Accepted Manuscript Characterization of 3-D creep constraint and creep crack growth rate in test specimens in ASTM-E1457 standard J.Z. He, G.Z. Wang, S.T. Tu, F.Z. Xuan PII: DOI: Reference:
S0013-7944(16)30493-3 http://dx.doi.org/10.1016/j.engfracmech.2016.10.009 EFM 5295
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
23 June 2016 15 October 2016 17 October 2016
Please cite this article as: He, J.Z., Wang, G.Z., Tu, S.T., Xuan, F.Z., Characterization of 3-D creep constraint and creep crack growth rate in test specimens in ASTM-E1457 standard, Engineering Fracture Mechanics (2016), doi: http://dx.doi.org/10.1016/j.engfracmech.2016.10.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Characterization of 3-D creep constraint and creep crack growth rate in test specimens in ASTM-E1457 standard J.Z. He, G.Z. Wang , S.T. Tu, F.Z. Xuan Key Laboratory of Pressure Systems and Safety, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China
Abstract Three-dimensional creep constraint of six specimen geometries in ASTM E1457 standard was characterized by using two constraint parameters, and creep crack growth (CCG) rates in these specimens were predicted by using numerical simulations. The results show that the specimens with different geometries and size ranges
ASTM-E1457 have different constraints, thus
produce different CCG rate data. With increasing specimen thickness and side-groove depth, the constraint and CCG rates of all specimens increase. It needs to consider constraint effects on CCG rate when predicting component life using laboratory data. The quantitative correlation formulas of CCG rate with creep constraint have been established. Keywords: Creep constraint, Creep crack growth rate, ASTM-E1457, Specimen geometry, Specimen size
Corresponding author Tel.: +86 21 64252681; fax: +86 21 64252681. E-mail address:
[email protected] (Guozhen Wang). 1
Nomenclature a a0 a a0
crack depth initial crack depth creep crack growth rate creep crack growth rate of the standard specimen
A
constant in Norton creep model constants in 2RN creep model, and A2 also is a constraint parameter unified characterization parameter of in-plane and out-of-plane creep constraint area surrounded by equivalent creep strain isoline area surrounded by equivalent creep strain isoline in a standard specimen specimen thickness net specimen thickness C* integral analogous to the J integral average C* integral C(t) integral Young’s modulus applied load stress triaxiality factor factor to estimate C* in experiment using load line displacement initial stress intensity factor characteristic length stress exponent in Norton creep model stress exponents in 2RN creep model constraint parameter under elastic-plastic or creep condition a parameter for correlating creep crack growth rate distance from a crack tip creep constraint parameter load-independent creep constraint parameter inner radius outer radius creep time creep redistribution time out-of-plane constraint parameter specimen width distance from specimen center along specimen thickness polar coordinate at the crack tip
A1, A2 Ac ACEEQ Aref B Bn C* C*avg C(t) E F h H Kin L n n1, n2 Q Q* r R R* Ri Ro t tred Tz W z θ εc ε
* f
creep strain rate multiaxial creep ductility 2
εf
uniaxial creep ductility
εc
equivalent creep strain normalizing stress, usually taken as yield stress opening stress opening stress of C(T) specimen under plane strain von Mises effective stress hydrostatic stress yield stress Poisson’s ratio damage parameter damage rate factor to estimate C*in experiment using load line displacement
σ0 σ22 σ22,CT σe σm σy v ω
ω η
load line displacement rate
Abbreviations 2-D 2RN 3-D C3D8 CPE4H CCG CCI CCP CEEQ CS(T) C(T) DEN(T) FEM M(T) PE PS SEN(B) SEN(T)
two-dimensional two-region Norton three-dimensional eight node brick elements four-node linear plane strain elements creep crack growth creep crack initiation centre crack panel equivalent creep strain in ABAQUS code C-shaped tension compact tension double edged notched tension finite element method middle cracked tension plane strain plane stress single-edge notched bend single-edge notched tension
1. Introduction It is a well-known fact that creep crack initiation (CCI) and creep crack growth (CCG) are the principle failure mechanism of in-service elevated temperature components. The CCG life prediction and assessments in components need the CCG rate data of materials. The recommended specimens 3
in the ASTM E1457 standard [1] for CCG testing include the compact tension (C(T)) specimen, single-edge notched bend (SEN(B)) specimen, C-shaped tension (CS(T)) specimen, single-edge notched tension (SEN(T)) specimen, double notched tension (DEN(T)) specimen and middle tension (M(T)) specimen. The configurations, size range of these specimens and CCG testing method are recommended in the ASTM E1457 standard, and it indicates that the CCG rate does not change within the range of normal data scatter examined in tests of these specimens with the size range for the relatively short term test durations [1]. However, many experimental, theoretical analyses and numerical simulations have shown that specimen geometries, sizes and loading configurations can affect creep crack-tip constraint and CCG rate. The CCG rates raise with the increase of crack depth [2] and specimen thickness [3-6]. The CCG rates in plane strain (PE) are generally faster than those in plane stress (PS) [7-10]. Yokobori et al. [11] and Tabuchi [12] used C(T) and circular notched specimens to examine the influence of specimen geometry on CCG rate, and the circular notched specimen showed the lower CCG rate than C(T) specimen due to the multiaxial stress conditions. Bettinson et al. [13] and Nikbin [14] investigated the effects of specimen type on CCG rates and demonstrated that the high constraint C(T) specimen has the fastest CCG rate and low constraint centre cracked panels (CCP) have the lowest CCG rate. It also has been indicated that the CCG rate examined in C(T) specimen is significantly faster than that of M(T) specimen at a given C* value for various steels [13, 15-20]. Some experiments also demonstrated that the CCG rate of C(T) specimen is considerably higher than those of SEN(T), SEN(B), DEN(T) and M(T) specimens [21, 22], and the long-term CCG rates of SEN(T) and M(T) specimens are lower than those of C(T), CS(T) and SEN(B) specimens [23]. Yokobori Jr. et al. [24-27] presented a parameter Q* to correlate CCG rate, and their work indicated 4
that the CCG rates of thick specimens are faster than those of thin specimens at a given Q* value [26]. The constraint effects on CCG rate were defined as "structural brittleness" [24]. The recent numerical simulations also showed that the creep constraints caused by specimen geometries and sizes have significant effects on CCG rates [18, 19]. For correctly and reasonably measuring the CCG rate data and using them in life predictions of elevated-temperature structures with different constraint levels, the creep crack-tip constraint in specimens and structures needs to be quantified. For this purpose, the two-parameter approach such as C*-Q [14, 28, 29], C*-Tz [30], C*-R [31], C*-R* [32] and three-parameter C(t)-Tz-Q [33] were used to describe the creep crack-tip stress and strain rate fields. The Q, R and R* are in-plane constraint parameters and the out-of-plane constraint was represented by the parameter Tz. The effects of the crack depth [31, 34], specimen thickness [35] and loading configuration [36] on the constraint parameter R have been studied. Based on the load-independent parameter R*, the characterization and correlation of creep constraint between axially cracked pipelines and test specimens were analyzed [37, 38]. The constraint parameter R* solutions for cracked pipes with different geometries and semi-elliptical surface crack sizes have been obtained [38, 39]. Base on C*-R* two-parameter approach, the creep crack growth prediction and assessment incorporating constraint effect for pressurized pipes have been done recently [40]. In recent work of Ma et al. [41-43], a unified creep constraint parameter Ac based on crack-tip equivalent creep strain also has been proposed. It has been indicated that the parameter Ac can characterize the overall constraint levels for specimens with various in-plane and out-of-plane constraints. In ASTM-E1457 standard, it indicates that for the recommended specimen geometries and size ranges, the CCG rates do not vary for a range of materials and loading conditions, and the 5
