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Residual stress and stress fields change around fatigue crack tip: Neutron diffraction measurement and finite element modeling
International Journal of Pressure Vessels and Piping 179 (2020) 104024
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Residual stress and stress fields change around fatigue crack tip: Neutron diffraction measurement and finite element modeling Wenchun Jiang a, *, Yue Yu a, Weiya Zhang a, Chengran Xiao a, Wanchuck Woo b a b
School of New Energy, China University of Petroleum (East China), Qingdao, 266555, PR China Neutron Science Division, Korea Atomic Energy Research Institute, 1045 Daedeok-daero, Yuseong-gu, Daejeon, 305-353, South Korea
A R T I C L E I N F O
A B S T R A C T
Keywords: Crack tip Residual stress Neutron diffraction Overload
The variation of residual stress and stress field induced by the irreversible deformation at the fatigue crack tip of compact tension (CT) specimen subjected to different loads are studied by finite element method (FEM) and neutron diffraction measurement (NDM). The finite element method adopts a fatigue damage model considering material hardening effect. The stress field induced by one static load after the cyclic loads and the corresponding residual stress field at fatigue crack tip are found to match the experimental measurement. The influence of the static load on the residual stress is discussed by investigating the three principal stresses, that is the longitudinal, the transverse and the normal stresses. The results show that the residual stress at crack tip is closely correlated to the static load magnitude when this magnitude is greater than the maximum fatigue load (Pmax), while this correlation does not exist for the load smaller than Pmax. Different from the residual stress, the stress field at the crack tip increases with the increase of static load. Especially when the load is less than 1 Pmax, the stress field at the crack tip mainly appears a compressive stress state. Meanwhile, the compressive residual stress ahead of the crack tip is generated after the release of the static load. Regardless of the loading history, this residual stress is dependent only on the loading magnitude.
1. Introduction Many engineering structures work under the cyclic loads. The mag nitudes of these cyclic loads are changing or even stochastic depending on different working conditions [1]. The stress fields and the fatigue crack propagation resulted from the variable amplitude cyclic loads are distinct from the stress field generated from the constant amplitude cyclic loads [2]. Therefore, the researches on constant amplitude cyclic loads are insufficient for the engineering applications. It is necessary to study the crack tip stress field and the crack growth rate under variable amplitude loads. In recent years, the effects of variable amplitude loads, such as overload and underload (Fig. 1), on fatigue crack have been extensively studied by both experiments and numerical models [3–6]. Recently, the influences of overload/underload on fatigue crack growths are intensively discussed. Sander et al. [7] found that the crack growth rates are significantly affected by the load sequences. When underload is applied before the tensile overload, the fatigue crack growth rates are not affected regardless of the amplitude. However, applying an underload after an overload accelerates crack propagation, which is related to the magnitude of the underload. Overload could
induce crack growth retardation because of the reduction of the crack driving force and the generation of large compressive residual stress [8]. In addition, large deformation occurred in the overload part and the crack side. The distribution of crack tip residual stress is mainly obtained by neutron diffraction measurement (NDM) and synchrotron X-ray diffraction measurement (XDM). The compressive residual stress created by a single overload is measured by using NDM and the delayed crack growth rate is identified [9–11]. Interestingly, as the crack continues to expand, the crack growth rate gradually increases to the rate before the overload is applied. In addition, the residual stress varies in the thick ness direction of the specimen [12]. Using the high energy XDM, the change in strain field after overload is discussed in depth. A single overload cycle can create a compressive strain field around the crack plane. However, the sample with a delay in the maximum crack growth rate after overload produced a tensile strain field at the crack tip [13]. Steuwer et al. [14] studied the stress distribution at the crack-tip sub jected to the overloads. A compressive residual stress was detected during unloading and there were tensile stresses of no more than 100 MPa in front of the crack (about 100 μm). In recent years, nano inden tation [15] and digital image correlation technique [16,17] are also used to measure the micro residual stresses and microstructure evolution.
* Corresponding author. E-mail address: [email protected] (W. Jiang). https://doi.org/10.1016/j.ijpvp.2019.104024 Received 20 August 2019; Received in revised form 20 November 2019; Accepted 24 November 2019 Available online 27 November 2019 0308-0161/© 2019 Elsevier Ltd. All rights reserved.
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
enhanced the effect of residual plastic wake. In recent years, the Extended Finite Element Method (XFEM) [21,22] and the damage me chanics method [23,24] are also used to predict the fatigue crack propagation and fatigue life under different amplitude load. Huffman [25] proposed a fatigue damage model based on strain energy to calculate the fatigue life and fatigue crack growth rate under strain amplitude. Unfortunately, the accuracy of the model needs to be further improved at very low strain amplitude. Benkabouche et al. [26] devel oped an improved nonlinear continuous damage mechanics (CDM) model to predict damage evolution and lifetime by considering the magnitude of the load and the load sequence. This model is suitable for multi-axis situations, while only for high cycle fatigue. CDM also showed good prediction results in the simulation of welded joints considering weld residual stress [27,28]. In addition, the CDM method also has been extended to multi-axis fatigue conditions to simulate the entire damage accumulation process reflecting material degradation [29,30]. Recently, Fazlali et al. [31] developed a CDM model that can distinguish the damage between fiber and matrix and predicted transverse and longi tudinal residual stiffness. The present study is an extension of the work described above. In this paper, fatigue crack growth was predicted through continuous damage mechanics model. The stress distribution ahead of the crack tip was measured by neutron diffraction measure ment. To compare the influence of static load on stress field, residual stresses before and after static load were also measured. The develop ment of crack-tip stresses during different static load and the influence of single overload following fatigue cyclic load on residual stress field around crack tip was revealed.
Nomenclature CT FEM NDM Pmax Pmin XDM CRS TRS XFEM CDM COD TD ND LD GV LRS NRS
compact tension finite element method neutron diffraction measurement the maximum fatigue load the minimum fatigue load X-ray diffraction measurement compressive residual stress tensile residual stress Extended Finite Element Method continuous damage mechanics crack-opening-displacement transverse direction normal direction longitudinal direction gauge volume longitudinal residual stress normal residual stress
2. Experimental 2.1. Fatigue test The fatigue crack growth tests are performed on an austenitic stainless steel 304L compact-tension specimen (CT specimen) with chemical components listed in Table 1 and geometry shown in Fig. 2. The CT specimens are prepared according to the ASTM Standards E647–99.22 with a notch length 10.16 mm, width 50.8 mm and thick ness 6.35 mm, respectively. Before performing the fatigue crack growth test, a pre-cracked length of 1.27 mm is made on the CT specimen. Then the stress-controlled crack propagation experiment is conducted under a constant load using the MTS Landmark 370.10 fatigue test system. The maximum fatigue load is 7400 N with load ratio R (Pmin/Pmax) equals 0.01 and load frequency equals 10 Hz. The crack length of the fatigue test is measured by crack-opening-displacement (COD) gauge. The fa tigue test is stopped after the crack length reaches 16 mm, and the re sidual stresses around the crack tip are measured.
Fig. 1. Different load types in fatigue experiments.
