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Universal characterization of three-dimensional creeping crack-front stress fields
Accepted Manuscript
Universal characterization of three-dimensional creeping crack-front stress fields Wanlin Guo , Zhiyuan Chen , Chongmin She PII: DOI: Reference:
S0020-7683(18)30252-X 10.1016/j.ijsolstr.2018.06.020 SAS 10030
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
1 December 2017 8 June 2018 18 June 2018
Please cite this article as: Wanlin Guo , Zhiyuan Chen , Chongmin She , Universal characterization of three-dimensional creeping crack-front stress fields, International Journal of Solids and Structures (2018), doi: 10.1016/j.ijsolstr.2018.06.020
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ACCEPTED MANUSCRIPT
Universal characterization of three-dimensional creeping crack-front stress fields Wanlin Guo*, Zhiyuan Chen, Chongmin She State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of
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Aeronautics and Astronautics, Nanjing 210016, China *Corresponding author, E-mail address: [email protected]
ABSTRACT
Creeping fracture in engineering always occurs in high temperature structures with
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complicated geometries and loading configurations, and how to characterize the threedimensional crack border stress fields is essentially important to design of the structures. By comprehensive finite element analyses of specimens with through-thethickness cracks and specimens with corner, surface and embedded elliptic cracks, we demonstrated that a three-parameter characterization based on C(t) integral, the out-
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of-plane stress constraint factor Tz and in-plane constraint coefficient Q* can be efficiently applied in all cases. It is shown that a two-parameter C(t)-Tz solution can
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provide efficient prediction for the stress field ahead of the crack under small scale creep condition. Under large scale creep conditions, it is found that Tz has nearly a unified distribution ahead of cracks, and the three-parameter C(t)-Tz-Q* solution can
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characterize the crack front stress fields efficiently. This universal characterization of the creeping crack front stress field should serve as a solid fundamental for three-
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dimensional damage tolerant design of high-temperature structures.
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Keywords:
Creeping crack border fields; Two-parameter description C(t)-Tz; Three-parameter description C(t)-Tz-Q*; Through-thickness cracks; Corner cracks; Surface cracks 1. Introduction Creeping fracture is vital in a wide range of engineering structures working at high temperature: turbine components, steam pipelines and chemical vessels, etc. The creep fracture mechanics has been developed in past three decades. However, most
ACCEPTED MANUSCRIPT researching works were performed within the frame of two-dimensional (2D) theory, although complicated three-dimensional (3D) geometry and loading conditions usually exist in high temperature structures. The crack border fields in power law creep solids have been described by Rice and Riedel (1980) with a single parameter C(t) under ideal plane stress and plane strain conditions, known as the RR solution. Recently, it is recognized that the RR solution must be improved to consider the constraint effects caused by 3D geometry of components and loading configurations
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(Budden and Ainsworth, 1999; Nguyen et al., 2000a, 2000b; Shlyannikov et al., 2009; Xiang et al., 2011; Liu et al., 2015; Shlyannikov et al., 2015; Ma et al., 2016a, 2016b). Budden and Ainsworth (1999) introduced the Q-parameter to investigate the in-plane constraint in creeping solids via analogy with power law plastic solids. Nguyen et al. (2000a, 2000b) proposed a three-term description
C A2 for
the crack tip fields in
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steady power law creeping solids. The parameters A2 and account for the constraint effect imposed by the specific geometry and loading configuration. By using the out-of-plane stress constraint factor Tz proposed by Guo (Guo,
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1993a; Guo, 1993b; Guo, 1995), Xiang et al. (2011) have derived out the 3D asymptotic fields near the border of mode-I through-the-thickness cracks in power law creeping solids and put forward the two-parameter C(t)-Tz solution and three-
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parameter C(t)-Tz-Q* solution for the tensile stress on the ligament ahead of the crack border under small scale and large scale creep conditions, respectively. They further
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formulize the 3D theoretical solutions into a set of empirical explicit formulae in whole range of out-of-plane stress constraint from Tz=0 at plane stress state to Tz=0.5
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at plane strain state and verified the formulation by 3D finite element analyses of single-edge-cracked plates under power law plastic and creeping conditions (Xiang
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and Guo, 2013). Then the out-of-plane constraint Tz is applied to conduct the 3D creep fracture by more and more researchers. Tan et al. (2012, 2013) also revealed influence of the crack and geometry configurations of the specimens on crack-front stress fields and creeping fracture of engineering materials. Matvienko et al.(2013) have investigated the influence of in-plane and out-of-plane constraint effects on crack-front stress fields under creep conditions and different specimen geometries using finite thickness boundary layer models. The in-plane and out-of-plane creep crack-tip constraints for different specimen geometries were also investigated by Liu et al. (2015) using 3D finite element analyses. Ma et al. (2016a, 2016b) developed a
ACCEPTED MANUSCRIPT unified correlation of in-plane and out-of-plane creep constraints with creep crack growth rate. They showed that the distributions of the constraint factor Tz and stress triaxiality change significantly with the thickness and configurations of the specimens. So the importance of two-parameter C(t)-Tz solution and three-parameter C(t)-Tz-Q* solution to describe the 3D stress fields of more general 3D cracks is obvious in the creep solids.
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In this paper, a universal solution of the 3D creeping crack-front stress field in specimens with various geometry and crack configurations is developed based on the dominance of the out-of-plane stress constraint factor Tz.
2. Three-parameter description of creep crack border fields
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The parameter C(t) characterizes the intensity of the near-crack-tip fields in elastic-nonlinear viscous materials in the same sense that the J-integral characterizes the crack-tip fields in elastic-plastic materials (Bassani and McClintock, 1981) and can be evaluated by
n C t n ij n1ui ,1 ds , Γ n 1 ij ij 1
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(1)
where is a vanishingly small counter-clockwise contour around the crack tip, ni is
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the unit outward normal to , ds is the length along and
ui ,1 is
the displacement
gradient rate with respect to x along the crack line. The C(t)-integral is path-
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independent within the creep zone, defined as the region where the equivalent creep strain c 2 3 ijc ijc exceeds the equivalent elastic strain e 2 3 ije ije .Due to the 12
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complexities of evaluating C(t)-integral at the crack tip, approximate solutions have been developed. Riedel and Rice (1980) obtained the relation between J-integral and
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C(t)-integral under small scale creep conditions as C t
1 v2 K I2 . J n 1 t n 1 Et
(2)
Under large scale creep conditions, the C(t)-integral gets close to a constant C*, C t C . When creeping deformation is between small scale and large scale creep
conditions, C(t) obeys the following formula suggested by Ehlers and Riedel (1981), t C t C T 1 , t
(3)
ACCEPTED MANUSCRIPT where tT is the transition time, which can be obtained by replacing C(t) in Eq.(2) with C*. The expression can be rearranged as
1 v K 2
tT
2 I
n 1 EC *
.
(4)
The effect of out-of-plane stress constraint has been explored in both elasticplastic solids and creeping solids. Tz is defined by Guo (1993a, 1993b, 1995) as
33 , 11 22
(5)
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Tz
where the subscripts 1, 2 and 3 stand for the Cartesian coordinate components x, y and z or cylindrical coordinate components r, θ and z respectively, with the x-y or r-θ plane in the normal plane of the crack front line, and z along the tangent line of the crack front line.
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Xiang et al. (2011, 2013) obtained the following two-parameter description of the in-plane fields near the crack border in 3D creeping solids, 1
n1 C t ij 0 ij n, , Tz , I T , n r 0 0 z
(6)
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where r and θ are polar coordinates centered at the crack border, θ=0 corresponds to the ligament directly ahead of the crack border. The two-parameter C(t)-Tz solution
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can provide efficient prediction for the tensile stress ahead of the crack border under small scale creep condition. Under large scale creep condition, a third parameter Q*
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which is similar to the Q item in the power law elastic-plastic solids should be introduced to take into account of the in-plane constraint. The three-parameter C(t)-
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Tz-Q* description was expressed as: 1
n1 C t ij 0 ij n, , Tz Q 0 ij , 0 0 I Tz , n r
(7)
where
Q
ij FE
ij
0
C t Tz
, 0, r 2 Λ .
(8)
The near tip stress is under the influence of finite geometrical deformation caused by crack border blunting. The Q term in elastic-plastic solids is calculated at r =2J/σ0. For the same consideration, the Q* parameter is calculated at r =2Λ. The normalized
ACCEPTED MANUSCRIPT length Λ, which is analogous to J/σ0 in power law plastic solids, has been defined by Xiang et al. (2011, 2013) as follows: C t t n 1 n 1 n t 1 n , small scale creep , 0 Λ C tt 1 n 0 , large scale creep , n 1 n 1 n C tT n 1 t ct 1 n 0 ， transition period ,
(9)
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where t is the dimensionless value of t. For convenience, the angular distribution of stress ij n, , Tz normalized by
n, 0, Tz is used in the following formulation. The Eq. (6) can be rearranged as 1
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n 1 C t ij 0 ij n, , Tz I T , n r 0 0 z
1
n 1 ij n, , Tz C t , 0 n 1 0 0 I Tz , n n, 0, Tz r n, 0, Tz
(10)
1
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n 1 C t 0 ij n, , Tz I T , n r 0 0 S z
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can be reduced to
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Obviously, the tensile stress on the ligament directly ahead of the crack border (θ=0)
C t 0 0 IS Tz , n r
0
1 n 1
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For some kind of creep materials, IS Tz , n
1 n 1
,
(11)
is the polynomial function of Tz. It
is efficient to evaluate the tensile stress. In the similarity, the three-parameter C(t)-Tz-
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Q* description can be expressed as: 1
n1 C t ij 0 ij n, , Tz Q 0 ij , 0 0 I S Tz , n r
(12)
where Q
ij FE
ij
C t TZ
0
, 0, r 2 Λ .
