NLNERICAL CONSIDERATIONS I N THE DEVELOP.?ENT

AND IXPLEMENTATION OF CONSTITUTIVE MODELS

W.E. Hais le r and P.K. Imbrie Aerospace Engineering Department

Texas A&M Univers i ty College S t a t i o n , Texas 77843

Severa l un i f i ed c o n s t i t u t i v e models were t e s t e d i n un iax ia l form by spec i fy ing input s t r a i n h i s t o r i e s and comparing output s t r e s s h i s t o r i e s . The purpose of t h e t e s t s was t o eva lua te s e v e r a l t ime i n t e g r a t i o n methods wi th regard t o accuracy, s t a b i l i t y , and computational economy. The sen- s i t i v i t y of t he models to s l i g h t changes i n input cons tan ts was a l s o in- ves t iga t ed . Resul t s a r e presented f o r IN100 a t 1350°F and k s t e l l o y - x a t 1800°F.

INTRODUCTION

The cha rac t e r i za t ion of t he c o n s t i t u t i v e behaviour of metals has i t s roo t s i n t h e e a r l y work of Tresca, Levy, vonMises, Hencky, P rand t l , Reuss, Prager , and Z ieg le r (Refs. 1-8). These e a r l y models a r e incremental i n na ture , a s s m e t h a t p l a s t i c i t y and creep can be separa ted , and they incor- porate a y i e l d func t ion , flow r u l e , and hardening r u l e t o d e f i n e t h e p l a s t i c s t r a i n increment. These o r i g i n a l incremental t heo r i e s have been =panded and modified by many researchers so that they provide adequate , and o f t e n very good p red ic t ions of rate-independent p l a s t i c flow ( see f o r example Refs. 9-10). However, they a r e sometimes c r i t i c i z e d a s having no formal micromechanical b a s i s upon which t o make t h e assumption of an uncoupling of t h e i n e l a s t i c s t r a i n i n t o rate-independent ( p l a s t i c ) and rate-dependent (creep) s t r a i n components. Nevertheless , t h e c l a s s i c a l incremental t h e o r i e s a r e widely used.

During t h e l a s t t e n yea r s , a number of un i f i ed c o n s t i u t i v e models have been proposed which r e t a i n t h e i n e l a s t i c s t r a i n a s a un i f i ed quan t i t y with- out a r i t i f i c a l s epa ra t ion i n t o p l a s t i c i t y and creep components. These in- c lude t h e models developed by Bodner (Refs. 11-13), S t o u f f e r (Refs. 14-15), Krieg (Ref. 16) , Mi l l e r (Ref. 17) , Walker (Refs. 18-19), Valanis (Refs. 20- 21), Krempl (Ref. 22), Cernocky (Ref. 23-24), Hart (Ref. 25) , Chaboche (Ref.. 26) , Robinson (Ref. 27), Kocks (Ref. 28'), and Cescotto and Leckie (Ref. 29). The a p p l i c a b i l i t y of t hese v i s c o p l a s t i c c o n s t i t u t i v e theo r i e s (mostly t o h igh temperature app l i ca t ions ) has been inves t iga t ed by seve ra l researchers . Walker (Ref. 19) compared t h e p r e d i c t i v e c a p a b i l i t y of s e v e r a l models (Walker, Mi l l e r and Krieg) f o r Hastelloy-X a t 1800°F. More r ecen t ly , Mil ly and Allen

https://ntrs.nasa.gov/search.jsp?R=19850023228 2020-03-23T13:37:34+00:00Z

(Ref.30) provideda qualitativeas well as quantitativecomparisonof themodels developedby Bodner,Krieg,Walker and Krempl for IN100. Both Refs.19 and 30 concludethat thesemodels generallyprovideadequateresultsfor elevatedisothermalconditions,they providepoor and overly-squareresultsat low temperature,the material constantsare often difficulttoobtain experimentally,the resultingrate equationsare "stiff"and sensi-tive to numericalintegration,and the models do not provideany satisfac-tory transienttemperaturecapability. Beek, Allen,.andMilly (Kef.31)have shown that all the unifiedviscoplasticmodelsmentionedabove can becast.ina functionallysimilarform (in terms of internalstate variables).

