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Approximate inelastic analysis of defective components
Nuclear Engineering and Design 133 (1992) 513-523 North-Holland
513
Approximate inelastic analysis of defective components R . A . A i n s w o r t h a n d P.J. B u d d e n
Nuclear Electric plc, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire GL13 9PB, United Kingdom Received 12 August 1991
In the inelastic analysis of defective components at high temperature, the crack tip characterising parameters are of interest for a range of loading conditions. In this paper, four situations are examined: (1) steady state creep under constant load; (2) the transitional behaviour under constant load prior to attainment of steady state conditions; (3) behaviour under displacement control; and (4) behaviour under displacement-controlled cycling with hold periods. For steady state creep, it is shown by comparison with finite-element solutions and experimental data that a reference stress approximation may be used to predict the parameter C * with reasonable accuracy. The transitional period prior to steady-state conditions is examined by finite-element calculations, and the time-dependent parameter C(t) [C(t)--, C* as t--,oo] is evaluated for a number of geometries under constant load. The results are predicted by extending the steady state reference stress approximation. Comparisons are made between C(t) and the C~ parameter of Saxena. Finite-element results are also reported for geometries creeping under a constant applied displacement. Under this loading, the parameter C(t) falls rapidly with time. The reference stress method is used to predict the rate at which the applied loading falls, and then to estimate the value of C(t). It is shown that the amount of crack growth associated with displacement-controlled loading is much less than under load control at the initial load. A simplified estimate is made of the total crack growth during a period of constant displacement, and compared with the experimental data for steels of both high and low creep ductility. Under displacement-controlled cycling, finite-element calculations involving hold periods would be extremely time consuming. It is shown that the simplified results developed for constant-displacement loading may also be applied to predict the experimental behaviour under cyclic loading provided the material creep data used in the estimates are obtained under relevant cyclically hardened or softened conditions.
I. Introduction For constant loading, developments in the understanding of defect behaviour at elevated temperature have been sufficient for Ainsworth et al. [1], Saxena [2] and Webster et al. [3] to produce procedures for assessing the failure of defective components. In such approaches, creep crack growth under steady state loading is evaluated by calculating the crack tip parameter, C * , which governs the amplitude of the near-tip stress and strain rate fields (e.g., Riedel [4]). C * may be evaluated by finite-element methods but the procedure of ref. [1] also embodies a reference stress approximation that enables simplified estimates to be made for a wide range of complex geometries, defect sizes and material laws. For component applications, it is often desirable to be able to predict crack growth under complex loading conditions. The finite-element method is a convenient approach for the examination of crack-tip parameters E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
under transient load conditions for simple geometries and material laws. In this paper, such analyses in conjunction with experimental data are used to provide guidance on the extension of approximate reference stress methods to these more complex loadings.
2. Steady state creep under constant load For materials in which creep strain rate ~c is related to stress ~ by the simple power law
~ = A o -n,
(I)
where A, n are constants, the stress and strain rate fields near a stationary crack tip in a body deforming in steady state creep are of the type derived by Hutchinson [5], and Rice and Rosengren [6]. The amplitude of these fields is described by the parameter C * , which may be evaluated from experimental measurements of
R.A. Ainsworth and P.,L Budden / Approximate inelastic analysis
514
the displacement rates in cracked test specimens and by finite-element methods. The analogy between power-law creep described by cq. (1) and power-law plasticity, enables solutions for J developed in handbook form by Kumar et al. [7] to be used to calculate C* for a range of geometries and crack sizes. An alternative approach to detailed finite-clement analysis is the reference stress method, which has been used by Ainsworth [8,9] to provide approximate estimates of both J and C*. The approximation in the creep case is cr*~, -
+ '. ,+r,.,+fE,.~,.R
(2)
where the reference stress Or++ is defined by Oref = PO'y/PL(O" v , a )
(3)
and R' is a length parameter: R' = K
:/~:~,.
(4)
Here P is the magnitude of the applied loading and Pt+ is the corresponding collapse load for a yield stress o-v and crack size, a; K is the linear elastic stress intensity factor, and ~ f is the creep strain rate from uniaxial creep data at the reference stress level. Miller and Ainsworth [10] have compared the approximation of eq. (2) with the detailed handbook solutions of ref. [7]. A typical result is shown in fig. 1 for a cylinder in tension with an external fully circumferential defect for a range of crack depths and stress indices, n. The results are presented in terms of a load factor, which is the ratio of the loads at which the
LOAD FACTOR
reference stress approximation and the handbook solution give the same value of C*: if the load factor is less than unity then the reference stress approximation is conservative relative to the handbook solution. It is apparent that the load factor is close to unity in this case and from an examination of a wide range of geometries, Miller and Ainsworth found that the reference stress approximation closely reproduced the finite-element results of ref. [7] in general, and on average corresponded to a conservatism of about 5% on load. The reference stress approach is particularly useful for practical applications because: (l) the only calculations required in eq. (2) are the stress intensity factor and the limit load, and such solutions are widely available (e.g., Miller [11]) because of their use in low-temperature fracture procedures such as R6 [12]. (2) Equation (2) is not restricted to secondary creep described by a power law but allows realistic creep data including primary, secondary and tertiary stages to be included in the creep strain rate at the reference stress. (3) Strain hardening rules may be used to allow for the increasing stress levels as a crack grows, by interpreting Ere "c f as the creep strain rate at the current reference stress and the creep strain accumulated under the reference stress history. The validity of using realistic creep laws and applying strain hardening rules has been checked for a range of materials by comparing eq. (2) with the test speci-
L
E X T E R N A LDEFECT Q/w=
1,OL /
%
1.2 |
O-8 / ~
3/4
0.6 0-4 0.2 n
0
3m
5i
1,0
j 16
Fig. l. Load factor for an externally circumferentially cracked cylinder in tension.
R.A. Ainsworth and P.J. Budden /Approximate inelastic analysis
515
the computed results are compared with approximate estimates. The analysis for a compact tension specimen is described in some detail to illustrate the mesh detail and numerical approach adopted.
10-3 A I .C
3.1. Finite-element analysis o f a compact tension specimen
10 "~
E 6
n
lo-s
10-6
10 . 7
10-7
I
I
I
I
10-6
lO-S
10-~
10-3
C:xp ( MPcl mh "1 ) Fig. 2. Comparison of value of C* predicted by eq. (2), Cref,* with that deduced from experimental data, Cexo, for a C-Mn weld metal under constant load.
men data for which C* can be estimated from experimental displacement rates [1]. Figure 2 shows a typical comparison for a C - M n weld metal for which creep data have been fitted by the 0-parameter approach of Evans et al. [13]. If secondary creep strain rates had been used to estimate C* then the experimental values of C* would have been significantly underestimated whereas, using the full creep curve, the agreement is much better and the predictions are conservative. Thus, for steady state creep under a constant load, reference stress estimates can be readily applied and have been validated both by comparison with finite-element results and experimental data. In the following sections the extension of the approach to more complex situations is considered and again both numerical and experimental validation is sought.
