modelling crack growth for creep and fatigue loading

Int. J. Pres. Ves. & Piping 50 (1992) 231-241

Modelling Crack Growth for Creep and Fatigue Loading

D. J. Smith & E. G. Ellison

Department of Mechanical Engineering, University of Bristol, Bristol, UK

A B S T R A C T

Estimates of creep crack growth in engineering components under steady load conditions are usually based on the application of fracture mechanics concepts. In particular the creep parameter C* has become widely used together with creep crack growth data obtained from laboratory tests. There are now a number of practical methods to utilise experimental data. For high temperature components, which are subjected to cyclic (fatigue) as well as creep loading, the estimation of the fracture mechanics parameters becomes much more difficult, and consequently the extent to which the growth of pre-existing cracks grow by creep and fatigue is difficult to quantify. In this paper the response of Type 316L stainless steel is examined. This material progressively strain hardens under reversed cyclic loading, and the creep behaviour also changes. Using uniaxial fatigue and creep results, fracture parameter maps are developed to establish the appropriate regimes for creep- fatigue crack growth. Using the maps a model is developed which can predict the combined effect of fatigue and creep on crack growth. The implications of the model are discussed in relation to the limitations of obtaining results from laboratory tests at short times, and the assessment of practical engineering components.

I N T R O D U C T I O N

In modern power plant operat ing at e levated temperatures , components can be working under condit ions when creep and fatigue processes are present. In addition to being able to predict the time

231 Int. J. Pres. Ves. & Piping 0308-0161/92/$05-00 ~ 1992 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland

232 D. J. Smith, E. G. Ellison

required to initiate a crack it is also essential that predictions of subsequent crack growth be available. This would enable the remaining life of components to be estimated when inspection during service operation reveals defects. In recent years there have been extensive investigations into the growth of cracks by creep.~-4 These studies have led to the widespread use of C* as a governing parameter for creep crack growth, though the vast majority of laboratory creep crack growth work has been under static load conditions.

However, in service the loading conditions may not be solely static, and an accurate and reliable calculation of C* may well be difficult since the material constitutive equations describing the creep and cyclic creep behaviour may change due to softening or hardening processes caused by the service load fluctuations.

In this paper the response of a Type 316L stainless steel at 550 °C when subjected to reversed continuous cyclic loading is examined. Using uniaxial fatigue and creep results, fracture parameter maps are developed to establish the appropriate regimes for creep-fat igue crack growth. Using the maps a model is developed which predicts the combined influence of creep and fatigue loading.

M A T E R I A L RESPONSE AND F R A C T U R E P A R A M E T E R MAP

Experimental work was carried out on Type 316L stainless steel. Fully reversed cyclic tests and uniaxial creep tests were carried out at 550 °C. From some of the specimens, which had been subjected to fully reversed continuous cyclic loads, uniaxial creep specimens were then extracted and subjected to loadings similar to the creep tests for the as-received material. The stabilised deformation behaviour for the continuously cycled tests are shown in Fig. 1, together with the stress-strain curve for the as-received material.

For monotonic loading of the as-received material, the total strain is described by:

(7 e = - + B o ( o ) r (1)

E

with B0 = 7.834 × 10 -26 and r = 10-42, with strain in absolute units and stress in MPa. The Young's Modulus at 550 °C is 150 GPa.

For the stabilised cyclic semi-stress, semi-strain curve, Fig. 1, the total semi-strain range is given by:

A e A o [ A o ~ r' - - - t- (2)

2 2E cr ~,~oo,/

Modelling crack growth for creep and fatigue loading 233

500

ZOO Stress or

semi -stress range, z~

2 MPa 300

20(

tO0

I I I i I I I I I

i

/1E= 150GPa Fit, r'= 5.0

ii I ! / . ~ ~experirnental

I I f cyclic. 100 cycles

i II . / . . . . . i . / / si?olation. N:I,

/ / / r '=5.0 _..

,>/ ._.2.-.----" /~ .~.~. . ' - -"-"- . _ .