[1] specimen geometries and sizes may have small influence on CCG rates. three-dimensional (3-D) creep constraints of the six kinds of specimen geometries in ASTM E1457 have not been quantitatively characterized by using suitable constraint parameters, and the effects of specimen geometries, size ranges and side-groove on CCG rates have not been systemically investigated in a wide-range C* region using constraint analyses. In this work, the 3-D creep constraints of the six specimens with different thicknesses and side-groove depths in ASTM E1457 were characterized by using two constraint parameters R* and Ac. The CCG rates in these specimens were predicted by numerical simulations based on ductility exhaustion damage model. The influences of the specimen geometry, thickness and side-groove on the 3-D creep constraints and CCG rates were analyzed, and the quantitative correlation of CCG rate with creep constraint was established. 2. Finite element simulation 2.1 Material A Cr-Mo-V steel was used in this study, and its Young’s Modulus E and yield stress σ y at 566℃ is 160GPa and 383MPa, respectively, and the Poisson’s ratio v is 0.3. The true stress-strain curve of this steel is shown in Fig.1 [44]. The creep strain rate can be expressed by two-region Norton (2RN) creep law and the different Norton model parameters of (A1,n1) and (A2,n2) are shown in Eq.(1) and Table 1 [18, 19, 44], respectively. The transition stress is about 250MPa. This 2RN behavior of the Cr-Mo-V steel is similar to that of the 316H steel [45, 46].
A1 n1 n2 A2
250MPa 250MPa
2.2 Specimen geometry and finite element model 6
2RN model
(1)
Six types of specimen geometries in ASTM E1457-15 [1], schematically illustrated in Fig.2, have been modeled by 3-D FE using ABAQUS code [47]. They include compact tension (C(T)) specimen, single-edge notched bend (SEN(B)) specimen, C-shaped tension (CS(T)) specimen, single-edge notched tension (SEN(T)) specimen, double notched tension (DEN(T)) specimen and middle tension (M(T)) specimen. Specimen dimensions in the FE models conforming to the ASTM E1457-15 standard are summarized in Table 2, where W is the specimen width of C(T), SEN(B) and CS(T) specimens and the half width of SEN(T), DEN(T) and M(T) specimens, L is the half length of SEN(B), SEN(T), DEN(T) and M(T) specimens, B is the thickness, Bn is the net thickness, a0 is the initial crack length, and Ri is the inner radius, Ro is the outer radius and X=2W for CS(T) specimen (Ri, Ro and X exist for CS(T) specimen geometry only). In Table 2, the thickness-to-width ratio (B/W) range recommended by ASTM E1457-15 is from 1/2 to 1/4 for the C(T), SEN(B) and SEN(T) specimens, and that is from 1 to 1/2 for the CS(T), DEN(T) and M(T) specimens. The C(T), SEN(B) and SEN(T) specimens with B/W = 1/2 and the CS(T), DEN(T) and M(T) specimens with B/W = 1 which have been modeled in this work which are described as thick specimens, and the specimens (C(T), SEN(B) and SEN(T)) with B/W = 1/4 and the specimens (CS(T), DEN(T) and M(T)) with B/W = 1/2 are described as thin specimens. To analyze the effects of side groove depth on 3-D constraint and CCG rate, the specimens with various Bn/B values (0.6, 0.8 and 1) in Table 2 were modeled. The 3-D FE models were built for each specimen in Fig.2 and Table 2 and illustrated in Fig. 3. The symmetry boundary condition were employed in the models, therefore only one-fourth FE models of all specimens were modeled. The load was imposed on the loading hole representing a distributed load for the C(T), CS(T), SEN(T), DEN(T) and M(T) specimens, and the load was 7
applied at the center of the compression face of the SEN(B) specimen by an analytical rigid circle. The loading hole diameter of all specimens is one-fourth of the specimen width W. The eight node brick elements (C3D8R) were applied. The 2-D standard C(T) specimen in plane strain (PE) condition with W=50mm and a/W=0.5 was chosen as the reference specimen for calculating constraint parameters, and its model contains 9500 four-node linear plane strain elements (CPE4H). The previous work shows that when the smallest element size near the crack-tip is under 50μm, the FEM results of the C*, stress and strain are independent of the element sizes [31]. Thus the smallest element size of 40 μm near the crack-tip was employed in this paper. 3. Calculations of creep crack-tip constraint parameters R* and Ac Recently, two new creep constraint parameters have been proposed by authors. One is the parameter R* based on crack-tip stress field [32, 37], which can incorporate in-plane constraint and partial out-of-plane constraint in specimens or components [48]. Another is the parameter Ac based on crack-tip equivalent creep strain [41-43], which can describe both in-plane and out-of-plane constraints and overall constraint levels in specimens or components. The 3-D FEM analyses were carried out to obtain the values of the two parameters R* and Ac for all specimens with different side groove depths (Bn/B) and thickness-to-width ratios (B/W) in Table 2. The load independent creep constraint parameter R* at steady-state creep has been investigated and expressed as follows [32, 37]:
R* 22 (
C1* n11 C* 1 ) 22,C(T) ( 2 ) n1 AL AL
at r = 0.2 mm, θ = 0, t/tred = 1
(2)
where σ22 and C1* is the opening stress and C* value in the evaluated specimen, respectively; the σ22,CT and C2* is the opening stress and C* value in the reference C(T) specimen in PE condition, respectively; A and n are material creep constants, and L is a characteristic length which is usually set to be 1m [37]. The increase in R* means the increase of constraint. In this work, the parameters R* 8
was calculated at r = 0.2 mm, θ=0 and t/tred = 1 by using Eq. (2). The constraint parameter Ac has been investigated and described as follows in the literature [41]: Ac
ACEEQ
at
Aref
t/tred = 1
(3)
where ACEEQ and Aref are the area surrounded by the equivalent creep strain εc (the εc is denoted as CEEQ in ABAQUS code) isoline ahead of crack tip in the evaluated specimen and the reference specimen, respectively. The t is creep time and tred is stress redistribution time. The increase in Ac means the decrease of constraint. In this work, the ACEEQ and Aref surrounded by εc = 0.01 isoline in Eq. (3) were obtained at the same creep time t/tred = 1 and a typical C* value of 1E-6MPamh-1 for all specimens in Table 2 (it has been shown that the Ac is independent of the section of the εc and C*values [41] ). The C(T) specimen with W =50mm and a/W=0.5 in PE condition was chosen as the standard reference specimen. 4. Creep damage model and creep crack growth simulations The creep damage accumulation has been modeled based on the ductility exhaustion model, which has been extensively applied on the CCG rate prediction. In this model, the accumulated creep damage parameter ω is expressed as follow: t
c dt 0 * f
dt 0
t
(4)
where ω is damage accumulation rate, εc is equivalent creep strain rate, ε*f is the multiaxial creep ductility. The damage parameter ω ranges from 0 to 1 with ω = 0 at time t = 0 and failure occurs when ω = 1. When the ω value at an element calculated from Eq. (4) reaches 1, the element is considered 9
fully creep damage, hence progressive cracking is simulated and the load-carrying capacity is reduced to a value of close to zero [49]. The multiaxial creep ductility can be estimated from the uniaxial creep ductility ε f by using the Cocks and Ashby model [50] in Eq(5).
