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Effect of constraint on creep crack initiation time in test specimens in ASTM-E1457 standard
Accepted Manuscript Effect of constraint on creep crack initiation time in test specimens in ASTME1457 standard J.Z. He, G.Z. Wang, S.T. Tu, F.Z. Xuan PII: DOI: Reference:
S0013-7944(16)30692-0 http://dx.doi.org/10.1016/j.engfracmech.2017.02.021 EFM 5417
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
4 December 2016 21 February 2017 22 February 2017
Please cite this article as: He, J.Z., Wang, G.Z., Tu, S.T., Xuan, F.Z., Effect of constraint on creep crack initiation time in test specimens in ASTM-E1457 standard, Engineering Fracture Mechanics (2017), doi: http://dx.doi.org/ 10.1016/j.engfracmech.2017.02.021
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Effect of constraint on creep crack initiation time 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 The creep crack initiation (CCI) time of six specimen geometries in ASTM E1457 standard has been predicted by using numerical simulations, and the constraint effect on CCI time was analyzed. The results show the specimens with different geometries, size ranges and side-grooves recommended by ASTM-E1457 produce different CCI time data due to different constraint levels. With increasing constraint levels in specimens, the CCI time is reduced. The quantitative correlation formulas of CCI time with creep constraint parameters have been established, and they may be applied in CCI life assessment of high-temperature components for accounting for constraint effect and improving accuracy.
Keywords: Constraint, Creep crack initiation, ASTM-E1457, Specimen geometry, Side groove
Corresponding author Tel.: +86 21 64252681; fax: +86 21 64252681. E-mail address: [email protected] (Guozhen Wang). 1
Nomenclature a a0 A A1, A2 Ac
B Bn C* C*avg E F H Kin n n1, n2 r Q* R* Ri Ro t ti t0.2 t0.5 tred W
crack depth initial crack depth constant in Norton creep model constants in 2RN creep model 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 Young’s modulus applied load factor to estimate C* in experiment using load line displacement initial stress intensity factor stress exponent in Norton creep model stress exponents in 2RN creep model distance from a crack tip a parameter to correlate creep crack growth rate load-independent creep constraint parameter inner radius outer radius creep time creep crack initiation time creep crack initiation time for a crack extension of 0.2mm creep crack initiation time for a crack extension of 0.5mm creep redistribution time specimen width
εc
creep strain rate
ε *f
multiaxial creep ductility
εf
uniaxial creep ductility
εc
equivalent creep strain
σ0 σ22 σ22, CT σy v
normalizing stress, usually taken as yield stress opening stress opening stress of C(T) specimen under plane strain yield stress Poisson’s ratio
ACEEQ Aref
2
ω
damage parameter
ω
damage rate
η
factor to estimate C* in experiment using load line displacement
load line displacement rate
Abbreviations 2-D 2RN 3-D C3D8R CPE4H CCG CCI CEEQ CS(T) C(T) DEN(T) FEM M(T) PE 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 equivalent creep strain in ABAQUS code C-shaped tension compact tension double edged notched tension finite element method middle cracked tension plane strain single-edge notched bend single-edge notched tension
3
1.
Introduction Creep crack initiation (CCI) and creep crack growth (CCG) are the principle failure
mechanism of high-temperature components [1]. The CCI time is defined as the time required to reach a small crack growth from a pre-existing crack. The small crack growth length is typically 0.2mm or 0.5mm, and the corresponding CCI time is denoted as t0.2 or t0.5, respectively [2-4]. Because the CCI time may take up to 80% of service lifetime of a component [1], it needs to be considered in life prediction and assessment of high-temperature components. The recommended specimens in the ASTM E1457 standard [2] for CCI and 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.
In Section 1.1.7 of ASTM E1457, it suggests that
the level of constraint, for the relatively short term test durations (less than one year), does not vary within the range of normal data scatter observed in tests of the recommended specimen geometries [2]. However, a lot of experiments and numerical simulations have shown that specimen geometries, crack sizes, test duration and loading modes can affect creep crack-tip stress state (crack-tip constraint level), which can subsequently influence the CCG rate [5-21] and CCI time [12, 22, 23] of materials. For a given C* value, the CCG rate increases with increasing crack depth [5] and specimen thickness [6-9]. Some experimental results have shown that the CCG rates of higher constraint C(T) specimens are significantly higher than those of the lower constraint M(T) specimen at a given C* value for various steels [10-14]. Yokobori et al.[15] proposed a parameter Q* to correlate CCG rate and their work indicated that the CCG rates of thick specimens are higher than those of thin specimens at a given Q* value. The constraint effects on CCG rate were regarded as 4
"structural brittleness"[16]. The numerical simulation results in some recent work [17-21] also have indicated that the creep constraints caused by specimen geometries and sizes have significant effects on CCG rates. As described above, a lot of studies [5-21] have been done for the constraint effects on the CCG behavior of steels, but the investigations of the constraint effects on CCI behavior in specimens with different geometries and sizes are very limited. Some experimental investigations [12, 22] and numerical simulations [23] have shown that the CCI time data obtained from low constraint M(T) specimens are longer than those from high constraint C(T) specimens for a given C* value for 316H stainless steel. It has also been shown that long term CCI time is shorter than the CCI time of prediction from short term tests for a given C* value [12]. Different approaches for estimation of CCI time have been developed, such as time-dependent failure assessment diagram (TDFAD) [24, 25], two-criteria diagram (2CD) [26-28] and Nikbin-Smith-Webster model (NSW model) [28]. In these approaches, the constraint effects caused by specimen or component geometries, crack sizes and loading configurations have not been considered, and the conservative or non-conservative CCI prediction results may be produced. In recent work of authors [21], the 3-D creep constraints have been characterized by the creep constraint parameters R* and Ac and the influences of constraint on CCG rates have been investigated for the recommended specimen geometries and size ranges in ASTM E1457 standard, and the quantitative correlation of CCG rate with creep constraint has also been established. However, the effects of constraint induced by specimen geometries, size ranges and side-groove in ASTM-E1457 on CCI time have not been systemically investigated in wide-range C* region, and the quantitative correlation of CCI time with constraint has not been established. In this work, the CCI time of the six specimens with different thicknesses and side-groove depths in ASTM-E1457 5
was predicted by numerical simulations based on ductility exhaustion damage model with stress dependent creep ductility and strain rate model. The main advances made in this paper are in the understanding of constraint effects on the CCI time in test specimens in ASTM-E1457 standard and the establishment of the quantitative correlation of CCI time with creep constraint. 2.
Finite element simulations
2.1 Material The material used in present work is a Cr-Mo-V steel (Chinese 25Cr2NiMo1V steel), and it is the same as that used in the previous work of authors [21, 29, 30]. The 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 creep behavior of the steel obeys two-regime Norton (2RN) creep model which has different Norton parameters of (A1,n1) in low stress regime and (A2,n2) in high stress regime, as shown in Eq.(1) and Table 1 [30]. This 2RN behavior of the Cr-Mo-V steel is similar to that of the 316H steel in the literature [18, 23, 31]. n A1σ 1 ε n2 A2σ
σ 250MPa σ 250MPa
2RN model
(1)
2.2 Specimen geometry and finite element model Six types of specimen geometries recommended in the ASTM-E1457 standard [2] for CCI and CCG testing (C(T), SEN(B), CS(T), SEN(T), DEN(T) and M(T)) have been modeled by ABAQUS code [32] in the previous work of authors [21]. The geometries of these specimens are shown in Fig.1. Specimen dimensions in the FE models conforming to the ASTM-E1457 are summarized in Table 2. The W is the specimen width of C(T), SEN(B) and CS(T) specimens and half width of SEN(T), DEN(T) and M(T) specimens. The L is the half length of SEN(B), SEN(T), DEN(T) and M(T) specimens, B is the specimen thickness (Bn is the net thickness), and a0 is the initial crack 6
length. The Ri is the inner radius, Ro is the outer radius and X=2W for CS(T) specimen. In Table 2, the thickness-to-width ratio (B/W) of C(T), SEN(B) and SEN(T) specimens is from 1/2 to 1/4, and that of CS(T), DEN(T) and M(T) specimens is from 1 to 1/2. 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 are denoted as thick specimens. The C(T), SEN(B) and SEN(T) specimens with B/W = 1/4 and the CS(T), DEN(T) and M(T) specimens with B/W = 1/2 are denoted as thin specimens. To analyze the effects of side groove depth on CCI time, the specimens with various Bn/B values (0.6, 0.8 and 1) in Table 2 were modeled. Due to geometrical symmetry, only one-fourth finite element models of all specimens were built. The FE models for simulating CCI in this work are the same as those for simulating CCG in the previous work of authors [21], which have been decribed in detail in the Section 2.2 of the previous paper [21]. The element size of 100μm which is similar to the average grain size of this steel [6] is regularly set in the crack growth zone for all specimens. 3.
Creep crack-tip constraint of test specimens in ASTM-E1457 In order to quantitatively analyze the constraint effects on CCI time in test specimens in
ASTM-E1457 standard, the values of creep crack-tip constraint parameters require to be calculated. Recently, the creep constraint parameter R* based on crack-tip stress field [33, 34] and Ac based on crack-tip equivalent creep strain [35-37] have been proposed. The load independent creep constraint parameter R* at steady-state creep has been investigated and can be calculated by Eq.(2) [34]:
C R* σ 22 AL * 1
1 n1
C σ 22,CT AL * 2
1 n1
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 with high constraint in PE condition, respectively; A and n are material creep constants, and L is a characteristic length 7
which is usually set to be 1m [34]. The unified creep constraint parameter Ac of in-plane and out-of-plane constraints has been defined as follows [35]: Ac
AC E E Q
at t/tred = 1
Ar e f
(3)
where ACEEQ and Aref are the area surrounded by the equivalent creep strain εc isoline ahead of crack tip in the evaluated specimen and the standard reference specimen with high constraint, respectively. The t is creep time and tred is stress redistribution time. In the recent work of authors [21], the parameters R* and Ac in all specimens in ASTM-E1457 with different side-grooved depths (different Bn/B) and different thickness-to-width ratios (B/W) in Table 2 have been calculated, and are shown in Figs.5-12 in the previous paper [21]. Fig.2 shows typical R* and Ac distributions along crack front of thin specimens with 20% side groove (Bn/B = 0.8) [21]. The z/Bn = 0 and z/Bn =0.5 are the central plane and surface of specimens, respectively. The results in the previous paper [21] and Fig.2 showed that 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 CCG rate increases. The parameter Ac has better capability for characterizing creep constraint than the parameter R* for the specimen geometries in ASTM-E1457. 4.
