Application of a Creep-Damage Constitutive Model for ...

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Jishen Jiang Key Laboratory of Power Machinery and Engineering, Gas Turbine Research Institute, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China e-mail: [email protected]

Weizhe Wang1 Key Laboratory of Power Machinery and Engineering, Gas Turbine Research Institute, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China e-mail: [email protected]

Nailong Zhao Key Laboratory of Power Machinery and Engineering, Gas Turbine Research Institute, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China e-mail: [email protected]

Peng Wang Key Laboratory of Power Machinery and Engineering, Gas Turbine Research Institute, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China e-mail: [email protected]

Application of a Creep-Damage Constitutive Model for the Rotor of a 1000 MW Ultrasupercritical Steam Turbine A damage-based creep constitutive model for a wide stress range is applied to the creep analysis of a 1000 MW ultrasupercritical steam turbine, the inlet steam of which reaches 600  C and 35 MPa. In this model, the effect of complex multiaxial stress and the nonlinear evolution of damage are considered. To this end, the model was implemented into the commercial software ABAQUS using a user-defined material subroutine code. The temperature-dependent material constants were identified from the experimental data of advanced heat resistant steels using curve fitting approaches. A comparison of the simulated and the measured results showed that they reached an acceptable agreement. The results of the creep analysis illustrated that the proposed approach explains the basic features of stress redistribution and the damage evolution in the steam turbine rotor over a wide range of stresses and temperatures. [DOI: 10.1115/1.4031323] Keywords: constitutive model, creep, damage, UMAT, steam turbine rotor

Yingzheng Liu Key Laboratory of Power Machinery and Engineering, Gas Turbine Research Institute, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China e-mail: [email protected]

Puning Jiang Shanghai Electric Power Generation Equipment Co., Ltd., 333 Jiangchuan Road, Shanghai 200240, China e-mail: [email protected]

Introduction 1 Corresponding author. Contributed by the Turbomachinery Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received July 23, 2015; final manuscript received August 3, 2015; published online September 7, 2015. Editor: David Wisler.

To improve the high thermal efficiency of power plants, supercritical steam turbines are widely used for increasing the steam pressure and temperature. However, the strength of rotors subjected to multiaxial creep under the high-temperature conditions may hinder the steam turbine design. Thus, understanding the multiaxial creep behavior of rotors under high temperature is of

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great practical significance to the design and operation of largecapacity steam turbines. Numerous investigations into the multiaxial creep behavior of high-temperature components have been performed. However, a reliable constitutive model is required for structural analysis [1]. In recent decades, the continuum damage mechanics (CDM) of Kachanov have been rapidly developed in a phenomenological manner to describe the multiaxial creep-damage behavior. The classic Kachanov–Rabotnov constitutive model [2,3] extended the power-law model to describe the tertiary creep stage, by introducing a scalar variable for damage measurement. The CDM-based method has been highlighted with the rapid development of finiteelement methods. Multiaxial creep-damage constitutive equations with a single damage variable, e.g., the Kachanov–Rabotnov constitutive equation and Lemaitre constitutive equation, have been generalized [4]. A triaxial factor, Rv , which reflects the effect of the stress state, was defined in the latter equation [5]. In addition, numerous applications of the nonlinear CDM models in the engineering structural analysis have been carried out. JianPing et al. [6] assessed the creep–fatigue life of a steam turbine rotor using a CDM model and discussed the advantages of the CDM model compared to the linear damage accumulation (LDM) theory. Kostenko et al. [7] incorporated a creep-damage constitutive model into the ABAQUS finite element code to elaborate the basic features of creep in thick-wall structural components. However, the mechanical tests disclosed that the materials exhibit stress-range dependence on creep behavior, demonstrating that the power-law creep for a certain stress range may change to the viscous-type creep with a decrease in the stress [8,9]. Furthermore, the ranges of low and moderate stresses are common to many steam turbine components under in-service loading conditions [10]. Therefore, the transition from the power law to the linear creep at relevant stress levels must be significantly considered in a creep constitutive model for a wide stress range. In this paper, a nonlinear creep-damage constitutive model for low and moderate stress is introduced and is implemented in the commercial software ABAQUS, using the user-defined material subroutine (UMAT). Identification of the material constants in the model is developed based on the experimental creep data for a wide loading. Comparison of the simulated and experimental results is performed to validate the model. Furthermore, the rotor of a 1000 MW ultrasupercritical steam turbine is chosen for study. The mechanical behavior of the rotor after 2651 hrs of steadystate running is analyzed in terms of temperature, von Mises stress, and damage.

