountered in elevated-temperaturetestingisthat,dependingontestconditions(testrate,time,temperatureand environment) and material, the identification of a governing failuremechanism may be obscured by the presence of possible multiplemechanisms, particu_y with a combination of SCG and creep [3-6].Thus, the determined SCG parameters cannot be soldy representative

'fatlgue" or 'suberitical crack growth') of inhenmt flaws can eccur until of one single process, slow crack growth, but a combination of the twoa critical size for catastrophic failure is attahe& Therefore, it,s competing mechanisms. They may also act in series, i.e., creepimportant to evaluate accurately slow crack growth (SCG) behaviorwith a specified loading condition so that reasonable life prcdiclinn ofceramic components is e_

There are severalmethods of determin_ SCG of advanced

ceramics.Typically,theSCG ofco'a_cs isdeterminedby applyingconstant _xe=B-rate (also c.BJ]ed"dynamic fatigue"), co_ant stress(abo caged "marie fatigue"or "stress r.ptmd') or cyclic stress (also.called "cyclic fi_t_,ue") to groundspecimensortoprecrackedfracture

followedby SCO. The underlyingbasisoftheaforementionedSCGtesting-constantstress-rate,constantsums andcyclic stress testing -is

thecrackvelocityformulationinwhich crackpropagationtypicallyfollowsapower-lawrelation.IftheSCG mechanismisdominantforagiven material/temperature/enviromentalsystem,then the SCOparameters obtained, in principle at least, should be in a reasonablerange ofaccuracy, resardle_ of test method. Fmltmmore, one mnst beable to predict life and/or strength from any loading history that could

"NASA geaior Reddem Research Sdentlst, Glenn ResearchCenter,Cleveland,OH44135 (All correspondence=to this address)

This is a preprint orreprintof a paper intendedfor presentation at a conference.Because changes may be made before formal publication, this is made available with theunderstanding that it will not be cited or reproduced without the permission of the author.

be a combination of consent s_ess-rate/constant s_s-rate, constantsb'ess/constent stress-rate, or cyclic stress/constant stress-rate loadingsequence_ There have been some _ental attempts to evaluatethe degree of crack growth or damage accumulation by determining'fast'-fracture "residual" strength of silicon nitride specimens that hadbeen subjected to and then interrupted from tensile cyclic loading atelevated temperature [7]. However, in general, both analytical workand systematic experimental data on this subject rarely exist in theliterature.

Consequently, the purpose of this work is to better understand howdamage (SCG, creep or both) was accumulated with time for givenloading history leading to failure of advanced structural ceramics atelevated tempemture_ Numerical sohfons of strength degradation inconjunction with crack growth were obtained for each loading historywith a major assumption that the governing failure mechanism wasslow crack growth. Included in the test matrix were two typical

_ethods of constant stress-rote ("dyna_c_ fatigue") and constant stress( static fatigue" or "s_ess rupture") tes_g. The SCG and relatedparameters were determined on the basis of these test results. Then, acombination of two different loading sequences was applied to testspecimens and the corresponding strengths were measured to see hows_ength degradation in the form of SCG/damage accumulation tookplace during the combined loading sequences. The combination ofloading used in this testing included slow test rate/fast test rate (whichis a combinallon of constant stress-rate/constant sVess-mte testing,called here Case I loading history) and static loading/fast test rate (acombination of static stress/constant s_ess-rate testing, called Case IIloading). The testing was interrupted after the fu'st loading sequence ata specified time, and then the specimens were fractured at the secondloading sequence using a fast test rate of typically 33 MPa/s. Theexperimental results were compared with the numerical solutions.

THEORETICAL BACKGROUND

Approach

Numerical solutions of strength, crack size and other requiredvariables for various loading histories are lXeSonted in this section. Theschematic loading history considered in this study is depicted in Fig. 1.The first one, Fig. l(a), called Case I loading, was a combination oftwo constant s_ss-rate testing with a fast test rate after a slow test rate.A specimen was subjected to a low stress rate. Then, the testin 8 wasinterrupted at a specified time J'muand resumed with a fast stress rateuntil the specimen fractured. The second loading history, Fig. l(b),called Case 17 loading, was a combination of constant stress andconstant stress-rote test_-_. The testing was started initially with aconstant _ inte_zupted at a specified test time 3.n and then resumedby applying a fast stress rate until the specimen broke. The thirdloading history, Fig. l(c), called Case 111 loading, which was acombination of cyclic stress and constant s_ess-rate testing, was simplya _placement of a static stress used in the Case II loading with cyclic_rem. The ratio (¢) of inten_tion time to time to failure is defined asfonows:

¢ =Jim (1)Ji

where J-, is the interruption time and J! is time to failure of a testspecimen, subjected to only the first loading sequence (without the

8

TIME, J

DV

8

CASE"n"w_o (b)

/l.t L

TIME, J

8

CASE "HI"LOADING

__ J'_'ij f

TIME, J

(c)

Fig. 1 Schematics of three loading histories considered: (a)Case I loading, (b) Case II loading and (c) Case III loading

second loading, i.e., ¢ = 1) which is either constant slress-r_e, constantstress or cyclic stress, see Fig. 1. The case for _ - 0 rcprcsmts thesecond loading sequ_ce with no the fu_t loading sequeace. Theintezrup6on time was chme_ such that the ratio ranged typically from p=Oto 90%.

