International Journal o f Plasticity, Vol. 4. pp. 355-370, 1988 0749-6419/88 $3,00 -t- .00 Printed in the U.S.A, Copyright © 1988 Pergamon Press pic

T H E C O N S T I T U T I V E R E P R E S E N T A T I O N O F

H I G H - T E M P E R A T U R E C R E E P D A M A G E

K.S. CHAN

Southwest Research Institute

(Communicated by David Rees, Brunel, University of West London)

Abs t rac t -The elastic-viscoplastic constitutive equations of Bodner-Partom were applied to modeling creep damage in a high temperature Ni-ailoy, BI900 + Hf. Both tertiary creep in bulk materials and creep crack growth in flawed materials were considered. In the latter case, the energy rate line integral C* was used for characterizing the crack driving force, and the rate of crack extension was computed using a local damage formulation that assumed fracture was controlled by cavitation occurring within the crack-tip process zone. The results of this inves- tigation were used to assess the evolution equation for isotropic damage utilized in the Bodner- Partom constitutive equations.

i. INTRODUCTION

Creep cavitation in structural alloys and ceramics at elevated temperatures generally occurs by nucleation, growth, and coalescence of cavities at grain boundaries subjected to local tensile stress. In most instances, the cavitation kinetics is controlled by the over- all rates of cavity nucleation, growth, and coalescence (PERRy [1974]). However, creep cavitation can sometimes be viewed as being either nucleation dominated or growth dominated by considering the role of individual mechanisms in the fracture process (PAGE ~, CI-L~q [1987]). In the former case, cavitation is controlled by continuous nucle- ation and coalescence of closely spaced cavities that are nucleated under high, transient local tensile stress and may not exhibit any significant growth after reaching a certain size when the transient stresses are removed by diffusion. On the other hand, cavity nucleation in the growth-dominated cavitation is relatively easy and rapid, but ceases to occur after the potential nucleation sites are exhausted. The rate of cavitation is then controlled by the growth and subsequent coalescence of the cavities. For widely spaced cavities, substantial cavity growth would be required before coalescence occurs.

Two different approaches are commonly used for modeling creep cavitation (LFCKm I-IA~augs'r [1977]); L~crd~ [1978]). In the mechanistic approach, the cavity nucleation,

growth, and coalescence processes that occur during creep cavitation are considered individually with kinetic expressions for individual mechanisms. In the continuum dam- age approach, cavitation is considered, without reference to particular mechanisms, as a damage process that is described in terms of internal variables with appropriate evo- lution equations. The damage variables are generally considered as a measure of the dis- tributed voids and microcracks in the deteriorated microstructure and are incorporated into a number of constitutive models as internal variables representing isotropic and directional softening. The internal variables for damage are generally inferred from experimental data and cannot be directly measured experimentally. Constitutive mod-

355

356 K.S . ('H.~u

els that incorporate a damage term include those of CHABOCHE [1979], BODNER & PAR- TOM [1975], BODNER [1987]), WILSON 8, WALKER [1984], and LECKm & HAYHURST [1977].

At a given temperature the evolutionary equation for isotropic damage, ~,, is gener- ally expressed as

& = Cof~(~) f2 (c~) (1)

in which d~ is the time rate of damage, Co is a material constant, f~ (w) is a function of the damage variable oJ, and f 2 ( a ) is a stress function. Table 1 summarizes the functions of f l (o~) and f2 (o) utilized in the damage models of RABOT~OV-KACnA~OV (RABOrNOV [1969], KACHANOV [1958]), CHABOCHE [1979], BODNER [1981,1987], and kvcKm & HAYHURST [1977]. These damage models are also contrasted to the cavity-growth models of COCKS ~, AsHaY [1982]. Table 1 indicates that there are considerable differences in the f l (oJ) functions among the various models; these differences are also depicted in Fig. 1. Note that despite the differences in the f~ (o~) functions, these damage models invariably lead to similar creep rupture life expressions, as shown in Table 1.