*f n 0.5 m 2 n 0.5 sin sinh 2 f 3 n 0.5 n 0.5 e
(5)
where ε f is uniaxial creep ductility, n is the creep exponent (for power law creep), and σ m σ e is the ratio between the mean stress and equivalent stress, which is often referred as stress triaxiality. The uniaxial creep ductility ε f in Eq. (5) is usually assumed to be a constant for a given temperature. However, it has been shown that the creep fracture mechanism depends on stress levels (strain rates), which leads to the stress-regime dependence of the creep ductility of materials [51, 52]. For the Cr-Mo-V steel used in this work, the estimated stress dependent creep ductility was described by Eq. (6) and shown in Fig. 4 [19].
0.5 f 1.14 1014 5.8 0.00537 1.6
225MPa 225 275MPa 275MPa
(6)
A user subroutine (USDFLD) has been defined in ABAQUS to calculate the CCG rates for the specimens with different geometries and sizes in Table 2. The FEM models for simulating CCG behavior are the same as those in Fig. 3. The element size of 100μm which is similar to the average grain size of this steel used in this paper is regularly set in the crack growth zone [18, 19, 44]. The initial load levels Kin for simulations for different specimens are listed in Table 2, and the values are lower for obtaining long-term CCG rate data. It has been demonstrated in recent work of Ma et al. [43] that for such lower initial load levels, the plasticity effect is small. The true stress-strain curve of this steel in Fig.1 and the 2RN creep model parameters in Table 1 were employed. The C* was calculated by using Eq. (7) in ASTM E1457[1]. 10
C*
F H Bn (W a)
(7)
where F is the applied load, is the creep load line displacement rate, Bn is the specimen net thickness, W is the specimen width and a is the current crack length. In Eq. (7), H and η are non-dimensional geometry dependent parameters and the solutions of which can be found in [1]. The criteria in ASTM E 1457 [1] was applied to ensure that C* is a valid parameter to describe the CCG rate. In the recent work of authors [18, 19, 44], Kim et al. [23] and Mehmanparast et al. [45], the CCG rate data predicted by FE simulations using the ductility exhaustion approach agree well with experimental data, thus the simulated method of creep crack growth used in this work has been validated. 5. Results and discussion 5.1 The 3-D creep constraint The 3-D creep constraint parameters R* and Ac in all specimens with different side-grooved depths (different Bn/B) and different thickness-to-width ratios (B/W) in Table 2 were calculated at an average C* value of C*avg ≈ 1E-6MPamh-1 along crack front (specimen thickness). The R* distributions along crack fronts for all specimens with different Bn/B (different side-groove depths) and different B/W (different thicknesses) are compared in Figs. 5-7. The z/Bn = 0 and z/Bn =0.5 are the central plane and surface of specimens, respectively. The Figs. 5-7(a) are the R* distributions of the thin specimens, and the Figs. 5-7(b) are those of the thick specimens. As seen in Figs.5-7, the center region of all specimen has higher constraint, and the surface region has lower constraint. The constraint level at the center region (z/Bn = 0 to 0.4) decreases in sequence from CS(T), C(T), SEN(B), SEN(T), DEN(T) to M(T) specimen,
and the sequence of constraint level in surface
region (z/Bn = 0.4 to 0.5) is irregular and have little difference. This results from the definition of the 11
parameter R* [32, 37]. The R* is based on the parameter R and represents opening stress difference between the evaluated specimen and the reference C(T) specimen in plane strain at the same C* value. The stress state at the center region (z/Bn = 0 to 0.4) of the specimens is usually close to the plane strain state [35], and that of the surface region (z/Bn = 0.4 to 0.5) is close to the plane stress state. The R* with plane strain reference field can characterize the constraint level at the center region caused by specimen geometries [37]. But at the surface regions of the specimens, the stress level is lower and the stress difference is small among different specimens. This may lead to the distribution forms of R* at the surface regions of the specimens in Figs. 5-7. According to Figs. 5-7, the CS(T), C(T) and SEN(B) specimens can be regarded as higher constraint specimens, and the SEN(T), DEN(T) and M(T) specimens can be regarded as lower constraint specimens. With increasing side groove depth from Bn/B = 1 (without side-groove) in Fig.5, through Bn/B = 0.8 (20%-side groove) in Fig.6 to Bn/B = 0.6 (40%-side groove) in Fig.7, the constraint distributions become more uniform and the center region size with plane strain high constraint (the R* value is around zero) increases. With increasing specimen thickness from Figs. 5-7(a) to Figs. 5-7(b), the constraint distributions also become more uniform and the center region size in plane strain also increases. Figs.5-7 also show that with increasing side groove depth and specimen thickness, the constraint difference between the higher constraint specimens (CS(T), C(T) and SEN(B)) becomes small, but that between the lower constraint specimens (SEN(T), DEN(T) and M(T)) still remains. This implies that the lower constraint specimen geometries are insensitive to the side groove and specimen thickness. For the thick specimens with side groove (Figs.6(b) and 7(b)), the three higher constraint specimens almost reach the same plane strain constraint level. The average constraint along 3-D crack front can characterize overall constraint level in a specimen. The average constraint R*avg was calculated for all specimens, as shown in Fig.8. It shows that the R*avg increases with increasing side groove depth (decreasing Bn/B). But for the higher constraint specimens (CS(T), C(T) and SEN(B)), with increasing side groove depth from 20% to 12
