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Numerical ductile fracture prediction of circumferential through-wall cracked pipes under very low cycle fatigue loading condition
Engineering Fracture Mechanics xxx (xxxx) xxx–xxx
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Numerical ductile fracture prediction of circumferential throughwall cracked pipes under very low cycle fatigue loading condition ⁎
Hyun-Suk Nama, Jong-Min Leea, Yun-Jae Kima, , Jin-Weon Kimb a b
Department of Mechanical Engineering, Korea University, Anam-Dong, Sungbuk-Ku, Seoul 136-701, Republic of Korea Department of Nuclear Engineering, Chosun University, Seosuk-Dong, Dong-Gu, Gwangju 501-759, Republic of Korea
AR TI CLE I NF O
AB S T R A CT
Keywords: Ductile crack growth simulation Multi-axial fracture strain energy Finite element damage analysis Pipe test Very low-cycle fatigue
In this paper, a method to simulate ductile crack growth under very low-cycle fatigue loading condition is given and simulation results are compared with published test data of compact tension specimens and circumferential through-wall cracked pipes. The damage model in simulation is based on the multi-axial fracture strain energy. Two parameters in the damage model are determined from tensile and fracture toughness data under the monotonic loading condition. The determined damage model is then used to simulate ductile crack growth in compact tension specimens subjected to cyclic loading with large-amplitudes and in full-scale through-wall cracked pipes subjected to monotonic and cyclic loading with two different load ratios. Predicted results show a good agreement with experimental results.
1. Introduction Structural integrity analysis of piping components in nuclear power plants under seismic loading conditions is a challenging issue. Seismic loading can be characterized by dynamic and cyclic loading. It has been experimentally shown that material tensile properties can be affected by dynamic (strain rate) effect [1–8]. However, it has been also shown that the strain rate effect is minimal on fracture toughness properties and piping component fracture behaviours [6–8]. For instance, In Refs. [6,7], it has been shown that the strain rate effect appears in tensile test results, but is not clearly seen in fracture toughness and full-scale pipe tests. In contrast, the cyclic loading effect on piping component fracture behaviour has been shown to be very significant. Specimen and full-scale pipe tests have been performed to quantify the cyclic effect on fracture toughness and on cracked full-scale pipe behaviours [9–14]. Test results showed that the maximum loads under reverse cyclic loading were lower than those under the monotonic loading condition. Furthermore, “apparent” fracture resistance curves (based on the positive load-load line displacement curve) were significantly decreased when subject to reversible cyclic loads, compared to that under monotonic loading condition. Seok [9] reported that the decrease in the J-R curve is related to the increased stain-hardening and generation of tensile stress at the crack tip during cyclic loading. Although most of existing works reported such phenomenon, reasons are still not clear. There has been an analytical attempt to quantify crack growth under reverse cyclic loading with large amplitude using combination of ductile tearing and low cycle fatigue crack growth [15,16]. Although experimental data can provide valuable information, testing under various cyclic loading conditions can be costly due to larger test matrices. Finite element (FE) damage simulation would be an efficient tool to perform failure analyses of complex geometries under various loading conditions. Some works for ductile crack growth simulation under cyclic loading conditions have been reported in literature [17–19]. Sherry and Wilkes [17] and Klingbeil et al. [18] reported ductile fracture simulation of compact
⁎
Corresponding author. E-mail address: [email protected] (Y.-J. Kim).
https://doi.org/10.1016/j.engfracmech.2018.02.025 Received 13 June 2017; Received in revised form 18 February 2018; Accepted 26 February 2018 0013-7944/ © 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: Nam, H., Engineering Fracture Mechanics (2018), https://doi.org/10.1016/j.engfracmech.2018.02.025
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Nomenclature
t Δa σe, σm
pipe thickness crack growth effective stress and mean normal stress, respectively σ1, σ2, σ3 principal stress components Δω, ω, ωc incremental, accumulated and critical damage, respectively δ displacement increment εpl equivalent plastic strain σy, σyo yield strength and initial yield strength
A, B, C
material constants in multi-axial facture strain energy locus, Eq. (2) crack length and its initial value a, a0 Ci, γi (i = 1–3) material constants in kinematic hardening model, Eq. (8) E elastic modulus J J-integral Le finite element size Wp equivalent plastic strain energy Wf fracture strain energy Pmax, Pmin maximum and minimum load in cyclic loading R load ratio = Pmin/Pmax Q, b material constants in isotropic hardening model, Eq. (7) rm, ro mean and outer radius of pipe
Abbreviation C(T) CMOD FE LLD
compact tension crack mouth opening displacement finite element load-line displacement
tension (C(T)) test under cyclic loading conditions based on the Gurson-based modeling. The model was determined using notch tensile and fracture toughness test under the monotonic loading condition. In Ref. [17], an isotropic hardening model was considered, whereas in Ref. [18], the kinematic hardening model was considered for cyclic hardening properties. Based on the determined damage model, the C(T) test under cyclic loading conditions could be simulated relatively well. However, the Gurson-based damage model has many parameters to be determined, which is often a difficult task. Khoei et al. [19] presented ductile crack growth simulation under cyclic and dynamic loading using a damage-viscoplasticity model. To define the cyclic hardening properties, ratedependent constitutive model and combined hardening model were considered. The damage model was determined from tensile and notch tensile test results under monotonic loading condition. Several tests using Arcan, C(T) and double-notched tests under dynamic and cyclic loading conditions were simulated using the determined damage model. Note also that the proposed damage-viscoplasticity model has many parameters and determination of those parameters requires many tests and elaborate calibration procedures. In this paper, ductile crack growth in C(T) specimens as well as in circumferential through-wall cracked pipes under very lowcycle fatigue loading condition is simulated using FE damage analysis and simulation results are compared with published experimental data. The damage model for simulation is based on the multi-axial fracture strain energy [20] which is an extension of the multi-axial fracture strain model previously used by the authors [8,21–23]. The damage model has only two parameters which can be determined from tensile and fracture toughness data under the monotonic loading condition. The determined damage model is then used to simulate ductile crack growth in C(T) specimens subjected to reverse cyclic loading with large-amplitudes and in circumferential through-wall cracked pipes subjected to monotonic and cyclic loading. Section 2 summarizes published mechanical test and circumferential through-wall cracked pipe test data under different load ratios. Section 3 introduces the energy-based damage model and explains how to determine the damage model and failure criterion for ductile crack growth simulation. In Section 4, simulation results of C(T) test under cyclic loading conditions are compared with experimental data. In Section 5, simulation results of full-scale pipe test under monotonic and cyclic loading conditions are compared with experimental data. Section 6 conclude the presented work.