3. Transitional behaviour under constant load Riedel and Rice [14] have shown that during the transition from initial loading to steady state creep, the near-tip stress and strain-rate fields are still of the type described by Hutchinson, Rice and Rosengren [5, 6] but have an amplitude C(t) that depends on time, t, and approaches C* as t ~ oo. In this section the finite-element program BERSAFE [15] is used to produce results for C(t) for a number of geometries and
A specimen of crack length to specimen width ratio a / w = 0.5 is analysed under plane strain conditions. Due to symmetry, only half of the specimen is modelled in the finite-element mesh shown in fig. 3. A constant unit point load per unit thickness is applied at the node corresponding to the centre of the pinhole, which is not modelled explicitly. Quadratic isoparametric mid-side node elements are used in the mesh, which contains 146 elements (triangles and quadrilaterals) and 467 nodes. The mesh construction is radial near the crack tip enabling the radial variation of the field quantities to be easily discerned. The innermost core of elements near the crack tip comprises triangles of radial size a/200. The stress intensity factor K was calculated on initial loading using both the virtual-crack extension (VCE) facility automated into BERSAFE and the J-integral method via post-processor contour-integral rou-
Ca)
½62 ~1/261 8
7
6
543 I
~
el=w/2
W
(b)
(c)
3 2 1 Fig. 3. Finite-element mesh of compact tension specimen showing the whole mesh, (a), and near-crack tip details, (b) and (c). The semicircular contours 1-8 are used for contour integral evaluation.
516
R.A. Ainsworth and P.J. Budden / A p p r o x i m a t e inelastic' analvsis
Table 1 Load-line and mouth opening displacements for CT specimen
tines. Over the eight contours shown in fig. 3, the scatter in J-values was 1.6% of the mean value, which corresponds to a stress intensity factor 1.4% lower than that given in the handbook of Tada et al. [16]. The V C E solution was within 0.1% of the handbook value. Although the mesh does not model the loading arrangement exactly, it is of interest to compare the computed and handbook [16] elastic load-line and crack mouth opening displacements, 6 2 and ~ respectively (see fig. 3). These are listed in Table 1. There is agreement to within 3% and 7%, respectively. Following initial loading, creep analysis was performed for the creep law of eq. (1) with n = 5 up to a dimensionless time r = 1 where ~" = E ' C * t / K
Displacement Mouth BERSAFEelastic~l,(~e(mm) Tada et al. [16] ~i, 62 (mm)
2.289x10 -4 1.655x10 4 2.457x 10 -4 1.706x l0 4
BERSAFE creep gt, 62 (mm/h) 6.356x 10 -t° 4.672x 10 ]0 Kumar et al. [7] 6t, 62 (mm/h)
5.996x t0- 11} 4.402 x 10- ii)
than the computed figures. These differences are not inconsistent with the elastic discrepancies noted above. B E R S A F E allows various J-integral parameters to be evaluated and the results are presented here for the mean values on the eight contours in fig. 3 of integrals denoted by Jw and Jw*. Jw is the R i c e - E s h e l b y definition of J; Jw* is defined as the limiting value of Jw I las the contour F shrinks onto the crack tip but is evaluated on other contours in conjunction with surface integrals defined by Green's theorem, and hence, to within numerical accuracy, is path independent by definition. The two integrals are the same when J~, is path independent. The scatter in the computed Jw and Jw* values about the mean was small, varying from 4
(5)
2.
Load line
H e r e E ' = E / ( 1 - u 2) where E is the Young's modulus and u is the Poisson's ratio. The finite-element results suggest that at time ~"= 1 the steady state has essentially been reached in the highly loaded areas. Hence, by differencing the computed nodal displacements, the steady state load line displacement and crack mouth opening displacement rates, 62 and 61 respectively, are obtained. The computed figures and the results of Kumar et al. [7] are listed in table 1. The latter figures are about 6% lower
3 E1j K2 Slope
E1Jw
/
0.99
~
. S ope o97
- ~ - = 1*%
1
,25
I
.5
I
.75
J
I
I
1.0 1.25 1.5 1.75 Non-climensionQI Time "[
I
2
I
2.25
I
2.5
Fig. 4. Variation of computed J-integral with time under both load and displacement controls,
I
2.75
l
3
R.A. Ainsworth and P.J. Budden /Approximate inelastic analysis
71
and 1.5% at z = 0 . 1 to 6 and 1%, respectively, at ~-= 1.0. The variations of the mean values with time are shown in fig. 4 and can be approximately described by E ' J / g 2 = 1 + ~'.
517
6
(6)
The value of C* used in the normalisation of eq. (5) has been taken from ref. [7] and hence the present results at r -- 1 are consistent with [7] as d J / d t --* C* as t ~ . BERSAFE computes contour integral approximations to C(t) using the J-integral routines following conversion of displacements and strains/strain energy to their respective rates of change by differencing output from successive time steps. Mean values of integrals denoted by Cw and C*, which are the results of using the Jw and J * algorithms on the rate output, are plotted in fig. 5. At short times Cw would not be expected to be path independent and, for example, was found to vary by 24% from its mean value at r = 0.1. Conversely, the scatter in Cw* was much less, ranging from 1.4% of the mean at ~"= 0.1 to 0.8% at r = 1.0. Some of the scatter at low z will be because of the differencing of variables at discrete time steps in a rapidly varying stress field to obtain rates. Because of
5C ('l:)/C ° Ct/C* 4
3
2
Ct / C"
o
I ,1
I ,2
I .3 "r
I .4
I .5
Fig. 6. Comparison of Ct and C(t) parameters for compact tension specimen under constant load.
the fine mesh detail near the crack tip, it was also possible to estimate C(t) by stress and displacement substitution methods. Displacement substitution was found to work better, particularly when the dominant displacement component along a particular ray from the crack tip was considered, and confirmed the accuracy of the C* algorithm. Saxena [2] has proposed the use of a parameter C t for correlating creep crack growth in the transient regime. Saxena and Liaw [17] have shown that C t may be estimated from the load-line displacement rate as
C(~) C °
F' P ( - ~ - [ ~ e ( t ) - ~ 2 ( o o ) ] +~7/~e(~)), c , = ~-w
C °
C
1.0 0
I .I
I .2
I .3
I .4
I .5
"t
Fig. 5. Compact tension values of C(t) under constant load.
(7)
where r/, F ' and F are geometry-dependent factors, P is the load and B is the thickness. Substituting values for the compact tension specimen, eq. (7) has been evaluated using the finite-element displacement values and the results are shown in fig. 6 and compared with C(t). Leung et al. [18] have similarly compared C t and C(t) and also found Ct << C(t) for ~"<< 1. Indeed Saxena [2] showed that C t does not have the 1 / t dependence on time that C(t) has as t ---,0, and the numerical results confirm that the displacement rates at short times are insufficient to define C(t) by simple expressions such as eq. (7).
