An Effective Continuum Damage Mechanics Model for Creep

International Journal of Pressure Vessels and Piping 80 (2003) 389–396 www.elsevier.com/locate/ijpvp

An effective continuum damage mechanics model for creep –fatigue life assessment of a steam turbine rotor Jing JianPinga,*, Meng Guanga, Sun Yib, Xia SongBob a

The State Key Laboratory of Vibration, Shock & Noise, ShangHai JiaoTong University, ShangHai 200030, China b School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China Received 30 December 2002; revised 8 April 2003; accepted 8 April 2003

Abstract A nonlinear Continuum Damage Mechanics model is proposed to assess the creep – fatigue life of a steam turbine rotor, in which the effects of complex multiaxial stress and the coupling of fatigue and creep are taken into account. The nonlinear evolution of damage is also considered. The model is applied to a 600 MW steam turbine under a practical start –stop operation. The results are compared with those from the linear accumulation theory that is dominant in life assessment of steam turbine rotors at present. The comparison show that the nonlinear continuum damage mechanics model describes the accumulation and development of damage better than the linear accumulation theory. q 2003 Elsevier Ltd. All rights reserved. Keywords: Creep–fatigue; Damage; Steam turbine rotor; Damage mechanics

1. Introduction The developments of modern industry require the parts of a steam turbine to operate under greater working loads and in a higher temperature environment. Much concern has been paid to fatigue and creep damage behaviour of turbine materials. Being an important part of the steam turbine, the rotor is often subject to high temperature and complex stress. Cracks are likely to initiate. Apart from fatigue, creep damage also plays an important role in the rotor damage. Generally, low cycle fatigue wears off seventy percent of the life of the rotor and creep accounts for the remaining thirty percent [1]. However, fatigue and creep always occur with each other. Therefore, coupling of fatigue and creep must be considered in the life prediction of a steam turbine rotor. At present, the Linear Damage Accumulation Theory (LDA) is widely used in the fatigue –creep life assessment of steam turbine parts. As it contains uniaxial assumptions and the effect of the coupling of fatigue and creep is ignored and the damage accumulation calculation * Corresponding author. E-mail address: [email protected]; [email protected] (J. JianPing). 0308-0161/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0308-0161(03)00070-X

is rather crude, the predicted results are usually quite different from practice. Continuum Damage Mechanics (CDM) developed in the past decades has not only a stronger theoretical foundation but also better research methods than LDA. It has been used successfully in many engineering fields such as creep, fatigue, ductile fracture and composite failure [2,3]. In this paper, the transient temperature and stress fields of a 600 MW steam turbine rotor are investigated. The nonlinear CDM model is employed to predict the creep– fatigue life of the steam turbine rotor and the results are compared with those from LDA theory. Finally, the advantages of the nonlinear CDM model are discussed.

2. Theoretical model 2.1. Linear Damage Accumulation theory At present, the LDA theory is mainly used in the fatigue – creep life assessment of steam turbine parts. The damage accumulation is considered as a linear process. The fatigue and creep damage are calculated separately and

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J. JianPing et al. / International Journal of Pressure Vessels and Piping 80 (2003) 389–396

Nomenclature

s0f b c Nf D1P Nfi tci n sH sij V; a1 ; g

fatigue strength coefficient fatigue strength exponent fatigue ductility exponent cycles to failure under fatigue plastic strain range cycles to failure under a load si cycles to failure under a creep load si Poisson’s ratio mean stress component of stress temperature dependent material constants under low cycle fatigue conditions A; B; l; a2 ; r temperature dependent material constants under creep conditions Dc creep damage Ds cyclic stress range K0 cyclic strength coefficient 1_pp strain rate of the secondary creep process

superposed linearly. The structure fails, as the damage amounts to 1. It can be expressed as follows X ti n þ i ¼1 ð1Þ tci Nfi The creep life tc can be obtained from the following equation seq B tc ¼ ð2Þ A where A and B are material constants. The fatigue life Nf is mainly obtained by the Manson– Coffin equation. D1e s0 ¼ f ð2Nf Þb 2 E D1p ¼ 10f ð2Nf Þc 2

