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A numerical creep analysis on the interaction of twin semi-elliptical cracks
ARTICLE IN PRESS
International Journal of Pressure Vessels and Piping 85 (2008) 459–467 www.elsevier.com/locate/ijpvp
A numerical creep analysis on the interaction of twin semi-elliptical cracks Jun Si, Fu-Zhen Xuan, Shan-Tong Tu School of Mechanical Engineering, East China University of Science and Technology, 130, Meilong Street, PO Box 402, Shanghai 200237, PR China Received 23 May 2007; received in revised form 19 January 2008; accepted 19 February 2008
Abstract The interaction and coalescence of multiple flaws will significantly influence the service life of components. In this paper, the interaction of two identical semi-elliptical cracks in a finite thickness plate subjected to the remote tension is investigated. The results indicated that interaction of multiple cracks is different between the time-dependent fracture characterized by C*-integral and linear elastic fracture noted by SIF. The magnifying factors of creep fracture are obviously larger than that of the linear elastic fracture cases. Therefore, the current interaction and coalescence rule developed from linear elastic fracture analysis may lead to a non-conservative result when it is used in the assessment of creep crack. At the end, an empirical equation is developed based on the numerical results. r 2008 Elsevier Ltd. All rights reserved. Keywords: Creep crack growth; Surface flaw; C* integral; Finite element analysis
1. Introduction Many high-temperature structures and components contain more than one crack-like defects and flaws. The interaction and coalescence of these cracks or flaws are therefore ineluctable during the evolution of crack and component failure. A good understanding of the behavior of crack interaction and coalescence will provide engineers with the quantitative tools to assess the structural integrity of high-temperature components [1]. Assessment of the structural integrity of components containing defects can be made using different fracture mechanics parameters corresponding to different fracture mechanisms. The stress intensity factor (SIF) is applicable for the linear elastic range. In the elastic–plastic case, the fracture parameter J-integral should be employed to accommodate the influence of plastic deformation. In the presence of creep at high temperature, however, ratedependent parameter of C*-integral [2,3] is developed to Corresponding author. Tel.: +86 21 64253513; fax: +86 21 64253425.
E-mail address: [email protected] (F.-Z. Xuan). 0308-0161/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2008.02.001
correlate the data of creep crack growth (CCG) and creep crack initiation (CCI). In addition, results indicate that C*integral is the most appropriate one among the existing fracture parameters for the interpretation of CCG and CCI. The interaction behavior of structural components with multiple semi-elliptic surface cracks under linear-elastic and elastic–plastic fracture mechanics regime has been widely studied [4–9], and some useful recommendations [10–12] have been proposed and applied for the integrity assessment of structures with multiple cracks. However, for the structure operated under creep regime, most of studies are focused on the creep analysis of single semi-elliptical surface crack [13–15], and very little attention has been paid on the interaction of multiple semi-elliptical surface cracks. In this study, a finite thickness plate containing two semi-elliptical surface cracks under a remote tension is considered, as shown in Fig. 1. The two coplanar surface cracks are assumed to be the same shape and size. The creep fracture parameter C*-integral involving wide ranges of crack configurations and material constants, i.e., 0.2pa/tp0.8, 0.2pa/cp0.8, 0.2pc/d p0.8, 0p2f/pp2,
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Nomenclature a A B, n c C* C*Normal C*Single C*Double d E H J K0 K MCreep ML PL PDouble L PSingle L Q
crack depth area enclosed by the contour material constants in the elastic-secondary creep constitutive relation half-width of semi-elliptic surface crack time-dependent fracture parameter a normalization of C* parameter time-dependent fracture parameter without the interaction influence time-dependent fracture parameter including the interaction influence distance between two cracks Young’s modulus length of plate elastic–plastic fracture parameter normalized stress intensity factor stress intensity factor interaction factor, seen in Eq. (4) dimensionless factor, seen in Eq. (10) plastic limit load plastic limit load referring to two cracks plastic limit load corresponding to one crack parameter of crack geometry
is calculated by using the finite element (FE) method. Here, a/t is the crack depth aspect ratio, a/c is the crack shape aspect ratio, c/d is the relative distance of two cracks, and 2f/p is the normalized location of the semi-elliptical surface crack front. Based on the calculated results, finally, an empirical expression of the interaction factor is proposed as a function of the crack configurations and material creep exponent. 2. Creep fracture parameter In the current study, mechanical properties of low alloy Cr–Mo steel [16] are used, which can be described by the elastic-secondary creep constitutive relation _ _ ¼ s=E þ Bsn
(1)
where _ denotes the uniaxial strain rate, s_ is the uniaxial stress rate, E is Young’s modulus, and B and n are the steady-state creep coefficient and exponent, respectively. This constitutive relationship has been widely used in the creep deformation analysis. The material constants employed in FE analysis are listed in Table 1. For three-dimensional (3D) crack situations, Kikuchi and Miyamoto [17] have defined the J-integral in terms of x component Z Z J ¼ ðW dy sij ui;x dsÞ ðsiz ui;x Þ;x dA (2) G
A
t u W W* G gSIF gCreep e,_ eref,_cref f n s,s_ sref sY sDouble ref sSingle ref
thickness of plate displacement width of plate strain energy density, seen in Eq. (3) integral path interaction factor under linear elastic fracture interaction factor, seen in Eq. (9) strain, strain rate reference strain, creep strain rate at the reference stress angular parameter defining the crack front position Poisson’s ratio stress, stress rate reference stress yield stress of the material reference stress referring to two cracks reference stress corresponding to one crack
Abbreviations CCG CCI FE SIF
creep crack growth creep crack initiation finite element stress intensity factor
here W* is the strain energy density defined by Z ij W ðij Þ ¼ sij dij
(3)
0
The contour, G, is contained in the xy plane normal to the z direction, A is the area enclosed by the contour G, and u is the displacement. According to the analogy between J and C* integral, expression of the C*-integral for 3D crack is thus obtained through replacing displacement and strain by displacement rate and strain rate in Eqs. (2) and (3). 3. FE analysis The FE code ABAQUS [18] was used in this analysis. In order to evaluate the interaction effect of multiple cracks, a number of FE computations were carried out for a plate with one semi-elliptical surface crack or two identical semielliptical surface cracks. The size of plate considered herein was H/W ¼ 2, W/c ¼ 10. To reduce the workload in construction of FE model, H, W, c was fixed. The depth of crack, a, and the thickness of the plate, t, varied with the ratio a/c and a/t. All the calculations in this study were subjected to a remote tension with the same value. The 3D 20-noded brick elements were adopted. Due to the symmetry, only a quarter of the plate was modeled with 10,000–35,000 elements in general. The total meshes varied in the different models, but the crack tip was modeled using
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2H
Y
2W
Z X
t Fig. 2. Three-dimensional finite element model of the semi-elliptical surface crack for the quarter of plate: (a) whole mesh; (b) crack tip mesh.
