Investigating crack propagation behavior and damage evolution in G115 steel under combined steady and cyclic loads

Theoretical and Applied Fracture Mechanics 100 (2019) 93–104

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Investigating crack propagation behavior and damage evolution in G115 steel under combined steady and cyclic loads

T

⁎

Zhengxin Tanga,b, Hongyang Jinga,b, Lianyong Xua,b,c, , Lei Zhaoa,b, Yongdian Hana,b, Bo Xiaoa,b, Yu Zhanga,b, Haizhou Lia,b a

School of Materials Science and Engineering, Tianjin University, Tianjin 300350, PR China Tianjin Key Laboratory of Advanced Joining Technology, Tianjin 300350, PR China c State Key Laboratory of Engines, Tianjin University, Tianjin 300072, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Finite element simulation Crack growth G115 steel Creep-fatigue

The creep-fatigue crack growth (CFCG) behavior and damage evolution of G115 steel were investigated based on modified multiaxial damage constitutive model. Multiaxial stress was taken into consideration in the fatigue damage model, and the simulated and experimental results were in close agreement, which verify the validity of the creep-fatigue model. Simulations of tests with different hold times ranging from 60 s to 7200 s were conducted and the parameters ΔK and (Ct )avg were utilized to characterize the crack propagation data. Results revealed that both frequency ranges of creep-fatigue interaction, and the frequency for the most significant interaction, shifted to higher frequencies as ΔK was increased. Meanwhile, for longer hold times or in higher (Ct )avg regions, variations of crack growth rate, (da/dt)avg became limited and CFCG data tended to pure creep crack propagation data, indicating domination of time-dependent creep. Additionally, the creep and fatigue damage at crack tip were extracted to investigate the damage evolution during crack growth process. It was verified further that the CFCG of G115 steel studied was a result of the effect of dominant creep damage, loworder fatigue, and creep-fatigue interactions, especially for longer hold time conditions.

1. Introduction With the aim of achieving environmental friendliness and improving thermal efficiency, the ultra-supercritical (USC) power plants have been employed [1-3]. Flexible operation (characterized by startups and shut-downs) of power plants is required, due to the needs for electricity [2,4]. Thus, the loading suffered by power plant components is a combination of thermal loads caused by temperature variations, and mechanical loads resulted from pressure variations. Therefore, the creep, fatigue, and creep-fatigue interaction damage can be generated, with the high-temperature load carrying capacity of these components being of significant importance for safety. Because of outstanding hightemperature mechanical properties, large thermal conductivity, excellent weldability, and low thermal expansion coefficient, 9%–12% Cr ferritic-martensitic steels are suitable candidate materials [2,5-8]. Notably, when steam temperature reaches 625–650 °C, ferritic-martensitic steels used previously such as P91, P92, and P122 exhibit significant degradation in material properties and are rendered unsuitable for further use. Recently, G115 steel has been developed and it displays excellent elevated-temperature properties (even under 650 °C), which

⁎

renders it suitable for USC power plants operated at 625–650 °C [3,913]. Meanwhile, it is difficult to detect sub-sized cracks in power plant components before they enlarge [14,15]. These defects need to be monitored during service, to prevent them from reaching a dangerous size under creep-fatigue loading. Determination of safe inspection intervals and the prediction of remaining life are usually conducted based on creep, fatigue, and CFCG tests. Creep-fatigue loading is simulated by adding various dwell times at the maximum load of a triangular wave under isothermal conditions in the laboratory. Numerous studies have been conducted to investigate the CFCG behavior of engineering alloys [2,5-7,14,16,17]. Narasimhachary et al. [2] performed CFCG tests on P91 steel under various hold times showing that the crack growth rate per cycle improved when dwell time increased. Xu et al. [5] studied crack growth behavior of P92 steel with various dwell times, crack depths, and specimen thicknesses. Results showed that CFCG was significantly dependent on the dwell time, and the crack growth rate became larger when the crack depth was deeper and the specimen was thicker. Shi et al. [6] and Bassi et al. [7] studied the CFCG of P92 and T/ P91 steels, respectively. These tests indicated that CFCG behavior is

Corresponding author at: School of Materials Science and Engineering, Tianjin University, Tianjin 300350, PR China. E-mail address: [email protected] (L. Xu).

https://doi.org/10.1016/j.tafmec.2019.01.009 Received 10 November 2018; Received in revised form 7 January 2019; Accepted 7 January 2019 Available online 08 January 2019 0167-8442/ © 2019 Elsevier Ltd. All rights reserved.

Theoretical and Applied Fracture Mechanics 100 (2019) 93–104

Z. Tang et al.

Nomenclature

ΔK stress intensity factor range W, B the width, thickness of CT specimens a crack length (da/dt)avg average time rate of crack growth (Ct)avg time average value of Ct th, tc hold and cycle time of the trapezoidal waveform total force-line displacement ΔV ΔVc , ΔVe creep and elastic part ofΔV η scale factor f frequency (da/dN)fatigue, (da/dN)creep cycle- and time-dependent crack growth rate C, m parameters according to the simulated FCG test C* creep fracture parameter P force applied to CT specimen Vc force-line displacement resulting from creep ȧNSWA crack growth rate predicted by NSWA model