Mathematical Model

(1)

where e_ cr is the steady-state creep rate, e_ cr v is the rate of viscous mechanism, and e_ cr pl is the rate of power-law mechanism. In addition, A1 ; A2 , and the stress exponent, n, are temperaturedependent material parameters, which can be determined from creep tests at a constant temperature. In the creep-damage mechanism, creep damage accelerates the consumption of the remaining life. Thus, Rabotnov and Kachanov [2,3] proposed a phenomenology method which assumes that the creep rate is additionally dependent on the current damage state. The constitutive model is described as follows: ( cr e_ ¼ e_ cr ðr; T; xÞ (2) _ x_ ¼ xðr; T; xÞ where x is the isotropic damage variable. The expression of the Rabotnov–Kachanov equation is given by 022606-2 / Vol. 138, FEBRUARY 2016

> rk > > _ x ¼ B > : ð1  xÞk

(3)

where A; B; n; r; and k are temperature-dependent material constants, which can be determined from uniaxial creep tension tests. The rupture time, t , based on Eq. (3), can be calculated by t ¼

1 Bð1 þ kÞrr

(4)

and the explicit form of the damage evolution equation can be developed by  1 t kþ1 x¼1 1  t

(5)

Most structural components are subjected to complex multiaxial loading, which can accelerate the damage behavior. The von Mises effective stress, rvM , and the maximum tensile stress, rT , could be used to determine the creep and damage accumulation under a multiaxial stress state. In addition, the creep damage under high loading is predominated by the ductile behavior and leads to the necking of a specimen due to the von Mises stress rvM , while the creep damage under the low loading is characterized by the nucleation and growth of intergranular cavities and microcracks. The minimal ductility leads to the degradation of the material’s microstructure due to the maximum tensile stress rT . Under moderate loading, the creep damage is dominated by both rvM and rT [8,12,13]. Thus, in the multiaxial stress state, Eq. (1) can be developed to satisfy the creep behavior within a wide loading range by taking into account a damage parameter and rational damage evolution equation. These equations can describe the transition from the brittle to the ductile damage mode. The damagebased creep constitutive model [1] is given by "  n # 3 rvM s (6a) e_ cr ¼ a rvM þ 2 rvM r0 ð1  xÞ

x_ ¼

b ð1  xÞl

rvM ¼

Constitutive Model. In this study, a double power-law stress response function [11] is used to describe the creep behavior cr n e_ cr ¼ e_ cr v þ e_ pl ¼ A1 rþA2 r

8  n > r > > _ e ¼ A > < 1x

"

rT r0

k

#   l þ 1 rvM n 1 þ nl n þ 1 r0 ð1  xÞ

rffiffiffiffiffiffiffiffiffiffiffiffi 3 1 s  s; s ¼ r  trrI; 2 3

1 rT ¼ ðrI þ jrI jÞ 2

(6b)

(6c)

where e_ cr is the creep strain rate tensor, s is the stress deviator, rvM is the von Mises stress, rT is the maximum tensile stress, rI is the first principal stress, I is the second rank unit tensor, and x is the damage parameter. a; b; r0 ; n; l; and k are the temperaturedependent material constants, which are identified from the experimental data using curve fitting approaches. For the ranges of low and moderate loadings, Eqs. (6a)–(6c) characterize not only the transition from the linear creep to the power-law creep but also the continuous transition from the pure brittle (rT -controlled) to the pure ductile (rvM -controlled) damage mode. Identification of Material Constants. Equations (6a)–(6c) can be applied in the creep analysis and residual life assessment of the components, which are exposed to high temperatures and low to moderate loadings. Reasonable predictions are strongly dependent on rational material constants in the constitutive models [14]. Various approaches [14–16] to creep material constant identification have been presented. The integral transformation is preferred to obtain explicit equations [14,16]. However, Eqs. (6a)–(6c) are too Transactions of the ASME