None of the publishedliteratureprovidesa thoroughevaluationof cur-rent viscoplasticconstitutivemodelswith comparisonto experimentalre-sponse for complexinput histories. Such an evaluationis difficultat pre-sent for many reasons,namely: i) Materialconstant_for most models areusuallyavailableonly for a singlematerialand often for a single temper-ature;2) The experimentalproceduresgiven by model developersfor deter-miningmaterial constantsfrom experimentaldata are often sketchyat best;3) Materialconstantsfor some models are often obtainedby trial-and-errorand are not based on experiments;and 4) There is a lack of good experimentaldata againstwhich the models can b_ evaluated(thatis, test data which issignificantlydifferentfrom that used to generatethe materialconstants).

The purposeof the presentpaper is to reportsome preliminaryevalu-ationsof severalof the unifiedviscoplasticmodels (Bodner,Krieg,Miller,and Walker). These four models are evaluatedwith regardto I) their sen-sitivityto numericalintegrationand 2) their sensitivityto slight changesin inputmaterialconstants.

CONSTITUTIVE MODELS CONSIDERED

The constitutive theories which have been studied to date include Bodner's

(Refs. II-15), Krleg's (Ref. 16), Miller's (Ref. 17), and Walker's (Refs.

18-19). These particular models were selected for this initial study be-causematerialconstantsfor Rastelloy-Xwere availablefor three of themodels. Other models are currently being considered as material constants

become available. Each model is listed below in uniaxial form using a con-sistent notation as presented by Beek, Allen and Milly (Ref. 31). In Ref.

31, it is shown that all of the currentviscop!astiz_cdels considered_ma_-be writtenin uniaxialform as

= E(£ - _1 - sT) (I)

where _ is stress,E is Young'smodulusm g is strain,eI is _he inelasticstrain (internalstate variable),and £x is the thermalstrain. Each vis-coplastictheorypostulatesa particulargrowthlaw for the internalstatevariable(s)and the inelasticstrainis obtainedby time integrationofthe growthlaw for el, i.e.

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t

=1 = f &1(t')dt' (z)--OO

where

de1

&l =d-_- = &i (g' T, _2' e3' " " " ' am) (3)

In equations (2) and (3), t is time, T is temperature, _2 is the back stress(related to the dislocation arrangement and produces kinematic hardening

or the Bauschinger effect), and e3 is the drag stress (which representsthe dislocation density and produces isotropic hardening).

Bodner's Theory

The growth law for the inelastic strain in Bodner's model may be writ-ten in uniaxlal form as

_2Do_p _ k2-_-/\_'J s_(a) (4)&l--/y

where

r

_3 = m(Zl - _3)Wp - AZI (_3 ZI ZI) (5)

Wp = u &i (6)

The quantities E, DO, n, m, ZI, _ Z 1 and r are material constants. Asnoted before, the variable m3 similar to the drag stress used inmany models (a measure of isotropic hardening or dislocation density).It is noted that the model contains no parameter representing the backstress and cannot account for the Bauschinger effect in kinematic harden-ing materials. The material constants are tabulated for INIO0 at 1350°F

(732°C) in Table 1 (taken from Ref. 14).

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Krie_'s Theoz7

The inelastic strain growth law for the model developed by Krieg andcoworkers may be written in terms of state variables rep=esenting backstress and drag stress:

I I_ --_21"_C2 sgn(_ - (7)&t--ci /

2 2_2 = C3 _i - C4 _2 [exp(C5_2 ) - i] sgn (a2) (8)

&3 : C61_II- C7(_3- _3 )n (9)o

The model contains ten constants (CI, C2 . . , C7, E, _ and n).