MCDF multiaxial creep ductility factor ω̇c creep damage rate ε ċ equivalent creep strain rate multiaxial and uniaxial creep ductility ε ∗f , εf A, n creep parameters σe , σm equivalent and mean (hydrostatic)stress dωf / dN fatigue damage accumulated per cycle fatigue parameters q, C1,β failure cycle Nf Δεp plastic strain range per cycle σmax , σmin ,Δσ the maximum, minimum stress, and stress range K ' , n' hardening parameters triaxial coefficient Rv v Poisson’s ratio ω , ωc ,ωf the total, creep and fatigue damage R force ratio Pmax, Pmin maximum and minimum force

mainly cycle-dependent under short hold times, but primarily timedependent under longer hold times. Lu et al. [14] studied the influence of temperature and hold time on the CFCG behavior of two nickel based superalloys, showing that the fracture mode was mainly intergranular with 2 min dwell times. However, testing requires high capital and time costs, especially for experiments with low load levels. Meanwhile, the hold times during the operation of power plants are usually relatively longer than these used in experiments [2,14,16-18], and conducting longer dwell time tests also requires much more costs. Therefore, increasing efforts have been devoted into finite element (FE) calculations and analytical model, to investigate the crack growth and damage process [5,19-29]. Gallo and Glinka et al. presented a unified model to estimate the stresses and strains at blunt V-notches under non-localized creep [21] which was further extended to sharp V-notches and cracks based on the concept of the strain energy density [22]. Based on damage mechanics, Hyde et al. [23,24] carried out FE simulations on parent and cross-weld specimens of P91 steel using the Liu and Murakami model [30] and their results agree with significant precision to the tests. Wen and Tu et al. proposed a novel multiaxial creep ductility factor (MCDF), which is the ratio of creep failure strain under multiaxial and uniaxial stress states [25]. They also developed a new creep-damage model which successfully predicts the creep crack growth (CCG) of 316H steel [26]. Jing et al. [27] developed a CFCG model including interactions of creep and fatigue, and investigated the CFCG behavior of P91 steel by FE calculations. A novel damage model, which considered the maximum stress effect, was proposed by Xu and Zhao et al. [28], and it was utilized to predict the CFCG behavior of P92 [5] and G115 steel [29]. However, up to now, studies on the CFCG properties of G115 steel are still limited and the effects of dwell time haven’t been investigated systematically, especially for long dwell time conditions. Therefore, a creep-fatigue damage model that considered the multiaxial stress in fatigue damage equation was initially established in this paper. Then, based on this model, FE simulations were conducted to study the CFCG behavior of G115 steel with various hold times. Simulated results were compared with existing test results, and they were in close agreement. Finally, crack growth behavior and damage evolution were analyzed exhaustively.

ω̇c =

ε ċ ε ∗f

(1)

where ε ċ denotes the equivalent creep strain rate, ε ∗f represents multiaxial creep ductility determined by the uniaxial failure strain and MCDF. Failure of elements around the crack tip occurs when accumulated creep strain attains the local ε ∗f [31]. The Norton power-law creep model is widely used to describe steady-state creep rate which can be expressed as follows:

the (εf ) the

εċ = Aσen

(2)

the

where A and n are material parameters that can be determined from uniaxial creep tests whereas σe is equivalent stress. According to the power-law creep controlled cavity growth theory, Wen and Tu et al. [25] proposed a MCDF that provides a more accurate ε ∗f than other MCDFs:

ε ∗f εf

2 n − 0.5 ⎞ ⎤ n − 0.5 ⎞ σm ⎤ /exp ⎡2 ⎛ = exp ⎡ ⎛ ⎢ ⎢ ⎥ 3 n 0.5 + ⎝ ⎠ ⎝ ⎣ ⎦ ⎦ ⎣ n + 0.5 ⎠ σe ⎥

(3)

where σm is the mean (hydrostatic) stress. Therefore, the creep damage (ωc ) can be obtained by: c

ωc =

∫ εε∗̇ dt f

=

Aσ n

e ∫ exp[2/3((n − 0.5)/(n + 0.5))]/exp[2( n − 0.5)/(n + 0.5) σm/σe ] εf

(4)

dt

Based on Lemaitre’s fatigue damage equations [32], Xu and Zhao et al. [28] presented a novel fatigue damage model in which both the maximum stress and plastic strain are considered:

dωf dN

=

(1 − ωf )−q Nf (Δεp)

=

(1 − ωf )−q C1 (σmax Δεp)−1/ β

(5)

where dωf / dN is fatigue damage accumulated per cycle; q, C1, and β are material parameters; Nf is the failure cycle which relies on the plastic strain range per cycle (Δεp ); σmax is the maximum stress in one cycle which is related to the cyclic stress–strain relation as follows: n′

2. Constitutive model for FE analysis

Δεp Δσ σmax − σmin ⎞ = = K′ ⎛ 2 2 ⎝ 2 ⎠

In this study, the creep ductility exhaustion approach is utilized to predict the creep damage accumulation near the crack tip. The creep damage rate (ω̇c ) is calculated by

where σmin and Δσ are minimum stress and the stress range, respectively; K ′ and n′ are material parameters obtained from fully reversed fatigue tests. Through Eq. (6), plastic strain range per cycle (Δεp ) can be calculated.

⎜

94

⎟

(6)

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this steel at the corresponding temperature, other parameters for the FE simulations (shown in Table 3) were obtained by non-linear fitting method. Based on these parameters, FE simulations have been conducted. Creep deformation behavior was calculated using the internal codes of ABAQUS/Standard. Damage accumulation and crack growth behavior were simulated by implementing Eqs. (4), (9), and (10) into the user subroutine USDFLD of ABAQUS. Fig. 5 shows a typical finite element damage contour of the CT specimen, in which the 1red area represents the damage zone. As time passed, the total damage of an integration point (represented by SDV2, which is the second solution dependent state variable) increased from 0 to a critical value 0.99, and local failure was assumed to occur. Then, the elastic modulus decreased to a small value and the failure element lost the load bearing ability. In this way, the crack extended and the crack length was obtained by averaging the number of completely damaged elements along three equally spaced lines ahead of the initial crack tip.