In many cases, slow crack growth of advanced ceramics undermode I loading above the fatigue limit, either by stress c(m_on atambimt temperature or by grain boundary sliding at elevatedteml_atmes , can be expressed by the following empirical power-lawrelation[g]

where v, a, t are crack velocity, crack size, and time, respectively. Aand n are the material/environment-dependent SCG peraraeters. Ki isthe mode I stress intmsity factor (SIF), and K,c is the critical SIF orfracture toughness of the _ subjected to mode I loading. Thesimplistic analytical solutions of strength in constant mess-rate testingand of time-to-faihue in constant stress and cyclic stress testing can be

approximated as follows [9-11]:

_r:= D_[a]vj'* (3)

t/,= D,[a]-* (4)

t_ = D_ [urn]-* (5)

where _/is the fracture strength corresponding to the applied stress

rate (_) in constant mess-rate testing, t_ is the time to failuresubjected to a constant applied mess (_-) in constant stress testing, and

t_ is the time to failure subjected to the maximum applied stress (am_)in cyclic mess testing. The parameters D's can be expressed as follows[9-11]:

D a = [B(n + l)Si"-2]11('0

D, = BSI "-2

Ir

0

(6)

where $_ is the inert strength andB = 2Kio/[Af(n-2)] with Y being thecrack geometry factor in the relation ofKi = Ycra In. j(t) is a periodicfunctionincyclicloadingspecifiedino(t)= triO(0 witharangeof0_

j_t)_l,and ris the period. The SCO perameters n andB (orA) can beobtained by a linear regression analysis with _ental data inconjunction with an appropriate equation, Eq. (3), (4) or (5), dependingon the type of loading.

To obtain more gene_diz_ convenient and accurate SCG analysis,several parameters that are commonly specified in the conventionalanalytical solutions (Eqs. 3 to 6) have to be minimized. This can bedone by using a normalization scheme, as used in the l_fevious stu_es[12,13]. The normalized variables utilized in the numerical approachwere as follows [13,14]:

Kt " j=At;_ C* a crK*=_, =_; _*K ;c a_ a.o = -_i;

(7)_. _,____"

where K*, J, C*, o*, o*mx and d * are, respectively, normalized stressintensity factor, normalized time, normalized crack size, normalizedapplied stress, normalized maximum applied stress (in cyclic loading)

and normalized mess rate. a_ is the critical crack s/ze in the inertcondition, or is the initial crack size. Using these variables, the crackpropegationrateofEq.(2)yields

The normalized SIF, K*, in constant mess-rate and cyclic (sinusoidal)mess testing can also be expressed, respectively [13,14]

g" = d*J[C*] 1/2 (9)

.I+R I-R . 07aK* = i---_---+--_-- sm[(T-_-):]}o*_ [C*]_2 (I0)

where R is the mess (or load) ratio, defined as R = _/_ with ¢_being the minimum applied mess in cyclic loading, and 07 is theangular velocity. The normalized SIF for constant s_ess loading issimply reduced to the case with bothR = 1.0 and o*m =o* in Eq. (10l

The differential equation Eq. (8), together with Eqs. (9) and (10),was solved numerically using a fourth-order Rungz-Kutla method for agiven loading history. The initial condition was C* ffi l.O at J ffiO. Theinstability conditions were K* = 1.0 and dK*ldC* > O. The solution incyclic loading was independent of frequency [14]; hence an arbitraryvalue of wa/A = 100 was used in the analysis. At interruption time,Jilt, the corresponding variables including crack Kze, mess, stressintensity factor and time were provided to the next (second) loadingsequence so that the requiredvariables were determined until aninstability condition was reached. Includedinthe typical input datawere n, R, d * and O*m_. The major assumption in the analysis wasthat only one mechanism, i._, slow crack growth, was associated withfailure.

Results of Numerical Solutions

Stren_hNormalized sVe.gth (o*/) as a function of percent of interruption

time (¢ = J'n/Jf) for the Case I loading history is shown in Fig. 2. Sixdifferent values of n tanging from n = 5 to 160 were employed. Foreach n value, the initial slow stress rate of _*= Ixl0 "swas followed

a*by the second loading which was chosen as " = lxl0 "4, lxl0 "3,lxl0 "2

and lx10 q. The choice of this range of d* was based on the typical

range of applied mess rates commonly used in the actual constantsUes,s-rate testing [1]. For the case ofn = 5 _ SCG susceptibilityis high,the strengthdegsdationwith respectto the strengthat 0,= 0depends on intetruptlon time, particularly with increasing stress rate.The nmximum strength degradation of about 16 % and 10 % occurredat ¢p-- 90 %, respectively, for the highest mess rate of d *= lxl0 "_andfor the lowest rate of dr*-- lxl0 "4. This i_cates that the initial

loadingup to _ ffi90 % resulted in somewhat appreciable en_growth/damage-accumulation. For n = I0, the nat_atum meogthdegradation was about 3 % both at d*= lxl0 "t and lx10 4. For the

case of higher resistance to SCG, n Z 20,themength degradation wasnegligible with less than 0.6 % at ¢ - 90 %, indicating that crackgrowth/damage-accumulation rarely occurred during the first loadingsequence. Therefore, R is concluded that for n > 20 which is file casefor most siliconniUidesand silicon carbides at devated temperaturesthe first loading sequence would not have any significant influence oncrack growth/damage, leadin8 to negligible strength degredation. Thesecond loading rote, which is at least one order of magnitude greaterthan the first one, contruls exclusively the strength via crack growth.