2. Summary of experiments 2.1. Material and tensile test results Published mechanical and full-scale cracked pipe test data under monotonic and cyclic loading conditions, performed in Battelle Memorial Institute and compiled in Ref. [6], are considered in this work. All tests were conducted at quasi-static rates at 288 °C. The material was A106 Gr. B carbon steel extracted from a pipe with the 152 mm nominal diameter, typically used in nuclear piping of Light Water Reactors. The chemical composition of A106 Gr. B steel is given in Table 1. Tensile properties at 288 °C were obtained using a smooth round bar specimen with the 5.08 mm diameter and 12.7 mm gauge Table 1 Chemical composition of A106 Gr. B [6]. (wt%) C
Mn
P
S
Si
Cu
Sn
Ni
Cr
Mo
Al
0.15
0.65
0.012
0.014
0.20
0.28
0.018
0.14
0.18
0.055
0.010
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Fig. 1. (a) Schematic illustration of smooth bar and (b) FE mesh of the specimen.
length (Fig. 1a). Tensile test was conducted at quasi-static rates and the axial displacement was measured using extensometer. Experimental engineering stress-strain curves are shown in Fig. 2a. The 0.2% proof (yield) strength, ultimate tensile strength and reduction of area were 320 MPa, 621 MPa and 34.4%, respectively. To characterize cyclic tensile properties of A106 Gr. B, low cycle fatigue test was conducted under the strain controlled condition (0.35% strain amplitude range). Test conditions such as the specimen geometry and loading rate are the same as those in monotonic tensile tests. Fig. 2b shows the stable hysteresis loop. 2.2. Monotonic and cyclic C(T) test results Three fracture toughness (J-resistance curve) tests were performed at 288 °C. In accordance with the ASTM E813-81 standard [24], the 0.5 T C(T) specimen with the thickness of 11.18 mm was used. The specimen were machined in the L-C direction and had no side groove. Fracture toughness tests were conducted under monotonic and cyclic loading conditions. For monotonic loading, monotonicallyincreasing tensile load was applied to the specimen. In cyclic tests, incremental cyclic load was applied to the specimen, as schematically shown in Fig. 3a. In the loading sequence, displacement was controlled at the loading step, whereas load was controlled at the unloading step to maintain constant load ratio (R = Pmin/Pmax). Two load ratios were used in the test; R = −0.8 and −1.0. During loading sequence, a constant displacement increment of 0.08 mm was applied after each cycle, as shown in Fig. 3a. Tests were performed in a servo-hydraulic machine with loading rate (1.5 mm/min) at 288 °C. During tests, the direct-current electric potential method was applied to measure crack initiation and subsequent growth. The monotonic J-R curve was calculated according to the ASTM E813-81 standard [6]. For cyclic tests, cyclic J-R curves were calculated using the area under load-load line displacement (LLD) curve above zero load, as shown Fig. 3b. More detailed procedure for evaluating the fracture toughness under the cyclic loading condition can be found in Ref. [6]. Fig. 4a shows experimental load versus load-line displacement (LLD) curves from fracture tests of two materials. Resulting J-R and LLD-Δa curves are shown in Fig. 4b and c. 2.3. Pipe test results Circumferential through-wall cracked pipe test results of A106 Gr. B under monotonic and cyclic loading condition were also obtained from Ref. [6]. Three pipes having identical pipe/crack dimensions were tested. The outer radius of the pipe was
Fig. 2. Comparison of experimental engineering stress–strain data with simulated results: (a) monotonic loading condition and (b) cyclic loading condition. Experimental data were taken from Ref. [6].
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Fig. 3. Schematic illustration of (a) cyclic loading sequence and (b) load-load line displacement (LLD) area to calculate cyclic J-integral.
Fig. 4. Monotonic and cyclic C(T) test results of A106 Gr. B: (a) load-LLD curves, (b) J-R curves and (c) LLD-Δa curves [6].
ro = ∼84 mm with the thickness of t = ∼14.0 mm. A circumferential through-wall crack of ∼37 percent of the pipe circumference was machined into the pipe specimen prior to fatigue pre-crack. The pipe test specimen is schematically shown in Fig. 5a. The cracked pipe was subjected to four-point (pure) bending at 288 °C, as schematically shown in Fig. 5b. Similarly to the fracture toughness test, monotonic and cyclic loading was applied to the cracked pipe. Monotonic loading test corresponds to the load ratio R = 1.0 (designated as the (1-2-7) test). Cyclic loading sequence was the same as that for the fracture toughness test. Two load ratios, R = 0.0 (designated as the (1-2-2) test) and R = −1.0 (fully reversed, designated as the (1-2-4) test), were tested. The test speed of all 4
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Fig. 5. (a) Schematics of the cracked pipe specimens (the x-point in the figure indicate the clip gage location for the crack mouth opening displacement (CMOD) measurement and (b) Schematics of the test equipment.
experiment was slower than 0.1 mm/s. Crack growth was estimated using direct-current electric potential probes located in the centerline. Crack growth was found to be out-of-plane and the average crack extension was projected back to the initial crack plane. Measured load-LLD curves and LLD-Δa curve are shown in Fig. 6a and 6b. Crack mouth opening displacement (CMOD) data were also measured at the center and tip of the through-wall crack, as indicated in Fig. 5(a) using the cross symbol. Measured CMOD-LLD curves are shown in Fig. 7. More detailed information of pipe tests can be found in Ref. [6].
3. Determintion of damage model for ductile crack growth simulation 3.1. Overview of the method In this paper, a strain-energy based ductile fracture model is used to simulate the monotonic and cyclic fracture toughness tests, where incremental damage is defined by 5
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Fig. 6. Measured pipe test results: (a) load-LLD curves and (b) LLD-Δa curves [6].
Fig. 7. Measured CMOD-LLD curves from pipe tests: (a) the (1-2-7) test (R = 1), (b) the (1-2-2) test (R = 0) and (c) the (1-2-4) test (R = −1) [6].
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Δω =
ΔWp Wf
(1)
In Eq. (1), Wp and Wf denotes equivalent plastic strain energy and the multi-axial fracture strain energy, respectively. The multi-axial fracture strain energy, Wf, is assumed to be given by
σ Wf = Aexp ⎛−C m ⎞ + B σe ⎠ ⎝ ⎜
⎟
(2)
where σm and σe denote the mean normal stress and the equivalent stress, respectively; A, B and C are material constants. To determine these three material constants, at least three tests with different triaxial stress states would be needed. When such test data are not available, Eq. (2) can be simplified as follows (see Appendix A)
σ Wf = Aexp ⎛−1.5(n + 1) m ⎞ + B σe ⎠ ⎝ ⎜
⎟
(3)
where n in Eq. (3) is the plastic strain hardening exponent. When accumulated damage becomes critical, i.e.,
ω=
∑ Δω = ωc
(4)
local ductile failure is assumed and incremental crack growth is simulated by reducing all stress components and the value of Young’s modulus at the gauss point where the failure condition is satisfied. This simulation technique has been implemented in the ABAQUS STANDARD commercial FE program [25] using user subroutines. Further details information on the failure simulation technique can be found in our previous works [8,20–23,26]. For ductile crack growth simulation under cyclic loading, this paper assumes that the multi-axial fracture strain energy, Wf, determined from monotonic loading can be applied to cyclic loading condition. Furthermore the critical damage ωc for monotonic loading is assumed to be also applied to cyclic loading conditions:
ω ⎛⎜= ∑ Δωmono = ⎝
∑ Δωcyclic = ∑
ΔWp ⎞ ⎟
Wf ⎠
= ωc
(5)
Above assumptions seem plausible in the sense that total energy for ductile fracture remains constant regardless of loading conditions. However, when Eq. (4) is applied to cyclic loading, one caution should be made in calculating incremental damage under compressive loading. This will be explained later in Section 4.2. 3.2. Determination of damage model under monotonic loading condition To determine the multi-axial fracture strain energy, Eq. (3) was considered in this work due to limited experimental test data. Tensile (stress–strain) curve gives n = 0.236 in Eq. (3). To determine two material constants in Eq. (3), A and B, elastic–plastic FE analysis of tensile test and FE damage analysis of the C(T) test were performed. Step-by-step procedures to determine these two constants are given below.
Fig. 8. (a) Variation of the stress triaxiality with equivalent plastic strain energy for A106 Gr. B and (b) multi-axial fracture strain energy locus for A106 Gr. B.
