Finite element analysis of creep using Theta projection data

Finite element analysis of creep using Theta projection data

M. Lawa,*, W. Paytenb, K. Snowdenc

aDepartment of Materials Science, University of Technology, PMB 1, Menai, Sydney, NSW 2234, AustraliabAustralian Nuclear Science and Technology Organisation (ANSTO), Sydney, AustraliacAustralian Nuclear Science and Technology Organisation (ANSTO), Sydney, Australia

Received 25 March 1998; accepted 4 April 1998

Abstract

The Theta projection creep algorithm has been implemented within a finite element code. This extends the predictive capability of Thetaprojection data to complex geometries, multiple material problems such as welded joints, and non-steady temperature conditions. Validationof the finite element methodology has been undertaken by re-modelling creep data based on the original Theta coefficients. The stressredistribution in a cylindrical pressure vessel was examined and compared to that predicted by the Norton equation. The effects oftemperature variation were also modelled.q 1998 Elsevier Science Ltd. All rights reserved

Keywords:finite element analysis; creep; theta projection data

1. Introduction

The Theta projection method [1] is being more widelyused as its advantages and flexibility become appreciated.The Theta projection equation is an expression of creepresponse over time

«c ¼ v1(1¹ e¹ v2t) þ v3(ev4t ¹ 1) (1)

where«c is the creep strain,t is the time, and thev i terms areexperimentally determined constants. This equation is anattempt to both empirically fit the strain–time behaviourof a material during creep, and to provide an insight intothe processes occurring during creep [2]. The equation mod-els the primary (the first group of terms) and tertiary (thesecond group of terms) stages of creep, and implies that anyconstant secondary creep region is an inflection on the curvewhere the effects of strain hardening and strain softening (ortertiary) processes are equal. The four Theta constants plotthe extent and curvature of the primary and tertiary stages.Each Theta term is itself a function of temperature andstress, of the form

Logvi ¼ Ai þ Bij þ CiT þ DijT (2)

wherej is the stress,T is the temperature andi ¼ 1, 2, 3, 4.The logv i values derived from testing vary approximately

linearly as functions of stress and temperature [3]. If thev i

functions can be defined on the basis of short term tests athigh temperature, then values for long term creep data canbe predicted [4–6]. With the Theta projection method, theshape of creep curves change (Fig. 1) as stress and tempera-ture are varied.

The Theta projection method is more useful for analysingcreep data than other current formulations for three reasons.Firstly, it can generate data that conveniently models stressand temperature changes. Secondly, the Theta projectionallows estimates of the minimum creep rate from thev i

values. Few other available equation systems describecreep curves accurately. Thirdly, the Theta projection canbe used to make a reliable estimate of the remaining life of amaterial undergoing creep. The present paper examines theimplementation of the Theta projection method within afinite element analysis code.

2. Methods

Thev parameters used in this work were obtained as partof an extensive creep testing program of ex-service 21/4 Cr–1 Mo steels. The experimental details have been reportedelsewhere [7]. Briefly, the creep specimens were tested attemperature in a vacuum to reduce possible oxidationeffects and their extensions were continuously monitoredand results stored in a computer memory file. The creep

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* Corresponding author. Tel.:0061 2 9717 9102; fax: 0061 2 9543 7179;e-mail: [email protected]

International Journal of Pressure Vessels and Piping 75 (1998) 437–442

curve information was fitted to the Theta equation using themethod of Evans and Wilshire [1].

3. Solution method

To implement the Theta equation within the FEA pro-gram (EMRC NISA II), a strain rate procedure was used.The first and second derivatives of the Theta equation wereused to calculate the creep rate, and hence the creep strain,for the timestep and element (which in turn depends on thetime, and the local stress and temperature). These are

d«=dt ¼ v1v2Exp( ¹ v2t) þ v3v4Exp(v4t) (3)

d2«=dt2 ¼ ( ¹ v1v2)2Exp( ¹ v2t) þ (v3v4)2Exp(v4t) (4)

4. Verification

Verification of the Theta modelling with FEA wasachieved by comparing Theta projection prediction of theoriginal creep data and strains predicted by the Theta pro-jection within the FEA code. This ensures that the modellingmethodology, the finite element program creep module, andthe creep data all return accurate values.

The creep specimen modelled was under a stress of40 MPa and a temperature of 6558C. For the Theta equation,the coefficients used are given in Table 1.

The Theta coefficients given in Table 1 were derivedfrom 8 creep tests at stresses from 40 to 60 MPa, and attemperatures from 6458C to 7158C using regression analysisand Eq. (2). The Theta projection calculates a materialresponse based on all 8 creep tests, and thus the originalcreep data may not be exactly replicated by the Thetacoefficients at the relevant temperature and stress. For thisreason the Theta/FEA procedure attempted to replicate theTheta prediction of creep at a specified temperature andstress.

The accuracy of the Theta/FEA remodelling of the origi-nal creep data is seen in Fig. 2. The maximum differencebetween the actual and the predicted creep response was lessthan 1.5%. This result implies that the Theta projection dataand equations may be used in the FEA code, and thatreliable predictions will be returned.