40%, their constraint level slightly increases. Fig. 8 also indicates that the average constraint of thin specimens is lower than that of thick specimens. The distributions of the constraint parameter Ac along crack fronts in Figs.9-11 are similar with the results in the previous studies [41, 42] which indicates that the center region of all specimens has high and uniform constraint, and the free surface (side-grooved plane for side-grooved specimen) region has lower constraint. This is consistent with the distributions of R*. Figs.9-11 also show that with increasing side-groove depth and specimen thickness, the constraint levels of all specimens increase (the Ac decreases) and the specimen sequence of constraint from high to low changes. For the specimens without side-groove (Bn/B = 1) in Fig.9, the C(T), SEN(B) and CS(T) are higher constraint specimens, and for those with 20% (Bn/B = 0.8 in Fig.10) and 40% side groove (Bn/B = 0.6 in Fig.11), the C(T), SEN(T) and DEN(T) are higher constraint specimens. For the thin specimens with side groove in Figs. 10 (a) and 11(a), except for the M(T) specimen with the lowest constraint, the difference in constraint between the other specimens is small. The average constraint Ac-avg along crack fronts for all specimens are given in Fig.12. It shows that compared with the non-side-grooved specimens (Bn/B = 1), the introduction of side groove leads to significant increase of constraint in specimens, but with increasing side-groove depth from 20% to 40%, the constraint only slightly increases. The constraint of thick specimens is higher than that of thin specimens. With increasing the side-groove depth from 0% to 40%, the difference in constraint between the specimens become more small and the constraint levels of SEN(T) and DEN(T) specimens have a significant increase. The constraint characterization results using the parameters R* in Figs.5-8 and Ac in Figs.9-12 are similar. The difference comes from their different capability for characterizing the in-plane and out-of-plane constraints. The R* can characterize in-plane constraint and partial out-of-plane 13
constraint [48], while Ac can characterize both in-plane and out-of-plane constraints and the overall constraint levels for specimens [41-43]. Figs.5-8 indicate that the parameter R* can characterize in-plane constraint caused by different specimen geometries, but it is not very sensitive to the out-of-plane constraint caused by side groove and specimen thickness. Figs.9-12 show that the parameter Ac is sensitive to both in-plane and out-of-plane constraint, which is consistent with the previous study of authors [42]. 5.2 Creep crack growth rates The calculated CCG rates from FE simulations for a variety of specimens in ASTM-E1457 are compared in Figs. 13-15 for a wide-range C* region. It can be seen that the constraint caused by specimen geometry has significant influence on CCG rates at low C* region, but it has small influence on CCG rates at high C* region. For the specimens without side-groove (Bn/B = 1) in Fig.13, the specimens (C(T), SEN(B) and CS(T)) with higher constraint have higher CCG rate at low C* region, and the specimens (DEN(T), SEN(T) and M(T)) with lower constraint have lower CCG rate. These simulation results agree with the experimental results in the literature [19, 20, 23]. The introduction of side groove results in the increase of CCG rate and the decrease in difference of CCG rates among different specimens. This comes from the increase of constraint and the reduction of constraint difference among different specimens. With increasing side-groove depth, the CCG rate of DEN(T) and SEN(T) specimens significantly increases. This is caused by the significant increase of their constraint (the decrease of Ac) in Figs.10-12. At the same C* value,the CCG rates of thick specimens are faster than that of thin specimens. The comparisons of CCG rates in Figs.13-15 with the constraint parameter R* in Figs.5-8 show that the CCG rates and their difference among different specimens are not very consistent with the constraint R* levels and their difference. Such as the CS(T) specimen has the highest constraint R*, but its CCG rate is not the highest, and all DEN(T) and SEN(T) specimens have lower constraint R*, but they have higher CCG rates in specimens with side grooves. The constraint R* has large difference among different specimens, but the difference in 14
CCG rates is not so large. However, by comparing CCG rates in Figs.13-15 with the constraint parameter Ac in Figs.9-12, it is interesting to find that the CCG rates and their difference among different specimens are consistent with the constraint Ac and their difference. The specimens with higher constraint (lower Ac) have higher CCG rate, and the larger difference in Ac among different specimens corresponds to the larger difference in CCG rates. Such as the larger difference in CCG rates in Fig.15(b) for the specimens with 40% side-groove comes from the larger difference in Ac in Fig.11(b). These results suggest that the parameter Ac has better capability for characterizing creep constraint than the parameter R* for various specimen geometries in ASTM E1457. This is due to that the parameter Ac can effectively and simultaneously incorporate both in-plane and out-of-plane constraints in the specimens [41-43]. To clearly show the effects of side groove depth, specimen thickness and load level on CCG rate, Figs. 16 and 17 give the CCG rates of all specimens at a low C* value of 1E-6MPam/h and a high C* value of C*=1E-3MPam/h, respectively. It can be seen from Figs. 16 and 17 that with increasing side-groove depth and specimen thickness, the CCG rate increases. An increase of side-groove depth from 20% to 40% cannot cause larger increase in constraint (Figs.9-12) and CCG rate. At low C*, specimen geometries and constraint have larger effect on CCG rate (Fig.16), but at high C*,they have small effect on CCG rate (Fig.17). 5.3 Correlation of CCG rate with creep constraint parameters R* and Ac In ASTM-E1457 standard, it has been shown that for the recommended specimen geometries and size ranges, the CCG rates are independent of materials and loading conditions, and the [1]
and CCG rates described in the Sections 5.1 and 5.2 have shown that the
specimen geometries, size range and
side groove by ASTM-E1457 standard [1] can induce different crack-tip constraints, thus produce 15
different CCG rates in low
which is relative to the service load levels of high temperature
components. In the case without side groove, the C(T), CS(T) and SEN(B) specimens have higher constraint and CCG rate, while the SEN(T), DEN(T) and M(T) specimens have lower constraint and CCG rate. The introduction of side groove is mainly for producing straight crack fronts [1]. The ASTM-E1457 indicates that in most cases 20% side-grooving is sufficient to meet crack front straightness requirements, but for extremely creep ductile materials, a total side-groove reduction of up to 40% may be needed to produce straight crack fronts [1]. The results in this work have shown that the introduction of side groove can significantly increase crack-tip constraint and CCG rates of all specimens, and with increasing side-groove depth from 20% to 40%, the constraint and CCG rates increase slowly. The constraint and CCG rates also increase with increasing specimen thickness. Therefore, it is an important issue to account for the constraint effects on CCG rate when predicting component life using laboratory data. In addition, because of the limitations of material availability and laboratory loading system, necessary to consider the constraint effects on CCG rate when
