Creep Damage Assessment and Remaining Life Evaluation

International Journal of Fracture97: 125–135, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

Creep damage assessment and remaining life evaluation

I. LE MAY 1 and H.C. FURTADO2

1Metallurgical Consulting Services, Box 5006, Saskatoon, S7K 4E3, Canada2Centro de Pesquisas de Energia Elétrica, C. P. 2754, 20001-970, Rio de Janeiro, Brazil

Received 24 March 1997; accepted in final form 23 April 1998

Abstract. The approach of Kachanov and Rabotnov to the assessment of damage has been related to the creep ofcomponents at elevated temperature. The relation between the damage function and void formation is discussed,together with recent developments in which remaining life can be estimated using available materials constantsand strain rate measurements.

Key words: Creep, damage, remaining life, damage mechanics, metallography.

Nomenclature

A parameter indicating the number fraction of cavitated grain boundariesAcr critical value ofA at failureAt the value ofA at timetA′ material constantB material constant in the damage accumulation relation at timet = 0c constant to account for deficiencies in the Norton exponent and microstructural factors

associated with stress change during creepe true straine0 true strain att = 0k exponent in the damage accumulation relation at timet = 0K material constant in the Norton equationL m/(λ− 1)m exponent in the Norton equationn power law exponent in the Kachanov damage accumulation equationp material constant in the Norton equation modified for damager exponent in the Rabotnov relation for damage accumulation ratet timetb time for ‘brittle’ failuretr time to rupturetrem remaining lifeα a proportion of the difference between yield and ultimate stress valuesβ a factor depending on the level of initial stress with respect to the average stress at

ruptureδ [1+ α(σu/σy − 1)]ε engineering strain

126 I. Le May and H.C. Furtado

ε0 strain att = 0εr strain at fractureεs εmtrεm secondary (minimum) creep rateη (1+ r)/(1+ r − p)λ εr/εsσ stressσ0 stress att = 0σr average stress attrσt stress at timetσu ultimate tensile strengthσy yield stressσ = σrψ Kachanov damage parameterω Rabotnov damage parameter(= 1− ψ)ωcr critical value ofω for failureω0 initial rate of damage accumulation att = 0�p m+ p + c

1. Introduction

There is increasing interest in estimating the remaining life of components that have operatedat elevated temperature for times in excess of that used in design. Microstructural changes andcreep may have occurred. In order to estimate the remaining life an assessment is required ofthe accumulated damage.

Various methods are in use to estimate ‘damage’, but it is appropriate to review the ideasof Kachanov (1986). The premise was that materials suffer a loss in strength as a result ofexposure to stress, temperature and resulting deformation and the parameterψ decreases from1 at the outset to 0 at failure. Damage does not need a direct physical model, although itcan be represented by loss in cross-section from the formation of voids and cracks on grainboundaries. This causes an increase in the ‘true’ stressσt under constant load. The rate ofdamage accumulation can be considered as a function of the initial stressσ0 and the damageand, using a power law:

dψ/dt = −A′(σ0/ψ)n, (1)

whereA′ andn are constants depending on temperature, althoughA′ may also depend onaging of the material. Integrating,

ψn+1 = 1− A′(n+ l)σ n0 t. (2)

At failure,ψ = 0 andt = tr , hencetr is given by

tr = 1/A′(n+ 1)σ n0 . (3)

Rabotnov (1969) modified the approach using the parameterω = 1− ψ , so thatω = 0 atthe start andω = 1 at the end of life. In many ways this is more satisfactory in that ‘damage’

Creep damage assessment and remaining life evaluation127

ω may be thought of as a concept without a strict physical meaning (loss of cross-section) asis implied byψ .

2. The physical nature of creep damage and the creep cavitation model

There has been much emphasis on grain boundary cavity formation as providing an indic-ation of creep damage, following the pioneering work of McLean (1957). Voids are classi-fied as being of two types, namely wedge cracks originating at triple points, and round orelliptically-shaped cavities. Formation of the former is facilitated by grain boundary sliding,while there has been much analysis of the growth of the latter by diffusion, involving vacancycondensation at grain boundaries.