Elevated temperature crack growth in materials under sustained load is characterized by prominent intergranular cavitation in the vicinity of the crack tip. For correlating creep crack propagation rate, net-section stress and fracture parameters such as the elas- tic stress intensity factor, K, and the energy rate integral, C*, have been used (VA~ LEEUWEN [1977], Fu [1980]). Since significant time-dependent deformation occurs dur- ing creep, the K approach should be limited to cases for which creep deformation is of small scale and contained within the crack tip plastic zone. The C* integral has been shown to characterize the crack tip behavior of materials undergoing steady-state creep (GoLD~a~ & HurcHn~sos [1975]), and it can be considered as the driving force for creep crack growth and an appropriate parameter for correlating the crack growth data (LA~DES & BEGLEY [1976], VAN LEEUW~N [1977], FU [1980], and SADANA.'qDA & SnA~N- ~AS [1983]).

Table 1. Summary of the evolution equations for damage: (o = Cofj ( w ) f , ( e )

Creep rupture Model f~ (,:) f_,(a) life expression

Rabotnov-Kachanov

Leckie and Hayhurst

Chaboche

Bodner

Cocks and Ashby ~ (creep controlled)

Cocks and Ashby" (diffusion controlled)

( 1 - ~:) -" o r tRo ~ = Ci

(1 -- w ) - " ae tRae = CI

( 1 - ~ : ) - " ~ ° ) e ~ t R o ~ = Ct

1 "

1 - ( 1 - ~)'+~ O v t R O ° = C I

(1 - w ) '

[ w -~'2 In o tRo ~ = C~

"Cavity growth models. c,~ = ~a~+.x + t~ 34~2 + "~J~. Co, C,, ~, and 3' are material constants; tR is the rupture life of creep specimens at unia:dal stress.

High-temperature creep damage 357

w

Fig. 1. Comparison of the damage functions, J, (w), in a number of evolution equations for damage pro- posed in the literature.

Although continuum damage models have been used extensively for describing creep damage in bulk materials, there has been little experience in using constitutive equations with damage terms in creep crack growth problems. Previous application of the Bodner- Partom equations in creep crack growth finite-element modeling was based on match- ing calculated load-line displacements with experimental data and did not include a dam- age term as a criterion for creep crack extension (HINNERICHS et al. [1982]). Since most of the damage functions have been developed based on observations of creep damage in bulk materials, it is not known whether these damage models would lead to reason- able creep crack growth rate behavior. The objective of this article is to assess a con- tinuum damage formulation for creep crack growth from its predictions of the creep crack growth rate. In particular, the Bodner-Partom constitutive equations were used

358 K . S . CHAN

for modeling work hardening, thermal recovery, and creep damage in a high tempera- ture Ni-alloy subject to either bulk cavitation or localized cavitation occurring ahead of the crack tip. In the latter case, a local damage approach was adapted for predict- ing crack growth under steady-state creep conditions. Important aspects of the crack growth model are the use of (1) the Bodner-Partom constitutive equations for repre- senting material inelastic response as well as creep damage within the fracture process zone, (2) the C* integral for representing the driving force for creep crack growth, and (3) a local damage formulation for describing crack extension.

!i. M ODE L ING A P P R O A C H E S

II. 1 Continuum damage modeling

The small strain constitutive equations of Bodner-Partom for elastic-viscoplastic materials are used to describe the material response in this investigation. A summary of the Bodner-Partom constitutive equations is presented in Appendix A. The Bodner- Partom equations are formulated within the framework of the unified approach and are intended for treating all aspect of inelastic deformation, including plasticity, creep, and stress relaxation. In addition, these equations do not require a yield criterion or load- ing and unloading conditions. The important features of the unified constitutive equa- tions in Appendix A are (1) a flow law, (2) a kinetic equation relating the inelastic strain rate to stress and the internal variables, and (3) evolution equations for the internal vari- ables representing either hardening or softening. In the present form (BoDNER [1987]), the Bodner-Partom equations are capable of representing strain rate dependent, inelastic deformation for multiaxial proportional and nonproportional loading. Appendix A shows that physical processes such as isotropic hardening, directional hardening, thermal recovery [second terms of eqns (A4) and (A5)], and damage are represented in the Bodner-Partom model.