constraint effects on CCG rates, it is necessary to establish the correlation of CCG rate with creep constraint parameters. The method for establishing the correlation of CCG rate with creep constraint parameters has been given in detail in the recent work of authors [32, 41]. Because the average constraint R*avg and Ac-avg can characterize the overall constraint levels in the specimens, they are used for establishing the correlation with CCG rate. The normalized CCG rate ratio a / a0 and the average constraint R*avg and Ac-avg were calculated at a lower C* value of C*=1E-6MPam/h. Figs. 18(a) and (b) show the linear relations on log-log scale between CCG rate ratio a / a0 and R*avg and Ac-avg, respectively. The correlation formulas of a / a0 and R*avg and Ac-avg are fitted as follows: 16
* a / a0 1.293(1 Ravg )1.381
(8)
a / a0 2.514 Ac avg 2.949
(9)
where a0 28.06C * is the CCG rate of the standard PE C(T) specimen with W=50mm and 0.83
a/W=0.5. It has been shown that the correlation lines and formulas of a / a0 with the constraint parameters R*avg and Ac-avg are independent of the load level C*, and are only dependent on the material [32, 41]. Fig. 18 implies that the two parameters both can characterize the constraints induced by specimen geometries, sizes and side grooves in ASTM-E1457. But the data scatter of Ac-avg seems to be smaller than that of R*avg, and the Ac-avg also can effectively characterize both in-plane and out-of-plane constraints. So the correlation formula of a / a0 with Ac-avg may be more accurate for describing the constraint effects on CCG rate. The correlation formulas in Eqs.(8) and (9) may be applied in CCG life predictions of elevated temperature structures for accounting for constraint effects and improving accuracy, and they also can be used to predict CCG rate data of various specimens. For actual components or specimens with a certain geometry and loading condition, so long as the constraint parameter Ac-avg or R*avg is obtained at a loading level C* using FE calculation, the constraint dependent CCG rates can be predicted by using Eqs. (8) or (9) based on the two-parameter C*- Ac-avg or C*- R*avg. 6. Conclusions The 3-D creep constraint of six kinds of specimen geometries with various thicknesses and side-groove depths in ASTM E1457 standard has been characterized by using two constraint parameters R* and Ac. The CCG rates in these specimens have been predicted by using stress dependent creep ductility and ductility exhaustion model in FE analyses. The effects of the specimen geometry, thickness and side-groove on 3-D creep constraints and CCG rates were analyzed, and the 17
quantitative correlation of CCG rate with creep constraint has been established. The primary results are summarized as follows: (1) For the specimens without side-groove (Bn/B = 1), the C(T), CS(T) and SEN(B) specimens have higher constraint, and the SEN(T), DEN(T) and M(T) specimens have lower constraint. With increasing side groove depth and specimen thickness, the constraint levels of all specimens increase and the constraint distributions become more uniform, and the center region size with high constraint increases. (2) The introduction of side groove results in the increase of CCG rate and the decrease in difference of CCG rates among different specimens. At the same C* value, the CCG rates of thick specimens are faster than that of thin specimens. (3) The difference in CCG rates among different specimens in ASTM E1457 comes from the difference in constraint levels. With increasing constraint level of specimens, the CCG rate increases. The parameter Ac has better capability for characterizing creep constraint than the parameter R* for a variety of specimen geometries in ASTM E1457. (4) It is necessary to account for the constraint effects on CCG rate when predicting component life using laboratory data examined by specimen geometries in ASTM E1457 and when
(5) Based on the average constraint parameters R*avg and Ac-avg along the crack fronts of all specimens, the quantitative correlation formulas of CCG rate with creep constraint have been established, which may be applied in CCG life assessments of elevated temperature structures for accounting for constraint effects and improving accuracy.
18
Acknowledgments This work was financially supported by the Projects of the National Natural Science Foundation of China (51375165, 51575184, 51325504).
Reference [1] ASTM-E1457. Standard test method for measurement of creep crack growth rates in metals. ASTM International. 2015. [2] Zhao L, Jing H, Xu L, Han Y, Xiu J. Evaluation of constraint effects on creep crack growth by experimental investigation and numerical simulation. Eng Fract Mech. 2012;96:251-66. [3] Tan JP, Tu ST, Wang GZ, Xuan FZ. Effect and mechanism of out-of-plane constraint on creep crack growth behavior of a Cr–Mo–V steel. Eng Fract Mech. 2013;99:324-34. [4] Tan JP, Wang GZ, Xuan FZ, Tu ST. Experimental Investigation of In-Plane Constraint and Out-of-Plane Constraint Effects on Creep Crack Growth. In: Proceedings of the ASME 2012 Pressure Vessels and Piping Conference. PVP 2012, Toronto, Ontario, Canada, (Paper No. PVP2012-78478). [5] Zhao L, Jing H, Xiu J, Han Y, Xu L. Experimental investigation of specimen size effect on creep crack growth behavior in P92 steel welded joint. Mater Des. 2014;57:736-43. [6] Yamamoto M, Miura N, Ogata T. Effect of Constraint on Creep Crack Propagation of Mod. 9Cr-1Mo Steel Weld Joint. In: Proceedings of the ASME 2009 Pressure Vessels and Piping Conference. PVP 2009, Prague, Czech Republic, (Paper No. PVP2009-77862). [7] Nikbin KM, Smith DJ, Webster GA. Prediction of Creep Crack Growth from Uniaxial Creep Data. Proceedings of the Royal Society of London A: Mathematical, Phys Eng Sci. 1984;396:183-97. 19
[8] Webster GA, R. A A. High temperature component life assessment: Springer; 1994. [9] Yatomi M, O’Dowd NP, Nikbin KM, Webster GA. Theoretical and numerical modelling of creep crack growth in a carbon–manganese steel. Eng Fract Mech. 2006;73:1158-75. [10] Nikbin KM, Smith DJ, Webster GA. An Engineering Approach to the Prediction of Creep Crack Growth. J Eng Mater Technol. 1986;108:186-91. [11] Yokobori AT, Sugiura R. Effects of component size, geometry, microstructure and aging on the embrittling behavior of creep crack growth correlated by the Q* parameter. Eng Fract Mech. 2007;74:898-911. [12] Tabuchi M, Adachi T, Yokobori AT, Fuji A, Ha J, Yokobori T. Evaluation of creep crack growth properties using circular notched specimens. Int J Pressure Vessels Piping. 2003;80:417-25. [13] Bettinson AD, O’Dowd NP, Nikbin KM, Webster GA. Experimental investigation of constraint effects on creep crack growth. In: Proceedings of the ASME 2002 Pressure Vessels and Piping Conference. PVP2002, Vancouver, BC, Canada. p. 143-50. [14] Nikbin K. Justification for meso-scale modelling in quantifying constraint during creep crack growth. Mater Sci Eng, A. 2004;365:107-13. [15] Ozmat B, Argon AS, Parks DM. Growth modes of cracks in creeping type 304 stainless steel. Mech Mater. 1991;11:1-17. [16] Bettinson AD, Nikbin KM, O’Dowd NP, Webster GA. The influence of constraint on the creep crack growth in 316H stainless steel. In: Proceedings of 5th International Conference Structural Integrity Assessment. Cambridge, UK, 2000. [17] Takahashi Y, Igari T, Kawashima F, Date S, Titoh NI, Noguchi Y, et al. High temperature crack 20