(13)
ACCEPTED MANUSCRIPT The transition time tT is calculated by Eq. (4). For the convenience, the parameter t* is introduced as: t*=t/tT, where t*=0.1 stands for small scale creep condition and t*=10 stands for large scale creep condition. All the stresses on the yaxis are normalized by the initial yield stress 0 . All the distances on the x-axis are normalized by Λ as defined by Eq. (9). 1 n 1
Empirical formulae for IS Tz , n
and ij n, , Tz fitted by Xiang (2013)
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greatly simplify the application of theoretical solutions. In this study, we focus on the stress fields on the ligament ahead of the crack border and the creep exponent of material is set at 5. Thus we derive out the following formulae, 1 n 1
I S Tz , n
-196.32Tz5 230.67Tz4 - 87.634Tz3 12.269Tz2 -1.7755Tz 1.0829 , (14)
(15)
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n, 0, Tz 1 ,
rr n, 0, Tz 0.301Tz 0.6086 .
1 n 1
When Tz=0 and 0.5, we can get the values of IS Tz , n
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and ij n, , Tz of RR
solutions in plane stress and plane strain conditions.
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On the other hand, it is reported that the stress triaxiality R is important to the failure of structures (Kong et al., 1995). So the R is also analyzed based on the 3D
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where
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with the Mises stress,
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solutions in the researches. The stress triaxiality R is the ratio of the mean stress
m
R
m , e
(17)
1 1 rr zz 1 Tz rr , 3 3
(18)
e 1 Tz Tz2 rr2 2 1 2Tz 2Tz2 rr 3 r2 rz2 2z . (19)
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3. Finite element models and numerical procedure In this work, an elastic-power law creep material model is used. The creep material parameters of the superalloy Inconel 800H at 1200℉ are E=154 GPa, v=0.33, A 1.348 10 16 MPa h -1 and n=5 (Yang et al., 1997). The initial yield stress is -n
ACCEPTED MANUSCRIPT 0 417 MPa and corresponding 0 A 0n 1.7 103 h -1 .The numerical simulation is
based on ABAQUS6.12. The geometry of crack specimens and the adopted coordinates are shown in Fig. 1 and Fig.2. The geometrical parameters of the specimen with a through-the-thickness crack are chosen as: a/W=0.5, H/W=2, W=20mm, the plate thickness B=2, 4, 10mm.To consider the detail of large deformation and blunting of crack tip, an initial notch with
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root radius ρ=0.001mm is adopted (Bettinson et al., 2002). The geometrical parameters of the specimen with an elliptic crack are chosen as: elliptical shape factor a/c=0.2,0.5,0.8,1. Ratio of crack length to width c/W=0.2; W=B=H=10mm. The direction angle of ellipse is ϕ. An initial notch with root radius ρ=0.001mm is adopted.
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The specimen is subjected to a constant remote tensile stress for all time, P=100MPa. The whole numerical procedure lasts for 5000h. The contour integral C(t) is evaluated by using the in-built ABAQUS routines. Twenty contours are set around the crack border. The opening stress and radial stress ahead of crack border are calculated by the two-parameter C(t)-Tz and three-parameters C(t)-Tz-Q* methods,
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and compared with different specimens. The C(t)-integral is defined by the maximum of twenty contour-integral results. The value of C* is defined as the C(t)-integral at
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t=5000h.
(b)
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PT
(a)
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2H
Fig. 1. Schematic illustration of a central through-the-thickness crack and the coordinate system: (a) the global geometry and the coordinate system of a crack, (b) the 3D FE model.
(e)
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(d)
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Fig. 2. Schematic illustration of the cracks and the coordinate systems: (a) the global geometry of
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a semi-elliptic surface crack, (b) Surface cracks with various a/c, (c) the local coordinate system and a normal sheet element of the crack front line, (d) the global geometry of a quarter elliptic
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corner crack, (e) the global geometry of an elliptic embedded crack.
The finite element mesh ahead of crack front is constructed with 20-node 3D
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quadratic brick elements. Since the crack-tip region contains steep stress and strain gradients, the mesh should be refined near the crack tip. In the plane (x-y plane) perpendicular to the crack front, the element size gradually increases with increasing radial distance from crack border. The identical planer mesh is repeated along the crack front line from the mid-plane to the free surface. In order to accommodate the strong variation of field quantities with respect to crack front, the thickness of successive element layers is gradually decrease toward the free surface and the point with minimum curvature.
ACCEPTED MANUSCRIPT Due to symmetry in geometries, one eighth of the through-the-thickness crack specimen is modeled. Fig. 1(b) shows the mesh sketch map of a through-the-thickness crack. Fig. 3 shows the mesh sketch map of an elliptic embedded crack. One eighth of the elliptic embedded crack specimen is modeled. For a surface crack specimen, one fourth of the specimen is modeled as shown in Fig. 4.
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For a corner crack specimen, half of the specimen is modeled as shown in Fig. 5.
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Fig. 3. The finite element model of a specimen with an embedded crack (1/8 model).
Fig. 4. The finite element model of specimen with a surface crack (1/4 model).
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Fig. 5. The finite element model of specimen with a corner crack (1/2 model).
4. Results and analyses
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4.1 Central through-the-thickness cracks
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The C(t)-integral characterizes the stress intensity of the crack-tip fields in creeping solids. The C(t)-integral decreases sharply at the beginning of creep deformation. As time goes on, the C(t)-integral gradually gets close to a constant. Fig. 6 shows the variations of C(t)-integral with time t for different thickness of plate. For a given time, the C(t)-integral decreases with increasing thickness. Fig. 7(a) shows the variations of Tz with radial distance r for different thickness at t*=0.1. It is found in Fig. 7(b) that the Tz curves will converge to a line with
PT
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increasing dimensionless distance r/B. For the large scale creep in Fig. 8, the Tz curves always converge to a line in whole dimensionless radial range. Fig. 9 shows the variations of Tz with time (a) t and (b) t* for different radial distance r/B.
C(t) (MPa.mm/h)
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0.0030
z/B=0 B=2mm B=4mm B=10mm
0.0025
0.0020
0.0015
0.0010
0
1000
2000
3000
4000
5000
t (h) Fig. 6.Variations of C(t)-integral with time t for different thickness at the mid plane of plate (z/B=0).
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0.6
t*=0.1, =0, z/B=0
(b) 0.6
0.5
0.5
0.4
0.4
0.3
0.3
Tz
Tz
(a)
0.2
0.2
B=2 mm B=4 mm B=10 mm
0.1
t*=0.1, =0, z/B=0
B=2 mm B=4 mm B=10 mm
0.1
0.0
0.0
1E-4
1E-3
0.01
0.1
1
10
1E-5 1E-4 1E-3 0.01
r (mm)
0.1
1
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r/B
Fig. 7.Variations of Tz with radial distance (a) r and (b) r/B at t*=0.1 for different thickness 0.6
t*=10, =0, z/B=0
(b) 0.6
0.5
0.5
0.4
0.4
0.3
0.3
Tz
0.2
B=2 mm B=4 mm B=10 mm
0.1 0.0 -0.1 1E-4
1E-3
0.01
t*=10, =0, z/B=0
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Tz
(a)
0.2 0.1
B=2 mm B=4 mm B=10 mm
0.0
-0.1
0.1
1
10
0.1
1
r/B
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r (mm)
1E-5 1E-4 1E-3 0.01
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Fig. 8.Variations of Tz with radial distance (a) r and (b) r/B at t*=10 for different thickness
(a) 0.8
r/B=0.01 r/B=0.11
PT
0.2
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0.0
0
Tz
0.4
r/B=0.01 r/B=0.11
0.6
B=4mm, z/B=0, θ=0
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Tz
0.6
(b) 0.8
r/B=0.05 r/B=0.24
r/B=0.05 r/B=0.24
B=4mm, z/B=0, θ=0
0.4 0.2
1000 2000 3000 4000 5000
t (h)
0.0
0
10
20
30
40
50
60
t*
Fig. 9.Variations of Tz with time (a) t and (b) t* for different radial distance r/B.
The parameters Q*, Λ and C* at the mid plane of plates with different thickness are shown in the table 1.
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Table 1. The parameters Q*, Λ and C* at the mid plane of plates with different thickness.
2 4 10 (a) 4
σrr -0.202 -0.223 -0.253
/0
C(t)-Tz-Q* FE results
2
RR plane strain 1
RR plane strain
1
RR plane stress 20
40
60
r/Λ
0
(d) 4
(c) 4
t*=0.1, =0, z/B=0, B=4mm
C(t)-Tz FE results
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2
20
40
r/Λ
C(t)-Tz-Q* FE results
2
RR plane strain
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1
RR plane strain
1
RR plane stress 0
0
20
40
RR plane stress 0 0
60
1
2
r/Λ
0
2
C(t)-Tz-Q* FE results
1
RR plane strain
RR plane strain
1 0
t*=10, =0, z/B=0, B=10mm Q*=-0.144
3
C(t)-Tz FE results
/0
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/0
AC
2
4
(f) 4
t*=0.1, =0, z/B=0, B=10mm
3
3
r/Λ
PT
(e) 4
60
t*=10, =0, z/B=0, B=4mm Q*=-0.114
3
3 /0
RR plane stress
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0
0
/0
0
1.51×10-3 1.44×10-3 1.29×10-3
t*=10, =0, z/B=0, B=2mm Q*=-0.101
3
C(t)-Tz FE results
2
1
t*=10 0.0116 0.0109 0.0112
(b) 4
t*=0.1, =0, z/B=0, B=2mm
3 /0
t*=0.1 0.00505 0.00479 0.00449
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σθθ -0.101 -0.114 -0.144
C*/MPa·mm·h-
Λ/mm
Q*
B/mm
RR plane stress 20
40
60
0
RR plane stress 0
20
40
60
r/Λ r/Λ Fig. 10. Comparisons of tensile stress obtained by the 3D Tz-based solutions, 3D FE results and
RR solutions, (a) B=2mm, t*=0.1; (b) B=2mm, t*=10; (c) B=4mm, t*=0.1; (d) B=4mm, t*=10; (e) B=10mm, t*=0.1; (f) B=10mm, t*=10.