Creep crack initiation simulations The creep damage model based on the ductility exhaustion approach was used to simulate CCI
in the specimens in Table 2. This model is the same as that for simulating CCG [19, 21] and CCI [29] in various specimens of Cr-Mo-V steel in the previous work of authors, and has been described 8
in detail in the previous papers [19, 21, 29]. The model is expressed by Eq.(4): t
t
0
0
ω ω dt
εc dt ε *f
(4)
where the ω is the accumulated creep damage parameter, ω is the creep damage rate, εc is equivalent creep strain rate, and ε *f is the multiaxial creep ductility. The value of damage parameter ω is from 0 to 1. When the ω value at an element calculated from Eq. (4) reaches 1, the element is considered fully creep damage and its load-carrying capacity is reduced to zero by reducing the elastic modulus to a very small value of 1MPa [17-21, 31]. The multiaxial creep ductility ε *f can be estimated from the uniaxial creep ductility ε f by using the Cocks and Ashby model [38] as follows:
n 0.5 σ m 2 n 0.5 sin sinh 2 εf 3 n 0.5 n 0.5 σ e ε *f
(5)
where n is the creep exponent (for power law creep), and m / e is the ratio between the mean stress and equivalent stress, which is defined 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 of materials depends on stress levels (strain rates), which leads to the stress-regime dependence of the creep ductility of materials [30, 31]. For the Cr-Mo-V steel used in this work, the estimated stress dependent creep ductility was described by Eq. (6) [19, 21, 29]. σ 225MPa
0.5 ε f 1.14 10 14 σ 5.8 0.00537 1.6
225 σ 275MPa σ 275MPa
(6)
The Eqs.(4), (5) and (6) were implemented in the ABAQUS code [32] by using the user subroutine USDFLD, and the CCI were predicted. The FEM models for simulating CCI behavior in the specimens in ASTM-E1457 in Table 2 are the same as those for simulating CCG behavior in the previous paper [21]. All calculations were carried out by using elastic-plastic-creep model. The true 9
stress-strain curve of the steel [21] and the 2RN creep model parameters in Table 1 were employed. For each specimen in Table 2, a wide range of load levels has been applied to produce a wide range of C*. The range of initial stress intensity factor Kin (load levels) for CCI simulations of all specimens is listed in Table 2. It has been demonstrated in recent work of Ma et al. [37] that for such lower initial load levels, the plasticity effect is small. The creep crack growth length was estimated by numbers of damage elements. The CCI time t0.2 for a crack extension of 0.2mm and t0.5 for a crack extension of 0.5mm were calculated. The CCI time can be correlated with C* parameter, and the C* was calculated by using Eq. (7) in ASTME1457 standard.
C*
FΔ Hη 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 their solutions can be found in ASTM-E1457 [2]. The C* validity criteria in ASTM-E 1457 were applied to ensure that C* is a valid parameter to describe the CCI time. The creep damage model based on the ductility exhaustion approach described above has been widely used in many studies for simulating CCG behavior in test specimens with different geometries for different steels, the CCG rate data predicted by the FE simulations agree well with the experimental data [18-20, 30, 31, 39]. In recent work of authors [29] and Quintero and Mehmanparast [23], the CCI time predicted by the FE simulations using the ductility exhaustion approach with stress dependent creep ductility agree well with the experimental data, thus the CCI simulation method used in this work has been validated. 10
5.
Results and discussion
5.1 Creep crack initiation time The FE simulations were performed on six specimens with different thickness-to-width ratios (B/W) and different side-grooved depths (different Bn/B) over a wide range of C* produced by various initial stress intensity factors Kin in Table 2. For each specimen, the initiation time was calculated for an average crack extension of 0.2mm and 0.5mm, i.e. t0.2 and t0.5 were predicted, respectively. The CCI times t0.2 and t0.5 are compared in Figs. 3-8 for a wide range of C*. The Figs. 3-8 (a) are the CCI trend of the thin specimens, and the Figs. 3-8(b) are those of the thick specimens. It can be observed that the simulated CCI times (t0.2 and t0.5) are inversely proportional to C* on log-log scale and the ti-C* curve of each specimen has two slope turning points (as marked by turning point 1 and 2 in Figs.3-8). The fitted curves have three CCI trend: a long-term trend at low C* region (below the turning point 1), a short-term trend at high C* region (above the turning point 2) and a transition behavior between short-term (high C*) and long-term (low C*) CCI trend in intermediate C* region (between the turning points 1 and 2), which are consistent with the experimental and simulated results in the literature [23, 29]. The slope turning points on the ti –C* curves are similar to those on da/dt–C* curves of CCG rate which are mainly caused by the stress dependent creep ductility [19, 20, 30]. Because the CCI time is defined based on crack extension length which is relevant to CCG rate, the slope turning points on the ti –C* curves may also be attributed to the stress dependent creep ductility. It has been shown in the previous work of authors [19] that the stress triaxiality between high C* region and low C* region is similar. But in high C* region, the equivalent stresses ahead of cracks in the specimens are above the transition stress of creep ductility, and the multiaxial creep ductility is determined by upper shelf value of uniaxial creep ductility ( ε f = 1.6 in Eq.(6)). In low C* region, the equivalent stresses ahead of cracks are below the transition stress of creep ductility, and the multiaxial creep ductility is determined by 11
lower shelf value of uniaxial creep ductility ( ε f = 0.5 in Eq.(6)). The results in Figs.3-8 suggest that if the extrapolation CCI time data from the transition or high C* region are used in CCI assessments of the components at lower C* region, excessive conservative or non-conservative results may be produced. Figs. 3-8 show that all specimens have similar CCI trends for both t0.2 and t0.5 in high, low and transition C* regions, and the CCI trend lines are parallel to each other for all specimens. For the specimens without side-groove (Bn/B = 1), the constraint caused by specimen geometry has significant influence on CCI time in low and transition C* region, but it has small influence on it in high C* region. The constraint effect is reduced in the high C* region may be due to plasticity causing loss of crack-tip constraint, as analyzed in the literature [12, 19]. The constraint level of the six specimen geometries in ASTM-E1457 has been characterized in the previous paper [21], and it decreases in sequence from CS(T), C(T), SEN(B), SEN(T), DEN(T) to M(T) specimen. The higher constraint specimens (CS(T), C(T) and SEN(B)) have shorter CCI time, and the lowest constraint M(T) specimen has the longest CCI time, and the CCI time of the SEN(T) and DEN(T) specimens with middle constraint are lie in between. These results suggest that if the CCI time data of high constraint specimens are used in CCI life assessment of components with lower constraint, excessive conservative results will be produced, and if the CCI time data of low constraint specimens are used in CCI life assessment of thick components with higher constraint, non-safety prediction will be produced. Figs.5-8 show that the introduction of side groove results in the reduction of CCI time and the decrease in difference of CCI time among different specimens. As analyzed by constraint in Ref. [21], this is caused by the increase of constraint and the reduction of constraint difference among different specimens. With increasing side groove depth from 0% to 40%, the CCI time decreases owing to the increase of constraint as analyzed in Ref. [21]. At the same C* value, the CCI time of thick specimens is shorter than that of thin specimens. In the previous paper [21], by comparing CCG rates with the constraint parameter Ac, it has been shown that the CCG rates and their 12
difference among different specimens in ASTM-E1457 are consistent with the parameter Ac and their difference. The specimens with higher constraint (lower Ac) have higher CCG rate, and the larger difference in CCG rates among different specimens corresponds to the larger difference in Ac . By comparing the CCI time (Figs.3-8) in this work with the constraint parameter Ac in Ref. [21], it also can be found that the CCI time and their difference among different specimens in ASTM-E1457 are consistent with the parameter Ac and their difference. The specimens with higher constraint have shorter CCI time, and the larger difference in CCI time among different specimens corresponds to the larger difference in Ac. These results imply that the difference in CCI time among different specimens in ASTM-E1457 is caused by different crack-tip constraint levels in different specimens, and the parameter Ac can accurately characterize creep constraint due to that it can effectively and simultaneously incorporate both in-plane and out-of-plane constraints in the specimens [35-37]. For clearly showing the effects of side groove depth and specimen thickness on CCI time, Figs. 9 and 10 give the change of the CCI times t0.2 and t0.5 with side groove depth at a low C* value of 1E-6MPam/h for all specimens. It can be seen that with increasing side-groove depth (decreasing Bn/B), the CCI times t0.2 and t0.5 decrease, and the higher constraint specimens (C(T), SEN(B) and CS(T)) have shorter CCI times t0.2 and t0.5. The t0.2 and t0.5 of lower constraint specimens (DEN(T), SEN(T) and M(T)) is more sensitive to side groove depth. This is consistent with the result of constraint analysis in Ref. [21] which shows that the constraint Ac of lower constraint specimens is more sensitive to side groove depth. Figs. 9 and 10 also show that the thick specimens with side groove have shorter CCI time due to their higher constraint [21]. It can be found from Figs. 9 and 10 that the scattering of creep crack initiation time due to mechanical constraint is about 3-4 times. Therefore, it is necessary to consider the constraint effect in CCI life assessment of high temperature components. 5.2 Correlation of CCI time with creep constraint parameters R* and Ac In Section 6.2.2 of ASTM E1457 standard, it suggests that the effects of crack tip constraint 13
arising from variations in specimen size, geometry and material ductility can influence t0.2 and da/dt. For example, crack growth rates at the same value of C*(t), Ct in creep-ductile materials generally increases with increasing thickness. It is therefore necessary to keep the component dimensions in mind when selecting specimen thickness, geometry and size for laboratory testing. These suggestions in ASTM E1457 are consistent with the calculated results in this paper. The results of CCI time described in Section 5.1 have been shown that the recommended specimen geometries, size range and side groove in ASTM-E1457 can significantly affect CCI time in low C* region which is relative to the service load levels of elevated temperature components. In the case without side groove, the higher constraint specimens (CS(T), C(T) and SEN(B)) have shorter CCI time, while the lower constraint specimens (SEN(T), DEN(T) and M(T)) have longer CCI time. The introduction of side groove can significantly decrease CCI time of lower constraint specimens (SEN(T), DEN(T) and M(T)). Therefore, it is an important issue to account for the constraint effects on CCI time when predicting component life using laboratory data. For accurately and quantitatively accounting for the constraint effects on CCI time, it is necessary to establish the correlation of geometry constraint with CCI time. By analogy to the method for establishing the correlation of CCG rate with creep constraint parameters R* and Ac which has been given in detail in the recent work of authors [21, 35], the quantitative correlation formulas of CCI times (t0.2 and t0.5) with creep constraint parameters R* and Ac may be established. Because the average constraint R*avg and Ac-avg can characterize the overall constraint levels in the specimens, they have been calculated for all specimen geometries in ASTM-E1457 in the previous work of authors [21]. In this work, the load-independent R*avg and Ac-avg data in Ref.[21] were used to establish their correlation with CCI time. The CCI times (t0.2)C(T)-PE and (t0.5)C(T)-PE calculated from the high constraint standard C(T) specimen with W=50mm and a0/W =0.5 under PE condition are taken to normalize CCI times. The normalized CCI time ratios (t0.2/(t0.2)C(T)-PE and t0.5/(t0.5)C(T)-PE) are calculated at two typical lower C* values (C*=1E-6MPam/h and C*=1E-7MPam/h) for all specimens, and the relations between the 14
CCI time ratios and constraint parameters R*avg and Ac-avg for the low C* region (the long-term trend below the turning point 1 in Figs.3-8) which is relative to the load levels of components are shown in Figs. 11 and 12, respectively. It can be seen that there exists a linear relation on log-log scale between the CCI time ratios and R*avg and Ac-avg regardless of the C* values (load levels). This implies that the linear relation is independent on the load levels. Figs. 13(a) and (b) show the linear relations on log-log scale between CCI time ratio ti/(ti)C(T)-PE (including both t0.2/(t0.2)C(T)-PE and t0.5/(t0.5)C(T)-PE) and R*avg and Ac-avg for two typical lower C* values (C*=1E-6MPam/h and C*=1E-7MPam/h), respectively. It is interesting to see that the data of the normalized CCI time ratios of t0.2/(t0.2)C(T)-PE and t0.5/(t0.5)C(T)-PE can form a sole line on log-log scale. This implies that the definition of CCI time (t0.2 or t0.5) does not influence the correlation line between the normalized CCI time ratio and constraint, and the correlation line may be only dependent on material. The correlation formulas of ti/(ti)C(T)-PE and R*avg and Ac-avg can be fitted from Fig.13 for the low C* region of C* < 2.3E-5MPam/h in Figs.3-8, and are shown in Eqs.(8) and (9), respectively. The (ti)C(T)-PE (for both t0.2 or t0.5) and its correlation with C* were calculated from the standard reference C(T) specimen with W=50mm and a/W=0.5 under PE condition, and shown in Eq.(10) for the low C* region.