R.A. Ainsworth and P.J. Budden / Approximate inelastic analysis
518
3.2. Estimates of C(t)
Ehlers and Riedel [19] suggested that C(t) may be estimated approximately as the sum of the short- and long-time limits, that is C(T)=C*
1+ (n+l)r
0
(8)
Ainsworth and Budden [20] derived a new estimate of C(r): (1 + r ) "+1
) (9)
C(r) =C* (1 + r ) " + J -
1
by using the J-estimate of eq. (6). Both eqs. (8) and (9) reduce to C(r) ~ C* and C(r) ~ C * / ( n + 1)r in the limits r ~ oo and r ~ 0, respectively, as derived by Riedel and Rice [14]. The finite-element results for the compact tension specimen and preliminary results for a single-edge notched tension specimen under tension and bending are compared with eqs. (8) and (9) in fig. 7. The results for the single-edge notched specimen have been normalised using the values of C* given by
Shih and Needleman [21]. It is apparent that when the finite-element results are normalised as in fig. 7, the values of C(t) can be reasonably approximated by cither eq. (8) or eq. (9), independently of the geometry. Although apparently morc complex than eq. (8), cq. (9) is more convenient to integrate and results in thc approximate estimate [22]
~1'C(t) n/(,,+l)d t = c,n/(,,+l)t[ 1 +~rcf/Eeref(t)]. (10) The integral in eq. (10) is proportional to the creep strain near the crack tip and eq. (10) shows that this creep strain is increased above that which would have been accumulated under steady state creep by the factor [1 + elastic strain at the reference stress/creep strain at the reference stress]. Ainsworth [22] has shown how this factor may be used to incorporate the effects of transitional creep on both crack initiation and growth. If the creep strains greatly exceed the elastic strains, then steady state solutions are applicable and the C* estimate of section 2 may be used without consideration of the redistribution effects.
30
C(~) C•
I 2.0
_-_ __ Eqn (8) 10
Eqn ( 9 ) I
I
05
1.0
I
T,
D
I 5
Fig. 7. Comparison of finite-element solutions for C(t) for a n u m b e r of specimens with the predictions of eqs. (8) and (9). All
solutions for n = 5.
R.A. Ainsworth and P.J. Budden / Approximateinelasticanalysis
(~)~-I
4. Creep under displacement control The finite-element analysis of a compact tension specimen described in section 3 was continued under fixed load-point displacement for times 1 ~
dref = -E~rCeJU,
4
-
1)] 1/4.
~
//
~ E Q N 3
(11)
= [1 + 2.4(7
FINITE
ELEMENT ~ / /
(12)
~/
2
,/
where p. is a factor that can be estimated from the elastic and creep compliances and p. = 2 for the compact tension specimen. Integrating eq. (11) for the creep laws of eq. (1) and inserting the constants used in the finite-element analysis predicts the load variation
P('r)/P(1)
519
"g
(12)
Fig. 8. Load variation for a compact tension specimen under constant-displacement creep.
Equation (12) is plotted in Fig. 8 in the form of [P(1)/P(r)] 4 - 1 against z, compared with the finiteelement results. From eq. (12) a straight line of slope 2.4 would be predicted. The finite-element results produce a straight line but with a slope of 2.55, which corresponds to a slightly more rapid fall in load. Ainsworth and Budden [20] also derived an estimate of J under constant applied displacement, assuming J to be path independent. For the particular case n = 5
and constant displacement from r = 1, the result reduces to
E'J(-r) K2
P(-r)( P(1)
P('r)) 3 - p~
(13) .
%
C(~) C*
\\
C
\
Estimate of [201
\\~//
. . . . . . . . . . . . . ~,
I I
1 ~.2
1 1.4
I 16
1 ~8
I 2
I 2.2
I 2~
I 2.6
l 28
Fig. 9. Varation of creep integrals, C(t), for a compact tension specimen under constant displacement.
I
3O
R.A. Ainsworth and P.J. Budden / Approximate inelastic analysis
520
Equation (13) is shown in fig. 4 and compared with the J-values from both the Jw and J* algorithms. The computed Jw values showed a greater path dependence than under load control with variations of 10% and 20% about the mean at r = 1.12, r = 3, respectively. However, the mean value of Jw follows eq. (13) well, with an error of 3.5% at r = 3. The mean values of the creep integrals C w and C* are shown in fig. 9 together with an approximation given in ref. [20] based on the J-estimate of eq. (13). The approximation predicts a rapid reduction in C(t), and the finite-element results confirm this rapid reduction, although falling more rapidly than predicted. The scatter in the computed integrals was found to be greater than in the case of a constant load, varying from 33 and 1.8% of the mean values at r = 1.12, for C w and C* respectively, to 17 and 2.6% at r = 3. The underestimation of the predicted value by the finiteelement analysis is qualitatively consistent with the underestimation of the load P ( r ) relative to eq. (12). Ainsworth and Budden [23] used the reference stress approximation of eq. (11) to derive an estimate of the total crack growth during a period of constant displacement from the load reduction Ap. Provided the crack growth is small so that the change in elastic compliance does not lead to a significant load reduction, the result is
Aa =A'Co *q ~
Eeref
where A', q are constants in the creep crack growth law a = A ' C *q, and the subscripts or superscripts "o" denote values at the start of the displacement hold. Where there is significant crack growth, the effect of the change in compliance can be included in eq. (14) using the result given by Ainsworth [24]. Equation (14) predicts that the crack extension is simply related to the reduction in load and is significantly less than the growth under a constant load Po. Crack growth data under displacement control have been presented by Neate [25] and by Nikbin and Webster [26]. Neate tested an untempered coarse-grained bainitic 0.5 CrMoV steel at 565°C. Uniaxial creep data have been fitted by a simple primary creep curve to evaluate ZIP via eq. (11). The corresponding predictions of eq. (14) are compared with the experimental data, which are taken from specimens of two thicknesses, in table 2. Nikbin and Webster [26] present data for a 2.25 CrlMo steel at 538°C and a low-ductility 0.5 CrMoV steel at 565°C. In the absence of uniaxial creep data, the experimentally measured values of
Table 2 Crack growth under displacement control Material
½CrMoV [25] for a range of initial loads
Experimental Aa (mm) 1.5- 4.0 ~t 7.0- 8.0 ~' 5.0- 8.0 ~' 12 -17 " 17 19 ~'
Predicted da (mm) eq. (14) 2.8 5 8 13 22
2.25Cr1Mo [26]
(1.4
0.22
0.5 Cr 0.5 Mo 0.25 V [26]
5
9
a Range of values from nominally identical tests.
load drop have been used in eq. (14) to obtain the predicted values of crack growth, which are compared with the experimental data in table 2. It is apparent from table 2 that the simple prediction of eq. (14) provides a reasonable prediction of crack growth under complex loading conditions for a range of materials.