growth behavior of high-chromium steels. In: Proceedings of 18th International Conference on Structural Mechanics in Reactor Technology. Beijing, China, 2005. p. 1904-15. [18] Zhang JW, Wang GZ, Xuan FZ, Tu ST. In-plane and out-of-plane constraint effects on creep crack growth rate in Cr-Mo-V steel for a wide range of C*. Mater High Temp. 2015;32:512-23. [19] Zhang JW, Wang GZ, Xuan FZ, Tu ST. Prediction of creep crack growth behavior in Cr–Mo–V steel specimens with different constraints for a wide range of C∗. Eng Fract Mech. 2014;132:70-84. [20] Zhao L, Xu L, Han Y, Jing H. Quantifying the constraint effect induced by specimen geometry on creep crack growth behavior in P92 steel. Int J Mech Sci. 2015;94-95:63-74. [21] Davies CM, Mueller F, Nikbin KM, O' Dowd NP, Webster GA. Analysis of Creep Crack Initiation and Growth in Different Geometries for 316H and Carbon Manganese. Journal of ASTM international. 2006. [22] Davies CM, Dean DW, Yatomi M, Nikbin KM. The influence of test duration and geometry on the creep crack initiation and growth behaviour of 316H steel. Mater Sci Eng, A. 2009;510-511:202-6. [23] Kim N-H, Oh C-S, Kim Y-J, Davies CM, Nikbin K, Dean DW. Creep failure simulations of 316H at 550°C: Part II – Effects of specimen geometry and loading mode. Eng Fract Mech. 2013;105:169-81. [24] Yokobori Jr AT, Sugiura R, Tabuchi M, Fuji A, Adachi T, T Y. The effect of multi-axial stress component on creep crack growth rate concerning structural brittleness. 2005. [25] Yokobori Jr AT, T. Y. New concept to crack growth at high temperature creep and creep-fatigue. 1989:1723-35. 21
[26] Yokobori Jr AT, Yokobori T, Nishihara T, T. Y. An alternative correlating parameter for creep crack growth rate and its application-Derivation of the parameter Q*. Mater High Temp. Materials at high temperature. 1992;10:109-18. [27] Yokobori AT, Jr., Yokobori T, Yamazaki K. A characterization of high temperature creep fracture life for ceramics. J Mater Sci Lett. 1996;15:2002-7. [28] Bettinson AD, O’Dowd NP, Nikbin K, Webster GA. Two Parameter Characterisation of Crack Tip Fields under Creep Conditions. In: Murakami S, Ohno N, editors. IUTAM Symposium on Creep in Structures: Springer Netherlands; 2001. p. 95-104. [29] Budden P, Ainsworth R. The effect of constraint on creep fracture assessments. Int J Fract. 1999;97:237-47. [30] Xiang M, Yu Z, Guo W. Characterization of three-dimensional crack border fields in creeping solids. Int J Solids Struct. 2011;48:2695-705. [31] Wang GZ, Liu XL, Xuan FZ, Tu ST. Effect of constraint induced by crack depth on creep crack-tip stress field in CT specimens. Int J Solids Struct. 2010;47:51-7. [32] Tan JP, Wang GZ, Tu ST, Xuan FZ. Load-independent creep constraint parameter and its application. Eng Fract Mech. 2014;116:41-57. [33] Xiang M, Guo W. Formulation of the stress fields in power law solids ahead of three-dimensional tensile cracks. Int J Solids Struct. 2013;50:3067-88. [34] Sun PJ, Wang GZ, Xuan FZ, Tu ST, Wang ZD. Quantitative characterization of creep constraint induced by crack depths in compact tension specimens. Eng Fract Mech. 2011;78:653-65. [35] Sun PJ, Wang GZ, Xuan FZ, Tu ST, Wang ZD. Three-dimensional numerical analysis of out-of-plane creep crack-tip constraint in compact tension specimens. Int J Pressure Vessels 22
Piping. 2012;96–97:78-89. [36] Wang GZ, Li BK, Xuan FZ, Tu ST. Numerical investigation on the creep crack-tip constraint induced by loading configuration of specimens. Eng Fract Mech. 2012;79:353-62. [37] Tan JP, Tu ST, Wang GZ, Xuan FZ. Characterization and correlation of 3-D creep constraint between axially cracked pipelines and test specimens. Eng Fract Mech. 2015;136:96-114. [38] Liu S, Wang GZ, Xuan FZ, Tu ST. Creep constraint analysis and constraint parameter solutions for axial semi-elliptical surface cracks in pressurized pipes. Eng Fract Mech. 2014;132:1-15. [39] Liu S, Wang GZ, Tu ST, Xuan FZ. Creep constraint analysis and constraint parameter solutions for circumferential surface cracks in pressurized pipes. Eng Fract Mech. 2015;148:1-14. [40] Liu S, Wang GZ, Tu ST, Xuan FZ. Creep crack growth prediction and assessment incorporating constraint effect for pressurized pipes with axial surface cracks. Eng Fract Mech. 2016;154:92-110. [41] Ma HS, Wang GZ, Xuan FZ, Tu ST. Unified characterization of in-plane and out-of-plane creep constraint based on crack-tip equivalent creep strain. Eng Fract Mech. 2015;142:1-20. [42] Ma HS, Wang GZ, Liu S, Tu ST, Xuan FZ. Three-dimensional analyses of unified characterization parameter of in-plane and out-of-plane creep constraint. Fatigue Fract Eng Mater Struct. 2016;39:251-63. [43] Ma HS, Wang GZ, Liu S, Tu ST, Xuan FZ. In-plane and out-of-plane unified constraint-dependent creep crack growth rate of 316H steel. Eng Fract Mech. 2016;155:88-101. [44] Zhang JW, Wang GZ, Xuan FZ, Tu ST. The influence of stress-regime dependent creep model and ductility in the prediction of creep crack growth rate in Cr–Mo–V steel. Mater Des. 2015;65:644-51. 23
[45] Mehmanparast A. Prediction of creep crack growth behaviour in 316H stainless steel for a range of specimen geometries. Int J Pressure Vessels Piping. 2014;120-121:55-65. [46] Mehmanparast A, Davies CM, Webster GA, Nikbin KM. Creep crack growth rate predictions in 316H steel using stress dependent creep ductility. Mater High Temp. 2014;31:84-94. [47] ABAQUS. Version 6.10. User's manual. Inc and Dassault Systemes. 2010. [48] Liu S, Wang GZ, Xuan FZ, Tu ST. Three-dimensional finite element analyses of in-plane and out-of-plane creep crack-tip constraints for different specimen geometries. Eng Fract Mech. 2015; 133: 264–280. [49] Oh C-S, Kim N-H, Kim Y-J, Davies C, Nikbin K, Dean D. Creep failure simulations of 316H at 550 °C: Part I – A method and validation. Eng Fract Mech. 2011;78:2966-77. [50] Cocks ACF, Ashby MF. Intergranular fracture during power-law creep under multiaxial stresses. Met Sci. 1980;14:395-402. [51] Samuel EI, Choudhary BK, Palaparti DPR, Mathew MD. Creep Deformation and Rupture Behaviour of P92 Steel at 923 K. Procedia Eng. 2013;55:64-9. [52] Takahashi Y, Shibamoto H, Inoue K. Long-term creep rupture behavior of smoothed and notched bar specimens of low-carbon nitrogen-controlled 316 stainless steel (316FR) and their evaluation. Nucl Eng Des. 2008;238:310-21.