* ti / ti CT PE 0.94 1 Ravg
0.95
ti / ti CT PE 0.63 Acavg
1.36
* 0.025C *0.86 0.048C 0.90
ti CT PE
Δa 0.2mm Δa 0.5mm
for for
C* < 2.3E-5MPam/h
(8)
C* < 2.3E-5MPam/h
(9)
for
C* < 2.3E-5MPa.m/h
(10)
Figs.11-13 imply that the two creep constraint parameters R* and Ac both can be used to correlate the CCI time with constraint induced by specimen geometries, sizes and side grooves. But the data scatter of Ac-avg in Figs.12 and 13(b) seems to be smaller than that of R*avg in Figs.11 and 13(a). This may be due to that the Ac-avg can effectively characterize both in-plane and out-of-plane constraints [35-37]. So the correlation formula of ti/(ti)C(T)-PE with Ac-avg may be more accurate for 15
describing the constraint effect on CCI time. The CCI time in structures (such as in pressurized pipes and vessels) may be assessed by the two-parameter C*-R*avg or C*- Ac-avg approach based on Eqs.(8) and (9) to incorporate crack-tip constraint effect. For actual structures with a certain geometry and loading condition, the creep fracture mechanics parameter C* can be obtained by reference stress method or FE calculations. And the corresponding constraint parameter Ac-avg or R*avg can be calculated at the loading level C* using FE analysis. Therefore, the constraint dependent CCI time in structures can be predicted by using Eq. (8) or (9) based on the two-parameter C*-R*avg or C*- Ac-avg. 6.
Conclusion The CCI time of six types of specimen geometries with different thicknesses and side-groove
depths in ASTM-E1457 standard has been predicted by using ductility exhaustion model with stress dependent creep ductility in FE analyses. The effects of the specimen geometry, thickness and side-groove on CCI time (t0.2 and t0.5) have been analyzed, and the quantitative correlation of CCI time with creep constraint has been established. The primary results are summarized as follows: (1) Short-term (high C* region) and long-term (low C* region) CCI trends (for both t0.2 and t0.5) with a transition region (intermediate C* region) in between have been predicted for specimen geometries with different thicknesses and side-groove depths in ASTM-E1457. In low and transition C* regions, the specimen geometries has significant influence on CCI time, but it has a small effect on it in high C* region. (2) The difference in CCI time among different specimens in ASTM-E 1457 comes from the difference in crack-tip constraint levels. The higher constraint specimens (CS(T), C(T) and SEN(B)) have shorter CCI time, and the lower constraint specimens (SEN(T), DEN(T) and M(T)) have longer CCI time. The scattering of creep crack initiation time due to mechanical constraint is about 3-4 times. 16
(3) With increasing side-groove depth (decreasing Bn/B), the CCI time decreases due to the increase of constraint. The CCI time of lower constraint specimens (DEN(T), SEN(T) and M(T)) is more sensitive to side groove depth. The thick specimens with side groove have shorter CCI time due to their higher constraint. (4) It is necessary to account for the constraint effects on CCI time when predicting component life using laboratory data measured by specimen geometries in ASTM-E1457 and when comparing CCI time data of the specimens with various geometries and sizes. (5) The quantitative correlation lines and formulas of the normalized CCI time with creep constraint parameters R* and Ac have been established, and they are independent on load level C* and definition of CCI time (t0.2 or t0.5). The correlation formulas may be applied in CCI life assessment of high-temperature components for accounting for constraint effects and improving accuracy. Acknowledgments This work was financially supported by the Projects of the National Natural Science Foundation of China (51375165, 51575184, 51325504).
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21
Figure and Table Captions Fig. 1 Schematic illustrations of specimens in ASTM E1457: (a) C(T), (b)SEN(B), (c) CS(T), (d) SEN(T), (e) DEN(T), (f) M(T) Fig. 2 Distributions of constraint parameter R* (a) and Ac (b) of six specimen geometries with 20%-side groove (Bn/B = 0.8) for thin specimens [21] Fig.3 Comparison of predicted CCI time of six specimen geometries without side groove (Bn/B = 1) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens Fig.4 Comparison of predicted CCI time of six specimen geometries without side groove (Bn/B = 1) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens Fig. 5 Comparison of predicted CCI time of six specimen geometries with 20%-side groove (Bn/B = 0.8) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens Fig. 6 Comparison of predicted CCI time of six specimen geometries with 20%-side groove (Bn/B = 0.8) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens Fig. 7 Comparison of predicted CCI time of six specimen geometries with 40%-side groove (Bn/B = 0.6) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens Fig. 8 Comparison of predicted CCI time of six specimen geometries with 40%-side groove (Bn/B = 0.6) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens Fig. 9 The change of CCI time with side groove depth at C*=1E-6MPam/h for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens Fig. 10 The change of CCI time with side groove depth at C*=1E-6MPam/h for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens Fig. 11 The relations between t0.2/(t0.2)C(T)-PE (a) and t0.5/(t0.5)C(T)-PE (b) and R*avg for all specimens Fig. 12 The relations between t0.2/(t0.2)C(T)-PE (a) and t0.5/(t0.5)C(T)-PE (b) and Ac-avg for all specimens 22
Fig.13 The relations between ti/(ti)C(T)-PE and R*avg (a) and Ac-avg (b) for all specimens at C*=1E-6MPam/h and C*=1E-7MPam/h value Table 1 The 2RN creep model parameters [30] Table 2 Detailed geometries and sizes of specimens
23
(a) C(T)
(b) SEN(B)
(d) SEN(T)
(c) CS(T)
(e) DEN(T)
(f) M(T)
Fig. 1 Schematic illustrations of specimens in ASTM E1457: (a) C(T), (b)SEN(B), (c) CS(T), (d) SEN(T), (e) DEN(T), (f) M(T) 24
2.00
0.5
a/W=0.5 Bn/B=0.8
0.0 1.75
-0.5 -1.5
Ac
R
*
-1.0 1.50
-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
1.25
a/W=0.5 Bn/B=0.8
0.2 0.3 z/Bn
1.00
0.4
0.5
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
0.3
0.4
0.5
z/Bn
(a)
(b)
Fig. 2 Distributions of constraint parameter R* (a) and Ac (b) of six specimen geometries with 20%-side groove (Bn/B = 0.8) for thin specimens [21]
26
6
6
10
10 a/W=0.5 Bn/B=1
5
10
4
3
Turning point 2
10
2
10
1
10
0
4
10
Turning point 1
t0.2,h
t0.2,h
10
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
Turning point 1
3
Turning point 2
10
2
10
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
a/W=0.5 Bn/B=1
5
10
1
10
0
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
*
0.1
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 3 Comparison of predicted CCI time of six specimen geometries without side groove (Bn/B = 1) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens
27
6
6
10
10 a/W=0.5 Bn/B=1
5
10
4
4
3
10
2
1
10
0
10
Turning point 1 Turning point 2 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
t0.5,h