Furthermore, multiaxial stress can influence fatigue life significantly [27,32-34], and Lemaitre [34,35] proposed a triaxial coefficient R ν to describe the triaxiality, which is given by: 2

Rν =

2 σ (1 + ν ) + 3(1 − 2ν ) ⎛ m ⎞ 3 ⎝ σe ⎠ ⎜

⎟

(7)

where ν is Poisson’s ratio and R ν = 1 for uniaxial load condition. Therefore, the influence of multiaxial stress on fatigue damage accumulation is considered in current study and subsequently the fatigue damage model is derived:

dωf

=

dN

R ν (1 − ωf )−q C1 (σmax Δεp)−1/ β

(8)

Thus, fatigue damage (ωf ) could be calculated by:

ωf =

R (1 − ω )−q

∫ C1 (νσmax Δεpf)−1/β dN

(9)

In addition to damage due to fatigue and creep, damage caused by creep-fatigue interactions should also be considered. Skelton et al. [36] proposed a coupled model to describe the non-liner damage accumulation as follows:

ω=

ωf ωc + 1 − ωf 1 − ωc

3.3. Determination of fracture mechanical parameters In the creep-fatigue environment, materials can be divided into two types: creep-ductile and creep-brittle materials [37]. Creep deformations at the crack tip dominate the CFCG in creep-ductile materials while limited creep strains occur during CFCG process of creep-brittle materials. The stress intensity factor range (ΔK ) is suitable to represent the CFCG behavior of creep-brittle materials and this parameter is given as follows:

(10)

Presently, the creep-fatigue damage constitutive model has been established, which consists of Eqs. (4), (9), and (10). 3. FE calculation procedure

ΔK = 3.1. Finite element model

(1 − R) Pmax F (a/ W ) B1/2W 1/2

(11)

2+a/W ⎤ (0.886 + 4.64(a/ W ) F (a/ W ) = ⎡ ⎣ (1 − a / W )3/2 ⎦ − 13.32(a/ W )2 + 14.72(a/ W )3 − 5.6(a/ W ) 4)

According to ASTM E2760-10 [37], a CT specimen with W = 30 mm was used, and in Fig. 1 the geometry and size of the employed CT specimen is shown. Because of the symmetry in loading and geometry, a one-quarter three-dimensional model was used to conduct the creepfatigue damage and crack growth FE analysis. In Fig. 2 meshing and boundary conditions of the FE model are shown, including 21,180 C3D8R (8-node linear brick, reduced integration) elements. To reduce computing time, the crack tip region is refined, while larger elements are utilized in other zones, as shown in Fig. 2. Furthermore, the surface of the hole in the CT is rigidly linked to the center point using a multipoint constraint (MPC), and therefore the load applied on this reference point is transferred to the specimen. Displacement and rotation of the reference point are constraint, except for displacement in the Y direction. As displayed in Fig. 2, the plane 1 possesses Z symmetric (ZSYMM) boundary condition, whereas a Y symmetric (YSYMM) boundary condition is applied on the crack growth plane (plane 2). Fig. 3 shows the cyclic load utilized for CFCG tests which is applied on the reference point of the FE model. The trapezoidal waveform has a load ratio R = Pmin/Pmax = 0.1, while a range of dwell times (th) from 60 to 7200 s are utilized to investigate the influence of dwell times on the CFCG and damage evolution.

(12)

where R and Pmax are the force ratio and the maximum force, respectively, and W, B, and a denote the width, and thickness of CT specimens, and crack length, respectively. It is recommended that crack growth rate, (da/ dt )avg is represented by (Ct )avg (the average value of Ct) for creep-ductile materials; (da/ dt )avg is obtained by:

(da/ dt )avg =

1 ⎛ da ⎞ th ⎝ dN ⎠

(13)

The force-line displacement obtained from tests or FE simulations can be used to calculate (Ct )avg through the following equation:

(Ct )avg =

ΔP ΔVc F ′ BWth F

(14)

where ΔVc is the force-line displacement resulted from creep deformation during hold time per cycle and can be determined by:

ΔVc = ΔV − ΔVe = ΔV −

th (da/ dt )avg Pmax

2ΔK 2 ⎤ B⎡ ⎥ ⎢ E′ ⎦ ⎣

(15)

where ΔV and ΔVe are the total force-line displacement and elastic part of ΔV per cycle, respectively. Here, E′ represents E /(1 − υ2) and E for plane strain and plane stress state, respectively. Term F ′/ F is calculated by:

3.2. Material parameters The material investigated in current study is G115 steel with the chemical composition (in wt.%) and heat treatment of the as-received steel being provided in Table 1. Fig. 4 shows the true stress-strain curve of G115 steel at 650 °C [9], which corresponds to the strain rate 5.2 × 10−5 s−1, and the data beyond the yield strength has been utilized as input for the FE calculations. Meanwhile, the corresponding material properties determined from the tensile test are shown in Table 2. Uniaxial failure strain, εf = 0.10 , was taken as the average value of the uniaxial failure creep strain under different load levels at 650 °C [12]. According to the uniaxial creep and strain-controlled fatigue tests of

F′ F

=⎡ ⎣

(

1 2+a/W

)+(

3 2(1 − a / W )

) ⎤⎦

4.64 − 26.64(a / W ) + 44.16(a / W )2 − 22.4(a / W )3

+⎡ ⎤ ⎣ 0.886 + 4.64(a / W ) − 13.32(a / W )2 + 14.72(a / W )3 − 5.6(a / W )4 ⎦

(16)

1 For interpretation of color in Figs. 5 and 6, the reader is referred to the web version of this article.

95

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Fig. 1. Geometry and size of the employed CT specimen.