Besides the experimental approaches, the numerical methods are also applied to investigate the fatigue crack propagation characters and to predict its remaining life. In the past several decades, considerable efforts have been devoted to numerically study the promotion/inhibi tion of fatigue crack propagation by overload/underload. For instance, based on blunting re-sharpening mechanism of crack tip, Viggo Tver gaard [18] found that the compression underload is beneficial to the propagation of cyclic cracks through re-meshing method, while a single overload delays the propagation of cracks. In addition, the crack growth rate is also related to the overload induced plastic deformation. Tensile overload increases the threshold value of long cracks because of the plastic deformation. The compression overload reduces the crack propagation resistance and increases the crack propagation threshold [6]. Recently, the influence of linear elastic stress field and residual plastic deformation on crack propagation has been further studied by Kumar et al. [19] and Antunes et al. [20]. For overload and low-high load sequences, residual plastic deformation accumulates, and the re sidual stress caused by it decreases the crack opening level, which leads to the passivation of the crack tip. However, the crack tip blunting is abruptly reduced when high-low load sequences are applied, which
2.2. Neutron diffraction measurement After the fatigue test, the residual stresses around the fatigue tip are measured by neutron diffraction instrument at Korea Atomic Energy Research Institute (KAERI). This method is based on Bragg theory: nλ ¼ 2d sin θ;
(1)
where n is an integer, 2θ is the diffraction angle, d is the lattices pacing, λ is the wavelength of neutron beam. The lattice strain is obtained from:
ε¼
Δd ¼ d
Δθ cot θ;
(2)
where Δd is the change of the lattice spacing, Δθ is the shift of the angular position. Three strain components (in transverse (TD), normal (ND) and lon gitudinal (LD) directions, respectively) are measured and the residual stress σi (i ¼ x, y or z) is calculated based on the measured strain: 2
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Table 1 Chemical composition of 304L stainless steel (%). Component
C
Mn
P
S
Si
Cu
Ni
Cr
Mo
N2
304L
0.029
1.86
0.029
0.004
0.37
0.02
10
18
0.04
0.056
3. Finite element analysis The residual stress and stress field changes around the fatigue crack tip are simulated by FEM. Firstly, the fatigue crack propagation is simulated based on the CDM. After the crack length reaches 16 mm, the fatigue load is released and the residual stress distribution is then ob tained. Then, static loads with different magnitudes are applied to determine the change of stress field and residual stress field. Finally, compare the simulation results with the measurement results to verify the current model. 3.1. Finite element model Considering the geometrical symmetry, as shown in Fig. 4, only one quarter of the specimen is modeled and meshed with 23557 elements and 29345 nodes. The element type is linear hexahedral elements named C3D8. In order to limit the rigid movement of the specimen, a reference point is set at the center of the loading hole to apply a constraint in the X direction. Corresponding to the modeling characteristics, symmetric boundary conditions are set on the intermediate surface of the CT sample. The maximum force is 7400 N and the stress ratio R is 0.01, which are identical to the experiment.
Fig. 2. Geometry of the CT specimen for 304L stainless steel.
σi ¼
E νE εi þ 1þν ð1 þ νÞð1
�
2νÞ
εx þ εy þ εz ;
(3)
3.2. Fatigue damage model
where E is the elastic modulus and ν is the Poisson’s ratio. The range from 0.5 mm to 22 mm at the tip of the crack in the crack-growth di rection is measured. The gauge volume (GV) is 1(x) � 1(y) � 1(z) mm3. The monochromatic beam with wavelengths at 2.39 Å and 1.55 Å for (110) and (211) on scattering angles of 72.1� and 82.9� are used, respectively. After the residual stress measurement, the loads with three distinct magnitudes are applied to the sample and the lattice strain evolution is measured in situ. The experimental installations are shown in Fig. 3.
The fatigue crack propagation is simulated based on the continuum damage mechanics. As proposed by Krajcinovic and Lemaître [28–30], the dissipate potential φ is described as: (4a)
φ ¼ φp þ φD þ φπ ; φp ¼
σeq 1
φD ¼
R D
(4b)
σY ;
b ðS0 þ 1Þð1
� DÞ
Y b
�S0 þ1 ;
(4c)
where φp is the plastic dissipated potential, φD is the damage dissipated potential, φπ is the micro-plastic dissipated potential, R is the isotropic hardening scalar variable associated with accumulated plastic strain, σY is the initial yield stress of the material, σ eq is the Von Mises equivalent stress, D is an isotropic damage scalar variable, and S0 and b are tem perature dependent material parameters representing unified damage law exponent and energetic damage law parameter, respectively. The effect of the micro-plastic dissipation potential is small and is not considered in this paper. The plastic strain tensor εpij equation coupled damage is derived from the plastic dissipated potential φp and the damage evolution equation is derived from the damage dissipated po tential. The strain tensor εij consists of two parts: the elastic strain tensor εeij and the plastic strain tensor εpij , These are:
εij ¼ εeij þ εpij ; ε_ pij ¼ λ_
p_ ¼ Fig. 3. Experimental setup by neutron diffraction. 3
λ_ ∂φ _ ∂φp 3 Sij ¼λ ¼ :; ∂σ ij ∂σ ij 2 ð1 DÞ σ eq
�1=2 � 2 p p ε_ ij ε_ ij 3
(5a) (5b) (5c)
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 4. The entire mesh (a) and local fined mesh (b) for CT specimen.
∂φ D_ ¼ λ_ D ; ∂Y
using kinematic hardening modulus (Ci) and dynamic rate (γi). The size change of the yield surface and its shift through the stress space are considered. Nonlinear kinematic hardening and isotropic hardening variable are combined to describe the cyclic plastic deformation. The kinematic hardening variable (X) is shown in Eqs. (11) and (12). Based on the Armstrong-Frederick and Chaboche model [32], the back stress is decomposed into n parts and each part consists of a linear reinforcement term and a dynamic recovery term:
(5d)
where εpij is the plastic strain rate, p_ is the accumulated plastic strain rate, λ_ is the non-negative proportion factor, σ ij is the stress tensor, Sij is the
stress deviator, D is the isotropic damage scalar variable ranging from 0 to 1, D_ is the fatigue damage rate, Y is the strain energy release rate and the function is:
σ2eq RV
Y¼
2Eð1
DÞ2
;
n X
(6)
X¼
where E is the Yong’s modules and RV is the stress triaxiality factor defined by: � �2 2 σH RV ¼ ð1 þ υÞ þ 3ð1 2υÞ ; (7) 3 σ eq
2 X_ i ¼ Ci ε_ p 3
λ_ 1
D
;
from Eq. (4c) one can easily get: � �S0 ∂φ Y p; _ D_ ¼ ¼ b ∂Y
γi αi p_
(12)
where Ck and γk are material parameters, and the back stress compo nents corresponding to different Ck and γ k can reflect the nonlinear behavior of the material in different plastic deformation stages. Four back stress kinematic hardening components are adopted to better describe the stress-strain behavior of materials. And the accumulated plastic strain is defined by Eq. (5c). The isotropic hardening is expressed as:
where σ H is the hydrostatic stress, σ eq is Von Mises equivalent stress. By substituting Eq. (5b) to Eq. (5c), the relationship between λ_ and accumulative plastic strain rate is obtained: p_ ¼
(11)
Xi 1
(8)
R_ ¼ bðQ
RÞp_
(13)
where Q is the saturation value of the isotropic deformation resistance, and b is the material constant, which controls the evolution rate of R. Based on the monotonic tensile curve at room temperature in Fig. 5, the initial yield strength can be estimated to be 153 MPa. Parameters used for this model can be determined by the stress vs. strain curve under stress-controlled fatigue text referring to the method provided by Lynda Djimli et al. [33] and the specific value is listed in Table 2.
(9)
substituting Eq. (6) into Eq. (9), the fatigue damage rate is expressed as: !S0 σ2eq RV _ p; _ (10) D¼ 2Ebð1 DÞ2
4. Results
The effect of temperature is not investigated in this study and both parameters are set to unit values. The fatigue damage model was compiled by the user subroutine named as USDFLD. The stiffness of the element decreases with the accumulation of damage. When the fatigue damage accumulates to the critical value (D ¼ 1), the stiffness of the element is reduced to a minimal value and the fatigue crack propagates to the next sequential element.
4.1. Distribution of residual stress Fatigue test is terminated after the crack length reaches 16 mm and the applied load reduced to 0. The value and direction of first principle are shown in Fig. 6. It can be clearly seen that the direction of first principle around crack tip are different, which is not vertical to the crack growth direction. To further compare the simulated and experimental results, contour plots of transverse (a), longitudinal (b) and normal (c) residual stress after fatigue load are shown in Fig. 7. And the distribu tions of residual stresses around the crack tip (along path P) are shown in Fig. 8(a)-(c). Clearly, the measurement results are in good agreement
3.3. Material hardening In this paper, the cyclic kinematic hardening plastic deformation is calculated by a non-linear kinematic law characterized by back stress 4
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
and then deceases to zero at 5 mm, after which the figure of normal residual stress does not experience any remarkable change. 4.2. Variation of the stress fields under static load Three different static loads (0.01 Pmax, 0.5 Pmax and 1 Pmax) were applied after terminating the fatigue load and measuring the residual stress for the first time. The direction of the static load is shown in Fig. 2. The distribution of stress field around the crack tip after applying static load is obtained by neutron diffraction method and finite element simulation, as shown in Fig. 9. The consistent results from experiment and FEM are observed. When the applied load is 0.01 Pmax, the maximum longitudinal compressive stress of 195 MPa is saw at 0 mm, nearly equal to the maximum longitudinal tensile stress of 182 MPa at 7.6 mm. The compressive residual stress is converted to tensile residual stress at about 2 mm. The rising trend is also seen in transverse stresses and normal stresses around the crack tip, but there is no tensile stress in transverse stress. With the static load increasing to 0.5 Pmax, the figure of longitudinal stresses climbs to 217 MPa at about 7 mm, despite longi tudinal tensile stress of 80 MPa is observed at 0 mm. The evolution of transverse and normal stresses with distance from crack tip are basically unchanged. In the contrast, there are opposite tendency in the evolution of the stresses fields when the applied load is 1 Pmax. The maximum longitudinal tensile stress of 369 MPa is observed at 0 mm, and the longitudinal stress decreases nonlinearly with the increase of the dis tance from the crack tip. Firstly, a sharp decline is observed at 0 mm–1mm. Then the stress enters a slowly decreasing stage, and the stress reaches a minimum of about 100 MPa at 10 mm. The distribution of the transverse and normal stress fields is similar to that of the longi tudinal stress field, with maximum values of 339 MPa and 160 MPa, respectively. This indicates that the static load is the main cause of fa tigue crack propagation, and when the static load is large enough, the crack will propagate.