t

%' 1.2

t 1.00.8

0.6

l 0.40.2

0.00

._, 1.2D

r 1.0

0.80.6

l 0.40.2

0.00

%" 1.2t:>

1.00.8

0.6

0.40.2

0.00

t • | • i • i •

n-5

°

• l s i J ¢ , i

20 40 60 80 100

%OF INTERRUPTIONTIME,_p

n-20

T T _" v

_ o

-_ ,_o' Ilx10"_[lxlff z [

--e-- lx10"1, I i I i l a I •

20 40 60 80 100

% OF INTERRUPTION TIME,

• i i • i • i •

n-80

i ./ t i . i , | i ,

20 40 60 80

%OF IN'IERRUPTIONTIME,_p

100

*" 1.2

i ].00.8

0.6,

l 0.40.2

0.00

• ! • ! • j • , •

n- 10

, I I I | l

20 40 60 80 100

% OF INTERRUPTfON TIME,

_ 1.2 . , . , , . , .

1.0 g _ g

0.2

0.0 • ' , t , l , , .0 20 40 60 80 100

% OF INTERRUFHON TIME,

_. 1.2

1.00.8

0.6

: I 0.40.2

0,00

w • i • m • ! •

n-160

--e-- lx104 I

1x104--e- lx1_--e-- lx10"_

• i . l , ! ,i I .

20 40 60 80 100

%OF INTERRUPTION TIME,

Fig. 2 Numerical results of normalized strength (o't) as i function of percent of Interruption time (p) for different values ofslow crack growth (SCG) parameter n in Case I loading

="

An analysis on how a crack grows under a given loading history will bepresented in a later section.

The results of strength as a function of percent of interruption time(= _v)for the Case H loading history, a combination of constant stressand constant stress-rate testing, Fig. l(b), is presented in Fig. 3. Two tothree different normalized applied stresses, ranging from cO = 0.2 to0.95 depending on n value, were used for each n value. Since thesusceptibilitytoSCG decreaseswithincreasingn, higher applied stresswithnarrowrangewas employedforhighern value.Two testrotesof

o*"= Ixl0"3and Ixl0q wereusedinthesecondloadingsequence.As

intheCaseIloading,forn < 10,strengthdegradationwas significantto

interruptiontime,particularlyatbothlowerapplieds_essand highers_essrateof b *= Ixl0q. Forn _ 20,strengthdegradationforagiven

valueof¢ was independentof eitherappliedconstantstress(oj) or

appliedstressrate.However,theoveralldegreeofstrengthdegradationoccurringforthewholerangeofn'swas greaterintheCase11loading

than in Case I loading. For n = 5, the maximum s_ength degradationof 42 % occurred at <p= 90 % with a loading combination of o* = 0.2and b * = lxl0 "l. For n =lO, about 20 percent strength degradation

was observed at _= 90 % for the combination of o_ = 0.3 andd'* ffi

Ixl0 "l. For n > 20, the maximum strength degradation of 10 %, 5 %, 3

% and 1% took place, independent of ci"*, for n =20, 40, gO and 160,rcspcctivdy.More _cy ofstrengthdegradationon interruption

time (_),compared withthe Case I loading,impliesthatthe first,constant-_ess loading sequence resulted in more crackgro_damage-accumulation,thusloadingtolowerstrengthwhen thedamaged sl>ecimen was subjected to the second constant stress-roteloading sequence.

The results of s_ength as a function of interruption time (_) for theCase HI loading, a combination of cyclic stress and comtant stress-rateloading, see Fig. 1(c), is shown in Fig. 4. The second constant stress-rote loading sequence was l_eceded by the first, sinusoidal cyclic stresswitha R-ratioofR = 0.I,untilthe specimenfailed.Two tothree

different normalized _um applied stzesses, ranging from o*_ ffi0.2 to 0.95, were used for each n value. Note that Cr*m_applied in the

Case M loading was identical in magnitude to o* applied in the Case lIloading.As inthecase]Iloading,two stressrotesof d *= Ixl0"3and

Ixl0q were alsousedinthesecondloadingsequence.Comparingtheresults in Figs. 3 and 4, it can be readily evident that for the given n ando*ma = c_ strengthdegradationintheCaseHI loadingwas almostthe

same asthatintheCase IIloading.Itshouldbe notedthatconstantstress(i.e.,R = 1.0)resultsinmuch longerlifethancyclicstresswithR

ffi0.1[I1,14].However,intermsofstrengthdegradationasa functionofpercentofinterruptiontime,eitherconstantstress(CaseIIloading)

orcyclicstress(CaseHIloading)yieldedthesameresult.

_IQw Crack GrowthTypical examples of crack growth/damage-accumulation subjected

to the three diffcrent loading histories are iTesented in Fig. 5 for both n=I0 and 20. The figuresshow how an initialcrackgrowswithtimeduring the whole loading history until failure occurs. The loadingcombinations employed for each n were as follows: d* ffi lxlO "5

(la)/d* = Ixl04 forCaseIloading0* = 0.5(la)/d*= Ixl0q (2e)

fcfCase IIloading;¢r*m_= 0.5(la)/d* = Ixl0"](2_) forCase HI

loading.Note againthatthecaseof<p= 1.0representsonlythefirst

loadingsequenceapplied.

a) For n =10 (/Zig.6A). For theCase I loading(Fig.5A(a)_aninitialcracksubjectedtoonlythefirstloadingsequenceofd* = Ixl0"s