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complicated to be simply integrated to obtain explicit forms. Thus, in the present paper, we propose the following method. To determine the material constants with the lack of creep experimental data under multiaxial loadings, the common practice is to obtain the uniaxial form of the constitutive equations and then to fit the equations to the experimental data, which are obtained in a uniaxial tensile test at steady temperature [15]. Equations (6a)–(6c) can be reduced to the uniaxial form "  n # r cr (7a) e_ ¼ a r þ r0 ð1  xÞ

x_ ¼

"  #   r k lþ1 r n 1 þ r0 n þ 1 r0 ð1  xÞnl ð1  xÞl b

(7b)

The material constants’ identification process of Eqs. (7a) and (7b) can be represented by the following steps: Step 1. As the damage parameter has little influence on the secondary creep stage, the minimum (secondary) creep rate, e_ cr min , can be expressed in terms of stress without the damage parameter, x, by transforming Eq. (7a) into "  n # r ¼ a r þ (8) e_ cr min r0 Using the simple linear curve fitting method, the data of the minimum creep rate e_ cr min can be derived from the secondary creep data taken from the uniaxial tests at various constant stresses. In addition, the set of a; n; r0 values can be identified by means of nonlinear curve fitting methods, which are for the most part supported by MATLAB. Step 2. The material constants b; k; and l strongly influence the damage evolution. As the complex Eq. (7b) cannot be integrated to the explicit formula, it is divided into two parts: one is the pure brittle (rT -controlled) damage mode under the low loading and the other one is the pure ductile (rvM -controlled) damage mode under the high loadings  k r under low loading ð1  xÞl r0

(9a)

 n lþ1 b r under high loading n n þ 1 ð 1  xÞ r 0

(9b)

x_ ¼

x_ ¼

b

The transition stress, r0 , which reflects the transition from the linear creep mechanism to the power-law creep mechanism, splits the wide stress range into the ranges of “low,” “moderate,” and “high” stress values. The experimental data after Ref. [17] show that the value of r0 declines with increased temperature. Step 3. According to step 2, Eqs. (9a) and (9b) can be, respectively, integrated into explicit forms to identify the material constants b; k; l of the damage evolution equation using the nonlinear curve fitting methods [14,15,18]. It is noteworthy that the material constants b; k; and l should be determined by fitting the creep data within the low and high stress ranges, as the moderate stress range is characterized by the transition from linear to power-law dependence.The predicted results, based on the above-mentioned fitting approaches, and the constitutive model are plotted in Figs. 1 and 2, and compared to the experimental results of 9Cr-1Mo-V-Nb (ASTM P91) steel [8,11,19,20] and %12Cr steel. As shown in the figures, the simulation and measurement results are in acceptable agreement. Finite Element Model. In the present study, the rotor of a 1000 MW ultrasupercritical steam turbine is chosen and a 2D 1:1 scaled axisymmetric finite element model is established. Figure 3 Journal of Engineering for Gas Turbines and Power

Fig. 1 Comparison of fitting curve and experimental data [8,11,19,20] for 9Cr-1Mo-V-Nb (ASTM P91) steel at 873 K: (a) minimum creep rate e_ min cr versus stress r and (b) creep ecr versus time t

shows the geometry and mesh of the model. The total number of computational elements for the rotor is 19,580. CAX8RT (eight-node biquadratic displacement, bilinear temperature, reduced integration)-type elements are adopted to combine temperature and displacement. The mechanical properties of %12Cr steel are listed in Table 1. The constitutive model (Eqs. (6a)–(6c)) is implemented into the commercial package ABAQUS, using the UMAT code [17], to numerically investigate the creep behavior. With regard to the boundary conditions, the axial displacement constraint is set at the front head surface of the rotor.

Fig. 2 Comparison of fitting curve and experimental data of creep ecr versus time t for %12Cr steel at 873 K

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Fig. 3 Finite element mesh of the high-temperature part of a 1000 MW ultrahigh-pressure rotor

Thermomechanical loadings are obtained from the steam turbine manufacturer. The rotor is subjected to the centrifugal force of itself due to high-speed rotation (3000 r/min), and the axial pressures from blades, which are applied on the contact surfaces of blade grooves, are in the form of equivalent loads to simulate the force of the blades on the rotor. Some areas of the rotor, especially the blade grooves, may endure remarkable local stress concentration, which may lead to a large amount of damage. The simulation is terminated after 2651 hrs when the amount of damage is close to a critical value eventually in the grooves of the first stage.