t0.5,h
10
10
a/W=0.5 Bn/B=1
5
10
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
3
10
2
10
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
Turning point 1
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
*
0.1
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 4 Comparison of predicted CCI time of six specimen geometries without side groove (Bn/B = 1) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens
28
6
6
10
10 a/W=0.5 Bn/B=0.8
5
10
4
4
10
10
1
10
0
Turning point 2 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
2
10
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
Turning point 1
3
10
t0.2,h
10
t0.2,h
10
Turning point 1
3
2
a/W=0.5 Bn/B=0.8
5
10
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
*
0.1
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 5 Comparison of predicted CCI time of six specimen geometries with 20%-side groove (Bn/B = 0.8) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens
29
6
6
10
10
a/W=0.5 Bn/B=0.8
5
10
4
4
10
10
t0.5,h
1
10
0
Turning point 2 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
2
10
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
3
10
t0.5,h
3
2
Turning point 1
Turning point 1
10 10
a/W=0.5 Bn/B=0.8
5
10
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
0.1
*
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 6 Comparison of predicted CCI time of six specimen geometries with 20%-side groove (Bn/B = 0.8) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens
30
6
6
10
10
a/W=0.5 Bn/B=0.6
5
10
4
4
3
Turning point 2
10
2
1
10
0
10
Turning point 1
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
t0.2,h
t0.2,h
10
10
a/W=0.5 Bn/B=0.6
5
10
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 * C ,MPam/h
10
2
10
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
Turning point 1
3
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 * C ,MPam/h
0.1
(a)
0.1
(b)
Fig. 7 Comparison of predicted CCI time of six specimen geometries with 40%-side groove (Bn/B = 0.6) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens
31
6
6
10
10 a/W=0.5 Bn/B=0.6
5
10
4
4
3
10 10
1
10
0
10
Turning point 1
Turning point 1
t0.5,h
t0.5,h
10
2
a/W=0.5 Bn/B=0.6
5
10
Turning point 2 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 * C ,MPam/h
3
10
2
10
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 * C ,MPam/h
0.1
(a)
0.1
(b)
Fig. 8 Comparison of predicted CCI time of six specimen geometries with 40%-side groove (Bn/B = 0.6) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens
32
30000
10000
20000
t0.2,h
t0.2,h
20000
30000
*
C =1E-6MPam/h a0/W=0.5
C(T) W=50mm, B/W=1/4 SEN(B) W=25mm, B/W=1/4 CS(T) W=25mm, B/W=1/2 SEN(T) W=25mm, B/W=1/4 DEN(T) W=12.5mm, B/W=1/2 M(T) W=12.5mm, B/W=1/2
0.6
0.7
0.8
0.9
1.0
C(T) W=50mm, B/W=1/2 SEN(B) W=25mm, B/W=1/2 CS(T) W=25mm, B/W=1 SEN(T) W=25mm, B/W=1/2 DEN(T) W=12.5mm, B/W=1 M(T) W=12.5mm, B/W=1
*
C =1E-6MPam/h a0/W=0.5
10000
0.6
0.7
0.8
Bn/B
Bn/B
(a)
(b)
0.9
1.0
Fig. 9 The change of CCI time with side groove depth at C*=1E-6MPam/h for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens
33
50000 40000 30000
C(T) W=50mm, B/W=1/4 SEN(B) W=25mm, B/W=1/4 CS(T) W=25mm, B/W=1/2 SEN(T) W=25mm, B/W=1/4 DEN(T) W=12.5mm, B/W=1/2 M(T) W=12.5mm, B/W=1/2
50000
*
C =1E-6MPam/h a0/W=0.5
40000 30000
*
C =1E-6MPam/h a0/W=0.5
t0.5,h
20000
t0.5,h
20000
C(T) W=50mm, B/W=1/2 SEN(B) W=25mm, B/W=1/2 CS(T) W=25mm, B/W=1 SEN(T) W=25mm, B/W=1/2 DEN(T) W=12.5mm, B/W=1 M(T) W=12.5mm, B/W=1
10000
10000
0.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
Bn/B
Bn/B
(a)
(b)
0.9
1.0
Fig. 10 The change of CCI time with side groove depth at C*=1E-6MPam/h for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens
34
100
10
10
1 *
C(T) C =1E-6MPam/h * SEN(B) C =1E-6MPam/h * CS(T) C =1E-6MPam/h * SEN(T) C =1E-6MPam/h * DEN(T) C =1E-6MPam/h * M(T) C =1E-6MPam/h
0.1
0.01 1.0
1.5
2.0
*
C(T) C =1E-7MPam/h * SEN(B) C =1E-7MPam/h * CS(T) C =1E-7MPam/h * SEN(T) C =1E-7MPam/h * DEN(T) C =1E-7MPam/h * M(T) C =1E-7MPam/h
2.5
t0.5/(t0.5)C(T)-PE
t0.2/(t0.2)C(T)-PE
100
*
0.1
0.01 1.0
3.0
C(T) C =1E-6MPam/h * SEN(B) C =1E-6MPam/h * CS(T) C =1E-6MPam/h * SEN(T) C =1E-6MPam/h * DEN(T) C =1E-6MPam/h * M(T) C =1E-6MPam/h
1.5
2.0
*
C(T) C =1E-7MPam/h * SEN(B) C =1E-7MPam/h * CS(T) C =1E-7MPam/h * SEN(T) C =1E-7MPam/h * DEN(T) C =1E-7MPam/h * M(T) C =1E-7MPam/h
2.5
3.0
*
*
1-R
1
1-R
avg
(a)
avg
(b)
Fig. 11 The relations between t0.2/(t0.2)C(T)-PE (a) and t0.5/(t0.5)C(T)-PE (b) and R*avg for all specimens
35
100
10
10
t0.5/(t0.5)C(T)-PE
t0.2/(t0.2)C(T)-PE
100
1 *
C(T) C =1E-6MPam/h * SEN(B) C =1E-6MPam/h * CS(T) C =1E-6MPam/h * SEN(T) C =1E-6MPam/h * DEN(T) C =1E-6MPam/h * M(T) C =1E-6MPam/h
0.1
0.01 1.0
1.2
1.4
1.6
1.8
2.0
*
C(T) C =1E-7MPam/h * SEN(B) C =1E-7MPam/h * CS(T) C =1E-7MPam/h * SEN(T) C =1E-7MPam/h * DEN(T) C =1E-7MPam/h * M(T) C =1E-7MPam/h
2.2
2.4
2.6
2.8
Ac-avg
1 *
C(T) C =1E-6MPam/h * SEN(B) C =1E-6MPam/h * CS(T) C =1E-6MPam/h * SEN(T) C =1E-6MPam/h * DEN(T) C =1E-6MPam/h * M(T) C =1E-6MPam/h
0.1
0.01 1.0
1.2
1.4
1.6
1.8
2.0
*
C(T) C =1E-7MPam/h * SEN(B) C =1E-7MPam/h * CS(T) C =1E-7MPam/h * SEN(T) C =1E-7MPam/h * DEN(T) C =1E-7MPam/h * M(T) C =1E-7MPam/h
2.2
2.4
Ac-avg
(a)
(b)
Fig. 12 The relations between t0.2/(t0.2)C(T)-PE (a) and t0.5/(t0.5)C(T)-PE (b) and Ac-avg for all specimens
36
2.6
2.8
100
10
10
1
0.1
0.01 1.0
1.5
C(T) t0.2/(t0.2)C(T)-PE
C(T) t0.5/(t0.5)C(T)-PE
SEN(B) t0.2/(t0.2)C(T)-PE
SEN(B) t0.5/(t0.5)C(T)-PE
CS(T) t0.2/(t0.2)C(T)-PE
CS(T) t0.5/(t0.5)C(T)-PE
SEN(T) t0.2/(t0.2)C(T)-PE
SEN(T) t0.5/(t0.5)C(T)-PE
DEN(T) t0.2/(t0.2)C(T)-PE
DEN(T) t0.5/(t0.5)C(T)-PE
M(T) t0.2/(t0.2)C(T)-PE
M(T) t0.5/(t0.5)C(T)-PE
2.0
2.5
ti/(ti)C(T)-PE
ti/(ti)C(T)-PE
100
1
0.1
0.01
3.0
1.2
1.4
C(T) t0.2/(t0.2)C(T)-PE
C(T) t0.5/(t0.5)C(T)-PE
SEN(B) t0.2/(t0.2)C(T)-PE
SEN(B) t0.5/(t0.5)C(T)-PE
CS(T) t0.2/(t0.2)C(T)-PE
CS(T) t0.5/(t0.5)C(T)-PE
SEN(T) t0.2/(t0.2)C(T)-PE
SEN(T) t0.5/(t0.5)C(T)-PE
DEN(T) t0.2/(t0.2)C(T)-PE
DEN(T) t0.5/(t0.5)C(T)-PE
M(T) t0.2/(t0.2)C(T)-PE
M(T) t0.5/(t0.5)C(T)-PE
1.6
*
1-R
1.8
2.0
2.2
Ac-avg
avg
(a)
(b)
Fig. 13 The relations between ti/(ti)C(T)-PE and R*avg (a) and Ac-avg (b) for all specimens at C*=1E-6MPam/h and C*=1E-7MPam/h value
37
2.4
Table 1 The 2RN creep model parameters [30]
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
38
Table 2 Detailed geometries and sizes of specimens
Specimen
W(mm)
H(L)/W
a0/W
B/W
Bn/B
Load levels Kin(Mpam1/2)
C(T)
50
1.2
0.5
1/2-1/4
0.6, 0.8, 1
13.66 ~ 42.85
SEN(B)
25
2
0.5
1/2-1/4
0.6, 0.8, 1
2.69 ~ 12.93
CS(T)
25
-
0.5
1-1/2
0.6, 0.8, 1
5.39 ~ 18.08
SEN(T)
25
2
0.5
1/2-1/4
0.6, 0.8, 1
6.39 ~ 27.16
DEN(T)
12.5
4
0.5
1-1/2
0.6, 0.8, 1
2.10 ~ 8.41
M(T)
12.5
4
0.5
1-1/2
0.6, 0.8, 1
5.91 ~ 16.04
39
Highlights The creep crack initiation (CCI) time in specimens in ASTM-E1457 were predicted. The specimens
ASTM-E1457 produce different CCI time data.
The CCI time decreases with increasing constraint levels in ASTM-E1457 specimens. The quantitative correlation formulas of CCI time with constraint were established. It needs to consider constraint effects on CCI time when predicting component life.