(i.e., q_= 1.0) remained almost unchanged in size for a long time, butstarted to grow very quickly atJ > 0.3848 xlO s until failure time of,//=0.3849 x l0 s. This indicates that the initial crack started to grow toinstability at a tine greater than 95 % of failure time. Therefore, anyinten'up6on of loading below ¢ ffi 95 % did not give any significantcrack growth so that the resulting strength al_ the second loadings_ucnce remained unchanged (compared with the strength at ¢ = 0),irrespective of intcm_ption time. This is also reflected as aninsignificant s_ength degradation with respect to the _ength at ¢ = 0,as shown in Fig. 2 for n = 10. Similar behavior as in the Case Iloading was aLso observed in the Case II loading (see Figs. 5ACo)).Most major crack growth occurred close to and/or at failure time.However, during the first static loading sequence, an initial crackstarted to grow earlier and greater in size than that of the Case Iloading. Hence, the resulting critical cracksizeafterthe second

loadingsequenceof b *= Ixl0"lwas increasedwith increasing

interruptiontime (_).As a result, strength degradationasa functionof

¢ became much greater, compared with that of the Case I loading (seeFig. 3 for n = 10)_ For the Case HI loading, crack growth behavior intermsofinterruptiontimewas almostidenticaltothatof theCase Hloading, as can be seen by compming Fig. 5ACo) with Fig. 5A(c). Theonly difference between Case II and HI loading lied in time to failure:Time to failure for a given n is always greater in cyclic (R = 0.1) thanin constant stress (R = 1.0) loading [11,14]. Therefore, the resulting

strength as a function of percent of interruption time (_) renmined thesame for either Case H (static) or Case HI (cyclic) loading history, asshown in Figs. 2and 3 forn = 10.

b) For n = 20 (Fig.5B). Forthecaseofn = 20,theoveralltrendin

crackgrowthbehaviorwas verysimilartothecaseofn = I0. Noteagainthataninitialcrackstartedtogrow closetoand/oratfailuretime,indelxazlent of the type of loading history. However, because ofhigherresistancetoSCG incaseof n = 20 the critical cracksizeat

instability between q ffi0 and 0.9 was all smaller (and less dependanton mtoxruption time) than that of the case for higher SCG susceptibilitywith n = 10. As a consequence, strength degradation as a function ofinterruption time (_v)was less significant compared with that ofn = 10(see Figs. 2 through 4 for n = 20).

Based on the numerical results on strm_ and crack growth, it can bemnmari_ that strength degmhtion due to crack growth or damageaccumulation as a result of the first loading sequence depends on

interruptiontime ¢ and SCG pmmnc_ n fora givenloadinghistory.The _ degradationcf degreeof crackgrowth/damage-accumulationasa functionofint_wptiontimewas significantforlower

SCG pmmnetem n _; 10, but became insignificantwith increasing SCGimmme_ of n > 20. This trend was observed more dominant for theCase I loading than the Case lI or HI loading history. The lmy factor thatgovernssuch strength degradation or crackgrowthbdmviorwas thataninitial crackstartedtogrowtypicallyclose toand/oratfailuretimeafterasubs_tisllylongincubationtime.Thislongincubationtime,uniquetoceramic materials exhibiting n > 20, was also a basis oftheaccelerating

test methodology in constant stressda_ testing where depending on nvalueapixopriateprcloadingcanbe _plied to testspecimens_or totesting thus saving a significant amount of test times [15].

_" 1.2 _ n=5" , ,"_ . ....

0.8

I 0.4 _ * Ix104o,Fl-.-:.:::I

r [-_- o.=0.7 I0.0 " " ' ' ' " ' " ' "

0 20 40 60 80 I00

% OF INTERRUPTION TIME, _p

_'- 1.2 • , .... - , •I n"lO

l.O I0.8 IxlO'_

I 0.40.2

0.00

lxlO 4

[E°'=°31°'=0"3 I0*=0.7 J

• e • I • s • i •

20 40 60 80 100

% OF INTERRUPTION TIME, _o

_" 1.21-n-20, .

_ 0.80.6

l 0.40.2

0.00

_" 1.2

1.0

0.80.6

0.40.2

0.00

I • i • I •

lxlO"

i xI 0.3_"'_--'q_.,-_

]_,.v_.._ °* =0"5o'_0.8 ]

I I , ! . 1 , I •

20 40 60 80

% OF INTERRUFI'ION TIME, _p

IOO

I t • e ! •

n-80 lxlO.i

ixl04

il°--,I_ o-=0.91

t I ' ! ' l , I ,

20 40 60 80 100

% OF INTERRUFI'ION TIME, _o

_'- 1.2

1.00.8'

r_ 0.6

l 0.40.2

0.00

•n-40! I |

lxlO "t

lxlO.S :

J._T._°"°-'Io.=0.7I I t I , u • i ,

20 40 60 80 I00

% OF INTERRUPTION TIME,

*_" 1.2

l°.4I !0.2 ' a,_.9o*=0.93

0.0 , t , , I , • •0 20 40 60 80 100

• I • t • ! !

n=160 lxlO"

IxlO 4

% OF INTERRUPTION TIME, 9

Fig. 3 Numerical results of normalized strength (ohf) as a function of percent of Interruption time (4P) for different values of

slow Prick growth (SCG) parameter n in Case II loading

_'- 1.21.0

io.