Only isotropic damage is considered in this investigation (see Appendix A). In the Bodner-Partom model, the evolution equation for isotropic damage, ~o, at a given tem- perature is represented by (BoDNER [1981])

J 3 (2)

where p and H are material constants, and Q is the stress term,

(3)

proposed by LECKIE & HAYHURST [1977] for damage growth under multiaxial stress states. In this expression, a+ax is the maximum principal stress, J~ is the first stress invariant with the symbol + indicating tensile stress, and c~,/3, 7, and v are material con- stants with ¢x + ~ + V = I. For constant stress loading, eqn (2) can be integrated to give

oxo(( ) ) ,,,

High-temperature creep damage 359

where

(2= A(o~)dt = f,.(o~)t

with oe given in Table 1. According to eqn (4), small value of damage occurs at low values of stress until a threshold is reached beyond which the damage function increases at a rate controlled by the parameter p.

II.2 Creep crack growth modeling

For creep crack growth modeling, it is envisioned that a mode I crack under plane stress is subjected to a remotely applied stress in a material undergoing steady-state creep. As illustrated in Fig. 2a, creep cavitation occurs within a circular process zone

~ a ,m.lm

Y t ~ process zone

,..,.ID X

- - b " -

(a)

i

~ . . - a - . . .m

J

Y o c )_., . , . / r ~TT( I

- 4 , j,,D.

(b)

Fig. 2. A creep crack in a material undergoing steady state creep. (a) Cavitation within the process zone; (b) Barenblatt-Dugdale representation of the process zone.

360 K. S, CHA.',

of size h ahead of the tip of the crack. Using the Barenblatt-Dugdale approach (BAREN- aLATT [1962]; DUGDALE [1960]), the process zone is replaced by a compressive creep stress ac and the length of the creep crack is extended to included the process zone as depicted in Fig. 2b. For steady-state creep, the creep crack driving force is generally rep- resented by the path-independent line-integral (GoLDM~'¢ 8, HUTCHINSON [1975]; LAZqDES ~, BEGLEY [1976])

C* = f r (Wnl - ounjtt,.~)cts (5)

where W, aij, t2i,/, and ds are the strain energy rate, stress, and velocity gradient and the incremental line segment length, respectively, and the line contour P is taken in the counterclockwise direction around the crack tip. Taking the contour F along the crack surface leads to

C* = ac6c (6)

where ac is the stress on the extended crack surface and the crack tip opening displace- ment rate, ~c, is related to the crack tip strain rate, ~, and the size of the process zone, h, by the expression

3 c = ~ c h (7)

for the Barenblatt-Dugdale crack. The inelastic strain rate corresponding to the con- stant stress of o¢ can be obtained from the Bodner-Partom constitutive equations by setting Z = Zs, ¢0 = 0, 3,/2 = o,r,' and D p = 3/4~,~ in eqn (A2) to give

~,. = -~- Doexp - ~ \ o,--~ / j (8)

in which Zs is the steady-state value of the hardening variable Z, and Do and n are material constants.