ð3aÞ

sr E 10f tc D1e ni ti RV seq Sij skk dij t D Df D sp n0 1_p Dr Yr

critical stress Young’s modulus fatigue ductility coefficient time to failure under creep elastic strain range cycles under a fatigue load si cycles under a creep load si triaxial coefficient effective stress component of deviatoric stress component of normal stress d function time total damage fatigue damage cyclic stress range at saturation cyclic strain hardening exponent strain rate of tertiary creep process critical damage value critical release rate of strain energy

rate of the secondary creep process and 1_p is the strain rate of the tertiary creep process, then !1=n 1_pp ð4Þ D¼12 1_p n is a material constant related to temperature. The definition of damage from Eq. (4) can be used to measure creep damage experimentally. Damage may also be related to the variation in density, resistivity or other material properties. In a fatigue process, during strain controlled cycling, if Ds is the cyclic stress range and Dsp the cyclic stress range at saturation, then the damage may be defined as [3] D¼12

ð3bÞ

The LDA theory is based on the following assumptions: (1) In every loading block, the load must be symmetric. (2) For any stress level, at the beginning or at the end, each creep process leads to the same damage. (3) Loading sequence does not affect life prediction. 2.2. Continuum Damage Mechanics theory For a nominal stress of s; the damage parameter D is zero for a material containing no cracks and unity when rupture takes place. Also s=ð1 2 DÞ; which is the ‘effective’ stress, takes into account the weakness of the material due to the presence of voids or micro-cracks. From a thermodynamic point of view, D is an internal variable of an irreversible damage process, if 1_pp is the strain

Ds D sp

ð5Þ

In terms of this, damage can be measured by strain controlled fatigue tests. Using Df to represent fatigue damage and Dc for creep damage, the incremental form of these two kinds of damage can be written as follows dDf ¼ fF ðDP; Df ; Dc ÞdN

ð6Þ

dDc ¼ fc ðseq ; Dc ; Df Þdt

ð7Þ

Although different defects in a material cannot be added directly, from the definition of effective stress in Damage Mechanics, the decrement of effective area made by different defects can be added. If it is assumed that D ¼ Df þ Dc ð8Þ


391

Eqs. (6) and (7) can be written as dDf ¼ fF ðDP; Df þ Dc ÞdN ¼ fF ðDp; DÞdN

ð9Þ

dDc ¼ fc ðseq ; Dc þ Df Þdt

ð10Þ

and then dD ¼ dDf þ dDc ¼ fF ðDP; DÞdN þ fc ðseq ; DÞdt

ð11Þ

Since both fF and fc are nonlinear, and relate to the variable D; the nonlinear coupling of fatigue– creep is therefore reflected in Eq. (11). The functions fF and fc based on thermodynamic principles have been proposed by Lemaitre [4] who took into account the effects of nonlinear damage accumulation and multiaxial stress RV D1gp þ1 dN Vðg þ 1Þ ð1 2 DÞa1 seq r RV dt dDc ¼ ð1 2 DÞa2 l dDf ¼

ð12Þ ð13Þ

where RV is the triaxial coefficient and is expressed as !2 2 sH ð14Þ RV ¼ ð1 þ nÞ þ 3ð1 2 2nÞ 3 seq and

seq ¼

3 2

Sij Sij

1=2

;

Sij ¼ sij 2 sH dij ;

sH ¼

1 3

skk

in which, the mean stress sH is included and the effects of triaxial load are considered. Many studies demonstrate that mean stress can have a significant influence on fatigue and creep life. Reduction of lives can be as large as an order of magnitude of substantial mean stress. Therefore, in the above damage models, the effects of multiaxial stress are considered by including RV ; and this will affect lives significantly. Under uni-axial load, RV ¼ 1: If Eqs. (12) and (13) are adopted, the incremental creep –fatigue damage Eq. (11) becomes seq r RV D1pgþ1 RV dD ¼ dN þ dt Vðg þ 1Þ ð1 2 DÞa1 ð1 2 DÞa2 l

ð15Þ

a1 ; a2 g; r and l are material constants adjusted to fit the experimental result. They can be obtained by uni-axial stress controlled creep tests and strain controlled fatigue tests. Note that in the present continuum damage model, the damage accumulates nonlinearly and the effects of multiaxial stress are considered. The effects of the nonlinear coupling of fatigue and creep are also taken into account.