0.6 _l 0.4

0.2

000 20 40 60 80 100

% OF INTERRUFrlON TIME,

_,- 1.2 . 0 . , .... •

1.0

0.80.6

0.40.2

0.0

n-lO

lxlO "l

Ixl04

o,_.o5Io._.o.,i

• I • I • I • I •

20 40 60 80 100

%OF IN'rERRUFTIONTIME, _p

_'... !.2 _ n'20'

0.6 _

lo..I,:-:,0.2 --e-- a*,,-0.5o,,I,'0.8

0.0 ;' 7 _ . ,

0 20 40 60

! ! ! ,

Ixl0"I

ixl04"

80 100

% OF INTERRUPTION TIME,

_" 1.2

1.0

0.80.6

l 0.40.2

0.00

I ! ! !

n'40lxlO "1

e

lxl04 :

°'-'°"1a*ma'0.7 I, I • I t 1 • I .

20 40 60 80 100

%OF I_ERRUPTION TIME,

*_'b- 1.2 ....... ,

' rt-801.0

0.80.6

0.40.2

0.00

Ixl04

lxlO 4 "--

-,- o._..o.,I°""0"9 I

• I , I t I • I .

20 40 60 80

% OF INTERRUPTION TIME,

I00

1.2b • , • , • ° • , •

_r n-1601,0 IxlO'St $

0.8 lx1040.6

l 0.40.2

0.00

R-0.1"

Io,_..o.,I, • | • I i | i I I

20 40 60 80 100

% OF INTERRUPTIONTIME,9

Fig. 4 Numerical results of normalized strength (o'f) as a function of percent of Interruption time (p) for different values ofslow crack growth (SCG) parameter n in Case Iil loading

4 n-iO; 10_/10 4

I

-----------_-- ,L

lip 101 102 103 104 11_ 106

NORMALIZED TIME, J

_ , | , , I

CASE "If"LOADING4 vr_10; 0.51104

l q_l.03

2 --_

I _ (b)1, I ] I I

10o 10 ! 102 103 104 l0 s 106

NORMALIZED TIME, J

C.) 5 ......

CASE "m" LOADING q

4 m-lO; R-0.1;

@=I.0 0.5/10"1 •3

1

1_ 10 ! 102 10s 104 10 _ 11_ 10 _

NOR/4ALIZED TIME, J

_-o.6-o.9.._

J(c)

[ i I ' [ J

U 5

m 4

r

CASE "I" LOADINGn=20; 10"5/10"l

_'I.0

I 2 _=0.6-0_

q,,.o

! _ (a}

10o 10 ! 10 2 l0 s 104 105 10 _

NORMALIZED TIME, J

D

0 5 , ....

CASE "H"LOADING4 n,-20; 0.5/10 "l

q_l.O I

3

l 2 _-0._.9_1 (b)I I I [ I

l0 ° 10' 10s 10_ 104 l0 s 10_

NORMALIZEDTIME,J

5 ......

CASE "I_'LOADING4 n=20; R-0.1; 0.5/10"'

q_l.0

3

_ 2 _.6._.9.

") ,o,I I I I I I

10e 10] 102 10 _ 104 l0 s 10• 107

NORMALIZEDTIME,J

[A]n = I0 [B] n = 20

Fig. 5 Nurnedcal results of normalized crack size (C*) ms • function of time (J) for different values of Interruption time (IP)In three Ioeding histodes: [A] For SCG parameter n ,, 10; [B] For SCG parameter n -" 20

EXPERIMENTAL PROCEDURE

In order to verify the numerical solutions, experiments to coverdifferent loading histories as specified in Fig.l, was conducted atelevated temperatures. The nominal dimensions of rec_mgular-beamtest specimens in accordance with test method ASTM C-1211 [16]were 3 mm by 4 mm by 50 ram, respectively, in height, width andleagth. Test sp_imens were subjected to appropriate flexural loadingdepending on the type of loading history using SiC four-point flexurefixtureswith 20-ram inner and 40-mm outer spans viaeleclromechanioaland serve-hydraulictestframes(ins_onModels

8562 and 8501). All test specimens were equilibratedat testtemperatures for about 20 min prior to testing. Four different materialsincluding 96 wt % alumina, NC132 silicon nitride, ASS00 siliconnitride,and Hexoloysiliconcarbidewere usedintheCase Iloading,while ouly 96 wt % alumina was used in the Case II loading. ThereasonforthechoiceofaluminainbothCase Iand IItestingwas that

unlike other materials, 96wt % alumina has exhibited a considerablysmall strength scatter with a Wdbull modulus typically greater than 20at tither ambient and elevated temperatures [6]. Hence, it would bepossible to see material's response to fife and strength more clearly andaccurately with even a small number (about 5 at each condition) of testspecimens. Also note that the alumim was very susceptible to SCG atelevated temperatures _ 800°C with significantly low values of SCGparameter ofn = 7-12 [17], so that it would be much easierusingthealuminatoscrutinizethe influence ofSCG/damage-accumulationonthecombined loadingseqtumcesmore accurately.The expedm_talwork for the Case IH loading was not conducted in this study, primarilydue to limited availability oftestspecimens.