5. Creep under displacement-controlled cycling Displacement-controlled cycling is a loading regime of practical importance, arising under conditions where displacements are imposed and where displacements are induced by thermal bending, for example. Finiteelement analyses to evaluate C(t) under such conditions would be extremely time consuming particularly given the complex materials behaviour under cyclic loading. Therefore, this loading condition has been examined by a number of authors [27,28] by performing displacement-controlled tests on compact tension specimens with creep occurring during hold periods. The values of C* during the hold periods have been evaluated in refs. [27,28] from experimental values of the load drop rate. These are compared with the predictions of eq. (2) in figs. 10 and 11 for two austenitic steels. In the reference stress approximation, creep data from material in the cyclically conditioned state have been used; had forward creep data from virgin material been used the good agreement in figs. 10 and 11 would not have been obtained. It can be seen from figs. 10 and 11 that straightforward use of the reference stress approximation, developed for constant load, allows C* to be predicted under these complex loading conditions.
R.A. Ainsworth and P.J. Budden / Approximate inelastic analysis
521
Acknowledgement
As well as monitoring the load drop to evaluate C*, Gladwin et al. [27,28] made detailed measurements of crack growth rates during the dwell periods. They found that the crack growth rate data correlated with C* in a similar manner to data collected under constant load. As a consequence it would be expected that eq. (14) could be used to predict the total crack growth during a displacement-controlled hold period just as it has been used to predict growth under constant displacement. This has been confirmed by analysis of the data on the 347 weld metal by Ainsworth and Budden [23]. Experimental data, therefore, suggest that the reference stress methods developed for constant load and constant displacement can be extended in an intuitive manner to predict creep crack growth under cyclic conditions. Of course, in the latter case it is also necessary to allow for fatigue crack growth due to the load changes in order to calculate the total crack growth per cycle. The manner in which this may be included has been described by Ainsworth [24].
This paper is published by permission of Nuclear Electric plc.
Nomenclature crack size constant in creep law of eq. (1) constant in crack growth rate law in eq. (14) specimen thickness steady state creep characterising parameter transient creep characterising parameter finite-element approximations to C(t) parameter of Saxena given by eq. (7) Young's modulus E/(1 v 2) functions used in C, estimate of eq. (7) elastic-plastic characterising parameter finite-element approximations to J
a A A' B C* C(t) C w, C* Ct E E' F, F ' J Jw, J*
-
10-3
A
V
h 0 Q.
ov
~7~ v/~
lO-S
y
(J
--- Key a V
/ Z ~ V
10-6
•
10 -7 10 -7
I 10-5
24h 96h
192h dwell
I 10 -s
C:x p
(MPomh
dwell dwell
I 10-~
I 10-3
"1)
Fig. 10. Comparison of value of C* predicted by eq. (2), Cre f* , with that deduced from experimental data, Cex p,* for a type 347 weld metal under displacement-controlled cycling.
522
R.A. Ainsworth and P.J. Budden / Appr(zrimate inelastic analysis 10-3 I
0 •
•
10 -L.
0
i !
E {3-
~E
v
I o
(,..) 0
0
10-s
l i i
•
10-6 10-6
A
A 192h d w e l l
A I
I
10-5
10-~
I
10-3
Ceexp ( M P Q m h "1 )
Fig. 11. Comparison of value of C* predicted by eq. (2), Cre 1,* with that deduced from experimental data, C~xp,* for a type 321 stainless steel under displacement-controlled cycling.
K n P Po
el. q R' t
w 61 , 62 AP ~c "c
Eref c Eref
tx O"
elastic stress intensity factor creep index in eq. (1) load value of P at start of displacement control period collapse load value of P constant in crack growth rate law in eq. (14) length parameter defined by eq. (4) time specimen width crack mouth opening and load line displacements, respectively load drop during period of constant displacement creep strain rate creep strain rate at reference stress accumulated creep strain at reference stress geometry-dependent factor in eq. (11) Poisson's ratio stress
O-r~f ~y r 77
reference stress defined by eq. (3) yield stress non-dimensional time of eq. (5) function used in C, estimate of eq. (7)
References [1] R.A. Ainsworth, G.G. Chell, M.C. Coleman, I.W. Goodall, D.J. Gooch, J.R. Haigh, S.T. Kimmins and G.J. Neate, CEGB assessment procedure for defects in plant operating in the creep range, Fatigue Fract. Eng. Mater. Struct. 10 (1987) 115-127. [2] A. Saxena, Creep crack growth under non-steady state conditions, ASTM STP 905 (ASTM, Philadelphia, 1986) pp. 185-201. [3] G.A. Webster, D.J. Smith and K.M. Nikbin, Prediction of failure of cracked components at elevated temperatures, Int. Conf. on Creep, JSME Tokyo, (1986) pp. 303-308. [4] H. Riedel, Fracture at High Temperatures (Springer, Berlin, 1986).
R.A. Ainsworth and P.J. Budden / Approximate inelastic analysis [51 J.W. Hutchinson, Singular behaviour at the end of a tensile crack in a hardening material, J. Mech. Phys. Solids 16 (1968) 13-31. [6] J.R. Rice and G.F. Rosengren, Plane strain deformation near a crack tip in a power law hardening material, J. Mech. Phys. Solids 16 (1968) 1-12. [7] V. Kumar, M.D. German and C.F. Shih, An engineering approach for elastic-plastic fracture analysis, EPRI Report NP-1931 (1981). [8] R.A. Ainsworth, Some observations on creep crack growth, Int. J. Fracture 20 (1982) 147-159. [9] R.A. Ainsworth, The assessment of defects in structures of strain hardening material, Eng. Fract. Mech. 19 (1984) 633-642. [10] A.G. Miller and R.A. Ainsworth, Consistency of numerical results for power-law hardening material and the accuracy of the reference stress approximation for J, Eng. Fract. Mech. 32 (1989) 233-247. [11] A.G. Miller, Review of limit loads of structures containing defects, Int. J. Pres. Ves. Piping 32 (1988) 197-327. [12] I. Milne, R.A. Ainsworth, A.R. Dowling and A.T. Stewart, Assessment of the integrity of structures containing defects, Int. J. Pres. Ves. Piping 32 (1988) 3-104. [13] R.W. Evans, J.D. Parker and B. Wilshire, An extrapolation procedure for long-term creep strain and creep life 1 I 1 prediction with special reference to ~CrTMozV ferritic steel, Recent Advances in Creep Fracture of Engineering Materials and Structures (Pineridge Press, Swansea, 1982). [14] H. Riedel and J.R. Rice, Tensile cracks in creeping solids, ASTM STP 700 (ASTM, Philadelphia, 1980) 112130. [15] T.K. Hellen and P.G. Harper, A user's guide to non-linear BERSAFE, Ph III, Level 3, BERSAFE Vol 1II, CEGB, UK (1986). [16] H. Tada, P.C. Paris and G.R. Irwin, The stress analysis of cracks handbook, 2nd edn (Paris Productions Inc, 1985). [17] A. Saxena and P.K. Liaw, Remanent life estimation of boiler pressure parts - crack growth studies, EPRI Report CS-4688 (1986).