ACCEPTED MANUSCRIPT The tensile stresses on the ligament are obtained by various approaches. Fig. 10 shows the comparisons of tensile stress obtained by the 3D Tz-based solutions, 3D FE results and RR solutions for different thickness of plate. It is found that the twoparameter C(t)-Tz solution can describe the crack-front stress field under small scale creep condition, and the three-parameter C(t)-Tz-Q* solution can describe the crackfront stress field under large scale creep condition. Good agreements are obtained between the 3D solutions and the FE results. (b) 6
t*=0.1, =0, z/B=0, B=2mm
5
RR plane strain
RR plane strain
4
2
R
C(t)-Tz FE results
3
C(t)-Tz-Q* FE results
3 2 1
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1
RR plane stress
0 0
10
20
30
40
0
50
(d) 6
t*=0.1, =0, z/B=0, B=4mm
5
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C(t)-Tz FE results
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3
1
PT
0
10
20
30
50
40
C(t)-Tz-Q* FE results
3
1
RR plane stress
0 50
0
10
20
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30
40
50
r/Λ (f) 6
t*=10, =0, z/B=0, B=10mm
5
RR plane strain
4
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RR plane strain
2
t*=0.1, =0, z/B=0, B=10mm
5
RR plane strain
4
C(t)-Tz FE results
3 2 1
R
R
40
t*=10, =0, z/B=0, B=4mm
r/Λ
(e) 6
30
4
RR plane stress
0
20
5
RR plane strain
4
2
10
r/Λ
r/Λ (c) 6
RR plane stress
0
R
R
t*=10, =0, z/B=0, B=2mm
5
4
R
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(a) 6
C(t)-Tz-Q* FE results
3 2 1
RR plane stress
0 0
10
20
30
r/Λ
RR plane stress
0
40
50
0
10
20
30
r/Λ
40
50
ACCEPTED MANUSCRIPT Fig. 11. Comparisons of the stress triaxiality R obtained by the 3D Tz-based solutions, 3D FE results and RR solutions, (a) B=2mm, t*=0.1; (b) B=2mm, t*=10; (c) B=4mm, t*=0.1; (d) B=4mm, t*=10; (e) B=10mm, t*=0.1; (f) B=10mm, t*=10.
The comparisons of the stress triaxiality R obtained by the 3D Tz-based solutions, 3D FE results and RR solutions are shown in Fig. 11. It is found that the 3D stress state between the plane stress and the plane strain states can be described
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accurately by the 3D Tz-based solutions under small scale creep and large scale creep conditions.
4.2 Semi-elliptic surface cracks
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Fig. 12(a) shows the variations of C(t)-integral with time t for different elliptical shape factors. The variations of C(t)-integral with elliptical parametrical angle ϕ for different time points t* are shown in Fig. 12(b).
=90
o
a/c=0.2 a/c=0.5 a/c=0.8 a/c=1.0
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C(t) (MPamm/h)
0.0008
0.0008 (b)
C(t) (MPa*mm/h)
0.0010
0.0006 0.0004
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(a)
0.0002
0.0004
0.0000
1000 2000 3000 4000 5000
t*=5 t*=10 t*=20 t*=50 t*=100
0.0002
PT
0
0.0006
t (h)
0
20
40 60 (degree)
80
100
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Fig. 12.Variations of C(t)-integral with (a) time t and (b) elliptical (a/c=1) parametrical angle ϕ.
(a) 0.6
(b) 0.5
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0.5
t*=0.1,
0.4
0.2 0.1 0.0
0.3
=5.6 o =26.3 o =43.1 o =70.5 o =90 o
Tz
Tz
0.3
0.2 0.1 0.0
0.01
0.1
r (mm)
t*=10,
0.4
1
=5.6 o =26.3 o =43.1 o =70.5 o =90 o
0.01
0.1
r (mm)
Fig. 13.Variations of Tz with radial distance at (a) t*=0.1 and (b) t*=10. a/c=1.
1
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0.1 0.0
Tensile stress Radial stress
t*=10, =0
Q*
-0.1 -0.2
-0.4 0
30
60 (degree)
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-0.3 90
Fig. 14.Variations of Q* with elliptical (a/c=1) parametrical angle
Fig. 13 shows the variations of Tz with radial distance r for different elliptical
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parametrical angle ϕ. Fig. 14 shows variations of Q* stress with elliptical parametrical angle ϕ.
The tensile stresses on the ligament ahead of the deepest point of surface crack are obtained by various approaches. Fig. 15 shows the comparisons of the tensile stress and radial stress from C(t)-Tz solution, RR solutions and 3D FE results under
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small scale creep condition. It is obvious that the C(t)-Tz solution is much close to the 3D FE results. For the large scale creep condition, the three-parameter C(t)-Tz-Q*
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solution is calculated as shown in Fig. 16, the C(t)-Tz and C(t)-Tz-Q*solutions for ϕ =5.6° are shown in Fig. 17. All these data are extracted from the specimen with a/c=1.
PT
The C(t)-Tz and C(t)-Tz-Q*solutions for ϕ =90°and a/c=0.2, 0.5, 0.8 are shown in Fig. 18. Fig. 19 Shows the variations of Tz with elliptical angle for (a) t*=0.1 and (b)
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t*=10 under different radial distance, Fig. 19(c) shows the variations of Tz with time t* for different r/c. The comparisons of stress triaxiality obtained by 3D methods C(t)TZ and C(t)-TZ-Q* with 3D FE results and RR solutions are shown in Fig. 20. Good
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agreements are obtained between the 3D solutions and the FE results.
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Fig. 16.Comparisons of tensile stress and radial stress obtained by C(t)-Tz-Q* method with 3D FE results and RR solutions (a/c=1), (a) tensile stress; (b) radial stress.
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Fig. 17. Comparisons of tensile stress and radial stress obtained by 3D methods C(t)-Tz and C(t)Tz-Q* with 3D FE results and RR solutions for ϕ =5.6° and a/c=1, (a) tensile stress at t*=0.1; (b) radial stress at t*=0.1; (c) tensile stress at t*=10; (d) radial stress at t*=10. (a) 4
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Fig.18. Comparisons of tensile stress obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions: (a) a/c=0.2, t*=0.1; (b) a/c=0.2, t*=10; (c) a/c=0.5, t*=0.1; (d) a/c=0.5, t*=10; (e) a/c=0.8, t*=0.1; (f) a/c=0.8, t*=10
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Fig. 19. Variations of Tz with elliptical angle for (a) t*=0.1, (b) t*=10; Variations of Tz with creep time t* for different r/c.
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Fig.20. Comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions: (a) a/c=0.2, t*=0.1; (b) a/c=0.2, t*=10; (c) a/c=0.5, t*=0.1; (d) a/c=0.5, t*=10; (e) a/c=0.8, t*=0.1; (f) a/c=0.8, t*=10; (g) a/c=1.0, t*=0.1; (h) a/c=1.0, t*=10
4.3 Quarter elliptic corner crack
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For the quarter circular corner crack (a/c=1), the variations of the C(t)-integral with elliptical parametrical angle are shown in Fig. 21, the crack-tip tensile stresses solved by C(t)-Tz solution, C(t)-Tz-Q* solution, RR solutions and 3D FE results for various elliptical parametrical angle ϕ under small scale creep and large scale conditions are
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shown in Fig. 22. Fig. 23 shows the variations of Tz with elliptical angle for (a) t*=0.1 and (b) t*=10, (c) variations of Tz with time t* for different r/c, (d) variations of Q* with elliptical angle for different stress components. The comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions are shown in Fig. 24.
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Fig. 21.Variations of C(t)-integral with elliptical parametrical angle. (b) 4
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Fig. 22. Comparisons of tensile stress obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions, (a) =6.1°, t*=0.1; (b) =6.1°, t*=10; (c) =11.9°, t*=0.1; (d) =11.9°,
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t*=10; (e) =23.2°, t*=0.1; (f) =23.2°, t*=10;(g) =45°, t*=0.1; (h) =45°, t*=10;
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Fig. 23. Variations of Tz with elliptical angle for (a) t*=0.1 and (b) t*=10, (c) variations of Tz with
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Fig. 24 Comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions, (a) =45°, t*=0.1; (b) =45°, t*=10;
4.4 Elliptic embedded crack For a circular embedded crack (a/c=1), the crack-tip tensile stresses obtained by C(t)Tz solution, RR solutions and 3D FE results under the small scale creep condition (t*=0.1) are shown in Fig. 25. The tensile stresses solved by C(t)-Tz-Q* solution, RR solutions and 3D FE results under the large scale creep condition (t*=10) are shown
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in Fig. 26. Fig. 27(a) shows the variations of Tz with time t* for different r/c. Fig. 27(b) shows the variations of C(t) with elliptical angle for different t*. Fig. 27(c) shows the variations of Q* with elliptical angle for different stress components. The comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions are shown in Fig. 28. (b) 4
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Fig. 25. Comparisons of (a) tensile stress and (b) radial stress obtained by 3D methods C(t)-Tz with 3D FE results and RR solutions for ϕ =0° and t*=0.1. (a) 2.5
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Fig. 26. Comparisons of (a) tensile stress and (b) radial stress obtained by 3D methods C(t)-Tz-Q*
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(b) 0.006 r/c=0.01 r/c=0.02 r/c=0.03
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Tensile stress Radial stress
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Fig. 27. (a) Variations of Tz with time t* for different r/c, (b) variations of C(t) with elliptical angle for different t*, (c) variations of Q* with elliptical angle for different stress components. (b) 5
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Fig. 28. Comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D
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FE results and RR solutions, (a) =0°, t*=0.1; (b) =0°, t*=10;
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5. Conclusions
The dominance of the two-parameter C(t)-Tz and three-parameter C(t)-Tz-Q*
solutions are validated for through-the-thickness cracks, surface cracks, corner cracks
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and embedded cracks by comprehensive numerical finite element simulations. It is shown that the two-parameter C(t)-Tz solution can provide efficient prediction for the stress field ahead of the cracks under small scale creep condition. Under large scale creep conditions, the three-parameter C(t)-Tz-Q* solution can characterize the crack front stress fields efficiently, and the factor Tz shows nearly a unified distribution ahead of the creep cracks. In all the simulated situations, the 3D solutions can provide a universal characterization of the crack tip fields efficiently while the two-
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Acknowledgments This work was supported by National Natural Science Foundation of China (51535005, 51472117), the Research Fund of State Key Laboratory of Mechanics and
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Control of Mechanical Structures (MCMS-0416K01, MCMS-0416G01), the Fundamental Research Funds for the Central Universities (NP2017101), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education
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Institutions.