a) Case I loadingIntheCaseI loadtesting,theloadinghistoryincludeda slowtest

rateof0.033MPa/s forthefastloadingsequenceand thena fasttest

rateof33.33MPa/s forthesecondloadingsequence.The percentage

of intmuption time (¢ = t._t/) ranged from ¢ = 70 to 90 %. Theaverage failure time (= t/) of test slx_imens only subjected to the fwstloading sequence (with 0.033 MPa/s) was detennin_ from theprevious studies [6,18], and used here as a reference value to calculatet_ for a given value of ¢. Four ceramics including 96 wt % alumina,NC132 siliconniU'ide,AS800 siliconhi,de, and Hexoloy siliconcarbideweretestedattemperaturesof1000,1100,1200 and 1371°C,

respectively.Typicallya totalof fivespecimens,dependingonmaterial,were used ateach valueof _. The majormechanicaland

physicalpropertiesof thetestmaterialssuch as Young'smodulus,density, fracture touglmess,strength and slow crack growth can befound elsewhere [18].

b) Case fl loadingConstantstress("static fatigue") testing for 96 wt % alumina was

first conducted in flexure at lOOff'C to determine SCG behavior and

thus to obtain the time-to-failure data_ Four different applied stressesranging from 50 to 100 MPa were employed,with a total of five to ninespecimenstestedat each applied stress. The Case IT loadin8 historyconsistedof a constant stress(for the fast loading sequence)anda faststressrate of 33.33 MPa/s (for the second sequence). Two appliedstressesof 50 and 65 MPa were used in the first loading sequence.Three different values of interruption time, _ = 60, 75 and 90 %, wereutilized at each applied stress, with a total three to five specime_ testedat each interruption time.

EXPERIMENTAL RESULTS AND DISCUSSION

(a) Case I LoadingThe resultsforthe Case I loadingtestsfor96wt % alumina, NC132

silicon niUide, ASS00 silicon nltride and Hexoloy silicon carbide aresummarized in Fig. 6. The figure incloded flexure strensth as afunction of percent of intetrup6on time ¢ for each material Thehorizontal line represents the strength de.mined with zero interruptiontime ¢ = 0, that is, the 'fast'-ffacture s_mgth evaluated at 33.33 MPa/s[6,18]. The three materials including NC132 and ASS00 siliconnitrides and Hexoloy silicon carbide exhibited a somewhat appreciablevariation (in average sense) in strength between ¢ = 0 and p ffi80 or 90

%. Itisbelieved that this was atlributedto the inherently largeslrength scatter, typical of advanced ceramics that ranges ommnonlyfrom 10 to 13 in We_ull modulus. By contrast, 96 wt % aluminaexhibited a very small scatter, thus re_ily concluding that thedifference in strength between _p = 0 and _ = 80 or 90 % wasinsignificant,

(b) Case II LoadingFigure 7 shows the results of constant stresstesting for 96 wt %

alumina at lO00°C. The slow crack growth parameters n and D, in Eq.(4) were determined as n = 9.8 and Ds = 4.69x102° with units of 'MPa'in stress and 'second' in time. Note thatSCO parameter n detenn_edfrom constant stress testing was in reasonable agreement with n = 8.3from constant s_ess-rate testin8 de,'mined from a previous study [6].The results oftheCaseII loadingtestswas presentedinFig. 8, wherestrengths determined at 33.33 MPa/S, after the first loading sequence ofconstant stress of 50 or 65 MPa, was plotted as a function of percent ofinterruptiontime (q_).As seeninthe figure, the slrength exhibitedasignificant scatter particularly at ¢ = 75 and 90 %. much greater thanthat exhibited in the Case I loadinghistoryfor the same aluminamaterial. It is believed that this was attributed to the fact that no exactfailure time of each individual test specimen subjected to the Case iIloading could be known and that as a result the actual correspondinginterruption time for each test specimen could not be determined. Thiswill be discussed in a later section.

(c) Compwison of Experimental Data with Numerical SolutionsThe comparison of mength as a function of _ between the

exl_-imental data andthe numerical solutions for each loading history• wasmadeandpresentedin Fiss. 9and I0. Thereducedslrensth(a,*)

used herewasdefined suchthat strengthdetermined at any given valueof ¢ was normalizedwith respectto the sUengthdeterminedat ¢= O,which is _ as follows"

a,*= o'f, (ll)

where o', is the strmgth at any given value of ¢, (which is determined

at a fast test rate of 33.33 MPa/s after the tim loading sequence) and

or4,.o is the strength determined at ¢ ffi0 (which is simply the 'fast'-

fracture strength determined at 33.33 MPa/s without any first loadingsequence).

i) Case I loading. As seenin Fig. 9, except for 96 wt % alumina,the discrepancybetweea the expedm_tal mean-strengthdata and thenumerical solutions wassomewl_t large. However, asaforementioned,considering low Wdbull modulus (10-13) typical of many advanced

6OO

i °300

2OO

00

' I • ! * ! i

96 wt % ALUMINA (a)1000°C

i ! . I • i i i

20 40 60 80 100

PERCENT OF INTERRUPTION TIME, 9 [%]

1000

D'-

t°700

0

' I . I ' I ! -

NC132 SLjN.;1100°C (b)

• ! , I I I I i

20 4O 6O 80 100

PERCENT OF INTERRUPTION TIME. 9 [%]

1000

_ 90O

o_

700

l °500

4OO

• i • ! ,

ASS00 Si:N.:1200°C!