Fig. 2. Meshing and boundary conditions of the finite element model.

simulation results and the tests results were conducted. The test conditions for comparisons taken from [17] are presented in Table. 4. The specimen named CT5, which has the same test conditions with CT4 in [17], is newly conducted. Fig. 6 displays a comparison between the predicted cycles by FE calculations and experimental cycles for G115 steel at 650 °C. The predicted cycles are obtained when the FE simulated crack propagation length reaches the crack growth length obtained from the corresponding hold time tests. The red solid line in Fig. 6 denotes an error band of factor 1.5 while the black solid line denotes an error band of factor 1.3, and if a point is located on the dashed line in Fig. 6, the FE predicted cycle is identical to the experimental cycle. It can be observed in Fig. 6 that the predicted cycles for 0, 60 and 180 s hold times are close to the test results which demonstrates the high degree of precision of the damage model. Fig. 7 displays the comparisons between the crack lengths obtained from FE simulations and experiments under various dwell time conditions. Crack growth duration increases as the hold time increases, and the FE results were nearly consistent with the experimental results, indicating that the damage model can accurately predict the CFCG behavior. It can be

Fig. 3. Cyclic load used for creep-fatigue crack growth tests.

4. Results and discussion 4.1. Comparison between results from FE and experiment To validate the creep-fatigue damage model and the corresponding material parameters determined, comparisons between the FE 96

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Z. Tang et al.

Table 1 Chemical composition (in wt.%) and heat treatment of the as-received G115 steel. Element

C

Cr

W

Co

Cu

Mn

Amount Heat treatment

0.08

8.8 2.8 3.0 1.0 0.5 (1) 1070 °C for 1.5 h, water cooled to room temperature; (2) 780 °C for 3 h, air cooled to room temperature.

Si

V

Nb

N

B

Fe

0.3

0.2

0.06

0.008

0.014

Bal.

slightly overestimates the crack growth rate, predicting a shorter crack growth life. However, the two results for 600 s hold time are still located within the error band in Fig. 6. As pointed in standard ASTM E2760 [37], the CFCG data exhibit scatter. The (da/dt)avg at a given (Ct)avg for creep-ductile materials or da/dN at a given ΔK for creepbrittle materials may vary by as much as a factor of 2 to 3 when all the conditions are kept constant. Therefore, the prediction for 600 s hold time condition is still reliable, and the slightly overestimation of the crack growth rate can ensure safety of components when conducting assessments. In addition, the simulated pure creep crack growth (CCG) behavior appears to agree with the test results, even though the test data are limited. Comparisons between the FE simulated crack growth rates and experimental crack growth rates for pure fatigue condition are displayed in Fig. 8. The fatigue crack growth (FCG) rate increased at a nearly constant rate, implying a linear relation between da/dN and ΔK for both the FE results and the test results. Meanwhile, FE derived data are approximating the experimental data, and the slope of the two data lines exhibit a good agreement. Therefore, maximum stress considered

Fig. 4. True stress-strain curve of G115 steel at 650 °C [9]. Table 2 Material properties of G115 steel at 650 °C [9]. Young’s modulus, E (MPa)

Poisson’s ratio, v

Yield strength, σys (MPa)

Tensile strength, σuts (MPa)

% Elongation

124,000

0.3

267

281

20

in the fatigue damage model by Xu and Zhao et al. [28] and multiaxial stress introduced in this study, can together adequately predict fatigue crack growth behavior. As mentioned in Section 3, the element failure technology is utilized to simulate the crack propagation behavior. In fact, boundary condi-

Table 3 Material parameters for the creep-fatigue damage model. A (MPa - nh - 1)

n

C1

β

q

K′(MPa)

n′

1.25E-26

9.36

2600

1.96

0.97

633

0.15

Fig. 5. Finite element damage contour of CT specimen.

tions still exist after crack propagation and the simulated crack can’t open as the real crack in test causing smaller force-line displacements [27]. As shown in Eqs. (14) and (15), the parameter (Ct )avg is dependent

observed in Figs. 6 and 7 that the difference in cycle/life between the predicted results and the experimental results is greater for the tests with 600 s dwell time than for any other condition. The FE simulation 97

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Table 4 Creep-fatigue crack growth test conditions. Specimen ID

Hold time(s)

Max. load(kN)

Initial ΔK (MPa m )

Final ΔK (MPa m )

Initial a(mm)

Final a(mm)

Nf cycles

CT1[17] CT2[17] CT3[17] CT4[17] CT5 CT6[17]

0 60 180 600 600 infinite

3.84 3.84 3.84 3.84 3.84 3.84

17 17 17 17 17 17

38.76 33.19 30.82 31.11 29.17 18.66

10.5 10.5 10.5 10.5 10.5 10.5

18.36 17.29 16.67 16.75 16.19 11.58

13,545 12,797 9345 9219 7717 1

Fig. 8. Comparisons between the FE simulated crack growth rates and experimental crack growth rates for pure fatigue condition.