40
S0013-7944(16)30692-0 http://dx.doi.org/10.1016/j.engfracmech.2017.02.021 EFM 5417
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
4 December 2016 21 February 2017 22 February 2017
Please cite this article as: He, J.Z., Wang, G.Z., Tu, S.T., Xuan, F.Z., Effect of constraint on creep crack initiation time in test specimens in ASTM-E1457 standard, Engineering Fracture Mechanics (2017), doi: http://dx.doi.org/ 10.1016/j.engfracmech.2017.02.021
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Effect of constraint on creep crack initiation time 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 The creep crack initiation (CCI) time of six specimen geometries in ASTM E1457 standard has been predicted by using numerical simulations, and the constraint effect on CCI time was analyzed. The results show the specimens with different geometries, size ranges and side-grooves recommended by ASTM-E1457 produce different CCI time data due to different constraint levels. With increasing constraint levels in specimens, the CCI time is reduced. The quantitative correlation formulas of CCI time with creep constraint parameters have been established, and they may be applied in CCI life assessment of high-temperature components for accounting for constraint effect and improving accuracy.
Keywords: Constraint, Creep crack initiation, ASTM-E1457, Specimen geometry, Side groove
Corresponding author Tel.: +86 21 64252681; fax: +86 21 64252681. E-mail address: [email protected] (Guozhen Wang). 1
Nomenclature a a0 A A1, A2 Ac
B Bn C* C*avg E F H Kin n n1, n2 r Q* R* Ri Ro t ti t0.2 t0.5 tred W
crack depth initial crack depth constant in Norton creep model constants in 2RN creep model 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 Young’s modulus applied load factor to estimate C* in experiment using load line displacement initial stress intensity factor stress exponent in Norton creep model stress exponents in 2RN creep model distance from a crack tip a parameter to correlate creep crack growth rate load-independent creep constraint parameter inner radius outer radius creep time creep crack initiation time creep crack initiation time for a crack extension of 0.2mm creep crack initiation time for a crack extension of 0.5mm creep redistribution time specimen width
εc
creep strain rate
ε *f
multiaxial creep ductility
εf
uniaxial creep ductility
εc
equivalent creep strain
σ0 σ22 σ22, CT σy v
normalizing stress, usually taken as yield stress opening stress opening stress of C(T) specimen under plane strain yield stress Poisson’s ratio
ACEEQ Aref
2
ω
damage parameter
ω
damage rate
η
factor to estimate C* in experiment using load line displacement
load line displacement rate
Abbreviations 2-D 2RN 3-D C3D8R CPE4H CCG CCI CEEQ CS(T) C(T) DEN(T) FEM M(T) PE 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 equivalent creep strain in ABAQUS code C-shaped tension compact tension double edged notched tension finite element method middle cracked tension plane strain single-edge notched bend single-edge notched tension
3
1.
Introduction Creep crack initiation (CCI) and creep crack growth (CCG) are the principle failure
mechanism of high-temperature components [1]. The CCI time is defined as the time required to reach a small crack growth from a pre-existing crack. The small crack growth length is typically 0.2mm or 0.5mm, and the corresponding CCI time is denoted as t0.2 or t0.5, respectively [2-4]. Because the CCI time may take up to 80% of service lifetime of a component [1], it needs to be considered in life prediction and assessment of high-temperature components. The recommended specimens in the ASTM E1457 standard [2] for CCI and 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.
In Section 1.1.7 of ASTM E1457, it suggests that
the level of constraint, for the relatively short term test durations (less than one year), does not vary within the range of normal data scatter observed in tests of the recommended specimen geometries [2]. However, a lot of experiments and numerical simulations have shown that specimen geometries, crack sizes, test duration and loading modes can affect creep crack-tip stress state (crack-tip constraint level), which can subsequently influence the CCG rate [5-21] and CCI time [12, 22, 23] of materials. For a given C* value, the CCG rate increases with increasing crack depth [5] and specimen thickness [6-9]. Some experimental results have shown that the CCG rates of higher constraint C(T) specimens are significantly higher than those of the lower constraint M(T) specimen at a given C* value for various steels [10-14]. Yokobori et al.[15] proposed a parameter Q* to correlate CCG rate and their work indicated that the CCG rates of thick specimens are higher than those of thin specimens at a given Q* value. The constraint effects on CCG rate were regarded as 4
"structural brittleness"[16]. The numerical simulation results in some recent work [17-21] also have indicated that the creep constraints caused by specimen geometries and sizes have significant effects on CCG rates. As described above, a lot of studies [5-21] have been done for the constraint effects on the CCG behavior of steels, but the investigations of the constraint effects on CCI behavior in specimens with different geometries and sizes are very limited. Some experimental investigations [12, 22] and numerical simulations [23] have shown that the CCI time data obtained from low constraint M(T) specimens are longer than those from high constraint C(T) specimens for a given C* value for 316H stainless steel. It has also been shown that long term CCI time is shorter than the CCI time of prediction from short term tests for a given C* value [12]. Different approaches for estimation of CCI time have been developed, such as time-dependent failure assessment diagram (TDFAD) [24, 25], two-criteria diagram (2CD) [26-28] and Nikbin-Smith-Webster model (NSW model) [28]. In these approaches, the constraint effects caused by specimen or component geometries, crack sizes and loading configurations have not been considered, and the conservative or non-conservative CCI prediction results may be produced. In recent work of authors [21], the 3-D creep constraints have been characterized by the creep constraint parameters R* and Ac and the influences of constraint on CCG rates have been investigated for the recommended specimen geometries and size ranges in ASTM E1457 standard, and the quantitative correlation of CCG rate with creep constraint has also been established. However, the effects of constraint induced by specimen geometries, size ranges and side-groove in ASTM-E1457 on CCI time have not been systemically investigated in wide-range C* region, and the quantitative correlation of CCI time with constraint has not been established. In this work, the CCI time of the six specimens with different thicknesses and side-groove depths in ASTM-E1457 5
was predicted by numerical simulations based on ductility exhaustion damage model with stress dependent creep ductility and strain rate model. The main advances made in this paper are in the understanding of constraint effects on the CCI time in test specimens in ASTM-E1457 standard and the establishment of the quantitative correlation of CCI time with creep constraint. 2.
Finite element simulations
2.1 Material The material used in present work is a Cr-Mo-V steel (Chinese 25Cr2NiMo1V steel), and it is the same as that used in the previous work of authors [21, 29, 30]. The 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 creep behavior of the steel obeys two-regime Norton (2RN) creep model which has different Norton parameters of (A1,n1) in low stress regime and (A2,n2) in high stress regime, as shown in Eq.(1) and Table 1 [30]. This 2RN behavior of the Cr-Mo-V steel is similar to that of the 316H steel in the literature [18, 23, 31]. n A1σ 1 ε n2 A2σ
σ 250MPa σ 250MPa
2RN model
(1)
2.2 Specimen geometry and finite element model Six types of specimen geometries recommended in the ASTM-E1457 standard [2] for CCI and CCG testing (C(T), SEN(B), CS(T), SEN(T), DEN(T) and M(T)) have been modeled by ABAQUS code [32] in the previous work of authors [21]. The geometries of these specimens are shown in Fig.1. Specimen dimensions in the FE models conforming to the ASTM-E1457 are summarized in Table 2. The W is the specimen width of C(T), SEN(B) and CS(T) specimens and half width of SEN(T), DEN(T) and M(T) specimens. The L is the half length of SEN(B), SEN(T), DEN(T) and M(T) specimens, B is the specimen thickness (Bn is the net thickness), and a0 is the initial crack 6
length. The Ri is the inner radius, Ro is the outer radius and X=2W for CS(T) specimen. In Table 2, the thickness-to-width ratio (B/W) of C(T), SEN(B) and SEN(T) specimens is from 1/2 to 1/4, and that of CS(T), DEN(T) and M(T) specimens is from 1 to 1/2. 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 are denoted as thick specimens. The C(T), SEN(B) and SEN(T) specimens with B/W = 1/4 and the CS(T), DEN(T) and M(T) specimens with B/W = 1/2 are denoted as thin specimens. To analyze the effects of side groove depth on CCI time, the specimens with various Bn/B values (0.6, 0.8 and 1) in Table 2 were modeled. Due to geometrical symmetry, only one-fourth finite element models of all specimens were built. The FE models for simulating CCI in this work are the same as those for simulating CCG in the previous work of authors [21], which have been decribed in detail in the Section 2.2 of the previous paper [21]. The element size of 100μm which is similar to the average grain size of this steel [6] is regularly set in the crack growth zone for all specimens. 3.
Creep crack-tip constraint of test specimens in ASTM-E1457 In order to quantitatively analyze the constraint effects on CCI time in test specimens in
ASTM-E1457 standard, the values of creep crack-tip constraint parameters require to be calculated. Recently, the creep constraint parameter R* based on crack-tip stress field [33, 34] and Ac based on crack-tip equivalent creep strain [35-37] have been proposed. The load independent creep constraint parameter R* at steady-state creep has been investigated and can be calculated by Eq.(2) [34]:
C R* σ 22 AL * 1
1 n1
C σ 22,CT AL * 2
1 n1
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 with high constraint in PE condition, respectively; A and n are material creep constants, and L is a characteristic length 7
which is usually set to be 1m [34]. The unified creep constraint parameter Ac of in-plane and out-of-plane constraints has been defined as follows [35]: Ac
AC E E Q
at t/tred = 1
Ar e f
(3)
where ACEEQ and Aref are the area surrounded by the equivalent creep strain εc isoline ahead of crack tip in the evaluated specimen and the standard reference specimen with high constraint, respectively. The t is creep time and tred is stress redistribution time. In the recent work of authors [21], the parameters R* and Ac in all specimens in ASTM-E1457 with different side-grooved depths (different Bn/B) and different thickness-to-width ratios (B/W) in Table 2 have been calculated, and are shown in Figs.5-12 in the previous paper [21]. Fig.2 shows typical R* and Ac distributions along crack front of thin specimens with 20% side groove (Bn/B = 0.8) [21]. The z/Bn = 0 and z/Bn =0.5 are the central plane and surface of specimens, respectively. The results in the previous paper [21] and Fig.2 showed that 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 CCG rate increases. The parameter Ac has better capability for characterizing creep constraint than the parameter R* for the specimen geometries in ASTM-E1457. 4.