These have been evaluated by Walker (Ref. 19) for Hastelloy-X at 1800°F

(982°C) and are tabulated in Table 2. It should be noted tha_ equations(7), (8) and (9) form a coupled set of ordinary differential equations.

Miller'sTheory

The growthlawsforMiller'smodelmay be writtenin uniaxialformas

hI = B@' inh _3 sgn(o - _2) (i0)

&2 = HI&I - HIB@' [sinh (A1 le2[)]n sgn(_2) (ii)

A2 3 3 n

_3 = H2 [hi[ 2 + I_21 - _ii_ - H2C2 B@' sinh 2_3 (12)

Miller's theory contains nine aonstants which are tabulated for Hastelloy-X at 1800°F (982°C) in Table 3 (see Ref. 19).

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Walker's Theory

Walker's nonlinear vlscoplastic theory can be cast in the followinguniaxial form

_i = _3 sgn(O - C_2) (13)

&2 = (nl + n2)&l - (_2 - °"2o" nlel) l&iI _ (n3 + n4R)Zn \ l+n6R + i

+ n7 I_2 - _2 Im-l} (14)O

&3 = n8 I_I I - n9 I&ll_3 - nl0 (_3 - e3 )q (15)o

where R is the cumulative inelastic strain

tt =ltR = / l_-{r_dr' (16)

O

The general model requires sixteen constants (E, n, m, q, nI, n2, • • • ,

nlO, _2o and _3(t=O). In determining the constants for Hastelloy-X at 1800°F(982°C), Walker made several simplifying assumptions [including e3 = cons-

tant _ _(t=O)] which reduces the number of parameters to those shown in Table 4(see efJ 19). Further, the constants reported in Ref. 19 were developed from tests

using strain rates in the range 10-3 to 10-6 sec -l and strain ranges of Z0.6%.

NUMERICAL TIME INTEGRATION STUDY

The integration of the constitutive relationship given by equations(i), (2) and (3) forms an integral and extremely important part in any nu-

merical solution of a nonlinear field problem. It has been observed by

many researchers that the coupled system of ordinary differential equations

defining the state variables may be locally "stiff" and thus are sensitiveto the time steF size and numerical algorithm. The accurate integration

oT these stiff equations can be accomplished by various means: use of small

time ste_s, higher-order or multi-point integration schemes, subincrementation

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procedures(Refs.33-35),"smart"algorithmswhich attemptto selectappro-priatetime steps in order to achieveaccuracyand stability(Refs.36,37),algorithmstailoredfor individualconstitutivetheories (Refs.32,37),orcombinationsof these approaches. In general,the computationtime requiredfor the accuratesolutionof materiallynonlinearproblemsis directlyre-lated to the numericalintegrationscheme used.

Regardingthe constitutivemodels reviewedherein,Walker (Ref.32)uses a stable,iterativeimplicitschemewhich takesadvantageof the func-tlonal'formof the integrandin the developmentof the reccurencerelation.Miller originallyused Gear'smethod (Ref.36) to integratethe stiff equa-tions in his theorybut later concludedin Ref. 37 thatan implicitback-ward differencemethod was more economicaland preferableto either Gear'smethod or the explicitEuler forwardintegrationmethod. The type of num-ericalintegrationschemeused by Bodnerand Krieg is not known.

The selectionof an appropriatetime integrationscheme co be used ina computercode is very importantbut is often based on the answersto suchquestionsas: "What is availablein the presentcode?","Whatwill workmost of the time?","What can we use thatmost users will understand?","What is the cheapestand easiestto use?", and the like. The usual re-sponsegiven is "it dependson the problembeing solved!"