Fig. 6. Comparison between the FE predicted cycles and experimental cycles for G115 steel at 650 °C.

follows:

ΔV ⎤ ⎛ e⎞ ΔVc = η (ΔV − ΔVe′) = η ⎡ ⎢ΔV − ⎝ ΔV ⎠avg ΔV ⎥ ⎣ ⎦ where η is a scale factor (taken as 3.3 in this study), and

(17)

( )

ΔVe ΔV avg

= 0.16

is the average value of ΔVe /ΔV taken from the CFCG tests with different hold times [17]. Fig. 9(a) provides a comparison of (da/ dt )avg and (Ct )avg based on two calculation methods under 180 s dwell time condition. It is evident that the differences between the FE results calculated from Eq. (17) and the experimental results are much smaller when compared with those between FE results obtained from Eq. (15) and test results. The adoption of Eq. (17) improves FE accuracy, compared with previous investigations [27,38]. In this study, all values of (Ct )avg used below are calculated based on Eq. (17). Fig. 9(b) shows the relationship between (da/ dt )avg and (Ct )avg for FE simulated results and test results. The development tendencies of the crack growth rates obtained from simulations are similar to that from experiments under all hold times. Meanwhile, a nearly linear relationship between (da/ dt )avg and (Ct )avg is shown and the differences in the CFCG rate (da/ dt )avg for various hold time tests are relatively small. Based on the comparisons above, it’s apparent that the creep-fatigue damage constitutive model described in Section 2 can adequately predict the CFCG behavior of G115 steel.

Fig. 7. Comparisons between the crack lengths obtained from FE simulation and experiment under various dwell time conditions.

on the force-line displacement, and it would be much smaller than test results at the same value of (da/ dt )avg , if we directly use the FE simulated force-line displacement for the calculation. To use the FE method more accurately, modification for the FE simulated force-line displacement is of great importance. Previous studies [1,2,17] have shown that the instantaneous elastic part of force-line displacement, ΔVe is much smaller than the creep part, ΔVc for creep-ductile material. Further, the difference of force-line displacement ratio ΔVc /ΔV among various hold time tests is small. Based on this, we calculated the creep part of force-line displacement as

4.2. Hold time effect on the CFCG As dwell times have great influence on the CFCG behavior, the effect of hold time should be investigated thoroughly. Therefore, as shown in Tables 4 and 5, CFCG FE analyses are conducted on specimens with various hold times in this section. Apart from differences in dwell times, all other conditions for these models are the same (W = 30 mm, a0/ W = 0.35, B = W/4, and ΔKin = 17 MPa m ). 98

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Fig. 9. (a) Comparison of (da/ dt )avg and (Ct )avg based on two calculation methods under 180 s dwell time condition. (b) Relationship between (da/ dt )avg and (Ct )avg for FE simulated results and test results.

in shorter hold time regions. To further investigate the influence of dwell time and ΔK on the CFCG behavior, the simulated results in Fig. 11(a) and (b) have been presented in Fig. 12(a) and (b), which shows da/ dN versus frequency and (da/ dt )avg versus frequency at three selected values of ΔK , respectively. The frequency ( f ) is calculated as follows:

Table 5 Finite element simulation conditions for other different hold times. Specimen ID

Hold time(s)

Max. load(kN)

Initial ΔK (MPa m )

Initial a/W

CT7 CT8 CT9 CT10

1200 2400 3600 7200

3.84 3.84 3.84 3.84

17 17 17 17

0.35 0.35 0.35 0.35

f = (tc )−1 = (th + 2 × 10)−1

(18)

where tc is the cycle time of the trapezoidal waveform, th is the hold time, and the 10 s represents the loading and unloading time, as shown in Fig. 3. Meanwhile, the CFCG rate is modeled by the analytical linear superposition damage model, which is given by:

In Fig. 10, the FE calculated crack growth length versus time for various dwell times is shown. The crack initiation stage, the steady crack growth period, and the accelerated crack growth phase are well predicted for all hold time conditions. It can be observed that CFCG life increases sharply from approximately 75 to 5900 h, when the hold time increases from 0 s to infinity. The relationship between da/ dN and ΔK , and the relationship between (da/ dt )avg and ΔK derived from FE calculations, are shown in Fig. 11(a) and (b), respectively. It is apparent that the crack growth rate can be divided into two regions as indicated in Fig. 11(a). In region I (lower ΔK region), the crack growth rate decreases with crack growth, and this trend appears to become more noticeable for longer hold time tests. This is the result of decreased stress relaxation and increased ΔK with crack growth [2,17]. It is known that creep deformation near the crack tip during hold times can greatly influence the CFCG behavior [2,5]. For longer dwell time conditions, the time for stress redistribution is longer, causing enough stress relaxation at the crack tip, which will significantly reduce the crack growth rate. In region II, the stress near the crack tip has reached a nearly steady state, and da/ dN is increased when ΔK is increased. As a result, a characteristic hook is present in both Fig. 11(a) and (b). Meanwhile, it can be found that da/ dN is increased when the hold time increases, and it is influenced by ΔK . The higher crack growth rate is caused by the greater creep damage during longer hold times. In Fig. 11 (a), it is shown that da/ dN for a specimen with 7200 s hold time is approximately four times that for a specimen with 60 s dwell time at ΔK = 20MPa m . For ΔK = 30MPa m , the CFCG rate for a specimen with 7200 s hold time is more than ten times higher than for the 60 s dwell time condition. Based on CFCG tests, Tang et al. [17] proved that increased dwell time has an accelerating effect on da/dN, especially for higher values of ΔK . As shown in Fig. 11(b), the average time rate of crack growth decreases with increasing hold times. It is interesting that the differences of (da/ dt )avg between various dwell times become larger

da da ⎞ da ⎞ =⎛ +⎛ dN ⎝ dN ⎠ fatigue ⎝ dN ⎠creep

(19)

da 1 ⎛ da ⎞ =⎛ ⎞ = C (ΔK )m dN dt 3600 f ⎝ ⎠ fatigue ⎝ ⎠ fatigue

(20)

da 1 ⎛ da ⎞ =⎛ ⎞ ⎝ dN ⎠creep ⎝ dt ⎠creep 3600f

(21)

