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Notch effect on creep damage for Hastelloy C276-BNi2 brazing joint
Materials and Design 84 (2015) 212–222
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Notch effect on creep damage for Hastelloy C276-BNi2 brazing joint Yun Luo a, Wenchun Jiang a,b,⁎, Weiya Zhang a, Y.C. Zhang c, W. Woo b, S.T. Tu c a b c
College of Chemical Engineering, China University of Petroleum (East China), Qingdao 266555, PR China Neutron Science Division, Korea Atomic Energy Research Institute, Daejeon 305-353, South Korea Key Laboratory of Pressure System and Safety (MOE), School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, PR China
a r t i c l e
i n f o
Article history: Received 22 April 2015 Received in revised form 3 June 2015 Accepted 12 June 2015 Available online 26 June 2015 Keywords: Brazing joint Creep damage Notch type Finite element analysis
a b s t r a c t The brazed structures have geometrical discontinuities like fillets working as notches. These notches have great effect on creep crack initiation and propagation. This paper studies the notch effect on creep damage for Hastelloy C276-BNi2 brazed joint, and the effects of notch type, notch radius and notch angle on creep damage have been investigated. The results show that the creep damage initiates in the filler metal. Different notch types bring different stress states, and generate different stress triaxialities and equivalent creep strains (CEEQs), leading to different creep damages. The maximum creep damage is generated in the notch tip for V-type notch, while the maximum creep damage is located at 0.4 mm away from the notch tip for C-type notch. For U-type notch, the location of the maximum creep damage moves from the notch tip to the inside gradually as the notch radius increases. With the increase of notch radius and notch angle, the failure time of creep damage increases for U-type and V-type notches, while it decreases for C-type notch. The creep failure is prone to happen to V-type notch because it belongs to sharp notch. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Brazed compact plate-fin heat exchangers (PFHE) with high efficiency are expected to be used as recuperator and intermediate heat exchanger for high temperature gas reactor (HTGR) [1,2]. The core of this heat exchanger is a porous structure with a number of brazed joints. It operates at high temperature and the creep strength should be ensured. Nobutada Ohno et al. [3] studied the elastic-viscoplastic behavior of plate-fin structures by a homogenous method. F. Kawashima et al. [4] studied the high temperature strength and inelastic behavior of platefin structures on the basis of equivalent-homogeneous-solid concept. In their models, the porous core is treated as an equivalenthomogeneous-solid plate. The creep strength is predicted by the stress–strain magnification factors and the stress analysis based on the equivalent-homogeneous-solid model [5]. Zhou and Tu [6] analyzed the time-dependent stresses in plate-fin structure by assuming the fins as elastic spring. The above work mainly focused on the macro-strength of platefin structure and the local stress concentration has not been considered. The plate fin structures always have geometrical
⁎ Corresponding author at: College of Chemical Engineering, China University of Petroleum (East China), Qingdao 266555, PR China. E-mail address: [email protected] (W. Jiang).
http://dx.doi.org/10.1016/j.matdes.2015.06.111 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
discontinuities like fillets working as notch. The creep cracks initiate in the fillet and propagate from these geometrical discontinuities [7]. As found by us, different brazing conditions lead to different geometrical morphologies of the fillet [8], generating different stress states. Loaded by the mechanical load and residual stress, the notch induces stress concentration and changes the stress state from uniaxial to multiaxial state. A study of notch effect on creep damage taking account of multiaxial stresses is very essential for the creep design of brazed joint, which has been addressed in this paper. Another issue is that residual stress is generated during the brazing and has a great effect on creep [9], which has also been considered in this paper. Wang et al. [10] demonstrated that the effects of residual stress on creep crack initiation time are related to the creep ductility and primary load level. Recently, Wang and Xuan et al. [11] made a deep investigation on the creep crack growth caused by stressregime dependent creep model, different constraints [12], the mismatching of creep properties [13] and initiation of crack positions [14]. Xu and Zhao et al. [15] investigated the effect of residual stress on creep crack growth behavior in ASME P92 steel, and found that a high tensile stress ahead of crack tip could stimulate the nucleation, growth and coalescence of creep voids. Sunil Goyal et al. [16,17] investigated the effect of multiaxial stress state on creep rupture behavior of 2.25Cr–1Mo steel and the rupture life of 9Cr–1Mo steel. Yu et al. [18] studied the notch effect on creep rupture of nickel-based single crystal superalloy by a double notched mini-specimen. Isobe N [19] discussed the effect of stress
Y. Luo et al. / Materials and Design 84 (2015) 212–222
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damage is clearly affected by the applied stress, mean stress and stress ratio. Most of the previous works is paid on the homogenous material or macro-welded joints, and very little attention has been paid on the brazed joints. The creep damage of brazed joints under multiaxial stress states is still unclear. The present objective is to study the notch effect on creep damage in a brazed joint by a continuum damage mechanics method, and the effect of notch type (U-type, C-type and V-type) on creep damage has been investigated. Emphasis is also placed on the effects of notch radius and angle on stress distribution and creep damage. 2. Analysis procedures In this analysis, four calculation steps were performed. The first step was to calculate the residual stress. The second step was to calculate the thermal stress when the temperature increases from 20 °C to 600 °C. In the third step, a mechanical load was applied and the total stresses were calculated. The fourth step performed a creep analysis induced by the total stresses. 2.1. Finite element model
Fig. 1. Geometry of the brazed specimen with notch.
Table 1 Chemical compositions of Hastelloy C276 (in wt.%). Element
C
Hastelloy C-276
0.02 0.08 1.00 0.03 0.03 55.49 5.5 15.5 16
Si
Mn
P
S
Ni
Fe
Cr
Mo V
Co
W
0.35 2.5 3.5
Table 2 Chemical compositions of BNi-2 (in wt.%). Element
C
Si
Ni
Fe
Cr
B
BNi-2
0.06
4.50
82.34
3.00
7.00
3.1
multiaxiality on creep damage assessment and found that the cracking area is closer to the surface in the specimen of smaller notch root radius. Jiang et al. [20,21] found that the development of creep
In this paper, the creep damage of a brazed joint with notch is studied under multiaxial stress state. Fig. 1 shows a sketching of the specimen with U-type, C-type and V-type notches. There is a critical dimension of radius R between U-type and C-type notches. If the ratio R/(b − a) N 1, the brazed joint is C-type joint. Otherwise it is U-type joint. In this model, the critical dimension is 1.0 mm. Two Hastelloy C-276 plates were brazed together by a nickel-based filler metal BNi-2. The chemical compositions of Hastelloy C-276 and BNi-2 are listed in Tables 1 [22] and 2 [23], respectively. The brazing components are heated to 500 °C at 10 °C/min and then hold about 60 min. Then it is heated to the brazing temperature 1050 °C and hold 25 min. At last, the assembly is cooled to the ambient temperature. The filler metal thickness is 100 μm. The specimen is round bar. Due to the structural symmetry, only a half of the model is built to save the computation time, and the half of the model was established by two-dimensional axisymmetric model. The finite element meshing of three different notches were shown in Fig. 2. The meshing is fine around the notch tip and then becomes coarse far away. The element type is four-node axisymmetric quadrilateral element CAX4. To study the elastic–plastic–creep behaviors of notched specimen, a tensile test of Hastelloy C276 at 600 °C is carried out and the stress–strain curve is shown in Fig. 3a. The stress–strain data of BNi-2 was calculated by linear interpolation based on Ref. [24] and is shown in Fig. 3b. The isotropic hardening law was assumed in the analysis. A nominal stress 5 MPa was loaded on the top surface
Fig. 2. Finite element meshing.