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The first step was to perform elastic–plastic FE analysis of smooth bar tensile test. For simulation, the FE mesh shown in Fig. 1b was used. Symmetric conditions were considered and the first order solid elements (C3D8 with in ABAQUS [25]) were used. The element size in the minimum cross section where the necking occurs was fixed to 0.1 mm but the use of the smaller element size gave the same results. The true stress-strain curve was used together with the large geometry change option and the J2 flow theory. Simulated results are compared with the experimental results in Fig. 2a, showing that FE results agree well with the experimental data. Variation of the stress triaixiality with the plastic strain energy, extracted from FE results up to the failure initiation point, is shown with solid line in Fig. 8a and the last point in Fig. 8a corresponds to the experimental failure point shown in Fig. 2a. The stress triaxiality increases with increasing plastic strain energy after necking. To reflect the history effect of stress and strain, an averaged stress triaxiality value was calculated, as shown using the dotted line in Fig. 8a. One point corresponding to the fracture strain energy versus the stress triaxiality is shown in Fig. 8b using open circle. The second step is to determine one more constant in Eq. (3) and the critical accumulated damage parameter ωc from FE damage simulation results of the fracture toughness test under monotonic loading condition. Three-dimensional (3-D) FE damage analysis was performed using a quarter model considering symmetrical conditions. The first order solid element with full integrations (C3D8 within ABAQUS [25]) was uniformly spaced in the crack propagation region, as shown in Fig. 9a. The element size in the crack propagation area was fixed to Le = 0.2 mm and the mesh had 12,478 elements and 14,640 nodes. For tensile properties, true stressstrain data were directly used. The large geometry change option was chosen with the J2 flow theory. The incremental damage was calculated using Eq. (1) with the assumed multi-axial fracture strain energy (the first trial is to assume the fracture strain energy value at σm/σe = 2.5 to be 10% of the uni-axial value). By matching experimental fracture initiation toughness, a proper value of the critical accumulated damage, ωc, was chosen. Then the value of the fracture strain energy at σm/σe = 2.5 was tuned to match the tearing modulus of the J-R curve. It would be useful to note that the predicted initiation toughness value was controlled mainly by the ωc value, whereas predicted tearing modulus was governed by the material constant B. A final expression of the multi-axial fracture strain energy was given by
J σ ⎤ = 599exp ⎛−1.854 m ⎞ + 29 Wf ⎡ σe ⎠ ⎣ mm3 ⎦ ⎝ ⎜
⎟
(6)
Fig. 9. Typical 3D FE meshes of simulate 0.5 T C(T) test: (a) the element size Le = 0.2 mm and (b) Le = 0.6 mm.
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Fig. 10. Comparison of experimental J-R curve with simulated results using two different element sizes, Le = 0.2 mm and Le = 0.6 mm.
Table 2 Parameters in the non-linear kinematic hardening model of A106 Gr. B. Material
Q (MPa)
b
σo (MPa)
C1
γ1
C2
γ2
C3
γ3
A106 Gr. b
150
6
150
110,000
1000
20,000
100
1000
10
with ωc = 0.97. This locus is shown in Fig. 8b and simulated J-R curves are compared with experimental J-R curve in Fig. 10. Simulated results agree relatively well with the experimental data. For practical application of ductile crack growth simulation of a large-scale piping component, the use of a small finite element can cause numerical difficulties. As damage parameters depend on the element size, they should be properly determined for a given element size. For the present model, it is assumed that the ωc value is set to be a function of an element size. To find an appropriate value of ωc for a different element size, the 0.6 mm element size in the crack propagation region was prepared, as shown in Fig. 9b. Then the same procedure as before was taken to determine the ωc value for Le = 0.6 mm, by matching predicted toughness value with experimental one. It was found that the appropriate value of ωc = 0.4 for Le = 0.6 mm, which is smaller than ωc = 0.97 for Le = 0.2 mm. Predicted J-R curves using Le = 0.6 mm with ωc = 0.4 are compared with experimental J-R curve in Fig. 10. 4. Ductile crack growth simulation of C(T) test under cyclic loading conditions 4.1. Cyclic material properties In this paper, a combined hardening model embedded within ABAQUS [25] was used to characterize cyclic material properties. In the model, isotropic hardening is considered by the amount of expansion taken to be a function of accumulated plastic strain: pl
σy = σyo + f (ε pl ) = σyo + Q (1−e−bε )
(7)
where σyo is the initial yield strength and f(ε ) denotes the isotropic hardening function with material constants Q and b. To consider kinematic hardening, the ‘decomposed’ nonlinear kinematic hardening rule proposed by Chaboche [27–29] is used, where the evolution of the back stress is given by pl
3
dα =
∑ i=1
1/2 2 2 Ci dε pl−γi αi dε pl where dε pl = |dε pl| = ⎡ dε pl·dε pl⎤ 3 3 ⎣ ⎦
(8)
In Eq. (8), C and γ are material constants; C is the initial kinematic hardening modulus and γ is the rate at which the kinematic hardening modulus decreases with increasing plastic deformation. Based on the cyclic tensile test data described in Section 2.1, the material parameters in the combined hardening model for A106 Gr. B were determined as follows. Firstly, the value of C3 was determined from the slope of the linear segment of hysteresis curve at
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Fig. 11. Comparison of C(T) test data with simulated results for A106 Gr. B: (a)–(b) load-LLD curves, (c) J-R curves and (d) LLD-Δa curves.
high strain ranges. Secondly, values of C1/γ1 and C2/γ2 were estimated. Note that the value of C1/γ1 should be very large to match the plastic modulus at yielding. The value of C2/α2 should be match the transient nonlinear portion of the stable hysteresis curve. For more detailed determination procedure, see Refs. [30,31]. Values of the material parameters are summarized in Table 2. For validation, cyclic tensile test was simulated using ABAQUS. Simulated results are compared with tests results in Fig. 2b, showing good agreement.
4.2. Simulation of C(T) test under cyclic loading conditions To perform ductile crack growth simulation under cyclic loading conditions, the damage model determined from mechanical test data under monotonic loading (the multi-axial fracture strain energy locus, Wf, and the critical accumulated damage parameter, ωc) was used according to the explanations given in Section 3.1. A caution should be made when calculating the incremental damage under reverse cyclic loading. Under the cyclic loading, equivalent plastic strain energy is positive even when the crack closes under compressive loading. In the present work, the incremental damage is calculated only when the maximum principal stress at the crack tip is positive. In simulation, the FE model used for the monotonic loading condition (Fig. 9) was used, having the 0.2 mm element size in crack propagation region. The multi-axial fracture strain energy locus given in Eq. (6) with ωc = 0.97 was used For strain hardening, combined hardening rule explained in Section 4.1 was used, with the parameters given in Table 2. Simulated load-LLD and J-R curves are compared with experimental data in Fig. 11. Note that J-R curves from FE simulations are calculated using the same method as in experiments. Comparison with experimental data shows overall good agreements.
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Fig. 12. 3-D FE mesh to simulate ductile crack growth in through-wall cracked pipe tests.
Fig. 13. Comparison of circumferential through-wall cracked pipe data under monotonic loading condition with simulated results: (a) load-LLD curve, (b) LLD-Δa curve and (c) CMOD-LLD curves.
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Fig. 14. Comparison of circumferential through-wall cracked pipe data under fully reversed cyclic loading condition (R = −1) with simulated results: (a) load-LLD curve, (b) LLD-Δa curve and CMOD-LLD curves (c) at crack center and (d) at crack tip.
