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Fatigue crack propagation behaviour of pressurised elbow pipes under cyclic
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DOI: 10.1016/j.tws.2020.106882
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Thin–Walled Structures 154 (2020) 106882
Contents lists available at ScienceDirect
Thin-Walled Structures
journal homepage: http://www.elsevier.com/locate/tws
Fatigue crack propagation behaviour of pressurised elbow pipes under
cyclic bending
Caiming Liu a, Bingbing Li a, Yebin Cai b, Xu Chen a, *
a
b
School of Chemical Engineering and Technology, Tianjin University, Tianjin, 300072, China
College of Mechanical and Electrical Engineering, Guangdong University of Petrochemical Technology, Maoming, 525000, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
Fatigue crack propagation
Crack growth rate
Finite element method
J-integral
Pressurised elbow
Fatigue crack propagation behaviours of Z2CND18.12N austenitic stainless steel pressurised elbow pipes were
studied under cyclic bending. The results show that the evolutions of crack depth with the number of cycles at
both intrados and crown present logarithmic growth, while the evolutions of crack length with the number of
cycles present exponential growth. The aspect ratios of crack increment, Δð2cÞ=Δa versus the number of cycles at
intrados and crown are linear. J-integrals of the corresponding crack size were solved by numerical method. The
da
evolutions of crack depth growth rate, dN
with J-integral range, ΔJ at intrados and crown conform to the unified
Paris law.
1. Introduction
Currently, the common piping system of China’s nuclear power
plants is mainly composed of Z2CND18.12N austenitic stainless steels.
The nuclear pipeline is served under high temperature and high pressure
for a long time, and it is affected by cyclic mechanical load, cyclic
thermal load, random load and transient impact. The defects such as
inclusions, bubbles and discontinuities usually exist in the processing
technology of nuclear pipeline system. Microcracks often emerge from
these imperfections and gradually grow up and expand under these
complex loads. The fatigue crack growth behaviour of pipeline is related
to the safety assessment and life prediction of nuclear power system,
which is worth studying.
So far, a lot of researches have focused on ratcheting behaviour of
internal pressurised elbow and straight pipes under cyclic bending
[1–10]. However, it is also important to study the fatigue crack growth
behaviour of nuclear power pipeline. Li et al. [11] proposed an analyt­
ical method to calculate the stress intensity factor of circumferential
surface crack in steel pipes subjected to fatigue bending. It was shown
that the stress intensity factor evaluated by this method matched well
with the test results and this analysis method was beneficial to fatigue
crack growth assessment and residual fatigue life prediction of cracked
steel pipes. Arora et al. [12] studied the fatigue crack growth behaviour
of austenitic stainless steel pipe welds. The fatigue crack growth rates at
Weld Centre Line (WCL), Heat Affected Zone (HAZ), HAZ-Fusion Line
(FL) interface and the parent material were compared. The results
showed that the parent material presented superior resistance to fatigue
crack growth. The influence of internal pressure, operation time and
shape imperfection of the elbow on fatigue crack growth rate was
investigated using FE-analysis and experiments at elevated temperature
[13,14]. It was found that loading history had a significant effect on the
fatigue crack growth rate. The fatigue crack growth rate was a function
of the critical zone position of the elbow and loading history. The fatigue
property of cracked aluminum-alloy pipe repaired by a shaped CFRP
patch [15] and the fatigue crack propagation behaviours of aluminum
pipe repaired by composite patch [16,17] were studied by experimental
method and the extended finite element method. Fatigue crack growth
behaviours of power plant piping materials under corrosive environ­
ment were studied by Vishnuvardhan et al. [18]. The results showed that
the corrosion environment and the concentration of corrosion solution
had a significant effect on the fatigue crack growth rate and fatigue life
of the material. Arora et al. [19] studied the fatigue crack growth
behaviour of austenitic stainless steel pipe and its weld by two different
treatment methods of stress intensity factor, and predicted the fatigue
crack growth life of the material. A 3D finite element model was
developed to determine stress intensity factors of surface cracked girth
welded pipes under membrane tension and bending [20]. The fatigue
lives of the straight pipe and weld toe with surface crack were predicted
and the reasonable results were obtained. Jeong et al. [21] gave an
engineering evaluation method of J-integral and crack opening
* Corresponding author.