Fig. 5. Stress-strain relationship of 304L at room temperature [34]. Table 2 Parameters of the CHABOCHE model for 304L stainless steel. Isotropic parameter: E ¼ 180.0 GPa, ν ¼ 0.3, Q ¼ 150 MPa, b ¼ 12 Kinematic hardening rule: C1 ¼ 95000, γ1 ¼ 7000, C2 ¼ 85325,γ2 ¼ 6500, C3 ¼ 35000, γ3 ¼ 500, C4 ¼ 9000, γ4 ¼ 500, n ¼ 4
4.3. Effect of static load on residual stress This section investigates how the residual stresses change when the static load is removed. The figure and direction of first principle around the crack tip after overload (2 Pmax) is shown in Fig. 10. Similar to the results in Fig. 6, the direction of first principle also are different so it is necessary to compare the stress in three different directions. Fig. 11 show the residual stress contour plots for after static load. Fig. 12 (a)-(c) show the effect of static load on residual stress. It is worth mentioning that regardless of the magnitude of the static load, compressive residual stress is generated in all three directions at 0 mm, and the absolute value of the residual stress increases as the static load increases. There is a large fluctuation in the longitudinal residual stress distributed around the crack tip, as shown in Fig. 11 (b) and Fig. 12 (b). For example, when the static load is 2 Pmax, the maximum compressive LRS of 656 MPa occurs at 0 mm. Meanwhile, the CRS rapidly converts to transverse re sidual stress at 4.5 mm and reaches a maximum tensile stress of 226 MPa at 8.4 mm. The fluctuation of the longitudinal residual stress along the crack exceeds 800 MPa. With the increase of static load, the devel opment trend of longitudinal residual stress becomes more gradual, and the position where compressive stress is converted into tensile stress is gradually away from the crack tip. Different from the distribution of the longitudinal residual stress, the evolution of the transverse residual stress is more gradual with the increase of static load. However, although the transverse residual stress gradually decreases along the path P, it is reduced to zero only away from the crack tip and there is no reverse increase in tensile residual stress. In addition, both the maximum compressive stress at the crack tip and the overall stress fluctuation of the transverse residual stress are smaller than the longitudinal residual stress. Undoubtedly, the changes in static load will also affect the evo lution of normal residual stress. As shown in Fig. 12 (c), when the static load is greater than 1 Pmax, the normal residual stress generated at 0 mm
Fig. 6. First principle around the crack tip after fatigue load.
with the finite element analysis results, which verifies the validity of the finite element model. The compressive transverse residual stresses are tested by NDM. The transverse residual stress reaches 306 MPa at the crack tip and gradually decreases to 0 MPa at approximately 7 mm (Fig. 7(a)). The figure of the longitudinal residual stress shows a similar pattern. The maximum compressive residual stress ( 200 MPa) occurs at 0 mm, and transitions to the tensile residual stress at about 2 mm. Then, the longitudinal residual stress appears a maximum tensile stress of 180 MPa at 7 mm. Finally, it is stable at zero after experiencing decrease and increase again. As shown in Fig. 8(c), the normal stress changes more gently around the crack tip comparing with the transverse and longitudinal residual stresses. There is CRS of 45 MPa at crack tip 5
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 7. Contour plots of transverse (a), longitudinal (b) and normal (c) residual stress around crack tip after fatigue load.
significantly increases as the static load increases. However, although the static load increases from 0 Pmax to 1 Pmax, there is no change in the transverse, normal and longitudinal residual stresses. This indicates that the static load less than the fatigue load has no effect on the distribution of residual stress. Interestingly, as the static load is larger than 1 Pmax, the compressive residual stresses are increased as the static load in creases from 2 Pmax to 3 Pmax, which means that overload is beneficial to the generation of compressive residual stress.
worth noting that compressive stresses appear at the crack tip when the static load is smaller than the maximum cyclic load, as shown in Fig. 13. This because compressive stress induced by inhomogeneous plastic deformation is larger than the tensile stress produced by the applied load. The rising trend can be seen in three stress directions as the static loads increase from 0.01 Pmax to 1 Pmax, where the increase of the lon gitudinal stress is the largest, which is more than twice the increase of normal stress. As shown in Fig. 9, the crack tip stress goes to tensile state once the tensile stress induced by the static load exceeds compressive stress induced by inhomogeneous plastic deformation. When the sub sequent static load exceeds 1 Pmax, stresses in three directions are greater than the yield limit, plastic deformation occurs permanently at the crack tip. The plastic zone is elongated and becomes larger than the previous cycle. Plastic deformation accumulates when static load is applied. After unloading, the stretched plastic zone will be compressed due to the restraining effect of the surrounding elastic zone. Similar to the obser vations of i.e. Thielena et al. [35], the irreversible plastic deformation induced by tensile overload leads to the occurrence of CRS. Meanwhile, the CRS near the crack tip offsets the TRS away from the crack tip. Each cyclic fatigue load produces plasticity, so the crack propagates in the plastic zone created by the previous cyclic load. When the static load is less than 1 Pmax, the crack tip is still in the plastic zone created by the fatigue load. There is no new plastic zone generated and the original plastic zone does not elongate. Therefore, after the static load is
5. Discussion 5.1. The reason for the evolution of stress and residual stress field As discussed above, the compressive residual stresses resulted from static load are generated after the fatigue test is terminated. When the fatigue load is applied, permanent plastic deformation accumulates at the crack tip. And such plastic deformation induced by fatigue and static load cannot be completely recovered after releasing the applied load. Due to the limitation of the elastic region around the plastic zone, re sidual stress is generated at the crack tip after the fatigue load or static load is reduced to zero. Heterogeneous plastic deformation of CT specimen with notch under fatigue loading and subsequent tensile loading is observed. As shown in Fig. 9, this leads to compressive/tensile stress around the crack. It is 6
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 9. Variation of transverse (a), longitudinal (b) and normal (c) stress field when loaded by 0.01 Pmax, 0.5 Pmax and 1 Pmax.
Fig. 8. Distribution of transverse (a), longitudinal (b) and normal (c) residual stress around crack tip.
larger plastic deformation zone. Likewise, larger CRS is generated dur ing the unloading because of the restraint of the surrounding elastic zone. Based on the above analysis, it can be inferred that the CRS near the crack tip after unloading are determined by the maximum value at uploading, independent of the loading history.
unloaded, the magnitude of the residual stress is the same as after the fatigue load is unloaded. This means that single static load does not produce new residual stress. Undoubtedly, when the tensile stress is greater than 1 Pmax, the deformation caused by the static load will pass through the plastic zone generated by the fatigue load, resulting in a 7
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 10. First principle around the crack tip after overload (2 Pmax).
Fig. 11. Contour plots of transverse (a), longitudinal (b) and normal (c) re sidual stress around crack tip after overload (2 Pmax). Fig. 12. Effect of static load on residual stress.