Fig. 1. (a) A finite thickness plate containing two semi-elliptical cracks under tension; (b) the size and parametric angle f for semi-elliptical surface cracks. Table 1 Material constants used in FE analysis E (MPa)
n
B (MPan/h)
n
200,000 158,000 144,000 140,100
0.3 0.3 0.3 0.3
5.0 1012 1.3375 1016 1.9 1019 1.462 1024
3 5 7 9
the same number of meshes for all cases considered herein. A typical mesh used in the current study is shown in Fig. 2. Extremely refined elements are doubtless generated in the crack tip zone, as shown in Fig. 2(b). Five different integral paths around the crack front were selected during the creep fracture analysis and the average value of calculated C*-integral was used at different points of f around the crack front. To quantify the interaction behavior of two cracks under creep regime, a ratio denoted with MCreep is proposed and defined by using the values of C*-integral between a plate with double cracks and with a single crack at the same location, subjected to the identical remote tension. M Creep ¼
C Double C Single
(4)
It is worth noting that the proposed factor MCreep should provide more accurate information on the interaction of multiple cracks than that of the C*-integral. The reason is that MCreep is extracted from the ratio of both C*Double and C*Single through the very similar FE model. Errors produced during the FE calculation of C*Double and C*Single are likely to be eliminated in terms of C*Double divided by C*Single and thereby diminished the total error. In order to provide the confidence on FE models employed in this study, the values of SIF were calculated by using the current models. The calculated SIFs were compared with the existing solutions of Raju and Newman [19] and the good result was obtained with the relative error o1%. Comparisons were also performed for the creep fracture parameter C* of the single crack. The fully plastic J-integral solutions are directly used for the C* calculation in consideration of the analogy of power-law relation between plasticity and creep. The FE results from this study were compared with the full plastic values derived from Yagawa [20], McClung [21], and Lei [22], as shown in Fig. 3. The normalizing parameter C*normal is defined by C normal ¼ s_c t c
(5)
where _ is the creep strain rate and t the thickness of plate. As expectedly, a good agreement was obtained, which provides us confidence on the FE modeling and interaction analysis of multiple cracks. As C*-integral is directly proportional to the creep coefficient B [23], the defined interaction factor MCreep is thus independent of the magnitude of B. Only the influence of the creep exponent n needs be considered in the following section. To obtain a comprehensive understanding on the influence of various parameters (a, c, d, t, f, n) of the interactive cracks, in this study, about 300 FE
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The reference stress is defined by [25] 1.5 Present FE NASA [21] Lei [22] Yagawa.et al[20]
a/c=0.2 a/t=0.2 n=5
sref ¼ sY
C*/C*normal
2 C ¼ Bsn1 ref K
0.5
0.2
0.4
0.6
0.8
1.0
2φ/π
2.5
C*/C*normal
2.0
Present FE NASA [21] Lei [22] Yagawa.et al[20]
(7)
where sY is the yield stress of the material, PL is the corresponding plastic limit load of the structure, and P is the applied load. For the power-law creep constitutive relation, Eq. (6) can be rearranged as
1.0
0.0 0.0
P PL
a/c=0.2 a/t=0.2 n=10
(8)
According to Eqs. (4), (7) and (8), therefore, the relationship between the proposed factors of MCreep and gSIF can be straightforwardly derived by !ðn1Þ=2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sDouble C Double ref gCreep ¼ M Creep ¼ ¼ C Single sSingle ref ! Single ðn1Þ=2 PL gSIF ¼ gSIF (9) PDouble L
1.0
where sDouble is the reference stress referring to two cracks, ref Single is the reference stress corresponding to one crack, sref is the plastic limit load referring to two cracks, and PDouble L is the plastic limit load corresponding to one crack. PSingle L The dimensionless factor, ML, is defined by
0.5
ML ¼
1.5
0.0 0.0
0.2
0.4
0.6
0.8
1.0
2φ/π Fig. 3. Comparisons of the C*/C*normal between the FE results and the full plastic solutions for pure tension: (a) a/c ¼ 0.2, a/t ¼ 0.2 and n ¼ 5; (b) a/c ¼ 0.2, a/t ¼ 0.2 and n ¼ 10.
models were analyzed with the nonlinear creep computation. The interaction factor defined by Eq. (4) at three extreme locations was mainly taken into account viz. the adjacent tip of two cracks (denoted by P1 as shown in Fig. 1), 2f/p ¼ 0, and the remote tip of two cracks (denoted by P2 as shown in Fig. 1), 2f/p ¼ 2, and the deepest point of crack (denoted by P3 as shown in Fig. 1), 2f/p ¼ 1. On the other hand, the interaction effect of multiple cracks under elastic–plastic regime was generally denoted by gSIF through the ratio of SIF. For the case under creep regime, a relationship had been established between the crack tip parameter C* and SIF according to the reference stress method [24] 2 K (6) C ¼ sref _cref sref where sref is the reference stress, _ cref is the creep strain rate at the reference stress, and K is the SIF at the crack-tip.
PSingle L PDouble L
(10)
It is worth noting that, the plastic limit loads of PSingle L and PDouble depend linearly on the yield stress sy of L material and thus the influence of yield stress is eliminated in Eq. (10). Therefore, ML is only a geometry-dependent function. As a consequence, the ratio of gCreep/gSIF is a function of geometry of structures and creep exponent of materials. On the other hand, the value of PDouble is L generally less than that of PSingle due to more cracks being L involved that leads to ML41. In consideration of n being in the range of 8–10 for the engineering alloys, we have gCreep ¼ ðM L Þðn1Þ=2 41 gSIF
(11)
To address the discrepancy of interaction behavior between creep cracks and elastic cracks quantitatively, the comprehensive FE analyses were performance and results were shown in Fig. 4. As expected, compared with that of elastic cases, more pronounced interaction is observed under the creep regime. For example, for the case of n ¼ 9, and a/c ¼ 0.6, a/t ¼ 0.6 and c/d ¼ 0.8, the enlarging factor gCreep/gSIF at the adjacent tip of interacted cracks is about 1.3, and is about 1.2 at the deepest point of crack front, respectively. That is, about 30% and 20% increments were observed for the creep crack interaction at 2f/p ¼ 0 and 1 compared with the elastic cases. On the other hand, for the given values of a/c, a/t, and c/d, the discrepancy between the interaction factors of gCreep and
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1.3 c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
Interaction factor MCreep
γCreep/γSIF
1.2
1.1
1.0
0.0
1.0 2φ/π
0.5
1.5
a/c=0.6 c/d=0.5 n=5
1.4
a/c=0.6 a/t=0.6 n=5
1.3
1.2
1.1
1.0 0.0
2.0
a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
0.5
1.0 2φ/π
1.5
2.0
1.4
γCreep/γSIF
1.3
2.0
a/c=0.6 a/t=0.6 c/d=0.8
Interaction factor MCreep
2φ/π=0 2φ/π=1 2φ/π=2
1.2
1.1
a/c=0.6 a/t=0.6 n=5
1.8
c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
1.6
1.4
1.2
1.0
2
4
6
8
10
n
Fig. 4. Comparisons of the interaction effect under the creep conditions and the elastic conditions: (a) interaction effect for different distances and angles; (b) interaction effect for different creep exponent n.
1.0 0.0
0.5
1.0 2φ/π
1.5
2.0
Fig. 5. Variations of the interaction factor with 2f/p: (a) a/c ¼ 0.6, c/d ¼ 0.5 and n ¼ 5; (b) a/c ¼ 0.6, a/t ¼ 0.6 and n ¼ 5.