523
[18] C.P. Leung, D.L. McDowell and A. Saxena, Consideration of primary creep at a stationary crack tip: implication for the C l parameter, Int. J. Fracture 36 (1988) 275-289. [19] R. Ehlers and H. Riedel, A finite-element analysis of creep deformation in a specimen containing a macroscopic crack, ICF 5, Vol. 2, Ed D. Francois (Pergamon, London, 1981) p. 691. [20] R.A. Ainsworth and P.J. Budden, Crack tip fields under non-steady creep conditions I: estimates of the amplitude of the fields, Fatigue Fract. Engrg. Mater. Struct. 13 (1990) 263-276. [21] C.F. Shih and A. Needleman, Fully plastic crack problems, part 1: solutions by a penalty method, J. Appl. Mech. 51 (1984) 48-56. [22] R.A. Ainsworth, Stress redistribution effects on creep crack growth, Proc. Conf. Mechanics of Creep Brittle Materials, Eds. A.C.F. Cocks and A.R.S. Ponter (Elsevier, London, 1989). [23] R.A. Ainsworth and P.J. Budden, Crack tip fields under non-steady creep conditions II: estimates of the associated crack growth, Fatigue Fract. Engrg. Mater. Struct. 13 (1990) 277-285. [24] R.A. Ainsworth, Defect assessment procedures at high temperature, Proc. SMiRT 10, Division L (1989). [25] G.J. Neate, Creep crack growth in ½CrMoV steel at 838 K, II: behaviour under displacement controlled cycling, Mater. Sci. Eng. 82 (1986) 77-84. [26] K.M. Nikbin and G.A. Webster, Prediction of crack growth under creep-fatigue loading conditions, ASTM STP 942 (ASTM, Philadelphia, 1988) pp. 281-292. [27] D.N. Gladwin, D.A. Miller, G.J. Neate and R.H. Priest, Creep, fatigue and creep-fatigue crack growth rates in parent and simulated HAZ type 321 stainless steel, Fatigue Fract. Eng. Mater. Struct. 11 (1988) 355-370. [28] D.N. Gladwin, D.A. Miller and R.H. Priest, Examination of fatigue and creep-fatigue crack growth behaviour of aged type 347 stainless steel weld metal at 650°C, Mater. Sci. Technol. 5 (1989) 40-51.
513
Approximate inelastic analysis of defective components R . A . A i n s w o r t h a n d P.J. B u d d e n
Nuclear Electric plc, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire GL13 9PB, United Kingdom Received 12 August 1991
In the inelastic analysis of defective components at high temperature, the crack tip characterising parameters are of interest for a range of loading conditions. In this paper, four situations are examined: (1) steady state creep under constant load; (2) the transitional behaviour under constant load prior to attainment of steady state conditions; (3) behaviour under displacement control; and (4) behaviour under displacement-controlled cycling with hold periods. For steady state creep, it is shown by comparison with finite-element solutions and experimental data that a reference stress approximation may be used to predict the parameter C * with reasonable accuracy. The transitional period prior to steady-state conditions is examined by finite-element calculations, and the time-dependent parameter C(t) [C(t)--, C* as t--,oo] is evaluated for a number of geometries under constant load. The results are predicted by extending the steady state reference stress approximation. Comparisons are made between C(t) and the C~ parameter of Saxena. Finite-element results are also reported for geometries creeping under a constant applied displacement. Under this loading, the parameter C(t) falls rapidly with time. The reference stress method is used to predict the rate at which the applied loading falls, and then to estimate the value of C(t). It is shown that the amount of crack growth associated with displacement-controlled loading is much less than under load control at the initial load. A simplified estimate is made of the total crack growth during a period of constant displacement, and compared with the experimental data for steels of both high and low creep ductility. Under displacement-controlled cycling, finite-element calculations involving hold periods would be extremely time consuming. It is shown that the simplified results developed for constant-displacement loading may also be applied to predict the experimental behaviour under cyclic loading provided the material creep data used in the estimates are obtained under relevant cyclically hardened or softened conditions.
I. Introduction For constant loading, developments in the understanding of defect behaviour at elevated temperature have been sufficient for Ainsworth et al. [1], Saxena [2] and Webster et al. [3] to produce procedures for assessing the failure of defective components. In such approaches, creep crack growth under steady state loading is evaluated by calculating the crack tip parameter, C * , which governs the amplitude of the near-tip stress and strain rate fields (e.g., Riedel [4]). C * may be evaluated by finite-element methods but the procedure of ref. [1] also embodies a reference stress approximation that enables simplified estimates to be made for a wide range of complex geometries, defect sizes and material laws. For component applications, it is often desirable to be able to predict crack growth under complex loading conditions. The finite-element method is a convenient approach for the examination of crack-tip parameters E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
under transient load conditions for simple geometries and material laws. In this paper, such analyses in conjunction with experimental data are used to provide guidance on the extension of approximate reference stress methods to these more complex loadings.
2. Steady state creep under constant load For materials in which creep strain rate ~c is related to stress ~ by the simple power law
~ = A o -n,
(I)
where A, n are constants, the stress and strain rate fields near a stationary crack tip in a body deforming in steady state creep are of the type derived by Hutchinson [5], and Rice and Rosengren [6]. The amplitude of these fields is described by the parameter C * , which may be evaluated from experimental measurements of
R.A. Ainsworth and P.,L Budden / Approximate inelastic analysis
514
the displacement rates in cracked test specimens and by finite-element methods. The analogy between power-law creep described by cq. (1) and power-law plasticity, enables solutions for J developed in handbook form by Kumar et al. [7] to be used to calculate C* for a range of geometries and crack sizes. An alternative approach to detailed finite-clement analysis is the reference stress method, which has been used by Ainsworth [8,9] to provide approximate estimates of both J and C*. The approximation in the creep case is cr*~, -
+ '. ,+r,.,+fE,.~,.R
(2)
where the reference stress Or++ is defined by Oref = PO'y/PL(O" v , a )
(3)
and R' is a length parameter: R' = K
:/~:~,.