In general,equation(3) may be integratedbetweentime t and t + Atby writing

_t+At

=J _i dt (17)J d_l _c+At.t t

or

t+At

AsI = _l(t +At) - _l(t) = f _i dt (18)t

where _i is defined by the particular constitutive theory being used. Thepresent investigation considers four integration schemes: explicit Eulerforward integration, implicit trapezoidal method, trapezoidal predictor-corrector (iterative) method, and Runge-Kutta 4th order method. The approx-

imations for each of these methods is given in Table 5.

Each of the integration schemes in Table 5 were used to obtain stress-

time and stress-strain responses for the four constitutive models considered

herein when subjected to the unlaxlal, alternating square-wave strain-rate

history shown in Fig. I. Figure I shows the 35 second response obtained

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by Krieg'stheoryfor Hastelloy-Xat 1800°Fusinga time step of 0.i sec-onds. For this time step, the Euler and trapezoidalpredictor-correctormethods provideessentiallythe same resultsand are virtuallyidenticalto that obtainedfor all methodsusing a time step of 0.005 seconds. The4thorder Runge-Kuttamethod generallyoverestimatesthe peak responsewhilethe trapezoidalmethod underestimatesthe response. Figure 2 presentsre-sults for three integrationmethodssuch that the total computationtimefor a 35 second responsesolutionis approximatelythe same. For equiva-lent computationtimes,the Eulermethod providesthe most accurateresultsalthoughsmallertime steps are required. Similarresultsare observedforMiller'smodel.

Figures3 and 4 illustratethat_variousconstitutivemodels may behaveappreciablydifferentusing the same integrationmethod (in this case theEuler method). In Fig. 3, Miller'stheory (forHastelloy-Xat 1800°F)givesconsiderableoscillatoryresponsefor a time step of 0.005 secondswhileWalker'stheory shown in Fig. 4 gives a much smootherresponsefor the sametime step. ComparingFigs. 3 and 4, it is seen that a smallertime stepis required (withEuler integration)in Miller'stheorythan in Walker'stheory.

Figure 5 presentsresultsfor IN100 at 1350°Fusing Bodner'smodel.Time steps were chosenfor each integrationscheme to obtainsolutionswhichrequiredapproximatelyequal computationtimes. These results,when com-pared to solutionswith much smallertime steps, indicatethat the Eulermethod providesthe most accurateresults. Again, the time step used issmallerthan that for the other methodsbut the computationtime is thesame (forintegratingthe constitutiveequations).

SENSITIVITYSTUDY FOR MATERIALCONSTANTS

In the previoussection,resultswere presentedwhich showedhow thenumericalintegrationmethod used to integratethe constitutiveequationscould affect the accuracyand computationtimes of predictedresultsforstress-timeand stress-strainresponses. In this section,we c6nsideranotherimportantparameterin the applicationof any constitutivetheory.Namely,"how does the accuracyto which materialconstantsare determinedfrom _xperimentaltest data affect the predictedresponse?"

Figures6 and 7 present resultsfor Walker'smodel (Hastelloy-Xat1800°Fsubjectedto an alternatingsquare-wavestrain-ratehistoryas shown)wherein specifiedinput materialconstantshave been adjustedby 5%. Fig-ure 6 shows the effectof a -5% change (error)in the stressexponentn(themost sensitiveparameter). Figure 7 shows that a +5% error in all testdata requiredto computematerialconstantsresultsin significantpredictedresponseerrors,up to 30% over-predictionin the stressat a time of 35seconds (duringthe relaxationperiod).

Figures8 and 9 presentsimilarresultsfor Krieg'smodel (Hastelloy-Xat 1800°F)and Bodner'smodel (INIO0at 1350°F),respectively. Both resultsindicatethat the most sensitiveparameteris the stressexponent"n" and

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that a 5% error in specifying n may produce significant errors in the pre-dicted response. Miller's model appears to be much less sensitive to er-rors in input material parameters.