5.2. Relationship between crack propagation and residual stress
the crack tip induced by a single overload changes the distribution of the residual stress field caused by the fatigue load. The change of residual stress field is an important consideration to understand the mechanism of crack propagation under variable amplitude loads. Undoubtedly, the CRS in the crack propagation direction and the tensile direction reduces the driving force of the crack tip, thereby significantly affecting the crack growth rate. Applying an overload after a series of fatigue cyclic loads causes the crack tip to open and become blunt, which reduce the
When exploring crack propagation under variable amplitude loads, there are two aspects that need to be taken into account: (i) the appli cation of single overload obviously affects the accumulation of irre versible plastic strain at the crack tip, which result in CRS field; (ii) under different amplitude loading, fatigue crack propagation is not only related to the currently applied load, but also relies on the previous loading amplitude, so the loading sequence has a non-negligible effect on crack propagation. The irreversible non-uniform deformation around 8
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 13. Stress values in three directions at the crack tip (0 mm).
Fig. 14. Residual stress values in three directions at the crack tip (0 mm).
opening stress and degrade the expansion of the crack to new crack tip plastic zone formed by overload. Schijve et al. [36] conducted a single overload test on aluminum alloy D16 with a single overload, and found that the crack appeared to be delayed during the expansion process. The delay of crack propagation may be due to the occurrence of large compressive residual stress. As shown in Fig. 14, after a single overload is applied, the change of compressive residual stress occurs in all three directions. The residual stress in the crack propagation direction changes the most, almost three times the residual stress after the fatigue load stops. At the same time, owing to the influence of CRS, crack closure may become more and more severe when the crack continues to propagate into the overloaded plastic zone, and therefore, the crack growth rate is decreased. When an overload is applied after the fatigue load, the CRS at the crack tip becomes larger, which contributes to a decrease in the crack growth rate, and it is worth noting that the mini mum crack growth rate does not occur immediately after the overload. However, when the static load is less than the fatigue load, the CRS does not change after unloading, which means that the plastic zone at the crack tip does not change, and the crack growth rate remains stable when the fatigue load continues to be applied. A number of experiments [29,37] on steel show that although a brief crack propagation acceleration occurs after a single overload, the crack growth rate is then significantly reduced. Therefore, the addition of a proper overload to the fatigue load causes a decrease in the average rate of crack propagation and an increase in the fatigue life of the structure. In order to avoid crack propagation during overload, the overload should not exceed the fracture toughness for safety reasons. When the overload reaches its maximum value, it should be unloaded immediately to prevent creep cracking. Applying overload provides a way to delay crack propagation in engineering by increasing compressive residual stress, but in practice it is necessary to monitor the actual condition of crack propagation in real time and consider the effect of crack propa gation on the static strength of the structure.
(1) Residual stresses around the fatigue tip have been measured by NDM. The FEM uses the fatigue damage model considering the hardening effect of the material. The calculated residual stress field and stress field at the fatigue crack tip agree well with the experimental measurements. (2) Compressive residual stresses attributed to tensile stress or compressive stress have been generated around the crack tip after the fatigue load is removed. The stress state at the crack tip obviously depends on the level of the static load. When the static load is less than the maximum fatigue load, the compressive stress field appears at the crack tip because the compressive stress induced by inhomogeneous plastic deformation is greater than the tensile stress produced by the static load. (3) When the static load is less than 1 Pmax, the transverse and lon gitudinal residual stress does not change although the static load increases. However, when the static load is greater than 1 Pmax, the compressive residual stresses increase as the static load in creases, revealing the larger CRS is induced by the overload of static stress after unloading, which is determined by the maximum value at uploading, independent of the loading history. Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. CRediT authorship contribution statement Wenchun Jiang: Conceptualization, Supervision. Yue Yu: Writing original draft, Data curation, Formal analysis. Weiya Zhang: Method ology, Software, Validation. Chengran Xiao: Writing - review & editing, Visualization. Wanchuck Woo: Resources.
6. Conclusions This paper studied the residual stress of CT specimens after removing static load by FEM and NDM. In order to predict the residual stress of CT specimens after undergoing fatigue cracking and overload, a simplified and accurate finite element method is presented. Based on the results, the following conclusions can be proposed:
Acknowledgments The authors gratefully acknowledge the support provided by the National Key R&D Program of China (2018YFC0808801), National Natural Science Foundation of China (51575531), Chang Jiang Scholars Program, Taishan Scholar Construction Funding (ts201511018), and 9
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
National Natural Science Foundation of China (51805543).
[17] M. Poncelet, G. Barbier, B. Raka, S. Courtin, R. Desmorat, J.C. Le-Roux, L. Vincent, Biaxial High Cycle Fatigue of a type 304L stainless steel: cyclic strains and crack initiation detection by digital image correlation, Eur. J. Mech. A Solid. 29 (5) (2010) 810–825. [18] V. Tvergaard, Effect of underloads or overloads in fatigue crack growth by crack-tip blunting, Eng. Fract. Mech. 73 (7) (2006) 869–879. [19] A. Kumar, A.R. Murthy, N.R. Iyer, Crack growth prediction under variable amplitude loading considering elastic-plastic stress field ahead of crack tip, Struct. Integr. 86 (2014) 645–652. [20] F.V. Antunes, F.A. Castanheira, R. Branco, A numerical analysis of the mechanisms behind plasticity induced crack closure: application to variable amplitude loadings, Int. J. Fatigue 89 (2016) 43–52. [21] H. Dirik, T. Yalçinkaya, Fatigue crack growth under variable amplitude loading through XFEM, Procedia Struct. Integr. 2 (2016) 3073–3080. [22] E. Ferrie, J.Y. Buffiere, W. Ludwig, A. Gravouil, L. Edwards, Fatigue crack propagation: in situ visualization using X-ray microtomography and 3D simulation using the extended finite element method, Acta Mater. 54 (4) (2006) 1111–1122. [23] Z.G. Zhou, L.J. Jia, Damage index for crack initiation of structural steel under cyclic loading, J. Constr. Steel Res. 114 (2015) 1–7. [24] J.H. You, Damage and fatigue crack growth of Eurofer steel first wall mock-up under cyclic heat flux loads. Part 2: finite element analysis of damage evolution, Fusion Eng. Des. 89 (4) (2014) 294–301. [25] P.J. Huffman, A strain energy based damage model for fatigue crack initiation and growth, Int. J. Fatigue 88 (2016) 197–204. [26] S. Benkabouche, H. Guechichi, A. Amrouche, M. Benkhettab, A modified nonlinear fatigue damage accumulation model under multiaxial variable amplitude loading, Int. J. Mech. Sci. 100 (2015) 180–194. [27] A. Ferjaoui, T. Yue, M.A. Wahab, R. Hojjati-Talemi, Prediction of fretting fatigue crack initiation in double lap bolted joint using continuum damage mechanics, Int. J. Fatigue 73 (2015) 66–76. [28] D. Krajcinovic, J. Lemaître, Continuum Damage Mechanics Theory and Application, International Centre for Mechanical Sciences, 1987, p. 295. [29] Z.Y. Ding, X.G. Wang, Z.L. Gao, S.Y. Bao, An experimental investigation and prediction of fatigue crack growth under overload/underload in Q345R steel, Int. J. Fatigue 98 (2017) 155–166. [30] M. Springer, H.E. Pettermann, Fatigue life predictions of metal structures based on a low-cycle, multiaxial fatigue damage model, Int. J. Fatigue 116 (2018) 355–365. [31] B. Fazlali, B. Mohammadi, A microscale energy-based fatigue damage model for unidirectional composites under multiaxial loading at different stress ratios, Eng. Fract. Mech. 205 (2019) 120–135. [32] J.L. Chaboche, A review of some plasticity and viscoplasticity constitutive theories, Int. J. Plast. 24 (10) (2008) 1642–1693. [33] L. Djimli, L. Taleb, S. Meziani, The role of the experimental data base used to identify material parameters in predicting the cyclic plastic response of an austenitic steel, Int. J. Press. Vessel. Pip. 87 (4) (2010) 177–186. [34] L. Taleb, A. Hauet, Multiscale experimental investigations about the cyclic behavior of the 304L SS, Int. J. Plast. 25 (7) (2009) 1359–1385. [35] M. Thielena, F. Schaefera, P. Gruenewalda, M. Lauba, M. Marxa, M. Meixnerb, M. Klausb, C. Motza, In situ synchrotron stress mappings to characterize overload effects in fatigue crack growth, Int. J. Fatigue 121 (2019) 155–162. [36] J. Schijve, M. Skorupa, A. Skorupa, T. Machniewicz, P. Gruszczynski, Fatigue crack growth in the aluminium alloy D16 under constant and variable amplitude loading, Int. J. Fatigue 26 (1) (2004) 1–15. [37] S. Kalnaus, F. Fan, A.K. Vasudevan, Y. Jiang, An experimental investigation on fatigue crack growth of AL6XN stainless steel, Eng. Fract. Mech. 75 (8) (2008) 2002–2019.