References
Bassani J.L., McClintock F.A., 1981. Creep relaxation of stress around a crack tip. Int. J. Solids Struct. 17, 479-492.
M
Bettinson A.D., O’Dowd N.P., Nikbin K.M., Webster G.A., 2002. Experimental investigation of constraint effects on creep crack growth. Pressure Vessels and Piping Conference. ASME. 434, 143-150.
ED
Budden P.J., Ainsworth R.A., 1999. The effect of constraint on creep fracture assessments. Int. J. Fract. 97, 237-247.
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Ehlers R., Riedel H., 1981. A finite element analysis of creep deformation in a specimen containing macroscopic crack. ICF5, Cannes (France).
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Guo W., 1993a. Elastoplastic three dimensional crack border field—I. Singular structure of the field. Eng. Fract. Mech. 46, 93-104.
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Guo W., 1993b. Elastoplastic three dimensional crack border field—II. Asymptotic solution for the field. Eng. Fract. Mech. 46, 105-113. Guo W., 1995. Elasto-plastic three-dimensional crack border field—III. Fracture parameters. Eng. Fract. Mech. 51, 51-71. Kong X., Schlüter N., Dahl W., 1995. Effect of triaxial stress on mixed-mode fracture. Eng. Fract. Mech. 52, 379-388. Liu S., Wang G.Z., Xuan F.Z., Tu S.T., 2015. Three-dimensional finite element analyses of in-plane and out-of-plane creep crack-tip constraints for different specimen geometries. Eng. Fract. Mech. 133, 264-280.
ACCEPTED MANUSCRIPT Ma H.S., Wang G.Z., Liu S., Tu S.T., Xuan F.Z., 2016a. In-plane and out-of-plane unified constraint-dependent creep crack growth rate of 316H steel. Eng. Fract. Mech. 155, 88-101. Ma H.S., Wang G.Z., Xuan F.Z., Tu S.T., 2016b. Unified correlation of in-plane and out-of-plane creep constraints with creep crack growth rate. Int. J. Pressure Vessels Piping. 139–140, 47-60.
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Matvienko Y.G., Shlyannikov V.N., Boychenko N.V., 2013. In-plane and out-ofplane constraint parameters along a three-dimensional crack-front stress field under creep loading. Fatigue Fract. Eng. Mat. Struct. 36, 14–24. Nguyen B.N., Onck P., Van Der Giessen E., 2000a. Crack-tip constraint effects on creep fracture. Eng. Fract. Mech. 65, 467-490. Nguyen B.N., Onck P., Van Der Giessen E., 2000b. On higher-order crack-tip fields in creeping solids. J. Appl. Mech. 67, 372-382.
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Rice J.R., Riedel H., 1980. Tensile cracks in creeping solids. Fracture Mechanics: 12th Conference. ASTM International. 700, 112-130. Shlyannikov VN, Tumanov AV, Boychenko NV. A creep stress intensity factor approach to creep-fatigue crack growth. Engng Fract Mech 2015; 142: 201-219. Shlyannikov VN, Ilchenko BV, Boychenko NV. Biaxial loading effect on higherorder crack tip parameters, ASTM STP 1508. Fatigue Fract Mech 2009; 36: 609–640.
M
Tan J.P., Wang G.Z., Xuan F.Z., Tu S.T., 2012. Correlation of creep crack-tip constraint between axially cracked pipelines and test specimens. Int. J. Pressure Vessels Piping. 98, 16-25.
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Tan J.P., Tu S.T., Wang G.Z., Xuan F.Z., 2013. Effect and mechanism of out-of-plane constraint on creep crack growth behavior of a Cr–Mo–V steel. Eng. Fract. Mech. 99, 324-334.
PT
Xiang M., Yu Z., Guo W., 2011. Characterization of three-dimensional crack border fields in creeping solids. Int. J. Solids Struct. 48, 2695-2705.
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Xiang M., Guo W., 2013. Formulation of the stress fields in power law solids ahead of three-dimensional tensile cracks. Int. J. Solids Struct. 50, 3067-3088.
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Yang L., Sutton M.A., Deng X., Lyons J.S., 1997. Finite element analysis of creep fracture initiation in a model superalloy material. Int. J. Fract. 81, 299-320.
Universal characterization of three-dimensional creeping crack-front stress fields Wanlin Guo , Zhiyuan Chen , Chongmin She PII: DOI: Reference:
S0020-7683(18)30252-X 10.1016/j.ijsolstr.2018.06.020 SAS 10030
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
1 December 2017 8 June 2018 18 June 2018
Please cite this article as: Wanlin Guo , Zhiyuan Chen , Chongmin She , Universal characterization of three-dimensional creeping crack-front stress fields, International Journal of Solids and Structures (2018), doi: 10.1016/j.ijsolstr.2018.06.020
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Universal characterization of three-dimensional creeping crack-front stress fields Wanlin Guo*, Zhiyuan Chen, Chongmin She State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of
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Aeronautics and Astronautics, Nanjing 210016, China *Corresponding author, E-mail address: [email protected]
ABSTRACT
Creeping fracture in engineering always occurs in high temperature structures with
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complicated geometries and loading configurations, and how to characterize the threedimensional crack border stress fields is essentially important to design of the structures. By comprehensive finite element analyses of specimens with through-thethickness cracks and specimens with corner, surface and embedded elliptic cracks, we demonstrated that a three-parameter characterization based on C(t) integral, the out-
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of-plane stress constraint factor Tz and in-plane constraint coefficient Q* can be efficiently applied in all cases. It is shown that a two-parameter C(t)-Tz solution can
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provide efficient prediction for the stress field ahead of the crack under small scale creep condition. Under large scale creep conditions, it is found that Tz has nearly a unified distribution ahead of cracks, and the three-parameter C(t)-Tz-Q* solution can
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characterize the crack front stress fields efficiently. This universal characterization of the creeping crack front stress field should serve as a solid fundamental for three-
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dimensional damage tolerant design of high-temperature structures.
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Keywords:
Creeping crack border fields; Two-parameter description C(t)-Tz; Three-parameter description C(t)-Tz-Q*; Through-thickness cracks; Corner cracks; Surface cracks 1. Introduction Creeping fracture is vital in a wide range of engineering structures working at high temperature: turbine components, steam pipelines and chemical vessels, etc. The creep fracture mechanics has been developed in past three decades. However, most
ACCEPTED MANUSCRIPT researching works were performed within the frame of two-dimensional (2D) theory, although complicated three-dimensional (3D) geometry and loading conditions usually exist in high temperature structures. The crack border fields in power law creep solids have been described by Rice and Riedel (1980) with a single parameter C(t) under ideal plane stress and plane strain conditions, known as the RR solution. Recently, it is recognized that the RR solution must be improved to consider the constraint effects caused by 3D geometry of components and loading configurations
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(Budden and Ainsworth, 1999; Nguyen et al., 2000a, 2000b; Shlyannikov et al., 2009; Xiang et al., 2011; Liu et al., 2015; Shlyannikov et al., 2015; Ma et al., 2016a, 2016b). Budden and Ainsworth (1999) introduced the Q-parameter to investigate the in-plane constraint in creeping solids via analogy with power law plastic solids. Nguyen et al. (2000a, 2000b) proposed a three-term description
C A2 for
the crack tip fields in
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steady power law creeping solids. The parameters A2 and account for the constraint effect imposed by the specific geometry and loading configuration. By using the out-of-plane stress constraint factor Tz proposed by Guo (Guo,
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1993a; Guo, 1993b; Guo, 1995), Xiang et al. (2011) have derived out the 3D asymptotic fields near the border of mode-I through-the-thickness cracks in power law creeping solids and put forward the two-parameter C(t)-Tz solution and three-
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parameter C(t)-Tz-Q* solution for the tensile stress on the ligament ahead of the crack border under small scale and large scale creep conditions, respectively. They further
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formulize the 3D theoretical solutions into a set of empirical explicit formulae in whole range of out-of-plane stress constraint from Tz=0 at plane stress state to Tz=0.5
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at plane strain state and verified the formulation by 3D finite element analyses of single-edge-cracked plates under power law plastic and creeping conditions (Xiang
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and Guo, 2013). Then the out-of-plane constraint Tz is applied to conduct the 3D creep fracture by more and more researchers. Tan et al. (2012, 2013) also revealed influence of the crack and geometry configurations of the specimens on crack-front stress fields and creeping fracture of engineering materials. Matvienko et al.(2013) have investigated the influence of in-plane and out-of-plane constraint effects on crack-front stress fields under creep conditions and different specimen geometries using finite thickness boundary layer models. The in-plane and out-of-plane creep crack-tip constraints for different specimen geometries were also investigated by Liu et al. (2015) using 3D finite element analyses. Ma et al. (2016a, 2016b) developed a
ACCEPTED MANUSCRIPT unified correlation of in-plane and out-of-plane creep constraints with creep crack growth rate. They showed that the distributions of the constraint factor Tz and stress triaxiality change significantly with the thickness and configurations of the specimens. So the importance of two-parameter C(t)-Tz solution and three-parameter C(t)-Tz-Q* solution to describe the 3D stress fields of more general 3D cracks is obvious in the creep solids.
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In this paper, a universal solution of the 3D creeping crack-front stress field in specimens with various geometry and crack configurations is developed based on the dominance of the out-of-plane stress constraint factor Tz.