Figure I0 provides a comparison of the Miller, Krieg, and Walker modelsfor the Hastelloy-X test at 1800°F (using constants obtained by Walker forall models). The Euler method was used with a time step of 0.0005 secondswhich provides a solution with no significant truncation error. The resultsobtained here show approximately 10-15% differences in peak stress ampli-tudes between the three constitutive models. Since no experimental resultsare available at this time, no conclusions can be drawn as to which _odelmore accurately represents observed test data. However, the results dopoint out that significant differences (greater than 15%) can be obtainedfor stress peaks and stress relation values through the use of differentconstitutive models.

CONCLUSIONS AND FUTURE WORK

The resultsof this study are not completesince only a portionof theavailableconstitutivemodelsand numericalintegrationschemeshave beenconsidered. However,some tentativeconclusionscan be reached. First,it appearsclear from the presentinvestigation,and the work of others,that simpleintegrationschemes (likethe Euler forwarddefferencemethod)are often preferableto more complexschemesfrom the standpointof accur-acy, computationtime, and ease of implementation.Althoughnot reportedherein,our work in progressindicatesthat Euler'smethodused with a simplesubincrementationstrategyprovidesthe most accurateand economicalsolu-tion for most constitutivemodels.

The sensitivitystudy on materialconstantsindicatesthat most visco-plasticconstitutivemodelsare significantlysensitiveto one or more mat-erial constantsderivedfrom laboratorytests. It has been shown that a5% "error"in laboratorymeasurementsmay lead to errors of 25%, or greater,in predictedstressresponses. Althoughmost model developershave fine-tuned their models and inputmaterialconstantsfor specificmaterial/temp-erature/strain-ratecombinations,it is not clear that end-userswill beable to do so when calledupon to developmaterialconstantsfor a new sit-uation. The problemcan be negatedto some extentby definingmore explicittestingproceduresfor obtainingmaterial constantsand by guidelinesde-finingwhich constantsare most sensitiveto experimentalerror.

Our currentand futurework concernsthe applicationof severalinte-grationschemesto the other constitutivetheories,investigationof sub-incrementalstrategies,and considerationof "smart"integrationmethodswhich detect local "stiffness"and adjusttime steps but withoutsignifi-cant computationalexpense. The materialparametersensitivitystudy willbe continuedby consideringother constitutivetheories,and more impor-tantly,by comparisonwith laboratorytestswhich involvecomplexthermo-mechanicalloadingsincludingtransienttemperatureinputs.

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ACKNOWLEDGEMENT

The authorsgratefullyacknowledgeahe financialsupportfor this re-searchby NASA Lewis ResearchCenter under Grant no. NAG3-491.

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15. Stouffer, D.C. and Bodner, S.R., "A Relationship Between Theory andExperiment for a State Variable Constitutive Equation," ASTM SpecialTechnica! Publication 785, 1982 (also AFWAL-TR-80-4194, 1981).

177

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tural Modeling with Advanced Time-Temperature Dependent ConstitutiveRelationships," PWA-5700-50, United Technologies Research Center,

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=oplasticity," Archives of Mechanics, Vol. 27, pp. 857-868.

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tutive Equation Incorporating Rate Effects, Creep and Relaxation,"International Journal of Nonlinear Mechanics, Vol. 14, pp. 183-205,1979.

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at High Temperatures," ORNL-TM-5969, October 1978.28. Kocks, U.F., "Laws for Work-Hardening and Low-Temperature Creep,"

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ing Large Deflections, Plasticity and Creep," in Constitutive Equations

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Table i. Material Constants Used in Bodner's Model

for IN100 at 1350°F (732=C)

Bodners notation Beek and Allen's notation Numerical Value

E E 21.3xi06 psin n 0.7

Z1 Z1 i.105x106 psim m 2.57xi03 psi-1Do DO i0_ sec-lA A l.gx10 -3 sec-Ir r 2.66

Z_ gI 0.6xlO 6 psi(t=O) _l(t=O) 0.0

Z° _3 (t:O) 0.915x106 psi

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Table 2. Material Constants Used in Krleg's Modelfor Hastelloy-X at 1800°F (982°C)