4.2. Influence of the relative crack depth a/t gSIF increases with the increasing creep exponent n as shown in Fig. 4(b). 4. Results and discussions 4.1. Influence of the angle 2f/p Fig. 5 shows the variation of interaction factors along the semi-elliptical crack front for various a/t and c/d with the given aspect ratio a/c and creep exponent n. It is observed that the interaction factor varies with different points along the crack front. The interaction factor MCreep decreases monotonously where the 2f/p varies from zero to unity. When 2f/p is in the range of 1p2f/pp2, however, MCreep increases with the decreasing thickness of plate, as shown in Fig. 5(a). The trend is the same for all the cases considered herein.
Fig. 6 depicts the influence of relative crack depth a/t on the interaction factor MCreep for a given crack location c/d and creep exponent n. It is worth noting that, for three points of 2f/p ¼ 0, 1, and 2 interested herein, MCreep increases with increasing the relative crack depth. As expected, the maximum of interaction factor MCreep is observed at the adjacent tip of interacted cracks. For the deepest point of crack front, the value of MCreep is less than that of other locations when the crack depth approaches to the thickness of the plate. The reason for this may be that the compressive stress field is developed because of the local bending stresses at the deepest point. 4.3. Influence of the crack aspect ratio a/c Fig. 7 shows the influence of the crack aspect ratio a/c on the interaction factor for a given the crack depth a/t and
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2.6
2.2 2.0
a/c=0.6, 2 a/c=0.6, 2 a/c=0.6, 2 a/c=0.8, 2 a/c=0.8, 2 a/c=0.8, 2
φ/π=0 φ/π=1 φ/π=2 φ/π=0 φ/π=1 φ/π=2
2.6
c/d=0.8 n=5
1.8 1.6 1.4 1.2 1.0 0.0
c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
2φ/π=0 a/t=0.6 n=5
2.4 Interaction factor MCreep
Interaction factor MCreep
2.4
2.2 2.0 1.8 1.6 1.4 1.2
0.2
0.4
0.6
0.8
1.0 0.0
1.0
0.2
0.4
a/t Fig. 6. Variations of the interaction factor with the crack depth a/t for c/d ¼ 0.8 and n ¼ 5.
1.5
The influence of n on interaction factors is shown in Fig. 9. It is clear that the interaction effect varies with creep exponent n for different cases analyzed herein. The slopes of these curves show a increasing trend when a/t and c/d are increased, as shown in Fig. 9(b) and (c). 5. Closed-form solution for creep interaction factor For the practical application, a closed solution is needed to estimate the interaction factor of multiple cracks under creep regime. By applying the least square method, the correlation of interaction factor with other influence parameters such as crack configurations and material properties is directly established in the range of
1.4
c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
2φ/π=1 a/t=0.6 n=5
1.3 1.2
1.0 0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
a/c 1.6 c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
2φ/π=2 a/t=0.6 n=5
1.5 Interaction factor MCreep
4.5. Influence of the creep exponent n
1.0
1.1
4.4. Influence of relative distance c/d The influence of c/d on the interaction factor for a typical value a/c ¼ 0.8, and n ¼ 5 is shown in Fig. 8. It is clear that, as the cracks approach to each other, i.e., c/d approaches to unity, the interaction factor is shown to increase. Nevertheless, the increasing trend of the adjacent tip P1 is more obvious than that of the other points. It is indicated that the cracks are likely to grow in the surface direction rather than grow in the depth direction. The trend is the same for all cases considered herein.
0.8
1.6
Interaction factor MCreep
creep exponent n. Unexpectedly, no significant changes were found for MCreep with the increasing a/c. However, it is observed that the interaction effect is different at three locations. The interaction effect enhances with the increase of a/c for the remote tip P2, whereas, for the adjacent tip P1, when the distance of cracks is very small, the interaction factor increases and closes the maximum firstly, and then decreases when a/c is increased.
0.6 a/c
1.4 1.3 1.2 1.1 1.0 0.0
0.2
0.4
0.6 a/c
Fig. 7. Variations of the interaction factor with the crack aspect ratio a/c for a/t ¼ 0.6 and n ¼ 5: (a) 2f/p ¼ 0; (b) 2f/p ¼ 1; (c) 2f/p ¼ 2.
0.2pa/tp0.8, 0.2pa/cp0.8, 0.2pc/dp0.8, and 3pnp9. The proposed solution for interaction factor of multiple cracks under creep regime is well described through the
ARTICLE IN PRESS J. Si et al. / International Journal of Pressure Vessels and Piping 85 (2008) 459–467
3.0 a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
2φ/π=0 a/c=0.8 n=5 2.0
Interaction factor MCreep
Interaction factor MCreep
2.5
1.5
1.0 0.0
0.2
0.4
0.6
0.8
2.0
1.5
1.0
1.0
4
2
6
8
10
6
8
10
6
8
10
n
1.5
3.0
Interaction factor MCreep
a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
2φ/π=1 a/c=0.8 n=5
1.4 Interaction factor MCreep
2φ/π=0 2φ/π=1 2φ/π=2
a/c=0.2 a/t=0.6 c/d=0.8
2.5
c/d
1.3
1.2
1.1
1.0 0.0
0.2
0.4
0.6
0.8
2φ/π=0 a/c=0.6 c/d=0.8
2.0
1.5
1.0
1.0
a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
2.5
c/d
4
2
n
1.6
3.0
1.4
a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
Interaction factor MCreep
2φ/π =2 a/c=0.8 n=5
1.5 Interaction factor MCreep
465
1.3 1.2 1.1 1.0 0.0
0.2
0.4
0.6
0.8
1.0
c/d Fig. 8. Variations of the interaction factor for the relative crack location c/d for a/c ¼ 0.8 and n ¼ 5: (a) 2f/p ¼ 0; (b) 2f/p ¼ 1; (c) 2f/p ¼ 2.
c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
2.5
2φ/π=0 a/c=0.6 a/t=0.8
2.0
1.5
1.0 2
4 n
Fig. 9. Variations of the interaction factor for the creep exponent n: (a) different location for a/c ¼ 0.2, a/t ¼ 0.6, and c/d ¼ 0.8; (b) different ratio of a/t for a/c ¼ 0.6, c/d ¼ 0.8, and 2f/p ¼ 0; (c) different ratio of c/d for a/c ¼ 0.6, c/d ¼ 0.8, and 2f/p ¼ 0.
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dimensionless form by aC 4 M Creep ¼ C 1 þ C 2 nC 3 f 1f 2 t
f 2 ¼ C8 þ C9
a c c
þ C7
(12)
a2
þ C 10
c c 2
(13) þ C 11
c 3
(14) d d d where C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, and C11 are constants. The corresponding values are presented in Table 2. To further validate the proposed solution of interaction factor, more creep FE computations were carried out again and comparisons between the predicted MCreep through Eq. (12) and FE data were shown in Fig. 10. As expected, excellent agreement was obtained between the predicted MCreep and FE results. The maximum difference between the estimated MCreep and FE results is less than 5% at the remote tip P2 and the deepest point P3, and less than 8% at the adjacent tip P1. Overall good agreement between the closed-form solution and the FE results provides us confidence in practical application. The proposed equation offers a significant advantage to evaluate the C* values of interacting cracks. According to the recommendation in some rules/procedures [10–12], two cracks should be characterized as a big semi-elliptical crack, when the relevant conditions are satisfied. However, such a simplified treatment is little accurate, especially these required conditions are not always satisfied in practice. In addition, this simple combination will introduce an unrealistic discontinuity in the process of crack growth. Accordingly, the proposed Eq. (12) can remove the insufficiency of the simple combination in these rules/ procedures, and thus improve the precision of the defect assessments.