214
(a)
Y. Luo et al. / Materials and Design 84 (2015) 212–222
700 600
True stress (MPa)
500 400 300 200 100 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
True strain
(b)
250
True stress (MPa)
200
Fig. 4. Contours of damage distribution for U (a), C (b) and V-type (c) notches.
150
of the model. All the nodes on the axisymmetric section were applied the symmetric boundary conditions in X-direction, and all the nodes on the bottom section were constrained in Y-direction.
100
50
0 0.000
0.001
0.002
0.003
0.004
0.005
Ture strain Fig. 3. The curve of stress–strain for Hastelloy C276 (a) and BNI-2 (b).
Table 3 Temperature-dependent mechanical properties of Hastelloy C276 and BNi-2 [24,25]. Temperature (°C)
α is coefficient thermal expansion (10−6 mm/mm/C), E is Young's Modulus (GPa), μ is Poisson's ratio, σ is yield strength (MPa) Hastelloy C276
20 100 200 300 400 500 600 700 800 1000 1100
BNi-2
α
E
μ
σ
α
E
μ
σ
10.8 11.3 12.1 12.7 13.1 13.4 14.3 15.2 16.1 17.9 18.2
206 201 195 190 183 178 163 138 114 100 73
0.3 0.29 0.322 0.298 0.256 0.23 0.228 0.227 0.226 0.224 0.224
388 343 303 278 264 256 241 233 229 102 73
13.4 14.1 15.1 16.0 16.8 17.5 18.2 18.6 19.9 25.3 35.3
205 201 195 190 184 178 172 169 161 40 17
0.30 0.30 0.30 0.30 0.30 0.30 0.31 0.32 0.32 0.37 0.47
300 280 260 240 220 206 193 180 160 90 76
Table 4 Creep constants at 600 °C [29,30]. Material
B/MPa−n h−1
n
εf
Hastelloy C276 BNi-2
1.29×10−18 8.75 × 10−40
5.83 14.75
0.08 0.0027
Fig. 5. Contours of CEEQ for U (a), C (b) and V-type (c) notches.
Y. Luo et al. / Materials and Design 84 (2015) 212–222
2.2. Brazed residual stress analysis
β¼
At high temperature brazing, the assembly is at stress-free state. Therefore, the as-brazed residual stress is simulated during the cooling from 1050 °C to 20 °C. The total strain rate contains elastic strain, plastic strain and thermal strain. Elastic strain is modeled using the isotropic Hooke's law with temperature-dependent Young's modulus and Poisson's ratio. The thermal strain is calculated using the temperaturedependent CTE. For the plastic strain, a rate-independent plastic model is employed with Von Mises yield surface, temperaturedependent mechanical properties and isotropic hardening model. Temperature-dependent material properties are considered, as listed in Table 3 [24,25].
2ρ ð2n þ 3Þρ2 ðn þ 3Þρ3 ðn þ 3Þρ4 þ þ þ 2 3 nþ1 nðn þ 1Þ 9nðn þ 1Þ 108nðn þ 1Þ4
(a)
200 180
" #nþ1 2 3 n−1 σ1 2 Bσ eq Si j 1 þ β 2 σ eq
U-type C-type V-type
140
Initial stress (MPa)
c
ð3Þ
160
In this paper, a continuum creep damage model proposed by Wen and Tu [26] is used, which reasonably reflects the effect of multi-axial stress on creep:
ð2Þ
2ðn þ 1Þ ρ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω3=2 π 1 þ 3=n
2.3. Creep damage analysis
εi j ¼
215
120 100 80 60
ð1Þ
40 20 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.4
1.6
1.4
1.6
Distance from notch tip (mm)
(b)
(a) 100
1.0
U-type C-type V-type
80
0.8 Stress triaxiality
Creep axial stress (MPa)
90
70 60 50 40
0.6
U-type C-type V-type
0.4 0.2
notch tip 0.0
30 20
1.2
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0
0.2
Distance from notch tip (mm)
(b)
0.4
0.6
0.8
1.0
1.2
Distance from notch tip (mm)
(c) 0.0045
1.2 notch tip
0.0040
U-type C-type V-type
1.0
0.0035 0.0030 CEEQ
Damage
0.8 0.6 0.4
U-type C-type V-type
0.0020
0.2 0.0
0.0025
0.0015 0.0010
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from notch tip (mm) Fig. 6. Creep axial stress (a) and damage (b) distribution in the notch section for different notch types.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Distance from notch tip (mm) Fig. 7. The initial stress (a), stress triaxiality (b) and CEEQ (c) distribution in the notch section for different notch types.
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Fig. 8. Contours of creep damage for U-type notch with different notch radii.
where β is a stress-dependent function reflecting the material behavior,
ω¼
εe εf
ð4Þ
where ω is the damage varies from 0–0.99, and the crack initiation occurs as damage is 0.99. ε e is the equivalent creep strain rate, and ε⁎f is the multi-axial creep failure strain described by Cocks and Ashby [28]:
2 n−0:5 n−0:5 σ m = sinh 2 ¼ sinh 3 n þ 0:5 n þ 0:5 σ eq εf εf
3. Results and discussion
c
ρ is the micro-crack damage parameter, εi j is the rate of creep strain tensor, σ1 is maximum principle stress, ω denotes the damage state parameter, σeq is the Von Mises equivalent stress, Sij is the deviatoric stress, and B and n are the material constants for creep. The creep damage accumulation and creep crack initiation (CCI) ahead of a notch is expressed by the ductility exhaustion approach [27]:
ð5Þ
where σm is the hydrostatic stress, and εf is the uniaxial creep failure strain. The continuum creep-damage model has been incorporated into ABAQUS by a user subroutine CREEP compiled by FORTRAN language. In order to obtain the variables σ1 and σm at each time increment, the USDFLD subroutine has also been embedded into the ABAQUS, and the creep damage is updated at the end of the each increment. Table 4 listed the creep constants required in the calculation. The creep parameters of Hastelloy C276 at 600° were obtained from Ref. [29] at 650°. The uniaxial creep failure strain was adopted from Refs. [29,30]. The creep time is 120 h.
3.1. Creep damage distribution As we all know, the presence of notch induces a stress concentration and changes the stress state from uniaxial to multiaxial. Different notch types produce different stress distributions in the notch tip, and induce different creep damages in brazed joint. Three different notch type (U-type: Ru = 0.2 mm, C-type: Rc = 2.0 mm, V-type: θ = 60°) specimens were studied to investigate the effect of notch type on creep damage. Fig. 4 shows the contours of creep damage distribution and the creep failure time (tD represents) for different notch types. The U-type and V-type models (see Figs. 4a and c) have a similar damage distribution, showing the maximum creep damage (i.e., initial creep failure location) at the notch tip. For C-type model, the maximum creep damage generates at about 0.4 mm away from the notch tip. The creep failure times for U-type, C-type and V-type were 4.83 h, 76.84 h and 0.48 h, respectively. Hence, the failure most likely happens to V-type notch, and the next is U-type notch. Fig. 5 shows the contours of the equivalent creep strain (CEEQ). The maximum CEEQs of U-type and V-type notches are 0.0033 and 0.0026, respectively, which all locate at the notch tip. For C-type notch, the maximum CEEQ is 0.0044 which is located at 0.4 mm away from the notch tip. The location of the maximum creep damage is the same as that of the maximum CEEQ for U-type and V-type notches. Therefore, the location of the larger CEEQ is the potential failure position. Fig. 6 presents the creep axial stress (σca) and damage distribution in the notch section with different notch types. See Fig. 6a, the axial stress increases gradually along the distance from the notch tip for C-type notch, while it decreases gradually for V-type notch. For U-type notch, the axial stress increases firstly and then decreases
Fig. 9. Contours of creep axial stress of U-type notch with notch radii Ru = 0.2 mm (a), 0.5 mm (b), 0.8 mm (c) and 1.0 mm (d).