References [1] H. Kovaci, A.F. Yetim, O. Baran, A. Celik, Fatigue crack growth analysis of plasma nitrided AISI 4140 low-alloy steel: Part 2-Variable amplitude loading and load interactions, Mater. Sci. Eng.-Struct. Mater. Prop. Microstruct. Process. 672 (2016) 265–275. [2] M. Fitzka, H. Mayer, Constant and variable amplitude fatigue testing of aluminum alloy 2024-T351 with ultrasonic and servo-hydraulic equipment, Int. J. Fatigue 91 (2016) 363–372. [3] H. Kovaci, A.F. Yetim, O. Baran, A. Celik, Fatigue crack growth behavior of DLC coated AISI 4140 steel under constant and variable amplitude loading conditions, Surf. Coat. Technol. 304 (2016) 316–324. [4] P. Lopez-Crespo, A. Steuwer, T. Buslaps, Y.H. Tai, A. Lopez-Moreno, J.R. Yates, P. J. Withers, Measuring overload effects during fatigue crack growth in bainitic steel by synchrotron X-ray diffraction, Int. J. Fatigue 71 (2015) 11–16 [1]. [5] N. Gates, A. Fatemi, Multiaxial variable amplitude fatigue life analysis including notch effects, Int. J. Fatigue 91 (2016) 337–351. [6] X. Zhou, H.P. Gaenser, R. Pippan, The effect of single overloads in tension and compression on the fatigue crack propagation behaviour of short cracks, Int. J. Fatigue 89 (2016) 77–86. [7] R. Sunder, A. Biakov, A. Eremin, S. Panin, Synergy of crack closure, near-tip residual stress and crack-tip blunting in crack growth under periodic overloads - a fractographic study, Int. J. Fatigue 93 (2016) 18–29. [8] E. Salvati, S. O’Connor, T. Sui, D. Nowell, A.M. Korsunsky, A study of overload effect on fatigue crack propagation using EBSD, FIB-DIC and FEM methods, Eng. Fract. Mech. 167 (2016) 210–223. [9] S.Y. Lee, P.K. Liaw, H. Choo, R.B. Rogge, A study on fatigue crack growth behavior subjected to a single tensile overload: Part I. An overload-induced transient crack growth micromechanism (vol 59, pg 485, 2011), Acta Mater. 59 (10) (2011), 42534253. [10] S.Y. Lee, H. Choo, P.K. Liaw, K. An, C.R. Hubbard, A study on fatigue crack growth behavior subjected to a single tensile overload: Part II. Transfer of stress concentration and its role in overload-induced transient crack growth (vol 59, pg 495, 2011), Acta Mater. 59 (10) (2011), 4254-4254. [11] S.Y. Lee, R.B. Rogge, H. Choo, P.K. Liaw, Neutron diffraction measurements of residual stresses around a crack tip developed under variable-amplitude fatigue loadings, Fatigue Fract. Eng. Mater. Struct. 33 (12) (2010) 822–831. [12] S. Seo, E.W. Huang, W. Woo, S.Y. Lee, Neutron diffraction residual stress analysis during fatigue crack growth retardation of stainless steel, Int. J. Fatigue 104 (2017) 408–415. [13] M. Croft, V. Shukla, N.M. Jisrawi, Z. Zhong, R.K. Sadangi, R.L. Holtz, P.S. Pao, K. Horvath, K. Sadananda, A. Ignatov, J. Skaritka, T. Tsakalakos, Mapping and load response of overload strain fields: synchrotron X-ray measurements, Int. J. Fatigue 31 (11–12) (2009) 1669–1677. [14] A. Steuwer, M. Rahman, A. Shterenlikht, M.E. Fitzpatrick, L. Edwards, P.J. Withers, The evolution of crack-tip stresses during a fatigue overload event, Acta Mater. 58 (11) (2010) 4039–4052. [15] L. Xiao, D.Y. Ye, C.Y. Chen, J.Z. Liu, L.N. Zhang, Instrumented indentation measurements of residual stresses around a crack tip under single tensile overloads, Int. J. Mech. Sci. 78 (2014) 44–51. [16] F. Yusof, P. Lopez-Crespo, P.J. Withers, Effect of overload on crack closure in thick and thin specimens via digital image correlation, Int. J. Fatigue 56 (2013) 17–24.
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International Journal of Pressure Vessels and Piping journal homepage: http://www.elsevier.com/locate/ijpvp
Residual stress and stress fields change around fatigue crack tip: Neutron diffraction measurement and finite element modeling Wenchun Jiang a, *, Yue Yu a, Weiya Zhang a, Chengran Xiao a, Wanchuck Woo b a b
School of New Energy, China University of Petroleum (East China), Qingdao, 266555, PR China Neutron Science Division, Korea Atomic Energy Research Institute, 1045 Daedeok-daero, Yuseong-gu, Daejeon, 305-353, South Korea
A R T I C L E I N F O
A B S T R A C T
Keywords: Crack tip Residual stress Neutron diffraction Overload
The variation of residual stress and stress field induced by the irreversible deformation at the fatigue crack tip of compact tension (CT) specimen subjected to different loads are studied by finite element method (FEM) and neutron diffraction measurement (NDM). The finite element method adopts a fatigue damage model considering material hardening effect. The stress field induced by one static load after the cyclic loads and the corresponding residual stress field at fatigue crack tip are found to match the experimental measurement. The influence of the static load on the residual stress is discussed by investigating the three principal stresses, that is the longitudinal, the transverse and the normal stresses. The results show that the residual stress at crack tip is closely correlated to the static load magnitude when this magnitude is greater than the maximum fatigue load (Pmax), while this correlation does not exist for the load smaller than Pmax. Different from the residual stress, the stress field at the crack tip increases with the increase of static load. Especially when the load is less than 1 Pmax, the stress field at the crack tip mainly appears a compressive stress state. Meanwhile, the compressive residual stress ahead of the crack tip is generated after the release of the static load. Regardless of the loading history, this residual stress is dependent only on the loading magnitude.
1. Introduction Many engineering structures work under the cyclic loads. The mag nitudes of these cyclic loads are changing or even stochastic depending on different working conditions [1]. The stress fields and the fatigue crack propagation resulted from the variable amplitude cyclic loads are distinct from the stress field generated from the constant amplitude cyclic loads [2]. Therefore, the researches on constant amplitude cyclic loads are insufficient for the engineering applications. It is necessary to study the crack tip stress field and the crack growth rate under variable amplitude loads. In recent years, the effects of variable amplitude loads, such as overload and underload (Fig. 1), on fatigue crack have been extensively studied by both experiments and numerical models [3–6]. Recently, the influences of overload/underload on fatigue crack growths are intensively discussed. Sander et al. [7] found that the crack growth rates are significantly affected by the load sequences. When underload is applied before the tensile overload, the fatigue crack growth rates are not affected regardless of the amplitude. However, applying an underload after an overload accelerates crack propagation, which is related to the magnitude of the underload. Overload could
induce crack growth retardation because of the reduction of the crack driving force and the generation of large compressive residual stress [8]. In addition, large deformation occurred in the overload part and the crack side. The distribution of crack tip residual stress is mainly obtained by neutron diffraction measurement (NDM) and synchrotron X-ray diffraction measurement (XDM). The compressive residual stress created by a single overload is measured by using NDM and the delayed crack growth rate is identified [9–11]. Interestingly, as the crack continues to expand, the crack growth rate gradually increases to the rate before the overload is applied. In addition, the residual stress varies in the thick ness direction of the specimen [12]. Using the high energy XDM, the change in strain field after overload is discussed in depth. A single overload cycle can create a compressive strain field around the crack plane. However, the sample with a delay in the maximum crack growth rate after overload produced a tensile strain field at the crack tip [13]. Steuwer et al. [14] studied the stress distribution at the crack-tip sub jected to the overloads. A compressive residual stress was detected during unloading and there were tensile stresses of no more than 100 MPa in front of the crack (about 100 μm). In recent years, nano inden tation [15] and digital image correlation technique [16,17] are also used to measure the micro residual stresses and microstructure evolution.
* Corresponding author. E-mail address: [email protected] (W. Jiang). https://doi.org/10.1016/j.ijpvp.2019.104024 Received 20 August 2019; Received in revised form 20 November 2019; Accepted 24 November 2019 Available online 27 November 2019 0308-0161/© 2019 Elsevier Ltd. All rights reserved.