The damage rate within the process zone is represented by the evolution equation for damage given in eqn (2). In general, there is coupling between the stress term and the damage rate in the process zone. For the present problem, an uncoupled form is assumed when cavitation is confined to the crack tip process zone whose size is small compared to the crack size and the uncracked ligament such that (1) the damage pro- cess is constrained by the surrounding uncavitated material; (2) C* is path independent along paths outside the process zone. Under these circumstances, the damage rate is con- trolled by C*, and the stress ac acting on the process zone remains constant during cav i ta t ion (HUTCHINSON [1983]). By substituting ae = oc, o~ = wc, and t = AtR into eqn (4), the time to rupture, AtR, of the material element located in the creep process zone is

H AtR = o~[ln(1Ao~)] i/p (9)

High-temperature creep damage 361

which leads to an approximate creep growth rate of

da h h H ~- = AtR = oc'[ln(l/o~c)] lip (10)

with we being the critical value of damage at creep rupture. Values in the range 0.1 wc -< 0.8 are examined in this article.

Iil . EXPERIMENTAL PROCEDURES

The material studied in the present investigation was the Ni-base alloy BI900 + Hf. The chemical composition of BI900 + Hf in weight percent is 7.7Cr, 9.9Co, 6.0Mo, 6.1Al, 4.1Ta, 1.0Ti, 1.19Hf, 0.09C, and balance Ni. The specimens were in the forms of round bars that were heat treated at 1080°C for 4 hours, air cooled, and aged at 900°C for 10 hours. A grain size of 0.8 mm (ASTM grain size number -2 to -3) was obtained in the gauge section of the specimens. The microstructure consists of 7' in a 3' matrix; the 7' size is approximately 0.9/~m in the fully heat-treated condition. The alloy derives its high-temperature strength from the gamma-prime precipitates.

All the creep tests were conducted isothermally under constant load conditions in a closed-loop, servohydraulic test machine equipped with an induction heating unit. Round specimens of 7.6 mm diameter and 25.4 mm gauge length were used, and they were soaked at the test temperature until an equilibrium temperature was reached; tem- perature variation over the specimen gauge section was less than 10°C, in accordance with the ASTM specification for short-term tests. Creep tests were performed at 649, 760, 871,982, and 1093°C under constant loads. In all cases, the applied loads were maintained throughout the tests and no significant load drifts were observed. Creep strains were measured as a function of time using an externally mounted extensometer with a quartz reach-rod attached to the specimens at two small indents that were spaced 25.4 mm apart. All the specimens failed at locations away from the small indents. The span of the stroke on the servohydraulic test machine was greater than 150 ram. For the small strains considered here, there were no axial constraints on the creep specimens due to the test machine. The creep curves generated by the servomachine should therefore be equivalent to those obtained using a deadweight lever machine.

It should be noted that the creep tests were a small portion of a large data acquistion effort at Southwest Research Institute and Pratt & Whitney Aircraft for the NASA HOST program. In this effort, the high temperature constitutive behavior of BI900 + Hf was characterized by performing tensile, creep, and cyclic tests at a number of tem- peratures and strain rates. Both uniaxial and biaxial strain paths under either propor- tional or nonproportional loading were examined. In addition, thermomechanical paths were also studied. Although this effort resulted in a larger constitutive database for B1900 + Hf under a variety of loading conditions, only creep damage is examined in this article.

IV. RESULTS

The Bodner-Partom model constants for BIg00 + Hf were determined from the ten- sile and the creep data. A summary of the material constants for B1900 + Hf is shown

362 K.S . CHAN

in Table 2. The material constants pertaining to the strain rate dependence of plastic flow, hardening, and thermal recovery were determined based on tensile and creep data using the procedures described elsewhere (LINDHOLM et al. [1985]). The material con- stants pertaining to the damage model were determined based on the time to creep rup- ture data (BoDt~R ~ CnAN [1986l). By applying the damage growth equations [(2) and (3)] to the case of constant uniaxial stress o, it can be readily shown that Q = to ~ would be constant (=QR) for all rupture times tR, if rupture is indicated by a certain level of damage. On the basis that

QR = IR Ol' = c o n s t a n t (11)

the parameter v was determined to be 8.3 from the slope of the straight line relations shown in Fig. 3. For the uniaxial stress condition considered in the model, a e equals tr (see Table l) and the sum of a , 3, and 3" in eqn (3) is unity. It is therefore not neces- sary to evaluate individual values for a , 3, and 3'. The present damage model is there- fore a special case that is applicable to creep crack growth under plane stress conditions. For creep under muttiaxial stress state, the distinction must be made between creep life of those materials that obey different stress criteria and experimental values of or, B, and 3' would be required accordingly.