E-mail address: xchen@tju.edu.cn (X. Chen).
https://doi.org/10.1016/j.tws.2020.106882
Received 18 October 2019; Received in revised form 10 March 2020; Accepted 30 May 2020
Available online 15 June 2020
0263-8231/© 2020 Elsevier Ltd. All rights reserved.
C. Liu et al.
Thin-Walled Structures 154 (2020) 106882
Fig. 1. True wall-thickness distribution and loading mode of the elbow: (a) true wall-thickness distribution in the middle cross-section, and (b) loading mode.
displacement (COD), and the plastic influence functions were proposed
to solve the J-integral of the pipe with complex crack [22]. Vormwald
et al. [23] and rabboliniet et al. [24] characterized the crack closure
effect by comparing the difference between the global cyclic stress-strain
curve and the local cyclic stress-strain curve of the specimen. They
provided a method for calculating the effective value of J-integral range.
A weight function was proposed for the pipe with high aspect ratio
semi-elliptical crack to calculate the stress intensity factor at the deepest
point of internal circumferential semi-elliptical crack [25]. The stress
intensity factors of semi-elliptical surface cracks with large aspect ratios
of pipes were calculated by finite element method [26]. Rahman et al.
[27] considered the influence of seismic load on the fatigue crack
growth behaviour of pipeline. The results showed that, for the low-cycle
fatigue crack growth, the linear elastic fracture mechanics under­
estimated the ability of crack growth, thus the elastic-plastic analysis
was needed. Taylor et al. [28] studied the fatigue crack propagation
behaviours of API 5L X-70 steel pipe with the crack in different orien­
tations of matrix and welding consumables. Fatigue crack growth rates
of x100 steel welds in high pressure hydrogen gas considering residual
stress effects were studied by Ronevich et al. [29]. It was emphasized
that the residual stress should be eliminated when measuring the fatigue
crack growth rate and the influence of residual stress should be
considered when conducting the structural assessment.
Generally, for complex structures, the crack periphery is subject to
stronger and more complex constraints and it is affected by the irregular
shape of elastic-plastic boundary. Thus, the fatigue crack propagation
behaviour of the structure is different from that of the material, and the
fracture features of the structure cannot be expressed by the fracture
mechanics properties of the material. Therefore, the significance of the
fatigue crack propagation behaviour of the nuclear power pipeline sys­
tem cannot be ignored. This study is devoted to the fatigue crack
propagation behaviour of Z2CND18.12N austenitic stainless steel elbow
pipe under constant internal pressure of 17.5 MPa and in-plane bending
load (the crack at intrados: 10~20 kN, and the crack at crown: -10~-20
kN). The fatigue crack propagation tests of the cracks at intrados and
crown of the elbow were designed under notch induction. The evolu­
tions of crack depth, crack length, crack depth growth rate and crack
Table 1
Material properties of Z2CND18.12N austenitic stainless
steel.
Young’s modulus，E
195 GPa
Yield strength，σy
300 MPa
Ultimate strength，σu
590 MPa
Elongation，ψ
52%
Poisson’s ratio，v
0.3
length growth rate with the number of cycles were evaluated. The
relation between crack depth and crack length and the macroscopic
fracture surfaces of cracks were analyzed. Based on the experimental
results, J-integrals of corresponding crack sizes were calculated ac­
cording to Ramberg-Osgood equation by numerical method. The evo­
da
lution laws of crack depth growth rate, dN
with J-integral range, ΔJ were
evaluated.
2. Specimen and experiment details
The elbow pipe specimen used in this study is shown in Fig. 1(a). It
consists of an elbow and two straight pipes. Its nominal cross-section
specification is ϕ76 � 4:5 mm. Actually, the wall-thickness of the
elbow is not uniform due to processing technology. In previous study
[6], the wall-thickness distribution of this elbow has been characterized
in detail. Fig. 1(a) shows the true wall-thickness distribution of the
elbow in the middle cross-section. During the experiment, the load was
applied at straight pipe ends through the connection block and loading
bar, as shown in Fig. 1(b).
The whole elbow pipe is made of Z2CND18.12N austenitic stainless
steel. It is a common material with higher yield strength of 300 MPa and
strength limit of 590 MPa. The basic mechanical properties of
Z2CND18.12N austenitic stainless steel are shown in Table 1. Table 2
shows its chemical composition.