Interaction factor MCreep
f 1 ¼ C5 þ C6
1.8
1.6
2φ/π=0 2φ/π=1 2φ/π=2 Approximation
a/c=0.4 a/t=0.4 n=5
1.4
1.2
1.0 0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
c/d
2.0 a/c=0.6 a/t=0.4 n=7
1.8 Interaction factor MCreep
466
2φ/π=0 2φ/π=1 2φ/π=2 Approximation
1.6
1.4
1.2
1.0 0.0
0.2
0.4
0.6 c/d
6. Conclusions
Fig. 10. Comparisons between the predicted MCreep and the FE solutions: (a) different location for a/c ¼ 0.4, a/t ¼ 0.4 and n ¼ 5; (b) different location for a/c ¼ 0.6, a/t ¼ 0.4 and n ¼ 7.
A closed-form solution for the prediction of interaction of two identical coplanar semi-elliptical cracks subjected to a remote tension at creep regime has been presented based
on the comprehensive FE creep analysis. The following conclusions are listed:
Table 2 Constants of the proposed equation at three locations Constants
2f/p ¼ 0
2f/p ¼ 1
2f/p ¼ 2
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11
0.9851 0.9103 0.6087 1.2213 0.5497 0.6969 0.7189 0.1508 0.6223 1.9227 0.3566
1.0036 0.8100 0.8561 1.1283 0.3104 0.6527 0.5498 0.0846 0.3478 1.1433 0.3452
1.0159 0.8086 0.8848 1.5923 0.5211 0.4169 0.2164 0.1013 0.3968 1.2104 0.5232
1. Compared with the elastic fracture analysis denoted by SIF, the interaction effect of multiple cracks at creep regime is obviously greater and implied that the current criteria for crack interaction should be modified in the defect assessment of high-temperature components. 2. For the bi-crack models considered herein, the interaction effect represented by C*-integral is influenced not only by the crack configurations but also by the material properties, especially the creep exponent n. 3. An empirical expression for the prediction of creep crack interaction factor is developed based on the comprehensive FE creep analyses. Excellent agreement between the closed-form solution and the FE results provides us confidence in practical application.
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Acknowledgments The authors are grateful for the supports provided by The National High Technology Research and Development Program of China (2006AA04Z413). FZ would also wish to thank the supports provided by Fok Ying Tung Education Foundation (101054). References [1] Liu X, Xuan FZ, Si J, Tu ST. Expert system for remnant life prediction of defected components under fatigue and creep-fatigue loadings. Expert Syst Appl 2008;34:222–30. [2] Riedel H, Rice JR. Tensile cracks in creeping solids, fracture mechanics. In: Proceedings of the 12th national symposium on fracture mechanics. Philadelphia, PA, USA, St. Louis, MO, USA: ASTM; 1980. p. 112–30. [3] Xuan FZ, Tu ST, Wang ZD. A modification of ASTM E 1457 C* estimation equation for compact tension specimen with a mismatched cross-weld. Eng Fract Mech 2005;72:2602–14. [4] Leek TH, Howard IC. An examination of methods of assessing interacting surface cracks by comparison with experimental data. Int J Press Vess Piping 1996;68:181–201. [5] Sethuraman R, Siva Sankara Reddy G, Thanga Ilango I. Finite element based evaluation of stress intensity factors for interactive semi-elliptic surface cracks. Int J Press Vess Piping 2003;80:843–59. [6] Carpinteri A, Brighenti R, Vantadori S. A numerical analysis on the interaction of twin coplanar flaws. Eng Fract Mech 2004;71:485–99. [7] Moussa WA, Bell R, Tan CL. Interaction of two parallel noncoplanar identical surface cracks under tension and bending. Int J Press Vess Piping 1999;76:135–45. [8] Tu ST. A study of the influence of irregularly oriented crack-like defects on the engineering structural integrity. PhD thesis, Department of Mechanical Engineering, Nanjing Institute of Chemical Technology, 1988 [in Chinese]. [9] Lo SH, Dong CY, Cheung YK. Integral equation approach for 3D multiple-crack problems. Eng Fract Mech 2005;72:1830–40. [10] API 579-1/ASME FFS-1. Fitness-For-Service, Section 9. American Society of Mechanical Engineers, 2007.
467
[11] BS7910. Guidance on methods of assessing the acceptability of flaws in metallic structures. London, UK: British Standard Institutions; 2005. [12] R6. Assessment of the integrity of structures containing defects, revision 4. Gloucester: British Energy Generation Ltd.; 2006. [13] Smith SD, Webster JJ, Hyde TH. Finite element simulation of elasticcreep growth of a semicircular surface crack. Eng Fract Mech 1988;31:947–57. [14] Biglari F, Nikbin KM, Goodall IW, Webster GA. Determination of fracture mechanics parameters J and C* by finite element and reference stress methods for a semi-elliptical flaw in a plate. Int J Press Vess Piping 2003;80:565–71. [15] Yoon KB, Park TG, Saxena A. Creep crack growth analysis of elliptic surface cracks in pressure vessels. Int J Press Vess Piping 2003;80:465–79. [16] Saxena A, Han J, Banerji K. Creep crack growth behavior in power plant boiler and steam pipe steels. J Press Vess Technol Trans ASME 1988;110:137–46. [17] Kikuchi M, Machida K, Miyamoto H. On the three dimensional J integral evaluation in the inelastic and transient stress fields. Modelling problems in crack tip mechanics. Dordrecht, Netherlands, Waterloo, Ontario, USA: Martinus Nijhoff; 1984. pp. 259–69. [18] ABAQUS User’s Manual. Version 6.7, Hibbit, Karlsson and Sorensen, 2006. [19] Raju IS, Newman Jr JC. Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Eng Fract Mech 1979;11:817–29. [20] Erratum 1997;74:77–80. [21] McClung RC, Chell GG, Lee Y-D, Russell DA, Orient GE. Development of a practical methodology for elastic–plastic and fully plastic fatigue crack growth. NASA Report NASA/CR-1999-209428, 1999. [22] Lei Y. J-integral and limit load analysis of semi-elliptical surface cracks in plates under tension. Int J Press Vess Piping 2004;81: 21–30. [23] Xuan FZ, Tu ST, Wang ZD. C* estimation for cracks in mismatched welds and finite element validation. Int J Fract 2004;126:267–80. [24] Ainsworth RA. The assessment of defects in structures of strain hardening materials. Eng Fract Mech 1984;19:633–42. [25] Sim RG. Evaluation of reference parameters for structures subject to creep. J Mech Eng Sci 1971;13:47–50.
