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Modeling of fatigue crack growth in a pressure vessel steel Q345R
Accepted Manuscript Modeling of fatigue crack growth in a pressure vessel steel Q345R Zhenyu Ding, Zengliang Gao, Xiaogui Wang, Yanyao Jiang PII: DOI: Reference:
S0013-7944(15)00012-0 http://dx.doi.org/10.1016/j.engfracmech.2015.01.011 EFM 4467
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
21 March 2014 9 January 2015 10 January 2015
Please cite this article as: Ding, Z., Gao, Z., Wang, X., Jiang, Y., Modeling of fatigue crack growth in a pressure vessel steel Q345R, Engineering Fracture Mechanics (2015), doi: http://dx.doi.org/10.1016/j.engfracmech. 2015.01.011
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Submitted to Engineering Fracture Mechanics March 2014 EFM-S-14-00137, Revised October, 2014
Modeling of fatigue crack growth in a pressure vessel steel Q345R Zhenyu Ding a, Zengliang Gao a*, Xiaogui Wang a, Yanyao Jiang b a
Zhejiang University of Technology, College of Mechanical Engineering, Hangzhou, Zhejiang 310032, China b
University of Nevada, Reno, Department of Mechanical Engineering Reno, NV 89557, USA
* Tel: +86-571-88320763 Fax: +86-571-88320842 E-mail: [email protected] ABSTRACT An approach developed earlier is used to predict the crack growth behavior of a pressure vessel steel. The approach consists of elastic-plastic finite element stress-strain analysis of a cracked component and application of a multiaxial fatigue damage criterion to access the crack growth. The computer simulations capture the experimentally observed insensitivity of crack growth to the Rratio. In particular, the models properly simulate the experimentally observed acceleration and retardation. Discussions are made to relate the characteristics of the crack growth behavior of the material to the cyclic deformation of the material and to the contact of cracked surfaces. Keywords: Multiaxial fatigue criterion, crack growth rate, mean stress relaxation, Q345R steel
1
Nomenclature
a
crack length from the notch root
an
initial crack length measured from the line of action of the applied load
aOL
crack length at overload applied
C
material constant
da / dN
crack growth rate
FCG
fatigue crack growth
K max
maximum stress intensity factor in a loading cycle,
K OL
stress intensity factor caused by overload
n
material constant
Pmax
maximum load in constant amplitude experiment
R
minimum load over maximum load in a loading cycle
r0
damage zone size in front of the crack tip
Dr
fatigue damage per loading cycle as a function of the location of the material point
Ni
crack initiation life
Di
fatigue damage per loading cycle
1. Introduction Engineering components often fail due to fatigue when subjected to cyclic loading. Significant efforts have been focused on studying fatigue crack initiation, early crack growth, and macro fatigue crack growth. For crack propagation, Paris law [1] has been widely applied to relate the crack growth rate ( da / dN ) to the stress intensity factor range due to its simplicity. Additional considerations are needed to include the R -ratio (minimum load over maximum load in a loading cycle) effect and the variable amplitude loading when using Paris law. For the R -ratio effect, Walker’s model [2] uses two parameters, R -ratio and K max (the maximum stress intensity factor in 2
a loading cycle). Kujawski [3,4] interpreted Walker’s model using a fatigue crack driving force concept considering only the positive part of the stress intensity factor range. Kujawski’s model improves the prediction of crack growth when the R -ratio is positive [5-7]. Sliva [8,9] studied fatigue crack propagation at negative stress ratios and suggested the importance of the compressive loading in a loading cycle. Noroozi et al [10] proposed a two-parameter model based on the crack tip plasticity. The model can be also used to consider the overload effect. Crack closure has been used to consider the R -ratio and overload effects since Elber [11-13]. A key is to find the crack opening load which is often determined by experimental measurement [14,15] and numerical analysis [16-28]. Due to different geometry of the testing specimen and crack length, the crack opening load could be different for an identical material [17]. There are some difficulties associated with the application of the crack closure concept. Jiang and co-workers [29,30] reexamined the plasticity-induced crack closure in fatigue crack propagation. The results show that when a crack closes during part of a loading cycle, plastic deformation in the material near the crack tip persists and is not negligible. The traditional concept of crack closure may overestimate the effect of crack closure, although contact of cracked surfaces does exist and influence crack growth rate. Generally, additional considerations are needed to account for the influences of variable amplitude loading. A complex stress field near the crack tip is induced by the variable amplitude loading, which enhances the difficulty and complexity of predictive models.
A single tensile
overload during constant amplitude loading is the simplest condition of variable amplitude loading. The most obvious feature of the influence of a single tensile overload on fatigue crack propagation is the retardation in crack growth rate [31-34]. The retardation effect increases with the overload ratio (the magnitude of the overload over the maximum load in the constant amplitude loading) [35]. Attempts have been made to explain the mechanisms involved in the retardation effect on the crack growth rate caused by the application of a single tensile overload. Residual stresses, crack tip blunting, crack front irregularities, and plasticity induced crack closure are possible mechanisms that contribute to the crack growth retardation [8,36]. Models were developed based on these mechanisms to predict the effect of overload on crack growth. Wheeler’s model [37] emphasizes the enlarged plastic zone ahead of the crack tip due to overloading. Several improved models [38,39] following Wheeler’s model were proposed for different materials with variable amplitude 3
loading. A modified Wheeler’s model was used to predict the crack growth rate of 7075-T651 aluminum alloy with success [5]. The crack tip blunting theory [40,41] assumes that a sharp crack becomes and remains blunt after the application of an overload, and the process leads to crack growth retardation. Such a theory is often related to the crack closure concept in an argument that crack tip blunting reduces the level of crack closure. The theory fails to explain the brief acceleration in crack growth observed right after overloading in certain materials. Crack initiation and propagation may have the same mechanisms. Oliva [42] applied the Manson-Coffin relation to predict the overload effect on crack growth based on the stresses and strains obtained from the finite element analysis implementing a kinematic hardening material model. Zhang et al. [43] studied crack propagation based on a finite element stress analysis coupled with a narrow band fast marching method for curvilinear fatigue crack growth and life predictions of metallic structures [44]. Richard and Sander studied the effect of variable amplitude loading on fatigue crack growth [45] and assessed different analytical and numerical methods [46]. Ghidini and Dalle Donne [47] developed a simplified engineering approach in which residual stresses and acceleration/retardation effects due to under-overloading were considered as the only factor influencing crack growth. Sun and Li [48] proposed a multi-scale fatigue damage evolution model for describing the short fatigue crack nucleation and growth in micro-scale and fatigue damage evolution.
Jiang and co-workers [49-51] introduced a continuum mechanics approach for crack
initiation and crack growth predictions. The approach consists of elastic-plastic stress-strain field analysis and application of a multiaxial fatigue damage criterion. The fatigue crack growth is considered as a continuous crack initiation process of the material at the crack tip. A fresh crack tip is created when the accumulative fatigue damage of the material point located at the crack tip reaches a critical damage.