Before the experiment, a total of two elbow pipe specimens were
pretreated. A longitudinal notch with the length, width and depth of 10
2
C. Liu et al.
Thin-Walled Structures 154 (2020) 106882
Table 2
Chemical composition of Z2CND18.12N (in wt %).
Chemical composition
C
Si
Mn
P
S
Ni
Cr
Mo
N
Cu
Co
%
0.025
0.43
1.211
0.021
0.003
12.07
17.517
2.388
0.07
0.075
0.035
Fig. 2. Notch specification of the elbow pipe: (a) at intrados, and (b) at crown.
Fig. 3. Fatigue crack propagation tests of the elbow pipes for the cracks at intrados and crown, respectively: (a) at intrados, and (b) at crown.
mm, 1 mm and 1 mm, respectively was machined at intrados of one
specimen, as shown in Fig. 2(a). While the other specimen was machined
a longitudinal notch with the length, width and depth of 20 mm, 1 mm
and 1 mm, respectively at crown, as shown in Fig. 2(b). During the
experiment, the stress concentration will occur at the notch front and the
initial crack will be induced to initiate from here.
Fatigue crack prefabrication and fatigue crack propagation tests
were carried out using fatigue test machine. Its load capacity is 10 tons,
and comes with data real-time monitoring system and hydraulic control
system. Fatigue crack propagation tests of the cracks at intrados and
crown are shown in Fig. 3(a) and (b), respectively. Both in the fatigue
crack prefabrication stage and the test stage, the internal pressure 17.5
MPa remained unchanged and was applied at the inner surface of the
elbow pipe by the internal pressure branch system. During the fatigue
crack prefabrication stage for the crack at intrados, in-plane cyclic
bending load of 10~20 kN was provided with the frequency of 1 Hz.
While for the crack at crown, in-plane cyclic bending load of -10~-20 kN
was provided with the same frequency of 1 Hz. The fatigue crack pre­
fabrication stage continued until an obvious macroscopic crack was
observed near the notch.
During the experiment in this study, the crack length can be
observed, as shown in Fig. 5, but the crack depth is not measurable. In
order to determine the size of crack depth under the corresponding
number of cycles. The “crack front marking technique” was adopted.
That is, first of all, a certain number of loading cycles are carried out
under design load (high cyclic load). Subsequently, keep the load ratio
unchanged, and reduce the load to 1/3–1/2 of the original load (low
cyclic load) and then loading for another certain number of cycles to
form “fatigue crack contour” in crack front. The experiment is alter­
nately conducted in this way until the end of the fatigue crack propa­
gation. After the experiment, the crack size can be obtained according to
these “fatigue crack contours” in the fracture surface. The “fatigue crack
contours” formed in Fig. 6. Therefore, the crack size will be conveniently
characterized. During the fatigue crack propagation test for the crack at
intrados of the elbow, the in-plane cyclic bending load of 10~20 kN of
20,000 cycles with the frequency of 1 Hz and the load of 5~10 kN of
40,000 cycles with the frequency of 2 Hz were alternately applied. While
for the crack at crown of the elbow, the in-plane cyclic bending load of
-10~-20 kN of 20,000 cycles with the frequency of 1 Hz and the load of
-5~-10 kN of 40,000 cycles with the frequency of 2 Hz were alternately
applied. The internal pressure of 17.5 MPa remained unchanged
throughout the whole test process. The design pressure of this elbow in
China’s nuclear power plant is 17.5 MPa, so this internal pressure is
used. Table 3 shows the details of notch and test condition. The
maximum loads of 20 kN for the crack at intrados and -20 kN for the
crack at crown were referred to Ref. [2–7]. The stress distributions at
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C. Liu et al.