(4)
Here P is the magnitude of the applied loading and Pt+ is the corresponding collapse load for a yield stress o-v and crack size, a; K is the linear elastic stress intensity factor, and ~ f is the creep strain rate from uniaxial creep data at the reference stress level. Miller and Ainsworth [10] have compared the approximation of eq. (2) with the detailed handbook solutions of ref. [7]. A typical result is shown in fig. 1 for a cylinder in tension with an external fully circumferential defect for a range of crack depths and stress indices, n. The results are presented in terms of a load factor, which is the ratio of the loads at which the
LOAD FACTOR
reference stress approximation and the handbook solution give the same value of C*: if the load factor is less than unity then the reference stress approximation is conservative relative to the handbook solution. It is apparent that the load factor is close to unity in this case and from an examination of a wide range of geometries, Miller and Ainsworth found that the reference stress approximation closely reproduced the finite-element results of ref. [7] in general, and on average corresponded to a conservatism of about 5% on load. The reference stress approach is particularly useful for practical applications because: (l) the only calculations required in eq. (2) are the stress intensity factor and the limit load, and such solutions are widely available (e.g., Miller [11]) because of their use in low-temperature fracture procedures such as R6 [12]. (2) Equation (2) is not restricted to secondary creep described by a power law but allows realistic creep data including primary, secondary and tertiary stages to be included in the creep strain rate at the reference stress. (3) Strain hardening rules may be used to allow for the increasing stress levels as a crack grows, by interpreting Ere "c f as the creep strain rate at the current reference stress and the creep strain accumulated under the reference stress history. The validity of using realistic creep laws and applying strain hardening rules has been checked for a range of materials by comparing eq. (2) with the test speci-
L
E X T E R N A LDEFECT Q/w=
1,OL /
%
1.2 |
O-8 / ~
3/4
0.6 0-4 0.2 n
0
3m
5i
1,0
j 16
Fig. l. Load factor for an externally circumferentially cracked cylinder in tension.
R.A. Ainsworth and P.J. Budden /Approximate inelastic analysis
515
the computed results are compared with approximate estimates. The analysis for a compact tension specimen is described in some detail to illustrate the mesh detail and numerical approach adopted.
10-3 A I .C
3.1. Finite-element analysis o f a compact tension specimen
10 "~
E 6
n
lo-s
10-6
10 . 7
10-7
I
I
I
I
10-6
lO-S
10-~
10-3
C:xp ( MPcl mh "1 ) Fig. 2. Comparison of value of C* predicted by eq. (2), Cref,* with that deduced from experimental data, Cexo, for a C-Mn weld metal under constant load.
men data for which C* can be estimated from experimental displacement rates [1]. Figure 2 shows a typical comparison for a C - M n weld metal for which creep data have been fitted by the 0-parameter approach of Evans et al. [13]. If secondary creep strain rates had been used to estimate C* then the experimental values of C* would have been significantly underestimated whereas, using the full creep curve, the agreement is much better and the predictions are conservative. Thus, for steady state creep under a constant load, reference stress estimates can be readily applied and have been validated both by comparison with finite-element results and experimental data. In the following sections the extension of the approach to more complex situations is considered and again both numerical and experimental validation is sought.
3. Transitional behaviour under constant load Riedel and Rice [14] have shown that during the transition from initial loading to steady state creep, the near-tip stress and strain-rate fields are still of the type described by Hutchinson, Rice and Rosengren [5, 6] but have an amplitude C(t) that depends on time, t, and approaches C* as t ~ oo. In this section the finite-element program BERSAFE [15] is used to produce results for C(t) for a number of geometries and
A specimen of crack length to specimen width ratio a / w = 0.5 is analysed under plane strain conditions. Due to symmetry, only half of the specimen is modelled in the finite-element mesh shown in fig. 3. A constant unit point load per unit thickness is applied at the node corresponding to the centre of the pinhole, which is not modelled explicitly. Quadratic isoparametric mid-side node elements are used in the mesh, which contains 146 elements (triangles and quadrilaterals) and 467 nodes. The mesh construction is radial near the crack tip enabling the radial variation of the field quantities to be easily discerned. The innermost core of elements near the crack tip comprises triangles of radial size a/200. The stress intensity factor K was calculated on initial loading using both the virtual-crack extension (VCE) facility automated into BERSAFE and the J-integral method via post-processor contour-integral rou-
Ca)
½62 ~1/261 8
7
6
543 I
~
el=w/2
W
(b)
(c)
3 2 1 Fig. 3. Finite-element mesh of compact tension specimen showing the whole mesh, (a), and near-crack tip details, (b) and (c). The semicircular contours 1-8 are used for contour integral evaluation.
516
R.A. Ainsworth and P.J. Budden / A p p r o x i m a t e inelastic' analvsis
Table 1 Load-line and mouth opening displacements for CT specimen
tines. Over the eight contours shown in fig. 3, the scatter in J-values was 1.6% of the mean value, which corresponds to a stress intensity factor 1.4% lower than that given in the handbook of Tada et al. [16]. The V C E solution was within 0.1% of the handbook value. Although the mesh does not model the loading arrangement exactly, it is of interest to compare the computed and handbook [16] elastic load-line and crack mouth opening displacements, 6 2 and ~ respectively (see fig. 3). These are listed in Table 1. There is agreement to within 3% and 7%, respectively. Following initial loading, creep analysis was performed for the creep law of eq. (1) with n = 5 up to a dimensionless time r = 1 where ~" = E ' C * t / K
Displacement Mouth BERSAFEelastic~l,(~e(mm) Tada et al. [16] ~i, 62 (mm)
2.289x10 -4 1.655x10 4 2.457x 10 -4 1.706x l0 4
BERSAFE creep gt, 62 (mm/h) 6.356x 10 -t° 4.672x 10 ]0 Kumar et al. [7] 6t, 62 (mm/h)
5.996x t0- 11} 4.402 x 10- ii)
than the computed figures. These differences are not inconsistent with the elastic discrepancies noted above. B E R S A F E allows various J-integral parameters to be evaluated and the results are presented here for the mean values on the eight contours in fig. 3 of integrals denoted by Jw and Jw*. Jw is the R i c e - E s h e l b y definition of J; Jw* is defined as the limiting value of Jw I las the contour F shrinks onto the crack tip but is evaluated on other contours in conjunction with surface integrals defined by Green's theorem, and hence, to within numerical accuracy, is path independent by definition. The two integrals are the same when J~, is path independent. The scatter in the computed Jw and Jw* values about the mean was small, varying from 4
(5)
2.
Load line
H e r e E ' = E / ( 1 - u 2) where E is the Young's modulus and u is the Poisson's ratio. The finite-element results suggest that at time ~"= 1 the steady state has essentially been reached in the highly loaded areas. Hence, by differencing the computed nodal displacements, the steady state load line displacement and crack mouth opening displacement rates, 62 and 61 respectively, are obtained. The computed figures and the results of Kumar et al. [7] are listed in table 1. The latter figures are about 6% lower
3 E1j K2 Slope
E1Jw
/
0.99
~
. S ope o97
- ~ - = 1*%
1
,25
I
.5
I
.75
J
I
I
1.0 1.25 1.5 1.75 Non-climensionQI Time "[
I
2
I
2.25
I
2.5
Fig. 4. Variation of computed J-integral with time under both load and displacement controls,
I
2.75
l
3
R.A. Ainsworth and P.J. Budden /Approximate inelastic analysis
71
and 1.5% at z = 0 . 1 to 6 and 1%, respectively, at ~-= 1.0. The variations of the mean values with time are shown in fig. 4 and can be approximately described by E ' J / g 2 = 1 + ~'.