5. Creep in a uniform cylindrical pressure vessel

A uniform cylindrical pressure vessel of internal diameter50 mm and wall thickness 25 mm was modelled withthe above material properties (Table 1) using the Thetaprojection.

The analysis was also carried out using the Norton equa-tion for the minimum creep rate:«min ¼ Kjn whereK andnare experimentally derived material constants. The analysisassumed constant secondary creep at the minimum creeprate. The temperature is 928 K and the average Lame stressfound in the model under elastic conditions is 32.08 MPa.UsingK ¼ 2.3995E¹ 20 andn ¼ 7, in uniaxial tension theNorton equation predicts a similar minimum creep rate tothat predicted by the Theta method (8.39E¹ 10 m m¹1 s¹1).

5.1. Results

The nature of the stress redistributions were slightly dif-ferent, with the Theta model redistributing more quicklythan the Norton model (Figs 3–6).

Table 2 shows that the Theta relaxed stresses are slightlydifferent (by up to 3.3%) compared to the correspondingNorton stresses. It is not expected that the Theta solutionshould be equal to the Norton-based relaxed stress state.The Norton model is based on the power law relationbetween stress and creep rate, whereas the Theta equationshows an increasing rate of minimum creep rate againststress in a log–log plot (Fig. 7). The Norton power

Fig. 1. Theta projection curves at different stresses and temperatures.

Table 1Theta coefficients

(v1) (v2) (v3) (v4)

A ¹ 106.55 18.966 ¹ 284.16 ¹ 63.664B (MPa) 2.7948 ¹ 2.754 3.6464 8.53E¹ 02C (K) 1.11E¹ 01 ¹ 4.45E¹ 02 2.79E¹ 01 4.88E¹ 02D (K MPa) ¹ 2.98E¹ 03 3.10E¹ 03 ¹ 3.66E¹ 03 1.77E¹ 05

438 M. Law et al./International Journal of Pressure Vessels and Piping 75 (1998) 437–442

Fig. 2. Comparison of original creep data and FEA re-modelling of data.

Fig. 3. Stress redistribution in a cylindrical pressure vessel.

Fig. 4. First principal stress in cylindrical pressure vessel during stressredistribution.

Fig. 5. First principal stress in cylindrical pressure vessel using Theta creepmodel.

439M. Law et al./International Journal of Pressure Vessels and Piping 75 (1998) 437–442

coefficient (n ¼ 7) has been obtained from testing across arange of stresses, and appears to be higher than what wouldapply in the region of interest in this modelling, the lowstress region in Fig. 7 wheren ¼ 4.975.

The explanation for the faster redistribution of the Thetamodel than the Norton model (even though they should pre-dict similar minium creep rates) depends on the fact that theseminimum creep rates only occur late in life in the Thetamodel (for the 32.08 average Lame stress, this is at 3.0Eþ

07 seconds). Thus the Theta modelling is in its primary sec-tion of the creep curve, is creeping more quickly than theNorton, and goes through its redistribution more quickly.

In other cases, higher stressed regions may enter their ter-tiary region more quickly than the lowly stressed regions inthe Theta model, while in the Norton model constant second-ary creep has been assumed across the full modelling region.

The later (.5Eþ 06 second) behaviour of the first princi-pal stresses in the Theta model (Fig. 5) can be explained byreference to the von-Mises stresses (Fig. 6, Fig. 8) in thismodel. In the finite element modelling, the driving force forcreep is the von-Mises stress. Although the first principalstress is highest at the outer wall, the von-Mises stress islowest there and thus the creep rate will drop. Conversely,the von-Mises stress is highest at the inner wall (Fig. 8), creepreaches the tertiary stage at approximately 1.2Eþ 07 sse-conds and the creep rate increases. The stresses reduce atthe inner wall and off-load to the outer wall (Fig. 6).

6. Modelling of creep under temperature variation

Theta projection and FEA were used to analyse the creepbehaviour of a cylindrical pressure vessel with the inner

wall 58C higher than the outer wall. The coupled tempera-ture and stress situation leads to the variation of creep ratesshown in Fig. 9. This case was similar to modelling per-formed by Loghman and Wahab [8] on a cylindrical pres-sure vessel which showed a significant variation of the von-Mises stresses with time. The closed form solutionemployed by them is not useable in complex geometriesor multiple material models such as welded joints. Thedetails and results are discussed below.

Fig. 6. von-Mises stress in cylindrical pressure vessel using Theta creepmodel.

Fig. 7. Comparison of minimum creep rates from Theta data, and Nortoncoefficient fits to portions of the graph.

Table 2Relaxed first principal stresses from Norton and Theta models

Inner wall (MPa) Outer wall (MPa)

Norton 27.2 35.2Theta (at 1Eþ 06 seconds) 27.8 (þ2.2%) 34.8 (¹1.1%)Theta (at 2Eþ 07 seconds) 26.3 (¹3.3%) 35.4 (þ0.5%)

Fig. 8. Stress and predicted creep rate at inner wall, long-term response.

Fig. 9. Effect of temperature and stress on minimum creep rate.