Metallographic methods have been used to determine the degree of void and microcrackformation in high temperature plant, the first systematic approach being that of Neubauerand Wedel (1983). Quantification has been made of cavity formation based on a model forconstrained cavity growth, the following relationship being predicted (Cane and Shammas,1984)

A = 1− (1− t/tr )(λ−1)/mλ, (4)

whereA is a parameter defined as the number fraction of cavitated grain boundaries, andλ = εr/εs, whereεr is the strain at rupture andεs is the secondary creep strain= εmtr (theMonkman–Grant constant),εm the minimum creep rate, andm is the exponent in the Nortonequation. The remaining lifetrem is

trem= t (tr/t − 1). (5)

From measured values ofA and (4) and (5)

trem= t{1/[1− (1− A)mλ/(λ−1)] − 1}. (6)

Taking conservative values ofm andλ as 3 and 1.5, respectively (Viswanathan, 1989)

trem= t{1/[1− (1− A)9] − 1}. (7)

When the remaining life becomes zerot = tr andA = 1.Various relationships between creep ‘damage’ω and theA-parameter have been derived,

the simplest procedure being to assume (Cane and Shammas, 1984; Viswanathan, 1989)

ω = A. (8)

Thus

ω/ωcr = A/Acr (9)

ωcr andAcr being the critical values at failure. These values are< 1, which is the limiting valueof ω orA, failure occurring when the loss of load carrying area produces an unacceptably highlocal strain rate, leading to fracture. Murakami et al. (1992) note that this relation can only be


considered valid when the cavity area fraction of grain boundary facets becomes large, that isin the last stages of life. Accordingly, they derived more precise relations betweenω andA.

For the situations where nucleation is complete on loading and for continuous nucleationthe relations derived are (Murakami et al., 1992)

ω/ωcr = (A/Acr)2 (nucleation att = 0) (10)

ω/ωcr = (A/Acr)2/(ε/εr) (continuous nucleation) (11)

whereε is the strain andεr is the strain at rupture. Equation (11) is likely to be more real-istic than either (9) or (10). Murakami et al. (1992) eliminated the creep strain from thisrelationship, as values ofε may not be available. The resulting expression is

(ω/ωcr){1− [1− (ω/ωcr)]L} = (A/Acr)2, (12)

whereL = m/(λ− 1). Equations (11) and (12) provide more rigorous relationships betweenω and theA-parameter. Murakami et al. (1992) point out that forL = 1, (12) reduces to (9),and forL = ∞, it reduces to (10). They note that a cavity growth mechanism of creep damageimplies a value ofL > 1, so the damage relationships fall between these two extremes. ForL→ 1, a ductile material is indicated, whileL→∞ implies a very brittle material.

Damage evolution is considered to be controlled by the relation,ω = 1− (1− t/tr )1/(1+r),and specifically the value of the material constantr. This is the same relation as shown forAfor constrained cavity growth in (4). Thus for constrained cavity growthA = ω as in (8). Thevalue ofr can be determined by fitting actual creep data curves. Leckie and Hayhurst (1974)derived the relation

r = [λm/(λ− 1)] − 1 (13)

andλ can be obtained from creep test data and the relation

λ = εr/εs = εr/εmtr . (14)

For remaining life estimation we can substitute the expression for damage evolution into (9),(10) and (12), and substitute forr from (13), to give

t/tr = 1− (1− A/Acr)λm/(λ−1), (15)

t/tr = 1− (1− A/Acr)2−λm/(λ−1), (16)

[1− (1− t/tr )(λ−1)λm][1− (1− t/tr )1/λ] = (A/Acr)2. (17)

2.1. PHYSICAL LIMITATIONS

Equations (15), (16) and (17) are plotted on the basis of life fraction versusA (Figure 1)with test data superimposed (Shammas, 1988). Note that the value ofA for CrMo steel is stillrelatively small(� 1) when the life is largely used up. Liu et al. (1994) note that the failurevaluesAcr for a number of materials can be 0.2 or less.