Fig. 10. FE calculated crack growth length versus time for various dwell times. 99

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Fig. 11. (a) Relationship between da/ dN and ΔK , and (b) relationship between (da/ dt )avg and ΔK obtained from FE simulations.

Fig. 12. (a) Crack growth rate per cycle da/ dN versus frequency, and (b) (da/ dt )avg versus frequency at three selected ΔK .

Moreover, in Fig. 12(a) it is shown that the CFCG behavior is both cycle- and time-dependent for frequencies between 10−4 and 0.004 Hz at ΔK = 23MPa m . The FE simulated crack growth rates are greater than the linear summation results. Similar results can also be seen in Fig. 12(b). The differences denote the interaction effect between creep and fatigue, which has also been reported in previous CFCG tests [15,39,40]. At frequencies above 0.004 Hz (the transition frequency), the crack growth is independent of frequency, indicating the domination of the cycle-dependent fatigue process, while the creep effect can be neglected. Notably, the da/ dN of the test with a 60 s dwell is slightly lower than pure FCG at ΔK = 23MPa m , which is identical to the test result in [17]. This phenomenon is a result of the significant stress relaxation during hold time, while creep damage is limited. Meanwhile, ΔK increases as the crack growth and the transition frequency shift from relatively low frequencies to higher ones. Yang and Bao et al. [39] have also shown that the most significant interaction occurs at low frequencies for lower ΔK , and the frequency changed to a higher region when ΔK increased. Thus, both the frequency range of creep-fatigue interaction, and the frequency for most significant interaction, are affected by ΔK . Notably, a hook shape occurs when ΔK was used to correlate the CFCG data, as mentioned above. Narasimhachary and Saxena [2] also

where (da/dN)fatigue and (da/dN)creep are the pure FCG rate represented by Paris law and the pure CCG rate, respectively. Terms C and m are parameters according to the simulated FCG test. As shown in Fig. 12, dots are FE simulated results and lines are derived from the analytical model provided above. The cycle-dependent fatigue cracking behavior and time-dependent creep cracking behavior are represented by dash-dot lines and dash lines in the different frequency ranges, respectively. The solid lines represent the CFCG rate obtained from Eq. (19), which is a linear summation of the pure creep and pure fatigue components. The pure CCG rate, (da/dt)creep, is constant at a specific stress level, which is shown in Fig. 12 (b). Therefore, the cyclic crack growth rate, (da/dN)creep, is inversely proportional to the frequency, and the slope of time-dependent creep cracking lines is −1 in log(da/dN) vs. log(f) curve, as indicated in Eq. (21). In Eq. (20), it is shown that (da/dN)fatigue is insensitive to frequency, and (da/ dt)fatigue is proportional to the frequency, causing the slope = 1 for pure fatigue cracking lines in Fig. 12(b). The crack growth rate, da/dN, decreases with increasing frequency, owing to the reduced creep effect for shorter hold times. However, the time rate of crack growth, (da/dt)avg, increases for higher frequencies, as a result of the significant fatigue effect. When ΔK increases, both da/dN and (da/dt)avg increase, which is identical to the conclusions in Fig. 11. 100

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longer existed strongly supporting that the (Ct )avg parameter is suitable for correlating the CFCG data. Meanwhile, (da/dt)avg reduces as dwell time increases, and the slope of fitting lines also gradually increases in this case. However, variations of (da/dt)avg and the slope are all limited when the hold time is larger than 1200 s. This originates from the fact that the creep deformation in the crack tip region dominates the CFCG behavior, with the influence of cycle-dependent fatigue gradually becoming negligible for longer dwell times. In the high (Ct )avg region, the differences of crack growth rate among various hold time tests are reduced, compared with lower (Ct )avg regions, and all the data collapse to the fitting line for pure CCG. This indicates that the creep deformation dominates the crack growth when the (Ct )avg is large enough. In fact, larger (Ct )avg corresponds to larger crack length and longer exposure to high temperature, which results in much more creep damage. It can be observed that the CFCG behaviors under various hold times can all be conservatively predicted by the PE NSWA model. The fitting line of pure CCG data is nearly parallel to the NSWA prediction lines, regardless of the tail part at the beginning of the test. For hold time tests, all data are located near the PE NSWA prediction line at relatively low values of (Ct )avg , and tend to the pure CCG data line at high values of (Ct )avg . The simulated crack growth data are all well bounded by the predictions of NSWA model, which is consistent with the experimental results for G115 steel [17]. Therefore, the time-dependent creep process has a major influence on the CFCG behavior of tests with hold times. In Fig. 13(b), it shows the comparisons between dominate damage and linear summation model. For clear distinguish, only the linear fitting results for three typical dwell times (180 s, 1200 s, 7200 s) are provided. In general, the (da/dt)avg calculated from the dominate damage model is larger than those based on linear summation model. The differences increase as the hold time decrease. Previous study [2] shows that the differences between the two approaches are small for P91 steel under tested hold times. However, the differences become large enough for 180 s hold time even though it is very small for test with 7200 s holding. It seems that the cycle-dependent crack growth occupies an important part of the total crack growth for short dwell times. As the hold time increases, the cycle-dependent effect gradually decreased and the CFCG rates blend into the CCG rates. In a previous study [17], we suggested that the CFCG is the comprehensive result of low-order fatigue and dominant creep. To further validate this, creep damage and fatigue damage evolution during crack growth period will be analyzed in next section.