6.1. Results

There is a thermally induced compressive hoop stress atthe inner wall; the combination of this, the pressure loading,and differing creep rates across the wall thickness due totemperature variation gives rise to the stress profiles throughtime illustrated in Fig. 10 and Fig. 11.

The final state of the von-Mises stresses is similar to thatreported by Loghman and Wahab [8], except for differencescaused by the relative vessel dimensions, material proper-ties, temperature response of the creep law, and magnitudeof the thermal and pressure stresses.

The stress redistributions show complex behaviour. Inaddition to the usual creep-induced redistribution, coupledwith a more complex initial stress state, there are two linkedfactors which add to the complex stress response. Firstly, thetemperature variation across the wall thickness alters thecreep rates so that, at the same stress, the inner wall experi-ences faster creep rates than that of the outer. Thus the innerwall relaxes; after this the outer wall completes its strainredistribution, and this in turn affects the inner wall so it issome time before the stresses stabilise. The difference inrelative creep rates between the inner and outer walls is afunction of temperature, local von-Mises stress (seen inFig. 11) and time.

Secondly, the temperature variation means the relativedifference in the creep rates may vary through time. For

example, the material in one part of the vessel may beentering its tertiary phase while another may still be exhi-biting a decreasing creep rate. These two factors are shownin Fig. 12 for the thermal case, calculated for the stress andtemperature at each timestep. The minimum creep rateoccurs at the inner wall at approximately 6Eþ 06 secondsand at the outer wall at 1.2Eþ 07 seconds. Tertiary creep isinitiated at the inner wall at approximately 8Eþ 06 seconds,and at the outer wall at 1.6Eþ 07 seconds.

6.2. Life consumption

The life consumption in the modelling was also plotted byusing a power-law life consumption equation

tfailure ¼ B:exp(QRT

)jn

WhereB ¼ 2.5E¹ 09, Q ¼ 450 000 J mol¹1, n ¼ ¹ 5.8,these were determined by regression from experimentaldata. Using a time-based Robertson fractional life rulebased on the von-Mises stress, failure was predicted asoccurring at the inner wall at 1.71Eþ 07 seconds (Fig. 13).

Thus for the material modelled, failure would initiatebefore stress redistribution was completed.

7. Conclusion

Modelling of creep behaviour using Theta projection data

Fig. 10. Theta projection modelling of thermal case, first principal stresses.

Fig. 12. Variation of creep rate at inner and outer walls.

Fig. 11. Theta Projection modelling of thermal cases, von-Mises stresses. Fig. 13. Life consumption in thermal case - power law failure criteria.

441M. Law et al./International Journal of Pressure Vessels and Piping 75 (1998) 437–442

has been implemented within an FEA code. Validation ofthe procedure was performed by remodelling the originalcreep data; agreement was achieved to within 1.5%. Thisresult allows confidence in the accuracy of the procedure.

The redistribution of stresses by creep in a pressure vesselwas modelled and compared to results predicted by theNorton equation. The differences found were attributed tothe more versatile features of the Theta projection.

Cases were modelled which included thermal variationacross a pressure vessel. The stress evolution was similar tothat found in previously publish worked. A complex inter-action was noted as a result of coupled thermal variation ofcreep rates and thermally induced stresses.

Acknowledgements

The authors are grateful to the CRC for Materials Weld-ing and Joining, Pacific Power, and the Australian Institutefor Nuclear Science and Engineering (AINSE) for supportfor this work.

References

[1] Evans, R., Wilshire, B. Creep of Metals and Alloys, Institute ofMetals, London, 1985.

[2] Ule, B., Rodi, T., Sustar, T. Modification ofv projection creep law byintroducing mean stress term. Materials Science and Technology1997:13:555-559.

[3] Parker, J. Prediction of Creep deformation and Failure for 1/2Cr-1/4Vand 2-1/4Cr-1Mo Steels. ASME J. Pres. Vessel Technology, 107:1985: 279-284.

[4] Viswanathan, R. Damage Mechanisms and Life Assessment of HighTemperature Components. ASM International, USA, p. 61.

[5] Snowden, K., Merhtens, E. Analysis of Creep Data for RemainingLife Estimation. Proc. Remaining Life Assessment on High Temp.Plant in Australia, Lucas Heights, 1990.

[6] Evans, R., Beden, I., Wilshire, B. Creep Life Prediction for 1/2 Cr 1/2

Mo 1/4 V Ferritic Steel. Creep and Fracture of Engineering Materialsand Structures, Pineridge Press, Swansea, UK, pp. 1277-1290.

[7] Ripley, M., Snowden, K. Remaining Creep Life Assessment of Weldsin Steel Pressure Vessels. Proc. Structural Integrity: Experiments,Models and Applications. 10th Biennial Conf. On Fracture, Vol.1,Berlin, Sept 1994, pp. 761-766.

[8] Loghman, A., Wahab, M. Creep damage simulation of thick walledtubes using thev projection concept, Int. J. Pres. Ves and Piping1996:67: 105-111.


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Finite element analysis of creep using Theta projection data