The computation of constant-load creep curves

Nuclear Engineering and Design 39 (1976) 289-292 © North-Holland Publishing Company

THE COMPUTATION OF CONSTANT-LOAD CREEP CURVES B.D. CLAY and H.E. EVANS Central Electricity Generating Board, Research Department, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire GL 13 9PB, U.K.

Received 26 January 1976 A numerical integration is used to compute the form of the strain-time relation under uniaxial constant-load conditions using data obtained from an equivalent constant-stress test. The method is applied to creep experiments on two types of stainless steel, intended for fuel cladding in the advanced gas-cooled reactors, where significant differences are observed in both the creep strength and stress dependence of secondary creep rates. The computed constant-load curves are in close agreement with the experimental data and show that the characteristic features of enhanced secondary creep rates, and reduced times to secondary creep and failure, may be accounted for satisfactorily. The results also emphasize the {mportance of using constant-stress creep equations as a basis for predicting cladding endurance..Since the primary and secondary creep stages observed in constant-load tests are largely artifacts, the associated values of stress exponent and activation energy need have no physical significance.

1. Introduction The successful and economic design of components for use at elevated temperatures requires that reliable equations are available to describe material creep behaviour under operational stresses. Through the years various empirical relationships have arisen to meet this need. The steady-state creep rate i s has been most successfully represented by the power lawequation i s =Ao n ,

(1)

where A and n are constants and o is the applied stress. The application by Garofalo [1 ] of an equation due to McVetty [2] for primary creep has also been shown by Amin et al. [3] to be applicable to a wide range of materials. This predicts the total strain e at time t as e = e 0 + e T [ 1 - e x p ( - m t ) ] + ~s t ,

(2)

where e 0 is the initial, time-independent strain, and and m are constants. Various workers have shown a proportionality between the rate constant m and the steady-state creep rate, thus,

6T

m = ~s,

(3)

where/~ is a constant of order 1 0 - 1 0 0 [3]. Whilst a number of detailed mechanisms have been proposed, e.g. Evans and Williams [4], to account for the Garofalo equation, it retains a physical significance only when obtained under conditions of constant stress. The commonly used constant-load creep test may lead to significant over-estimates in the true creep rate, and accordingly, under-estimates in the time to rupture. The extent of the errors involved have been appreciated for a considerable time Andrade [5] but have been difficult to quantify in the absence of a computational method to allow transposition from the constant-stress state. The success of a recent attempt by Haberlin et al. [6] was limited by the exclusion of primary creep strain. This paper describes a method which overcomes this restriction by a numerical integration of eq. (2). Predicted behaviour is compared with experiments on two types of stainless steel, having different creep exponents n which are intended as cladding materials for the nuclear fuel in the advanced gas-cooled reactors. The possible errors introduced by the use of constant-load data in predictions of cladding performance are also discussed.

290

B.D. Clay, H.E. Evans / Constant-load creep curt'es

2. Computational method

Table 1, Composition of the steels, wt/;~.

The constant-load creep curve is generated from the constant-stress creep equations (1)-(3). Basically the initial strain rate is calculated from the initial stress and, at a suitably small increment of strain, the new stress appropriate to the reduced section is evaluated. Assuming no change in structure ('strain hardening'), the new strain rate is obtained from the equations and the stress recalculated after a further strain increment, and so on. The assumption of a constant structure is reasonable provided the strain increments are small. The values of the primary creep parameters given in eqs. (2) and (3) were obtained by a C o n w a y Mullikin plot [7] of the difference between the transient strain and the extrapolated secondary creep strain. It is well known that the Garofalo equation under-estimates the strain rate in the early stages of primary creep and this is thought to be due to the concurrent variation of the creep work-hardening coefficient [4]. There are computational methods to take this variation into account but these are, at present, empirical. Since the intention of this paper is to demonstrate the errors introduced by constantload creep equations at large strains, the deviation from the Garofalo relation (eq. (2)) at small strains is accommodated here in the parameter %. The values sub~quently quoted for this are artificially high, having been optimized to produce a reasonable fit between eq. (2) and the observed creep curve.

Steel Cr A B

Ni

Ti

Si

19.8 25.2 1.55 0.92 20.2 24.6 0.49

Mn

Nb

C

Fe

0.65 0.40

0.21 0.015 balance 0.41 0.023 balance

also at 1423 K. The final average grain size, in this case, was about 100/Jm. Creep tests were performed at 1023 K in air on a Denison T48 creep machine fitted with a profiled cam whose movement maintained constant-stress testing conditions. Constant-load conditions were obtained on the same machine through the use of a microswitch operated motor which maintained the cam in a constant-lever position. Specimen extensions were obtained from Rank capacitance transducers arranged to measure the displacement at the tips of the specimen grips. 3.2. Results

A comparison of results for the two stress states, at an initial value of 140 Nmm -2 (20 kpsi), is shown in fig. 1 for the nitrided steel and in fig. 2 for steel B. The obvious features are the increased endurances under constant stress conditions, the protraction of the primary stage, and postponement of the advent of tertiary creep. The enhancement in apparent secondary creep rate under constant-load conditions is a factor of about 10 for steel A and about 5 for steel B.