517
6
(6)
The value of C* used in the normalisation of eq. (5) has been taken from ref. [7] and hence the present results at r -- 1 are consistent with [7] as d J / d t --* C* as t ~ . BERSAFE computes contour integral approximations to C(t) using the J-integral routines following conversion of displacements and strains/strain energy to their respective rates of change by differencing output from successive time steps. Mean values of integrals denoted by Cw and C*, which are the results of using the Jw and J * algorithms on the rate output, are plotted in fig. 5. At short times Cw would not be expected to be path independent and, for example, was found to vary by 24% from its mean value at r = 0.1. Conversely, the scatter in Cw* was much less, ranging from 1.4% of the mean at ~"= 0.1 to 0.8% at r = 1.0. Some of the scatter at low z will be because of the differencing of variables at discrete time steps in a rapidly varying stress field to obtain rates. Because of
5C ('l:)/C ° Ct/C* 4
3
2
Ct / C"
o
I ,1
I ,2
I .3 "r
I .4
I .5
Fig. 6. Comparison of Ct and C(t) parameters for compact tension specimen under constant load.
the fine mesh detail near the crack tip, it was also possible to estimate C(t) by stress and displacement substitution methods. Displacement substitution was found to work better, particularly when the dominant displacement component along a particular ray from the crack tip was considered, and confirmed the accuracy of the C* algorithm. Saxena [2] has proposed the use of a parameter C t for correlating creep crack growth in the transient regime. Saxena and Liaw [17] have shown that C t may be estimated from the load-line displacement rate as
C(~) C °
F' P ( - ~ - [ ~ e ( t ) - ~ 2 ( o o ) ] +~7/~e(~)), c , = ~-w
C °
C
1.0 0
I .I
I .2
I .3
I .4
I .5
"t
Fig. 5. Compact tension values of C(t) under constant load.
(7)
where r/, F ' and F are geometry-dependent factors, P is the load and B is the thickness. Substituting values for the compact tension specimen, eq. (7) has been evaluated using the finite-element displacement values and the results are shown in fig. 6 and compared with C(t). Leung et al. [18] have similarly compared C t and C(t) and also found Ct << C(t) for ~"<< 1. Indeed Saxena [2] showed that C t does not have the 1 / t dependence on time that C(t) has as t ---,0, and the numerical results confirm that the displacement rates at short times are insufficient to define C(t) by simple expressions such as eq. (7).
R.A. Ainsworth and P.J. Budden / Approximate inelastic analysis
518
3.2. Estimates of C(t)
Ehlers and Riedel [19] suggested that C(t) may be estimated approximately as the sum of the short- and long-time limits, that is C(T)=C*
1+ (n+l)r
0
(8)
Ainsworth and Budden [20] derived a new estimate of C(r): (1 + r ) "+1
) (9)
C(r) =C* (1 + r ) " + J -
1
by using the J-estimate of eq. (6). Both eqs. (8) and (9) reduce to C(r) ~ C* and C(r) ~ C * / ( n + 1)r in the limits r ~ oo and r ~ 0, respectively, as derived by Riedel and Rice [14]. The finite-element results for the compact tension specimen and preliminary results for a single-edge notched tension specimen under tension and bending are compared with eqs. (8) and (9) in fig. 7. The results for the single-edge notched specimen have been normalised using the values of C* given by
Shih and Needleman [21]. It is apparent that when the finite-element results are normalised as in fig. 7, the values of C(t) can be reasonably approximated by cither eq. (8) or eq. (9), independently of the geometry. Although apparently morc complex than eq. (8), cq. (9) is more convenient to integrate and results in thc approximate estimate [22]
~1'C(t) n/(,,+l)d t = c,n/(,,+l)t[ 1 +~rcf/Eeref(t)]. (10) The integral in eq. (10) is proportional to the creep strain near the crack tip and eq. (10) shows that this creep strain is increased above that which would have been accumulated under steady state creep by the factor [1 + elastic strain at the reference stress/creep strain at the reference stress]. Ainsworth [22] has shown how this factor may be used to incorporate the effects of transitional creep on both crack initiation and growth. If the creep strains greatly exceed the elastic strains, then steady state solutions are applicable and the C* estimate of section 2 may be used without consideration of the redistribution effects.
30
C(~) C•
I 2.0
_-_ __ Eqn (8) 10
Eqn ( 9 ) I
I
05
1.0
I
T,
D
I 5
Fig. 7. Comparison of finite-element solutions for C(t) for a n u m b e r of specimens with the predictions of eqs. (8) and (9). All
solutions for n = 5.
R.A. Ainsworth and P.J. Budden / Approximateinelasticanalysis
(~)~-I
4. Creep under displacement control The finite-element analysis of a compact tension specimen described in section 3 was continued under fixed load-point displacement for times 1 ~
dref = -E~rCeJU,
4
-
1)] 1/4.
~
//
~ E Q N 3
(11)
= [1 + 2.4(7
FINITE
ELEMENT ~ / /
(12)
~/
2
,/
where p. is a factor that can be estimated from the elastic and creep compliances and p. = 2 for the compact tension specimen. Integrating eq. (11) for the creep laws of eq. (1) and inserting the constants used in the finite-element analysis predicts the load variation
P('r)/P(1)
519
"g
(12)
Fig. 8. Load variation for a compact tension specimen under constant-displacement creep.
Equation (12) is plotted in Fig. 8 in the form of [P(1)/P(r)] 4 - 1 against z, compared with the finiteelement results. From eq. (12) a straight line of slope 2.4 would be predicted. The finite-element results produce a straight line but with a slope of 2.55, which corresponds to a slightly more rapid fall in load. Ainsworth and Budden [20] also derived an estimate of J under constant applied displacement, assuming J to be path independent. For the particular case n = 5
and constant displacement from r = 1, the result reduces to
E'J(-r) K2
P(-r)( P(1)
P('r)) 3 - p~
(13) .
%
C(~) C*
\\
C
\
Estimate of [201
\\~//
. . . . . . . . . . . . . ~,
I I
1 ~.2
1 1.4
I 16
1 ~8
I 2
I 2.2
I 2~
I 2.6
l 28
Fig. 9. Varation of creep integrals, C(t), for a compact tension specimen under constant displacement.