3. Temperature and stress field analysis of a 600 MW steam turbine rotor In the control stage and first compressor stage of the High-Pressure (HP) rotor of a 600 MW steam turbine,

Fig. 1. Finite element mesh of high pressure rotor of 600 MW steam turbine.

the temperature of the steam is much higher than other areas. Large temperature gradients and thermal stress usually occur. Therefore, the life of a turbine is mainly dependent on the damage evolution in these stages. In the present paper, the transient temperature and thermal stress field under a practical start– stop operation are calculated by the finite element software ADINA. The finite element model is shown in Fig. 1. The start and stop curves are shown in Fig. 2 [5]. In the calculations all related parameters are considered as functions of time and temperature, they are given in Table 1. The convection coefficients from Westinghouse [6] are adopted. Note that using the monotonic stress – strain constitutive relation to analyze the stress and strain of cyclically softened material would result in an underestimation of the real damage of the material. Therefore the cyclic stress– strain relation of rotor material 30Cr1Mo1V is employed in the stress and strain analysis. The cyclic stress – strain relation is written as 0

Ds=2 ¼ K 0 ðD1p =2Þn

the parameter n0 is the cyclic strain hardening exponent and K 0 is the cyclic strength coefficient. If N0 is the number of cycles at which the drop of peak-stresses become sharp, not slow as before, the cyclic stress – strain relation is selected at N0 =2 cycles. Generally, it is considered that N0 < Nf [7]. The parameters related to the cyclic property of 30Cr1Mo1V are given in Table 2. The monotonic and cyclic stress –strain relations at several temperatures for 30Cr1Mo1V are shown in Fig. 3 [7]. Fig. 3 shows that 30Cr1Mo1V is a cyclic softening material. Therefore, it is dangerous to use the monotonic stress – strain relation to predict the life of a steam turbine. In the analysis, the centrifugal force of the rotor is also taken into account. The calculated results reveal that in the start and stop process, some areas become plastic; the accumulated plastic strain and the triaxial coefficients of these yield locations are shown in Table 3. The effective stress fields at peak strain amplitude under start and stop operations are shown in Figs. 4 and 5. As the load of the steam turbine approaches the maximum rating, the steam temperature also nearly reaches the rated value. Creep usually comes into function at this

392


Fig. 2. Practical start and stop operation curves.

time. The maximum stress and temperature of the rotor at the rated load are given in Table 4. The corresponding temperature field and effective stress field are shown in Figs. 6 and 7. Table 1 Parameters at several temperatures

4. Damage and life analysis and model comparison The traditional creep life prediction equation of 30Cr1Mo1V rotor steel at 525 8C is given by [8]:

tr ¼

Temperature (8C)

100

200

300

400

500

600

Specific heat (J/kg 8C) Dilation coefficient (1026/8C) Heat conduction W/(m k)

487.4

507.6

565.2

622.8

669.6

716.4

seq 381:6

28:596

ð16Þ

Table 2 Parameters of the cyclic property of 30Cr1Mo1V at several temperatures

11.49

12.03

12.43

12.80

13.23

13.32

Temperature (8C) s0f =E

38.9

38.1

33.9

33.1

30.1

26.4

25 510 538

b

10f

c

n0

k0

0.00541 20.0804 1.823 20.825 749.4 0.0505 0.00466 20.0856 0.5635 20.716 592.7 0.0648 0.00375 20.0697 0.8176 20.755 494.7 0.0458


393

Fig. 8 shows that although the LDA model Eq. (16) and the CDM model Eq. (17) are both obtained by one uniaxial creep test, the above CDM model fits well the test data, for which the experimental damage parameter D is defined by Eq. (4). In Fig. 8, the curves show that the damage evolution from the uniaxial CDM model appears nonlinearly. This coincides with the real creep damage evolution process of a rotor material. The linear accumulation law overestimates the creep damage evolution values. From the low cycle test data of 30Cr1Mo1V rotor steel at 510 8C [7], the Manson –Coffin equation and multiaxial fatigue damage equation can be, respectively, written as D1p ¼ 0:5635ð2Nf Þ20:716