The approach has been successfully used to predict the crack
propagation behavior of several materials with considerations of the effects of the R -ratio, notch influence, and single tensile overload [51-53]. None of the approaches developed so far can properly predict the brief acceleration in crack growth right after the application of an overload experimentally observed on one material and at the same time predict commonly observed crack growth retardation due to overloading for some other materials. The current effort is to fulfill such a task. The approach for fatigue crack initiation and 4
crack growth developed by Jiang and co-workers [49-51] is used to predict the crack growth rate and fatigue life of a pressure vessel steel, Q345R. The crack growth behavior of the material is characterized with insensitivity to R -ratio and a significant influence of overloading. The crack growth rate is greatly increased right after the application of an overload, followed by a gradual decrease in crack growth rate. 2. Modeling of Fatigue Crack Growth The approach developed by Jiang and co-workers [49-51] to consider both fatigue crack initiation and crack growth is used for Q345R, a pressure vessel steel, in the present investigation. Q345R, whose old designation is 16MnR, is one of the most widely used pressure vessel steels in China. It is similar to St52-3 in Germany, ASME SA302B, and JIS SM490A [54]. The chemical composition of the Q345R steel is listed in Table 1[54]. Crack growth is modeled based on the macroscopic continuum mechanics concepts. A basic assumption of this approach is that both crack initiation and crack growth are governed by the same fatigue damage mechanism. Fatigue crack initiation occurs when the accumulative fatigue damage of a material point reaches the critical damage, D0 , and a fresh crack is formed. The approach consists of two steps: 1) elastic-plastic stress-strain analysis of a cracked or notched component and, 2) determination of the fatigue crack growth rate by applying a multiaxial fatigue damage criterion using the outputted stress-strain from the numerical stress analysis. 2.1 Analysis of elastic-plastic stress-strain Axial-torsion fatigue experiments of Q345R were investigated earlier [54] and the results are used to determine the model constants in the current study. The cyclic plasticity model developed by Jiang and Sehitoglu [55,56] is used in the current work for the elastic-plastic stress-strain analysis of a notched/cracked component. The choice of the cyclic plasticity model is based on its capability to describe the cyclic material behavior including cyclic strain ratcheting and stress relaxation of the material [57]. The plasticity model adopts an Armstrong-Frederick type kinematic hardening rule. All the parameters in the cyclic plasticity model are determined by the fatigue experimental data of the smooth testing specimens. A detailed description of the cyclic plasticity model together with the procedure for the determination of material constants can be found in literature [55,56]. The cyclic plasticity model in Table 2 was implemented into the general-purpose finite element (FE) code ABAQUS as a user defined subroutine UMAT. The material constants used in the cyclic plasticity model were determined from the cyclic deformation experiments [54] and are listed in Table 3. 5
The experimental crack growth results of the Q345R material are taken from previous publications [34,59] where compact tension (CT) specimens were used to determine the crack growth behavior under constant amplitude loading with different R -ratios and a single tensile overload during constant amplitude loading. The geometry and dimensions of the CT specimens are shown in Fig. 1. The thickness of each CT specimen is 3.6 mm. Due to the small thickness, the plane-stress condition is considered to simulate the CT specimen. The two-dimensional FE mesh model is created considering the symmetry of geometry and loading condition, and it is shown in Fig. 2. Four-node plane-stress elements (CPS4) are used in FE mesh model. To simulate the high stress and strain gradients, fine mesh element are used for the region near the crack tip. The stress and strain results near the crack tip obtained from the elastic-plastic FE analysis of a cracked component is sensitive to the FE size near the crack tip. When the mesh size is extremely small for the material near the crack tip, the stresses and strains at the crack tip obtained from the FE analysis will be unrealistically high, which will result in a very high predicted crack growth rate. According to preliminary studies [51], the current FE simulations use an element size of 60 m near the crack tip. To consider the symmetry of the analytical model, appropriate boundary condition needs to be applied. Referring to the x-y coordinates system shown in Fig. 2, the displacement in the y direction is constrained for all the nodes located on the y 0 symmetric plane ahead of the notch root before crack propagation. The nodal displacement in the x -direction is constrained for each node located at the upper top of the pin loading hole of the specimen. The applied loading is uniformly distributed over 17 nodes of the loading hole in the y -direction. Sinusoidal waveform is used in the application of the external load. Surface contact pairs are used to simulate the contact between the upper and lower crack surfaces. Due to the symmetry in the specimen geometry and loading, a rigid surface is set on the y 0 plane to serve as the master surface, and the upper cracked surface within the crack length is modeled as the slaved surface. The possible contact surface does not include the machined slot in the ST specimen since the surfaces of the slot will not contact during a loading cycle. There is no tendency of relative slip between the contact surfaces and therefore, the friction coefficient is set to be zero. The FE starts from a notched specimen. Five constant amplitude loading cycles are simulated on the notched specimen and the stress-strain results of the material at the notch of the fifth loading cycle are used to determine the crack initiation from the notch. In the current FE study, crack extension is simulated by debonding in the mesh model. In all the simulation cases, debonding 6
occurs at the maximum load. At a given crack length, five constant amplitude loading cycles are simulated in order to obtain the stabilized cyclic stress-strain response. The stress-strain results of the fifth loading cycle are used for the prediction of the crack growth rate. The crack tip is debonded by a distance of one element size and a new crack is formed. The procedure repeats till a designed crack length is reached. It should be noticed that the debonding method is not directly related to the crack growth rate. Its purpose is to use one mesh model to simulate the stress-strain responses at different crack lengths. The simulation method before overload is the same as the constant amplitude loading. To simulate the stress-strain at overloading, when the crack length grows to the overloading location, five constant amplitude loading cycles are applied before the application of the overload. The stress-strain results obtained from the fifth loading cycles are used to determine the crack growth rate right before the application of the overload. This is followed by the application of a single tensile overload. After the application of the overload, five loading cycles are simulated and the stress-strain results of the fifth loading cycle are used to determine the crack growth rate of the material right after the application of the overload.
No crack extension is assumed during
overloading. The debonding method introduced in the previous paragraph is used to obtain the stress-strain responses at different crack lengths after overloading. 2.2 Multiaxial fatigue and crack growth rate After the elastic-plastic stress analysis of a notched or cracked component, the stress-strain results are used to determine crack initiation and the crack growth using a multiaxial fatigue damage criterion [50]. The criterion has an incremental form that incorporates cyclic plasticity, critical plane concept, and material memory. The critical plane is defined as the material plane where the accumulative fatigue damage reaches a critical damage, D0 . The criterion takes the following form,
D
mr / 0 1 (1 / f ) b d p m
cycle
1 b d p 2
(1)
where D represents the fatigue damage on a material plane, which is a function of the orientation of the material plane. mr is the maximum equivalent von Mises stress in a loading cycle. 0 is a material constant reflecting the fatigue endurance limit and f is the true fracture stress of the material. m and b are material constants. is the normal stress, is the shear stress, and p and
p represent plastic strains corresponding to and , respectively, on a material plane. The symbols
are the MaCauley brackets (i.e. x 0.5( x x ) ). The material constants involved in 7
the fatigue model, Eq.(1), for Q345R steel were determined from a previous multiaxial fatigue study of the identical material and they are 0 240MPa , f 632.2MPa , D0 3200MJ/m3 ,
m 0.85 and b 0.38 [54]. The approach predicts Mode I crack growth for the CT specimen subjected to axial loading [51]. Substituting detailed stress-strain results of any material point into Eq. (1) and integrating over a loading cycle, the distribution of fatigue damage per loading cycle along a given direction radiated from crack tip, Dr , can be determined.
The following formula was derived to
determine the fatigue crack growth rate along a given radial direction for constant amplitude loading [51], da dN
r0
0
D(r )dr
(2)
D0
where da dN is the crack growth rate, r0 is the damage zone size in front of the crack tip, and
Dr is the fatigue damage per loading cycle as a function of the location of the material point. Fig. 3 shows the distribution of Dr along the x-direction for Specimen CP17S ( R 0.1, P 3.15kN ) when the crack length is 9.48mm. The symbol A denotes the area enclosed by the r Dr curve. To consider the influence of the fatigue damage accumulated during crack initiation, Eq. (2) is modified into the following form to obtain the crack growth rate near the notch [52], da 0 D(r )dr dN D0 N i Di r0
(3)
where N i represents the crack initiation life for the notched specimen and Di is the fatigue damage per loading cycle for the material at the notch during crack initiation. Eq. (3) is only applied for the location of a given material point near the notch in the notch influencing zone. The crack initiation life, N i , is determined by the following formula,
Ni
D0 Dir
(4)
where Dir is the fatigue damage per loading cycle for the material at the notch root. 3. Results and discussion 3.1. Constant amplitude loading with different R -ratios The crack propagation experiments with different R -ratios and single overloads during 8
constant amplitude loading were conducted earlier [34,59]. The crack growth rates predicted and experimental results obtained from the constant amplitude loading are shown in Fig. 4. The markers in Fig. 4 denote the experimental results under constant amplitude loading with R -ratios ranging from -1 to 0.75. Predictions of the crack growth rates are presented with lines in the same figure. All the crack growth results are presented using the traditional plot with the crack growth rate da dN versus the stress intensity factor range, K , in the log-log scale. When R 0 , the stress intensity factor range is taken to be the maximum stress intensity factor. With such a commonly used treatment, there is no obvious R -ratio effect on the crack growth rate from experiments. The model captures the insensitivity of the R-ratio effect on crack growth. In addition, the models properly mimic early crack growth from notches. The material is obviously different from some other metallic materials such as aluminum alloy [3], 304L stainless steel [53], and 1070 steel [51]. In the fatigue prediction with the approach, there are no any additional assumptions about the influence of different R -ratio.