Y. Luo et al. / Materials and Design 84 (2015) 212–222
gradually. See Fig. 6b, the damage failure is located at the notch tip for U and V-type notches. For C-type notch, the maximum damage is located at 0.4 mm away from notch tip. According to Eqs (4) and (5), the creep damage is determined by CEEQ and stress triaxiality. In order to further explain this result, the initial stress, stress triaxiality and CEEQ distribution in the notch section for different notch types are shown in Fig. 7. See Fig. 7a, the initial stresses are
(a)
1.2 1.0
Ru=0.2 mm Ru=0.5 mm Ru=0.8 mm Ru=1.0 mm
Damage
0.8
0.4 0.2 0.0 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from notch tip (mm)
(b)
1.8 1.6 Ru=0.2 mm Ru=0.5 mm Ru=0.8 mm Ru=1.0 mm
1.4 Stress triaxiality
1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
concentrated at the notch tip and then decrease gradually far away. The peak stresses for U-type and V-type are 170 MPa, while the peak stresses for C-type is 130 MPa. For stress triaxiality (see Fig. 7b), the maximum stress triaxialities of U-type and V-type models are very closed to the notch tip, and the maximum CEEQ is also present at the notch tip (see Fig. 7c). Therefore, the creep failure is located at the notch tip for U-type and V-type notches. For C-type notch, the maximum stress triaxiality and CEEQ are located at 0.5 mm and 0.4 mm away from the notch tip, respectively, which generates a corresponding failure. It concludes that different notches bring different creep damages, because it generates different stress states, CEEQs and stress triaxialities.
3.2. Effect of notch radius for U-type notch
0.6
0.0
217
1.6
Keeping other parameters constant, four different U-type (Ru = 0.2, 0.5, 0.8 and 1.0 mm) specimens were studied to investigate the effect of notch radius on creep damage, and their damage contours are shown in Fig. 8. With the increase of notch radius, the failure location moves from the notch tip to the inside gradually, and creep failure time also increases from 4.83 h to 112.36 h. That is to say, smaller notch radius can induce the creep failure easier. Fig. 9 shows contours of creep axial stress distribution of U-type notch with different notch radii. With the increase of notch radius, the maximum axial stress decreases from 71.9 to 59.8 MPa and the stresses become more uniform near the notch tip, which has a great effect on the creep failure location and time. Fig. 10 shows the distributions of creep damage, stress triaxiality and CEEQ in the notch section with different notch radii. It obviously shows that different notch radii generate different stress triaxialities and CEEQ distributions. When the radius is smaller than 0.5 mm, the maximum stress triaxiality is closed to the notch tip and the maximum CEEQ is also located at the notch tip. As a result, the maximum creep damage is generated at the notch tip. When the radius is larger than 0.5 mm, the maximum stress triaxiality and CEEQ are located at 0.5 mm and 0.4 mm away from the notch tip, respectively, and therefore the maximum creep damage is located at 0.4 mm away from the notch tip. Fig. 11 shows the creep damage evolution with time for U-type notch with different notch radii. The creep failure time increases as the notch radius increases. When the notch radius is 0.2 mm, the creep failure quickly increases to failure value 0.99 within 5 h. While the notch radius is 1.0 mm, the failure time has increased to 112 h. For larger radius, the damage curve obviously contains three typical stages: first stage, steady state and tertiary state, corresponding to the three creep stages.
Distance from notch tip (mm)
(c)
0.0040 1.0
0.0035
Ru=0.2 mm Ru=0.5 mm Ru=0.8 mm Ru=1.0 mm
0.8
0.6
0.0025
Damage
CEEQ
0.0030
0.0020 0.0015 0.0010
0.4
Ru= 0.2 mm Ru= 0.5 mm Ru= 0.8 mm Ru= 1.0 mm
0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from notch tip (mm)
0.0 0
20
40
60
80
100
120
Time (h) Fig. 10. Creep damage (a), stress triaxiality (b) and CEEQ (c) distributions in the notch section for U-type notch with different notch radii.
Fig. 11. Damage evolution with time for U-type notch with different notch radii.
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Y. Luo et al. / Materials and Design 84 (2015) 212–222
Fig. 12. Contours of creep damage for C-type notch with different notch radii.
Both the failure time and creep damage have proportional relationship with notch radius for U-type notch. The smaller the notch radius, the smaller the creep failure time and the easier the creep failure happens.
the steady creep state, the larger notch radius has a bigger damage rate than the smaller notch radius, leading to the decrease of creep failure time. 3.4. Effect of notch angle for V-type notch
3.3. Effect of notch radius for C-type notch Keeping other parameters constant, four different C-type specimens with Rc = 1.2, 1.5, 1.8 and 2.0 mm were studied, and their contours of creep damage are presented in Fig. 12. It shows that the notch radius has little effect on the location of creep failure, and almost all the maximum creep damages generate at 0.4 mm away from the notch tip. The creep failure time decreases from 94.02 to 76.34 h as the notch radius increases from 1.2 to 2.0 mm. Fig. 13 shows the distribution of creep axial stress of C-type notch with different notch radii. The magnitude and distribution of creep axial stress are similar for different notch radii. Contrary to U-type notch, the maximum creep axial stress increases with the increase of notch radius, leading to the decrease of creep failure time. Similarly, the effect of notch radius on creep damage, stress triaxiality and CEEQ is shown in Fig. 14. From 0 to 0.5 mm, the damage distribution is almost the same and its peak is located at 0.4 mm away from the notch tip. But away from 0.5 mm, the damage increases with the increase of notch radius because the stress triaxiality and CEEQ increase. The maximum of stress triaxiality increases and moves away from the notch tip as notch radius increases, as shown in Fig. 14b. And the maximum CEEQ also increases as the notch radius increases, as shown in Fig. 14c. The damage evolution with time for C-type model with different radii is shown in Fig. 15. There is little difference at the first stage. When the creep enters
Keeping other parameters constant, four different V-type models with angles θ = 15°, 30°, 45° and 60° were studied, and their damage contours are shown in Fig. 16. The location of the maximum creep damage moves gradually from notch inside to the notch tip, and creep failure time increases from 0.1 to 0.48 h when notch angle increases. The smaller the notch angle, the shorter the creep failure time. Fig. 17 presents the distribution of creep axial stress with different notch angles. There is a stress concentration at the notch tip. The maximum axial stress decreases from 115 to 96 MPa as the notch angle increases from 15° to 60°. As a result, the creep failure time increases. Fig. 18 shows the creep damage, stress triaxiality and CEEQ distribution in the notch section with different notch angles. All the maximum creep damages generate at the notch tip. In total, the damage increases with the increase of the notch angle. Fig. 18b shows that the maximum stress triaxiality increases and moves to near the notch tip with the increase of the notch radius. The CEEQ also increases with the increase of the notch angle, and the peak value is located at the notch tip, as shown in Fig. 18c. There is a competition between stress triaxiality and CEEQ which ultimately determines the damage distribution. Obviously, the initiation of creep failure is in the notch tip because of the leading factor of CEEQ, as shown in Fig. 18a and c. Fig. 19 presents the damage evolution with time for different notch angles. Compared to the U-type and C-type notches,
Fig. 13. Contours of creep axial stress of C-type notch with notch radii Rc = 1.2 mm (a), 1.5 mm (b), 1.8 mm (c) and 2.0 mm (d).