Figure 1. Lifetime fractiont/tr versus theA-parameter. The plots are of (15), (16) and (17), and are comparedwith test data from Shammas, 1988. After Liu et al. (1994).

Figure 2. Section adjacent to the fracture and at the surface of a CrMo tube that ruptured after 70,000 h at anominal temperature of 490◦C; (a) polished and etched to emphasize grain boundary cavities; (b) polished andetched carefully to show grain boundaries. Nital etch.

Figure 3. Plots of theA-parameter across the wall of the CrMo tube of Figure 2. The plots are heavy etching (A)and careful etching (B).


Figure 4. Microhardness across the wall of the material of Figure 2.

The methodology depends on observation of voids and determination ofA to estimate re-maining life. But, asAcr need not be large for many steels, void formation or surface crackingon base metal is not observed until very near the end of life (Westwood, 1994), and damageis often very local (Le May et al., 1994), there is much uncertainty. Also, the observation ofvoids and their apparent size depends greatly on polishing and etching procedures (da Silveiraand Le May, 1992; Samuels et al., 1992). The best procedures to avoid enlargement of cavitiesmay not disclose them, detection depending on their being enlarged. These factors make theA-parameter approach a dubious one.

To illustrate the uncertainties in computing damage based on measured values of theA-parameter, a section of CrMo steel tube that failed at a bend by longitudinal cracking after70,000 h at a nominal temperature of 490◦C is shown (Figure 2). A specimen adjacent to thefracture was polished and etched to emphasize the damage in the form of apparent voids (Fig-ure 2a): the same specimen was also polished and etched carefully to show grain boundaries(Figure 2b). TheA-parameter values across the wall differ for the two cases (Figure 3). Theyare dependent on preparation methods, and vary through the wall thickness.

Hardness measurements and, specifically, changes in hardness, can provide an indicationof exposure to temperature and may be correlated with damage (Viswanathan, 1989). Accord-ingly, microhardness measurements were also made across the wall of the CrMo tube: theyare shown in Figure 4. As with the computedA-parameter values, they vary greatly across thewall thickness. Both sets of data (Figures 3 and 4) suggest that damage is significantly greaternearer to the outside surface.

3. Practical approaches based on damage accumulation

3.1. THE APPROACH OF PENNY

Penny (1974; 1996) extended the Kachanov–Rabotnov (KR) approach. Integrating the Rabot-nov relationω = ω0(1− ω)r for temperatureT = constant, whereω0 = Bσk0 , B andk beingmaterial constants

(1− ω)(1+r) = 1− B(1+ r)σ k0 t. (18)


The stress at timet is

σt = σ0A0/At = σ0/(1− ω) (19)

takingω to represent an effective loss of area. Attr , ω = ωcr and the average stress isσr(≡ σ),reached betweenσy (yield) andσu (UTS). Thus

σ0/(1− ωcr) = σ = σy + α(σu − σy), (20)

whereα is a proportion aboveσy, and

σ = δσy, (21)

where

δ = [1+ α(σu/σy − 1)]. (22)

δ = 1 in the extreme case for nonhardening materials, but a reasonable estimate of a boundfor ductile materials has been given asδ = m/(m+ 1) (Penny, 1996). The failure condition isbounded bym/(m+ 1) 6 δ 6 σu/σy. In all casesωcr < 1. Substituting from (20) in (18),

tr = [1/B(1+ r)σ k0 ][1− (σ0/σ )(1+r)] (23)

or

tr = βtb, (24)

where

tb = [1/B(1+ r)σ k0 ] (25)

which is the result for ‘brittle’ failure, and

β = [1− (σ0/σ )(1+r)] (26)

β is a factor depending on the level ofσ0 with respect toσ (Figure 5).In Penny’s (1974) approach to creep strain versus damage, we substitute for(1− ω) from

(18) into the Norton equation in the form

ε = Kσm0 /(1− ω)p (27)

wherep is a material constant relating to microstructural damage. Integrating,

ε/ε0tb = η[1− (1− t/tb)1/η], (28)

where

η = (1+ r)/(1+ r − p). (29)


Figure 5. The modified Kachanov rupture curve. After Penny (1996).