5. Ductile crack growth simulation of circumferential through-wall cracked pipe test under monotonic and cyclic loading conditions To simulate ductile crack growth in circumferential through-wall cracked pipe tests under monotonic and cyclic loading conditions, explained in Section 2.3, 3-D FE damage analysis was performed. To reflect symmetric conditions, a quarter model was used and the first order solid elements of Le = 0.6 mm (C3D8 within ABAQUS [25]) were uniformly spaced in the cracked section. To accommodate crack closure, the contact surface were added in the symmetric plane, as shown in Fig. 12. For the damage model, the fracture strain energy given in Eq. (6) with ωc = 0.4 (for t Le = 0.6 mm) was used. The FE mesh had 43,190 elements and 48,900 nodes. For tensile properties, the true stress+strain curve shown in Fig. 2(a) was used for the monotonic loading case and the combined hardening model given in Section 4.1 was used for the cyclic loading cases. The non-linear geometry change option was used with the J2 flow theory. In Fig. 13, simulated results for the monotonic loading case are compared with experimental data; the load versus LLD, crack extension versus LLD and CMOD versus LLD curves. Predicted results agree overall well with experimental data. Corresponding results for cyclic pipe test results of the R = −1 (fully reversed loading) case are compared with experimental data in Fig. 14. Predicted results also agree overall well with experimental data. Finally corresponding results for the R = 0 case are shown in Fig. 15. Predicted load versus LLD results agree well with experimental data. For crack extension versus LLD results, good agreement can be seen up to 30 mm crack growth. Predicted CMOD versus LLD curves for the crack tip are in good agreement but for the crack center are much lower than experimental data. On the other hand, when experimental data for R = 0 are compared with those for R = 1.0
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Fig. 15. Comparison of circumferential through-wall cracked pipe data under cyclic loading condition (R = 0) with simulated results: (a) load-LLD curve, (b) LLD-Δa curve and CMOD-LLD curves (c) at the crack center and (d) at the crack tip.
and R = -1.0, it can be seen that the results for R = 0 are unusually large, suggesting possible error in measurement during the test. Fig. 16 compares fracture surfaces at the maximum load point in each test, resulting from FE simulations. The cracked surface is shown using the darker color. In all cases, crack tunneling can be clearly seen. 6. Conclusion This paper presents ductile crack growth simulation in compact tension specimens and in cracked pipes under very low-cycle fatigue loading condition and results are compared with published experimental data. For experimental data, four-point bending tests of circumferential through-wall cracked A106 Gr. B pipes at 288 °C, performed by Battelle institute, were taken. For simulation, the damage model based on the multi-axial fracture strain, previously proposed by the authors [8,21–23], is extended to a model based on the multi-axial fracture strain energy. The damage model has two parameters which are determined from tensile and fracture toughness data under the monotonic loading condition. Due to such simplicity of the proposed model, it can be easily applied to practical situations where material data for calibrating parameters in the damage model are limited. The determined damage model is then used to simulate ductile crack growth in compact tension specimens subjected to cyclic loading with large-amplitudes and in full-scale through-wall cracked pipes subjected to monotonic and cyclic loading with two different load ratios. To improve computational efficiency and numerical stability, relatively large element size is used in cracked pipe simulation. The element size effect on damage simulation is accommodated by the critical damage value in the present damage model. Predicted results show a good agreement with experimental results.
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Fig. 16. Simulated crack surface at the maximum load: (a) monotonic loading condition, (b) cyclic loading condition with R = 0 and (c) cyclic loading condition with R = -1.
Acknowledgement This research was supported by National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B2009759) and by the Nuclear Power Core Technology Development Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), the Ministry of Trade, Industry & Energy, Republic of Korea (No. 20141520100860). Appendix A. Simplified method to determine the material constants for multi-axial fracture strain energy It is well known that the fracture strain for ductile fracture depends strongly on the triaxial stress states [32–35]. The triaxial stress dependence on fracture strain (the multi-axial fracture strain, ɛf) is typically quantified by the following exponential equation
σ εf = α exp ⎛−γ m ⎞ + β σe ⎠ ⎝ ⎜
⎟
(A1)
where α, β and γ are material constants determined from several tests with different triaxial stress states. When test data are not sufficient, Eq. (A1) can be simplified by assuming some of constants. For instance, Rice and Tracey suggested that the constant γ is approximately 1.5 [32], leading to
σ εf = α exp ⎛−1.5 m ⎞ + β σe ⎠ ⎝ ⎜
⎟
(A2)
For power-law plastic materials, the plastic strain energy Wp up to the fracture strain εf can be written as
Wp =
∫ σ0 (εp)ndεp =
σ0 (εp)n + 1 n+1
εf
= 0
σ0 (εf )n + 1 n+1
(A3)
This suggests that the multi-axial fracture strain energy equivalent to Eq. (A2) is given by
σ Wf = Aexp ⎛−1.5(n + 1) m ⎞ + B σe ⎠ ⎝ ⎜
⎟
(A4) 14
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References [1] Anvari M, Liu J, Thaulow C. Dynamic ductile fracture in aluminum round bars: experiments and simulation. Int J Fract 2007;143:317–32. [2] Anvari M, Scheider I, Thaulow C. Simulation of dynamic ductile crack growth using strain-rate and triaxiality-dependent cohesive model. Engng Fract Mech 2006;73:2210–28. [3] Ren ZJ, Ru CQ. Numerical investigation of speed dependent dynamic fracture toughness of line pipe steels. Engng Fract Mech 2013;99:214–22. [4] Yu PS, Ru CQ. Strain rate effects on dynamics fracture of pipeline steels: finite element simulation. Int J Pres Ves Pip 2015;126–127:1–7. [5] Scheider I, Nonn A, Volling A, Mondry A, Kalwa C. A damage mechanics based evaluation of dynamic fracture resistance in gas pipelines. Proc Mater Sci 2014;3:1956–64. [6] Pipe Fracture Encyclopedia, vol. 3. Pipe fracture test data. Columbus (Battelle, OH, USA); 1997. [7] Kim JW, Choi MR, Kim YJ. Effect of loading rate on the fracture behavior of nuclear piping materials under cyclic loading condition. Nucl Eng Tech 2016;48:1376–86. [8] Nam HS, Kim JS, Ryu HW, Kim YJ, Kim JW. Numerical ductile crack growth simulation of circumferential cracked pipe test under dynamic loading condition. Nucl Eng Tech 2016;48:1252–63. [9] Seok CS, Murty KL. A study on the decrease of fracture resistance curve under reversed cyclic loading. Int J Pres Ves Pip 2000;77:303–11. [10] Singh PK, Ranganath VR, Tarafder S, Pradad P, Bhasin V, Vaze KK, et al. Effect of cyclic loading on elastic-plastic fracture resistance of PHT system piping material of PHWR. Int J Pres Ves Pip 2003;80:745–52. [11] Roy H, Sivaprasad S, Tarafder S, Ray KK. Monotonic vis-à-vis cyclic fracture behavior of AISI 304LN stainless steel. Eng Fract Mech 2009;76:1822–32. [12] Chowdhury T, Sivaprasad S, Bar HN, Tarafder S, Bandyopadhyay NR. Cyclic fracture behavior of 20MnMoNi55 steel at room and elevated temperatures. Fatig Fract Eng Mater Struct 2015;38:813–27. [13] Mogami