Y. Luo et al. / Materials and Design 84 (2015) 212–222
(a)
1.0
1.2 1.0
Damage
Damage
0.8
Rc=1.2 mm Rc=1.5 mm Rc=1.8 mm Rc=2.0 mm
0.8 0.6
0.6
0.4
Rc=1.2 mm Rc=1.5 mm Rc=1.8 mm Rc=2.0 mm
0.4
0.2
0.2
0.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0
1.4 Rc=1.2 mm Rc=1.5 mm Rc=1.8 mm Rc=2.0 mm
Stress triaxiality
1.0
0.6 0.4 0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from notch tip (mm)
(c) 0.0048 0.0044
Rc=1.2 mm Rc=1.5 mm Rc=1.8 mm Rc=2.0 mm
0.0040 0.0036 CEEQ
60
80
100
4. Discussion
0.8
0.0032 0.0028 0.0024 0.0020
40
Fig. 15. Damage evolution with time for C-type notch with different notch radii.
1.2
0.0
20
Time (h)
Distance from notch tip (mm)
(b)
219
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from notch tip (mm) Fig. 14. Creep damage (a), stress triaxiality (b) and CEEQ (c) distributions in the notch section for C-type notch with different notch radii.
the creep failure time is very small for V-type notch. In other words, the creep failure is easy to happen in V-type notch. The creep failure time with 15° notch angle is the smallest. The creep failure time increases with the increase of notch angle. That is to say, the smaller notch angle generates the failure easier than the bigger notch angle. Therefore, the creep failure time increases with the increase of the notch angle and we can deduce that the brazed joint with higher stress concentration generates creep failure easier.
Based on the above results, it is found that different notch types, radii and angles bring different stress distributions, leading to the different distributions of creep damage. This conclusion is consistent with the results found by Jiang et al. [20,21]. The creep damage failure is mainly caused by the CEEQ and stress triaxiality. The initiation of failure location is dependent on which is the lead factor. The creep failure generates in brazed joint, which is different from the homologous specimen. This is because nickel-based filler metal has different creep parameters from the parent metal, as shown in Table 3. Tu et al. [31, 32] defined a mismatching factor to quantify the mismatching effect in creep properties and studied the influence of material mismatching on the evaluation of the time-dependent fracture mechanics parameters C(t) and C*. In the case of same creep parameter n, as the creep parameter B of weld metal is smaller than that of base metal, the weld metal is defined as creep-hard material [31]. The brazed joint is a hard joint because the creep strain rate of nickel-based filler is lower than the Hastelloy C-276. As the creep time increases, the creep strain rate of nickel-based filler is always lower than the Hastelloy C-276, leading that the stress concentration moving from base metal to filler metal. Therefore the maximum axial stress is located in brazed joint after creep, as shown in Figs. 9, 13 and 17. The brazed joint with larger stress is easy to generate failure. Therefore, the larger stress concentration on the brazed joint accelerates the creep damage. From the above analysis, the brazed joint with V-type notch is very easy to generate creep failure compared to U and C-type notches. This is because the V-type notch resembles the crack shape and lager stresses distribute at the crack tip, which favors the crack propagation [33] and the lager stress triaxiality and CEEQ are generated in the notch tip. Therefore the failure generates at the notch tip very easily. Meanwhile, the creep failure time is also very small. The notch tip acts as a stress concentrator, causing the different behaviors of creep damage in different notch types. A decrease of notch radius implies an increase of stress concentration at the notch tip. Compared to U-type and C-type notches, a larger stress concentrates at the notch tip for V-type notch, as shown in Fig. 6a. The V-type notch with smaller angle belongs to sharp crack [34]. After creep, a larger stress distributes around the crack tip, and then some areas around the crack tip are fully plasticized and yielded, and therefore the failure is generated. If the crack is much sharper, the larger stress generates at the notch tip, and then the creep failure time is much lower. The U and C-type notches are blunt notch with a relatively smaller stress at the notch tip, and the filler metal around the notch tip is not easily plasticized and yielded. Therefore, the failure is not easy to occur relatively.
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Fig. 16. Contours of creep damage for V-type notch with different notch angles.
Fig. 17. Contours of creep axial stress of V-type notch with different notch angles θ = 15°(a), 30°(b), 45°(c) and 60°(d).
Sunil Goyal et al. [17] found that the notch acuity ratio has a great influence on creep cavitation and rupture behavior. The creep damage initiated at the center of notch plane for shallow notches and at the notch root for sharp notches. In our study, the U-type notch with smaller radius has larger notch acuity ratio, and it belongs to sharp notches. Therefore, the stress concentration at the notch tip leads that the damage initiates at the notch tip. The stress triaxiality accelerates intergranular creep cavitation and reduces the creep rupture ductility, which leads that the creep failure happens more easily and even generates brittle creep rupture [17]. The C-type notch has a relative lower acuity ratio, which generates intergranular ductile failure [17]. The ductile failure is mainly determined by the plasticity, which is different from intergranular brittle failure determined by creep cavitation. Compared to C-type notch with bigger acuity ratio, the creep failure is easily generated in U-type notch with smaller acuity ratio. The brazed structures always have geometrical discontinuities like fillets working as notches, which have a great effect on creep crack initiation and propagation. Jiang et al. [8] found that different filler thicknesses generate different fillet patterns. When the filler metal is very thin, the fillet quality is very bad and the flaws are produced [8]. The brazed joint with thin filler metal would generate smaller notch acuity ratio, and then results in serious creep damage. In order to reduce the creep damage and ensure the integrity of brazed structure, a good brazing process should be promised to avoid the sharp notch.
the notch type, notch radius and notch angle have been also studied. The following conclusions have been drawn: (1) The creep damage initiates in the filler metal. Different notch types generate different stress distributions, different stress triaxialities and CEEQs, which lead to the different creep damages. (2) Compared to U and C-type notches, the creep failure is very easy to happen to V-type notch because it is sharp crack. The larger stress concentration in sharp crack tip favors the crack propagation. Therefore, the maximum damage is located at the notch tip for V-type notch, while it is located at 0.4 mm away from the notch tip for C-type notch. However, for U-type notch, the creep failure location is dependent on the notch radius. It is located at the notch tip for smaller notch radius, while it is 0.4 mm away from the notch tip for bigger notch radius. (3) Different notch radii lead to the different distributions of creep damage. For U-type notch, with the increase of notch radius, the creep failure time increases and the maximum axial stress decreases gradually. For C-type notch, the creep failure time decreases and the maximum axial stress increases with the increase of notch radius. For V-type notch, the creep failure time increases and the maximum axial stress decreases with the notch angle increase, which is similar to U-type notch.
Acknowledgments 5. Conclusions This paper discusses the notch effect on creep damage for brazed joints by using a continuum damage mechanics method. The effects of
The authors gratefully acknowledge the support provided by the National Natural Science Foundation of China (11372359), the Natural Science Foundation for Distinguished Young Scholars of Shandong
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1.0
1.2 1.0
θ=15° θ=30° θ=45° θ=60°
Damage
0.8
0.8
Damage
(a)
221
0.6
0.6
0.4
θ=15° θ=30° θ=45° θ=60°
0.4 0.2
0.2 0.0
0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.1
0.2
0.3
0.4
0.5
Time (h)
Distance from notch tip (mm)
(b)
Fig. 19. Damage evolution with time for V-type notch with different notch angles.
0.8 θ=15° θ=30° θ=45° θ=60°
Stress triaxiality
0.6 0.4
[3]
[4]
0.2 [5]
0.0 [6]
-0.2 [7]
-0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from notch tip (mm)
[8]
(c) 0.0028
[9]
θ=15° θ=30° θ=45° θ=60°
0.0024
CEEQ
0.0020
[10]
[11]
[12]
0.0016
[13]
0.0012
0.0008
[14]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from notch tip (mm) Fig. 18. Creep damage (a), stress triaxiality (b) and CEEQ (c) distributions in the notch section for V-type notch with different notch angles.