Thin-Walled Structures 154 (2020) 106882
Fig. 4. Stress distribution at intrados and crown: (a) longitudinal stress at intrados, (b) circumferential stress at intrados, (c) longitudinal stress at crown, and (d)
circumferential stress at crown.
intrados and crown under the maximum load were also given in the
following section. Low loads of 5~10 kN for the crack at intrados and
-5~-10 kN for the crack at crown were determined by the
pre-experiment. Under low loads, the crack did not propagate during the
experiment by the observation of crack length, and its purpose was only
to form “fatigue crack contours” on the fracture surface, as shown in
Fig. 6. Therefore, the load ratio of 0.5 was used. It was time-consuming
to study the fatigue crack growth behaviuor of the elbow. Thus, the
loading frequency was expected to be higher during the experiment. The
loading frequencies were determined by considering the loading ca­
pacity of the testing machine, and 1 Hz for high loads and 2 Hz for low
loads were used.
The notch form and loading method are mainly based on the
following finite element analysis results, which are from Refs. [6], as
shown in Fig. 4. It can be observed from Fig. 4(a) and (b) that, at intrados
(under 20 kN and 17.5 MPa), the circumferential stress level at outer
surface of the elbow is higher than the longitudinal stress level. The
maximum longitudinal stress does not appear at intrados. However, the
maximum circumferential stress appears on the outer surface of the
elbow, and decreases from the outer surface to the inner surface. Simi­
larly, it can be observed from Fig. 4(c) and (d) that, at crown (under -20
kN and 17.5 MPa), the circumferential stress level at outer surface of the
elbow is higher than the longitudinal stress level, and the maximum
longitudinal stress does not appear at crown. Also, the maximum value
of circumferential stress appears on the outer surface of the elbow, and it
decreases from the outer surface to the inner surface, and the inner
surface is compressive stress. Thus, the longitudinal outer surface
notches are used in both intrados and crown under corresponding loads.
3. Experimental results and discussion
The fatigue crack propagation tests of the longitudinal crack at
intrados and crown continued until the crack penetrated the inner sur­
face of the elbow, and the hydraulic oil leaked out from the notch. Fig. 5
(a) and (b) show the ultimate test stages of the crack at intrados and
crown, respectively. During the experiment, the crack depth is not
measurable, but the crack length can be observed, as shown in Fig. 5.
Through the observation of the crack length, the crack does not extend
under low cyclic load. Furthermore, it can be seen from Fig. 6 that the
“fatigue crack contour” does not form a wide band, but a very thin line.
It can also be determined that the low cyclic load has little contribution
to the fatigue crack growth. Thus, the lower load of 5~10 kN (intrados)
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C. Liu et al.
Thin-Walled Structures 154 (2020) 106882
Fig. 5. The ultimate test stage of the crack at (a) intrados and (b) crown, respectively.
and -5~-10 kN (crown) can not be accounted for the post-processing of
crack propagation, which are only aimed at forming the “fatigue crack
contours” in the fracture surface, as shown in Fig. 6. In other words, the
higher load of 10~20 kN (intrados) and -10~-20 kN (crown) dominate
the propagation of fatigue crack. At intrados, the fatigue crack propa­
gation process experienced 160,000 low load cycles and 79,980 high
load cycles. At crown, the test experienced 400,000 low load cycles and
198,000 high load cycles.
extending to the middle of the section are also obvious, which are easy to
be inferred. According to the trend of the “fatigue crack contours”, the
contours at the middle section are inferred, as shown in Fig. 6(b2).
Compared with the macroscopic fracture surface at intrados, the fracture
surface at crown is rough and not smooth, which is the mixed fracture
modes (mode I and mode II). As shown in Fig. 4(c) and (d), although the
circumferential stress is greater than the longitudinal stress at crown and
it is the dominant stress driving crack growth. However, due to the crack
at crown are asymmetric with respect to the elbow, as shown in Fig. 2
(b). Thus, its cracking mode is not purely opening mode (mode I), but
affected by some sliding (mode II), which is a mixed fracture mode. As
indicated by a red circle in Fig. 6(b1), the inner surface of the elbow also
occurs a small area crack growth zone. This is also due to the crack
initiation on the inner surface of the elbow at the later stage of fatigue
crack growth.
3.1. Morphology of fracture surface
After the tests, the fatigue crack propagation region was cut off from
the elbow. Fracture surface was separated after liquid nitrogen freezing
treatment. The macroscopic fracture surfaces at intrados and crown
were obtained, as shown in Fig. 6(a) and (b), respectively. At intrados
shown in Fig. 6(a), fatigue crack prefabricated zone and “fatigue crack
contours” are clearly visible. The morphology of fracture surface is
smooth without wrinkle and dislocation, which is the opening fracture
mode (mode I). Firstly, the crack at intrados is symmetrical with respect
to the elbow, as shown in Fig. 2(a). Besides, as shown in Fig. 4(a) and
(b), the circumferential stress is greater than the longitudinal stress at
intrados, which is the dominant stress driving crack growth. Therefore,
that causes an opening fracture mode (mode I) for the crack at intrados.