In general, the constants p and H in eqn (2) can be determined from a limited set of creep test results with the values used subsequently for predictive purposes. Due to the limited creep test data, it was sufficient to set p = 1 for all temperatures, which is equiv- alent to using a simple exponential function for damage growth. Since an initial porosity existed in the BI900 + Hf specimens, an initial value of o~, ~0 = 1 × 10 -9 , was assumed.

Table 2. Bodner-Partom model constants for BI900 + Hf

Temperature, °C

T ~ 760°C 871°C 982°C 1093°C

Temperat ure-independent constants mj = 0.270 M P a - ' m 2 -- 1.52 MPa -I Zi = 3000 MPa Z 3 = 1150 MPa r l = r : = 2

D o = 1 x 104sec -I p = l e = 8.34

COo = l x 10 -9 oq = 0

Temperature-dependent constants n 1.055 1.03 .850 0.70 Zo (MPa) 2700 2400 1900 1200 AI = A2 (sec - t ) 0 0.0055 .02 0.25 Zz (=Zo) (MPa) 2700 2400 1900 1200 H((MPa) 'sec) 2 x 10 z7 4 x 1024 5 x 20 =o

Elastic modulus for BI900 + Hf E = 1.987 x 105 + 1 6 . 7 8 7 - .10347-" + 1.143 x 10 -5 T 3 MPa with T i n °C

High-temperature creep damage 363

5000

1000

500

100

50

10

BIg00 + Hf

760°C 871°C 982°C

Q A O Onset of Tertiary Creep

• • • Fracture

I I , I w 10 102

Time to Rupture, h

Fig. 3. Time to rupture data of BI900 + Hf.

10 3

The parameter H was then estimated from eqn (9) based on the rupture life at a given applied stress. The value of H was further refined by fitting the Bodner-Partom model to the experimental creep curves until good overall agreement between theory and exper- iment was obtained. Using the material constants shown in Table 2, the Bodner-Partom model was used to calculate the creep response of BIg00 + Hf at 871,982, and 1093°C. Those creep calculations were performed by specifying a constant stress value and by numerically integrating eqns (AI), (A2), and (A4) to (A6) as a function of time using an implicit integration method (LnsDICOLM et al. [1985]). At each time step, the values of i, ,~i Be, and 6: were computed and multiplied by a small time increment, 4t , to give the incremental changes in strain ( 4 0 and the internal variables ( 4 Z t, zIZ t), and ,4co). The current values of strain and internal variables (Z t, Z D, and {o) were then updated by adding their incremental changes to their values in the previous time step. Comparison of the calculated and experimental creep curves for BIg00 + Hf at 871, 892, and 1093°C is shown with the corresponding damage values in Figs. 4, 5, and 6, respectively. It is worthwhile to note that the calculated creep curves show all three stages of creep (primary, secondary, and tertiary) because work hardening, thermal recovery, and softening due to damage are represented in the Bodner-Partom model. In the primary creep region, material response is dominated by hardening [the first terms in eqns (A4) and (AS)]. Secondary creep is predicted when the hardening terms in eqns (A4) and (AS) are balanced by the thermal recovery terms [second terms in

364 K . S . CHAN

3 g

c

u~

-081 .04

.06

.04

.02

0 0

'•I ~ = 417 MPa

517 MPa

Big00 + Hf

871°C

m m Experiment

- - B o d n ~ r t o m

o = 517 MPa

/ I - z = 427 MPa

/ . .