(c)

I I i l i I i I I ,

0 20 4O 60 SO 100

PERCENT OF INTERRUPTION TIME, 9 [%]

t°3OO

2OO

00

' I ! , ! , ! ,

HEXOLOY (SIC):1371"C (d)

, I • i | I I I

20 40 60 80 IOO

PERCENT OF INTERRUPTION TIME. 9 [%]

Fig. 6 Experimental results of flexural strength as a function of percent of Interruption time (9) for Case I loading tests,determined from 96 wt % alumina, NC132 and AS800 silicon nitddea, and Hexoloy silicon carbide at elevatedtemperatures. Each solid line represents the mean strength at p=O.

o

200

8O7O

6O

5O

4O

3O

]oo

I | I I

96 wt % ALUMINA;IOO0°C

Prediction

(_ eon_ stre_ tom

Best-tit _

EXP. DATA I

I l I I

101 lO2 10_ 104

TO FAILURE, tr[s]

iOs

Fig. 7 Experimental results of constant streu ("static fstigue") testing for 96 wt % alumina at 100000. A prediction madefrom the constant stress-rate ("dynamic fatigue") testing data [6] was Included as a dotted line.

t

°

6OO

o_40O

30O

200I

loo

00

I ' I I

96wt % _,UMINA;1000°C

Cuen_

IX&,65 MPa

i I E 1 i I z I l

20 40 60 80 loo

PERCENT OF INTERRUPTION TIME, (p [%]

Fig. 8 Experimental results of flexure strength as a function of percent of Interruption time (_) for Case !1 loading tests,determined from 96 wt % alumina at 1000°C. Two different constant stresses of S0 and 60 MPa were employed. The solidline represents the mean strength at p=0.

1.6

1.4

1.21.0

0.80.6

0.40

96 wt % ALUMINA;1000°C (a)

Theoretical

1.6_" [ AS800' Si3N4;f200°C

: .0_ 1"4I_ 1.2

LoI \l o.s Theoretical0.6

0.40

1.6 [ , I , I , I ' t ° -

_" _ NC132 Si3N4;1100°C (b)

I_ 1.2

l.o \ g

0.8

_ 0.6

0.40

Theoretical

• ' - I . I . I = . i I . l . i • l

20 40 60 80 100 20 40 60 80 I00

% OF ]_TW_.RRUP'_ON T2_JE, cp % OF INTERRUPTION _, (p

(c) HEXOLOY (SiC);1371"C (d)

1.6

1.4

1.21.0,

0.80.6

0.40

{

20 40 60 80 100 20 40 60 80 100

%OF INTERRUFHONTIME,cp %OF INTERRUFHONTIME,q_

Fig. 9 Comparison of experimental data with numerical solutions ("theoretical") in Case I loading: (a) 96 wt % alumina,

(b) NC132 silicon nltride, (c) AS800 silicon nitride, and (d) Hexoloy silicon carbide. Error bars represents +1.0 standarddeviation, normalized with respect to a mean strength at p=0.

ceramicsincludingthetestmaterials(except96 wt % alumina),thediscrepancyis believedratherstatisticallyinsignificant The_tal datafor96wt % aluminawhichexhibitedaconsiderably

small strength scarer were in good agreement with the theoreticalprediction. Based on the results shown in Fig. 9, several conclusionswere made. First, the governing mechanism associated with failure forthese test _ was slow crack growth. Some materials such asalumina and NCl32 silicon nitride exhibited some degroe of creepdeformation (_ 0.2 % creep strain) at a lower test rate of 0.033 MPa/s_Despite such creep mechanism presented, the agreement betweenexperimental data and prediction was still reasonable, implying SCG tobe a dominant failure mechanism. Second, the phenomenon typical of

advanced ceramics, which showed numerically that for n > 10 inconstant stress-rate condition an initial crack started to grow at and/orclose to failme time a_er a long incubation time, was verified. Thiswas verified from the results that percent of interruption time up to q =90 % did not show any significant crack growth or damageaccumulation, as reflected in insignificant strength degradation. This'long-incubation' phenomenon was also validated previously by theacceleratingtesttechniquedevelopedin constantstress-ratetesting[15]. Finally,the slow crackgrowth formulationof Eq. (2) welldescribed the actual SCG behavior of the test materials at elevatedtemperatures. Note thatthenumericalsolutionwas made exclusively

basedonsuchSCG formulationsothatifpooragreementwould exist,then it would be indicative of inapplicability of the SCG formulation tothe actual material behavior.

ii) Csse IIloading. The reduced s_'ength as a function of pe_entof interruption time 0P) for 96 wt % alumina is presented in Fig. 10.Unlike the Case I loading history (Fig. 9), the difference between the_tal mean-strength data (with 'triangle' symbols) and thetheoretical prediction was amplified even for the same aluminamateriaL The reason for this discrepancy can be reasoned as follows.Because of the two combined loading sequences, no exact failure timeof each individual test specimen exclusively subjected to the firstloading sequence (constant stress) could be known so that the actualinten'uption time for each test specimen can never be deterrained. Notethatthe(nominal)time-to-failureateach appliedstress was taken as anaverage failure time determined from the specimens subjected to 50MPa or 65 MP& The corresponding interruptiontimewas calculated

simply from a relation of t_ = _ t/for a given value of ¢_. Fm_'rmore,as seen in Fig. 7, the scatter in time to failure was greater than that ofs_ength typically observed in constant stress-rate testing [6], thusfurthor increasing the unc_adnty in failure time. Therefore, somespecimens would have been actually subjected to greater interruptiontime than the nominal interruption time, while other specimens to lessinten'ulXion time. As a consequence, this would have resulted indifference in SCG/damage-accumulation even with the same nominalq_, thereby resulting in a wide scatter in strength. The discrepancywould be small if the strength at lower end of data points, which wouldbe close to the actual failure-time data, was used, as shown in the figurewith the 'circle' symbol. Certainly, the discrepancy will be dindn_edif a large number of test spechnmm m'eused.