pointed out that CFCG rate for P91 steel can not uniquely correlate with ΔK even if linear elastic conditions dominate. Though ΔK can correlate the crack growth data well for creep-brittle materials [39,40], it seems that another parameter is needed to establish better correlation for creep-ductile material. The parameter (Ct )avg is a candidate parameter because it relies on both th and ΔK [2]. The relationship between (da/ dt )avg and (Ct )avg obtained from FE simulations is shown in Fig. 13. The linear summation and the dominate damage model [2] have been used to determine (da/ dt )avg . For linear summation approach, based on the relation in Eq. (19) and (da/ dt )avg =

1 th

( )

da , dN creep

(da/ dt )avg is calcu-

lated by:

(da/ dt )avg =

1 ⎡ ⎛ da ⎞ ⎛ da ⎞ ⎤ − ⎥ th ⎢ ⎣ ⎝ dN ⎠ ⎝ dN ⎠ fatigue ⎦

(22)

For the dominate damage method, CFCG rate depends on the larger one of the cycle-dependent and time-dependent crack propagation rate:

da da ⎞ ⎤ ⎛ da ⎞ ,⎛ = max ⎡ ⎢ ⎝ dN ⎠ ⎥ dN dN ⎝ ⎠ fatigue creep ⎣ ⎦

(23)

where (da/ dt )avg is estimated by

(da/ dt )avg =

1 th

( )

da dN creep

(24)

In Fig. 13(a), the pure CCG rate, da/dt, is correlated with the C* parameter, which is close to Ct [14]. Under extensive creep regimes, C* is identical to Ct , and the C* value is determined by:

C ∗ = Ct =

PVċ F ′ BW F

(25)

where P denotes the force applied to CT specimen, and Vc denotes the force-line displacement resulting from creep. In addition, the approximate NSW model (NSWA) is suitable for predicting the steady state CCG rate [41,42] as follows:

̇ aNSWA =

3C ∗0.85 ε ∗f

(26) (ε ∗f )

is recommended to be the rewhere the multiaxial creep ductility duction in area, εf = 73% [12], in the plane stress (PS) state, while εf /30 is recommended in the plane strain (PE) state [42,43]. Generally, the dominate damage model is more conservative to analyze the CFCG data and the corresponding results are presented in Fig. 13(a). As shown in Fig. 13(a), the hooks observed in Fig. 11 no

Fig. 13. Relationship between (da/ dt )avg and (Ct )avg obtained from FE simulations: (a) dominate damage model, and (b) comparisons between dominate damage and linear summation model. 101

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4.3. Damage evolution during crack growth procedure In Fig. 14(a) and (b), the variations of the accumulated creep damage, and the accumulated fatigue damage during crack growth process under various dwell times, are shown, respectively. The damage values were extracted from the elements at the crack tip, whose total damage reached the critical damage (0.99). Accumulation trends of fatigue and creep damage are similar for different dwell times. Creep damage sharply increases in the initial period, as shown in Fig. 14(a). Further, when the crack length reaches approximately 0.7–1 mm, accumulated creep damage becomes relatively stable for all hold time conditions, with a slight increase occurring in the following crack propagation period. In contrast, fatigue damage is initially reduced significantly, and subsequently this decrease becomes relatively limited with crack growth, which can be observed in Fig. 14(b). It is evident that creep damage is greater than fatigue damage for all hold time conditions, indicating the domination of time-dependent creep process in CFCG. Differences between creep damage and fatigue damage become greater with crack propagation, due to increased creep and decreased fatigue during the crack propagation procedure. In addition, hold time influences damage significantly. The value of creep damage improved, while fatigue damage reduced, when the hold time increased. This indicates the enhanced effect of creep deformation. The maximum creep damage increased from approximately 0.57–0.95 as the hold time varied from 60 s to 7200 s. Meanwhile, minimum fatigue damage reduced from approximately 0.15 to 0.002. It is interesting that the effect of hold time is much more significant in shorter hold times. Within the dwell time of 60–1200 s, variations of hold time alter the creep and fatigue damage components significantly. However, for a hold time greater than 1200 s, variations of damage are relatively limited. This is consistent with the limited variations of (da/dt)avg and the slope in Fig. 13 for tests with longer hold times. The interaction diagram of the accumulated fatigue and creep damage for different hold time conditions is shown in Fig. 15. Because only the critical value for Grade 91 steel can be found, and G115 steel is similar to Grade 91 steel, the critical damage lines provided by RCC-MR design codes [44] and ASME Section III [45] for Grade 91 steel are also included in this figure. If damage values are located below a critical line, then the component is judged to be safe according to that standard. Additionally, Takahashi [46] proposed an equation to describe the evolution of creep and fatigue damage of high-chromium steels:

Fig. 15. Interaction diagram of accumulated fatigue and creep damage for different hold time conditions.