In addition, it can be observed from the fracture surface as indicated by a
red circle in Fig. 6(a), the inner surface of the elbow also occurs crack
initiation, which results in a small area crack growth zone. However,
because the growth area is very small, so it is speculated that which
occurs in the later stage of fatigue crack growth.
At crown shown in Fig. 6(b1), the “fatigue crack contours” at the
middle section of the fracture surface are not clear and defective. While
the “fatigue crack contours” at both ends of the fracture surface can be
clearly distinguished by naked eye observation. One of the most
reasonable explanations is that the crack at crown includes mode II
cracking, which causes the “fatigue crack contours” to be worn. In fact,
first of all, in the middle of the fracture surface, the curvatures of the
“fatigue crack contours” are relatively small, basically close to the
straight-line segments, which are easy to be described. Moreover, the
trends of the “fatigue crack contours” at both ends of the section
3.2. Evolution of crack size
The evolution of crack size with the number of cycles can be obtained
from the “fatigue crack contours” on the fracture surface, as shown in
Fig. 6. Fig. 7(a) shows the evolution of crack depth with the number of
cycles. The crack depth at intrados starts at 1.85 mm, and eventually
evolves to 4.41 mm after 79,980 cycles. Also, the crack depth at crown
starts at 2.48 mm, and eventually evolves to 4.02 mm after 198,000
cycles. It is obvious that the crack depth at intrados develops faster than
that at crown. However, their evolution trends are consistent and the
crack depth growth rate decreases with the number of cycles. The
change trends of crack depth present the following logarithmic growth
patterns.
�
a ¼ A1 InðN þ B1 Þ þ C1
(1)
a0 ¼ A1 InB1 þ C1
where a is crack depth (mm), N is the number of cycles (10,000 cycles),
and a0 is the initial crack depth (mm). A1 , B1 and C1 are constants and
they are obtained by fitting the experimental data. At intrados, A1 ¼
1:3650, B1 ¼ 1:3645 and C1 ¼ 1:4258. At crown, A1 ¼ 0:4594, B1 ¼
0:7173 and C1 ¼ 2:6326.
Fig. 7(b) shows the evolution of crack length with the number of
5
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Thin-Walled Structures 154 (2020) 106882
Fig. 6. Macroscopic fracture surface: (a) intrados, (b1) original at crown, and (b2) the result inferred.
A2 , B2 and C2 are constants and they are obtained by fitting the exper­
imental data. At intrados, A2 ¼ 10:1098, B2 ¼ 0:1538 and C2 ¼
0:3519. At crown, A2 ¼ 6:7746 � 106 , B2 ¼ 1:9215 � 10 7 and C2 ¼
6:7746 � 106 .
Table 3
Details of notch and test condition.
No.
Notch
location
Notch size
(L � W �
D)
Alternate
load
Load
ratio
Cycles
Frequency
(Hz)
1
Intrados
10 � 1 � 1
0.5
2
Crown
20 � 1 � 1
10~20 kN
5~10 kN
-10~-20
kN
-5~-10 kN
20,000
40,000
20,000
1
2
1
40,000
2
0.5
3.3. Evolution of crack growth rate
Crack growth rate can be calculated by the derivative of Eq. (1) and
Eq. (2). The evolutions of crack growth rate with the number of cycles
are shown in Fig. 8. Fig. 8(a) shows the crack depth growth rate, and it is
indicated that the crack depth growth rates at both intrados and crown
of the elbow gradually decreases with the number of cycles. The change
trends of two locations are almost consistent. In the early stage of crack
propagation, the crack depth growth rate decreases sharply in the short
term. In the later stage, the crack depth growth rate decreases slowly
with the number of cycles. Fig. 8(b) shows the change of crack length
growth rate with the number of cycles, and it is observed that the evo­
lutions of the cracks at intrados and crown are obviously different. The
crack length growth rate at intrados rapidly increases with the number
of cycles while the crack length growth at crown is close to uniform rate.