(d) Comparison of SCG Behavior between Constant Stress-Rateand Constant StressTesting

As .q_atedbefore, the SCG parameter n for 96 wt % alumina was inreasonable agreement between constant s_ess-rate and constant stresstesting with the respective values ofn = 8.3 and 9.8. It is possible toonvert the SCG data from one test method to another by using the

1.6

"" 1.41.2

1.0

0.8

0.6

• I • ! • I ' I

96 wt %ALUMINA;1000°C

0.40 I00

Theoretical

O XZ

o=50Mr.od==) 1 _'_0--65 MP=0d=m) I

y 0,-50 MI'= (Low=-e_) IA

• I = I = I , I I

20 40 60 80

% OF INTERRUPTION THvIE, q_

Fig. 10 Comparison of experimental data with numericalsolutions ("theoretical") in Case II loading tests for 96 wt %alumina at t000"O. Each data point with 'triangle' symbolrepre=ent_ an overall mean strength value (normalized).The data points with 'triangle's' symbols (Inner cross-marked) represent the lower-end strength values(normalized) from the data In Fig. 8. Error barn were omittedfrom plot= for clarity.

appropriate relations shown in Eqs. (3) to (6). The resultingcomparison is depicted in Fig. 7,where the prediction from constantsUess-rate ('dynamic fatigue") [6] to constant stress Cstatic fatigue")testing was includedas a dotted line. Considering the inherent scatterin time to failure, reasonable agreement was found between the twodata, implying that the mechanism associated with failure for bothcases was presumably slow crack growth. It was observed that creepdeformation inconstantstresstestingwas much greater than that in

constant stress-rate testing, since test time was much longer in constantscess testing. A maximum creep m-sin of about 0.4 % was found forthe test specimens subjectedtoanappliedstressof50 MI'a. Inspiteofsuchappreciablecreepdeformation,overallagreementbetweenthetwo

testmethodswas reasonable,againindicativeofSCG asan operativefailuremechanism involvedinbothconstants_ess-rateand constant

s_esstestingforthismaterialsystem.However,some effectby creepdeformation, particularly in constant s_.ss ("static fatigue"), should notbe overlooked, since some slJess redistribution would be erpected fortestspecimenssubjectedtolong-term,constantstresstesting. Thiscreepeffect,ofcourse,becomesmore dominantathighertemperatures,resulting in more deviation beaten the two test methods,

Since the predictionsofstrengthwere in reasonable agreement with_tal data, independent ofthetypeofIondinghistoryorthetype of testing, it is concluded that slow crack growth was a uniquemechanism associated with failure for 96 wt % alumina as well as forother test materials.Therefore, it is feasible in principle _ thenumerics] analysis developed in this work to predict life and/or strengthdegradationforany given simple or complexloadinghistaryaslongasan explicit mathematical expression of load history can be made. Moreimportantly, the analysis can be used in conjunction with approp_e_3edments -usingone of three loadinghistories-topromptly assess a

r

governing failure mechanism involved in the conventional life-prediction testing, which could be either constant stress-rate, constantstress, or cyclic stress testing. F_ore, the analysis would bepossible to be used at least as a quantitative tool for damage assessmentby estimating/predicting crack growth/damage-accumulation of astructmal component in service. A verification rt_ for the numericalanalysis for the Case M loading history was not made in this work, soit will be a subject of future study.

CONCLUSIONS

1) For n < 10, simulated strength degradation as a function ofpercent of intezruption time was dgnificant for all the CaseI, ]I and Illloading histories. For the given interruption time _ and n, thedegradation was greater in constant or cyclic stress (Case U or m) thanin constant stress-rate loading (Case I). By contrast, for n > 20, thedegradation became negligible either for the Case I, lI or Ill loadinghistory.

2) The numerical solutions of strength degradation were examinedusing the experimental data determined at elevated temperatures fromfour different advanced ceramics -two silicon nitrides, one siliconcarbide and one alumina- for the Case I loading history, and fromalumina for the Case II loading. The experimental data was inreasonable agreement with the numerical solutions for both loadinghistories Also the phenomenon typical of advanced ceramics, whichhas shown that in constant _-mte condition an initial crack starts togrow at and/or close to failure time after a long incubation time, wasverified

3) The numerical analysis that assumedslow crack growth as agoverning failure mechanism was in good agreement with theexpo'in_tal data Despite some degree of creep deformationpresented, slow crack growth was presumably a significantly operatingfaiktre mechanism involved in all the test materials, regardless of typeof loading history or type of test method. Notwithstanding somedeviation, there was also reasonable agreement in SCG behaviorbetween constant stress-rate testing and constant stress testing for 96 wt% alumina. This suHx_ that the widely utilized SCG formulation ofv =A[Kt/K¢_' described reasonably the slow crack growth behavior ofthe test materials.

4) R appears that the analysis in conjunction with properexperiments, using one of three loadin$ histories (Case I preferred),may be utilized as a tool for damage (SCG and creep) assessment oftest specimens subjected to convontional life l_ediction _.However, care must be exercised when several failure mechanisms

such as SCO, creep and oxidation are actively operating in series athigher temlxratmes A convenient analytical tool(s) accessible to suchcombined conditions, currently not available though, is inevitable.

Acknowledgementa

The authors _e thankful to R. Pawtik of NASA Glenn for theexperimental work. This work was sponsored in part by the H1TEMPpcoject, NASA Glenn Research Center, Cleveland, OH.

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