[15]
[16] [17]
[18]
Province (JQ201417), the Fundamental Research Funds for the Central Universities (12CX02006A, 14CX05036A and 15Cx08006A) and the Innovation Project Foundation for Graduate Student in China University of Petroleum (YCX2014034).
[19]
[20]
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[23] Wenchun Jiang, J.M. Gong, S.T. Tu, Effect of holding time on vacuum brazing for a stainless steel plate-fin structure, Mater. Des. 31 (2010) 2157–2162. [24] K.S. Weil, B.J. Koeppel, Comparative finite element analysis of the stress–strain states in three different bonded solid oxide fuel cell seal designs, J. Power Sources 180 (2008) 343–353. [25] Guo Yuquan, Wu Dongjiang, Ma Guangyi, Guo Dongming, Numerical simulation and experimental investigation of residual stresses and distortions in pulsed laser welding of Hastelloy C-276 thin sheets, Rare Met. Mater. Eng. 43 (2014) 2663–2668. [26] J.F. Wen, S.T. Tu, X.L. Gao, J.N. Reddy, Simulations of creep crack growth in 316 stainless steel using a novel creep-damage model, Eng. Fract. Mech. 98 (2013) 169–184. [27] M. Yatomi, K.M. Nikbin, N.P. O' Dowd, Creep crack growth prediction using a damage based approach, Int. J. Pres. Ves. Pip. 80 (2003) 573–583. [28] A.F. Cocks, M.F. Ashby, Intergranular fracture during power-law creep under multiaxial stresses, Met. Sci. 8 (1980) 395–402. [29] M.A.O. Xueping, G.U.O. Qi, H.U. Suyang, Z.H.A.N.G. Shengyuan, L.U. Daogang, Creep behaviors of Ni-based alloy C276 at high temperature, Proceed the CSEE 32 (2012) 100–105 (in Chinese).
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Materials and Design journal homepage: www.elsevier.com/locate/jmad
Notch effect on creep damage for Hastelloy C276-BNi2 brazing joint Yun Luo a, Wenchun Jiang a,b,⁎, Weiya Zhang a, Y.C. Zhang c, W. Woo b, S.T. Tu c a b c
College of Chemical Engineering, China University of Petroleum (East China), Qingdao 266555, PR China Neutron Science Division, Korea Atomic Energy Research Institute, Daejeon 305-353, South Korea Key Laboratory of Pressure System and Safety (MOE), School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, PR China
a r t i c l e
i n f o
Article history: Received 22 April 2015 Received in revised form 3 June 2015 Accepted 12 June 2015 Available online 26 June 2015 Keywords: Brazing joint Creep damage Notch type Finite element analysis
a b s t r a c t The brazed structures have geometrical discontinuities like fillets working as notches. These notches have great effect on creep crack initiation and propagation. This paper studies the notch effect on creep damage for Hastelloy C276-BNi2 brazed joint, and the effects of notch type, notch radius and notch angle on creep damage have been investigated. The results show that the creep damage initiates in the filler metal. Different notch types bring different stress states, and generate different stress triaxialities and equivalent creep strains (CEEQs), leading to different creep damages. The maximum creep damage is generated in the notch tip for V-type notch, while the maximum creep damage is located at 0.4 mm away from the notch tip for C-type notch. For U-type notch, the location of the maximum creep damage moves from the notch tip to the inside gradually as the notch radius increases. With the increase of notch radius and notch angle, the failure time of creep damage increases for U-type and V-type notches, while it decreases for C-type notch. The creep failure is prone to happen to V-type notch because it belongs to sharp notch. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Brazed compact plate-fin heat exchangers (PFHE) with high efficiency are expected to be used as recuperator and intermediate heat exchanger for high temperature gas reactor (HTGR) [1,2]. The core of this heat exchanger is a porous structure with a number of brazed joints. It operates at high temperature and the creep strength should be ensured. Nobutada Ohno et al. [3] studied the elastic-viscoplastic behavior of plate-fin structures by a homogenous method. F. Kawashima et al. [4] studied the high temperature strength and inelastic behavior of platefin structures on the basis of equivalent-homogeneous-solid concept. In their models, the porous core is treated as an equivalenthomogeneous-solid plate. The creep strength is predicted by the stress–strain magnification factors and the stress analysis based on the equivalent-homogeneous-solid model [5]. Zhou and Tu [6] analyzed the time-dependent stresses in plate-fin structure by assuming the fins as elastic spring. The above work mainly focused on the macro-strength of platefin structure and the local stress concentration has not been considered. The plate fin structures always have geometrical
⁎ Corresponding author at: College of Chemical Engineering, China University of Petroleum (East China), Qingdao 266555, PR China. E-mail address: [email protected] (W. Jiang).
http://dx.doi.org/10.1016/j.matdes.2015.06.111 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
discontinuities like fillets working as notch. The creep cracks initiate in the fillet and propagate from these geometrical discontinuities [7]. As found by us, different brazing conditions lead to different geometrical morphologies of the fillet [8], generating different stress states. Loaded by the mechanical load and residual stress, the notch induces stress concentration and changes the stress state from uniaxial to multiaxial state. A study of notch effect on creep damage taking account of multiaxial stresses is very essential for the creep design of brazed joint, which has been addressed in this paper. Another issue is that residual stress is generated during the brazing and has a great effect on creep [9], which has also been considered in this paper. Wang et al. [10] demonstrated that the effects of residual stress on creep crack initiation time are related to the creep ductility and primary load level. Recently, Wang and Xuan et al. [11] made a deep investigation on the creep crack growth caused by stressregime dependent creep model, different constraints [12], the mismatching of creep properties [13] and initiation of crack positions [14]. Xu and Zhao et al. [15] investigated the effect of residual stress on creep crack growth behavior in ASME P92 steel, and found that a high tensile stress ahead of crack tip could stimulate the nucleation, growth and coalescence of creep voids. Sunil Goyal et al. [16,17] investigated the effect of multiaxial stress state on creep rupture behavior of 2.25Cr–1Mo steel and the rupture life of 9Cr–1Mo steel. Yu et al. [18] studied the notch effect on creep rupture of nickel-based single crystal superalloy by a double notched mini-specimen. Isobe N [19] discussed the effect of stress
Y. Luo et al. / Materials and Design 84 (2015) 212–222
213
damage is clearly affected by the applied stress, mean stress and stress ratio. Most of the previous works is paid on the homogenous material or macro-welded joints, and very little attention has been paid on the brazed joints. The creep damage of brazed joints under multiaxial stress states is still unclear. The present objective is to study the notch effect on creep damage in a brazed joint by a continuum damage mechanics method, and the effect of notch type (U-type, C-type and V-type) on creep damage has been investigated. Emphasis is also placed on the effects of notch radius and angle on stress distribution and creep damage. 2. Analysis procedures In this analysis, four calculation steps were performed. The first step was to calculate the residual stress. The second step was to calculate the thermal stress when the temperature increases from 20 °C to 600 °C. In the third step, a mechanical load was applied and the total stresses were calculated. The fourth step performed a creep analysis induced by the total stresses. 2.1. Finite element model
Fig. 1. Geometry of the brazed specimen with notch.