Overall, the crack growth rates at intrados maintain a higher level both
in depth direction and length direction compared with the crack at
cycles. The crack length at intrados starts at 10.00 mm, and eventually
evolves to 34.15 mm after 79,980 cycles. Also, the crack length at crown
starts at 22.01 mm, and eventually evolves to 47.93 mm after 198,000
cycles. It is observed that their evolution trends are slightly different.
The crack at intrados presents accelerated growth while the crack at
crown presents uniform change. The evolution of crack length can be
expressed in the following exponential growth patterns.
2c ¼ A2 eB2 N þ C2
(2)
where 2c is crack length (mm), N is the number of cycles (10,000 cycles).
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Thin-Walled Structures 154 (2020) 106882
Fig. 7. Evolutions of crack size with the number of cycles: (a) crack depth, and (b) crack length.
Fig. 8. Evolutions of crack growth rate with the number of cycles: (a) crack depth growth rate, and (b) crack length growth rate.
crown.
the initial crack size, as shown in Fig. 9(b). They conform to the
following power equation.
3.4. Relation between crack depth and crack length
Δð2cÞ ¼ A4 ðΔaÞB4
Generally, for a component, if there is surface crack, it will extend in
the length direction and depth direction. The crack length is easy to be
found and observed, but the crack depth is difficult to be detected. If the
relation between crack depth and crack length is obtained, the crack
depth can be deduced according to the measured crack length, thus the
failure time of the component can be evaluated. So it is necessary to
study the relation between crack depth and crack length. Fig. 9(a) gives
the relations between crack length and crack depth at intrados and
crown, and they conform to the following power equation.
2c ¼ A3 aB3 þ C3
(4)
where Δð2cÞ is crack length increment (mm) and Δa is crack depth
increment (mm). A4 and B4 are constants. At intrados, A4 ¼ 0:5928 and
B4 ¼ 3:9068. At crown, A4 ¼ 7:3577 and B4 ¼ 2:8784.
In addition, the crack increment aspect ratio, Δð2cÞ=Δa is calculated,
and the relations between the increment aspect ratio and the number of
cycles are evaluated, as shown in Fig. 9(c). The increment aspect ratios
versus the number of cycles at intrados and crown are unified and they
are consistent with linear growth.
4. Numerical process and results
(3)
In order to explain the crack growth law and evaluate the fracture
characteristics of the elbow pipes, the stress analysis and fracture
toughness were evaluated by finite element method. The large-scale
finite element software ANSYS 15.0 was used [30].
where 2c is crack length (mm) and a is crack depth (mm). A3 , B3 and C3
are constants and they are obtained by fitting the experimental data. At
intrados, A3 ¼ 1:8458 � 10 4 , B3 ¼ 7:8997 and C3 ¼ 10:8704. At
crown, A3 ¼ 2:4496 � 10 4 , B3 ¼ 8:3274 and C3 ¼ 21:5129.
Actually, the relations between the crack depth and length described
above are affected by the initial crack size. In order to avoid this effect,
the relations between crack depth increment, Δa and crack length
increment, Δð2cÞ need to be evaluated. That is, all crack sizes subtract
4.1. Ramberg-Osgood equation
The stress-strain curve of the material was obtained from the uniaxial
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Thin-Walled Structures 154 (2020) 106882
Fig. 9. Relation between crack depth and length: (a) crack size, (b) crack size increment, and (c) increment aspect ratio with the number of cycles.
tensile test of the tensile specimen that cut off from the straight pipe
section of the elbow pipe. Ramberg-Osgood equation is widely used to
describe the stress-strain behaviour of the material and solve the frac­
ture toughness of the components. Its definition is as follows [31]:
σ � σ �1n
ε¼ þ
E
K
(5)
where E is the Young’s modulus. σ and ε are stress and strain variables
respectively. K and n are hardening coefficient and hardening exponent,
respectively. They can be determined by fitting the experimental data of
uniaxial tensile curve of the material and K ¼ 671:745 (MPa), n ¼
0:112, respectively. Fig. 10 shows the stress-strain behaviour of
Z2CND18.12N austenitic stainless steel and Ramberg-Osgood relation­
ship fitting. It is observed that Ramberg-Osgood equation can describe
the uniaxial tensile property of this material. For fatigue crack growth,
transient cyclic stress-strain curve is better for accurate numerical
analysis. But there is no such method available in ANSYS. It should be
noted that, Z2CND18.12N austenitic stainless steel is a slightly cyclic
hardening material at small strain [32], thus, under the same strain, the
stress level of tensile stress-strain curve is slightly lower than that of
transient cyclic stress-strain curve.