Creep crack initiation simulations The creep damage model based on the ductility exhaustion approach was used to simulate CCI
in the specimens in Table 2. This model is the same as that for simulating CCG [19, 21] and CCI [29] in various specimens of Cr-Mo-V steel in the previous work of authors, and has been described 8
in detail in the previous papers [19, 21, 29]. The model is expressed by Eq.(4): t
t
0
0
ω ω dt
εc dt ε *f
(4)
where the ω is the accumulated creep damage parameter, ω is the creep damage rate, εc is equivalent creep strain rate, and ε *f is the multiaxial creep ductility. The value of damage parameter ω is from 0 to 1. When the ω value at an element calculated from Eq. (4) reaches 1, the element is considered fully creep damage and its load-carrying capacity is reduced to zero by reducing the elastic modulus to a very small value of 1MPa [17-21, 31]. The multiaxial creep ductility ε *f can be estimated from the uniaxial creep ductility ε f by using the Cocks and Ashby model [38] as follows:
n 0.5 σ m 2 n 0.5 sin sinh 2 εf 3 n 0.5 n 0.5 σ e ε *f
(5)
where n is the creep exponent (for power law creep), and m / e is the ratio between the mean stress and equivalent stress, which is defined 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 of materials depends on stress levels (strain rates), which leads to the stress-regime dependence of the creep ductility of materials [30, 31]. For the Cr-Mo-V steel used in this work, the estimated stress dependent creep ductility was described by Eq. (6) [19, 21, 29]. σ 225MPa
0.5 ε f 1.14 10 14 σ 5.8 0.00537 1.6
225 σ 275MPa σ 275MPa
(6)
The Eqs.(4), (5) and (6) were implemented in the ABAQUS code [32] by using the user subroutine USDFLD, and the CCI were predicted. The FEM models for simulating CCI behavior in the specimens in ASTM-E1457 in Table 2 are the same as those for simulating CCG behavior in the previous paper [21]. All calculations were carried out by using elastic-plastic-creep model. The true 9
stress-strain curve of the steel [21] and the 2RN creep model parameters in Table 1 were employed. For each specimen in Table 2, a wide range of load levels has been applied to produce a wide range of C*. The range of initial stress intensity factor Kin (load levels) for CCI simulations of all specimens is listed in Table 2. It has been demonstrated in recent work of Ma et al. [37] that for such lower initial load levels, the plasticity effect is small. The creep crack growth length was estimated by numbers of damage elements. The CCI time t0.2 for a crack extension of 0.2mm and t0.5 for a crack extension of 0.5mm were calculated. The CCI time can be correlated with C* parameter, and the C* was calculated by using Eq. (7) in ASTME1457 standard.
C*
FΔ Hη 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 their solutions can be found in ASTM-E1457 [2]. The C* validity criteria in ASTM-E 1457 were applied to ensure that C* is a valid parameter to describe the CCI time. The creep damage model based on the ductility exhaustion approach described above has been widely used in many studies for simulating CCG behavior in test specimens with different geometries for different steels, the CCG rate data predicted by the FE simulations agree well with the experimental data [18-20, 30, 31, 39]. In recent work of authors [29] and Quintero and Mehmanparast [23], the CCI time predicted by the FE simulations using the ductility exhaustion approach with stress dependent creep ductility agree well with the experimental data, thus the CCI simulation method used in this work has been validated. 10
5.
Results and discussion
5.1 Creep crack initiation time The FE simulations were performed on six specimens with different thickness-to-width ratios (B/W) and different side-grooved depths (different Bn/B) over a wide range of C* produced by various initial stress intensity factors Kin in Table 2. For each specimen, the initiation time was calculated for an average crack extension of 0.2mm and 0.5mm, i.e. t0.2 and t0.5 were predicted, respectively. The CCI times t0.2 and t0.5 are compared in Figs. 3-8 for a wide range of C*. The Figs. 3-8 (a) are the CCI trend of the thin specimens, and the Figs. 3-8(b) are those of the thick specimens. It can be observed that the simulated CCI times (t0.2 and t0.5) are inversely proportional to C* on log-log scale and the ti-C* curve of each specimen has two slope turning points (as marked by turning point 1 and 2 in Figs.3-8). The fitted curves have three CCI trend: a long-term trend at low C* region (below the turning point 1), a short-term trend at high C* region (above the turning point 2) and a transition behavior between short-term (high C*) and long-term (low C*) CCI trend in intermediate C* region (between the turning points 1 and 2), which are consistent with the experimental and simulated results in the literature [23, 29]. The slope turning points on the ti –C* curves are similar to those on da/dt–C* curves of CCG rate which are mainly caused by the stress dependent creep ductility [19, 20, 30]. Because the CCI time is defined based on crack extension length which is relevant to CCG rate, the slope turning points on the ti –C* curves may also be attributed to the stress dependent creep ductility. It has been shown in the previous work of authors [19] that the stress triaxiality between high C* region and low C* region is similar. But in high C* region, the equivalent stresses ahead of cracks in the specimens are above the transition stress of creep ductility, and the multiaxial creep ductility is determined by upper shelf value of uniaxial creep ductility ( ε f = 1.6 in Eq.(6)). In low C* region, the equivalent stresses ahead of cracks are below the transition stress of creep ductility, and the multiaxial creep ductility is determined by 11
lower shelf value of uniaxial creep ductility ( ε f = 0.5 in Eq.(6)). The results in Figs.3-8 suggest that if the extrapolation CCI time data from the transition or high C* region are used in CCI assessments of the components at lower C* region, excessive conservative or non-conservative results may be produced. Figs. 3-8 show that all specimens have similar CCI trends for both t0.2 and t0.5 in high, low and transition C* regions, and the CCI trend lines are parallel to each other for all specimens. For the specimens without side-groove (Bn/B = 1), the constraint caused by specimen geometry has significant influence on CCI time in low and transition C* region, but it has small influence on it in high C* region. The constraint effect is reduced in the high C* region may be due to plasticity causing loss of crack-tip constraint, as analyzed in the literature [12, 19]. The constraint level of the six specimen geometries in ASTM-E1457 has been characterized in the previous paper [21], and it decreases in sequence from CS(T), C(T), SEN(B), SEN(T), DEN(T) to M(T) specimen. The higher constraint specimens (CS(T), C(T) and SEN(B)) have shorter CCI time, and the lowest constraint M(T) specimen has the longest CCI time, and the CCI time of the SEN(T) and DEN(T) specimens with middle constraint are lie in between. These results suggest that if the CCI time data of high constraint specimens are used in CCI life assessment of components with lower constraint, excessive conservative results will be produced, and if the CCI time data of low constraint specimens are used in CCI life assessment of thick components with higher constraint, non-safety prediction will be produced. Figs.5-8 show that the introduction of side groove results in the reduction of CCI time and the decrease in difference of CCI time among different specimens. As analyzed by constraint in Ref. [21], this is caused by the increase of constraint and the reduction of constraint difference among different specimens. With increasing side groove depth from 0% to 40%, the CCI time decreases owing to the increase of constraint as analyzed in Ref. [21]. At the same C* value, the CCI time of thick specimens is shorter than that of thin specimens. In the previous paper [21], by comparing CCG rates with the constraint parameter Ac, it has been shown that the CCG rates and their 12
difference among different specimens in ASTM-E1457 are consistent with the parameter Ac and their difference. The specimens with higher constraint (lower Ac) have higher CCG rate, and the larger difference in CCG rates among different specimens corresponds to the larger difference in Ac . By comparing the CCI time (Figs.3-8) in this work with the constraint parameter Ac in Ref. [21], it also can be found that the CCI time and their difference among different specimens in ASTM-E1457 are consistent with the parameter Ac and their difference. The specimens with higher constraint have shorter CCI time, and the larger difference in CCI time among different specimens corresponds to the larger difference in Ac. These results imply that the difference in CCI time among different specimens in ASTM-E1457 is caused by different crack-tip constraint levels in different specimens, and the parameter Ac can accurately characterize creep constraint due to that it can effectively and simultaneously incorporate both in-plane and out-of-plane constraints in the specimens [35-37]. For clearly showing the effects of side groove depth and specimen thickness on CCI time, Figs. 9 and 10 give the change of the CCI times t0.2 and t0.5 with side groove depth at a low C* value of 1E-6MPam/h for all specimens. It can be seen that with increasing side-groove depth (decreasing Bn/B), the CCI times t0.2 and t0.5 decrease, and the higher constraint specimens (C(T), SEN(B) and CS(T)) have shorter CCI times t0.2 and t0.5. The t0.2 and t0.5 of lower constraint specimens (DEN(T), SEN(T) and M(T)) is more sensitive to side groove depth. This is consistent with the result of constraint analysis in Ref. [21] which shows that the constraint Ac of lower constraint specimens is more sensitive to side groove depth. Figs. 9 and 10 also show that the thick specimens with side groove have shorter CCI time due to their higher constraint [21]. It can be found from Figs. 9 and 10 that the scattering of creep crack initiation time due to mechanical constraint is about 3-4 times. Therefore, it is necessary to consider the constraint effect in CCI life assessment of high temperature components. 5.2 Correlation of CCI time with creep constraint parameters R* and Ac In Section 6.2.2 of ASTM E1457 standard, it suggests that the effects of crack tip constraint 13
arising from variations in specimen size, geometry and material ductility can influence t0.2 and da/dt. For example, crack growth rates at the same value of C*(t), Ct in creep-ductile materials generally increases with increasing thickness. It is therefore necessary to keep the component dimensions in mind when selecting specimen thickness, geometry and size for laboratory testing. These suggestions in ASTM E1457 are consistent with the calculated results in this paper. The results of CCI time described in Section 5.1 have been shown that the recommended specimen geometries, size range and side groove in ASTM-E1457 can significantly affect CCI time in low C* region which is relative to the service load levels of elevated temperature components. In the case without side groove, the higher constraint specimens (CS(T), C(T) and SEN(B)) have shorter CCI time, while the lower constraint specimens (SEN(T), DEN(T) and M(T)) have longer CCI time. The introduction of side groove can significantly decrease CCI time of lower constraint specimens (SEN(T), DEN(T) and M(T)). Therefore, it is an important issue to account for the constraint effects on CCI time when predicting component life using laboratory data. For accurately and quantitatively accounting for the constraint effects on CCI time, it is necessary to establish the correlation of geometry constraint with CCI time. By analogy to the method for establishing the correlation of CCG rate with creep constraint parameters R* and Ac which has been given in detail in the recent work of authors [21, 35], the quantitative correlation formulas of CCI times (t0.2 and t0.5) with creep constraint parameters R* and Ac may be established. Because the average constraint R*avg and Ac-avg can characterize the overall constraint levels in the specimens, they have been calculated for all specimen geometries in ASTM-E1457 in the previous work of authors [21]. In this work, the load-independent R*avg and Ac-avg data in Ref.[21] were used to establish their correlation with CCI time. The CCI times (t0.2)C(T)-PE and (t0.5)C(T)-PE calculated from the high constraint standard C(T) specimen with W=50mm and a0/W =0.5 under PE condition are taken to normalize CCI times. The normalized CCI time ratios (t0.2/(t0.2)C(T)-PE and t0.5/(t0.5)C(T)-PE) are calculated at two typical lower C* values (C*=1E-6MPam/h and C*=1E-7MPam/h) for all specimens, and the relations between the 14
CCI time ratios and constraint parameters R*avg and Ac-avg for the low C* region (the long-term trend below the turning point 1 in Figs.3-8) which is relative to the load levels of components are shown in Figs. 11 and 12, respectively. It can be seen that there exists a linear relation on log-log scale between the CCI time ratios and R*avg and Ac-avg regardless of the C* values (load levels). This implies that the linear relation is independent on the load levels. Figs. 13(a) and (b) show the linear relations on log-log scale between CCI time ratio ti/(ti)C(T)-PE (including both t0.2/(t0.2)C(T)-PE and t0.5/(t0.5)C(T)-PE) and R*avg and Ac-avg for two typical lower C* values (C*=1E-6MPam/h and C*=1E-7MPam/h), respectively. It is interesting to see that the data of the normalized CCI time ratios of t0.2/(t0.2)C(T)-PE and t0.5/(t0.5)C(T)-PE can form a sole line on log-log scale. This implies that the definition of CCI time (t0.2 or t0.5) does not influence the correlation line between the normalized CCI time ratio and constraint, and the correlation line may be only dependent on material. The correlation formulas of ti/(ti)C(T)-PE and R*avg and Ac-avg can be fitted from Fig.13 for the low C* region of C* < 2.3E-5MPam/h in Figs.3-8, and are shown in Eqs.(8) and (9), respectively. The (ti)C(T)-PE (for both t0.2 or t0.5) and its correlation with C* were calculated from the standard reference C(T) specimen with W=50mm and a/W=0.5 under PE condition, and shown in Eq.(10) for the low C* region.