K, Hayashi T, Ando K, Ogura N. Elastic-plastic fatigue-crack growth and tearing instability behavior under cyclic loads. Int J Pres Ves Pip 1990;44:85–97. [14] Tranchand B, Chapuliot S, Aubin V, Marie S, Bourgeois M. Ductile fracture analysis under large amplitude cycles. In: Proceedings of the 2014 ASME pressure vessels and piping conference, paper PVP2014-28426; 2014. [15] Yamaguchi Y, Katsuyama J, Onizawa K, Sugino H, Li Y. Investigation on evaluation method based on J-integral for retardation of crack growth due to excessive loading beyond small scale yielding condition, Proceedings of the 2010 ASME Pressure Vessels and Piping Conference, Paper PVP2010-22563; 2010. [16] Yamaguchi Y, Katsuyama J, Onizawa K, Li Y. Evaluation of excessive loading effect on fatigue crack growth behavior based on crack blunting and stress distribution in front of the crack tip. In: Proceedings of the 2013 ASME pressure vessels and piping conference, paper PVP2013-97939; 2013. [17] Sherry AH, Wilkes MA. Numerical simulation of tearing-fatigue interactions in 316l(N) austenitic stainless steel. Int J Pres Ves Pip 2005;82:905–16. [18] Klingbeil D, Svendsen B, Reusch F. Gurson-based modelling of ductile damage and failure during cyclic loading processes at large deformation. Engng Fract Mech 2016;160:95–123. [19] Khoei AR, Eghbalian M, Azadi H, Saffar H. Numerical simulation of ductile crack growth under cyclic and dynamic loading with a damage-viscoplasticity model. Eng Fract Mech 2013;99:169–90. [20] Nam HS, Kim YJ, Kim JW, Kim JS. Energy-based numerical modelling of the strain rate effect on fracture toughness of SA508 Gr. 1a. J Strain Anal 2017;52:177–89. [21] Nam HS, Oh YR, Kim YJ, Kim JS, Miura N. Application of engineering ductile tearing simulation method to CRIPEI pipe test. Eng Fract Mech 2016;153:128–42. [22] Ryu HW, Bae KD, Kim YJ, Han JJ, Kim JS, Budden PJ. Ductile tearing simulation of Battelle pipe test using simplified stress-modified fracture strain concept. Fatig Fract Eng Mater Struct 2016;39:1391–406. [23] Bae KD, Ryu HW, Kim YJ, Kim JS. Comparison of ductile tearing simulation with complex cracked pipe test data. Int J Pres Ves Pip 2017;139. (011203-1). [24] ASTM Standards E813e81 Standard test method for J, a measure of fracture toughness. In: Annual book of ASTM standards part 10. West Conshohocken (PA, USA); 1983. [25] ABAQUS Version 6.13. User’s manual. Inc., and Dassault Systemes; 2013. [26] Oh CS, Kim NH, Kim YJ, Baek JH, Kim YP, Kim WS. A finite element ductile failure simulation method using stress-modified fracture strain model. Eng Fract Mech 2011;78:124–37. [27] Chaboche JL, Dang-Van K, Cordier G. Modelization of the strain memory effect on the cyclic hardening of 316 stainless steel. In: Proceedings of the 5th international conference on SMiRT, Div. L; 1979. [28] Chaboche JL. On some modificiations of kinematic hardening to improve the description of ratcheting effects. Int J Plastic 1991;7:661–7. [29] Chaboche JL. Modeling of ratcheting: evaluation of various approaches. Euro J Mech A/Sol 1994;13:501–18. [30] Bari S, Hassan T. An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation. Int J Plast 2002;18:873–94. [31] Bari S, Hassan T. Anatomy of coupled constitutive models for ratcheting simulation. Int J Plast 2000;16:381–409. [32] Rice JR, Tracey DM. On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Sol 1969;17:201–17. [33] McClintock FA. A criterion for ductile fracture by the growth of holes. J Appl Mech 1968;35:363–71. [34] Hancock JW, Mackenzie AC. On the mechanisms of ductile failure in high-strength steels subject to multi-axial stress states. J Mech Phys Sol 1976;24:147–60. [35] Mackenzie A, Hancock J, Brown D. On the influence of state of stress on ductile failure initiation in high strength steels. Eng Fract Mech 1977;9:167–88.
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Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Numerical ductile fracture prediction of circumferential throughwall cracked pipes under very low cycle fatigue loading condition ⁎
Hyun-Suk Nama, Jong-Min Leea, Yun-Jae Kima, , Jin-Weon Kimb a b
Department of Mechanical Engineering, Korea University, Anam-Dong, Sungbuk-Ku, Seoul 136-701, Republic of Korea Department of Nuclear Engineering, Chosun University, Seosuk-Dong, Dong-Gu, Gwangju 501-759, Republic of Korea
AR TI CLE I NF O
AB S T R A CT
Keywords: Ductile crack growth simulation Multi-axial fracture strain energy Finite element damage analysis Pipe test Very low-cycle fatigue
In this paper, a method to simulate ductile crack growth under very low-cycle fatigue loading condition is given and simulation results are compared with published test data of compact tension specimens and circumferential through-wall cracked pipes. The damage model in simulation is based on the multi-axial fracture strain energy. Two parameters in the damage model are determined from tensile and fracture toughness data under the monotonic loading condition. The determined damage model is then used to simulate ductile crack growth in compact tension specimens subjected to cyclic loading with large-amplitudes and in full-scale through-wall cracked pipes subjected to monotonic and cyclic loading with two different load ratios. Predicted results show a good agreement with experimental results.
1. Introduction Structural integrity analysis of piping components in nuclear power plants under seismic loading conditions is a challenging issue. Seismic loading can be characterized by dynamic and cyclic loading. It has been experimentally shown that material tensile properties can be affected by dynamic (strain rate) effect [1–8]. However, it has been also shown that the strain rate effect is minimal on fracture toughness properties and piping component fracture behaviours [6–8]. For instance, In Refs. [6,7], it has been shown that the strain rate effect appears in tensile test results, but is not clearly seen in fracture toughness and full-scale pipe tests. In contrast, the cyclic loading effect on piping component fracture behaviour has been shown to be very significant. Specimen and full-scale pipe tests have been performed to quantify the cyclic effect on fracture toughness and on cracked full-scale pipe behaviours [9–14]. Test results showed that the maximum loads under reverse cyclic loading were lower than those under the monotonic loading condition. Furthermore, “apparent” fracture resistance curves (based on the positive load-load line displacement curve) were significantly decreased when subject to reversible cyclic loads, compared to that under monotonic loading condition. Seok [9] reported that the decrease in the J-R curve is related to the increased stain-hardening and generation of tensile stress at the crack tip during cyclic loading. Although most of existing works reported such phenomenon, reasons are still not clear. There has been an analytical attempt to quantify crack growth under reverse cyclic loading with large amplitude using combination of ductile tearing and low cycle fatigue crack growth [15,16]. Although experimental data can provide valuable information, testing under various cyclic loading conditions can be costly due to larger test matrices. Finite element (FE) damage simulation would be an efficient tool to perform failure analyses of complex geometries under various loading conditions. Some works for ductile crack growth simulation under cyclic loading conditions have been reported in literature [17–19]. Sherry and Wilkes [17] and Klingbeil et al. [18] reported ductile fracture simulation of compact
⁎
Corresponding author. E-mail address: [email protected] (Y.-J. Kim).