4.2. Finite element model
Fig. 10. Stress-strain behaviour of Z2CND18.12 N austenitic stainless steel and
Ramberg-Osgood fitting.
The three-dimensional finite element model of the elbow was
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Thin-Walled Structures 154 (2020) 106882
Fig. 11. Finite element models of the crack at intrados and crown: (a) crack at intrados, and (b) crack at crown.
Fig. 12. Stress distributions of the first group of crack sizes at intrados: (a) longitudinal stress, (b) circumferential stress, and (c) von-Mises stress.
carefully designed before stress analysis and fracture parameter solu­
tion. Due to the symmetry of the crack with respect to the elbow, only
the 1/4 model was established for the elbow with the crack at intrados,
and 1/2 model was established for the elbow with the crack at crown.
Fig. 11(a) and (b) show their finite element models respectively. Sin­
gular mesh was adopted at crack front, and which can more precisely
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Thin-Walled Structures 154 (2020) 106882
Fig. 13. Stress distributions of the first group of crack sizes at crown: (a) longitudinal stress, (b) circumferential stress, and (c) von-Mises stress.
describe the singularity of stress and strain at crack front. The loose
regular hexahedral mesh was adopted far away from the crack front. The
mesh independence was verified in the simulation process. Considering
the calculation efficiency, the element size of crack front at intrados was
set to 0.04, and the element size of crack front at crown was set to 0.2.
The symmetrical constraints were set on the symmetrical surface of the
elbow, and a constant internal pressure of 17.5 MPa was applied on the
inner surface of the elbow pipe. The load was applied at one end of the
elbow. Then the stress analysis was carried out and the J-integral range
was obtained by subtracting the minimum value of J-integral from the
maximum value of J-integral.
where σij is stress tensor, uj is displacement vector, w is strain energy
density, δji is kronecker symbol, xi is coordinate system, q is crack
propagation vector, ne is number of integrated elements, wiw is
weighting function, Aie is the area of element ie. The calculation is
realized by the convert to integral (CINT) command in ANSYS.
For the crack at intrados, a total of four groups of crack sizes are used
to solve the J-integrals, which are (length, depth): (10.00, 1.85), (13.00,
3.00), (18.35, 3.81) and (25.36, 4.20), respectively. For the crack at
crown, a total of five groups of crack sizes are used, which are (length,
depth): (22.01, 2.48), (26.98, 3.37), (33.31, 3.63), (37.99, 3.80) and
(42.66, 3.92), respectively. The final crack sizes at intrados and crown
which have penetrated the inner surface of the elbow are not included.
Fig. 12(a), (b) and (c) give the longitudinal stress, circumferential
stress and von-Mises stress distributions of the first group of crack sizes
at intrados under the maximum load (20 kN and 17.5 MPa), respec­
tively. Fig. 13(a), (b) and (c) show the longitudinal stress, circumfer­
ential stress and von-Mises stress distributions of the first group of crack
sizes at crown under the maximum load (-20 kN and 17.5 MPa). It can be
observed that there is an obvious stress concentration in crack front both
at intrados and crown, and the circumferential stress level is higher than
the other two stresses. In fact, circumferential stress plays one of the
most important roles in the process of crack growth. So it is necessary to
analyze it more deeply. Fig. 14(a) gives the evolutions of circumferential
4.3. Numerical results and analysis
Crack front contains multiple nodes for the three-dimensional crack,
and the J-integral varies along the crack front. The J-integral of the
deepest point is evaluated here, which is also the maximum value of the
J-integral [33]. The discrete form of domain integral is used to solve
J-integral in ANSYS [34], as follows
�
ne �
X
∂u
∂q
J¼
σ ij j wδji
wiw Aie
(6)
∂xi
∂xi
ie¼1
10
C. Liu et al.
Thin-Walled Structures 154 (2020) 106882
Fig. 14. (a) Evolutions of circumferential stress at the deepest point (Dep. P) and the end point (End. P) of the crack with the number of cycles, and (b) numerical
results of the relation between J-integral range and crack depth growth rate.