Early experimental work on Q345R (16MnR) under strain-
controlled experiment with a mean strain [34] suggests that the insensitivity of crack growth rate to R -ratio can be attributed to the fast mean stress relaxation behavior of the material. In the cyclic
plasticity model used in the current study, (i ) (i=1, 2, …, m) control ratcheting behavior and the mean stress relaxation behavior of a material [56]. The experimental data of mean stress relaxation for Q345R steel obtained from a strain-controlled experiment with a mean strain (strain amplitude=0.0016 and mean strain=0.0084), as given in Fig. 5, is used to calibrate the model parameter (i ) (i=1, 2, …, m). Fig. 5 shows simulation results of the mean stress relaxation with
(i ) (i=1, 2, …, m)=1, 5, 10, 20 and comparison with the experimental observations. It is clear from Fig.5 that when (i ) (i=1, 2, …, m)=1 the predicted mean stress relaxation under the straincontrolled uniaxial loading condition with a mean strain is in good agreement with the experimental results. Therefore, (i ) (i=1, 2, …, m)=1 is used in the cyclic plasticity model to describe the mean stress relaxation behavior of the material Q345R. As a comparison, (i ) (i=1, 2, …, m)=5 for 1070 steel and (i ) (i=1, 2, …, m)=8 for 304L stainless steel were determined [55]. Since the (i ) (i=1, 2, …, m) values for Q345R are much smaller than those for 1070 steel and 304L, a much faster mean stress relaxation in the asymmetric strain-controlled experiment for Q345R is expected. Consequently, under a similar loading condition in the fatigue crack growth experiment, the mean stress relaxation in the material near the crack tip is much faster in Q345R steel than that in 1070 steel and in 304L stainless steel. 9
3.2. Single tensile overload during constant amplitude loading The crack growth behavior is disturbed after the application of a single tensile overload. In Fig. 6, crack growth rates predicted are compared with experimental results obtained from an earlier presentation [34]. The circles denote the experimental results and the lines are the predicted crack growth rates. The post-overload crack propagation behavior of the three specimens is predicted successfully and the crack growth rates predicted are very close to the experimental results. In particular, the check mark shaped curves with instantaneous crack growth acceleration right after overloading are captured in the prediction. The yy yy hysteresis loops of the material at the Gauss point closest to the crack tip for Specimen CP17S ( R 0.1, P 3.15kN, POL 6.0kN at a 9.48mm ) before and after the application of the overload are shown in Fig. 7. The specimen was subjected to a constant load range of 3.15kN with R 0.1 . An overload of 6.0 kN was applied when the crack length was 9.48 mm. The FE stress analysis provides detailed stress, strain, and displacement at any node and Gauss point at any given time during a loading cycle, and all the stress, strain, displacement results can be outputted into data file for further process. The results shown in Fig.7 are the y component of the stress and the y component of the strain at the Gauss point closet to the crack tip (refer to Fig. 2 for the coordinates system). There are five yy yy hysteresis loops shown in Fig. 7: one for the moment right before overloading, one right after overloading, and the other three for the crack extension after overloading. Point A and point B on the yy yy loops represent the moments when the cracked surfaces start to contact and are fully open, respectively. The simulation results shown in Fig. 7 suggest that contact of the cracked surfaces occurs before the application of the overload. Right after the application of the overload, no crack surface contact occurs during a loading cycle. With further extension after overloading, the cracked surfaces are gradually in contact. The size of the yy yy loop is the largest right after the application of the overload (a=9.48mm) and becomes smaller with crack extension (a=9.54mm). The size of the yy yy loop reaches the smallest when a=10.68mm. Further extension of the crack results in an increased
yy yy loop (a=13.5 mm). Noticing that, in general, Dr is proportional to the size of the yy yy loops of the material near the crack tip, and the crack growth rate is linearly proportional to Dr , the evolution of the yy yy loop before and after the application of the overload reflects the observed increase-decrease-increase (check mark shape) in the da / dN K curve. Details of the crack profiles with loading history at the application of the overload can further 10
explain the increased stress-strain hysteresis loop of the material point near the crack tip right after the application of the overload.
Fig. 8 shows the crack profiles for Specimen CP17S
( R 0.1, P 3.15kN, POL 6.0kN at a 9.48mm ). The displacements at any given nodes and Gauss points in the component are obtained from the FE analysis and can be outputted into data file for further analysis. The results shown in Fig. 8 are the displacement in the y-direction at the nodes on the cracked surface at the peak loads before and after the application of the overload (refer to Fig. 2 for the coordinates system).
Contact of cracked surfaces occurs at the minimum external load
(line 2) before overloading. Right after the application of the overload, the crack keeps open at the minimum external load (lines 4 and 6).
Another observation is that the crack tip opening
displacement is larger after overloading than that before overloading. The absence of cracked surface contact in the wake zone increases the crack growth rate instantaneously. However, the residual compressive stresses created by the overload ahead of the crack tip usually retards the crack growth. A competing mechanism exists between the absence of cracked surface contact and the enlarged compressive stress field due to overloading ahead of crack tip. The result from the competition determines the instantaneous acceleration in the crack growth rate after overloading and retardation subsequently. In addition to low sensitivity of the crack growth rate to the R -ratio, the material under investigation is characterized by the great increase in crack growth rate right after the application of an overload. The check marker feature of the overload effect on crack growth has never been properly modeled. It will be revealed that such a characteristic of overload effect can be also related to the fast mean stress relaxation behavior of the material, or low (i ) (i=1, 2, …, m) values in the cyclic plasticity model. The enlarged stress-strain hysteresis loop after overloading shown in Fig. 7 and non-contact of the cracked surfaces right after overloading shown in Fig. 8 are due to the small (i ) (i=1,2, …, m) values for the material that result in fast mean stress relaxation in the material near the crack tip. When (i ) (i=1, 2, …, m)=5 were used for 1070 steel, the yy yy loop at a material point right after the application of the overload was predicted to be much smaller by using the same plasticity model [51].
Consequently, the approach predicts crack growth retardation right after the
application of the overload. To affirm the influence of (i ) (i=1, 2, …, m) on the overall predictions of the crack growth after the application of the overload, Fig. 9 shows the predicted crack growth for Specimen CP18S ( R 0.1, P 3.15kN, POL 6.0kN at a 10.32mm ). With otherwise identical material constants in the cyclic plasticity model and the fatigue model, higher (i ) (i=1, 2, …, m) 11
values result in higher predicted crack growth rate. More importantly, the approach with high (i ) (i=1, 2, …, m) values predict small or no instantaneous crack growth increase right after the application of the overload. With small (i ) (i=1, 2, …, m) values, the approach can predict the check marker shaped crack growth curve for the overload influence. The approach can not only predict crack growth with the overload effect, it can also predict the crack growth with the underload effect. A CT specimen was first subjected to constant amplitude loading with a load range of 3.6 kN at R=0.1. After the crack length reached a=4.254mm, an underload (compressive load) of 6.0 kN was applied. Constant amplitude loading resumed after underloading. A clear acceleration of crack growth was observed right after the application of the underload. The crack growth went back to the stable crack growth expected from the constant amplitude loading after a crack extension of approximately 0.991mm from the moment of underload application. When the crack length, a, reached 9.488 mm, an overload (tensile) of 6.0 kN followed by an underload (compressive) of 6.0 kN was applied. Constant amplitude loading with a load range of 3.6 kN at R=0.1 resumed after the application of the combined overload and underload. Clearly, an acceleration in the crack growth rate was observed right after the application of the combined overload and underload. In addition, the influencing zone size due to the combined overload and underload is large.
The crack length corresponding to the influence zone is
approximately 2.921 mm (the difference in crack length between the moment of the application of the combined overload and underloading and the moment when the crack growth rate returned to the stable state expected under constant amplitude loading). The results shown in Fig. 10 indicate that the approach discussed in the current work can properly describe the underload effect. It should be reiterated that no special material or fitting constants are introduced in the prediction of the overload and the underload effects. 3.3. Life predictions After the crack growth rates are obtained for different crack lengths, the a-N relationship can be determined by using a numerical integration. The predicted fatigue life corresponding to a given crack length, a1 , is obtained using the following equation [52],
N Ni
a1
0
da f a
(5)
where N is the number of loading cycles corresponding a crack length, a1 . The crack length, a , is measured from the notch root of the CT specimen. N i is the crack initiation life. f a is the crack 12
growth rate as a function of the crack length obtained from the approach being discussed. Eq. (5) is integrated numerically and is applicable after crack initiation. The crack initiation life, N i , is determined by Eq. (4). Fig. 11 shows the experimentally obtained a-N relationship for four specimens of Q345R steel and predictions by using the approach discussed. A close agreement between the predictions and the experimental observations can be found. The retardation effect due to overloading can be observed in the a - N curves for those specimens experienced overloading. 4. Further Discussions Difference between the traditional crack closure concept and the influence of contact of cracked surfaces on crack growth should be noticed. With the application of the FE methods for the elasticplastic stress analysis of the cracked specimen, the detailed contact of the cracked surfaces is simulated automatically and therefore, the influence of the cracked surface contact is considered in the approach naturally. No crack closure assumption regarding an effective range of stress intensity factor calculation ( ΔKeff K max Kop ) is utilized in modeling. Earlier studies reveal that when a crack closes during a part of a loading cycle, plastic deformation in the material near the crack tip persists and is not negligible [29,30]. In theory, the crack growth predictions using the current approach do not make use of any experimentally obtained crack growth results. This feature is fundamentally different from the methods where a stress intensity factor or J-integral is used. However, an inherent difficulty in the elastic-plastic FE stress analysis of a cracked component is the influence of the element size on the simulated stress and strain of the material near the crack tip. While there is no definite way to determine the element size for the stress analysis of a cracked component, an early study indicates that a proper selection of the element size is a few times the size of the average grain in a polycrystalline material [51]. The selection of the element size near the crack tip in the current study followed such a “rule of thumb.” In addition to element size, the way of debonding in the FE analysis also influences the predicted crack growth rate when using the approach adopted in the current study. A parametric study was conducted to reveal a proper selection of a debonding method in the FE analysis. Such a study was based on a fixed element size and element type. Fig. 12 shows a comparison of the experimental crack growth data of Specimen CP18S ( R 0.1, P 3.6kN, POL 6.0kN at a 10.32mm ) and the predicted crack growth results obtained from using four different debonding methods: one node after every 10 loading cycles, one node 13
after every five loading cycles, one node after every two loading cycles, and three nodes after every five loading cycles. Clearly, with four-node plane-stress elements and an element size of 60 m near the crack tip, the crack growth results obtained from debonding a node after every 5 or 10 loading cycles agree well with the experimental observations. These two debonding methods result in similar predicted crack growth rates. There is a tendency in the predicted crack growth results shown in Fig. 12 that a slower debonding scenario provides predicted results in better agreement with the experimental observations. However, slow debonding requires significant computation time to simulate a certain range of crack length. For the current investigation, debonding one node for every five loading cycles provides satisfactory predictions of crack growth for the material under investigation with realistic computation time. 5. Conclusions An approach developed earlier by Jiang et al. [49-51] was applied to predict the fatigue crack growth of Q345R pressure vessel steel under constant amplitude loading and a single tensile overload during constant amplitude loading. The crack growth of the material under investigation is characterized by the insensitivity of the crack growth to the R -ratio and an instantaneous acceleration of crack growth rate right after the application of an overload with a clear check mark in the da / dN K curve.