I
3O
R.A. Ainsworth and P.J. Budden / Approximate inelastic analysis
520
Equation (13) is shown in fig. 4 and compared with the J-values from both the Jw and J* algorithms. The computed Jw values showed a greater path dependence than under load control with variations of 10% and 20% about the mean at r = 1.12, r = 3, respectively. However, the mean value of Jw follows eq. (13) well, with an error of 3.5% at r = 3. The mean values of the creep integrals C w and C* are shown in fig. 9 together with an approximation given in ref. [20] based on the J-estimate of eq. (13). The approximation predicts a rapid reduction in C(t), and the finite-element results confirm this rapid reduction, although falling more rapidly than predicted. The scatter in the computed integrals was found to be greater than in the case of a constant load, varying from 33 and 1.8% of the mean values at r = 1.12, for C w and C* respectively, to 17 and 2.6% at r = 3. The underestimation of the predicted value by the finiteelement analysis is qualitatively consistent with the underestimation of the load P ( r ) relative to eq. (12). Ainsworth and Budden [23] used the reference stress approximation of eq. (11) to derive an estimate of the total crack growth during a period of constant displacement from the load reduction Ap. Provided the crack growth is small so that the change in elastic compliance does not lead to a significant load reduction, the result is
Aa =A'Co *q ~
Eeref
where A', q are constants in the creep crack growth law a = A ' C *q, and the subscripts or superscripts "o" denote values at the start of the displacement hold. Where there is significant crack growth, the effect of the change in compliance can be included in eq. (14) using the result given by Ainsworth [24]. Equation (14) predicts that the crack extension is simply related to the reduction in load and is significantly less than the growth under a constant load Po. Crack growth data under displacement control have been presented by Neate [25] and by Nikbin and Webster [26]. Neate tested an untempered coarse-grained bainitic 0.5 CrMoV steel at 565°C. Uniaxial creep data have been fitted by a simple primary creep curve to evaluate ZIP via eq. (11). The corresponding predictions of eq. (14) are compared with the experimental data, which are taken from specimens of two thicknesses, in table 2. Nikbin and Webster [26] present data for a 2.25 CrlMo steel at 538°C and a low-ductility 0.5 CrMoV steel at 565°C. In the absence of uniaxial creep data, the experimentally measured values of
Table 2 Crack growth under displacement control Material
½CrMoV [25] for a range of initial loads
Experimental Aa (mm) 1.5- 4.0 ~t 7.0- 8.0 ~' 5.0- 8.0 ~' 12 -17 " 17 19 ~'
Predicted da (mm) eq. (14) 2.8 5 8 13 22
2.25Cr1Mo [26]
(1.4
0.22
0.5 Cr 0.5 Mo 0.25 V [26]
5
9
a Range of values from nominally identical tests.
load drop have been used in eq. (14) to obtain the predicted values of crack growth, which are compared with the experimental data in table 2. It is apparent from table 2 that the simple prediction of eq. (14) provides a reasonable prediction of crack growth under complex loading conditions for a range of materials.
5. Creep under displacement-controlled cycling Displacement-controlled cycling is a loading regime of practical importance, arising under conditions where displacements are imposed and where displacements are induced by thermal bending, for example. Finiteelement analyses to evaluate C(t) under such conditions would be extremely time consuming particularly given the complex materials behaviour under cyclic loading. Therefore, this loading condition has been examined by a number of authors [27,28] by performing displacement-controlled tests on compact tension specimens with creep occurring during hold periods. The values of C* during the hold periods have been evaluated in refs. [27,28] from experimental values of the load drop rate. These are compared with the predictions of eq. (2) in figs. 10 and 11 for two austenitic steels. In the reference stress approximation, creep data from material in the cyclically conditioned state have been used; had forward creep data from virgin material been used the good agreement in figs. 10 and 11 would not have been obtained. It can be seen from figs. 10 and 11 that straightforward use of the reference stress approximation, developed for constant load, allows C* to be predicted under these complex loading conditions.
R.A. Ainsworth and P.J. Budden / Approximate inelastic analysis
521
Acknowledgement
As well as monitoring the load drop to evaluate C*, Gladwin et al. [27,28] made detailed measurements of crack growth rates during the dwell periods. They found that the crack growth rate data correlated with C* in a similar manner to data collected under constant load. As a consequence it would be expected that eq. (14) could be used to predict the total crack growth during a displacement-controlled hold period just as it has been used to predict growth under constant displacement. This has been confirmed by analysis of the data on the 347 weld metal by Ainsworth and Budden [23]. Experimental data, therefore, suggest that the reference stress methods developed for constant load and constant displacement can be extended in an intuitive manner to predict creep crack growth under cyclic conditions. Of course, in the latter case it is also necessary to allow for fatigue crack growth due to the load changes in order to calculate the total crack growth per cycle. The manner in which this may be included has been described by Ainsworth [24].
This paper is published by permission of Nuclear Electric plc.
Nomenclature crack size constant in creep law of eq. (1) constant in crack growth rate law in eq. (14) specimen thickness steady state creep characterising parameter transient creep characterising parameter finite-element approximations to C(t) parameter of Saxena given by eq. (7) Young's modulus E/(1 v 2) functions used in C, estimate of eq. (7) elastic-plastic characterising parameter finite-element approximations to J
a A A' B C* C(t) C w, C* Ct E E' F, F ' J Jw, J*
-
10-3
A
V
h 0 Q.
ov
~7~ v/~
lO-S
y
(J
--- Key a V
/ Z ~ V
10-6
•
10 -7 10 -7
I 10-5
24h 96h
192h dwell
I 10 -s
C:x p
(MPomh
dwell dwell
I 10-~
I 10-3
"1)
Fig. 10. Comparison of value of C* predicted by eq. (2), Cre f* , with that deduced from experimental data, Cex p,* for a type 347 weld metal under displacement-controlled cycling.
522
R.A. Ainsworth and P.J. Budden / Appr(zrimate inelastic analysis 10-3 I
0 •
•
10 -L.
0
i !
E {3-
~E
v
I o
(,..) 0
0
10-s
l i i
•
10-6 10-6
A
A 192h d w e l l
A I
I
10-5
10-~
I
10-3
Ceexp ( M P Q m h "1 )
Fig. 11. Comparison of value of C* predicted by eq. (2), Cre 1,* with that deduced from experimental data, C~xp,* for a type 321 stainless steel under displacement-controlled cycling.
K n P Po
el. q R' t
w 61 , 62 AP ~c "c
Eref c Eref
tx O"
elastic stress intensity factor creep index in eq. (1) load value of P at start of displacement control period collapse load value of P constant in crack growth rate law in eq. (14) length parameter defined by eq. (4) time specimen width crack mouth opening and load line displacements, respectively load drop during period of constant displacement creep strain rate creep strain rate at reference stress accumulated creep strain at reference stress geometry-dependent factor in eq. (11) Poisson's ratio stress
O-r~f ~y r 77
reference stress defined by eq. (3) yield stress non-dimensional time of eq. (5) function used in C, estimate of eq. (7)
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