, I , I , I , 4 8 12

Time, h

Fig. 4. Comparison of calculated and experimental creep curves of B I900 + Hf at 871°C with [he correspond- ing calculated damage value.

3

g

.2

.l

0

.lO

.08

.06

• 0 4

.02

0

/- c = 283 MPa =

Big00 + Hf

982°C

Bodner-Partom Mode]

mm Experiment ~ = 283 MPa

~ " I I I 5 I0 15 20

Time, h

Fig. 5. Comparison of calculated and experimental creep curves of BIgO0 + Hf at 982°C with the correspond- ing calculated damage value.

High-temperature creep damage 36. ~

.050

3

~ .025

0 1.0

.8

.6

.4

.2

_ ; = 79.3 MPa /

2/ I I i, I I, 1 2 3 4 5

Time, h

Fig. 6. Comparison of calculated and experimental creep curve of BIgO0 + Hf at 1093°C with the correspond- ing calculated damage value.

eqns (A4) and (AS)] so that both Zt and ~ij are zero. Tertiary creep occurs when soft- ening due to damage becomes significant. Note that in all cases the onset of tertiary creep occurs at a small value of the damage parameter ~0.

The creep growth rate was predicted as a function of the C °-integral using the equa- tions developed in the previous section and the material constants listed in Ta- ble 2. The procedures for computing the creep crack growth rate and C" are as follow: (1) assigRing a value for o¢, (2) computing the corresponding values of ~c and Zs using the Bodner-Partom model, (3) computing C ' / h using eqn (6), (4) calculating the rup- ture life, 4tR, of the material element located within the process zone, and (5) comput- ing the normalized creep growth rate ~t/h, using eqns (6) and (7). Figure 7 shows the calculated creep crack growth rate for Big00 + Hf at 871, 982, and 1093°C, using tar = 0.1 as the failure criterion. The calculated creep crack growth rate depends on the

366 K.S . CHau

i0 "I

I

- I 10

10"2

BI9OO+Hf

i093°C

871°C

cO = .i C

10"31 , , I , ~ ' ~ , ! , :

io-~ io-4 Io-3 Io-2 i,3-1

C*lh, KJlm h

Fig. 7. Calculated creep crack growth curves of BI900 + Hf at three temperatures.

failure criterion. Figure 8 shows that a larger value of ~c results in a lower crack growth rate.

The grain size of the BI900 + Hf alloy is approximately 0.8 mm. Metallography examinations have revealed that creep cavitation in this alloy occurred mostly at grain boundaries. The minimum value of the fracture process zone is thus in the order of the grain size (i.e., = 1 mm). Taking h to be 1 mm, the computed creep crack growth results in Figs. 7 and 8 are in qualitative agreement with the experimental trends observed in a number of structural alloys (SADANANDA a SH-ANINIAr~ [1983]), even though creep crack growth data of B1900 + Hf is not available. Both the computed C ° and t~ are in the ranges observed experimentally for Ni alloys. At low values of C*,a is approxi- mately linear with C* in a log-log plot, indicating a power-law relationship between d and C*, as observed experimentally. At higher values of C °, a increases rapidly with C ° , and the power-law dependence breaks down, primarily due to unstable creep crack growth. Comparison of the calculated curves with the experimental result for Udimet 700 at 850"C (SAD,~a~,~TDA a Sm,~INtm,~ [1983]) in Fig. 8 indicates good agreement for the case in which o~c is equal to 0.1. Although it is not a direct test of the present ap- proach, the good agreement between theory and experiment suggests that the Bodner damage model can predict realistic creep crack growth behavior when appropriate values of too, toc, and the process zone size are used:. Further evaluation o f the Bodner dam- age model against experimental results is needed to establish the accuracy of the model and the relevant values of tOo, tOc, and the process zone size for B1900 + Hf.

High-temperature creep damage 367

10

'7 e -

10 "I

10 -2

10 -3

10-5 I0 "4 10"3 10 ~2 10 "I

C*/h, KJ/m h.