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
enhanced the effect of residual plastic wake. In recent years, the Extended Finite Element Method (XFEM) [21,22] and the damage me chanics method [23,24] are also used to predict the fatigue crack propagation and fatigue life under different amplitude load. Huffman [25] proposed a fatigue damage model based on strain energy to calculate the fatigue life and fatigue crack growth rate under strain amplitude. Unfortunately, the accuracy of the model needs to be further improved at very low strain amplitude. Benkabouche et al. [26] devel oped an improved nonlinear continuous damage mechanics (CDM) model to predict damage evolution and lifetime by considering the magnitude of the load and the load sequence. This model is suitable for multi-axis situations, while only for high cycle fatigue. CDM also showed good prediction results in the simulation of welded joints considering weld residual stress [27,28]. In addition, the CDM method also has been extended to multi-axis fatigue conditions to simulate the entire damage accumulation process reflecting material degradation [29,30]. Recently, Fazlali et al. [31] developed a CDM model that can distinguish the damage between fiber and matrix and predicted transverse and longi tudinal residual stiffness. The present study is an extension of the work described above. In this paper, fatigue crack growth was predicted through continuous damage mechanics model. The stress distribution ahead of the crack tip was measured by neutron diffraction measure ment. To compare the influence of static load on stress field, residual stresses before and after static load were also measured. The develop ment of crack-tip stresses during different static load and the influence of single overload following fatigue cyclic load on residual stress field around crack tip was revealed.
Nomenclature CT FEM NDM Pmax Pmin XDM CRS TRS XFEM CDM COD TD ND LD GV LRS NRS
compact tension finite element method neutron diffraction measurement the maximum fatigue load the minimum fatigue load X-ray diffraction measurement compressive residual stress tensile residual stress Extended Finite Element Method continuous damage mechanics crack-opening-displacement transverse direction normal direction longitudinal direction gauge volume longitudinal residual stress normal residual stress
2. Experimental 2.1. Fatigue test The fatigue crack growth tests are performed on an austenitic stainless steel 304L compact-tension specimen (CT specimen) with chemical components listed in Table 1 and geometry shown in Fig. 2. The CT specimens are prepared according to the ASTM Standards E647–99.22 with a notch length 10.16 mm, width 50.8 mm and thick ness 6.35 mm, respectively. Before performing the fatigue crack growth test, a pre-cracked length of 1.27 mm is made on the CT specimen. Then the stress-controlled crack propagation experiment is conducted under a constant load using the MTS Landmark 370.10 fatigue test system. The maximum fatigue load is 7400 N with load ratio R (Pmin/Pmax) equals 0.01 and load frequency equals 10 Hz. The crack length of the fatigue test is measured by crack-opening-displacement (COD) gauge. The fa tigue test is stopped after the crack length reaches 16 mm, and the re sidual stresses around the crack tip are measured.
Fig. 1. Different load types in fatigue experiments.
Besides the experimental approaches, the numerical methods are also applied to investigate the fatigue crack propagation characters and to predict its remaining life. In the past several decades, considerable efforts have been devoted to numerically study the promotion/inhibi tion of fatigue crack propagation by overload/underload. For instance, based on blunting re-sharpening mechanism of crack tip, Viggo Tver gaard [18] found that the compression underload is beneficial to the propagation of cyclic cracks through re-meshing method, while a single overload delays the propagation of cracks. In addition, the crack growth rate is also related to the overload induced plastic deformation. Tensile overload increases the threshold value of long cracks because of the plastic deformation. The compression overload reduces the crack propagation resistance and increases the crack propagation threshold [6]. Recently, the influence of linear elastic stress field and residual plastic deformation on crack propagation has been further studied by Kumar et al. [19] and Antunes et al. [20]. For overload and low-high load sequences, residual plastic deformation accumulates, and the re sidual stress caused by it decreases the crack opening level, which leads to the passivation of the crack tip. However, the crack tip blunting is abruptly reduced when high-low load sequences are applied, which
2.2. Neutron diffraction measurement After the fatigue test, the residual stresses around the fatigue tip are measured by neutron diffraction instrument at Korea Atomic Energy Research Institute (KAERI). This method is based on Bragg theory: nλ ¼ 2d sin θ;
(1)
where n is an integer, 2θ is the diffraction angle, d is the lattices pacing, λ is the wavelength of neutron beam. The lattice strain is obtained from:
ε¼
Δd ¼ d
Δθ cot θ;
(2)
where Δd is the change of the lattice spacing, Δθ is the shift of the angular position. Three strain components (in transverse (TD), normal (ND) and lon gitudinal (LD) directions, respectively) are measured and the residual stress σi (i ¼ x, y or z) is calculated based on the measured strain: 2
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Table 1 Chemical composition of 304L stainless steel (%). Component
C
Mn
P
S
Si
Cu
Ni
Cr
Mo
N2
304L
0.029
1.86
0.029
0.004
0.37
0.02
10
18
0.04
0.056
3. Finite element analysis The residual stress and stress field changes around the fatigue crack tip are simulated by FEM. Firstly, the fatigue crack propagation is simulated based on the CDM. After the crack length reaches 16 mm, the fatigue load is released and the residual stress distribution is then ob tained. Then, static loads with different magnitudes are applied to determine the change of stress field and residual stress field. Finally, compare the simulation results with the measurement results to verify the current model. 3.1. Finite element model Considering the geometrical symmetry, as shown in Fig. 4, only one quarter of the specimen is modeled and meshed with 23557 elements and 29345 nodes. The element type is linear hexahedral elements named C3D8. In order to limit the rigid movement of the specimen, a reference point is set at the center of the loading hole to apply a constraint in the X direction. Corresponding to the modeling characteristics, symmetric boundary conditions are set on the intermediate surface of the CT sample. The maximum force is 7400 N and the stress ratio R is 0.01, which are identical to the experiment.
Fig. 2. Geometry of the CT specimen for 304L stainless steel.
σi ¼
E νE εi þ 1þν ð1 þ νÞð1
�
2νÞ
εx þ εy þ εz ;
(3)
3.2. Fatigue damage model
where E is the elastic modulus and ν is the Poisson’s ratio. The range from 0.5 mm to 22 mm at the tip of the crack in the crack-growth di rection is measured. The gauge volume (GV) is 1(x) � 1(y) � 1(z) mm3. The monochromatic beam with wavelengths at 2.39 Å and 1.55 Å for (110) and (211) on scattering angles of 72.1� and 82.9� are used, respectively. After the residual stress measurement, the loads with three distinct magnitudes are applied to the sample and the lattice strain evolution is measured in situ. The experimental installations are shown in Fig. 3.
The fatigue crack propagation is simulated based on the continuum damage mechanics. As proposed by Krajcinovic and Lemaître [28–30], the dissipate potential φ is described as: (4a)
φ ¼ φp þ φD þ φπ ; φp ¼
σeq 1
φD ¼
R D
(4b)
σY ;
b ðS0 þ 1Þð1
� DÞ
Y b
�S0 þ1 ;
(4c)
where φp is the plastic dissipated potential, φD is the damage dissipated potential, φπ is the micro-plastic dissipated potential, R is the isotropic hardening scalar variable associated with accumulated plastic strain, σY is the initial yield stress of the material, σ eq is the Von Mises equivalent stress, D is an isotropic damage scalar variable, and S0 and b are tem perature dependent material parameters representing unified damage law exponent and energetic damage law parameter, respectively. The effect of the micro-plastic dissipation potential is small and is not considered in this paper. The plastic strain tensor εpij equation coupled damage is derived from the plastic dissipated potential φp and the damage evolution equation is derived from the damage dissipated po tential. The strain tensor εij consists of two parts: the elastic strain tensor εeij and the plastic strain tensor εpij , These are:
εij ¼ εeij þ εpij ; ε_ pij ¼ λ_
p_ ¼ Fig. 3. Experimental setup by neutron diffraction. 3
λ_ ∂φ _ ∂φp 3 Sij ¼λ ¼ :; ∂σ ij ∂σ ij 2 ð1 DÞ σ eq
�1=2 � 2 p p ε_ ij ε_ ij 3
(5a) (5b) (5c)
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 4. The entire mesh (a) and local fined mesh (b) for CT specimen.