The approach can satisfactorily describe these crack growth
characteristics observed in experiments. It was found that the influence of an overload on the crack growth behavior is closely related to the strain ratcheting or mean stress relaxation behavior of the material. With the selection of the material constants in the plasticity model that properly reflects the ratcheting or mean stress relaxation behavior of the material, the approach is able to predict both crack growth acceleration and crack growth retardation after overloading.
6. Acknowledge The research work was supported by the National Natural Science Foundation of China (51175469, 50975260) and Key Projects in the National Science & Technology Pillar Program during the Twelfth Five-year Plan Period (2011BAK06B03). Dr. Qiu Baoxiang assisted in the finite modeling part of the work. References [1] Paris P. C., Erdogan F. A critical analysis of crack propagation laws. ASME J Basic Engng 1963;85:528-34. [2] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 707514
T6 aluminum. In: effect of environment and complex load history for fatigue life. ASTM STP 1970;462:1-14. [3] Kujawski D. A fatigue crack driving force parameter with load ratio effects. Int J Fatigue 2001;23:S239-46. [4] Kujawski D. A new (K Kmax )0.5 driving force parameter for crack growth in aluminum alloys. Int J Fatigue 2001;23:733-40. [5] Zhao T., Zhang J., Jiang Y. A study of fatigue crack growth of 7075-T651 aluminum alloy. Int J Fatigue 2008;30:1169-80. [6] Donald K., Paris P. C. An evaluation of K eff estimation procedures on 6061-T6 and 2024-T3 aluminum alloys. Int J Fatigue 1999;21:S47-57. [7] Dinda S., Kujawski D. Correlation and prediction of fatigue crack growth for different R -ratios using K max and K parameters. Engng Fract Mech 2004;71:1779-90. [8] Silva F. S. The importance of compressive stresses on fatigue crack propagation rate. Int J Fatigue 2005;27:1441-52. [9] Silva F. S. Fatigue crack propagation after overloading and under loading at negative stress ratios. Int J Fatigue 2009;29:1757-71. [10] Noroozi A. H., Glinka G., Lambert S. Prediction of fatigue crack growth under constant amplitude loading and a single overload based on elasto-plastic crack tip stresses and strains. Engng Fract Mech 2008;75:188-206. [11] Elber W. Fatigue crack closure under cyclic tension. Engng Fract Mech 1970;2:37-45. [12] Elber W. The significance of fatigue crack closure. ASTM STP 1971;486:230-42. [13] Schijve J. Some formulas for the crack opening stress level. Engng Fract Mech 1981;14:461-5. [14] Macha D. E., Corbly D. M, Jones J. W. On the variation of fatigue-crack-opening load with measurement location. Exp Mech 1979;19:207-13. [15] Shin C. S, Smith R. A. Fatigue crack growth from sharp notches. Int J Fatigue 1985;7:87-93. [16] Ling M. R, Schijve J. The effect of intermediate heat treatments on overload induced retardation during fatigue crack growth in an Al-alloy. Fatigue Fract Engng Mater Struct 15
1992;15:421-30. [17] Wei L. W., James M. N. A study of fatigue crack closure in polycarbonate CT specimens. Engng Fract Mech 2000;66:223-42. [18] Ohji K., Ogura K., Ohkubo Y. On the closure of fatigue cracks under cyclic tensile loading. Int J Fract 1974;10:123-24. [19] Ohji K., Ogura K., Ohkubo Y. Cyclic analysis of a propagating crack and its correlation with fatigue crack growth. Engng Fract Mech 1975;7:457-64. [20] Newman JC., Armen H. Elastic-plastic analysis of a propagating crack under cyclic loading. AIAA Journal 1975;13:1017-23. [21] Fleck N. A. Shercliff H. R, Overload retardation due to plasticity-induced crack closure. In: Proceedings of the 7th International Conference on Fracture ICF7, Houston, Texas, USA. 1989; 1405-15. [22] Shercliff H. R, Fleck N. A. Effect of specimen geometry on fatigue crack growth in plane strain-II. Overload response. Fatigue Fract Engng Mater Struct 1990;3:297-310. [23] Zhang J. Z., Halliday M. D., Bowen P., Poole P. Three dimensional elastic-plastic finite element modelling of small fatigue crack growth under a single tensile overload. Engng Fract Mech 1999;63:229-51. [24] Dougherty J. D, Padovan J., Srivatsan T. S. Fatigue crack propagation and closure behavior of modified 1070 steel: finite element study. Engng Fract Mech 1997;56:189-212. [25] Ellyin F., Wu J. A numerical investigation on the effect of an overload on fatigue crack opening and closure behaviour. Fatigue Fract Engng Mater Struct 1999;22:835-847. [26] Chermahini R. G, Shivakumar K. N, Newman Jr. JC. Three-dimensional finite-element simulation of fatigue crack growth and closure. ASTM STP 1988;982:398-413. [27] Gonzalez-Herrera A., Zapatero J. Tri-dimensional numerical modelling of plasticity induced fatigue crack closure. Engng Fract Mech 2008;75:4513-28. [28] McClung R. C., Sehitoglu H. On the finite element analysis of fatigue crack closure-1. Basic modeling issues. Engng Fract Mech 1989;33:237-52. [29] Jiang Y., Feng M. A reexamination of plasticity-induced crack closure in fatigue crack 16
propagation. Int J Plasticity 2005;21:1720-40. [30] Feng M., Ding F., Jiang Y. A study of crack growth retardation due to artificially induced crack surface contact. Int J Fatigue 2005;27:1319-27. [31] Corlby D. M., Packman P. F. On the influence of single and multiple peak overloads on fatigue crack propagation in 7075-T6511 aluminum. Engng Fract Mech 1973;5:479-97. [32] Ward-Close C. M., Ritchie R. O. On the role of crack closure mechanisms in influencing fatigue crack growth following tensile overloads in a Titanium alloy: Near Threshold Versus Higher ΔK Behavior. ASTM STP 1988;982:93-111. [33] Rao K. T. V, Ritchie R. O. Mechanisms for the retardation of fatigue cracks following single tensile overloads: behavior in aluminum-lithium alloys. Acta Metall 1988;36:2849-62. [34] Wang X., Gao Z., Zhao T., Jiang Y. An experimental study of the crack growth behavior of 16MnR pressure vessel steel. J Pressure Vessel Technol 2009;131(2): 021402-1-9. [35] Borrego L. P., Ferreira J. M., Cruz J. M. P., Costa J. M. Evaluation of overload effects on fatigue crack growth and closure. Engng Fract Mech 2003;70:1379-97. [36] Skorupa M. Load interaction effects during fatigue crack growth under variable amplitude loading-a literature review. Part II: qualitative interpretation. Fatigue Fract Engng Mater Struct 1999;22:905-26. [37] Wheeler O. E. Spectrum loading and crack growth. J Basic Engng 1972;94:181-86. [38] Kim K. S., Kim K. S., Shim C. S., Cho H. M. A study on the effect of overload ratio on fatigue crack growth. Key Engng Mats 2004;261-263:1159-68. [39] Yuen B. K. C, Taheri F. Proposed modifications to the Wheeler retardation model for multiple overloading fatigue life prediction. Int J Fatigue 2006;28:1803-19. [40] Makabe C., Purnowidodo A., McEvily A. J. Effects of surface deformation and crack closure on fatigue crack propagation after overloading and underloading. Int J Fatigue 2004;26:1341-48. [41] Tvergaard V. On fatigue crack growth in ductile materials by crack–tip blunting. J Mech Phys Solids 2004;52:2149-66 [42] Oliva V., Cseplo L., Materna A., Blahova L. FEM simulation of fatigue crack growth. Mats Sci and Engng 1997; 234-6:517-20. 17
[43] Zhang J. Z., Zhang J. Z., Du S. Y. Elastic-plastic finite element analysis and experimental study of short and long fatigue crack growth. Engng Fract Mech 2001;68:1591-605. [44] Shi J., Chopp D., Lua J., Sukumar N., Belytchko T. Abaqus implementation of extended finite element method using a level set representation for three-dimensional fatigue crack growth and life predictions. Engng Fract Mech 2010;77:2840-63. [45] Sander M., Richard H. A. Fatigue crack growth under variable amplitude loading Part I: experimental investigations. Fatigue Fract Engng Mater Struct 2006;29:291-301. [46] Sander M., Richard H. A. Fatigue crack growth under variable amplitude loading Part II: analytical and numerical investigations. Fatigue Fract Engng Mater Struct 2006;29:303-19. [47] Ghidini T., Dalle Donne C. Fatigue crack propagation assessment based on residual stresses obtained through cut-compliance technique. Fatigue Fract Engng Mater Struct 2007;30:214-22. [48] Sun B., Li Z. A multi-scale damage model for fatigue accumulation due to short cracks nucleation and growth. Engng Fract Mech 2014;127:280-95. [49] Jiang Y., Ding F., Feng M. An approach for fatigue life prediction. ASME J Engng Mater Tech 2007;129:182-89. [50] Jiang Y. A fatigue criterion for general multiaxial loading. Fatigue Fract Engng Mater Struct 2000;23:19-32. [51] Jiang Y., Feng M. Modeling of fatigue crack propagation. ASME J Eneng Mater Tech 2004;126:77-86. [52] Ding F,. Feng M., Jiang Y. Modeling of fatigue crack growth from a notch. Int J Plasticity 2007;23:1167-88. [53] Fan F., Kalnaus S., Jiang Y. Modelling of fatigue crack growth of stainless steel 304L. Mech Mater 2008;40:961-73. [54] Gao Z., Zhao T., Wang X., Jiang Y. Multiaxial fatigue of 16MnR steel. J Pressure Vessel Technol 2009;131(2):021403-1-9. [55] Jiang Y., Sehitoglu H. Modeling of cyclic ratchetting plasticity, Part I: Development of constitutive relations. J Appl Mech 1996;63:720-25. [56] Jiang Y., Sehitoglu H. Modeling of cyclic ratchetting plasticity, Part II: Comparison of model 18
simulations with experiments. J Appl Mech 1996;63:726-33. [57] Jiang Y., Kurath, P. Characteristics of the Armstrong-Frederick type plasticity models. Int J Plasticity 1996;12:387-415. [58] Jiang Y., Xu B., Sehitoglu H. Three-dimensional elastic-plastic stress analysis of rolling contact. ASME J Tribol 2002;124:699-708. [59] Wang X., Yin D., Xu F., Qiu B., Gao Z. Fatigue crack initiation and growth of 16MnR steel with stress ratio effects. Int J Fatigue 2012;35:10-15.