Fig. 8. The dependence ofthe calculated creep crack growth curves on the value of w,.in thelocalfailure cri- terion.

Y. DISCUSSION AND SUMMARY

The results of the present investigation demonstrated that the Bodner-Partom model with a damage term is a viable approach for modeling creep damage and crack growth. In particular, the isotropic damage function proposed by Bodner has yielded fairly accu- rate tertiary creep behavior for bulk materials under sustained loads and reasonable pre- dictions of creep crack growth rate as a function of C*. The predicted dependence of creep crack growth rate with temperature under plane stress condition is also encour- aging. These results suggested that once the material constants are evaluated from per- tinent experimental data, the Bodner-Partom constitutive equations with the damage variable can give good representation of the bulk creep and the crack growth behavior of a cavity-forming material. The phenomenological nature of this model, however, requires not only the determination of a number of material constants, but also the need for assuming values of damage parameters ~ initially and at creep rupture, leading to a computed creep crack growth that is dependent on the local failure criterion.

Acknowledgements--This research was supported in part by the National Aeronautics and Space Adminis- tration through NASA Contract NAS3-23925, and partially by the Internal Research Program of Southwest Research Institute.

1958

REFERENCES

KACttANOV, M.L. "Time of the Fracture Process Under Creep Conditions," lzv. Akad. Nauk. SSSR O.T.N. Tekh. Nauk., 8, 26.

368 K.S. CHAN

1960 DUGDALE, D.S., "Yielding of Steel Sheets Containing Slits," J. Mech. Phys. Solids, 8, 100-104. 1962 BARENBLATT, G.l., "The Mathematical Theory of Equilibrium of Cracks in Brittle Fracture," Advan.

Appl. Mech., 7, 55-129. 1969 RAaOTNOV, Y.N., Creep Problems in Structural Members, North-Holland, Amsterdam. 1974 PERRY, A.J., "Cavitation in Creep," J. Mat. Sci., 9, 1016-1039. 1975 BOONER, S.R. and PARTOM, Y., "Constitutive Equations for Elastic-Viscoplastic Strain Hardening

Materials," ASME J. Appl. Mech., 42, 385-389. 1975 GOLDMAN, N.L. and HUTCHINSON, J.W., "Fully Plastic Crack Problem: The Center-Cracked Strip

Under Plane Strain," Int. J. Solids Structures, I1, 575-591. 1976 LANOES, J.D. and BEGLEY, J.A., "A Fracture Mechanics Approach to Creep Crack Growth,"

Mechanics of Crack Growth, ASTM STP 590, American Society for Testing and Materials, 128-148. 1977 LECKIE, F.A. and HAYHURST, D., "Constitutive Equations for Creep Rupture," Acta Metall., 25,

1059-1070. 1977 VAN LEEUWEN, H.P., "The Application of Fracture Mechanics to Creep Crack Growth," Eng. Fract.

Mech., 9, 951-974. 1978 LECKIE, F.A., "The Constitutive Equations of Continuum Creep Damage Mechanics," Phil. Trans.

R. Soc. London A., 288, 27-47. 1979 CaAaOCHE, J.L., "Thermodynamic and Phenomenological Description of Cyclic Viscoplasticity with

Damage," Translation of Publication No. 1978-3 of the Office National d'Etudes et de Researches Aerospatiales, France, European Space Agency Technical Translation Service, Publication No. ESA-TT-548.

1980 Fu, L.S., "Creep Crack Growth in Technical Alloys at Elevated Temperature-A Reviex~," Eng. Fract. Mech., 13, 307-330.

1981 Boom:R, S.R., "A Procedure for Including Damage in Constitutive Equations for Elastic-Viscoplastic Work-Hardening Materials," Proc. IUTAM Symposium on Physical Nonlinearities in Structural Anal- ysis, ed. by HULT, J. and LEMAITRE, J., Springer-Verlag, New York, pp. 21-28.