2. Three-parameter description of creep crack border fields
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The parameter C(t) characterizes the intensity of the near-crack-tip fields in elastic-nonlinear viscous materials in the same sense that the J-integral characterizes the crack-tip fields in elastic-plastic materials (Bassani and McClintock, 1981) and can be evaluated by
n C t n ij n1ui ,1 ds , Γ n 1 ij ij 1
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(1)
where is a vanishingly small counter-clockwise contour around the crack tip, ni is
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the unit outward normal to , ds is the length along and
ui ,1 is
the displacement
gradient rate with respect to x along the crack line. The C(t)-integral is path-
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independent within the creep zone, defined as the region where the equivalent creep strain c 2 3 ijc ijc exceeds the equivalent elastic strain e 2 3 ije ije .Due to the 12
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complexities of evaluating C(t)-integral at the crack tip, approximate solutions have been developed. Riedel and Rice (1980) obtained the relation between J-integral and
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C(t)-integral under small scale creep conditions as C t
1 v2 K I2 . J n 1 t n 1 Et
(2)
Under large scale creep conditions, the C(t)-integral gets close to a constant C*, C t C . When creeping deformation is between small scale and large scale creep
conditions, C(t) obeys the following formula suggested by Ehlers and Riedel (1981), t C t C T 1 , t
(3)
ACCEPTED MANUSCRIPT where tT is the transition time, which can be obtained by replacing C(t) in Eq.(2) with C*. The expression can be rearranged as
1 v K 2
tT
2 I
n 1 EC *
.
(4)
The effect of out-of-plane stress constraint has been explored in both elasticplastic solids and creeping solids. Tz is defined by Guo (1993a, 1993b, 1995) as
33 , 11 22
(5)
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Tz
where the subscripts 1, 2 and 3 stand for the Cartesian coordinate components x, y and z or cylindrical coordinate components r, θ and z respectively, with the x-y or r-θ plane in the normal plane of the crack front line, and z along the tangent line of the crack front line.
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Xiang et al. (2011, 2013) obtained the following two-parameter description of the in-plane fields near the crack border in 3D creeping solids, 1
n1 C t ij 0 ij n, , Tz , I T , n r 0 0 z
(6)
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where r and θ are polar coordinates centered at the crack border, θ=0 corresponds to the ligament directly ahead of the crack border. The two-parameter C(t)-Tz solution
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can provide efficient prediction for the tensile stress ahead of the crack border under small scale creep condition. Under large scale creep condition, a third parameter Q*
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which is similar to the Q item in the power law elastic-plastic solids should be introduced to take into account of the in-plane constraint. The three-parameter C(t)-
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Tz-Q* description was expressed as: 1
n1 C t ij 0 ij n, , Tz Q 0 ij , 0 0 I Tz , n r
(7)
where
Q
ij FE
ij
0
C t Tz
, 0, r 2 Λ .
(8)
The near tip stress is under the influence of finite geometrical deformation caused by crack border blunting. The Q term in elastic-plastic solids is calculated at r =2J/σ0. For the same consideration, the Q* parameter is calculated at r =2Λ. The normalized
ACCEPTED MANUSCRIPT length Λ, which is analogous to J/σ0 in power law plastic solids, has been defined by Xiang et al. (2011, 2013) as follows: C t t n 1 n 1 n t 1 n , small scale creep , 0 Λ C tt 1 n 0 , large scale creep , n 1 n 1 n C tT n 1 t ct 1 n 0 ， transition period ,
(9)
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where t is the dimensionless value of t. For convenience, the angular distribution of stress ij n, , Tz normalized by
n, 0, Tz is used in the following formulation. The Eq. (6) can be rearranged as 1
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n 1 C t ij 0 ij n, , Tz I T , n r 0 0 z
1
n 1 ij n, , Tz C t , 0 n 1 0 0 I Tz , n n, 0, Tz r n, 0, Tz
(10)
1
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n 1 C t 0 ij n, , Tz I T , n r 0 0 S z
PT
can be reduced to
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Obviously, the tensile stress on the ligament directly ahead of the crack border (θ=0)
C t 0 0 IS Tz , n r
0
1 n 1
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For some kind of creep materials, IS Tz , n
1 n 1
,
(11)
is the polynomial function of Tz. It
is efficient to evaluate the tensile stress. In the similarity, the three-parameter C(t)-Tz-
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Q* description can be expressed as: 1
n1 C t ij 0 ij n, , Tz Q 0 ij , 0 0 I S Tz , n r
(12)
where Q
ij FE
ij
C t TZ
0
, 0, r 2 Λ .
(13)
ACCEPTED MANUSCRIPT The transition time tT is calculated by Eq. (4). For the convenience, the parameter t* is introduced as: t*=t/tT, where t*=0.1 stands for small scale creep condition and t*=10 stands for large scale creep condition. All the stresses on the yaxis are normalized by the initial yield stress 0 . All the distances on the x-axis are normalized by Λ as defined by Eq. (9). 1 n 1
Empirical formulae for IS Tz , n
and ij n, , Tz fitted by Xiang (2013)
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greatly simplify the application of theoretical solutions. In this study, we focus on the stress fields on the ligament ahead of the crack border and the creep exponent of material is set at 5. Thus we derive out the following formulae, 1 n 1
I S Tz , n
-196.32Tz5 230.67Tz4 - 87.634Tz3 12.269Tz2 -1.7755Tz 1.0829 , (14)
(15)
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n, 0, Tz 1 ,
rr n, 0, Tz 0.301Tz 0.6086 .
1 n 1
When Tz=0 and 0.5, we can get the values of IS Tz , n
(16)
and ij n, , Tz of RR
solutions in plane stress and plane strain conditions.
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On the other hand, it is reported that the stress triaxiality R is important to the failure of structures (Kong et al., 1995). So the R is also analyzed based on the 3D
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where
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with the Mises stress,
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solutions in the researches. The stress triaxiality R is the ratio of the mean stress
m
R
m , e
(17)
1 1 rr zz 1 Tz rr , 3 3
(18)
e 1 Tz Tz2 rr2 2 1 2Tz 2Tz2 rr 3 r2 rz2 2z . (19)
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3. Finite element models and numerical procedure In this work, an elastic-power law creep material model is used. The creep material parameters of the superalloy Inconel 800H at 1200℉ are E=154 GPa, v=0.33, A 1.348 10 16 MPa h -1 and n=5 (Yang et al., 1997). The initial yield stress is -n
ACCEPTED MANUSCRIPT 0 417 MPa and corresponding 0 A 0n 1.7 103 h -1 .The numerical simulation is
based on ABAQUS6.12. The geometry of crack specimens and the adopted coordinates are shown in Fig. 1 and Fig.2. The geometrical parameters of the specimen with a through-the-thickness crack are chosen as: a/W=0.5, H/W=2, W=20mm, the plate thickness B=2, 4, 10mm.To consider the detail of large deformation and blunting of crack tip, an initial notch with
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root radius ρ=0.001mm is adopted (Bettinson et al., 2002). The geometrical parameters of the specimen with an elliptic crack are chosen as: elliptical shape factor a/c=0.2,0.5,0.8,1. Ratio of crack length to width c/W=0.2; W=B=H=10mm. The direction angle of ellipse is ϕ. An initial notch with root radius ρ=0.001mm is adopted.
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The specimen is subjected to a constant remote tensile stress for all time, P=100MPa. The whole numerical procedure lasts for 5000h. The contour integral C(t) is evaluated by using the in-built ABAQUS routines. Twenty contours are set around the crack border. The opening stress and radial stress ahead of crack border are calculated by the two-parameter C(t)-Tz and three-parameters C(t)-Tz-Q* methods,
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and compared with different specimens. The C(t)-integral is defined by the maximum of twenty contour-integral results. The value of C* is defined as the C(t)-integral at
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t=5000h.
(b)
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PT
(a)
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2H
Fig. 1. Schematic illustration of a central through-the-thickness crack and the coordinate system: (a) the global geometry and the coordinate system of a crack, (b) the 3D FE model.
(e)
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(d)
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Fig. 2. Schematic illustration of the cracks and the coordinate systems: (a) the global geometry of
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a semi-elliptic surface crack, (b) Surface cracks with various a/c, (c) the local coordinate system and a normal sheet element of the crack front line, (d) the global geometry of a quarter elliptic
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corner crack, (e) the global geometry of an elliptic embedded crack.
The finite element mesh ahead of crack front is constructed with 20-node 3D
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quadratic brick elements. Since the crack-tip region contains steep stress and strain gradients, the mesh should be refined near the crack tip. In the plane (x-y plane) perpendicular to the crack front, the element size gradually increases with increasing radial distance from crack border. The identical planer mesh is repeated along the crack front line from the mid-plane to the free surface. In order to accommodate the strong variation of field quantities with respect to crack front, the thickness of successive element layers is gradually decrease toward the free surface and the point with minimum curvature.
ACCEPTED MANUSCRIPT Due to symmetry in geometries, one eighth of the through-the-thickness crack specimen is modeled. Fig. 1(b) shows the mesh sketch map of a through-the-thickness crack. Fig. 3 shows the mesh sketch map of an elliptic embedded crack. One eighth of the elliptic embedded crack specimen is modeled. For a surface crack specimen, one fourth of the specimen is modeled as shown in Fig. 4.
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For a corner crack specimen, half of the specimen is modeled as shown in Fig. 5.
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PT
Fig. 3. The finite element model of a specimen with an embedded crack (1/8 model).
Fig. 4. The finite element model of specimen with a surface crack (1/4 model).
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Fig. 5. The finite element model of specimen with a corner crack (1/2 model).
4. Results and analyses
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4.1 Central through-the-thickness cracks
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The C(t)-integral characterizes the stress intensity of the crack-tip fields in creeping solids. The C(t)-integral decreases sharply at the beginning of creep deformation. As time goes on, the C(t)-integral gradually gets close to a constant. Fig. 6 shows the variations of C(t)-integral with time t for different thickness of plate. For a given time, the C(t)-integral decreases with increasing thickness. Fig. 7(a) shows the variations of Tz with radial distance r for different thickness at t*=0.1. It is found in Fig. 7(b) that the Tz curves will converge to a line with
PT
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increasing dimensionless distance r/B. For the large scale creep in Fig. 8, the Tz curves always converge to a line in whole dimensionless radial range. Fig. 9 shows the variations of Tz with time (a) t and (b) t* for different radial distance r/B.