∂φ D_ ¼ λ_ D ; ∂Y
using kinematic hardening modulus (Ci) and dynamic rate (γi). The size change of the yield surface and its shift through the stress space are considered. Nonlinear kinematic hardening and isotropic hardening variable are combined to describe the cyclic plastic deformation. The kinematic hardening variable (X) is shown in Eqs. (11) and (12). Based on the Armstrong-Frederick and Chaboche model [32], the back stress is decomposed into n parts and each part consists of a linear reinforcement term and a dynamic recovery term:
(5d)
where εpij is the plastic strain rate, p_ is the accumulated plastic strain rate, λ_ is the non-negative proportion factor, σ ij is the stress tensor, Sij is the
stress deviator, D is the isotropic damage scalar variable ranging from 0 to 1, D_ is the fatigue damage rate, Y is the strain energy release rate and the function is:
σ2eq RV
Y¼
2Eð1
DÞ2
;
n X
(6)
X¼
where E is the Yong’s modules and RV is the stress triaxiality factor defined by: � �2 2 σH RV ¼ ð1 þ υÞ þ 3ð1 2υÞ ; (7) 3 σ eq
2 X_ i ¼ Ci ε_ p 3
λ_ 1
D
;
from Eq. (4c) one can easily get: � �S0 ∂φ Y p; _ D_ ¼ ¼ b ∂Y
γi αi p_
(12)
where Ck and γk are material parameters, and the back stress compo nents corresponding to different Ck and γ k can reflect the nonlinear behavior of the material in different plastic deformation stages. Four back stress kinematic hardening components are adopted to better describe the stress-strain behavior of materials. And the accumulated plastic strain is defined by Eq. (5c). The isotropic hardening is expressed as:
where σ H is the hydrostatic stress, σ eq is Von Mises equivalent stress. By substituting Eq. (5b) to Eq. (5c), the relationship between λ_ and accumulative plastic strain rate is obtained: p_ ¼
(11)
Xi 1
(8)
R_ ¼ bðQ
RÞp_
(13)
where Q is the saturation value of the isotropic deformation resistance, and b is the material constant, which controls the evolution rate of R. Based on the monotonic tensile curve at room temperature in Fig. 5, the initial yield strength can be estimated to be 153 MPa. Parameters used for this model can be determined by the stress vs. strain curve under stress-controlled fatigue text referring to the method provided by Lynda Djimli et al. [33] and the specific value is listed in Table 2.
(9)
substituting Eq. (6) into Eq. (9), the fatigue damage rate is expressed as: !S0 σ2eq RV _ p; _ (10) D¼ 2Ebð1 DÞ2
4. Results
The effect of temperature is not investigated in this study and both parameters are set to unit values. The fatigue damage model was compiled by the user subroutine named as USDFLD. The stiffness of the element decreases with the accumulation of damage. When the fatigue damage accumulates to the critical value (D ¼ 1), the stiffness of the element is reduced to a minimal value and the fatigue crack propagates to the next sequential element.
4.1. Distribution of residual stress Fatigue test is terminated after the crack length reaches 16 mm and the applied load reduced to 0. The value and direction of first principle are shown in Fig. 6. It can be clearly seen that the direction of first principle around crack tip are different, which is not vertical to the crack growth direction. To further compare the simulated and experimental results, contour plots of transverse (a), longitudinal (b) and normal (c) residual stress after fatigue load are shown in Fig. 7. And the distribu tions of residual stresses around the crack tip (along path P) are shown in Fig. 8(a)-(c). Clearly, the measurement results are in good agreement
3.3. Material hardening In this paper, the cyclic kinematic hardening plastic deformation is calculated by a non-linear kinematic law characterized by back stress 4
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
and then deceases to zero at 5 mm, after which the figure of normal residual stress does not experience any remarkable change. 4.2. Variation of the stress fields under static load Three different static loads (0.01 Pmax, 0.5 Pmax and 1 Pmax) were applied after terminating the fatigue load and measuring the residual stress for the first time. The direction of the static load is shown in Fig. 2. The distribution of stress field around the crack tip after applying static load is obtained by neutron diffraction method and finite element simulation, as shown in Fig. 9. The consistent results from experiment and FEM are observed. When the applied load is 0.01 Pmax, the maximum longitudinal compressive stress of 195 MPa is saw at 0 mm, nearly equal to the maximum longitudinal tensile stress of 182 MPa at 7.6 mm. The compressive residual stress is converted to tensile residual stress at about 2 mm. The rising trend is also seen in transverse stresses and normal stresses around the crack tip, but there is no tensile stress in transverse stress. With the static load increasing to 0.5 Pmax, the figure of longitudinal stresses climbs to 217 MPa at about 7 mm, despite longi tudinal tensile stress of 80 MPa is observed at 0 mm. The evolution of transverse and normal stresses with distance from crack tip are basically unchanged. In the contrast, there are opposite tendency in the evolution of the stresses fields when the applied load is 1 Pmax. The maximum longitudinal tensile stress of 369 MPa is observed at 0 mm, and the longitudinal stress decreases nonlinearly with the increase of the dis tance from the crack tip. Firstly, a sharp decline is observed at 0 mm–1mm. Then the stress enters a slowly decreasing stage, and the stress reaches a minimum of about 100 MPa at 10 mm. The distribution of the transverse and normal stress fields is similar to that of the longi tudinal stress field, with maximum values of 339 MPa and 160 MPa, respectively. This indicates that the static load is the main cause of fa tigue crack propagation, and when the static load is large enough, the crack will propagate.
Fig. 5. Stress-strain relationship of 304L at room temperature [34]. Table 2 Parameters of the CHABOCHE model for 304L stainless steel. Isotropic parameter: E ¼ 180.0 GPa, ν ¼ 0.3, Q ¼ 150 MPa, b ¼ 12 Kinematic hardening rule: C1 ¼ 95000, γ1 ¼ 7000, C2 ¼ 85325,γ2 ¼ 6500, C3 ¼ 35000, γ3 ¼ 500, C4 ¼ 9000, γ4 ¼ 500, n ¼ 4
4.3. Effect of static load on residual stress This section investigates how the residual stresses change when the static load is removed. The figure and direction of first principle around the crack tip after overload (2 Pmax) is shown in Fig. 10. Similar to the results in Fig. 6, the direction of first principle also are different so it is necessary to compare the stress in three different directions. Fig. 11 show the residual stress contour plots for after static load. Fig. 12 (a)-(c) show the effect of static load on residual stress. It is worth mentioning that regardless of the magnitude of the static load, compressive residual stress is generated in all three directions at 0 mm, and the absolute value of the residual stress increases as the static load increases. There is a large fluctuation in the longitudinal residual stress distributed around the crack tip, as shown in Fig. 11 (b) and Fig. 12 (b). For example, when the static load is 2 Pmax, the maximum compressive LRS of 656 MPa occurs at 0 mm. Meanwhile, the CRS rapidly converts to transverse re sidual stress at 4.5 mm and reaches a maximum tensile stress of 226 MPa at 8.4 mm. The fluctuation of the longitudinal residual stress along the crack exceeds 800 MPa. With the increase of static load, the devel opment trend of longitudinal residual stress becomes more gradual, and the position where compressive stress is converted into tensile stress is gradually away from the crack tip. Different from the distribution of the longitudinal residual stress, the evolution of the transverse residual stress is more gradual with the increase of static load. However, although the transverse residual stress gradually decreases along the path P, it is reduced to zero only away from the crack tip and there is no reverse increase in tensile residual stress. In addition, both the maximum compressive stress at the crack tip and the overall stress fluctuation of the transverse residual stress are smaller than the longitudinal residual stress. Undoubtedly, the changes in static load will also affect the evo lution of normal residual stress. As shown in Fig. 12 (c), when the static load is greater than 1 Pmax, the normal residual stress generated at 0 mm
Fig. 6. First principle around the crack tip after fatigue load.
with the finite element analysis results, which verifies the validity of the finite element model. The compressive transverse residual stresses are tested by NDM. The transverse residual stress reaches 306 MPa at the crack tip and gradually decreases to 0 MPa at approximately 7 mm (Fig. 7(a)). The figure of the longitudinal residual stress shows a similar pattern. The maximum compressive residual stress ( 200 MPa) occurs at 0 mm, and transitions to the tensile residual stress at about 2 mm. Then, the longitudinal residual stress appears a maximum tensile stress of 180 MPa at 7 mm. Finally, it is stable at zero after experiencing decrease and increase again. As shown in Fig. 8(c), the normal stress changes more gently around the crack tip comparing with the transverse and longitudinal residual stresses. There is CRS of 45 MPa at crack tip 5
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 7. Contour plots of transverse (a), longitudinal (b) and normal (c) residual stress around crack tip after fatigue load.