24
Figure and Table Captions Fig. 1 True stress-strain curve of the Cr-Mo-V steel at 566℃[44] Fig. 2 Schematic illustrations of specimens: (a) C(T), (b)SEN(B), (c) CS(T), (d) SEN(T), (e) DEN(T), (f) M(T) Fig.3 Mesh design of specimen geometries: (a) C(T), (b) SEN(B), (c) CS(T), (d) SEN(T), (e) DEN(T), (f) M(T) Fig.4 The estimated stress dependent creep ductility for Cr-Mo-V steel [19] Fig. 5 Comparison of constraint parameter R* of six specimen geometries without side groove (Bn/B = 1), (a) thin specimens, (b) thick specimens Fig. 6 Comparison of constraint parameter R* of six specimen geometries with 20%-side groove (Bn/B = 0.8), (a) thin specimens, (b) thick specimens Fig. 7 Comparison of constraint parameter R* of six specimen geometries with 40%-side groove (Bn/B = 0.6), (a) thin specimens, (b) thick specimens Fig. 8 The values of R*avg of different specimen geometries with different Bn/B and B/W at C*avg ≈ 1E-6MPamh-1, (a) thin specimens, (b) thick specimens Fig. 9 Comparison of constraint parameter Ac of six specimen geometries without side groove (Bn/B = 1), (a) thin specimens, (b) thick specimens Fig. 10 Comparison of constraint parameter Ac of six specimen geometries with 20%-side groove (Bn/B = 0.8), (a) thin specimens, (b) thick specimens Fig. 11 Comparison of constraint parameter Ac of six specimen geometries with 40%-side groove (Bn/B = 0.6), (a) thin specimens, (b) thick specimens 25
Fig. 12 The values of Ac-avg of different specimen geometries with different Bn/B and B/W at C*avg ≈ 1E-6MPamh-1,(a) thin specimens, (b) thick specimens Fig. 13 Comparison of the predicted CCG rate of six specimen geometries without side groove (Bn/B = 1), (a) thin specimens, (b) thick specimens Fig. 14 Comparison of the predicted CCG rate of six specimen geometries with 20%-side groove (Bn/B = 0.8), (a) thin specimens, (b) thick specimens Fig. 15 Comparison of the predicted CCG rate of six specimen geometries with 40%-side groove (Bn/B = 0.6), (a) thin specimens, (b) thick specimens Fig. 16 The change of CCG rate da/dt with side-groove depth at C*=1E-6MPamh-1, (a) thin specimens, (b) thick specimens Fig. 17 The change of CCG rate da/dt with side-groove depth at C*=1E-3MPamh-1, (a) thin specimens, (b) thick specimens Fig. 18 The relation between a / a0 and R*avg (a) and Ac-avg (b) for all specimens Table 1 The 2RN creep model parameters [19] Table 2 Detailed geometries and sizes of specimens
26
Fig. 1 True stress-strain curve of the Cr-Mo-V steel at 566℃[44]
27
(a) C(T)
(b) SEN(B)
(d) SEN(T)
(c) CS(T)
(e) DEN(T)
(f) M(T)
Fig. 2 Schematic illustrations of specimens: (a) C(T), (b)SEN(B), (c) CS(T), (d) SEN(T), (e) DEN(T), (f) M(T) 28
(a) C(T)
(d) SEN(T)
(b) SEN(B)
(e) DEN(T)
(c) CS(T)
(f) M(T)
Fig. 3 Mesh design of specimen geometries: (a) C(T), (b) SEN(B), (c) CS(T), (d) SEN(T), (e) DEN(T), (f) M(T)
29
Fig.4 The estimated stress dependent creep ductility for Cr-Mo-V steel [19]
30
0.5
-0.5
-1.0
-1.0
-1.5
-1.5
*
-0.5
-2.0 -2.5 -3.0 -3.5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.0
W=50mm,B/W=1/4 W=25mm,B/W=1/4 W=25mm,B/W=1/2 W=25mm,B/W=1/4 W=12.5mm,B/W=1/2 W=12.5mm,B/W=1/2
0.1
0.2
-2.0 -2.5 -3.0
0.3
0.4
a/W=0.5 Bn/B=1
0.0
R
*
0.0
R
0.5
a/W=0.5 Bn/B=1
-3.5
0.5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.0
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
0.1
0.2
0.3
z/Bn
z/Bn
(a)
(b)
0.4
0.5
Fig. 5 Comparison of constraint parameter R* of six specimen geometries without side groove (Bn/B = 1), (a) thin specimens, (b) thick specimens
31
0.0
0.0
-0.5
-0.5
-1.0
-1.0
-1.5
-1.5
*
0.5
R
*
R
0.5
-2.0 -2.5 -3.0 -3.5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.0
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
0.1
-2.0 -2.5
a/W=0.5 Bn/B=0.8
0.2 0.3 z/Bn
0.4
-3.0 -3.5
0.5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.0
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
0.1
a/W=0.5 Bn/B=0.8
0.2
0.3
0.4
0.5
z/Bn
(a)
(b)
Fig. 6 Comparison of constraint parameter R* of six specimen geometries with 20%-side groove (Bn/B = 0.8), (a) thin specimens, (b) thick specimens
32
0.0
0.0
-0.5
-0.5
-1.0
-1.0
-1.5
-1.5
*
0.5
R
*
R
0.5
-2.0 -2.5 -3.0 -3.5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.0
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
0.1
-2.5
a/W=0.5 Bn/B=0.6
0.2
0.3
0.4
-2.0 -3.0 -3.5
0.5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.0
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
0.1
z/Bn
0.2
a/W=0.5 Bn/B=0.6 0.3
0.4
0.5
z/Bn
(a)
(b)
Fig. 7 Comparison of constraint parameter R* of six specimen geometries with 40%-side groove (Bn/B = 0.6), (a) thin specimens, (b) thick specimens
33
0.5
0.5
-1
0.0
a/W=0.5
-0.5
-0.5
-1.0
-1.0 Ravg
-1.5 -2.0 -2.5 -3.0
*
*
Ravg
0.0
*
Cavg=1E-6MPamh
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.6
0.7
0.8 Bn/B
-2.5
0.9
-3.0
1.0
-1
a/W=0.5
-1.5 -2.0
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
*
Cavg=1E-6MPamh
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
0.6
(a)
0.7
0.8 Bn/B
0.9
1.0
(b)
Fig. 8 The values of R*avg of different specimen geometries with different Bn/B and B/W at C*avg ≈ 1E-6MPamh-1 (a) thin specimens, (b) thick specimens
34
2.75
2.50 a/W=0.5 Bn/B=1
2.50
2.25
2.25
2.00
2.00 Ac
Ac
2.75
1.75 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
1.50 1.25 1.00
0.0
0.1
0.2 0.3 z/Bn
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
0.4
0.5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
1.75 1.50 a/W=0.5 Bn/B=1
1.25 1.00
0.0
0.1
0.2
0.3
0.4
0.5
z/Bn
(a)
(b)
Fig. 9 Comparison of constraint parameter Ac of six specimen geometries without side groove (Bn/B = 1), (a) thin specimens, (b) thick specimens
35
2.00
2.00
1.75
1.75