https://doi.org/10.1016/j.engfracmech.2018.02.025 Received 13 June 2017; Received in revised form 18 February 2018; Accepted 26 February 2018 0013-7944/ © 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: Nam, H., Engineering Fracture Mechanics (2018), https://doi.org/10.1016/j.engfracmech.2018.02.025
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Nomenclature
t Δa σe, σm
pipe thickness crack growth effective stress and mean normal stress, respectively σ1, σ2, σ3 principal stress components Δω, ω, ωc incremental, accumulated and critical damage, respectively δ displacement increment εpl equivalent plastic strain σy, σyo yield strength and initial yield strength
A, B, C
material constants in multi-axial facture strain energy locus, Eq. (2) crack length and its initial value a, a0 Ci, γi (i = 1–3) material constants in kinematic hardening model, Eq. (8) E elastic modulus J J-integral Le finite element size Wp equivalent plastic strain energy Wf fracture strain energy Pmax, Pmin maximum and minimum load in cyclic loading R load ratio = Pmin/Pmax Q, b material constants in isotropic hardening model, Eq. (7) rm, ro mean and outer radius of pipe
Abbreviation C(T) CMOD FE LLD
compact tension crack mouth opening displacement finite element load-line displacement
tension (C(T)) test under cyclic loading conditions based on the Gurson-based modeling. The model was determined using notch tensile and fracture toughness test under the monotonic loading condition. In Ref. [17], an isotropic hardening model was considered, whereas in Ref. [18], the kinematic hardening model was considered for cyclic hardening properties. Based on the determined damage model, the C(T) test under cyclic loading conditions could be simulated relatively well. However, the Gurson-based damage model has many parameters to be determined, which is often a difficult task. Khoei et al. [19] presented ductile crack growth simulation under cyclic and dynamic loading using a damage-viscoplasticity model. To define the cyclic hardening properties, ratedependent constitutive model and combined hardening model were considered. The damage model was determined from tensile and notch tensile test results under monotonic loading condition. Several tests using Arcan, C(T) and double-notched tests under dynamic and cyclic loading conditions were simulated using the determined damage model. Note also that the proposed damage-viscoplasticity model has many parameters and determination of those parameters requires many tests and elaborate calibration procedures. In this paper, ductile crack growth in C(T) specimens as well as in circumferential through-wall cracked pipes under very lowcycle fatigue loading condition is simulated using FE damage analysis and simulation results are compared with published experimental data. The damage model for simulation is based on the multi-axial fracture strain energy [20] which is an extension of the multi-axial fracture strain model previously used by the authors [8,21–23]. The damage model has only two parameters which can be determined from tensile and fracture toughness data under the monotonic loading condition. The determined damage model is then used to simulate ductile crack growth in C(T) specimens subjected to reverse cyclic loading with large-amplitudes and in circumferential through-wall cracked pipes subjected to monotonic and cyclic loading. Section 2 summarizes published mechanical test and circumferential through-wall cracked pipe test data under different load ratios. Section 3 introduces the energy-based damage model and explains how to determine the damage model and failure criterion for ductile crack growth simulation. In Section 4, simulation results of C(T) test under cyclic loading conditions are compared with experimental data. In Section 5, simulation results of full-scale pipe test under monotonic and cyclic loading conditions are compared with experimental data. Section 6 conclude the presented work.
2. Summary of experiments 2.1. Material and tensile test results Published mechanical and full-scale cracked pipe test data under monotonic and cyclic loading conditions, performed in Battelle Memorial Institute and compiled in Ref. [6], are considered in this work. All tests were conducted at quasi-static rates at 288 °C. The material was A106 Gr. B carbon steel extracted from a pipe with the 152 mm nominal diameter, typically used in nuclear piping of Light Water Reactors. The chemical composition of A106 Gr. B steel is given in Table 1. Tensile properties at 288 °C were obtained using a smooth round bar specimen with the 5.08 mm diameter and 12.7 mm gauge Table 1 Chemical composition of A106 Gr. B [6]. (wt%) C
Mn
P
S
Si
Cu
Sn
Ni
Cr
Mo
Al
0.15
0.65
0.012
0.014
0.20
0.28
0.018
0.14
0.18
0.055
0.010
2
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Fig. 1. (a) Schematic illustration of smooth bar and (b) FE mesh of the specimen.
length (Fig. 1a). Tensile test was conducted at quasi-static rates and the axial displacement was measured using extensometer. Experimental engineering stress-strain curves are shown in Fig. 2a. The 0.2% proof (yield) strength, ultimate tensile strength and reduction of area were 320 MPa, 621 MPa and 34.4%, respectively. To characterize cyclic tensile properties of A106 Gr. B, low cycle fatigue test was conducted under the strain controlled condition (0.35% strain amplitude range). Test conditions such as the specimen geometry and loading rate are the same as those in monotonic tensile tests. Fig. 2b shows the stable hysteresis loop. 2.2. Monotonic and cyclic C(T) test results Three fracture toughness (J-resistance curve) tests were performed at 288 °C. In accordance with the ASTM E813-81 standard [24], the 0.5 T C(T) specimen with the thickness of 11.18 mm was used. The specimen were machined in the L-C direction and had no side groove. Fracture toughness tests were conducted under monotonic and cyclic loading conditions. For monotonic loading, monotonicallyincreasing tensile load was applied to the specimen. In cyclic tests, incremental cyclic load was applied to the specimen, as schematically shown in Fig. 3a. In the loading sequence, displacement was controlled at the loading step, whereas load was controlled at the unloading step to maintain constant load ratio (R = Pmin/Pmax). Two load ratios were used in the test; R = −0.8 and −1.0. During loading sequence, a constant displacement increment of 0.08 mm was applied after each cycle, as shown in Fig. 3a. Tests were performed in a servo-hydraulic machine with loading rate (1.5 mm/min) at 288 °C. During tests, the direct-current electric potential method was applied to measure crack initiation and subsequent growth. The monotonic J-R curve was calculated according to the ASTM E813-81 standard [6]. For cyclic tests, cyclic J-R curves were calculated using the area under load-load line displacement (LLD) curve above zero load, as shown Fig. 3b. More detailed procedure for evaluating the fracture toughness under the cyclic loading condition can be found in Ref. [6]. Fig. 4a shows experimental load versus load-line displacement (LLD) curves from fracture tests of two materials. Resulting J-R and LLD-Δa curves are shown in Fig. 4b and c. 2.3. Pipe test results Circumferential through-wall cracked pipe test results of A106 Gr. B under monotonic and cyclic loading condition were also obtained from Ref. [6]. Three pipes having identical pipe/crack dimensions were tested. The outer radius of the pipe was
Fig. 2. Comparison of experimental engineering stress–strain data with simulated results: (a) monotonic loading condition and (b) cyclic loading condition. Experimental data were taken from Ref. [6].
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Fig. 3. Schematic illustration of (a) cyclic loading sequence and (b) load-load line displacement (LLD) area to calculate cyclic J-integral.
Fig. 4. Monotonic and cyclic C(T) test results of A106 Gr. B: (a) load-LLD curves, (b) J-R curves and (c) LLD-Δa curves [6].
ro = ∼84 mm with the thickness of t = ∼14.0 mm. A circumferential through-wall crack of ∼37 percent of the pipe circumference was machined into the pipe specimen prior to fatigue pre-crack. The pipe test specimen is schematically shown in Fig. 5a. The cracked pipe was subjected to four-point (pure) bending at 288 °C, as schematically shown in Fig. 5b. Similarly to the fracture toughness test, monotonic and cyclic loading was applied to the cracked pipe. Monotonic loading test corresponds to the load ratio R = 1.0 (designated as the (1-2-7) test). Cyclic loading sequence was the same as that for the fracture toughness test. Two load ratios, R = 0.0 (designated as the (1-2-2) test) and R = −1.0 (fully reversed, designated as the (1-2-4) test), were tested. The test speed of all 4
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Fig. 5. (a) Schematics of the cracked pipe specimens (the x-point in the figure indicate the clip gage location for the crack mouth opening displacement (CMOD) measurement and (b) Schematics of the test equipment.
experiment was slower than 0.1 mm/s. Crack growth was estimated using direct-current electric potential probes located in the centerline. Crack growth was found to be out-of-plane and the average crack extension was projected back to the initial crack plane. Measured load-LLD curves and LLD-Δa curve are shown in Fig. 6a and 6b. Crack mouth opening displacement (CMOD) data were also measured at the center and tip of the through-wall crack, as indicated in Fig. 5(a) using the cross symbol. Measured CMOD-LLD curves are shown in Fig. 7. More detailed information of pipe tests can be found in Ref. [6].