* ti / ti CT PE 0.94 1 Ravg
0.95
ti / ti CT PE 0.63 Acavg
1.36
* 0.025C *0.86 0.048C 0.90
ti CT PE
Δa 0.2mm Δa 0.5mm
for for
C* < 2.3E-5MPam/h
(8)
C* < 2.3E-5MPam/h
(9)
for
C* < 2.3E-5MPa.m/h
(10)
Figs.11-13 imply that the two creep constraint parameters R* and Ac both can be used to correlate the CCI time with constraint induced by specimen geometries, sizes and side grooves. But the data scatter of Ac-avg in Figs.12 and 13(b) seems to be smaller than that of R*avg in Figs.11 and 13(a). This may be due to that the Ac-avg can effectively characterize both in-plane and out-of-plane constraints [35-37]. So the correlation formula of ti/(ti)C(T)-PE with Ac-avg may be more accurate for 15
describing the constraint effect on CCI time. The CCI time in structures (such as in pressurized pipes and vessels) may be assessed by the two-parameter C*-R*avg or C*- Ac-avg approach based on Eqs.(8) and (9) to incorporate crack-tip constraint effect. For actual structures with a certain geometry and loading condition, the creep fracture mechanics parameter C* can be obtained by reference stress method or FE calculations. And the corresponding constraint parameter Ac-avg or R*avg can be calculated at the loading level C* using FE analysis. Therefore, the constraint dependent CCI time in structures can be predicted by using Eq. (8) or (9) based on the two-parameter C*-R*avg or C*- Ac-avg. 6.
Conclusion The CCI time of six types of specimen geometries with different thicknesses and side-groove
depths in ASTM-E1457 standard has been predicted by using ductility exhaustion model with stress dependent creep ductility in FE analyses. The effects of the specimen geometry, thickness and side-groove on CCI time (t0.2 and t0.5) have been analyzed, and the quantitative correlation of CCI time with creep constraint has been established. The primary results are summarized as follows: (1) Short-term (high C* region) and long-term (low C* region) CCI trends (for both t0.2 and t0.5) with a transition region (intermediate C* region) in between have been predicted for specimen geometries with different thicknesses and side-groove depths in ASTM-E1457. In low and transition C* regions, the specimen geometries has significant influence on CCI time, but it has a small effect on it in high C* region. (2) The difference in CCI time among different specimens in ASTM-E 1457 comes from the difference in crack-tip constraint levels. The higher constraint specimens (CS(T), C(T) and SEN(B)) have shorter CCI time, and the lower constraint specimens (SEN(T), DEN(T) and M(T)) have longer CCI time. The scattering of creep crack initiation time due to mechanical constraint is about 3-4 times. 16
(3) With increasing side-groove depth (decreasing Bn/B), the CCI time decreases due to the increase of constraint. The CCI time of lower constraint specimens (DEN(T), SEN(T) and M(T)) is more sensitive to side groove depth. The thick specimens with side groove have shorter CCI time due to their higher constraint. (4) It is necessary to account for the constraint effects on CCI time when predicting component life using laboratory data measured by specimen geometries in ASTM-E1457 and when comparing CCI time data of the specimens with various geometries and sizes. (5) The quantitative correlation lines and formulas of the normalized CCI time with creep constraint parameters R* and Ac have been established, and they are independent on load level C* and definition of CCI time (t0.2 or t0.5). The correlation formulas may be applied in CCI life assessment of high-temperature components for accounting for constraint effects and improving accuracy. Acknowledgments This work was financially supported by the Projects of the National Natural Science Foundation of China (51375165, 51575184, 51325504).
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21
Figure and Table Captions Fig. 1 Schematic illustrations of specimens in ASTM E1457: (a) C(T), (b)SEN(B), (c) CS(T), (d) SEN(T), (e) DEN(T), (f) M(T) Fig. 2 Distributions of constraint parameter R* (a) and Ac (b) of six specimen geometries with 20%-side groove (Bn/B = 0.8) for thin specimens [21] Fig.3 Comparison of predicted CCI time of six specimen geometries without side groove (Bn/B = 1) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens Fig.4 Comparison of predicted CCI time of six specimen geometries without side groove (Bn/B = 1) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens Fig. 5 Comparison of predicted CCI time of six specimen geometries with 20%-side groove (Bn/B = 0.8) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens Fig. 6 Comparison of predicted CCI time of six specimen geometries with 20%-side groove (Bn/B = 0.8) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens Fig. 7 Comparison of predicted CCI time of six specimen geometries with 40%-side groove (Bn/B = 0.6) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens Fig. 8 Comparison of predicted CCI time of six specimen geometries with 40%-side groove (Bn/B = 0.6) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens Fig. 9 The change of CCI time with side groove depth at C*=1E-6MPam/h for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens Fig. 10 The change of CCI time with side groove depth at C*=1E-6MPam/h for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens Fig. 11 The relations between t0.2/(t0.2)C(T)-PE (a) and t0.5/(t0.5)C(T)-PE (b) and R*avg for all specimens Fig. 12 The relations between t0.2/(t0.2)C(T)-PE (a) and t0.5/(t0.5)C(T)-PE (b) and Ac-avg for all specimens 22
Fig.13 The relations between ti/(ti)C(T)-PE and R*avg (a) and Ac-avg (b) for all specimens at C*=1E-6MPam/h and C*=1E-7MPam/h value Table 1 The 2RN creep model parameters [30] Table 2 Detailed geometries and sizes of specimens
23
(a) C(T)
(b) SEN(B)
(d) SEN(T)
(c) CS(T)
(e) DEN(T)
(f) M(T)
Fig. 1 Schematic illustrations of specimens in ASTM E1457: (a) C(T), (b)SEN(B), (c) CS(T), (d) SEN(T), (e) DEN(T), (f) M(T) 24
2.00
0.5
a/W=0.5 Bn/B=0.8
0.0 1.75
-0.5 -1.5
Ac
R
*
-1.0 1.50
-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
1.25
a/W=0.5 Bn/B=0.8
0.2 0.3 z/Bn
1.00
0.4
0.5
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
0.3
0.4
0.5
z/Bn
(a)
(b)
Fig. 2 Distributions of constraint parameter R* (a) and Ac (b) of six specimen geometries with 20%-side groove (Bn/B = 0.8) for thin specimens [21]
26
6
6
10
10 a/W=0.5 Bn/B=1
5
10
4
3
Turning point 2
10
2
10
1
10
0
4
10
Turning point 1
t0.2,h
t0.2,h
10
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
Turning point 1
3
Turning point 2
10
2
10
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
a/W=0.5 Bn/B=1
5
10
1
10
0
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
*
0.1
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 3 Comparison of predicted CCI time of six specimen geometries without side groove (Bn/B = 1) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens
27
6
6
10
10 a/W=0.5 Bn/B=1
5
10
4
4
3
10
2
1
10
0
10
Turning point 1 Turning point 2 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
t0.5,h
t0.5,h
10
10
a/W=0.5 Bn/B=1
5
10
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
3