Walker's notation Beek and A11'en's notation Numerical Value

for Krieg's constants

CI 1.0n C2 4.49AI C3 1.0xl06 psiA2 C_ 6.21xi0-_ psi-I sec-IA3 Cs 4.027xI0-7 psi-2A_ Cb i00 psi secI/nAs C7 4.365 psil-n secI/n-2E E 13.2x106 psi

Ko _3o 59,292 psi secI/nn n 4.49c (t=O) c_ (t:O) O. 0

(t=O) c_2(t:O) O.0K(t=0) a3 (_=0) 59,292 psi

Table 3. Material Constants Used in Miller's Model

for Hastelloy-X at 1800°F (982°C)

Miller's notation Beek and Allen's notation Numerical Value

n n 2.363BS' B8' 2.616xi0-s sec-IHI HI Ixl06 psiAI A z 1.4053xi0-3 _si-zHz H2 i00 psi secz/nC= C2 5,000 psiA2 A2 4.355xi0-Iz psi-3E E 13.2xi06 psi

(t:O) _i (t:O) O. 0R (t=0) _2(t=0) 0.0Do e_ (t=0) 8,642 psi

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Table 4. Material Constants Used in Walker's Model

for Hastelloy-X at 1800°F (982°C)

Walker'snotation Beek and Allen'snotation NumericalValue

_z -1,200psi

nl nl 0 psi (notused)n2 n2 Ixl0 6 psin9 * 312.5n7 n7 2.73xi0r_ psil-m sec-In n 4.49m m 1.16

E E 13.2x10 6 psi

¢(t=0) _(t=0) 0.0_(t=0) a2(t=0) 0.0K(t=0) a_(t=0) 59,292 psi

ns,ng,n10,q 0 (not used)

* = _ (n3 + n4R) £n \[-_6 R +

Table 5. NumericalIntegrationApproximationfor haI = t/t+At_ldt

Method Approximation

Euler Forward Difference A_= At _I (t)

Air"

Trapezoidal Rule AC_z=_-Le I(t) + al (t+At)]

Trapezoidal Predictor-Corrector Same as trapezoidal except iterate

Runge-Kutta 4th Order AaI = i(_ + 2K2 + 2K3 + K4 )

KI = At &l(t,_1(t))

K2 = At al(t+At/2,el(t)+_/2)

K3 = At gl(t+At/2,_(t)+K2/2)

K4 = At _l(t+At,_l(t)+K 3)

181

- A; = 0 . 1 0 o o 0 0 SEC KRIEG MOOEL IEULERI

20.0 . . . . . . . . . - AC = 0 . 1 0 o o 00 SEC RRIEC ~ O O E L l T R R P E Z O l O R L l -.-.-.-. AL = o . 1 0 0 0 00 SEC KRIEC ~ O O E L IPREOICTOR-CORRECTORI -..-..-. At - 0 . 1 0 0 0 0 0 SEC R R I E G ROOEL l 4 T H ORDER R U N G E - K U T T R I

12.0

4 .0

0 -4 .0

Time ( s e c )

- 1 2.0

Fig. 1 Comparison of i n t e g r a t i o n methods f o r Kr ieg ' s theory (Hastelloy-X a t 1800°F)

-. :, - 0: - :.I:::-:? :I: .--- : : . I? . . . . . . . . . . 3: - 3.5500-2: SLC *ILL:P ROCfL I C U L E I I . - DL - 0 . 1 0 0 0 :: S L C R I L L I R 1JCLL I L u L E R ~

. . . - . . - . 0: - O. ;ZSJ c5 ~ L C R!L::~ 1 . 3 3 : ~ , E ~ L ~ I I

Fig. 2 Comparison of i n t e g r a t i o n methods Fig. 3 S t a b i l i t y and accuracy of f o r Kreigls thoery w i th equa l com- Eu le r ' s method f o r M i l l e r ' s put 'ation t ime allowed f o r each method theory (Hastelloy-X a t 1800°F) (Hastelloy-X a t 1800°F)

- &. :,::::-:: ::: .:,.;, .--x .,. . . . . . .. .. . . - .. ... i* .