International Journal of Pressure Vessels and Piping 85 (2008) 459–467 www.elsevier.com/locate/ijpvp
A numerical creep analysis on the interaction of twin semi-elliptical cracks Jun Si, Fu-Zhen Xuan, Shan-Tong Tu School of Mechanical Engineering, East China University of Science and Technology, 130, Meilong Street, PO Box 402, Shanghai 200237, PR China Received 23 May 2007; received in revised form 19 January 2008; accepted 19 February 2008
Abstract The interaction and coalescence of multiple flaws will significantly influence the service life of components. In this paper, the interaction of two identical semi-elliptical cracks in a finite thickness plate subjected to the remote tension is investigated. The results indicated that interaction of multiple cracks is different between the time-dependent fracture characterized by C*-integral and linear elastic fracture noted by SIF. The magnifying factors of creep fracture are obviously larger than that of the linear elastic fracture cases. Therefore, the current interaction and coalescence rule developed from linear elastic fracture analysis may lead to a non-conservative result when it is used in the assessment of creep crack. At the end, an empirical equation is developed based on the numerical results. r 2008 Elsevier Ltd. All rights reserved. Keywords: Creep crack growth; Surface flaw; C* integral; Finite element analysis
1. Introduction Many high-temperature structures and components contain more than one crack-like defects and flaws. The interaction and coalescence of these cracks or flaws are therefore ineluctable during the evolution of crack and component failure. A good understanding of the behavior of crack interaction and coalescence will provide engineers with the quantitative tools to assess the structural integrity of high-temperature components [1]. Assessment of the structural integrity of components containing defects can be made using different fracture mechanics parameters corresponding to different fracture mechanisms. The stress intensity factor (SIF) is applicable for the linear elastic range. In the elastic–plastic case, the fracture parameter J-integral should be employed to accommodate the influence of plastic deformation. In the presence of creep at high temperature, however, ratedependent parameter of C*-integral [2,3] is developed to Corresponding author. Tel.: +86 21 64253513; fax: +86 21 64253425.
E-mail address: [email protected] (F.-Z. Xuan). 0308-0161/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2008.02.001
correlate the data of creep crack growth (CCG) and creep crack initiation (CCI). In addition, results indicate that C*integral is the most appropriate one among the existing fracture parameters for the interpretation of CCG and CCI. The interaction behavior of structural components with multiple semi-elliptic surface cracks under linear-elastic and elastic–plastic fracture mechanics regime has been widely studied [4–9], and some useful recommendations [10–12] have been proposed and applied for the integrity assessment of structures with multiple cracks. However, for the structure operated under creep regime, most of studies are focused on the creep analysis of single semi-elliptical surface crack [13–15], and very little attention has been paid on the interaction of multiple semi-elliptical surface cracks. In this study, a finite thickness plate containing two semi-elliptical surface cracks under a remote tension is considered, as shown in Fig. 1. The two coplanar surface cracks are assumed to be the same shape and size. The creep fracture parameter C*-integral involving wide ranges of crack configurations and material constants, i.e., 0.2pa/tp0.8, 0.2pa/cp0.8, 0.2pc/d p0.8, 0p2f/pp2,
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Nomenclature a A B, n c C* C*Normal C*Single C*Double d E H J K0 K MCreep ML PL PDouble L PSingle L Q
crack depth area enclosed by the contour material constants in the elastic-secondary creep constitutive relation half-width of semi-elliptic surface crack time-dependent fracture parameter a normalization of C* parameter time-dependent fracture parameter without the interaction influence time-dependent fracture parameter including the interaction influence distance between two cracks Young’s modulus length of plate elastic–plastic fracture parameter normalized stress intensity factor stress intensity factor interaction factor, seen in Eq. (4) dimensionless factor, seen in Eq. (10) plastic limit load plastic limit load referring to two cracks plastic limit load corresponding to one crack parameter of crack geometry
is calculated by using the finite element (FE) method. Here, a/t is the crack depth aspect ratio, a/c is the crack shape aspect ratio, c/d is the relative distance of two cracks, and 2f/p is the normalized location of the semi-elliptical surface crack front. Based on the calculated results, finally, an empirical expression of the interaction factor is proposed as a function of the crack configurations and material creep exponent. 2. Creep fracture parameter In the current study, mechanical properties of low alloy Cr–Mo steel [16] are used, which can be described by the elastic-secondary creep constitutive relation _ _ ¼ s=E þ Bsn
(1)
where _ denotes the uniaxial strain rate, s_ is the uniaxial stress rate, E is Young’s modulus, and B and n are the steady-state creep coefficient and exponent, respectively. This constitutive relationship has been widely used in the creep deformation analysis. The material constants employed in FE analysis are listed in Table 1. For three-dimensional (3D) crack situations, Kikuchi and Miyamoto [17] have defined the J-integral in terms of x component Z Z J ¼ ðW dy sij ui;x dsÞ ðsiz ui;x Þ;x dA (2) G
A
t u W W* G gSIF gCreep e,_ eref,_cref f n s,s_ sref sY sDouble ref sSingle ref
thickness of plate displacement width of plate strain energy density, seen in Eq. (3) integral path interaction factor under linear elastic fracture interaction factor, seen in Eq. (9) strain, strain rate reference strain, creep strain rate at the reference stress angular parameter defining the crack front position Poisson’s ratio stress, stress rate reference stress yield stress of the material reference stress referring to two cracks reference stress corresponding to one crack
Abbreviations CCG CCI FE SIF
creep crack growth creep crack initiation finite element stress intensity factor
here W* is the strain energy density defined by Z ij W ðij Þ ¼ sij dij
(3)
0
The contour, G, is contained in the xy plane normal to the z direction, A is the area enclosed by the contour G, and u is the displacement. According to the analogy between J and C* integral, expression of the C*-integral for 3D crack is thus obtained through replacing displacement and strain by displacement rate and strain rate in Eqs. (2) and (3). 3. FE analysis The FE code ABAQUS [18] was used in this analysis. In order to evaluate the interaction effect of multiple cracks, a number of FE computations were carried out for a plate with one semi-elliptical surface crack or two identical semielliptical surface cracks. The size of plate considered herein was H/W ¼ 2, W/c ¼ 10. To reduce the workload in construction of FE model, H, W, c was fixed. The depth of crack, a, and the thickness of the plate, t, varied with the ratio a/c and a/t. All the calculations in this study were subjected to a remote tension with the same value. The 3D 20-noded brick elements were adopted. Due to the symmetry, only a quarter of the plate was modeled with 10,000–35,000 elements in general. The total meshes varied in the different models, but the crack tip was modeled using
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2H
Y
2W
Z X
t Fig. 2. Three-dimensional finite element model of the semi-elliptical surface crack for the quarter of plate: (a) whole mesh; (b) crack tip mesh.