3. Experimental 4. Discussion 3.1. Materials and techniques

The steels used were based on the 20w/oCr-25w/oNi composition, table I. Both were supplied as a 30% cold worked strip from which creep specimens having gauge dimensions 25.4 X 6.35 X 0.38 mm were stamped. The titanium-free steel (steel B) was annealed in a static vacuum for 1 hr at 1223 K prior to testing to give a recrystallized grain size of about 25/am. Steel A was nitrided at 1423 K for 13 hr in an atmosphere of 95 vol.% N2/5 vol.% H 2 to produce a dispersion of titanium nitride particles [8], and then outgassed, to remove excess nitrogen, in pure hydrogen for 1~ hr

For the constant stress tests shown in figs. 1 and 2, a best fit was obtained to eq. (2) using the C o n w a y Mullikin method, the appropriate values of the constants of which are given in table 2. The constant-load curves were computed using these parameters with a stress exponent for secondary creep (eq. (1)) of 9.7 for the nitrided steel (Evans, unpublished work) and 5.6 for the conventional steel (Clay, unpublished work). The curves obtained are shown in figs. 1 and 2 and are in satisfactory agreement with the constant-load data. In particular, the reduction in the duration of primary creep and the enhancement of primary and secondary

30

I

I

Fracture

I

I

I

I

I

I

I

Fract.u~

o

Load

25

o o o

A

20,

0

I

I

Constant Stress Test

Constant

Load Test

Fitted Constant Stress Curve

o o

0 X

D

C

' a- 15 L

m

Experi mental Data 14ON/ram 2, 7 5 0 ° C Steel A = 20/25/Ti O Constant Load • Constant Stress

~ ~o

I-

aT O

I

I

I

200

I

I

400

I

I

I

6OO Time Hours

800

I

I

IOOO

I

12OO

Fig. 1. Comparison of the computed constant-stress and constant-load curves with the experimental data for nitrided stainless steel. t

40

I

,~

I

I

I

Fracture

351 83C

Constant Computed~ / Load Curve ] o

]

/

oo

DiscontinuedTest

Constant

-

~x to 25

Stress Test

C

[,

2c

Fitted Constant Stress Curve

u= i

J

Experimental Data 14ON/rum2 750°C Steel B = 2 0 / 2 5 / N b o Constant Load • Constant Stress

IO

0,1

O

-

I

Ioo

I

I

200 300 Time Minutes

I

400

• soo

Fig. 2. Comparison of the computed constant-stress and constant-load curves with the experimental data for 20/25/Nb stainless steel.

292

B.D. Clay, 11.1:2 Evans/Constant-load creep curves

Table 2. Creep constants from the constant-stress tests. Steel

Tensile stress (Nmm -2)

Temp. (K)

eo (X 100)

eT (X 100)

m (sec-1 )

~s (sec-1 )

A B

140 140

t023 1023

4.0 3.0

8.0 8.8

1.5 X 10 -6 2.1 X 10 .4

3.3 × 10 .8 5.2 X 10 .6

creep rates are accounted for. The generated curve for the nitrided steel, for example (fig. 1), gives a minimum creep rate 12 times that in the constant stress test which compares favourably with the factor of about 10 observed. This contrasts with the predicted enhancement of about 4 obtained by ignoring the contribution of primary creep. The computations made here also show that the early commencement of tertiary creep, characteristic of constant-load testing, does not necessarily reflect plastic instability ~s postulated by Burke and Nix [9]. In the presentcase, it is due simply to the increasing strain rate associated with a uniform reduction in the specimen cross section. The severity of the errors introduced in practice by the use of constant-load data in fundamental creep equations depends on the detailed application. Provided that small strains, say less than 1%, are involved and the important parameter is the time to reach this strairL then the error should be small (cf. figs. 1 and 2). This is probably the situation during creep-down of fuel cladding. On the other hand, significant errors could arise when the strains involved are large, e.g. in tube rupture behaviour of ductile materials and, possibly, fuel end cap deformation. The application of uniaxial test data to fuel pin deformation under depressurization is a particular instance where care must be exercised and the fundamental, constant-stress creep equations used as the starting point of the calculations. It should be noted also that since the secondary creep rate observed under constant load is an artifact having no relationship with the true steady state, there is no obvious reason why the values of stress exponent or activation energy obtained under such conditions should have any physical significance. This associated uncertainty exists whenever extrapolation beyond the test conditions is required, even for applications involving small strains.

45 40

5. Conclusions The characteristic features of constant load testing, namely enhanced secondary creep rates and reduction in the times to secondary creep, tertiary creep and failure, may be accounted for satisfactorily using a Garofalo-type creep equation and uniform specimen extension. It is not necessary to postulate plastic instability. The secondary creep stage observed in constant load tests is an artifact and the associated values of stress exponent and activation energy need have no physical significance.

Acknowledgements The authors express their thanks to Mr. K. Swallow for performing the creep tests. This paper is published by permission o f the Central Electricity Generating Board.

References [1 ] F. Garofalo, Fundamentals of creep and creep-rupture in metals, Macmillan, New York (1965). [2] P.G. McVetty, Mech. Eng. 56 (1934) 149. [3] K.E. Amin, A.K. Mukherjee and J.E. Dorn, J. Mech. Phys. Solids 18 (1970)413. [4] H.E. Evans and K.R. Williams, Phil. Mag. 25 (1972) 1339. [5] E.N. daC. Andrade, Proc. Roy. Soc. 84 (1910) 1. [6] M.M. Haberlin, B. Tomkins and Z. Szkopiak, (1976) to be published. [7] J.B. Conway and M.J. Mullikin, Trans. AIME 236 (1966) 946. [8] H.E. Evans, D. Raynor, A.C. Roberts and J.M. Silcock, Proc. Third Int. Conf. on Strength of Metals and Alloys, Cambridge (;1973). [9] M.A. Burke and W.D. Nix, Acta. Met. 23 (1975) 793.