Table 1 Chemical compositions of Hastelloy C276 (in wt.%). Element
C
Hastelloy C-276
0.02 0.08 1.00 0.03 0.03 55.49 5.5 15.5 16
Si
Mn
P
S
Ni
Fe
Cr
Mo V
Co
W
0.35 2.5 3.5
Table 2 Chemical compositions of BNi-2 (in wt.%). Element
C
Si
Ni
Fe
Cr
B
BNi-2
0.06
4.50
82.34
3.00
7.00
3.1
multiaxiality on creep damage assessment and found that the cracking area is closer to the surface in the specimen of smaller notch root radius. Jiang et al. [20,21] found that the development of creep
In this paper, the creep damage of a brazed joint with notch is studied under multiaxial stress state. Fig. 1 shows a sketching of the specimen with U-type, C-type and V-type notches. There is a critical dimension of radius R between U-type and C-type notches. If the ratio R/(b − a) N 1, the brazed joint is C-type joint. Otherwise it is U-type joint. In this model, the critical dimension is 1.0 mm. Two Hastelloy C-276 plates were brazed together by a nickel-based filler metal BNi-2. The chemical compositions of Hastelloy C-276 and BNi-2 are listed in Tables 1 [22] and 2 [23], respectively. The brazing components are heated to 500 °C at 10 °C/min and then hold about 60 min. Then it is heated to the brazing temperature 1050 °C and hold 25 min. At last, the assembly is cooled to the ambient temperature. The filler metal thickness is 100 μm. The specimen is round bar. Due to the structural symmetry, only a half of the model is built to save the computation time, and the half of the model was established by two-dimensional axisymmetric model. The finite element meshing of three different notches were shown in Fig. 2. The meshing is fine around the notch tip and then becomes coarse far away. The element type is four-node axisymmetric quadrilateral element CAX4. To study the elastic–plastic–creep behaviors of notched specimen, a tensile test of Hastelloy C276 at 600 °C is carried out and the stress–strain curve is shown in Fig. 3a. The stress–strain data of BNi-2 was calculated by linear interpolation based on Ref. [24] and is shown in Fig. 3b. The isotropic hardening law was assumed in the analysis. A nominal stress 5 MPa was loaded on the top surface
Fig. 2. Finite element meshing.
214
(a)
Y. Luo et al. / Materials and Design 84 (2015) 212–222
700 600
True stress (MPa)
500 400 300 200 100 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
True strain
(b)
250
True stress (MPa)
200
Fig. 4. Contours of damage distribution for U (a), C (b) and V-type (c) notches.
150
of the model. All the nodes on the axisymmetric section were applied the symmetric boundary conditions in X-direction, and all the nodes on the bottom section were constrained in Y-direction.
100
50
0 0.000
0.001
0.002
0.003
0.004
0.005
Ture strain Fig. 3. The curve of stress–strain for Hastelloy C276 (a) and BNI-2 (b).
Table 3 Temperature-dependent mechanical properties of Hastelloy C276 and BNi-2 [24,25]. Temperature (°C)
α is coefficient thermal expansion (10−6 mm/mm/C), E is Young's Modulus (GPa), μ is Poisson's ratio, σ is yield strength (MPa) Hastelloy C276
20 100 200 300 400 500 600 700 800 1000 1100
BNi-2
α
E
μ
σ
α
E
μ
σ
10.8 11.3 12.1 12.7 13.1 13.4 14.3 15.2 16.1 17.9 18.2
206 201 195 190 183 178 163 138 114 100 73
0.3 0.29 0.322 0.298 0.256 0.23 0.228 0.227 0.226 0.224 0.224
388 343 303 278 264 256 241 233 229 102 73
13.4 14.1 15.1 16.0 16.8 17.5 18.2 18.6 19.9 25.3 35.3
205 201 195 190 184 178 172 169 161 40 17
0.30 0.30 0.30 0.30 0.30 0.30 0.31 0.32 0.32 0.37 0.47
300 280 260 240 220 206 193 180 160 90 76
Table 4 Creep constants at 600 °C [29,30]. Material
B/MPa−n h−1
n
εf
Hastelloy C276 BNi-2
1.29×10−18 8.75 × 10−40
5.83 14.75
0.08 0.0027
Fig. 5. Contours of CEEQ for U (a), C (b) and V-type (c) notches.
Y. Luo et al. / Materials and Design 84 (2015) 212–222
2.2. Brazed residual stress analysis
β¼
At high temperature brazing, the assembly is at stress-free state. Therefore, the as-brazed residual stress is simulated during the cooling from 1050 °C to 20 °C. The total strain rate contains elastic strain, plastic strain and thermal strain. Elastic strain is modeled using the isotropic Hooke's law with temperature-dependent Young's modulus and Poisson's ratio. The thermal strain is calculated using the temperaturedependent CTE. For the plastic strain, a rate-independent plastic model is employed with Von Mises yield surface, temperaturedependent mechanical properties and isotropic hardening model. Temperature-dependent material properties are considered, as listed in Table 3 [24,25].
2ρ ð2n þ 3Þρ2 ðn þ 3Þρ3 ðn þ 3Þρ4 þ þ þ 2 3 nþ1 nðn þ 1Þ 9nðn þ 1Þ 108nðn þ 1Þ4
(a)
200 180
" #nþ1 2 3 n−1 σ1 2 Bσ eq Si j 1 þ β 2 σ eq
U-type C-type V-type
140
Initial stress (MPa)
c
ð3Þ
160
In this paper, a continuum creep damage model proposed by Wen and Tu [26] is used, which reasonably reflects the effect of multi-axial stress on creep:
ð2Þ
2ðn þ 1Þ ρ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω3=2 π 1 þ 3=n
2.3. Creep damage analysis
εi j ¼
215
120 100 80 60
ð1Þ
40 20 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.4
1.6
1.4
1.6
Distance from notch tip (mm)
(b)
(a) 100
1.0
U-type C-type V-type
80
0.8 Stress triaxiality
Creep axial stress (MPa)
90
70 60 50 40
0.6
U-type C-type V-type
0.4 0.2
notch tip 0.0
30 20
1.2
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0
0.2
Distance from notch tip (mm)
(b)
0.4
0.6
0.8
1.0
1.2
Distance from notch tip (mm)
(c) 0.0045
1.2 notch tip
0.0040
U-type C-type V-type
1.0
0.0035 0.0030 CEEQ
Damage
0.8 0.6 0.4
U-type C-type V-type
0.0020
0.2 0.0
0.0025
0.0015 0.0010
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from notch tip (mm) Fig. 6. Creep axial stress (a) and damage (b) distribution in the notch section for different notch types.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Distance from notch tip (mm) Fig. 7. The initial stress (a), stress triaxiality (b) and CEEQ (c) distribution in the notch section for different notch types.
216
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Fig. 8. Contours of creep damage for U-type notch with different notch radii.
where β is a stress-dependent function reflecting the material behavior,
ω¼
εe εf
ð4Þ
where ω is the damage varies from 0–0.99, and the crack initiation occurs as damage is 0.99. ε e is the equivalent creep strain rate, and ε⁎f is the multi-axial creep failure strain described by Cocks and Ashby [28]:
2 n−0:5 n−0:5 σ m = sinh 2 ¼ sinh 3 n þ 0:5 n þ 0:5 σ eq εf εf
3. Results and discussion
c
ρ is the micro-crack damage parameter, εi j is the rate of creep strain tensor, σ1 is maximum principle stress, ω denotes the damage state parameter, σeq is the Von Mises equivalent stress, Sij is the deviatoric stress, and B and n are the material constants for creep. The creep damage accumulation and creep crack initiation (CCI) ahead of a notch is expressed by the ductility exhaustion approach [27]:
ð5Þ
where σm is the hydrostatic stress, and εf is the uniaxial creep failure strain. The continuum creep-damage model has been incorporated into ABAQUS by a user subroutine CREEP compiled by FORTRAN language. In order to obtain the variables σ1 and σm at each time increment, the USDFLD subroutine has also been embedded into the ABAQUS, and the creep damage is updated at the end of the each increment. Table 4 listed the creep constants required in the calculation. The creep parameters of Hastelloy C276 at 600° were obtained from Ref. [29] at 650°. The uniaxial creep failure strain was adopted from Refs. [29,30]. The creep time is 120 h.