1.50
1.50
1.25 1.00
Ac
Ac
a/W=0.5 Bn/B=0.8
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.0
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
0.1
0.2
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
1.25 a/W=0.5 Bn/B=0.8
0.3
0.4
1.00
0.5
0.0
0.1
0.2
0.3
z/Bn
z/Bn
(a)
(b)
0.4
0.5
Fig. 10 Comparison of constraint parameter Ac of six specimen geometries with 20%-side groove (Bn/B = 0.8), (a) thin specimens, (b) thick specimens
36
2.00
2.00 1.75
1.75
1.50
1.50
1.25 1.00
Ac
Ac
a/W=0.5 Bn/B=0.6
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.0
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
0.1
0.2
0.3
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
1.25 a/W=0.5 Bn/B=0.6 0.4
1.00
0.5
0.0
0.1
z/Bn (a)
0.2 0.3 z/Bn
0.4
0.5
(b)
Fig. 11 Comparison of constraint parameter Ac of six specimen geometries with 40%-side groove (Bn/B = 0.6), (a) thin specimens, (b) thick specimens
37
2.50 2.25
2.50
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
*
2.25 Cavg=1E-6MPamh 2.00
1.75
Ac-avg
Ac-avg
2.00
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
-1
a/W=0.5
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
1.75 1.50
1.50 *
Cavg=1E-6MPamh
1.25
-1
1.25
a/W=0.5 1.00
0.6
0.7
0.8 Bn/B
0.9
1.00
1.0
0.6
0.7
(a)
0.8 Bn/B
0.9
(b)
Fig. 12 The values of Ac-avg of different specimen geometries with different Bn/B and B/W at C*avg ≈ 1E-6MPamh-1, (a) thin specimens, (b) thick specimens
38
1.0
0.1
0.1
0.01 da/dt,mm/h
da/dt,mm/h
0.01
a/W=0.5 Bn/B=1
a/W=0.5 Bn/B=1
1E-3
1E-3 1E-4 1E-5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
1E-4
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
1E-6 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 * C ,MPam/h
1E-5
0.01
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
1E-6 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 * C ,MPam/h
(a)
0.01
(b)
Fig. 13 Comparison of the predicted CCG rate of six specimen geometries without side groove(Bn/B = 1), (a) thin specimens, (b) thick specimens
39
0.1 a/W=0.5 Bn/B=0.8
0.01
1E-3 1E-4
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
1E-5
da/dt,mm/h
da/dt,mm/h
0.01
0.1
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
1E-6 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3
a/W=0.5 Bn/B=0.8
1E-3 1E-4
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
1E-5
0.01
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
1E-6 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3
*
0.01
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 14 Comparison of the predicted CCG rate of six specimen geometries with 20%-side groove (Bn/B = 0.8), (a) thin specimens, (b) thick specimens
40
0.1
0.1
da/dt,mm/h
Bn/B=0.6
0.01
1E-3 1E-4
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
1E-5
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
1E-6 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3
da/dt,mm/h
a/W=0.5
0.01
a/W=0.5 Bn/B=0.6
1E-3 1E-4 1E-5
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
1E-6 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 * C ,MPam/h
0.01
*
C ,MPam/h (a)
0.01
(b)
Fig. 15 Comparison of the predicted CCG rate of six specimen geometries with 40%-side groove (Bn/B = 0.6), (a) thin specimens, (b) thick specimens
41
5E-4 4E-4
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
da/dt,mm/h
da/dt,mm/h
3E-4
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
2E-4
1E-4 *
C =1E-6MPamh
-1
a/W=0.5 0.6
0.7
0.8 Bn/B
0.9
5E-4 4E-4
C =1E-6MPamh
3E-4
a/W=0.5
*
2E-4
1E-4
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.6
1.0
W=50mm, B/W=1/2 W=25mm, B/W=1/2 W=25mm, B/W=1 W=25mm, B/W=1/2 W=12.5mm, B/W=1 W=12.5mm, B/W=1
0.7
0.8
0.9
Bn/B (a)
(b)
Fig. 16 The change of CCG rate da/dt with side-groove depth at C*=1E-6MPamh-1 (a) thin specimens, (b) thick specimens
42
1.0
-1
da/dt,mm/h
0.12 0.08
0.04
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.2 0.16
W=50mm, B/W=1/4 W=25mm, B/W=1/4 W=25mm, B/W=1/2 W=25mm, B/W=1/4 W=12.5mm, B/W=1/2 W=12.5mm, B/W=1/2
*
C =1E-3MPamh
0.08
0.04
-1
*
C =1E-3MPamh
W=50mm,B/W=1/2 W=25mm,B/W=1/2 W=25mm,B/W=1 W=25mm,B/W=1/2 W=12.5mm,B/W=1 W=12.5mm,B/W=1
-1
a/W=0.5
a/W=0.5 0.6
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
0.12 da/dt,mm/h
0.2 0.16
0.7
0.8 Bn/B
0.9
1.0
0.6
(a)
0.7
0.8 Bn/B
0.9
(b)
Fig. 17 The change of CCG rate da/dt with side-groove depth at C*=1E-3MPamh-1 (a) thin specimens, (b) thick specimens
43
1.0
100
100 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
10
1
a/a0
a/a0
10
0.1 0.01
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
1 0.1
1
1.5
2 2.5 * 1-Ravg
3
3.5
(a)
0.01 1.2
1.6 Ac-avg (b)
Fig. 18 The relation between a / a0 and R*avg (a) and Ac-avg (b) for all specimens
44
2
2.4
Table 1 The 2RN creep model parameters [19]
Stress
A(MPa-nh-1)
n
σ ≤ 250MPa
A1=7.26×10-26
n1=8.75
σ > 250MPa
A2=3.53×10-36
n2=13.08
45
Table 2 Detailed geometries and sizes of specimens
Specimen
W(mm)
H(L)/W
a0/W
B/W
Bn/B
C(T)
50
1.2
0.5
1/2-1/4
0.6, 0.8, 1
15.45
19.01
27.65
SEN(B)
25
2
0.5
1/2-1/4
0.6, 0.8, 1
2.69
5.39
8.08
CS(T)
25
-
0.5
1-1/2
0.6, 0.8, 1
4.81
8.65
11.54
SEN(T)
25
2
0.5
1/2-1/4
0.6, 0.8, 1
1.79
3.13
4.47
DEN(T)
12.5
4
0.5
1-1/2
0.6, 0.8, 1
3.87
6.20
9.30
M(T)
12.5
4
0.5
1-1/2
0.6, 0.8, 1
6.33
8.44
10.55
46
Load levels Kin(MPam1/2)
Highlights
The 3-D creep constraint of specimens in ASTM E1457 standard was characterized. The creep crack growth rate in the specimens were predicted by numerical simulations. The specimens
ASTM-E1457 have different constraints and CCG rates.
Larger specimen thickness and deeper side-groove induce higher constraint and CCG rate It needs to consider constraint effects on CCG rate when predicting component life.
47