stress at the deepest point (Dep. P) and the end point (End. P) of the
crack with the number of cycles. It can be seen that the circumferential
stress at intrados is greater than that at crown, no matter at the deepest
point or at the end point of the crack. This is also a main reason why the
crack growth rates at intrados are faster than that at crown, as shown in
Fig. 8(a) and (b). Furthermore, at intrados, the circumferential stress at
the deepest point of the crack decreases with the increase of the number
of cycles (crack size), while the circumferential stress at the end point of
the crack increases with the increase of the number of cycles (crack size).
This is an important reason that the crack depth growth rate at intrados
is getting smaller and smaller, while the crack length growth rate is
getting larger and larger, as shown in Fig. 8(a) and (b). At crown, the
circumferential stress at the deepest point of the crack decreases with
the increase of the number of cycles (crack size), while the circumfer­
ential stress at the end point of the crack is basically stable with the
increase of the number of cycles (crack size). This is a main reason for
the decrease in crack depth growth rate and for the inapparent change in
crack length growth rate, as shown in Fig. 8(a) and (b).
Fig. 14(b) gives the numerical results of the relation between J-in­
tegral range and crack depth growth rate. The evolution of crack depth
da
growth rate, dN
with J-integral range, ΔJ conforms to the Paris Law as
follows:
da
¼ CðΔJÞm
dN
Paris Law can reasonably describe fatigue crack growth law of this
elbow.
5. Conclusions
In this study, the fatigue crack propagation behaviours of
Z2CND18.12N austenitic stainless steel elbow pipe were investigated
experimentally. Based on the test results, the stress analysis and J-in­
tegrals of the elbow were numerically evaluated by finite element
method. The conclusions are as follows:
(1) Both the cracks at intrados and crown, the evolutions of crack
depth with the number of cycles present logarithmic growth. The
evolutions of crack length with the number of cycles present
exponential growth. The crack depth growth rates of these two
locations gradually decrease with the increase of the number of
cycles. The crack length growth rate at intrados gradually in­
creases with the increase of the number of cycles, while the crack
length growth rate at crown remains unchanged. The relations of
crack depth with crack length at both intrados and crown are
consistent with power variation, while the aspect ratios of crack
increment, Δð2cÞ=Δa versus the number of cycles at intrados and
crown are unified and consistent with linear growth.
(2) There is a significant difference between the fracture surface of
the cracks at the intrados and crown. The fracture surface at
intrados is smooth and presents a typical opening fracture mode
(mode I). While the fracture surface at crown is rough and pre­
sents a mixed fracture mode (mode I and mode II).
(3) The relations of crack depth growth rate and J-integral range at
da
intrados and crown conform to the unified Paris law, dN
¼
CðΔJÞm .
(7)
where C ¼ 1:7781 � 10 8 and m ¼ 2:1177, respectively.
The results show that although the cracks are located in different
locations of the elbow, the crack growth laws are unified. As the driving
energy of crack growth, J-integral can unify the crack growth laws of
different locations on the same elbow. The crack growth law shown in
Fig. 14(b) is different from that in Fig. 8. This is because the crack
growth law given in Fig. 8 is the change of crack growth rate with the
number of cycles, in which the crack driving energy at intrados and
crown are different, thus resulting in different crack growth laws for the
cracks at two different locations. If the differences in driving energy are
characterized at intrados and crown, the crack depth growth rates versus
J-integral ranges are obtained, as shown in Fig. 14(b). Thus the crack
growth laws at intrados and crown are unified. It can be found that the
Declaration of competing interest
None.
11
C. Liu et al.
Thin-Walled Structures 154 (2020) 106882
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Caiming Liu: Writing - original draft, Data curation, Software.
Bingbing Li: Investigation. Yebin Cai: Validation. Xu Chen: Concep­
tualization, Writing - review & editing, Supervision.
Acknowledgment
The authors gratefully acknowledge the National Key Research and
Development Program of China (No. 2018YFC0808600).
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://doi.
org/10.1016/j.tws.2020.106882.
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