significantly increases as the static load increases. However, although the static load increases from 0 Pmax to 1 Pmax, there is no change in the transverse, normal and longitudinal residual stresses. This indicates that the static load less than the fatigue load has no effect on the distribution of residual stress. Interestingly, as the static load is larger than 1 Pmax, the compressive residual stresses are increased as the static load in creases from 2 Pmax to 3 Pmax, which means that overload is beneficial to the generation of compressive residual stress.
worth noting that compressive stresses appear at the crack tip when the static load is smaller than the maximum cyclic load, as shown in Fig. 13. This because compressive stress induced by inhomogeneous plastic deformation is larger than the tensile stress produced by the applied load. The rising trend can be seen in three stress directions as the static loads increase from 0.01 Pmax to 1 Pmax, where the increase of the lon gitudinal stress is the largest, which is more than twice the increase of normal stress. As shown in Fig. 9, the crack tip stress goes to tensile state once the tensile stress induced by the static load exceeds compressive stress induced by inhomogeneous plastic deformation. When the sub sequent static load exceeds 1 Pmax, stresses in three directions are greater than the yield limit, plastic deformation occurs permanently at the crack tip. The plastic zone is elongated and becomes larger than the previous cycle. Plastic deformation accumulates when static load is applied. After unloading, the stretched plastic zone will be compressed due to the restraining effect of the surrounding elastic zone. Similar to the obser vations of i.e. Thielena et al. [35], the irreversible plastic deformation induced by tensile overload leads to the occurrence of CRS. Meanwhile, the CRS near the crack tip offsets the TRS away from the crack tip. Each cyclic fatigue load produces plasticity, so the crack propagates in the plastic zone created by the previous cyclic load. When the static load is less than 1 Pmax, the crack tip is still in the plastic zone created by the fatigue load. There is no new plastic zone generated and the original plastic zone does not elongate. Therefore, after the static load is
5. Discussion 5.1. The reason for the evolution of stress and residual stress field As discussed above, the compressive residual stresses resulted from static load are generated after the fatigue test is terminated. When the fatigue load is applied, permanent plastic deformation accumulates at the crack tip. And such plastic deformation induced by fatigue and static load cannot be completely recovered after releasing the applied load. Due to the limitation of the elastic region around the plastic zone, re sidual stress is generated at the crack tip after the fatigue load or static load is reduced to zero. Heterogeneous plastic deformation of CT specimen with notch under fatigue loading and subsequent tensile loading is observed. As shown in Fig. 9, this leads to compressive/tensile stress around the crack. It is 6
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 9. Variation of transverse (a), longitudinal (b) and normal (c) stress field when loaded by 0.01 Pmax, 0.5 Pmax and 1 Pmax.
Fig. 8. Distribution of transverse (a), longitudinal (b) and normal (c) residual stress around crack tip.
larger plastic deformation zone. Likewise, larger CRS is generated dur ing the unloading because of the restraint of the surrounding elastic zone. Based on the above analysis, it can be inferred that the CRS near the crack tip after unloading are determined by the maximum value at uploading, independent of the loading history.
unloaded, the magnitude of the residual stress is the same as after the fatigue load is unloaded. This means that single static load does not produce new residual stress. Undoubtedly, when the tensile stress is greater than 1 Pmax, the deformation caused by the static load will pass through the plastic zone generated by the fatigue load, resulting in a 7
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 10. First principle around the crack tip after overload (2 Pmax).
Fig. 11. Contour plots of transverse (a), longitudinal (b) and normal (c) re sidual stress around crack tip after overload (2 Pmax). Fig. 12. Effect of static load on residual stress.
5.2. Relationship between crack propagation and residual stress
the crack tip induced by a single overload changes the distribution of the residual stress field caused by the fatigue load. The change of residual stress field is an important consideration to understand the mechanism of crack propagation under variable amplitude loads. Undoubtedly, the CRS in the crack propagation direction and the tensile direction reduces the driving force of the crack tip, thereby significantly affecting the crack growth rate. Applying an overload after a series of fatigue cyclic loads causes the crack tip to open and become blunt, which reduce the
When exploring crack propagation under variable amplitude loads, there are two aspects that need to be taken into account: (i) the appli cation of single overload obviously affects the accumulation of irre versible plastic strain at the crack tip, which result in CRS field; (ii) under different amplitude loading, fatigue crack propagation is not only related to the currently applied load, but also relies on the previous loading amplitude, so the loading sequence has a non-negligible effect on crack propagation. The irreversible non-uniform deformation around 8
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International Journal of Pressure Vessels and Piping 179 (2020) 104024
Fig. 13. Stress values in three directions at the crack tip (0 mm).
Fig. 14. Residual stress values in three directions at the crack tip (0 mm).
opening stress and degrade the expansion of the crack to new crack tip plastic zone formed by overload. Schijve et al. [36] conducted a single overload test on aluminum alloy D16 with a single overload, and found that the crack appeared to be delayed during the expansion process. The delay of crack propagation may be due to the occurrence of large compressive residual stress. As shown in Fig. 14, after a single overload is applied, the change of compressive residual stress occurs in all three directions. The residual stress in the crack propagation direction changes the most, almost three times the residual stress after the fatigue load stops. At the same time, owing to the influence of CRS, crack closure may become more and more severe when the crack continues to propagate into the overloaded plastic zone, and therefore, the crack growth rate is decreased. When an overload is applied after the fatigue load, the CRS at the crack tip becomes larger, which contributes to a decrease in the crack growth rate, and it is worth noting that the mini mum crack growth rate does not occur immediately after the overload. However, when the static load is less than the fatigue load, the CRS does not change after unloading, which means that the plastic zone at the crack tip does not change, and the crack growth rate remains stable when the fatigue load continues to be applied. A number of experiments [29,37] on steel show that although a brief crack propagation acceleration occurs after a single overload, the crack growth rate is then significantly reduced. Therefore, the addition of a proper overload to the fatigue load causes a decrease in the average rate of crack propagation and an increase in the fatigue life of the structure. In order to avoid crack propagation during overload, the overload should not exceed the fracture toughness for safety reasons. When the overload reaches its maximum value, it should be unloaded immediately to prevent creep cracking. Applying overload provides a way to delay crack propagation in engineering by increasing compressive residual stress, but in practice it is necessary to monitor the actual condition of crack propagation in real time and consider the effect of crack propa gation on the static strength of the structure.
(1) Residual stresses around the fatigue tip have been measured by NDM. The FEM uses the fatigue damage model considering the hardening effect of the material. The calculated residual stress field and stress field at the fatigue crack tip agree well with the experimental measurements. (2) Compressive residual stresses attributed to tensile stress or compressive stress have been generated around the crack tip after the fatigue load is removed. The stress state at the crack tip obviously depends on the level of the static load. When the static load is less than the maximum fatigue load, the compressive stress field appears at the crack tip because the compressive stress induced by inhomogeneous plastic deformation is greater than the tensile stress produced by the static load. (3) When the static load is less than 1 Pmax, the transverse and lon gitudinal residual stress does not change although the static load increases. However, when the static load is greater than 1 Pmax, the compressive residual stresses increase as the static load in creases, revealing the larger CRS is induced by the overload of static stress after unloading, which is determined by the maximum value at uploading, independent of the loading history. Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. CRediT authorship contribution statement Wenchun Jiang: Conceptualization, Supervision. Yue Yu: Writing original draft, Data curation, Formal analysis. Weiya Zhang: Method ology, Software, Validation. Chengran Xiao: Writing - review & editing, Visualization. Wanchuck Woo: Resources.
6. Conclusions This paper studied the residual stress of CT specimens after removing static load by FEM and NDM. In order to predict the residual stress of CT specimens after undergoing fatigue cracking and overload, a simplified and accurate finite element method is presented. Based on the results, the following conclusions can be proposed:
Acknowledgments The authors gratefully acknowledge the support provided by the National Key R&D Program of China (2018YFC0808801), National Natural Science Foundation of China (51575531), Chang Jiang Scholars Program, Taishan Scholar Construction Funding (ts201511018), and 9
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National Natural Science Foundation of China (51805543).
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