3.1. Creep damage distribution As we all know, the presence of notch induces a stress concentration and changes the stress state from uniaxial to multiaxial. Different notch types produce different stress distributions in the notch tip, and induce different creep damages in brazed joint. Three different notch type (U-type: Ru = 0.2 mm, C-type: Rc = 2.0 mm, V-type: θ = 60°) specimens were studied to investigate the effect of notch type on creep damage. Fig. 4 shows the contours of creep damage distribution and the creep failure time (tD represents) for different notch types. The U-type and V-type models (see Figs. 4a and c) have a similar damage distribution, showing the maximum creep damage (i.e., initial creep failure location) at the notch tip. For C-type model, the maximum creep damage generates at about 0.4 mm away from the notch tip. The creep failure times for U-type, C-type and V-type were 4.83 h, 76.84 h and 0.48 h, respectively. Hence, the failure most likely happens to V-type notch, and the next is U-type notch. Fig. 5 shows the contours of the equivalent creep strain (CEEQ). The maximum CEEQs of U-type and V-type notches are 0.0033 and 0.0026, respectively, which all locate at the notch tip. For C-type notch, the maximum CEEQ is 0.0044 which is located at 0.4 mm away from the notch tip. The location of the maximum creep damage is the same as that of the maximum CEEQ for U-type and V-type notches. Therefore, the location of the larger CEEQ is the potential failure position. Fig. 6 presents the creep axial stress (σca) and damage distribution in the notch section with different notch types. See Fig. 6a, the axial stress increases gradually along the distance from the notch tip for C-type notch, while it decreases gradually for V-type notch. For U-type notch, the axial stress increases firstly and then decreases
Fig. 9. Contours of creep axial stress of U-type notch with notch radii Ru = 0.2 mm (a), 0.5 mm (b), 0.8 mm (c) and 1.0 mm (d).
Y. Luo et al. / Materials and Design 84 (2015) 212–222
gradually. See Fig. 6b, the damage failure is located at the notch tip for U and V-type notches. For C-type notch, the maximum damage is located at 0.4 mm away from notch tip. According to Eqs (4) and (5), the creep damage is determined by CEEQ and stress triaxiality. In order to further explain this result, the initial stress, stress triaxiality and CEEQ distribution in the notch section for different notch types are shown in Fig. 7. See Fig. 7a, the initial stresses are
(a)
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concentrated at the notch tip and then decrease gradually far away. The peak stresses for U-type and V-type are 170 MPa, while the peak stresses for C-type is 130 MPa. For stress triaxiality (see Fig. 7b), the maximum stress triaxialities of U-type and V-type models are very closed to the notch tip, and the maximum CEEQ is also present at the notch tip (see Fig. 7c). Therefore, the creep failure is located at the notch tip for U-type and V-type notches. For C-type notch, the maximum stress triaxiality and CEEQ are located at 0.5 mm and 0.4 mm away from the notch tip, respectively, which generates a corresponding failure. It concludes that different notches bring different creep damages, because it generates different stress states, CEEQs and stress triaxialities.
3.2. Effect of notch radius for U-type notch
0.6
0.0
217
1.6
Keeping other parameters constant, four different U-type (Ru = 0.2, 0.5, 0.8 and 1.0 mm) specimens were studied to investigate the effect of notch radius on creep damage, and their damage contours are shown in Fig. 8. With the increase of notch radius, the failure location moves from the notch tip to the inside gradually, and creep failure time also increases from 4.83 h to 112.36 h. That is to say, smaller notch radius can induce the creep failure easier. Fig. 9 shows contours of creep axial stress distribution of U-type notch with different notch radii. With the increase of notch radius, the maximum axial stress decreases from 71.9 to 59.8 MPa and the stresses become more uniform near the notch tip, which has a great effect on the creep failure location and time. Fig. 10 shows the distributions of creep damage, stress triaxiality and CEEQ in the notch section with different notch radii. It obviously shows that different notch radii generate different stress triaxialities and CEEQ distributions. When the radius is smaller than 0.5 mm, the maximum stress triaxiality is closed to the notch tip and the maximum CEEQ is also located at the notch tip. As a result, the maximum creep damage is generated at the notch tip. When the radius is larger than 0.5 mm, the maximum stress triaxiality and CEEQ are located at 0.5 mm and 0.4 mm away from the notch tip, respectively, and therefore the maximum creep damage is located at 0.4 mm away from the notch tip. Fig. 11 shows the creep damage evolution with time for U-type notch with different notch radii. The creep failure time increases as the notch radius increases. When the notch radius is 0.2 mm, the creep failure quickly increases to failure value 0.99 within 5 h. While the notch radius is 1.0 mm, the failure time has increased to 112 h. For larger radius, the damage curve obviously contains three typical stages: first stage, steady state and tertiary state, corresponding to the three creep stages.
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(c)
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Time (h) Fig. 10. Creep damage (a), stress triaxiality (b) and CEEQ (c) distributions in the notch section for U-type notch with different notch radii.
Fig. 11. Damage evolution with time for U-type notch with different notch radii.
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Fig. 12. Contours of creep damage for C-type notch with different notch radii.
Both the failure time and creep damage have proportional relationship with notch radius for U-type notch. The smaller the notch radius, the smaller the creep failure time and the easier the creep failure happens.
the steady creep state, the larger notch radius has a bigger damage rate than the smaller notch radius, leading to the decrease of creep failure time. 3.4. Effect of notch angle for V-type notch
3.3. Effect of notch radius for C-type notch Keeping other parameters constant, four different C-type specimens with Rc = 1.2, 1.5, 1.8 and 2.0 mm were studied, and their contours of creep damage are presented in Fig. 12. It shows that the notch radius has little effect on the location of creep failure, and almost all the maximum creep damages generate at 0.4 mm away from the notch tip. The creep failure time decreases from 94.02 to 76.34 h as the notch radius increases from 1.2 to 2.0 mm. Fig. 13 shows the distribution of creep axial stress of C-type notch with different notch radii. The magnitude and distribution of creep axial stress are similar for different notch radii. Contrary to U-type notch, the maximum creep axial stress increases with the increase of notch radius, leading to the decrease of creep failure time. Similarly, the effect of notch radius on creep damage, stress triaxiality and CEEQ is shown in Fig. 14. From 0 to 0.5 mm, the damage distribution is almost the same and its peak is located at 0.4 mm away from the notch tip. But away from 0.5 mm, the damage increases with the increase of notch radius because the stress triaxiality and CEEQ increase. The maximum of stress triaxiality increases and moves away from the notch tip as notch radius increases, as shown in Fig. 14b. And the maximum CEEQ also increases as the notch radius increases, as shown in Fig. 14c. The damage evolution with time for C-type model with different radii is shown in Fig. 15. There is little difference at the first stage. When the creep enters
Keeping other parameters constant, four different V-type models with angles θ = 15°, 30°, 45° and 60° were studied, and their damage contours are shown in Fig. 16. The location of the maximum creep damage moves gradually from notch inside to the notch tip, and creep failure time increases from 0.1 to 0.48 h when notch angle increases. The smaller the notch angle, the shorter the creep failure time. Fig. 17 presents the distribution of creep axial stress with different notch angles. There is a stress concentration at the notch tip. The maximum axial stress decreases from 115 to 96 MPa as the notch angle increases from 15° to 60°. As a result, the creep failure time increases. Fig. 18 shows the creep damage, stress triaxiality and CEEQ distribution in the notch section with different notch angles. All the maximum creep damages generate at the notch tip. In total, the damage increases with the increase of the notch angle. Fig. 18b shows that the maximum stress triaxiality increases and moves to near the notch tip with the increase of the notch radius. The CEEQ also increases with the increase of the notch angle, and the peak value is located at the notch tip, as shown in Fig. 18c. There is a competition between stress triaxiality and CEEQ which ultimately determines the damage distribution. Obviously, the initiation of creep failure is in the notch tip because of the leading factor of CEEQ, as shown in Fig. 18a and c. Fig. 19 presents the damage evolution with time for different notch angles. Compared to the U-type and C-type notches,
Fig. 13. Contours of creep axial stress of C-type notch with notch radii Rc = 1.2 mm (a), 1.5 mm (b), 1.8 mm (c) and 2.0 mm (d).