At - 0 . I300 -02 3% UaLxCa .IOOCL :C"LL.I - - -.- a - 3.::OC DO SCC ~ $ : n f I -:EEL IE>LCII ...-..-. bt - 0.1250 :O SEC ~ D L I . C R ~ O D C L : EULER I

Fig. 4 S tab i l i ty and accuracy of Fig. 5 Comparison of integration methods ~ u l e r ' s method for Walker's for Bodner's theory with equal theory (Hastelloy-X a t 1800°F) computation time allowed for each

method (IN100 a t 1350°F)

20-0 r A t - O.SO00-02 SEE URLKLR IlOOCL (EULERI CORRECT SOLUTION ...-- ..... ~t - O.SOOO-02 scc URLKCR n o o n ILULLRI VALUE 0 1 n ROJUSICD 01 -SX -.-.-.- bL 0.5000-02 SLC URLI(CR nOOCL I CULLRl VRLUC 0 1 I( ROJUSTLO 01 -5%

I 36.0 40.0

-4.0 - Time (sec)

-12.0 - -

Fig. 6 Sensi t iv i ty of Walker's theory to -5% change i n input constants (Hastelloy-X a t 1800°F)

&t - 0.5000°02 SEC WALKERM00EL {EULERI CORRECT SOLUTION.......... At - O.$000°0Z SEC WRLKERMODEL IEULER! ALL TEST O_,A AOJUSIEO BY ./5Z

20.0 -

..............,. ---........... °.............

GO 4-.0

"_ ' I,¢, I I f I t I t ! I I I I , I

0 4.0 12.0 1 .( 24.0 2B.0 32.0 36.0 40.0P -_.o !

Time (see)

-_2.0 °°q-! "-"•U' _0-U

-20.0

Fig. 7 Sensitivity of Walker's theory to 5% change in experimental test

data used to generate constants (Hastelloy-X at 1800°F)

_AL - 0.5000-02 SEC KRIEG _0OEL {(ULERI CORRECT SOLUTION

.......... At - 0-5000-0Z SEC _RIEG MODELIEULERI V_LUE 0F N ROJU_TEO BT -SZ

........ At - 0.5000-02 SEE _RIEO MOOELIEULERI V_LUE OF R nOJUSTEO BY -5%

20.0r

12.0

-,2.o %°'°°'F-1F-1F! . .=ooo,LL._I.II ,oo

?1_ (it,)

-20.0

Fig. 8 Sensitivity of Krieg's theory Co -5% change in input constants(Hastelloy-X at 1800°F)

184

- At - 0.4000-02 SEC aOONER MOOEL IEULERI CORRECT SOLUTION

. . . . . . . . . . AL = 0.4000-02 SEC BOONER MOOEL IEULERI VRLUE OF N AOJUSIEO 87 - 5 %

. At - 0.4000-02 SEC BOONER MOOEL IEULERI VRLUE OF ZI.ROJUS1EO BY -5%

Fig. 9 S e n s i t i v i t y of Bodner's theory t o -5% change i n i npu t cons tan ts (IN1 00 a t 1350 OF)

- At = 0.5000-03 SEC MILLER IIOOEL IEULERI . . . . . . . . . . AL - 0 . ~ 0 0 0 - 0 3 SEC KRIEG MODEL I EULER 1 - -.-.-. At 0 -5000-03 SEC YRLKER MOOEL I EULER I

Fig . 10 Comparison of s t ress - t ime p red i c t i ons f o r M i l l e r , Krieg and Walker t h e o r i e s ( ~ a s t e l l o ~ - ~ a t 1800°F)