ð18Þ

and D11:3966 dD RV p ¼ dN 1:5554 ð1 2 DÞ6:0881

Fig. 3. Stress–strain relation of 30Cr1Mo1V rotor steel. Table 3 Strain and triaxial coefficient of yielded points of HP rotor under start and stop Dangerous points

Front foot of control stage

Back foot of control stage

Sealing of control stage

Root of first stage

Root of second stage

Start D1p Start RV Stop D1p Stop RV

0.00048 1.94 0.00051 2.21

0.00164 2.13 0.00172 2.20

0.000156 1.59 0.000279 1.61

0.00063 1.64 0.00079 1.68

0.00014 1.60 0.00018 1.71

In Eq. (16), the units of tr is hour, and the units of seq is MPa. In terms of the CDM theory (Section 2) and the test data from Ref. [8], the multiaxial creep damage equation of 30Cr1Mo1V rotor steel at 525 8C is written from Eq. (13) as seq 8:569 RV dDc ¼ dt ð17Þ ð1 2 DÞ6:02 478:8

ð19Þ

By using the traditional creep life prediction of Eq. (16) and the CDM model of Eq. (17), the creep life predictions of a high-pressure rotor of a 600 MW steam turbine at the rated load under start – stop operation are calculated. The results are listed in Table 5. It is shown that the life predictions of LDA model coincide with those of an uniaxial CDM model, but are quite different from the results of a multiaxial CDM model. The inherent weakness of the LDA model in not dealing well with the multiaxial problem is reflected. The triaxial coefficient is adopted in the multiaxial CDM model, and the effects of stress state are considered. It is therefore expected that the multiaxial CDM model could predict the life and damage evolution more accurately [9]. In the life predictions of the multiaxial CDM model, the triaxial coefficient affects the results greatly. In Fig. 9, the triaxial coefficient fields corresponding to the stress fields in Fig. 7 are plotted. The results show that the creep life predictions in turbine rotor design may change greatly if the effects of multiaxial loading are considered.

Fig. 4. Stress field of high pressure rotor of 600 MW steam turbine under start operation.

Fig. 5. Stress field of high pressure rotor of 600 MW steam turbine under stop operation.

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Table 4 Effective stress, temperature and triaxial coefficients at dangerous points at rated load Dangerous points

s (MPa)

T (8C)

RV ð/Þ

Front root of control stage Back root of control stage

65.3 70.4

519.3 509.6

1.53 1.64

In Fig. 10 the damage evolution from a multiaxial CDM model is compared with that from a uniaxial CDM model. The results show that the multiaxial complex stress accelerates the damage accumulation greatly. Therefore, in damage analysis of a steam turbine rotor, the effects of multiaxial complex stress must be considered. Similar conclusions have been drawn in fatigue analysis and the details are given in Ref. [10]. From the combination of Eqs. (17) and (19), the CDM model of creep– fatigue can be written as RV dD ¼ ð1 2 DÞ6:02

seq 478:8

D11:3966 p dN ð1 2 DÞ6:0881

8:569

RV dt þ 1:5554 ð20Þ

The average design life of a steam turbine is usually 30 years. Assume that the frequencies of the start and stop operations are three times per year, and the stable running time is 100 days [11] during each start – stop process. Finally, the total damages in 30 years of the back root of the control stage, which is most dangerous, are calculated by

Fig. 6. Temperature field of high pressure rotor of 600 MW steam turbine at rated load.

Fig. 8. Damage evolution for creep of 30Cr1Mo1V at 525 8C and s ¼ 100 MPa:

applying the LDA theory and the multiaxial CDM model. The results are shown in Table 6. In both CDM and LDA theory, the structure fails when damage amounts to 1, i.e. the critical damage Dr ¼ 1: However, tests show that when damage is still at a low value, cracks have initiated and the structure will fracture in very much fewer cycles. Therefore, in CDM, the critical damage is not set to 1. Many tests show that 0:2 # Dr # 0:8: The critical value can be obtained by uniaxial tests as follow [3] pffiffiffiffiffiffi Dr ¼ 1 2 sr = 2EYr

ð21Þ

where sr ; Yr are the critical stress and critical release rate of strain energy when structure rupture. In this study, Dr ¼ 0:24 is obtained by uniaxial test [7]. Fig. 10 shows that the life of the back root of the control stage impeller under multiaxial load has nearly been exhausted when the damage is as low as 0.24. It demonstrates that because damage evolves nonlinearly, damage can be low when there is little remanent life. If life-used rate is written as d ¼ D=Dr ; because the difference of the fracture criteria of CDM and LDA (Dr ¼ 0:24 for CDM in this study and Dr ¼ 1 for LDA), although the estimated damage from the multiaxial CDM model is

Table 5 Creep life at dangerous points (h)

Fig. 7. Stress field of high pressure rotor of 600 MW steam turbine at rated load.