C(t) (MPa.mm/h)
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0.0030
z/B=0 B=2mm B=4mm B=10mm
0.0025
0.0020
0.0015
0.0010
0
1000
2000
3000
4000
5000
t (h) Fig. 6.Variations of C(t)-integral with time t for different thickness at the mid plane of plate (z/B=0).
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0.6
t*=0.1, =0, z/B=0
(b) 0.6
0.5
0.5
0.4
0.4
0.3
0.3
Tz
Tz
(a)
0.2
0.2
B=2 mm B=4 mm B=10 mm
0.1
t*=0.1, =0, z/B=0
B=2 mm B=4 mm B=10 mm
0.1
0.0
0.0
1E-4
1E-3
0.01
0.1
1
10
1E-5 1E-4 1E-3 0.01
r (mm)
0.1
1
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r/B
Fig. 7.Variations of Tz with radial distance (a) r and (b) r/B at t*=0.1 for different thickness 0.6
t*=10, =0, z/B=0
(b) 0.6
0.5
0.5
0.4
0.4
0.3
0.3
Tz
0.2
B=2 mm B=4 mm B=10 mm
0.1 0.0 -0.1 1E-4
1E-3
0.01
t*=10, =0, z/B=0
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Tz
(a)
0.2 0.1
B=2 mm B=4 mm B=10 mm
0.0
-0.1
0.1
1
10
0.1
1
r/B
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r (mm)
1E-5 1E-4 1E-3 0.01
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Fig. 8.Variations of Tz with radial distance (a) r and (b) r/B at t*=10 for different thickness
(a) 0.8
r/B=0.01 r/B=0.11
PT
0.2
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0.0
0
Tz
0.4
r/B=0.01 r/B=0.11
0.6
B=4mm, z/B=0, θ=0
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Tz
0.6
(b) 0.8
r/B=0.05 r/B=0.24
r/B=0.05 r/B=0.24
B=4mm, z/B=0, θ=0
0.4 0.2
1000 2000 3000 4000 5000
t (h)
0.0
0
10
20
30
40
50
60
t*
Fig. 9.Variations of Tz with time (a) t and (b) t* for different radial distance r/B.
The parameters Q*, Λ and C* at the mid plane of plates with different thickness are shown in the table 1.
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Table 1. The parameters Q*, Λ and C* at the mid plane of plates with different thickness.
2 4 10 (a) 4
σrr -0.202 -0.223 -0.253
/0
C(t)-Tz-Q* FE results
2
RR plane strain 1
RR plane strain
1
RR plane stress 20
40
60
r/Λ
0
(d) 4
(c) 4
t*=0.1, =0, z/B=0, B=4mm
C(t)-Tz FE results
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2
20
40
r/Λ
C(t)-Tz-Q* FE results
2
RR plane strain
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1
RR plane strain
1
RR plane stress 0
0
20
40
RR plane stress 0 0
60
1
2
r/Λ
0
2
C(t)-Tz-Q* FE results
1
RR plane strain
RR plane strain
1 0
t*=10, =0, z/B=0, B=10mm Q*=-0.144
3
C(t)-Tz FE results
/0
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/0
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2
4
(f) 4
t*=0.1, =0, z/B=0, B=10mm
3
3
r/Λ
PT
(e) 4
60
t*=10, =0, z/B=0, B=4mm Q*=-0.114
3
3 /0
RR plane stress
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0
0
/0
0
1.51×10-3 1.44×10-3 1.29×10-3
t*=10, =0, z/B=0, B=2mm Q*=-0.101
3
C(t)-Tz FE results
2
1
t*=10 0.0116 0.0109 0.0112
(b) 4
t*=0.1, =0, z/B=0, B=2mm
3 /0
t*=0.1 0.00505 0.00479 0.00449
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σθθ -0.101 -0.114 -0.144
C*/MPa·mm·h-
Λ/mm
Q*
B/mm
RR plane stress 20
40
60
0
RR plane stress 0
20
40
60
r/Λ r/Λ Fig. 10. Comparisons of tensile stress obtained by the 3D Tz-based solutions, 3D FE results and
RR solutions, (a) B=2mm, t*=0.1; (b) B=2mm, t*=10; (c) B=4mm, t*=0.1; (d) B=4mm, t*=10; (e) B=10mm, t*=0.1; (f) B=10mm, t*=10.
ACCEPTED MANUSCRIPT The tensile stresses on the ligament are obtained by various approaches. Fig. 10 shows the comparisons of tensile stress obtained by the 3D Tz-based solutions, 3D FE results and RR solutions for different thickness of plate. It is found that the twoparameter C(t)-Tz solution can describe the crack-front stress field under small scale creep condition, and the three-parameter C(t)-Tz-Q* solution can describe the crackfront stress field under large scale creep condition. Good agreements are obtained between the 3D solutions and the FE results. (b) 6
t*=0.1, =0, z/B=0, B=2mm
5
RR plane strain
RR plane strain
4
2
R
C(t)-Tz FE results
3
C(t)-Tz-Q* FE results
3 2 1
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1
RR plane stress
0 0
10
20
30
40
0
50
(d) 6
t*=0.1, =0, z/B=0, B=4mm
5
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C(t)-Tz FE results
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3
1
PT
0
10
20
30
50
40
C(t)-Tz-Q* FE results
3
1
RR plane stress
0 50
0
10
20
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30
40
50
r/Λ (f) 6
t*=10, =0, z/B=0, B=10mm
5
RR plane strain
4
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RR plane strain
2
t*=0.1, =0, z/B=0, B=10mm
5
RR plane strain
4
C(t)-Tz FE results
3 2 1
R
R
40
t*=10, =0, z/B=0, B=4mm
r/Λ
(e) 6
30
4
RR plane stress
0
20
5
RR plane strain
4
2
10
r/Λ
r/Λ (c) 6
RR plane stress
0
R
R
t*=10, =0, z/B=0, B=2mm
5
4
R
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(a) 6
C(t)-Tz-Q* FE results
3 2 1
RR plane stress
0 0
10
20
30
r/Λ
RR plane stress
0
40
50
0
10
20
30
r/Λ
40
50
ACCEPTED MANUSCRIPT Fig. 11. Comparisons of the stress triaxiality R obtained by the 3D Tz-based solutions, 3D FE results and RR solutions, (a) B=2mm, t*=0.1; (b) B=2mm, t*=10; (c) B=4mm, t*=0.1; (d) B=4mm, t*=10; (e) B=10mm, t*=0.1; (f) B=10mm, t*=10.
The comparisons of the stress triaxiality R obtained by the 3D Tz-based solutions, 3D FE results and RR solutions are shown in Fig. 11. It is found that the 3D stress state between the plane stress and the plane strain states can be described
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accurately by the 3D Tz-based solutions under small scale creep and large scale creep conditions.
4.2 Semi-elliptic surface cracks
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Fig. 12(a) shows the variations of C(t)-integral with time t for different elliptical shape factors. The variations of C(t)-integral with elliptical parametrical angle ϕ for different time points t* are shown in Fig. 12(b).
=90
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a/c=0.2 a/c=0.5 a/c=0.8 a/c=1.0
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C(t) (MPamm/h)
0.0008
0.0008 (b)
C(t) (MPa*mm/h)
0.0010
0.0006 0.0004
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(a)
0.0002
0.0004
0.0000
1000 2000 3000 4000 5000
t*=5 t*=10 t*=20 t*=50 t*=100
0.0002
PT
0
0.0006
t (h)
0
20
40 60 (degree)
80
100
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Fig. 12.Variations of C(t)-integral with (a) time t and (b) elliptical (a/c=1) parametrical angle ϕ.
(a) 0.6
(b) 0.5
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0.5
t*=0.1,
0.4
0.2 0.1 0.0
0.3
=5.6 o =26.3 o =43.1 o =70.5 o =90 o
Tz
Tz
0.3
0.2 0.1 0.0
0.01
0.1
r (mm)
t*=10,
0.4
1
=5.6 o =26.3 o =43.1 o =70.5 o =90 o
0.01
0.1
r (mm)
Fig. 13.Variations of Tz with radial distance at (a) t*=0.1 and (b) t*=10. a/c=1.
1
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0.1 0.0
Tensile stress Radial stress
t*=10, =0
Q*
-0.1 -0.2
-0.4 0
30
60 (degree)
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-0.3 90
Fig. 14.Variations of Q* with elliptical (a/c=1) parametrical angle
Fig. 13 shows the variations of Tz with radial distance r for different elliptical
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parametrical angle ϕ. Fig. 14 shows variations of Q* stress with elliptical parametrical angle ϕ.
The tensile stresses on the ligament ahead of the deepest point of surface crack are obtained by various approaches. Fig. 15 shows the comparisons of the tensile stress and radial stress from C(t)-Tz solution, RR solutions and 3D FE results under
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small scale creep condition. It is obvious that the C(t)-Tz solution is much close to the 3D FE results. For the large scale creep condition, the three-parameter C(t)-Tz-Q*
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solution is calculated as shown in Fig. 16, the C(t)-Tz and C(t)-Tz-Q*solutions for ϕ =5.6° are shown in Fig. 17. All these data are extracted from the specimen with a/c=1.
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The C(t)-Tz and C(t)-Tz-Q*solutions for ϕ =90°and a/c=0.2, 0.5, 0.8 are shown in Fig. 18. Fig. 19 Shows the variations of Tz with elliptical angle for (a) t*=0.1 and (b)
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t*=10 under different radial distance, Fig. 19(c) shows the variations of Tz with time t* for different r/c. The comparisons of stress triaxiality obtained by 3D methods C(t)TZ and C(t)-TZ-Q* with 3D FE results and RR solutions are shown in Fig. 20. Good
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agreements are obtained between the 3D solutions and the FE results.