Results and Discussion Temperature and Stress Field of the Rotor. First, Fig. 4 displays the overall temperature distribution. As seen from the figure, the maximum temperature of 597.5  C is located at the steam inlet notch zone, and the overview of the temperature distribution shows that the temperature decreases in the axial direction. The minimum temperature of 433.3  C is found at the right side of the balance piston, which has insignificant creep behavior. In the rotor grooves, the temperature exceeds 550  C locally, meaning they are susceptible to creep-damage mechanisms. Further, these grooves experience the centrifugal force from the blades during operation, which would contribute substantially to the damage accumulation. The stress distribution in the rotors is plotted in Fig. 5 to illustrate the creep stress behavior. Figure 5 shows the von Mises equivalent stress distributions of the rotor at different times. The figure shows that the larger von Mises stress is mainly concentrated on the surface, e.g., the blade grooves and the steam inlet notch zone. At the blade groove (point A) of the third stage, the von Mises stress reaches the maximum value of 298.9 MPa at the beginning of the operation period. The von Mises stress at the blade groove (point B) of the first stage reaches 269.8 MPa. Although this is less than that at point A, the temperature at point B has the higher value of 583  C. Thus, the combination of the blade centrifugal force and the temperature could aggravate the creep deformation at point B. To further demonstrate the influence of the creep deformation on stress, Fig. 5 also shows the development of the maximum von Mises stress at I and III zones. Experiencing 2651 hrs of creep behavior, the maximum von Mises stress at point A decreases from 298.9 MPa to 222.8 MPa, and the amplitude of the decrease

reaches 76.1 MPa. For the results at point B, we can see the decrease in amplitude is 77.4 MPa, from 269.8 MPa to 192.4 MPa. Corresponding to the results in other zones, the von Mises stress at points A and B shows significant change due to the influence of the creep deformation. This demonstrates that the combination of the blade centrifugal forces and the high temperature accelerates the creep deformation, which could lead to a significant increase in the local damage. Creep-Damage Analysis of the Rotor. Figure 6 shows the damage distribution at t ¼ 2651 hrs. The overall distribution illustrates that the damage in most areas is maintained at 0. The significant damage behavior exists at points A and B, with point B having the greatest damage, reaching 0.412. Although the stress at point A is larger than that at point B, the damage at point A only reaches 0.14, which is much lower than for point B. The creep damage is strongly related to the temperature and the stress. In particular, the temperature has a significant influence on the creep deformation. In Eqs. (6a) and (6b), the values of the material constants identified from the creep data may change greatly at different temperatures. The constants describing the damage evolution can be expressed in simple temperature-dependent form using an Arrhenius-type function, which reveals that temperature may have an exponential contribution to damage. As shown in Fig. 4, the temperature at point B reaches 590  C, which is higher than 556  C at point A. Thus, the combination of temperature and stress induces more significant damage at point B. Figure 7 shows the plot of the stress evolution at point B. The von Mises stress and the maximum tensile stress decline sharply at the beginning of the creep period due to a significant primary creep rate, and then enter a smooth failing period for the most part. At the end of the creep period, stresses are significantly reduced because of the large creep strain and damage at the tertiary creep stage. The damage evolution at point B is plotted in Fig. 8. The damage rate in Fig. 8 increases initially due to the high initial stress state. The initial relaxation of stress then decreases the damage accumulation rate. Following this, the gradual decrease in stress causes the slow evolution of damage through the bulk of the component’s life. At the tertiary creep stage, the continually accumulating cracks or creep voids aggravate the growth of damage. Equation (6b) also explains this acceleration. The damage rate has

Table 1 Mechanical properties under different temperatures Temperature ( C) 20 200 400 600

Young’s modulus (MPa)

Poisson’s ratio

Thermal conductivity (mW mm1 K1)

2.100  105 1.988  105 1.818  105 1.372  105

0.288 0.289 0.299 0.314

26.3 26.5 26.6 27.7 Fig. 4

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Temperature distribution of rotor in operation

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Fig. 5

von Mises stress distributions of rotor in operation

Fig. 6 Damage distribution of the rotor at t 5 2651 hrs

close relations with the damage, the von Mises stress, and the maximum tensile stress. The gradually increasing level of damage has a positive role in promoting the damage rate, so that the damage is ascending with an increasing rate of growth even though the stress exhibits a sharp decrease in the end.