. . . . 7 - i / s,mS:!!on. l . f expirirnental r--10.Z2

monotonic

I I I I

o:, o2 & 0.5 o7 o.s og Strain or semi-strain range A._~¢

2

Fig. 1. Deformation response of Type 316L stainless steel at 550 °C.

where r ~ = 5, a: = 5 x 10 -4 and o0 is 280 MPa. During cyclic hardening we would expect the material constant o0 and r 1 to be functions of the cycle number N. However , for convenience we assume that r 1 is independent of N, and only Oo is a function of N. A fit to the experimental results for hardening gives:

oo[N] = 156{1 - exp ( -0 -04N)} 4- 124 (3)

The predicated monotonic (N = 1) s t ress-strain curve using eqns (2) and (3) is shown in Fig. 1. As might be expected with r 1 < r the extent of hardening is predicted to be higher than observed experimentally.

An example of the creep response at s teady load is shown in Fig. 2 for the as-received and ha rdened material. For times up to about 400 h it is observed that the creep strains are similar. At longer times the strains and strain rates for the ha rdened material are significantly lower. From creep tests over a range of stresses on the as-received material at 550 °C the creep strain can be described by

ec = A o " t u(p+l) + Bo~t (4)

where A = 2-92 x 10 -14, p = 1-374, m = 4.18, B -- 5.29 x 10 -26 and n = 8-2. The creep behaviour of the ha rdened material could be described


Creep strain

%

Fig. 2.

(23

0.2

OA

stress t75 o"

time

Stress t75 MPa

• As received

0 ~Cyclically strain C3J hardened

0 ~ I I [ I I I I ,ooo ~ooo doo doo 5o00

Time, hours

Comparison of creep strain from as-received and hardened Type 316L stainless steel at 550 °C.

by the first term of the right hand side (the primary creep term) of eqn (4), and it is assumed that secondary creep did not take place.

To develop the fracture parameter maps we begin by establishing simplified estimates of the fracture mechanics parameters for a crack in an infinite plate subjected to a remote stress tr. In the elastic regime the stress intensity factor K is appropriate, for the plastic regime the plastic J-integral Jp, for primary creep and hardening creep integral C~', and finally for secondary creep the parameter C* is appropriate. The characteristic time from elastic-plastic (ep) to primary creep (pr) l'z is:

1 tep,pr- (m + 1------~ (J/C*)~+P (5)

where J = KZ/E + Jp (6)

At low stresses when Jp << KZ/E the characteristic time is tep,p r.

The characteristic time from primary (pr) to secondary creep (s) is: 1'2

tpr,s = {C~/(1 + p ) C * } (p+wv (7)

The characteristic times are shown in Fig. 3 for different remote stresses. For static loading and for stresses below the yield stress, Fig. 3(a), the characteristic time increases with decreasing stress. At stresses above the yield stress, the characteristic time increases with increasing


300

20O

I00

V3

50

3CI01 a)

400

300

200

I00

Fig. 3.

Compact tension

% \

\ \

\ \

Jp \ \ r = 5 \

\

\

I0.4

C ~

Je

I I tO 2 103

N= 100 \

Jp N = 100

(a)

I I I

10 k 105 106 Time (hours)

2O

10

#o." N = I

20 10 5 N = I

c'g

Je

301~ t a I I I = I I I0 0 tO 1 10 2 10 3 104 105 106

Time, hours (b)

Fracture mechanics parameter map for Type 316L stainless steel at 550 °C; (a) as-received, (b) during cyclic loading.


stress. This effect at high stress is a consequence of the power law stress exponent for plasticity being greater than the stress exponent for primary or secondary creep i.e. r > n > m. The implications of this are discussed later. In contrast, if the cylic stress exponent r I is used to describe the monotonic plasticity, the extent of the region dominated by Jp is reduced, Fig. 3(a).

The parameter map for cyclic loading is shown in Fig. 3(b). As hardening progresses the plasticity regime is reduced and the transition time to primary creep is shortened. In the hardened material it was shown earlier that creep was principally dominated by a primary component, and secondary creep did not take place. The hatched line in Fig. 3(b) for the transition from primary to secondary in the hardened material is shown to indicate a region where it is assumed that no further creep occurs beyond that boundary.