3. Determintion of damage model for ductile crack growth simulation 3.1. Overview of the method In this paper, a strain-energy based ductile fracture model is used to simulate the monotonic and cyclic fracture toughness tests, where incremental damage is defined by 5
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Fig. 6. Measured pipe test results: (a) load-LLD curves and (b) LLD-Δa curves [6].
Fig. 7. Measured CMOD-LLD curves from pipe tests: (a) the (1-2-7) test (R = 1), (b) the (1-2-2) test (R = 0) and (c) the (1-2-4) test (R = −1) [6].
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Δω =
ΔWp Wf
(1)
In Eq. (1), Wp and Wf denotes equivalent plastic strain energy and the multi-axial fracture strain energy, respectively. The multi-axial fracture strain energy, Wf, is assumed to be given by
σ Wf = Aexp ⎛−C m ⎞ + B σe ⎠ ⎝ ⎜
⎟
(2)
where σm and σe denote the mean normal stress and the equivalent stress, respectively; A, B and C are material constants. To determine these three material constants, at least three tests with different triaxial stress states would be needed. When such test data are not available, Eq. (2) can be simplified as follows (see Appendix A)
σ Wf = Aexp ⎛−1.5(n + 1) m ⎞ + B σe ⎠ ⎝ ⎜
⎟
(3)
where n in Eq. (3) is the plastic strain hardening exponent. When accumulated damage becomes critical, i.e.,
ω=
∑ Δω = ωc
(4)
local ductile failure is assumed and incremental crack growth is simulated by reducing all stress components and the value of Young’s modulus at the gauss point where the failure condition is satisfied. This simulation technique has been implemented in the ABAQUS STANDARD commercial FE program [25] using user subroutines. Further details information on the failure simulation technique can be found in our previous works [8,20–23,26]. For ductile crack growth simulation under cyclic loading, this paper assumes that the multi-axial fracture strain energy, Wf, determined from monotonic loading can be applied to cyclic loading condition. Furthermore the critical damage ωc for monotonic loading is assumed to be also applied to cyclic loading conditions:
ω ⎛⎜= ∑ Δωmono = ⎝
∑ Δωcyclic = ∑
ΔWp ⎞ ⎟
Wf ⎠
= ωc
(5)
Above assumptions seem plausible in the sense that total energy for ductile fracture remains constant regardless of loading conditions. However, when Eq. (4) is applied to cyclic loading, one caution should be made in calculating incremental damage under compressive loading. This will be explained later in Section 4.2. 3.2. Determination of damage model under monotonic loading condition To determine the multi-axial fracture strain energy, Eq. (3) was considered in this work due to limited experimental test data. Tensile (stress–strain) curve gives n = 0.236 in Eq. (3). To determine two material constants in Eq. (3), A and B, elastic–plastic FE analysis of tensile test and FE damage analysis of the C(T) test were performed. Step-by-step procedures to determine these two constants are given below.
Fig. 8. (a) Variation of the stress triaxiality with equivalent plastic strain energy for A106 Gr. B and (b) multi-axial fracture strain energy locus for A106 Gr. B.
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The first step was to perform elastic–plastic FE analysis of smooth bar tensile test. For simulation, the FE mesh shown in Fig. 1b was used. Symmetric conditions were considered and the first order solid elements (C3D8 with in ABAQUS [25]) were used. The element size in the minimum cross section where the necking occurs was fixed to 0.1 mm but the use of the smaller element size gave the same results. The true stress-strain curve was used together with the large geometry change option and the J2 flow theory. Simulated results are compared with the experimental results in Fig. 2a, showing that FE results agree well with the experimental data. Variation of the stress triaixiality with the plastic strain energy, extracted from FE results up to the failure initiation point, is shown with solid line in Fig. 8a and the last point in Fig. 8a corresponds to the experimental failure point shown in Fig. 2a. The stress triaxiality increases with increasing plastic strain energy after necking. To reflect the history effect of stress and strain, an averaged stress triaxiality value was calculated, as shown using the dotted line in Fig. 8a. One point corresponding to the fracture strain energy versus the stress triaxiality is shown in Fig. 8b using open circle. The second step is to determine one more constant in Eq. (3) and the critical accumulated damage parameter ωc from FE damage simulation results of the fracture toughness test under monotonic loading condition. Three-dimensional (3-D) FE damage analysis was performed using a quarter model considering symmetrical conditions. The first order solid element with full integrations (C3D8 within ABAQUS [25]) was uniformly spaced in the crack propagation region, as shown in Fig. 9a. The element size in the crack propagation area was fixed to Le = 0.2 mm and the mesh had 12,478 elements and 14,640 nodes. For tensile properties, true stressstrain data were directly used. The large geometry change option was chosen with the J2 flow theory. The incremental damage was calculated using Eq. (1) with the assumed multi-axial fracture strain energy (the first trial is to assume the fracture strain energy value at σm/σe = 2.5 to be 10% of the uni-axial value). By matching experimental fracture initiation toughness, a proper value of the critical accumulated damage, ωc, was chosen. Then the value of the fracture strain energy at σm/σe = 2.5 was tuned to match the tearing modulus of the J-R curve. It would be useful to note that the predicted initiation toughness value was controlled mainly by the ωc value, whereas predicted tearing modulus was governed by the material constant B. A final expression of the multi-axial fracture strain energy was given by
J σ ⎤ = 599exp ⎛−1.854 m ⎞ + 29 Wf ⎡ σe ⎠ ⎣ mm3 ⎦ ⎝ ⎜
⎟
(6)
Fig. 9. Typical 3D FE meshes of simulate 0.5 T C(T) test: (a) the element size Le = 0.2 mm and (b) Le = 0.6 mm.
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Fig. 10. Comparison of experimental J-R curve with simulated results using two different element sizes, Le = 0.2 mm and Le = 0.6 mm.
Table 2 Parameters in the non-linear kinematic hardening model of A106 Gr. B. Material
Q (MPa)
b
σo (MPa)
C1
γ1
C2
γ2
C3
γ3
A106 Gr. b
150
6
150
110,000
1000
20,000
100
1000
10
with ωc = 0.97. This locus is shown in Fig. 8b and simulated J-R curves are compared with experimental J-R curve in Fig. 10. Simulated results agree relatively well with the experimental data. For practical application of ductile crack growth simulation of a large-scale piping component, the use of a small finite element can cause numerical difficulties. As damage parameters depend on the element size, they should be properly determined for a given element size. For the present model, it is assumed that the ωc value is set to be a function of an element size. To find an appropriate value of ωc for a different element size, the 0.6 mm element size in the crack propagation region was prepared, as shown in Fig. 9b. Then the same procedure as before was taken to determine the ωc value for Le = 0.6 mm, by matching predicted toughness value with experimental one. It was found that the appropriate value of ωc = 0.4 for Le = 0.6 mm, which is smaller than ωc = 0.97 for Le = 0.2 mm. Predicted J-R curves using Le = 0.6 mm with ωc = 0.4 are compared with experimental J-R curve in Fig. 10. 4. Ductile crack growth simulation of C(T) test under cyclic loading conditions 4.1. Cyclic material properties In this paper, a combined hardening model embedded within ABAQUS [25] was used to characterize cyclic material properties. In the model, isotropic hardening is considered by the amount of expansion taken to be a function of accumulated plastic strain: pl
σy = σyo + f (ε pl ) = σyo + Q (1−e−bε )
(7)
where σyo is the initial yield strength and f(ε ) denotes the isotropic hardening function with material constants Q and b. To consider kinematic hardening, the ‘decomposed’ nonlinear kinematic hardening rule proposed by Chaboche [27–29] is used, where the evolution of the back stress is given by pl
3
dα =
∑ i=1
1/2 2 2 Ci dε pl−γi αi dε pl where dε pl = |dε pl| = ⎡ dε pl·dε pl⎤ 3 3 ⎣ ⎦
(8)
In Eq. (8), C and γ are material constants; C is the initial kinematic hardening modulus and γ is the rate at which the kinematic hardening modulus decreases with increasing plastic deformation. Based on the cyclic tensile test data described in Section 2.1, the material parameters in the combined hardening model for A106 Gr. B were determined as follows. Firstly, the value of C3 was determined from the slope of the linear segment of hysteresis curve at
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Fig. 11. Comparison of C(T) test data with simulated results for A106 Gr. B: (a)–(b) load-LLD curves, (c) J-R curves and (d) LLD-Δa curves.
high strain ranges. Secondly, values of C1/γ1 and C2/γ2 were estimated. Note that the value of C1/γ1 should be very large to match the plastic modulus at yielding. The value of C2/α2 should be match the transient nonlinear portion of the stable hysteresis curve. For more detailed determination procedure, see Refs. [30,31]. Values of the material parameters are summarized in Table 2. For validation, cyclic tensile test was simulated using ABAQUS. Simulated results are compared with tests results in Fig. 2b, showing good agreement.