Fig. 8 Evolution of damage, x, at point B during the whole operation

It is noteworthy that the dangerous zone with the highest damage value may not occur where there is the highest stress state (the maximum value of the von Mises stress and the maximum tensile stress is at point A, while the maximum value of damage is at point B), because the effect of temperature may surpass the effect of stress in the creep regime. According to the above-mentioned results, the significant damage at point B may generate cracks, which are harmful to the service life of the rotor. This should be taken into account during the design and the operation of the rotor.

Conclusion

Fig. 7 Evolution of von Mises stress, rvM , and maximum tensile stress, rT , at the key point B

Journal of Engineering for Gas Turbines and Power

In this paper, a nonlinear damage-based creep constitutive model was established to predict the creep damage for a rotor of a 1000 MW ultrasupercritical steam turbine. This model was implemented into the commercial software ABAQUS using the UMAT code. The validation of the model was carried out based on the experimental data. Further analysis of the creep behavior of the rotor FEBRUARY 2016, Vol. 138 / 022606-5

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was performed in terms of the temperature, von Mises stress, and damage. The comparison of the simulated and measured results showed an acceptable agreement. The analysis of the creep behavior in the rotor showed that the maximum stress was located at the blade groove of the third stage. However, the maximum damage existed at the blade groove of the first stage. Analysis revealed that the combination of high temperature and the blade centrifugal forces induced the maximum damage of 0.412 at point B. This should be taken into account during the design and operation of the rotor and gives an indication of the precautions that should be taken, providing input for future improvements in high-temperature component design.

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[8] Gorash, Y., 2008, “Development of a Creep-Damage Model for NonIsothermal Long-Term Strength Analysis of High-Temperature Components Operating in a Wide Stress Range,” Ph.D. thesis, Martin-Luther-Universit€at Halle-Wittenberg, Halle an der Saale, Germany. [9] Naumenko, K., 2006, “Modeling of High-Temperature Creep for Structural Analysis Applications,” Habilitationschrift, Mathematisch-NaturwissenschaftlichTechnische Fakult€at, Martin-Luther-Universit€at Halle-Wittenberg, pp. 17–63. [10] Naumenko, K., and Altenbach, H., 2007, Modeling of Creep for Structural Analysis, Springer, Berlin. [11] Kloc, L., and Sklenicˇka, V., 1997, “Transition From Power-Law to Viscous Creep Behaviour of P-91 Type Heat-Resistant Steel,” Mater. Sci. Eng., A, 234–236, pp. 962–965. [12] Niu, L. B., Kobayashi, M., and Takaku, H., 2002, “Creep Rupture Properties of an Austenitic Steel With High Ductility Under Multi-Axial Stresses,” ISIJ Int., 42(10), pp. 1156–1161. [13] Lee, J. S., Armaki, H. G., Maruyama, K., Muraki, T., and Asahi, H., 2006, “Causes of Breakdown of Creep Strength in 9Cr–1.8 W–0.5 Mo–VNb Steel,” Mater. Sci. Eng., A, 428(1), pp. 270–275. [14] Hyde, T. H., Sun, W., and Tang, A., 1998, “Determination of Material Constants in Creep Continuum Damage Constitutive Equations,” Strain, 34(3), pp. 83–90. [15] Dunne, F. P. E., Othman, A. M., Hall, F. R., and Hayhurst, D. R., 1990, “Representation of Uniaxial Creep Curves Using Continuum Damage Mechanics,” Int. J. Mech. Sci., 32(11), pp. 945–957. [16] Othman, A. M., and Hayhurst, D. R., 1990, “Multi-Axial Creep Rupture of a Model Structure Using a Two Parameter Material Model,” Int. J. Mech. Sci., 32(1), pp. 35–48. [17] ABAQUS, 2010, ABAQUS V. 6.10—User Subroutines Reference Manual, Dassault Syste`mes Simulia Corp., Providence, RI. [18] Shi, D. Q., and Yang, X. G., 2004, “Application of the Time-Hardening Creep Law Coupling Damage,” J. Aerosp. Power, 19(1), pp. 12–16. [19] Kloc, L., and Sklenicˇka, V., 2004, “Confirmation of Low Stress Creep Regime in 9% Chromium Steel by Stress Change Creep Experiments,” Mater. Sci. Eng., A, 387–389, pp. 633–638. [20] Kimura, K., Kushima, H., and Sawada, K., 2009, “Long-Term Creep Deformation Property of Modified 9Cr–1Mo Steel,” Mater. Sci. Eng., A, 510, pp. 58–63.

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