MODELLING CRACK GROWTH

In the following, fatigue and creep crack growth is modelled as two independent processes. The model assumes that fully reversed cyclic loading results only in fatigue cracking, while a dwell period at maximum load leads to crack growth by creep. The load cycle is assumed to be load controlled, with the near crack tip stress and strain fields sufficiently contained such that these are subjected to displacement control from the surrounding elastic field. In the case of cyclic loading this results in local cyclic hardening behaviour, and during the dwell local stress relaxation from the peak stress occurs with the near crack tip stress field described by the parameter C[t] described later.

Reversed cyclic fatigue

To predict crack growth by fatigue the model developed by Wareing 5 is used. The crack growth per cycle is:

dNda _aAEp{SeC(JrOmax~_\__~O~ / 1} (8)

where AEp is the total plastic strain range, and Omax is the peak tensile stress and o. the ultimate stress, which for Type 316L stainless steel at 550 °C is 375 MPa. In the following fatigue crack growth for fully reversed cyclic loading under load control is assumed to be governed by the tensile part of the cycle. During cyclic loading as the material hardens the plastic strain range decreases, and the increment of crack


Fig. 4.

I0 ~"

tOC ~ AK=53MPa

30

Iu

£

o

"a

"6 I1:

I

I0 100 Number of cycles, N

Predicted fatigue crack growth rates in Type 316L stainless steel.

growth per cycle decreases. The ratio of da /dN at a given cycle to that for the fully hardened material is shown in Fig. 4.

Creep

If a dwell period, th, is introduced at the peak tensile stress, crack growth by creep would occur. At high stresses Fig. 3(a) illustrates the crack growth which may occur with Jp dominating the behaviour in the as-received material. When the response is predominantly elastic-creep, crack growth by creep in the as-received material can be expressed 8 by:

A a = a d t = D C [ t ] ~' d t (9)

where D and ~ are experimental constants which, for Type 316L stainless steel at 550 °C, are 5 x 10 -4 and 0.76 respectively, 7 with units of ti in m/h and C[t] in MPa/m2h. Careful analysis of the creep crack growth data from Ref. 7 indicates that these constants may contain an

238 D.J. Smith, E. G. EUison

element of plasticity as well as pure creep. For the moment we shall assume that they are valid creep constants.

The fracture mechanics parameter C[t] is given by: 2"8

J C~t -p/(,+o C[ t ] -I + C* (10)

(m + 1)t (1 + p )

Figure 5 illustrates creep crack growth increments Aa, for a constant stress of 100MPa applied to the infinite centre cracked plate (see Appendix) containing a crack of length 2a = 20 mm, such that K m a x =

17.7 MPaV~. Crack growth is initially rapid. This is because of the strong influence of initial rapid redistribution and primary creep. When the material hardens through cyclic loading the extent of crack growth diminishes because the redistribution time from elastic to primary creep is shorter. The first term on the right hand side of eqn (10) is no longer dominant at short times for the hardened material compared with the as-received material.

Creep and fatigue

A simple model for creep and fatigue crack growth is:

d ~ [Total] + [Creep] da Aa

= ~ [Fatigue] dN (11)

Fig. 5.

E E

o

i j

0.1

15 3

i i

Kmax= IZ 7 MPa Fatigue cycles

N = l

N = 5 N~ 100

2'o0 660 86o ,ooo Hold time, hours

Predicted creep crack growth at constant load following cyclic loading.

Fig. 6.


i0-2

td 3

E lg 4 E ~f

# (j 5

,~ 16e

#d 7

~6 8

Cycle No. N = I N ~ 100 'stabilised)

' / . / / I / i I th I " 1 / / / " ,oo11//k'

17,o // I/ /'/'hour, // // /./ / /

, / ; / /

/ K D ~ I 140 : aD ~J~O a o = lOmrn

aO=2e -yy

10 I00 Krnax , MPa

Predicted fatigue and creep crack growth rate per cycle in Type 316L stainless steel at 550 °C.