ACCEPTED MANUSCRIPT (b) 4
(a) 4 t*=0.1, =0, =90
t*=0.1, =0, =90
o
C(t)-Tz FE results
3
C(t)-Tz FE results
2
2
rr/0
/0
3
RR plane strain
RR plane strain
1
1
RR plane stress 0
o
0
10
RR plane stress
20
30
40
0
50
0
10
20
30
40
50
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r/ r/ Fig. 15. Comparisons of tensile stress and radial stress obtained by C(t)-Tz method with 3D FE
results and RR solutions (a/c=1), (a) tensile stress; (b) radial stress.
(b) 2.5
(a) 2.5 t*=10, =0, =90
2.0
o
0.5 0.0
10
1.0
RR plane strain
0.5
RR plane stress 0
rr/0
RR plane strain
C(t)-Tz-Q* FE results
1.5
RR plane stress
20
30
r/
40
0.0
50
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/0
1.0
o
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C(t)-Tz-Q* FE results
1.5
t*=10, =0, =90
2.0
0
10
20
30
40
50
r/
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Fig. 16.Comparisons of tensile stress and radial stress obtained by C(t)-Tz-Q* method with 3D FE results and RR solutions (a/c=1), (a) tensile stress; (b) radial stress.
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t*=0.1, =0, =5.6
2
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/0
3
(b) 4
o
t*=0.1, =0, =5.6
RR plane strain
C(t)-Tz FE results
2
RR plane strain
1
1
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RR plane stress
0
0
10
20
30
r/
40
o
3
C(t)-Tz FE results rr/0
(a) 4
50
0
RR plane stress 0
10
20
30
r/
40
50
ACCEPTED MANUSCRIPT (c) 2.5
(d) 2.5 t*=10, =0, =5.6
2.0
2.0
C(t)-Tz-Q* FE results RR plane strain
0.5 0.0
10
C(t)-Tz-Q* FE results
1.0
RR plane strain
0.5
RR plane stress 0
o
1.5 rr/0
/0
1.5 1.0
t*=10, =0, =5.6
o
20
30
40
0.0
50
RR plane stress 0
10
r/
20
30
40
50
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r/
Fig. 17. Comparisons of tensile stress and radial stress obtained by 3D methods C(t)-Tz and C(t)Tz-Q* with 3D FE results and RR solutions for ϕ =5.6° and a/c=1, (a) tensile stress at t*=0.1; (b) radial stress at t*=0.1; (c) tensile stress at t*=10; (d) radial stress at t*=10. (a) 4
(b) 4
RR plane strain 1
0
10
20
30
40
(c) 4
PT
AC
10
20
30
r/
40
30
40
2
C(t)-Tz-Q* FE results
1
RR plane strain
RR plane stress
0
20
50
a/c=0.5, c/W=0.2, W=10mm o t*=10, =0, =90
3
1 0
10
r/
C(t)-Tz FE results
RR plane strain
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2
0
(d) 4
a/c=0.5, c/W=0.2, W=10mm o t*=0.1, =0, =90
3
0
50
ED
r/
RR plane strain
RR plane stress
M
0
2 1
RR plane stress
/0
/0
2
3
/0
/0
3
a/c=0.2, c/W=0.2, W=10mm o t*=10, =0, =90 C(t)-Tz-Q* FE results
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a/c=0.2, c/W=0.2, W=10mm o t*=0.1, =0, =90 C(t)-Tz FE results
50
0
RR plane stress 0
10
20
30
r/
40
50
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(e) 4
/0
3 2
/0
a/c=0.8, c/W=0.2, W=10mm o t*=0.1, =0, =90 C(t)-Tz FE results RR plane strain
1
a/c=0.8, c/W=0.2, W=10mm o 3 t*=10, =0, =90 C(t)-Tz-Q* FE results 2 RR plane strain
1
RR plane stress
RR plane stress 0
0
10
20
30
40
0
50
0
10
r/
20
30
40
50
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r/
Fig.18. Comparisons of tensile stress obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions: (a) a/c=0.2, t*=0.1; (b) a/c=0.2, t*=10; (c) a/c=0.5, t*=0.1; (d) a/c=0.5, t*=10; (e) a/c=0.8, t*=0.1; (f) a/c=0.8, t*=10
0.4
r/ r/ r/
a/c=1, =0, t*=0.1
Tz
0.1 0.0 0
30
45 60 (degree)
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(c) 0.5 0.4
CE
0.3
Tz
75
90
ED
15
M
Tz
0.2
a/c=1, =0, =90
0.1
0
10
20
30
r/ r/ r/
0.2 0.1
0.0 0
a/c=1, =0, t*=10 15
30
45
60
75
90
(degree)
r/c=0.01 r/c=0.02 r/c=0.03
0.2
AC
0.4 0.3
0.3
0.0
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(b) 0.5
(a) 0.5
o
40
50
60
t*
Fig. 19. Variations of Tz with elliptical angle for (a) t*=0.1, (b) t*=10; Variations of Tz with creep time t* for different r/c.
ACCEPTED MANUSCRIPT (a) 5
(b) 5 RR plane strain
RR plane strain 4
a/c=0.2, t*=0.1, =0, =90
2 1 0
a/c=0.2, t*=10, =0, =90
3
C(t)-Tz FE results
R
R
3
4
o
C(t)-Tz-Q* FE results
2 1
RR plane stress 0
10
20
30
40
0
50
RR plane stress 0
r/
o
C(t)-Tz-Q* FE results
2
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RR plane stress 0
10
20
30
40
0
50
RR plane stress
0
10
20
30
40
50
r/
(f) 5
M
(e) 5 RR plane strain 4
a/c=0.8, t*=0.1, =0, =90
3
R
ED PT
1
20
CE
10
30
40
AC
C(t)-Tz-Q* FE results
2
0
50
RR plane stress 0
10
20
r/
30
40
50
r/
(g) 5 4
o
1
RR plane stress
0
a/c=0.8, t*=10, =0, =90
3
C(t)-Tz FE results
2
RR plane strain 4
o
(h) 5
RR plane strain
RR plane strain
a/c=1, t*=0.1, =0, =90
3
4
o
C(t)-Tz FE results
2
a/c=1, t*=10, =0, =90
3
R
R
50
1
r/
R
40
a/c=0.5, t*=10, =0, =90
3
R
C(t)-Tz FE results
1
R
4
o
2
o
C(t)-Tz-Q* FE results
2 1
1 0
30
RR plane strain
a/c=0.5, t*=0.1, =0, =90
3
0
20
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(d) 5
RR plane strain 4
10
r/
(c) 5
0
o
RR plane stress 0
10
20
30
r/
40
50
0
RR plane stress 0
10
20
30
r/
40
50
ACCEPTED MANUSCRIPT
Fig.20. Comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions: (a) a/c=0.2, t*=0.1; (b) a/c=0.2, t*=10; (c) a/c=0.5, t*=0.1; (d) a/c=0.5, t*=10; (e) a/c=0.8, t*=0.1; (f) a/c=0.8, t*=10; (g) a/c=1.0, t*=0.1; (h) a/c=1.0, t*=10
4.3 Quarter elliptic corner crack
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For the quarter circular corner crack (a/c=1), the variations of the C(t)-integral with elliptical parametrical angle are shown in Fig. 21, the crack-tip tensile stresses solved by C(t)-Tz solution, C(t)-Tz-Q* solution, RR solutions and 3D FE results for various elliptical parametrical angle ϕ under small scale creep and large scale conditions are
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shown in Fig. 22. Fig. 23 shows the variations of Tz with elliptical angle for (a) t*=0.1 and (b) t*=10, (c) variations of Tz with time t* for different r/c, (d) variations of Q* with elliptical angle for different stress components. The comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions are shown in Fig. 24.
-4
M
10 8 6
ED
C(t) (MPa*mm/h)
x10 12
4
t*=1, 2, 5, 10, 20, 50, 100
PT
2
CE
increasing t*
0
30 60 (degree)
90
Fig. 21.Variations of C(t)-integral with elliptical parametrical angle. (b) 4
AC
(a) 4
a/c=1, t*=0.1, =0, =6.1
o
a/c=1, t*=10, =0, =6.1
C(t)-Tz FE results
2
/0
/0
3
RR plane strain
1
2
C(t)-Tz-Q* FE results
1
RR plane strain
RR plane stress 0
0
o
3
10
20
30
r/
RR plane stress 40
50
0
0
10
20
30
r/
40
50
ACCEPTED MANUSCRIPT (d) 4
(c) 4 a/c=1, t*=0.1, =0, =11.9
2
RR plane strain
C(t)-Tz-Q* FE results
2
RR plane strain
1
1
RR plane stress 0
0
10
20
RR plane stress 30
40
0
50
0
10
r/
/0
RR plane strain
2
30
40
r/
(g) 4
a/c=1, t*=0.1, =0, =45
0
50
ED
RR plane strain
1
0
PT
0
10
20
30
r/
40
20
30
40
50
r/ o
3 2
C(t)-Tz-Q* FE results
1
RR plane strain
RR plane stress 0
10
a/c=1, t*=10, =0, =45
C(t)-Tz FE results
2
o
(h) 4
M
3
o
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RR plane stress 20
50
C(t)-Tz-Q* FE results
1
1
/0
3
2
10
40
a/c=1, t*=10, =0, =23.2
/0
/0
o
C(t)-Tz FE results
0
30
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a/c=1, t*=0.1, =0, =23.2
3
0
20
r/
(f) 4
(e) 4
o
3
C(t)-Tz FE results /0
/0
3
a/c=1, t*=10, =0, =11.9
o
RR plane stress 50
0
0
10
20
30
40
50
r/
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Fig. 22. Comparisons of tensile stress obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions, (a) =6.1°, t*=0.1; (b) =6.1°, t*=10; (c) =11.9°, t*=0.1; (d) =11.9°,
AC
t*=10; (e) =23.2°, t*=0.1; (f) =23.2°, t*=10;(g) =45°, t*=0.1; (h) =45°, t*=10;
ACCEPTED MANUSCRIPT
(b) 0.5
(a) 0.5 a/c=1, =0, t*=0.1
0.4 0.3
Tz
Tz
0.3 0.2
0.1
0.1 0.0 0
15
30
45 60 (degree)
75
0.0 0
90
r/ r/ r/
0.2
a/c=1, =0, t*=10 15
(d) 0.2
(c) 0.5
0.0
0.1 0.0
a/c=1, =0, =45 0
10
20
Q* -0.2
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Tz
0.3
r/c=0.01 r/c=0.02 r/c=0.03
45 60 (degree)
75
90
Tensile stress Radial stress
0.4
0.2
30
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0.4
r/ r/ r/
a/c=1, t*=10,
o
30
40
50
-0.4 0
60
15
30
45
(degree)
t*
Fig. 23. Variations of Tz with elliptical angle for (a) t*=0.1 and (b) t*=10, (c) variations of Tz with
(a) 5 RR plane strain 4
C(t)-Tz FE results
1
CE
20
30
o
C(t)-Tz-Q* FE results
2
40
50
0
0
10
r/
RR plane stress 20 30
40
50
r/
AC
10
a/c=1, t*=10, =0, =45
1
RR plane stress
0
RR plane strain
3
R
PT
R
2
5 4
o
ED
a/c=1, t*=0.1, =0, =45
3
0
M
time t* for different r/c, (d) Variations of Q* with elliptical angle for different stress components.