Y. Luo et al. / Materials and Design 84 (2015) 212–222
(a)
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4. Discussion
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40
Fig. 15. Damage evolution with time for C-type notch with different notch radii.
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0.0
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Distance from notch tip (mm) Fig. 14. Creep damage (a), stress triaxiality (b) and CEEQ (c) distributions in the notch section for C-type notch with different notch radii.
the creep failure time is very small for V-type notch. In other words, the creep failure is easy to happen in V-type notch. The creep failure time with 15° notch angle is the smallest. The creep failure time increases with the increase of notch angle. That is to say, the smaller notch angle generates the failure easier than the bigger notch angle. Therefore, the creep failure time increases with the increase of the notch angle and we can deduce that the brazed joint with higher stress concentration generates creep failure easier.
Based on the above results, it is found that different notch types, radii and angles bring different stress distributions, leading to the different distributions of creep damage. This conclusion is consistent with the results found by Jiang et al. [20,21]. The creep damage failure is mainly caused by the CEEQ and stress triaxiality. The initiation of failure location is dependent on which is the lead factor. The creep failure generates in brazed joint, which is different from the homologous specimen. This is because nickel-based filler metal has different creep parameters from the parent metal, as shown in Table 3. Tu et al. [31, 32] defined a mismatching factor to quantify the mismatching effect in creep properties and studied the influence of material mismatching on the evaluation of the time-dependent fracture mechanics parameters C(t) and C*. In the case of same creep parameter n, as the creep parameter B of weld metal is smaller than that of base metal, the weld metal is defined as creep-hard material [31]. The brazed joint is a hard joint because the creep strain rate of nickel-based filler is lower than the Hastelloy C-276. As the creep time increases, the creep strain rate of nickel-based filler is always lower than the Hastelloy C-276, leading that the stress concentration moving from base metal to filler metal. Therefore the maximum axial stress is located in brazed joint after creep, as shown in Figs. 9, 13 and 17. The brazed joint with larger stress is easy to generate failure. Therefore, the larger stress concentration on the brazed joint accelerates the creep damage. From the above analysis, the brazed joint with V-type notch is very easy to generate creep failure compared to U and C-type notches. This is because the V-type notch resembles the crack shape and lager stresses distribute at the crack tip, which favors the crack propagation [33] and the lager stress triaxiality and CEEQ are generated in the notch tip. Therefore the failure generates at the notch tip very easily. Meanwhile, the creep failure time is also very small. The notch tip acts as a stress concentrator, causing the different behaviors of creep damage in different notch types. A decrease of notch radius implies an increase of stress concentration at the notch tip. Compared to U-type and C-type notches, a larger stress concentrates at the notch tip for V-type notch, as shown in Fig. 6a. The V-type notch with smaller angle belongs to sharp crack [34]. After creep, a larger stress distributes around the crack tip, and then some areas around the crack tip are fully plasticized and yielded, and therefore the failure is generated. If the crack is much sharper, the larger stress generates at the notch tip, and then the creep failure time is much lower. The U and C-type notches are blunt notch with a relatively smaller stress at the notch tip, and the filler metal around the notch tip is not easily plasticized and yielded. Therefore, the failure is not easy to occur relatively.
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Fig. 16. Contours of creep damage for V-type notch with different notch angles.
Fig. 17. Contours of creep axial stress of V-type notch with different notch angles θ = 15°(a), 30°(b), 45°(c) and 60°(d).
Sunil Goyal et al. [17] found that the notch acuity ratio has a great influence on creep cavitation and rupture behavior. The creep damage initiated at the center of notch plane for shallow notches and at the notch root for sharp notches. In our study, the U-type notch with smaller radius has larger notch acuity ratio, and it belongs to sharp notches. Therefore, the stress concentration at the notch tip leads that the damage initiates at the notch tip. The stress triaxiality accelerates intergranular creep cavitation and reduces the creep rupture ductility, which leads that the creep failure happens more easily and even generates brittle creep rupture [17]. The C-type notch has a relative lower acuity ratio, which generates intergranular ductile failure [17]. The ductile failure is mainly determined by the plasticity, which is different from intergranular brittle failure determined by creep cavitation. Compared to C-type notch with bigger acuity ratio, the creep failure is easily generated in U-type notch with smaller acuity ratio. The brazed structures always have geometrical discontinuities like fillets working as notches, which have a great effect on creep crack initiation and propagation. Jiang et al. [8] found that different filler thicknesses generate different fillet patterns. When the filler metal is very thin, the fillet quality is very bad and the flaws are produced [8]. The brazed joint with thin filler metal would generate smaller notch acuity ratio, and then results in serious creep damage. In order to reduce the creep damage and ensure the integrity of brazed structure, a good brazing process should be promised to avoid the sharp notch.
the notch type, notch radius and notch angle have been also studied. The following conclusions have been drawn: (1) The creep damage initiates in the filler metal. Different notch types generate different stress distributions, different stress triaxialities and CEEQs, which lead to the different creep damages. (2) Compared to U and C-type notches, the creep failure is very easy to happen to V-type notch because it is sharp crack. The larger stress concentration in sharp crack tip favors the crack propagation. Therefore, the maximum damage is located at the notch tip for V-type notch, while it is located at 0.4 mm away from the notch tip for C-type notch. However, for U-type notch, the creep failure location is dependent on the notch radius. It is located at the notch tip for smaller notch radius, while it is 0.4 mm away from the notch tip for bigger notch radius. (3) Different notch radii lead to the different distributions of creep damage. For U-type notch, with the increase of notch radius, the creep failure time increases and the maximum axial stress decreases gradually. For C-type notch, the creep failure time decreases and the maximum axial stress increases with the increase of notch radius. For V-type notch, the creep failure time increases and the maximum axial stress decreases with the notch angle increase, which is similar to U-type notch.
Acknowledgments 5. Conclusions This paper discusses the notch effect on creep damage for brazed joints by using a continuum damage mechanics method. The effects of
The authors gratefully acknowledge the support provided by the National Natural Science Foundation of China (11372359), the Natural Science Foundation for Distinguished Young Scholars of Shandong
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Fig. 19. Damage evolution with time for V-type notch with different notch angles.
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[3]
[4]
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Distance from notch tip (mm) Fig. 18. Creep damage (a), stress triaxiality (b) and CEEQ (c) distributions in the notch section for V-type notch with different notch angles.
[15]
[16] [17]
[18]
Province (JQ201417), the Fundamental Research Funds for the Central Universities (12CX02006A, 14CX05036A and 15Cx08006A) and the Innovation Project Foundation for Graduate Student in China University of Petroleum (YCX2014034).
[19]
[20]
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