4.2. Simulation of C(T) test under cyclic loading conditions To perform ductile crack growth simulation under cyclic loading conditions, the damage model determined from mechanical test data under monotonic loading (the multi-axial fracture strain energy locus, Wf, and the critical accumulated damage parameter, ωc) was used according to the explanations given in Section 3.1. A caution should be made when calculating the incremental damage under reverse cyclic loading. Under the cyclic loading, equivalent plastic strain energy is positive even when the crack closes under compressive loading. In the present work, the incremental damage is calculated only when the maximum principal stress at the crack tip is positive. In simulation, the FE model used for the monotonic loading condition (Fig. 9) was used, having the 0.2 mm element size in crack propagation region. The multi-axial fracture strain energy locus given in Eq. (6) with ωc = 0.97 was used For strain hardening, combined hardening rule explained in Section 4.1 was used, with the parameters given in Table 2. Simulated load-LLD and J-R curves are compared with experimental data in Fig. 11. Note that J-R curves from FE simulations are calculated using the same method as in experiments. Comparison with experimental data shows overall good agreements.
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Fig. 12. 3-D FE mesh to simulate ductile crack growth in through-wall cracked pipe tests.
Fig. 13. Comparison of circumferential through-wall cracked pipe data under monotonic loading condition with simulated results: (a) load-LLD curve, (b) LLD-Δa curve and (c) CMOD-LLD curves.
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Fig. 14. Comparison of circumferential through-wall cracked pipe data under fully reversed cyclic loading condition (R = −1) with simulated results: (a) load-LLD curve, (b) LLD-Δa curve and CMOD-LLD curves (c) at crack center and (d) at crack tip.
5. Ductile crack growth simulation of circumferential through-wall cracked pipe test under monotonic and cyclic loading conditions To simulate ductile crack growth in circumferential through-wall cracked pipe tests under monotonic and cyclic loading conditions, explained in Section 2.3, 3-D FE damage analysis was performed. To reflect symmetric conditions, a quarter model was used and the first order solid elements of Le = 0.6 mm (C3D8 within ABAQUS [25]) were uniformly spaced in the cracked section. To accommodate crack closure, the contact surface were added in the symmetric plane, as shown in Fig. 12. For the damage model, the fracture strain energy given in Eq. (6) with ωc = 0.4 (for t Le = 0.6 mm) was used. The FE mesh had 43,190 elements and 48,900 nodes. For tensile properties, the true stress+strain curve shown in Fig. 2(a) was used for the monotonic loading case and the combined hardening model given in Section 4.1 was used for the cyclic loading cases. The non-linear geometry change option was used with the J2 flow theory. In Fig. 13, simulated results for the monotonic loading case are compared with experimental data; the load versus LLD, crack extension versus LLD and CMOD versus LLD curves. Predicted results agree overall well with experimental data. Corresponding results for cyclic pipe test results of the R = −1 (fully reversed loading) case are compared with experimental data in Fig. 14. Predicted results also agree overall well with experimental data. Finally corresponding results for the R = 0 case are shown in Fig. 15. Predicted load versus LLD results agree well with experimental data. For crack extension versus LLD results, good agreement can be seen up to 30 mm crack growth. Predicted CMOD versus LLD curves for the crack tip are in good agreement but for the crack center are much lower than experimental data. On the other hand, when experimental data for R = 0 are compared with those for R = 1.0
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Fig. 15. Comparison of circumferential through-wall cracked pipe data under cyclic loading condition (R = 0) with simulated results: (a) load-LLD curve, (b) LLD-Δa curve and CMOD-LLD curves (c) at the crack center and (d) at the crack tip.
and R = -1.0, it can be seen that the results for R = 0 are unusually large, suggesting possible error in measurement during the test. Fig. 16 compares fracture surfaces at the maximum load point in each test, resulting from FE simulations. The cracked surface is shown using the darker color. In all cases, crack tunneling can be clearly seen. 6. Conclusion This paper presents ductile crack growth simulation in compact tension specimens and in cracked pipes under very low-cycle fatigue loading condition and results are compared with published experimental data. For experimental data, four-point bending tests of circumferential through-wall cracked A106 Gr. B pipes at 288 °C, performed by Battelle institute, were taken. For simulation, the damage model based on the multi-axial fracture strain, previously proposed by the authors [8,21–23], is extended to a model based on the multi-axial fracture strain energy. The damage model has two parameters which are determined from tensile and fracture toughness data under the monotonic loading condition. Due to such simplicity of the proposed model, it can be easily applied to practical situations where material data for calibrating parameters in the damage model are limited. The determined damage model is then used to simulate ductile crack growth in compact tension specimens subjected to cyclic loading with large-amplitudes and in full-scale through-wall cracked pipes subjected to monotonic and cyclic loading with two different load ratios. To improve computational efficiency and numerical stability, relatively large element size is used in cracked pipe simulation. The element size effect on damage simulation is accommodated by the critical damage value in the present damage model. Predicted results show a good agreement with experimental results.
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Fig. 16. Simulated crack surface at the maximum load: (a) monotonic loading condition, (b) cyclic loading condition with R = 0 and (c) cyclic loading condition with R = -1.
Acknowledgement This research was supported by National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B2009759) and by the Nuclear Power Core Technology Development Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), the Ministry of Trade, Industry & Energy, Republic of Korea (No. 20141520100860). Appendix A. Simplified method to determine the material constants for multi-axial fracture strain energy It is well known that the fracture strain for ductile fracture depends strongly on the triaxial stress states [32–35]. The triaxial stress dependence on fracture strain (the multi-axial fracture strain, ɛf) is typically quantified by the following exponential equation
σ εf = α exp ⎛−γ m ⎞ + β σe ⎠ ⎝ ⎜
⎟
(A1)
where α, β and γ are material constants determined from several tests with different triaxial stress states. When test data are not sufficient, Eq. (A1) can be simplified by assuming some of constants. For instance, Rice and Tracey suggested that the constant γ is approximately 1.5 [32], leading to
σ εf = α exp ⎛−1.5 m ⎞ + β σe ⎠ ⎝ ⎜
⎟
(A2)
For power-law plastic materials, the plastic strain energy Wp up to the fracture strain εf can be written as
Wp =
∫ σ0 (εp)ndεp =
σ0 (εp)n + 1 n+1
εf
= 0
σ0 (εf )n + 1 n+1
(A3)
This suggests that the multi-axial fracture strain energy equivalent to Eq. (A2) is given by
σ Wf = Aexp ⎛−1.5(n + 1) m ⎞ + B σe ⎠ ⎝ ⎜
⎟
(A4) 14
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