Figure 6 illustrates each component on the RHS of eqn (11). The fatigue component d a / d N is shown for the as-received (N = 1) and the hardened material (N = 100). Also shown is the increment Aa per cycle for hold times t~ = 10 and 100 h for the as-received material. These results when compared with the fatigue crack growth indicate that the creep crack increment is, in general, substantially higher.

DISCUSSION

The combination of the fatigue and creep crack growth models described above suggest that at Kmax = 25 MPaX/-m, crack growth by creep is dominant , and there is no creep and fatigue interaction.

For initial crack sizes of 2a = 20 mm, and at a design stress of ]cry (with ay = 120 MPa) the maximum stress intensity factor at the design condition is about 9 M P a V ~ . Crack growth is predicted to be dominated by the creep component . For cyclic creep loading with a dwell period of 100 h, where it is assumed that primary creep is regenerated at each cycle, and integrating eqn (9), the extent of crack growth for the lifetime of 20 years from the initial defect size of 20 mm is about 2 mm. If the dwell period is reduced to 10 h and again for a lifetime of 20 years, the crack extension 2Aa is about 10 mm.


The above has been estimated on the basis that load levels are below try. In the region where plasticity dominates the condition r > m or n indicates that the stress field characterising creep crack growth will theoretically be dominated by Jp and not the creep parameter C~ and C*. Therefore, for fracture mechanics tests at reference stresses above yield, it would not be appropriate to correlate creep crack growth rates with the measured creep parameter C* (estimated from the load-line displacement rates). Consequently there is concern regarding the generation of creep crack growth data in regimes outside regions pertinent to component operating conditions. It is not clear whether it is acceptable to extrapolate creep crack growth rates which are generated at high stresses. The fracture mechanics parameter map for the as-received material, Fig. 3(a), illustrates that at stresses below yield C~ and C* are the appropriate parameters, but only at very long times.

For cyclic loading the influence of hardening on the fracture mechanics parameter map, Fig. 3(b), has been accounted for in an approximate manner. With an increase in hardening the C* and C* regimes extend to high stresses. However, it is apparent from the uniaxial tests that creep is reduced with an increase in hardening. Hence creep crack growth is reduced.

CONCLUSIONS

Maps have been developed for Type 316L stainless steel at 550°C which define the regimes for different fracture mechanics parameters to describe cracking behaviour. The maps have been drawn for the as-received material and for conditions when strain hardening occurs. In the former case the map has shown that it appears inappropriate to use the creep parameters C~' and C* at stresses about yield. With hardening the C~ and C* regimes extend to higher stresses, but the extent of creep is limited. Models for fatigue and creep crack growth have been adopted to provide predictions for conditions near to design (approximately 23Oy). The results have shown that crack growth is dominated by creep even for short dwells (<10 hours).

REFERENCES

1. Ellison, E. G., Musicco, G. G. & Pineau, A., State of the art report on fracture mechanics--fracture in the creep regime. Final Report CEC report RAP-053-UK, 1985.


2. Riedel, H., Fracture at High Temperatures. Springer-Verlag, Berlin, 1987. 3. Nikbin, K. M., Smith, D. J. & Webster, G. A., Jnl of Engineering

Materials Technology, 108 (1986) 106-91. 4. Leung, C.-P. McDowell, D. L. & Saxena, A., Jnl of Testing and

Evaluation, 18(1) (1990) 25-37. 5. Wareing, J., In Fatigue at High Temperatures, ed. R. P. Skelton. Applied

Science Publishers, Chapter 4. 6. He, M. Y. & Hutchinson, J. W., In Elastic-Plastic Fracture, ed. C. F. Shih

& J. P. Gudas. ASTM 803, 1983, vol. 1,277-90. 7. Huthmann, H., Private communication 1989. 8. Smith, D. J., Journal of Strain Analysis, 26 (1991) 111-22.

APPENDIX

For plane stress, estimates of the fracture mechanics parameters for an infinite centre-cracked plate (CCP) subjected to a remote stress o are: 6

K = o V r ~ (A1)

Jp = OEpJga ~//r (A2)

C~ = A a" +'zayr-m (A3)

C* = Bo"+'staVn (A4)

modelling crack growth for creep and fatigue loading

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