ωc =1 − exp(− ωc / ωf ) for α = 1

As shown in Fig. 15, the corresponding curve of Eq. (27) with α = 3.7 is provided, which matches well with the FE results. It appears that the creep and fatigue damage accumulation behaviors are predicted by FE simulations with great precision. Because of the neglect of creep-fatigue interaction, the linear summation method can be dangerous in practical application. The critical line provided by ASME Section III is too conservative, while the critical value provided by RCCMR design codes agrees well with the simulated damage. Meanwhile, it is obvious that fatigue damage accounts only for a small part of the total damage, while creep damage accounts for more than 50% of the total damage under all hold times conditions. Combined with the analyses in Fig. 13(a) and 14, a comprehensive effect of dominant creep damage and low-order fatigue in CFCG of G115 steel is therefore verified. Besides, the crack growth and damage evolution behavior presented above were all under constant amplitude loads while the amplitude of loads suffered by power plant and other elevated-temperature components is not constant. For the final application, CFCG and damage evolution of specimens with different geometries, pipes, and other structures under loads with constant and variable amplitude should be investigated further. Meanwhile, the interaction between creep and fatigue is complex and significant, which also demands further study.

1/(α − 1)

1 ⎞ ωc = 1 − ⎜⎛ ⎟ ⎝ 1 + (α − 1) ωc / ωf ⎠

(28)

for α ≠ 1 (27)

Fig. 14. Variations of (a) accumulated creep damage, and (b) accumulated fatigue damage during crack growth process under various dwell times. 102

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5. Conclusions

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Based on the creep-fatigue damage constitutive model, the influence of dwell time on the CFCG of G115 steel at 650 °C was investigated by FE simulations. Simulated CFCG behavior and damage accumulation were analyzed systematically. The conclusions are: 1. The fatigue damage model, considering multiaxial stress, was combined with the Norton power-law creep and the non-linear interaction model, to provide the constitutive model for CFCG. Modification of the FE simulated force-line displacement was carried out to give a more accurate (Ct )avg . All FE results were in good agreement with the tests results, which demonstrates the reliability of the damage model. 2. The CFCG behavior was greatly influenced by hold time and ΔK . The da/dN reduced with increasing frequency, due to the reduced creep effect for shorter hold times. However, (da/dt)avg increased for higher frequencies, because of the significant fatigue effect. Both frequency ranges of creep-fatigue interaction and the frequency for most significant interaction shifted to higher frequencies as ΔK was increased. 3. The (da/dt)avg at the same (Ct)avg decreased as dwell time increased and the slope of fitting lines gradually increased with increasing dwell time. However, for longer hold times or higher (Ct )avg , variations of (da/dt)avg became limited and CFCG data tended to the pure CCG data, indicating domination of time-dependent creep. 4. During the crack growth process, creep damage increased sharply initially, and eventually slightly increased. Concurrently, fatigue damage initially decreased sharply, then smoothly. Meanwhile, creep damage was greater than fatigue damage, and the differences became more pronounced as the hold time increased. It was verified that the CFCG of G115 steel studied was a result of the effect of dominate creep damage, low-order fatigue, and creep-fatigue interactions, especially for longer hold time conditions. Acknowledgments The authors wish to acknowledge the financial support provided by the Project of the National Natural Science Foundation of China (Grant number 51475326) and the Demonstration project of national marine economic innovation (BHSF2017-22). Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.tafmec.2019.01.009. References [1] L. Zhao, H. Jing, L. Xu, Y. Han, J. Xiu, Analysis of creep crack growth behavior of P92 steel welded joint by experiment and numerical simulation, Mater. Sci. Eng. A 558 (2012) 119–128. [2] S.B. Narasimhachary, A. Saxena, Crack growth behavior of 9Cr-1Mo (P91) steel under creep–fatigue conditions, Int J Fatigue 56 (2013) 106–113. [3] B. Xiao, L. Xu, L. Zhao, H. Jing, Y. Han, K. Song, Transient creep behavior of a novel tempered martensite ferritic steel G115, Mater. Sci. Eng. A 716 (2018) 284–295. [4] T. Fischer, B. Kuhn, Frequency and hold time influence on crack growth behavior of a 9–12% Cr ferritic martensitic steel at temperatures from 300 °C to 600 °C in air, Int J Fatigue 112 (2018) 165–172. [5] L. Xu, L. Zhao, Y. Han, H. Jing, Z. Gao, Characterizing crack growth behavior and damage evolution in P92 steel under creep-fatigue conditions, Int. J. Mech. Sci. (2017). [6] K.X. Shi, F.S. Lin, H.B. Wan, Y.F. Wang, Crack growth behaviour of P92 steel under creep and creep–fatigue conditions, Mater. High Temp. 31 (4) (2014) 343–347. [7] F. Bassi, S. Foletti, Conte A. Lo, Creep fatigue crack growth and fracture mechanisms of T/P91 power plant steel, Mater. High Temp. 32 (3) (2015) 250–255. [8] N. Ab Razak, C.M. Davies, K.M. Nikbin, Creep-fatigue crack growth behaviour of P91 steels, Procedia Struct. Integrity 2 (2016) 855–862. [9] B. Xiao, L. Xu, L. Zhao, H. Jing, Y. Han, Tensile mechanical properties, constitutive equations, and fracture mechanisms of a novel 9% chromium tempered martensitic steel at elevated temperatures, Mater. Sci. Eng. A 690 (2017) 104–119.

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