Fig. 24 Comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions, (a) =45°, t*=0.1; (b) =45°, t*=10;
4.4 Elliptic embedded crack For a circular embedded crack (a/c=1), the crack-tip tensile stresses obtained by C(t)Tz solution, RR solutions and 3D FE results under the small scale creep condition (t*=0.1) are shown in Fig. 25. The tensile stresses solved by C(t)-Tz-Q* solution, RR solutions and 3D FE results under the large scale creep condition (t*=10) are shown
ACCEPTED MANUSCRIPT
in Fig. 26. Fig. 27(a) shows the variations of Tz with time t* for different r/c. Fig. 27(b) shows the variations of C(t) with elliptical angle for different t*. Fig. 27(c) shows the variations of Q* with elliptical angle for different stress components. The comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D FE results and RR solutions are shown in Fig. 28. (b) 4
(a) 4
t*=0.1, =0 , =0 o
o
3
C(t)-Tz FE results rr/0
/0
3 2
RR plane strain
C(t)-Tz FE results
2
RR plane strain
1
1
RR plane stress 0
0
10
20
30
o
40
0
50
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t*=0.1, =0 , =0 o
RR plane stress 0
10
20
30
40
50
r/
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r/
Fig. 25. Comparisons of (a) tensile stress and (b) radial stress obtained by 3D methods C(t)-Tz with 3D FE results and RR solutions for ϕ =0° and t*=0.1. (a) 2.5
(b) 2.5
t*=10, =0 , =0 o
t*=10, =0 , =0 o
2.0
1.0
RR plane strain
1.0
RR plane strain
RR plane stress 0.0
0
10
20
30
C(t)-Tz-Q* FE results
0.5
ED
0.5
o
1.5
rr/0
1.5
C(t)-Tz-Q* FE results
M
/0
2.0
o
40
0.0
50
RR plane stress 0
10
r/
20
30
40
50
PT
r/
Fig. 26. Comparisons of (a) tensile stress and (b) radial stress obtained by 3D methods C(t)-Tz-Q*
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with 3D FE results and RR solutions for ϕ =0° and t*=10. (a) 0.5
(b) 0.006 r/c=0.01 r/c=0.02 r/c=0.03
Tz
0.3 0.2 0.1 0.0
0
a/c=1, =0 , =0 o
10
20
30
t*
o
40
50
60
C(t) (MPa*mm/h)
AC
0.4
t*=0.1 t*=10 t*=100
0.005
0.0005
0.0004
0
15 30 (degree)
45
ACCEPTED MANUSCRIPT (c) 0.0 t*=10, =0
Tensile stress Radial stress
o
Q*
-0.1 -0.2 -0.3
0
15 30 (degree)
45
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-0.4
Fig. 27. (a) Variations of Tz with time t* for different r/c, (b) variations of C(t) with elliptical angle for different t*, (c) variations of Q* with elliptical angle for different stress components. (b) 5
(a) 5
RR plane strain
RR plane strain
C(t)-Tz FE results
3
R
4
o
2
o
o
C(t)-Tz-Q* FE results
2 1
1
RR plane stress 0
10
20
30
r/
40
50
M
0
t*=10, =0 , =0
3
R
t*=0.1, =0 , =0 o
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4
0
RR plane stress
0
10
20
30
40
50
r/
ED
Fig. 28. Comparisons of stress triaxiality obtained by 3D methods C(t)-Tz and C(t)-Tz-Q* with 3D
PT
FE results and RR solutions, (a) =0°, t*=0.1; (b) =0°, t*=10;
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5. Conclusions
The dominance of the two-parameter C(t)-Tz and three-parameter C(t)-Tz-Q*
solutions are validated for through-the-thickness cracks, surface cracks, corner cracks
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and embedded cracks by comprehensive numerical finite element simulations. It is shown that the two-parameter C(t)-Tz solution can provide efficient prediction for the stress field ahead of the cracks under small scale creep condition. Under large scale creep conditions, the three-parameter C(t)-Tz-Q* solution can characterize the crack front stress fields efficiently, and the factor Tz shows nearly a unified distribution ahead of the creep cracks. In all the simulated situations, the 3D solutions can provide a universal characterization of the crack tip fields efficiently while the two-
ACCEPTED MANUSCRIPT dimensional solution fails. Creep fracture criterions should be developed on the basis of the 3D C(t)-Tz or C(t)-Tz-Q* solutions in the future.
Acknowledgments This work was supported by National Natural Science Foundation of China (51535005, 51472117), the Research Fund of State Key Laboratory of Mechanics and
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Control of Mechanical Structures (MCMS-0416K01, MCMS-0416G01), the Fundamental Research Funds for the Central Universities (NP2017101), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education
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Institutions.
References
Bassani J.L., McClintock F.A., 1981. Creep relaxation of stress around a crack tip. Int. J. Solids Struct. 17, 479-492.
M
Bettinson A.D., O’Dowd N.P., Nikbin K.M., Webster G.A., 2002. Experimental investigation of constraint effects on creep crack growth. Pressure Vessels and Piping Conference. ASME. 434, 143-150.
ED
Budden P.J., Ainsworth R.A., 1999. The effect of constraint on creep fracture assessments. Int. J. Fract. 97, 237-247.
PT
Ehlers R., Riedel H., 1981. A finite element analysis of creep deformation in a specimen containing macroscopic crack. ICF5, Cannes (France).
CE
Guo W., 1993a. Elastoplastic three dimensional crack border field—I. Singular structure of the field. Eng. Fract. Mech. 46, 93-104.
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Guo W., 1993b. Elastoplastic three dimensional crack border field—II. Asymptotic solution for the field. Eng. Fract. Mech. 46, 105-113. Guo W., 1995. Elasto-plastic three-dimensional crack border field—III. Fracture parameters. Eng. Fract. Mech. 51, 51-71. Kong X., Schlüter N., Dahl W., 1995. Effect of triaxial stress on mixed-mode fracture. Eng. Fract. Mech. 52, 379-388. Liu S., Wang G.Z., Xuan F.Z., Tu S.T., 2015. Three-dimensional finite element analyses of in-plane and out-of-plane creep crack-tip constraints for different specimen geometries. Eng. Fract. Mech. 133, 264-280.
ACCEPTED MANUSCRIPT Ma H.S., Wang G.Z., Liu S., Tu S.T., Xuan F.Z., 2016a. In-plane and out-of-plane unified constraint-dependent creep crack growth rate of 316H steel. Eng. Fract. Mech. 155, 88-101. Ma H.S., Wang G.Z., Xuan F.Z., Tu S.T., 2016b. Unified correlation of in-plane and out-of-plane creep constraints with creep crack growth rate. Int. J. Pressure Vessels Piping. 139–140, 47-60.
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Matvienko Y.G., Shlyannikov V.N., Boychenko N.V., 2013. In-plane and out-ofplane constraint parameters along a three-dimensional crack-front stress field under creep loading. Fatigue Fract. Eng. Mat. Struct. 36, 14–24. Nguyen B.N., Onck P., Van Der Giessen E., 2000a. Crack-tip constraint effects on creep fracture. Eng. Fract. Mech. 65, 467-490. Nguyen B.N., Onck P., Van Der Giessen E., 2000b. On higher-order crack-tip fields in creeping solids. J. Appl. Mech. 67, 372-382.
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Rice J.R., Riedel H., 1980. Tensile cracks in creeping solids. Fracture Mechanics: 12th Conference. ASTM International. 700, 112-130. Shlyannikov VN, Tumanov AV, Boychenko NV. A creep stress intensity factor approach to creep-fatigue crack growth. Engng Fract Mech 2015; 142: 201-219. Shlyannikov VN, Ilchenko BV, Boychenko NV. Biaxial loading effect on higherorder crack tip parameters, ASTM STP 1508. Fatigue Fract Mech 2009; 36: 609–640.
M
Tan J.P., Wang G.Z., Xuan F.Z., Tu S.T., 2012. Correlation of creep crack-tip constraint between axially cracked pipelines and test specimens. Int. J. Pressure Vessels Piping. 98, 16-25.
ED
Tan J.P., Tu S.T., Wang G.Z., Xuan F.Z., 2013. Effect and mechanism of out-of-plane constraint on creep crack growth behavior of a Cr–Mo–V steel. Eng. Fract. Mech. 99, 324-334.
PT
Xiang M., Yu Z., Guo W., 2011. Characterization of three-dimensional crack border fields in creeping solids. Int. J. Solids Struct. 48, 2695-2705.
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Xiang M., Guo W., 2013. Formulation of the stress fields in power law solids ahead of three-dimensional tensile cracks. Int. J. Solids Struct. 50, 3067-3088.
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Yang L., Sutton M.A., Deng X., Lyons J.S., 1997. Finite element analysis of creep fracture initiation in a model superalloy material. Int. J. Fract. 81, 299-320.