10
2
10
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
Turning point 1
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
*
0.1
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 4 Comparison of predicted CCI time of six specimen geometries without side groove (Bn/B = 1) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens
28
6
6
10
10 a/W=0.5 Bn/B=0.8
5
10
4
4
10
10
1
10
0
Turning point 2 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
2
10
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
Turning point 1
3
10
t0.2,h
10
t0.2,h
10
Turning point 1
3
2
a/W=0.5 Bn/B=0.8
5
10
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
*
0.1
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 5 Comparison of predicted CCI time of six specimen geometries with 20%-side groove (Bn/B = 0.8) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens
29
6
6
10
10
a/W=0.5 Bn/B=0.8
5
10
4
4
10
10
t0.5,h
1
10
0
Turning point 2 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
2
10
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
3
10
t0.5,h
3
2
Turning point 1
Turning point 1
10 10
a/W=0.5 Bn/B=0.8
5
10
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01
0.1
0.1
*
*
C ,MPam/h
C ,MPam/h
(a)
(b)
Fig. 6 Comparison of predicted CCI time of six specimen geometries with 20%-side groove (Bn/B = 0.8) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens
30
6
6
10
10
a/W=0.5 Bn/B=0.6
5
10
4
4
3
Turning point 2
10
2
1
10
0
10
Turning point 1
C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
t0.2,h
t0.2,h
10
10
a/W=0.5 Bn/B=0.6
5
10
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 * C ,MPam/h
10
2
10
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
Turning point 1
3
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 * C ,MPam/h
0.1
(a)
0.1
(b)
Fig. 7 Comparison of predicted CCI time of six specimen geometries with 40%-side groove (Bn/B = 0.6) for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens
31
6
6
10
10 a/W=0.5 Bn/B=0.6
5
10
4
4
3
10 10
1
10
0
10
Turning point 1
Turning point 1
t0.5,h
t0.5,h
10
2
a/W=0.5 Bn/B=0.6
5
10
Turning point 2 C(T) SEN(B) CS(T) SEN(T) DEN(T) M(T)
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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 * C ,MPam/h
3
10
2
10
1
10
0
Turning point 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
10 1E-8 1E-7 1E-6 1E-5 1E-4 1E-3 0.01 * C ,MPam/h
0.1
(a)
0.1
(b)
Fig. 8 Comparison of predicted CCI time of six specimen geometries with 40%-side groove (Bn/B = 0.6) for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens
32
30000
10000
20000
t0.2,h
t0.2,h
20000
30000
*
C =1E-6MPam/h a0/W=0.5
C(T) W=50mm, B/W=1/4 SEN(B) W=25mm, B/W=1/4 CS(T) W=25mm, B/W=1/2 SEN(T) W=25mm, B/W=1/4 DEN(T) W=12.5mm, B/W=1/2 M(T) W=12.5mm, B/W=1/2
0.6
0.7
0.8
0.9
1.0
C(T) W=50mm, B/W=1/2 SEN(B) W=25mm, B/W=1/2 CS(T) W=25mm, B/W=1 SEN(T) W=25mm, B/W=1/2 DEN(T) W=12.5mm, B/W=1 M(T) W=12.5mm, B/W=1
*
C =1E-6MPam/h a0/W=0.5
10000
0.6
0.7
0.8
Bn/B
Bn/B
(a)
(b)
0.9
1.0
Fig. 9 The change of CCI time with side groove depth at C*=1E-6MPam/h for a crack extension of 0.2mm, (a) thin specimens, (b) thick specimens
33
50000 40000 30000
C(T) W=50mm, B/W=1/4 SEN(B) W=25mm, B/W=1/4 CS(T) W=25mm, B/W=1/2 SEN(T) W=25mm, B/W=1/4 DEN(T) W=12.5mm, B/W=1/2 M(T) W=12.5mm, B/W=1/2
50000
*
C =1E-6MPam/h a0/W=0.5
40000 30000
*
C =1E-6MPam/h a0/W=0.5
t0.5,h
20000
t0.5,h
20000
C(T) W=50mm, B/W=1/2 SEN(B) W=25mm, B/W=1/2 CS(T) W=25mm, B/W=1 SEN(T) W=25mm, B/W=1/2 DEN(T) W=12.5mm, B/W=1 M(T) W=12.5mm, B/W=1
10000
10000
0.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
Bn/B
Bn/B
(a)
(b)
0.9
1.0
Fig. 10 The change of CCI time with side groove depth at C*=1E-6MPam/h for a crack extension of 0.5mm, (a) thin specimens, (b) thick specimens
34
100
10
10
1 *
C(T) C =1E-6MPam/h * SEN(B) C =1E-6MPam/h * CS(T) C =1E-6MPam/h * SEN(T) C =1E-6MPam/h * DEN(T) C =1E-6MPam/h * M(T) C =1E-6MPam/h
0.1
0.01 1.0
1.5
2.0
*
C(T) C =1E-7MPam/h * SEN(B) C =1E-7MPam/h * CS(T) C =1E-7MPam/h * SEN(T) C =1E-7MPam/h * DEN(T) C =1E-7MPam/h * M(T) C =1E-7MPam/h
2.5
t0.5/(t0.5)C(T)-PE
t0.2/(t0.2)C(T)-PE
100
*
0.1
0.01 1.0
3.0
C(T) C =1E-6MPam/h * SEN(B) C =1E-6MPam/h * CS(T) C =1E-6MPam/h * SEN(T) C =1E-6MPam/h * DEN(T) C =1E-6MPam/h * M(T) C =1E-6MPam/h
1.5
2.0
*
C(T) C =1E-7MPam/h * SEN(B) C =1E-7MPam/h * CS(T) C =1E-7MPam/h * SEN(T) C =1E-7MPam/h * DEN(T) C =1E-7MPam/h * M(T) C =1E-7MPam/h
2.5
3.0
*
*
1-R
1
1-R
avg
(a)
avg
(b)
Fig. 11 The relations between t0.2/(t0.2)C(T)-PE (a) and t0.5/(t0.5)C(T)-PE (b) and R*avg for all specimens
35
100
10
10
t0.5/(t0.5)C(T)-PE
t0.2/(t0.2)C(T)-PE
100
1 *
C(T) C =1E-6MPam/h * SEN(B) C =1E-6MPam/h * CS(T) C =1E-6MPam/h * SEN(T) C =1E-6MPam/h * DEN(T) C =1E-6MPam/h * M(T) C =1E-6MPam/h
0.1
0.01 1.0
1.2
1.4
1.6
1.8
2.0
*
C(T) C =1E-7MPam/h * SEN(B) C =1E-7MPam/h * CS(T) C =1E-7MPam/h * SEN(T) C =1E-7MPam/h * DEN(T) C =1E-7MPam/h * M(T) C =1E-7MPam/h
2.2
2.4
2.6
2.8
Ac-avg
1 *
C(T) C =1E-6MPam/h * SEN(B) C =1E-6MPam/h * CS(T) C =1E-6MPam/h * SEN(T) C =1E-6MPam/h * DEN(T) C =1E-6MPam/h * M(T) C =1E-6MPam/h
0.1
0.01 1.0
1.2
1.4
1.6
1.8
2.0
*
C(T) C =1E-7MPam/h * SEN(B) C =1E-7MPam/h * CS(T) C =1E-7MPam/h * SEN(T) C =1E-7MPam/h * DEN(T) C =1E-7MPam/h * M(T) C =1E-7MPam/h
2.2
2.4
Ac-avg
(a)
(b)
Fig. 12 The relations between t0.2/(t0.2)C(T)-PE (a) and t0.5/(t0.5)C(T)-PE (b) and Ac-avg for all specimens
36
2.6
2.8
100
10
10
1
0.1
0.01 1.0
1.5
C(T) t0.2/(t0.2)C(T)-PE
C(T) t0.5/(t0.5)C(T)-PE
SEN(B) t0.2/(t0.2)C(T)-PE
SEN(B) t0.5/(t0.5)C(T)-PE
CS(T) t0.2/(t0.2)C(T)-PE
CS(T) t0.5/(t0.5)C(T)-PE
SEN(T) t0.2/(t0.2)C(T)-PE
SEN(T) t0.5/(t0.5)C(T)-PE
DEN(T) t0.2/(t0.2)C(T)-PE
DEN(T) t0.5/(t0.5)C(T)-PE
M(T) t0.2/(t0.2)C(T)-PE
M(T) t0.5/(t0.5)C(T)-PE
2.0
2.5
ti/(ti)C(T)-PE
ti/(ti)C(T)-PE
100
1
0.1
0.01
3.0
1.2
1.4
C(T) t0.2/(t0.2)C(T)-PE
C(T) t0.5/(t0.5)C(T)-PE
SEN(B) t0.2/(t0.2)C(T)-PE
SEN(B) t0.5/(t0.5)C(T)-PE
CS(T) t0.2/(t0.2)C(T)-PE
CS(T) t0.5/(t0.5)C(T)-PE
SEN(T) t0.2/(t0.2)C(T)-PE
SEN(T) t0.5/(t0.5)C(T)-PE
DEN(T) t0.2/(t0.2)C(T)-PE
DEN(T) t0.5/(t0.5)C(T)-PE
M(T) t0.2/(t0.2)C(T)-PE
M(T) t0.5/(t0.5)C(T)-PE
1.6
*
1-R
1.8
2.0
2.2
Ac-avg
avg
(a)
(b)
Fig. 13 The relations between ti/(ti)C(T)-PE and R*avg (a) and Ac-avg (b) for all specimens at C*=1E-6MPam/h and C*=1E-7MPam/h value
37
2.4
Table 1 The 2RN creep model parameters [30]
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
38
Table 2 Detailed geometries and sizes of specimens
Specimen
W(mm)
H(L)/W
a0/W
B/W
Bn/B
Load levels Kin(Mpam1/2)
C(T)
50
1.2
0.5
1/2-1/4
0.6, 0.8, 1
13.66 ~ 42.85
SEN(B)
25
2
0.5
1/2-1/4
0.6, 0.8, 1
2.69 ~ 12.93
CS(T)
25
-
0.5
1-1/2
0.6, 0.8, 1
5.39 ~ 18.08
SEN(T)
25
2
0.5
1/2-1/4
0.6, 0.8, 1
6.39 ~ 27.16
DEN(T)
12.5
4
0.5
1-1/2
0.6, 0.8, 1
2.10 ~ 8.41
M(T)
12.5
4
0.5
1-1/2
0.6, 0.8, 1
5.91 ~ 16.04
39
Highlights The creep crack initiation (CCI) time in specimens in ASTM-E1457 were predicted. The specimens
ASTM-E1457 produce different CCI time data.
The CCI time decreases with increasing constraint levels in ASTM-E1457 specimens. The quantitative correlation formulas of CCI time with constraint were established. It needs to consider constraint effects on CCI time when predicting component life.
40