19
Table1 Chemical composition of Q345R (wt %) Element
C
Al
Ni
Cr
Mn
Co
Nb
Cu
Ti
Si
P
V
Fe
Weight
0.203
0.035
0.003
0.008
1.535
0.004
0.039
0.010
0.002
0.236
0.009
0.003
Balance
Table 2 Jiang-Sehitoglu cyclic plasticity model used in the finite element simulations ~ S deviatoric stress tensor ~ ~ f (S ~) : (S ~) 2k 2 0
Yield Function
~ backstress k yield stress in shear
n~ normal of yield surface
d~ p
Flow law
1 ~ ~ ~ dS : n n h
h plastic modulus
~ p plastic strain
~
M
~
(i )
~ i ith backstress part
i 1
Hardening Rule
d~ (i )
~ (i ) (i ) (i ) ~ c r n ( i ) r
( i ) 1
~ (i ) dp ~ (i )
M number of backstress parts dp equivalent plastic strain increment
c (i ) , r (i ) and (i ) material constants
(i 1,2,...,M )
Table 3 Material constants of Q345R in Jiang-Sehitoglu cyclic plasticity model E 212.5GPa c1 221.0 r 1 83.0
0.31 c2 46.0
r 2 91.0
k 14.5GPa c3 14.5 r 3 89.0
c4 5.8
c5 2.721
r 4 88.0
r 5 212.0
1 2 3 4 5 1
20
63.5 W=50.8 F 30.5 12.7
r0 =0.1,1.0,2.0
14 0.2
a
an
14 30.5 F
Fig. 1 Standard compact tension specimen (mm) (a)
y
x a Crack growth direction (b)
Notch Root
Fig. 2 Mesh model for CT specimen: (a) overall model; (b) Fine mesh near crack tip
21
D, Fatigue Damage, MJ/m
3
8 Damaging Zone
6 4 A
Specimen CP17S R=0.1, Pmax=3.5kN a=9.48mm
2 0 0.0
0.2
0.4
0.6
0.8
1.0
r, Distance From Crack Tip, mm
Fig. 3 Distribution of fatigue damage per loading cycle along the x-direction
22
10
10
-3 -3
da/dN, Crack Growth Rate, mm/cycle
da/dN, Crack Growth Rate, mm/cycle
10
-4
-5
R=-1.0, Pmax=2.7kN Prediction
10
-6
10
-4
10
-5
10
R=0.1, Pmax=4.0kN Prediction -6
2
3
4
5
10
6 7 8 9
10 100 1/2 K, Stress Intensity Factor Range, MPa m
2
3
4
5
6 7 8 9
10 100 1/2 K, Stress Intensity Factor Range, MPa m 2
10
da/dN, Crack Growth Rate, mm/cycle
da/dN, Crack Growth Rate, mm/cycle
2
-3 8 6 4
2
10
-4 8 6
R=0.2, Pmax=8.0kN Prediction
4
10
-3 8 6 4
2
10
-4 8 6 4
2
2
10
10
-5 1
4
5
6
R=0.3, Pmax=6.9kN Prediction
-5 2
3
4
5
6 7 8 9
10
7
3x10 1/2 K, Stress Intensity Factor Range, MPa m
100
K, Stress Intensity Factor Range, MPa m
1/2
da/dN, Crack Growth Rate, mm/cycle
2
da/dN, Crack Growth Rate, mm/cycle
-3
10
-4
10
-5
10
R=0.5, Pmax=6.0kN Prediction
8 6 4
2
-5
10
8 6 4
2
R=0.75, Pmax=10.0kN Prediction
-6
10
-6
10
-4
10
2
3
4
5
6 7 8 9
10 100 1/2 K, Stress Intensity Factor Range, MPa m
1
3
4
5
2x10 10 1/2 K, Stress Intensity Factor Range, MPa m
Fig. 4 R -ratio effect on crack growth at constant amplitude loading (experimental data taken from [34,59])
23
mean, Mean Axial Stress, MPa
300
250
200
150 Experiment 100
(i)
(i=1,2,3,4,5)=1 (i)
(i=1,2,3,4,5)=5 (i)
50
/2 = 0.16%, mean = 0.84%
(i=1,2,3,4,5)=10 (i)
(i=1,2,3,4,5)=20 0
2
3
4
5 6 7 8 9
1
2
3
4
5 6 7 8 9
10
100
2
3
4
5 6 7 8 9
1000
N, Loading Cycles, cycle
Fig. 5 Mean stress relaxation under strain-controlled uniaxial loading with a mean strain (experimental data taken from [34])
24
-3
10 -3 9 8 7 6 5
da/dN, Crack Growth Rate, mm/cycle
4
Specimen CP17S R=0.1 Pmax=3.5kN POL=6.0kN
Specimen CP18S R=0.1 Pmax=4.0kN POL=6.0kN
4
3
2
-4
10
9 8 7 6 5
da/dN, Crack Growth Rate, mm/cycle
10
9 8 7 6 5 4
3
2
-4
10
9 8 7 6 5 4
3
3
2
2
Experiment Prediction
Experiment Prediction
-5
-5
10
1
10
3
2x10
4
K, Stress Intensity Factor Range, MPa m
10
5
1
3
4
2x10 1/2 K, Stress Intensity Factor Range, MPa m
1/2
5
-3
10
8 7 6 5
Specimen CP08N R=0.1 Pmax=2.5kN POL=4.5kN
4
da/dN, Crack Growth Rate, mm/cycle
3 2
-4
10
8 7 6 5 4 3 2
-5
10
8 7 6 5 4 3
Experiment Prediction
2
-6
10
1
3
2x10
10
K, Stress Intensity Factor Range, MPa m
1/2
Fig. 6 Overload effect on crack growth (experimental data taken from [34]): (a) CP17S ( R =0.1, P =3.15 kN, POL =6.0 kN, a=9.48mm); (b) CP18S ( R =0.1, P =3.15 kN, POL =6.0 kN, a=10.32
mm); (c) CP08N ( R =0.1, P =2.25 kN, POL =4.5 kN, a=9.42mm).
25
1000
Specimen CP17S R=0.1 Pmax=3.5kN POL=6.0kN
a=13.5mm a=9.54mm
yy, y-Direction Normal Stress, MPa
a=10.68mm
500
0 B
B
B B
-500 A
A
A
A
a=9.48mm (After overload) a=9.48mm (Befor overload)
-1000 0.00
0.02
0.04 yy, y-Direction Normal Strain
0.06
0.08
Fig. 7 yy yy hysteresis loops of a material point near the crack tip
0.04
Specimen CP17S R=0.1 Pmax=3.5kN POL=6.0kN
3 5
y,y-Direction Displacement,mm
1
0.03 2
4
6
3
0.02
5
1
0.01
4 6 2
0.00 symmetrical plane
Crack tip -0.01 0.0
0.5
1.0 1.5 r, Distance to Crack Tip,mm
2.0
Fig. 8 y-direction displacements of nodes in the crack wake zone
26
2.5
Fig. 9 Influence of parameters (i ) on the predicted crack growth rate
27
-3
10
9 8 7 6
Specimen DJ07 R=0.1 Pmax=4.0kN
5
da/dN, Crack Growth Rate, mm/cycle
4
Overload/Underload PUL=-6.0kN POL=6.0kN
3
2
-4
10
9 8 7 6 5 4
Underload PUL=-6.0kN
3
2
Experiment Prediction -5
10
1
2x10
3
4
K, Stress Intensity Factor Range, MPa m
5 1/2
Fig. 10 Crack growth with the underload effect
28
20 Constant amplitude loading CP02N Experiment Prediction CP04N Experiment Overload influence zone Prediction
Crack Length, mm
15
10
Overload CP17S Pmax=3.5kN POL=6.0kN Experiment Prediction CP18S Pmax=4.0kN POL=6.0kN Experiment Prediction
5
0 2
3
4
5 6 7 8 9
4
10
2
3
4
5 6 7 8 9
5
10 Number of cycle, cycle
2
3
4
5 6 7 8 9
6
2
10
Fig. 11 Crack length versus number of loading cycles for four specimens (experimental data taken from [34])
Fig. 12 Influence of debonding method on predicted crack growth rate 29
Highlights 1. The crack growth behavior with a single tensile overload is properly predicted. 2. The low sensitivity of R-ratio is attributed to the fast mean stress relaxation. 3. The retarding effect is the competing result of different influence factors. 4. Different material constants in plasticity model used in modeling are discussed.