Dangerous points

LDA theory

Uniaxial CDM model

Multiaxial CDM model

Front root of control stage Back root of control stage

3.8441 £ 106

3.8509 £ 106

2.5169 £ 106

2.0407 £ 106

2.0442 £ 106

1.3361 £ 106


395

Fig. 9. Distribution of triaxial coefficient of 600 MW steam turbine at rated load.

0.0741 shown in Table 6, at this time 30.88% ðd ¼ 0:0741=0:24Þ of the life has been used. It is obviously more reasonable than the result from LDA theory of 21.60%. ðd ¼ 0:216=1Þ It is suggested that although the LDA theory may overestimate the damage evolution, it is not always a conservative estimation in life prediction. This should be important in practical rotor design. The results from CDM model also show that the damage under creep– fatigue coupling function is not equal to the damage of creep and fatigue added directly as the LDA theory does. It shows that the coupling function accelerates the damage evolution significantly. The above analyses show that the nonlinear CDM model cannot only takes into account the effects of multiaxial complex stress but also the coupling function of creep and fatigue.

5. Conclusions In this paper, a nonlinear CDM model is employed to predict the creep – fatigue damage and life of a 600 MW stream turbine high pressure rotor, and the results are compared with those from a LDA. The conclusions drawn from the study can be summarized as follows:

1. Since the LDA used a uniaxial assumption both in creep and fatigue analysis, the effects of multiaxial complex stress are ignored. The life is over evaluated. Multiaxial complex stress makes the life shorter and accelerates damage accumulation. Therefore, in the creep – fatigue analysis of a steam turbine rotor, the effects of multiaxial complex stress must be considered 2. The nonlinear coupling function of creep and fatigue accelerates the damage evolution significantly. It must be taken into account in the life assessment of steam turbine rotor. 3. Because damage evolves nonlinearly, damage can be low when there is little remanent life. It is suggested that although the LDA theory may overestimate the damage evolution, it is not always a conservative estimation in life prediction. This could be important in the life prediction of steam turbine rotors.

Acknowledgements The present work is supported by the National High-Tech Research and Development Fund (No. 2002AA412410).

Fig. 10. Creep damage comparison of uniaxial and multiaxial CDM model.

Table 6 Total damage of back root of control stage in 30 years Damage theory Fatigue damage Creep damage Creep– fatigue damage LDA theory CDM Model

0.1102 0.0346

0.1058 0.0230

0.216 0.0741

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[5] Zhang Y. The reliability analysis of the 600 MW steam turbines of the East China electrical net. Elect Power East China 1996;25(11): 45–51. [6] Zhang BH. Life management and load-peak adjustment running of large capacity fossil power generator. China: Water Power Electric Power Publishing Company; 1988. p. 19–27. [7] Li YW, Li CB. Investigation on low-cycle performance of 30Cr1Mo1V rotor steel and damage evolution. Turbine Technol 1998;40(4):248–50.

[8] Li J, Zhou JL. Investigation on creep performance of 30Cr1Mo1V rotor steel. Phys Chem Test—Phys Fascicule 2000;36(1):10–12. [9] Chaboche JL. Continuum damage mechanics. Part 2. Trans ASME, J Appl Mech 1988;55(1):65–72. [10] JianPing J, Yi S. A continuum damage mechanics model on low cycle fatigue life assessment of steam turbine rotor. Int J Pressure Vessels Piping 2001;78(1):59–64. [11] Wang XJ, Shi KZ. Life prediction of 600 MW steam turbine under cold start. Turbine Technol 1998;40(5):269–71.

An Effective Continuum Damage Mechanics Model for Creep

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