Fig. 1. (a) A finite thickness plate containing two semi-elliptical cracks under tension; (b) the size and parametric angle f for semi-elliptical surface cracks. Table 1 Material constants used in FE analysis E (MPa)
n
B (MPan/h)
n
200,000 158,000 144,000 140,100
0.3 0.3 0.3 0.3
5.0 1012 1.3375 1016 1.9 1019 1.462 1024
3 5 7 9
the same number of meshes for all cases considered herein. A typical mesh used in the current study is shown in Fig. 2. Extremely refined elements are doubtless generated in the crack tip zone, as shown in Fig. 2(b). Five different integral paths around the crack front were selected during the creep fracture analysis and the average value of calculated C*-integral was used at different points of f around the crack front. To quantify the interaction behavior of two cracks under creep regime, a ratio denoted with MCreep is proposed and defined by using the values of C*-integral between a plate with double cracks and with a single crack at the same location, subjected to the identical remote tension. M Creep ¼
C Double C Single
(4)
It is worth noting that the proposed factor MCreep should provide more accurate information on the interaction of multiple cracks than that of the C*-integral. The reason is that MCreep is extracted from the ratio of both C*Double and C*Single through the very similar FE model. Errors produced during the FE calculation of C*Double and C*Single are likely to be eliminated in terms of C*Double divided by C*Single and thereby diminished the total error. In order to provide the confidence on FE models employed in this study, the values of SIF were calculated by using the current models. The calculated SIFs were compared with the existing solutions of Raju and Newman [19] and the good result was obtained with the relative error o1%. Comparisons were also performed for the creep fracture parameter C* of the single crack. The fully plastic J-integral solutions are directly used for the C* calculation in consideration of the analogy of power-law relation between plasticity and creep. The FE results from this study were compared with the full plastic values derived from Yagawa [20], McClung [21], and Lei [22], as shown in Fig. 3. The normalizing parameter C*normal is defined by C normal ¼ s_c t c
(5)
where _ is the creep strain rate and t the thickness of plate. As expectedly, a good agreement was obtained, which provides us confidence on the FE modeling and interaction analysis of multiple cracks. As C*-integral is directly proportional to the creep coefficient B [23], the defined interaction factor MCreep is thus independent of the magnitude of B. Only the influence of the creep exponent n needs be considered in the following section. To obtain a comprehensive understanding on the influence of various parameters (a, c, d, t, f, n) of the interactive cracks, in this study, about 300 FE
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The reference stress is defined by [25] 1.5 Present FE NASA [21] Lei [22] Yagawa.et al[20]
a/c=0.2 a/t=0.2 n=5
sref ¼ sY
C*/C*normal
2 C ¼ Bsn1 ref K
0.5
0.2
0.4
0.6
0.8
1.0
2φ/π
2.5
C*/C*normal
2.0
Present FE NASA [21] Lei [22] Yagawa.et al[20]
(7)
where sY is the yield stress of the material, PL is the corresponding plastic limit load of the structure, and P is the applied load. For the power-law creep constitutive relation, Eq. (6) can be rearranged as
1.0
0.0 0.0
P PL
a/c=0.2 a/t=0.2 n=10
(8)
According to Eqs. (4), (7) and (8), therefore, the relationship between the proposed factors of MCreep and gSIF can be straightforwardly derived by !ðn1Þ=2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sDouble C Double ref gCreep ¼ M Creep ¼ ¼ C Single sSingle ref ! Single ðn1Þ=2 PL gSIF ¼ gSIF (9) PDouble L
1.0
where sDouble is the reference stress referring to two cracks, ref Single is the reference stress corresponding to one crack, sref is the plastic limit load referring to two cracks, and PDouble L is the plastic limit load corresponding to one crack. PSingle L The dimensionless factor, ML, is defined by
0.5
ML ¼
1.5
0.0 0.0
0.2
0.4
0.6
0.8
1.0
2φ/π Fig. 3. Comparisons of the C*/C*normal between the FE results and the full plastic solutions for pure tension: (a) a/c ¼ 0.2, a/t ¼ 0.2 and n ¼ 5; (b) a/c ¼ 0.2, a/t ¼ 0.2 and n ¼ 10.
models were analyzed with the nonlinear creep computation. The interaction factor defined by Eq. (4) at three extreme locations was mainly taken into account viz. the adjacent tip of two cracks (denoted by P1 as shown in Fig. 1), 2f/p ¼ 0, and the remote tip of two cracks (denoted by P2 as shown in Fig. 1), 2f/p ¼ 2, and the deepest point of crack (denoted by P3 as shown in Fig. 1), 2f/p ¼ 1. On the other hand, the interaction effect of multiple cracks under elastic–plastic regime was generally denoted by gSIF through the ratio of SIF. For the case under creep regime, a relationship had been established between the crack tip parameter C* and SIF according to the reference stress method [24] 2 K (6) C ¼ sref _cref sref where sref is the reference stress, _ cref is the creep strain rate at the reference stress, and K is the SIF at the crack-tip.
PSingle L PDouble L
(10)
It is worth noting that, the plastic limit loads of PSingle L and PDouble depend linearly on the yield stress sy of L material and thus the influence of yield stress is eliminated in Eq. (10). Therefore, ML is only a geometry-dependent function. As a consequence, the ratio of gCreep/gSIF is a function of geometry of structures and creep exponent of materials. On the other hand, the value of PDouble is L generally less than that of PSingle due to more cracks being L involved that leads to ML41. In consideration of n being in the range of 8–10 for the engineering alloys, we have gCreep ¼ ðM L Þðn1Þ=2 41 gSIF
(11)
To address the discrepancy of interaction behavior between creep cracks and elastic cracks quantitatively, the comprehensive FE analyses were performance and results were shown in Fig. 4. As expected, compared with that of elastic cases, more pronounced interaction is observed under the creep regime. For example, for the case of n ¼ 9, and a/c ¼ 0.6, a/t ¼ 0.6 and c/d ¼ 0.8, the enlarging factor gCreep/gSIF at the adjacent tip of interacted cracks is about 1.3, and is about 1.2 at the deepest point of crack front, respectively. That is, about 30% and 20% increments were observed for the creep crack interaction at 2f/p ¼ 0 and 1 compared with the elastic cases. On the other hand, for the given values of a/c, a/t, and c/d, the discrepancy between the interaction factors of gCreep and
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1.3 c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
Interaction factor MCreep
γCreep/γSIF
1.2
1.1
1.0
0.0
1.0 2φ/π
0.5
1.5
a/c=0.6 c/d=0.5 n=5
1.4
a/c=0.6 a/t=0.6 n=5
1.3
1.2
1.1
1.0 0.0
2.0
a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
0.5
1.0 2φ/π
1.5
2.0
1.4
γCreep/γSIF
1.3
2.0
a/c=0.6 a/t=0.6 c/d=0.8
Interaction factor MCreep
2φ/π=0 2φ/π=1 2φ/π=2
1.2
1.1
a/c=0.6 a/t=0.6 n=5
1.8
c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
1.6
1.4
1.2
1.0
2
4
6
8
10
n
Fig. 4. Comparisons of the interaction effect under the creep conditions and the elastic conditions: (a) interaction effect for different distances and angles; (b) interaction effect for different creep exponent n.
1.0 0.0
0.5
1.0 2φ/π
1.5
2.0
Fig. 5. Variations of the interaction factor with 2f/p: (a) a/c ¼ 0.6, c/d ¼ 0.5 and n ¼ 5; (b) a/c ¼ 0.6, a/t ¼ 0.6 and n ¼ 5.