30
S0013-7944(15)00012-0 http://dx.doi.org/10.1016/j.engfracmech.2015.01.011 EFM 4467
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
21 March 2014 9 January 2015 10 January 2015
Please cite this article as: Ding, Z., Gao, Z., Wang, X., Jiang, Y., Modeling of fatigue crack growth in a pressure vessel steel Q345R, Engineering Fracture Mechanics (2015), doi: http://dx.doi.org/10.1016/j.engfracmech. 2015.01.011
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Submitted to Engineering Fracture Mechanics March 2014 EFM-S-14-00137, Revised October, 2014
Modeling of fatigue crack growth in a pressure vessel steel Q345R Zhenyu Ding a, Zengliang Gao a*, Xiaogui Wang a, Yanyao Jiang b a
Zhejiang University of Technology, College of Mechanical Engineering, Hangzhou, Zhejiang 310032, China b
University of Nevada, Reno, Department of Mechanical Engineering Reno, NV 89557, USA
* Tel: +86-571-88320763 Fax: +86-571-88320842 E-mail: [email protected] ABSTRACT An approach developed earlier is used to predict the crack growth behavior of a pressure vessel steel. The approach consists of elastic-plastic finite element stress-strain analysis of a cracked component and application of a multiaxial fatigue damage criterion to access the crack growth. The computer simulations capture the experimentally observed insensitivity of crack growth to the Rratio. In particular, the models properly simulate the experimentally observed acceleration and retardation. Discussions are made to relate the characteristics of the crack growth behavior of the material to the cyclic deformation of the material and to the contact of cracked surfaces. Keywords: Multiaxial fatigue criterion, crack growth rate, mean stress relaxation, Q345R steel
1
Nomenclature
a
crack length from the notch root
an
initial crack length measured from the line of action of the applied load
aOL
crack length at overload applied
C
material constant
da / dN
crack growth rate
FCG
fatigue crack growth
K max
maximum stress intensity factor in a loading cycle,
K OL
stress intensity factor caused by overload
n
material constant
Pmax
maximum load in constant amplitude experiment
R
minimum load over maximum load in a loading cycle
r0
damage zone size in front of the crack tip
Dr
fatigue damage per loading cycle as a function of the location of the material point
Ni
crack initiation life
Di
fatigue damage per loading cycle
1. Introduction Engineering components often fail due to fatigue when subjected to cyclic loading. Significant efforts have been focused on studying fatigue crack initiation, early crack growth, and macro fatigue crack growth. For crack propagation, Paris law [1] has been widely applied to relate the crack growth rate ( da / dN ) to the stress intensity factor range due to its simplicity. Additional considerations are needed to include the R -ratio (minimum load over maximum load in a loading cycle) effect and the variable amplitude loading when using Paris law. For the R -ratio effect, Walker’s model [2] uses two parameters, R -ratio and K max (the maximum stress intensity factor in 2
a loading cycle). Kujawski [3,4] interpreted Walker’s model using a fatigue crack driving force concept considering only the positive part of the stress intensity factor range. Kujawski’s model improves the prediction of crack growth when the R -ratio is positive [5-7]. Sliva [8,9] studied fatigue crack propagation at negative stress ratios and suggested the importance of the compressive loading in a loading cycle. Noroozi et al [10] proposed a two-parameter model based on the crack tip plasticity. The model can be also used to consider the overload effect. Crack closure has been used to consider the R -ratio and overload effects since Elber [11-13]. A key is to find the crack opening load which is often determined by experimental measurement [14,15] and numerical analysis [16-28]. Due to different geometry of the testing specimen and crack length, the crack opening load could be different for an identical material [17]. There are some difficulties associated with the application of the crack closure concept. Jiang and co-workers [29,30] reexamined the plasticity-induced crack closure in fatigue crack propagation. The results show that when a crack closes during part of a loading cycle, plastic deformation in the material near the crack tip persists and is not negligible. The traditional concept of crack closure may overestimate the effect of crack closure, although contact of cracked surfaces does exist and influence crack growth rate. Generally, additional considerations are needed to account for the influences of variable amplitude loading. A complex stress field near the crack tip is induced by the variable amplitude loading, which enhances the difficulty and complexity of predictive models.
A single tensile
overload during constant amplitude loading is the simplest condition of variable amplitude loading. The most obvious feature of the influence of a single tensile overload on fatigue crack propagation is the retardation in crack growth rate [31-34]. The retardation effect increases with the overload ratio (the magnitude of the overload over the maximum load in the constant amplitude loading) [35]. Attempts have been made to explain the mechanisms involved in the retardation effect on the crack growth rate caused by the application of a single tensile overload. Residual stresses, crack tip blunting, crack front irregularities, and plasticity induced crack closure are possible mechanisms that contribute to the crack growth retardation [8,36]. Models were developed based on these mechanisms to predict the effect of overload on crack growth. Wheeler’s model [37] emphasizes the enlarged plastic zone ahead of the crack tip due to overloading. Several improved models [38,39] following Wheeler’s model were proposed for different materials with variable amplitude 3
loading. A modified Wheeler’s model was used to predict the crack growth rate of 7075-T651 aluminum alloy with success [5]. The crack tip blunting theory [40,41] assumes that a sharp crack becomes and remains blunt after the application of an overload, and the process leads to crack growth retardation. Such a theory is often related to the crack closure concept in an argument that crack tip blunting reduces the level of crack closure. The theory fails to explain the brief acceleration in crack growth observed right after overloading in certain materials. Crack initiation and propagation may have the same mechanisms. Oliva [42] applied the Manson-Coffin relation to predict the overload effect on crack growth based on the stresses and strains obtained from the finite element analysis implementing a kinematic hardening material model. Zhang et al. [43] studied crack propagation based on a finite element stress analysis coupled with a narrow band fast marching method for curvilinear fatigue crack growth and life predictions of metallic structures [44]. Richard and Sander studied the effect of variable amplitude loading on fatigue crack growth [45] and assessed different analytical and numerical methods [46]. Ghidini and Dalle Donne [47] developed a simplified engineering approach in which residual stresses and acceleration/retardation effects due to under-overloading were considered as the only factor influencing crack growth. Sun and Li [48] proposed a multi-scale fatigue damage evolution model for describing the short fatigue crack nucleation and growth in micro-scale and fatigue damage evolution.
Jiang and co-workers [49-51] introduced a continuum mechanics approach for crack
initiation and crack growth predictions. The approach consists of elastic-plastic stress-strain field analysis and application of a multiaxial fatigue damage criterion. The fatigue crack growth is considered as a continuous crack initiation process of the material at the crack tip. A fresh crack tip is created when the accumulative fatigue damage of the material point located at the crack tip reaches a critical damage.