1982 CocKs, A.C.F. and ASHaY, M.F., "On Creep Fracture by Void Growth," Progress in Mat. Sci., 27, 189-244.

1982 HINNERICHS, T., NICHOLAS, T. and PALAZOTTO, A., "Hybrid Experimental-Numerical Procedure for Determining Creep Crack Growth Rates," Eng. Fract. Mech., 16, 265-277.

1983 HUTCHINSON, J.W., "Constitutive Behavior and Crack Tip Fields for Materials Undergoing Creep- Constrained Grain Boundary Cavitation," Acta Metallurgica, 31, 1079-1088.

1983 SADANANDA, K. and SHANINIAN, P., "Creep Crack Growth Behavior of Several Structural Alloys," Metall. Trans., 14A, 1467-1480.

1984 WILSON, D.A. and WALKER, K.P., "'Constituti',e Modeling of Engine Materials," AFWAL-TR-84- 4073, Air Force Wright Aeronautical Laboratories.

1985 LINDHOLM, U.S., CHAN, K.S., BODNER, S.R., WEBER, R.M., WALKER, K.P. and CASSENTI, B.N., "Constitutive Modeling for Isotropie Materials (HOST)," Second Annual Status Report, NASA-CR 174980.

1986 BODNER, S.R, and C,AN. K.S., "Modeling of Continuum Damage for Application in Elastic- Viscoplastic Constitutive Equations," Eng, Fract. Mech., 25, 705-712.

1987 BOONER, S.R., "Review of a Unified Elastic-Viscoplastic Theory," in MILLER, A.K. (ed.), Unified Constitutive Equations for Plastic Deformation and Creep of Engineering Alloys, Elsevier Appl. Sci. Pub., New York, pp. 273-301.

1987 PAGE, R.A. and CriAN, K.S., "Stochastic Aspects of Creep Cavitation in Ceramics," Metall. Trans., lgA, 1843-1854.

Southwest Research Institute San Antonio, TX 78284, U.S.A.

(Received 10 January 1987; in final revised form 15 October 1987)

APPENDIX A. A SUMMARY OF THE BODNER-PARTOM MODEL

WITH THE CONTINUUM DAMAGE TERM

1. Flow Law

~ij = ~ + ~ (A1)

= = o

High-temperature ¢i'eep damage 369

with

2. Kinetic Equation

with

1 Su = o~j - ~ au.**

D p = Do2exp[ r Z2(I -- - -[ 3,/2 °J)2] "]

Z = Z I + Z D

1 .p .p Df = ~ ~ijEij

1 A = ~ susu

A 2 = D£/J2

3. Evolution Equations of Internal Variables a. Isotropic Hardening

where

&' = m2(oqsin 0 -- ~ ' ) W p

0 = cos - ' ( vu~ A

Vii ~ - - ~i j / (~k l~kl )l/2;

llij = a i j / ( tTkl Okl )l/2;

with

or 0 = cos-l(uu~u)

uij = Oij/(O,i dkl)lJ2

z'(o) = Zo; W. = a,.#~; w~(o) = o; ~'(o) - o

b. Directional Hardening

Bu = m2(z3u, j - ~u )W. - A . Z , Z, J ~

(A2)

(A3)

(A4)

(A5)

370 K.S. CHAN

with

c. Isotropic Damage

with

Z D .~ ~ij l l i j; z D ( o ) = O, ~t j (O) = 0

Q = (C~O'+ax + ~ 3 ~ 2 -'k 7Jl+)";

Material Constants:

Do, Zo, ZI, Z2, Z3, nil, In_,, o~l

Ai ,A,.,rl,r,.,n,p,H,u,c~,~3,%wo

and elastic constants

In most.cases can set: rt = r2, A~ = A2, Zo = Z2

~(0) = Wo

(A6)