4.2. Influence of the relative crack depth a/t gSIF increases with the increasing creep exponent n as shown in Fig. 4(b). 4. Results and discussions 4.1. Influence of the angle 2f/p Fig. 5 shows the variation of interaction factors along the semi-elliptical crack front for various a/t and c/d with the given aspect ratio a/c and creep exponent n. It is observed that the interaction factor varies with different points along the crack front. The interaction factor MCreep decreases monotonously where the 2f/p varies from zero to unity. When 2f/p is in the range of 1p2f/pp2, however, MCreep increases with the decreasing thickness of plate, as shown in Fig. 5(a). The trend is the same for all the cases considered herein.
Fig. 6 depicts the influence of relative crack depth a/t on the interaction factor MCreep for a given crack location c/d and creep exponent n. It is worth noting that, for three points of 2f/p ¼ 0, 1, and 2 interested herein, MCreep increases with increasing the relative crack depth. As expected, the maximum of interaction factor MCreep is observed at the adjacent tip of interacted cracks. For the deepest point of crack front, the value of MCreep is less than that of other locations when the crack depth approaches to the thickness of the plate. The reason for this may be that the compressive stress field is developed because of the local bending stresses at the deepest point. 4.3. Influence of the crack aspect ratio a/c Fig. 7 shows the influence of the crack aspect ratio a/c on the interaction factor for a given the crack depth a/t and
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2.6
2.2 2.0
a/c=0.6, 2 a/c=0.6, 2 a/c=0.6, 2 a/c=0.8, 2 a/c=0.8, 2 a/c=0.8, 2
φ/π=0 φ/π=1 φ/π=2 φ/π=0 φ/π=1 φ/π=2
2.6
c/d=0.8 n=5
1.8 1.6 1.4 1.2 1.0 0.0
c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
2φ/π=0 a/t=0.6 n=5
2.4 Interaction factor MCreep
Interaction factor MCreep
2.4
2.2 2.0 1.8 1.6 1.4 1.2
0.2
0.4
0.6
0.8
1.0 0.0
1.0
0.2
0.4
a/t Fig. 6. Variations of the interaction factor with the crack depth a/t for c/d ¼ 0.8 and n ¼ 5.
1.5
The influence of n on interaction factors is shown in Fig. 9. It is clear that the interaction effect varies with creep exponent n for different cases analyzed herein. The slopes of these curves show a increasing trend when a/t and c/d are increased, as shown in Fig. 9(b) and (c). 5. Closed-form solution for creep interaction factor For the practical application, a closed solution is needed to estimate the interaction factor of multiple cracks under creep regime. By applying the least square method, the correlation of interaction factor with other influence parameters such as crack configurations and material properties is directly established in the range of
1.4
c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
2φ/π=1 a/t=0.6 n=5
1.3 1.2
1.0 0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
a/c 1.6 c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
2φ/π=2 a/t=0.6 n=5
1.5 Interaction factor MCreep
4.5. Influence of the creep exponent n
1.0
1.1
4.4. Influence of relative distance c/d The influence of c/d on the interaction factor for a typical value a/c ¼ 0.8, and n ¼ 5 is shown in Fig. 8. It is clear that, as the cracks approach to each other, i.e., c/d approaches to unity, the interaction factor is shown to increase. Nevertheless, the increasing trend of the adjacent tip P1 is more obvious than that of the other points. It is indicated that the cracks are likely to grow in the surface direction rather than grow in the depth direction. The trend is the same for all cases considered herein.
0.8
1.6
Interaction factor MCreep
creep exponent n. Unexpectedly, no significant changes were found for MCreep with the increasing a/c. However, it is observed that the interaction effect is different at three locations. The interaction effect enhances with the increase of a/c for the remote tip P2, whereas, for the adjacent tip P1, when the distance of cracks is very small, the interaction factor increases and closes the maximum firstly, and then decreases when a/c is increased.
0.6 a/c
1.4 1.3 1.2 1.1 1.0 0.0
0.2
0.4
0.6 a/c
Fig. 7. Variations of the interaction factor with the crack aspect ratio a/c for a/t ¼ 0.6 and n ¼ 5: (a) 2f/p ¼ 0; (b) 2f/p ¼ 1; (c) 2f/p ¼ 2.
0.2pa/tp0.8, 0.2pa/cp0.8, 0.2pc/dp0.8, and 3pnp9. The proposed solution for interaction factor of multiple cracks under creep regime is well described through the
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3.0 a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
2φ/π=0 a/c=0.8 n=5 2.0
Interaction factor MCreep
Interaction factor MCreep
2.5
1.5
1.0 0.0
0.2
0.4
0.6
0.8
2.0
1.5
1.0
1.0
4
2
6
8
10
6
8
10
6
8
10
n
1.5
3.0
Interaction factor MCreep
a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
2φ/π=1 a/c=0.8 n=5
1.4 Interaction factor MCreep
2φ/π=0 2φ/π=1 2φ/π=2
a/c=0.2 a/t=0.6 c/d=0.8
2.5
c/d
1.3
1.2
1.1
1.0 0.0
0.2
0.4
0.6
0.8
2φ/π=0 a/c=0.6 c/d=0.8
2.0
1.5
1.0
1.0
a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
2.5
c/d
4
2
n
1.6
3.0
1.4
a/t=0.2 a/t=0.4 a/t=0.6 a/t=0.8
Interaction factor MCreep
2φ/π =2 a/c=0.8 n=5
1.5 Interaction factor MCreep
465
1.3 1.2 1.1 1.0 0.0
0.2
0.4
0.6
0.8
1.0
c/d Fig. 8. Variations of the interaction factor for the relative crack location c/d for a/c ¼ 0.8 and n ¼ 5: (a) 2f/p ¼ 0; (b) 2f/p ¼ 1; (c) 2f/p ¼ 2.
c/d=0.2 c/d=0.4 c/d=0.5 c/d=0.625 c/d=0.8
2.5
2φ/π=0 a/c=0.6 a/t=0.8
2.0
1.5
1.0 2
4 n
Fig. 9. Variations of the interaction factor for the creep exponent n: (a) different location for a/c ¼ 0.2, a/t ¼ 0.6, and c/d ¼ 0.8; (b) different ratio of a/t for a/c ¼ 0.6, c/d ¼ 0.8, and 2f/p ¼ 0; (c) different ratio of c/d for a/c ¼ 0.6, c/d ¼ 0.8, and 2f/p ¼ 0.
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dimensionless form by aC 4 M Creep ¼ C 1 þ C 2 nC 3 f 1f 2 t
f 2 ¼ C8 þ C9
a c c
þ C7
(12)
a2
þ C 10
c c 2
(13) þ C 11
c 3
(14) d d d where C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, and C11 are constants. The corresponding values are presented in Table 2. To further validate the proposed solution of interaction factor, more creep FE computations were carried out again and comparisons between the predicted MCreep through Eq. (12) and FE data were shown in Fig. 10. As expected, excellent agreement was obtained between the predicted MCreep and FE results. The maximum difference between the estimated MCreep and FE results is less than 5% at the remote tip P2 and the deepest point P3, and less than 8% at the adjacent tip P1. Overall good agreement between the closed-form solution and the FE results provides us confidence in practical application. The proposed equation offers a significant advantage to evaluate the C* values of interacting cracks. According to the recommendation in some rules/procedures [10–12], two cracks should be characterized as a big semi-elliptical crack, when the relevant conditions are satisfied. However, such a simplified treatment is little accurate, especially these required conditions are not always satisfied in practice. In addition, this simple combination will introduce an unrealistic discontinuity in the process of crack growth. Accordingly, the proposed Eq. (12) can remove the insufficiency of the simple combination in these rules/ procedures, and thus improve the precision of the defect assessments.