The approach has been successfully used to predict the crack
propagation behavior of several materials with considerations of the effects of the R -ratio, notch influence, and single tensile overload [51-53]. None of the approaches developed so far can properly predict the brief acceleration in crack growth right after the application of an overload experimentally observed on one material and at the same time predict commonly observed crack growth retardation due to overloading for some other materials. The current effort is to fulfill such a task. The approach for fatigue crack initiation and 4
crack growth developed by Jiang and co-workers [49-51] is used to predict the crack growth rate and fatigue life of a pressure vessel steel, Q345R. The crack growth behavior of the material is characterized with insensitivity to R -ratio and a significant influence of overloading. The crack growth rate is greatly increased right after the application of an overload, followed by a gradual decrease in crack growth rate. 2. Modeling of Fatigue Crack Growth The approach developed by Jiang and co-workers [49-51] to consider both fatigue crack initiation and crack growth is used for Q345R, a pressure vessel steel, in the present investigation. Q345R, whose old designation is 16MnR, is one of the most widely used pressure vessel steels in China. It is similar to St52-3 in Germany, ASME SA302B, and JIS SM490A [54]. The chemical composition of the Q345R steel is listed in Table 1[54]. Crack growth is modeled based on the macroscopic continuum mechanics concepts. A basic assumption of this approach is that both crack initiation and crack growth are governed by the same fatigue damage mechanism. Fatigue crack initiation occurs when the accumulative fatigue damage of a material point reaches the critical damage, D0 , and a fresh crack is formed. The approach consists of two steps: 1) elastic-plastic stress-strain analysis of a cracked or notched component and, 2) determination of the fatigue crack growth rate by applying a multiaxial fatigue damage criterion using the outputted stress-strain from the numerical stress analysis. 2.1 Analysis of elastic-plastic stress-strain Axial-torsion fatigue experiments of Q345R were investigated earlier [54] and the results are used to determine the model constants in the current study. The cyclic plasticity model developed by Jiang and Sehitoglu [55,56] is used in the current work for the elastic-plastic stress-strain analysis of a notched/cracked component. The choice of the cyclic plasticity model is based on its capability to describe the cyclic material behavior including cyclic strain ratcheting and stress relaxation of the material [57]. The plasticity model adopts an Armstrong-Frederick type kinematic hardening rule. All the parameters in the cyclic plasticity model are determined by the fatigue experimental data of the smooth testing specimens. A detailed description of the cyclic plasticity model together with the procedure for the determination of material constants can be found in literature [55,56]. The cyclic plasticity model in Table 2 was implemented into the general-purpose finite element (FE) code ABAQUS as a user defined subroutine UMAT. The material constants used in the cyclic plasticity model were determined from the cyclic deformation experiments [54] and are listed in Table 3. 5
The experimental crack growth results of the Q345R material are taken from previous publications [34,59] where compact tension (CT) specimens were used to determine the crack growth behavior under constant amplitude loading with different R -ratios and a single tensile overload during constant amplitude loading. The geometry and dimensions of the CT specimens are shown in Fig. 1. The thickness of each CT specimen is 3.6 mm. Due to the small thickness, the plane-stress condition is considered to simulate the CT specimen. The two-dimensional FE mesh model is created considering the symmetry of geometry and loading condition, and it is shown in Fig. 2. Four-node plane-stress elements (CPS4) are used in FE mesh model. To simulate the high stress and strain gradients, fine mesh element are used for the region near the crack tip. The stress and strain results near the crack tip obtained from the elastic-plastic FE analysis of a cracked component is sensitive to the FE size near the crack tip. When the mesh size is extremely small for the material near the crack tip, the stresses and strains at the crack tip obtained from the FE analysis will be unrealistically high, which will result in a very high predicted crack growth rate. According to preliminary studies [51], the current FE simulations use an element size of 60 m near the crack tip. To consider the symmetry of the analytical model, appropriate boundary condition needs to be applied. Referring to the x-y coordinates system shown in Fig. 2, the displacement in the y direction is constrained for all the nodes located on the y 0 symmetric plane ahead of the notch root before crack propagation. The nodal displacement in the x -direction is constrained for each node located at the upper top of the pin loading hole of the specimen. The applied loading is uniformly distributed over 17 nodes of the loading hole in the y -direction. Sinusoidal waveform is used in the application of the external load. Surface contact pairs are used to simulate the contact between the upper and lower crack surfaces. Due to the symmetry in the specimen geometry and loading, a rigid surface is set on the y 0 plane to serve as the master surface, and the upper cracked surface within the crack length is modeled as the slaved surface. The possible contact surface does not include the machined slot in the ST specimen since the surfaces of the slot will not contact during a loading cycle. There is no tendency of relative slip between the contact surfaces and therefore, the friction coefficient is set to be zero. The FE starts from a notched specimen. Five constant amplitude loading cycles are simulated on the notched specimen and the stress-strain results of the material at the notch of the fifth loading cycle are used to determine the crack initiation from the notch. In the current FE study, crack extension is simulated by debonding in the mesh model. In all the simulation cases, debonding 6
occurs at the maximum load. At a given crack length, five constant amplitude loading cycles are simulated in order to obtain the stabilized cyclic stress-strain response. The stress-strain results of the fifth loading cycle are used for the prediction of the crack growth rate. The crack tip is debonded by a distance of one element size and a new crack is formed. The procedure repeats till a designed crack length is reached. It should be noticed that the debonding method is not directly related to the crack growth rate. Its purpose is to use one mesh model to simulate the stress-strain responses at different crack lengths. The simulation method before overload is the same as the constant amplitude loading. To simulate the stress-strain at overloading, when the crack length grows to the overloading location, five constant amplitude loading cycles are applied before the application of the overload. The stress-strain results obtained from the fifth loading cycles are used to determine the crack growth rate right before the application of the overload. This is followed by the application of a single tensile overload. After the application of the overload, five loading cycles are simulated and the stress-strain results of the fifth loading cycle are used to determine the crack growth rate of the material right after the application of the overload.
No crack extension is assumed during
overloading. The debonding method introduced in the previous paragraph is used to obtain the stress-strain responses at different crack lengths after overloading. 2.2 Multiaxial fatigue and crack growth rate After the elastic-plastic stress analysis of a notched or cracked component, the stress-strain results are used to determine crack initiation and the crack growth using a multiaxial fatigue damage criterion [50]. The criterion has an incremental form that incorporates cyclic plasticity, critical plane concept, and material memory. The critical plane is defined as the material plane where the accumulative fatigue damage reaches a critical damage, D0 . The criterion takes the following form,
D
mr / 0 1 (1 / f ) b d p m
cycle
1 b d p 2
(1)
where D represents the fatigue damage on a material plane, which is a function of the orientation of the material plane. mr is the maximum equivalent von Mises stress in a loading cycle. 0 is a material constant reflecting the fatigue endurance limit and f is the true fracture stress of the material. m and b are material constants. is the normal stress, is the shear stress, and p and
p represent plastic strains corresponding to and , respectively, on a material plane. The symbols
are the MaCauley brackets (i.e. x 0.5( x x ) ). The material constants involved in 7
the fatigue model, Eq.(1), for Q345R steel were determined from a previous multiaxial fatigue study of the identical material and they are 0 240MPa , f 632.2MPa , D0 3200MJ/m3 ,
m 0.85 and b 0.38 [54]. The approach predicts Mode I crack growth for the CT specimen subjected to axial loading [51]. Substituting detailed stress-strain results of any material point into Eq. (1) and integrating over a loading cycle, the distribution of fatigue damage per loading cycle along a given direction radiated from crack tip, Dr , can be determined.
The following formula was derived to
determine the fatigue crack growth rate along a given radial direction for constant amplitude loading [51], da dN
r0
0
D(r )dr
(2)
D0
where da dN is the crack growth rate, r0 is the damage zone size in front of the crack tip, and
Dr is the fatigue damage per loading cycle as a function of the location of the material point. Fig. 3 shows the distribution of Dr along the x-direction for Specimen CP17S ( R 0.1, P 3.15kN ) when the crack length is 9.48mm. The symbol A denotes the area enclosed by the r Dr curve. To consider the influence of the fatigue damage accumulated during crack initiation, Eq. (2) is modified into the following form to obtain the crack growth rate near the notch [52], da 0 D(r )dr dN D0 N i Di r0
(3)
where N i represents the crack initiation life for the notched specimen and Di is the fatigue damage per loading cycle for the material at the notch during crack initiation. Eq. (3) is only applied for the location of a given material point near the notch in the notch influencing zone. The crack initiation life, N i , is determined by the following formula,
Ni
D0 Dir
(4)
where Dir is the fatigue damage per loading cycle for the material at the notch root. 3. Results and discussion 3.1. Constant amplitude loading with different R -ratios The crack propagation experiments with different R -ratios and single overloads during 8
constant amplitude loading were conducted earlier [34,59]. The crack growth rates predicted and experimental results obtained from the constant amplitude loading are shown in Fig. 4. The markers in Fig. 4 denote the experimental results under constant amplitude loading with R -ratios ranging from -1 to 0.75. Predictions of the crack growth rates are presented with lines in the same figure. All the crack growth results are presented using the traditional plot with the crack growth rate da dN versus the stress intensity factor range, K , in the log-log scale. When R 0 , the stress intensity factor range is taken to be the maximum stress intensity factor. With such a commonly used treatment, there is no obvious R -ratio effect on the crack growth rate from experiments. The model captures the insensitivity of the R-ratio effect on crack growth. In addition, the models properly mimic early crack growth from notches. The material is obviously different from some other metallic materials such as aluminum alloy [3], 304L stainless steel [53], and 1070 steel [51]. In the fatigue prediction with the approach, there are no any additional assumptions about the influence of different R -ratio.