Substituting (24) in (28),

ε/ε0tr = (η/β)[1− (1− βt/tr )1/η] (30)

At t = tr ,εr/ηε0tr = [1− (1− β)1/η/β]. (31)

The termε0tr can be considered as the Monkman–Grant constant, butε0 is not the minimumcreep rate in a creep test involving primary and tertiary creep (at least): it is the initial creeprate upon loading withσ0.

An alternative derivation (Penny, 1996) for strain variation provides a useful result. Fromconstant volume considerations

dLt/Lt = −dAt/At = −d(1− ω)/(1− ω) (32)

or

(1+ ε)(1− ω) = 1. (33)

Based on true straine = ln(1+ ε), and from (18), (24) and (25):

e = − ln[1− βt/tr ]1/(1+r). (34)

Differentiating,

[(tr/β)− t]de/dt = 1/(1+ r). (35)

This shows that remaining life times the current (true) strain rate is a constant. If strain ratecan be measured in service, the remaining life can be computed, asr can be determined fromcreep rupture tests andβ can be estimated, being∼ 1 at low stresses.


3.2. THE MATERIALS PROPERTIES COUNCIL(MPC) OMEGA METHOD

The omega method was developed from the KR approach to assess in-service components(Prager, 1995). Starting with the KR equation for strain rate in the form

e = e0(σ/σ0)m[1/(1− ω)p], (36)

wheree is the true strain, and substituting for(σ/σ0) = exp(e) from constant volume consid-erations,

e = e0 exp(me)[1/(1− ω)p]. (37)

This is not easily integrated, so is rewritten as an exponential function, noting that there arethree factors, namely increasing stress, increasing damage, and microstructural changes notrelated to damage. Hence,

e = e0 exp(me)[1/ exp(−pe)][1/ exp(−ce)], (38)

wherec is a constant that accounts for deficiencies in Norton’s exponent and other microstruc-tural factors associated with stress change. Thus,

e = e0 exp[(m+ p + c)e]. (39)

Integrating

[1/e(m+ p + c)]{1− exp[−(m+ p + c)e]} = t, (40)

e = −[1/(m+ p + c)] ln[1− e0(m+ p + c)t]. (41)

For values of(m+ p + c)e� 2 or 3, exp[−(m+ p + c)e] is negligible, thus at failure

1/e0(m+ p + c) = tr = 1/e0�p, (42)

where

�p = m+ p + c. (43)

From (39),

d ln e/de = m+ p + c = �p, (44)

�p may be determined by plotting lne versuse and taking the slope. From (40), at fracture

tr = (1/e�p)[1− exp(−�per)] (45)

and the remaining life is

tr − t = (1/e�p)[exp(−�pe)− exp(−�per)] (46)


Now,

e = e0 exp(�pe). (47)

Neglectinger in (46) and substituting from (47)

tr − t ∼= 1/e�p (48)

or

(tr − t)de/dt = 1/�p. (49)

As with Penny’s analysis ((35), there is constancy of the product of remaining life and (true)strain rate during service.

4. Concluding remarks

The uncertainties of computing damage based on measured values of theA-parameter havebeen discussed. It has been demonstrated that metallographic preparation is extremely im-portant in determining the value of theA-parameter in creep-damaged material. There is notnecessarily a ‘right’ or a ‘wrong’ procedure, but a standardized procedure to display creepdamage is required (Samuels et al., 1992). The apparent variation in creep damage throughthe wall of a steam tube is interesting, suggesting that observations at the outside surface mayoverestimate the creep damage in the bulk material.