Interaction factor MCreep
f 1 ¼ C5 þ C6
1.8
1.6
2φ/π=0 2φ/π=1 2φ/π=2 Approximation
a/c=0.4 a/t=0.4 n=5
1.4
1.2
1.0 0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
c/d
2.0 a/c=0.6 a/t=0.4 n=7
1.8 Interaction factor MCreep
466
2φ/π=0 2φ/π=1 2φ/π=2 Approximation
1.6
1.4
1.2
1.0 0.0
0.2
0.4
0.6 c/d
6. Conclusions
Fig. 10. Comparisons between the predicted MCreep and the FE solutions: (a) different location for a/c ¼ 0.4, a/t ¼ 0.4 and n ¼ 5; (b) different location for a/c ¼ 0.6, a/t ¼ 0.4 and n ¼ 7.
A closed-form solution for the prediction of interaction of two identical coplanar semi-elliptical cracks subjected to a remote tension at creep regime has been presented based
on the comprehensive FE creep analysis. The following conclusions are listed:
Table 2 Constants of the proposed equation at three locations Constants
2f/p ¼ 0
2f/p ¼ 1
2f/p ¼ 2
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11
0.9851 0.9103 0.6087 1.2213 0.5497 0.6969 0.7189 0.1508 0.6223 1.9227 0.3566
1.0036 0.8100 0.8561 1.1283 0.3104 0.6527 0.5498 0.0846 0.3478 1.1433 0.3452
1.0159 0.8086 0.8848 1.5923 0.5211 0.4169 0.2164 0.1013 0.3968 1.2104 0.5232
1. Compared with the elastic fracture analysis denoted by SIF, the interaction effect of multiple cracks at creep regime is obviously greater and implied that the current criteria for crack interaction should be modified in the defect assessment of high-temperature components. 2. For the bi-crack models considered herein, the interaction effect represented by C*-integral is influenced not only by the crack configurations but also by the material properties, especially the creep exponent n. 3. An empirical expression for the prediction of creep crack interaction factor is developed based on the comprehensive FE creep analyses. Excellent agreement between the closed-form solution and the FE results provides us confidence in practical application.
ARTICLE IN PRESS J. Si et al. / International Journal of Pressure Vessels and Piping 85 (2008) 459–467
Acknowledgments The authors are grateful for the supports provided by The National High Technology Research and Development Program of China (2006AA04Z413). FZ would also wish to thank the supports provided by Fok Ying Tung Education Foundation (101054). References [1] Liu X, Xuan FZ, Si J, Tu ST. Expert system for remnant life prediction of defected components under fatigue and creep-fatigue loadings. Expert Syst Appl 2008;34:222–30. [2] Riedel H, Rice JR. Tensile cracks in creeping solids, fracture mechanics. In: Proceedings of the 12th national symposium on fracture mechanics. Philadelphia, PA, USA, St. Louis, MO, USA: ASTM; 1980. p. 112–30. [3] Xuan FZ, Tu ST, Wang ZD. A modification of ASTM E 1457 C* estimation equation for compact tension specimen with a mismatched cross-weld. Eng Fract Mech 2005;72:2602–14. [4] Leek TH, Howard IC. An examination of methods of assessing interacting surface cracks by comparison with experimental data. Int J Press Vess Piping 1996;68:181–201. [5] Sethuraman R, Siva Sankara Reddy G, Thanga Ilango I. Finite element based evaluation of stress intensity factors for interactive semi-elliptic surface cracks. Int J Press Vess Piping 2003;80:843–59. [6] Carpinteri A, Brighenti R, Vantadori S. A numerical analysis on the interaction of twin coplanar flaws. Eng Fract Mech 2004;71:485–99. [7] Moussa WA, Bell R, Tan CL. Interaction of two parallel noncoplanar identical surface cracks under tension and bending. Int J Press Vess Piping 1999;76:135–45. [8] Tu ST. A study of the influence of irregularly oriented crack-like defects on the engineering structural integrity. PhD thesis, Department of Mechanical Engineering, Nanjing Institute of Chemical Technology, 1988 [in Chinese]. [9] Lo SH, Dong CY, Cheung YK. Integral equation approach for 3D multiple-crack problems. Eng Fract Mech 2005;72:1830–40. [10] API 579-1/ASME FFS-1. Fitness-For-Service, Section 9. American Society of Mechanical Engineers, 2007.
467
[11] BS7910. Guidance on methods of assessing the acceptability of flaws in metallic structures. London, UK: British Standard Institutions; 2005. [12] R6. Assessment of the integrity of structures containing defects, revision 4. Gloucester: British Energy Generation Ltd.; 2006. [13] Smith SD, Webster JJ, Hyde TH. Finite element simulation of elasticcreep growth of a semicircular surface crack. Eng Fract Mech 1988;31:947–57. [14] Biglari F, Nikbin KM, Goodall IW, Webster GA. Determination of fracture mechanics parameters J and C* by finite element and reference stress methods for a semi-elliptical flaw in a plate. Int J Press Vess Piping 2003;80:565–71. [15] Yoon KB, Park TG, Saxena A. Creep crack growth analysis of elliptic surface cracks in pressure vessels. Int J Press Vess Piping 2003;80:465–79. [16] Saxena A, Han J, Banerji K. Creep crack growth behavior in power plant boiler and steam pipe steels. J Press Vess Technol Trans ASME 1988;110:137–46. [17] Kikuchi M, Machida K, Miyamoto H. On the three dimensional J integral evaluation in the inelastic and transient stress fields. Modelling problems in crack tip mechanics. Dordrecht, Netherlands, Waterloo, Ontario, USA: Martinus Nijhoff; 1984. pp. 259–69. [18] ABAQUS User’s Manual. Version 6.7, Hibbit, Karlsson and Sorensen, 2006. [19] Raju IS, Newman Jr JC. Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Eng Fract Mech 1979;11:817–29. [20] Erratum 1997;74:77–80. [21] McClung RC, Chell GG, Lee Y-D, Russell DA, Orient GE. Development of a practical methodology for elastic–plastic and fully plastic fatigue crack growth. NASA Report NASA/CR-1999-209428, 1999. [22] Lei Y. J-integral and limit load analysis of semi-elliptical surface cracks in plates under tension. Int J Press Vess Piping 2004;81: 21–30. [23] Xuan FZ, Tu ST, Wang ZD. C* estimation for cracks in mismatched welds and finite element validation. Int J Fract 2004;126:267–80. [24] Ainsworth RA. The assessment of defects in structures of strain hardening materials. Eng Fract Mech 1984;19:633–42. [25] Sim RG. Evaluation of reference parameters for structures subject to creep. J Mech Eng Sci 1971;13:47–50.