It has been shown that Kachanov based procedures exist that can be used to estimateremaining life. The procedure of Penny (1974; 1996) and the later MPC one have strong simil-arities. Penny (1996) indicated how remaining life can be estimated by in-situ measurement ofthe bulk strain rate of a component, and substituting this and available materials constants in(35). Similarly, Prager (1995) discussed the application of the omega model by running testson damaged material samples to obtain initial strain rate and the rate of change of strain ratewith strain. This allows determination of�p and from this and an estimation (or measurement)of the strain rate in service, the remaining life is estimated. Thus, tools are available forremaining life estimation of creeping components on a reasonable and fundamental basis.The Kachanov based methods are also likely to provide better estimates of the damage in acomponent, particularly if the section thickness is large, than can be obtained with surfaceobservations alone. Combining them with microstructural evaluations provides an appropriateapproach to the evaluation of aging high temperature plant.

References

Cane, B.J. and Shammas, M.S. (1984).A Method for Remanent Life Estimation by Quantitative Assessment ofCreep Cavitation on Plant: Report TPRD/L/2645/N84.Central Electricity Generating Board, Leatherhead.

Da Silveira, T.L. and Le May, I. (1992). Effects of metallographic preparation procedures on creep damageassessment.Materials Characterization28, 75–85.

Kachanov, L.M. (1986).Introduction to Continuum Damage Mechanics.Martinus Nijhoff, Dordrecht.Liu, Y., Murakami, S. and Sugita, Y. (1994). Identification of creep damage variable from A-parameter by a

stochastic analysis.International Journal of Pressure Vessels and Piping59, 149–159.


Le May, I., Da Silveira, T.L. and Cheung-Mak, S.K.P. (1994). Uncertainties in the evaluation of high temperaturedamage in power stations and petrochemical plant.International Journal of Pressure Vessels and Piping59,335–343.

Leckie, F.A. and Hayhurst, D.R. (1974). Creep rupture of structures.Proceedings of the Royal Society, Series A340, 323–347.

McLean, D. (1957).Grain Boundaries in Metals.Oxford University Press, Oxford.Murakami, S., Liu, Y. and Sugita, Y. (1992). Interrelation between damage variables of continuum damage

mechanics and metallographic parameters in creep damage.International Journal of Damage Mechanics1,172–190.

Neubauer, B. and Wedel, V. (1983). Restlife estimation of creeping components by means of replicas.ASME Inter-national Conference on Advances in Life Prediction Methods(Edited by D.A. Woodford and T.R. Whitehead),American Society of Mechanical Engineers. New York, 353–356.

Penny, R.K. (1974). The usefulness of engineering damage parameters during creep.Metals and Materials8,278–283.

Penny, R.K. (1996). The use of damage concepts in component life assessment.International Journal of PressureVessels and Piping66, 263–280.

Prager, M. (1995). Development of the MPC omega method for life assessment in the creep range.Journal ofPressure Vessel Technology117, 95–103.

Rabotnov, Yu. N. (1969). A mechanism of a long time failure.Creep Problems in Structural Members.NorthHolland, Amsterdam.

Samuels, L.E., Coade, R.W. and Mann, S.D. (1992). Prepacking structures in a creep-ruptured low-carbonCr-Mo steel: their nature and detection by light microscopy and scanning electron microscopy.MaterialsCharacterization29, 343–363.

Shammas, M.S. (1988). Metallographic methods for predicting the remanent life of ferritic coarse-grained weldheat affected zones subject to creep cavitation.International Conference on Life Assessment. Electric PowerResearch Institute, Palo Alto, 238–244.

Viswanathan, R. (1989).Damage Mechanisms and Life Assessment of High Temperature Components. ASMInternational, Metals Park, Ohio.

Westwood, H.J. (1994). Applications of quantitative metallography in creep life assessment.Materials Perform-ance, Maintenance and Plant Life Assessment(Edited by I. Le May, P. Mayer, P.R. Roberge and V.S. Sastri).The Metallurgical Society of the Canadian Institute of Mining, Metallurgy and Petroleum, Montreal, 57–70.

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Creep Damage Assessment and Remaining Life Evaluation