Early experimental work on Q345R (16MnR) under strain-
controlled experiment with a mean strain [34] suggests that the insensitivity of crack growth rate to R -ratio can be attributed to the fast mean stress relaxation behavior of the material. In the cyclic
plasticity model used in the current study, (i ) (i=1, 2, …, m) control ratcheting behavior and the mean stress relaxation behavior of a material [56]. The experimental data of mean stress relaxation for Q345R steel obtained from a strain-controlled experiment with a mean strain (strain amplitude=0.0016 and mean strain=0.0084), as given in Fig. 5, is used to calibrate the model parameter (i ) (i=1, 2, …, m). Fig. 5 shows simulation results of the mean stress relaxation with
(i ) (i=1, 2, …, m)=1, 5, 10, 20 and comparison with the experimental observations. It is clear from Fig.5 that when (i ) (i=1, 2, …, m)=1 the predicted mean stress relaxation under the straincontrolled uniaxial loading condition with a mean strain is in good agreement with the experimental results. Therefore, (i ) (i=1, 2, …, m)=1 is used in the cyclic plasticity model to describe the mean stress relaxation behavior of the material Q345R. As a comparison, (i ) (i=1, 2, …, m)=5 for 1070 steel and (i ) (i=1, 2, …, m)=8 for 304L stainless steel were determined [55]. Since the (i ) (i=1, 2, …, m) values for Q345R are much smaller than those for 1070 steel and 304L, a much faster mean stress relaxation in the asymmetric strain-controlled experiment for Q345R is expected. Consequently, under a similar loading condition in the fatigue crack growth experiment, the mean stress relaxation in the material near the crack tip is much faster in Q345R steel than that in 1070 steel and in 304L stainless steel. 9
3.2. Single tensile overload during constant amplitude loading The crack growth behavior is disturbed after the application of a single tensile overload. In Fig. 6, crack growth rates predicted are compared with experimental results obtained from an earlier presentation [34]. The circles denote the experimental results and the lines are the predicted crack growth rates. The post-overload crack propagation behavior of the three specimens is predicted successfully and the crack growth rates predicted are very close to the experimental results. In particular, the check mark shaped curves with instantaneous crack growth acceleration right after overloading are captured in the prediction. The yy yy hysteresis loops of the material at the Gauss point closest to the crack tip for Specimen CP17S ( R 0.1, P 3.15kN, POL 6.0kN at a 9.48mm ) before and after the application of the overload are shown in Fig. 7. The specimen was subjected to a constant load range of 3.15kN with R 0.1 . An overload of 6.0 kN was applied when the crack length was 9.48 mm. The FE stress analysis provides detailed stress, strain, and displacement at any node and Gauss point at any given time during a loading cycle, and all the stress, strain, displacement results can be outputted into data file for further process. The results shown in Fig.7 are the y component of the stress and the y component of the strain at the Gauss point closet to the crack tip (refer to Fig. 2 for the coordinates system). There are five yy yy hysteresis loops shown in Fig. 7: one for the moment right before overloading, one right after overloading, and the other three for the crack extension after overloading. Point A and point B on the yy yy loops represent the moments when the cracked surfaces start to contact and are fully open, respectively. The simulation results shown in Fig. 7 suggest that contact of the cracked surfaces occurs before the application of the overload. Right after the application of the overload, no crack surface contact occurs during a loading cycle. With further extension after overloading, the cracked surfaces are gradually in contact. The size of the yy yy loop is the largest right after the application of the overload (a=9.48mm) and becomes smaller with crack extension (a=9.54mm). The size of the yy yy loop reaches the smallest when a=10.68mm. Further extension of the crack results in an increased
yy yy loop (a=13.5 mm). Noticing that, in general, Dr is proportional to the size of the yy yy loops of the material near the crack tip, and the crack growth rate is linearly proportional to Dr , the evolution of the yy yy loop before and after the application of the overload reflects the observed increase-decrease-increase (check mark shape) in the da / dN K curve. Details of the crack profiles with loading history at the application of the overload can further 10
explain the increased stress-strain hysteresis loop of the material point near the crack tip right after the application of the overload.
Fig. 8 shows the crack profiles for Specimen CP17S
( R 0.1, P 3.15kN, POL 6.0kN at a 9.48mm ). The displacements at any given nodes and Gauss points in the component are obtained from the FE analysis and can be outputted into data file for further analysis. The results shown in Fig. 8 are the displacement in the y-direction at the nodes on the cracked surface at the peak loads before and after the application of the overload (refer to Fig. 2 for the coordinates system).
Contact of cracked surfaces occurs at the minimum external load
(line 2) before overloading. Right after the application of the overload, the crack keeps open at the minimum external load (lines 4 and 6).
Another observation is that the crack tip opening
displacement is larger after overloading than that before overloading. The absence of cracked surface contact in the wake zone increases the crack growth rate instantaneously. However, the residual compressive stresses created by the overload ahead of the crack tip usually retards the crack growth. A competing mechanism exists between the absence of cracked surface contact and the enlarged compressive stress field due to overloading ahead of crack tip. The result from the competition determines the instantaneous acceleration in the crack growth rate after overloading and retardation subsequently. In addition to low sensitivity of the crack growth rate to the R -ratio, the material under investigation is characterized by the great increase in crack growth rate right after the application of an overload. The check marker feature of the overload effect on crack growth has never been properly modeled. It will be revealed that such a characteristic of overload effect can be also related to the fast mean stress relaxation behavior of the material, or low (i ) (i=1, 2, …, m) values in the cyclic plasticity model. The enlarged stress-strain hysteresis loop after overloading shown in Fig. 7 and non-contact of the cracked surfaces right after overloading shown in Fig. 8 are due to the small (i ) (i=1,2, …, m) values for the material that result in fast mean stress relaxation in the material near the crack tip. When (i ) (i=1, 2, …, m)=5 were used for 1070 steel, the yy yy loop at a material point right after the application of the overload was predicted to be much smaller by using the same plasticity model [51].
Consequently, the approach predicts crack growth retardation right after the
application of the overload. To affirm the influence of (i ) (i=1, 2, …, m) on the overall predictions of the crack growth after the application of the overload, Fig. 9 shows the predicted crack growth for Specimen CP18S ( R 0.1, P 3.15kN, POL 6.0kN at a 10.32mm ). With otherwise identical material constants in the cyclic plasticity model and the fatigue model, higher (i ) (i=1, 2, …, m) 11
values result in higher predicted crack growth rate. More importantly, the approach with high (i ) (i=1, 2, …, m) values predict small or no instantaneous crack growth increase right after the application of the overload. With small (i ) (i=1, 2, …, m) values, the approach can predict the check marker shaped crack growth curve for the overload influence. The approach can not only predict crack growth with the overload effect, it can also predict the crack growth with the underload effect. A CT specimen was first subjected to constant amplitude loading with a load range of 3.6 kN at R=0.1. After the crack length reached a=4.254mm, an underload (compressive load) of 6.0 kN was applied. Constant amplitude loading resumed after underloading. A clear acceleration of crack growth was observed right after the application of the underload. The crack growth went back to the stable crack growth expected from the constant amplitude loading after a crack extension of approximately 0.991mm from the moment of underload application. When the crack length, a, reached 9.488 mm, an overload (tensile) of 6.0 kN followed by an underload (compressive) of 6.0 kN was applied. Constant amplitude loading with a load range of 3.6 kN at R=0.1 resumed after the application of the combined overload and underload. Clearly, an acceleration in the crack growth rate was observed right after the application of the combined overload and underload. In addition, the influencing zone size due to the combined overload and underload is large.
The crack length corresponding to the influence zone is
approximately 2.921 mm (the difference in crack length between the moment of the application of the combined overload and underloading and the moment when the crack growth rate returned to the stable state expected under constant amplitude loading). The results shown in Fig. 10 indicate that the approach discussed in the current work can properly describe the underload effect. It should be reiterated that no special material or fitting constants are introduced in the prediction of the overload and the underload effects. 3.3. Life predictions After the crack growth rates are obtained for different crack lengths, the a-N relationship can be determined by using a numerical integration. The predicted fatigue life corresponding to a given crack length, a1 , is obtained using the following equation [52],
N Ni
a1
0
da f a
(5)
where N is the number of loading cycles corresponding a crack length, a1 . The crack length, a , is measured from the notch root of the CT specimen. N i is the crack initiation life. f a is the crack 12
growth rate as a function of the crack length obtained from the approach being discussed. Eq. (5) is integrated numerically and is applicable after crack initiation. The crack initiation life, N i , is determined by Eq. (4). Fig. 11 shows the experimentally obtained a-N relationship for four specimens of Q345R steel and predictions by using the approach discussed. A close agreement between the predictions and the experimental observations can be found. The retardation effect due to overloading can be observed in the a - N curves for those specimens experienced overloading. 4. Further Discussions Difference between the traditional crack closure concept and the influence of contact of cracked surfaces on crack growth should be noticed. With the application of the FE methods for the elasticplastic stress analysis of the cracked specimen, the detailed contact of the cracked surfaces is simulated automatically and therefore, the influence of the cracked surface contact is considered in the approach naturally. No crack closure assumption regarding an effective range of stress intensity factor calculation ( ΔKeff K max Kop ) is utilized in modeling. Earlier studies reveal that when a crack closes during a part of a loading cycle, plastic deformation in the material near the crack tip persists and is not negligible [29,30]. In theory, the crack growth predictions using the current approach do not make use of any experimentally obtained crack growth results. This feature is fundamentally different from the methods where a stress intensity factor or J-integral is used. However, an inherent difficulty in the elastic-plastic FE stress analysis of a cracked component is the influence of the element size on the simulated stress and strain of the material near the crack tip. While there is no definite way to determine the element size for the stress analysis of a cracked component, an early study indicates that a proper selection of the element size is a few times the size of the average grain in a polycrystalline material [51]. The selection of the element size near the crack tip in the current study followed such a “rule of thumb.” In addition to element size, the way of debonding in the FE analysis also influences the predicted crack growth rate when using the approach adopted in the current study. A parametric study was conducted to reveal a proper selection of a debonding method in the FE analysis. Such a study was based on a fixed element size and element type. Fig. 12 shows a comparison of the experimental crack growth data of Specimen CP18S ( R 0.1, P 3.6kN, POL 6.0kN at a 10.32mm ) and the predicted crack growth results obtained from using four different debonding methods: one node after every 10 loading cycles, one node 13
after every five loading cycles, one node after every two loading cycles, and three nodes after every five loading cycles. Clearly, with four-node plane-stress elements and an element size of 60 m near the crack tip, the crack growth results obtained from debonding a node after every 5 or 10 loading cycles agree well with the experimental observations. These two debonding methods result in similar predicted crack growth rates. There is a tendency in the predicted crack growth results shown in Fig. 12 that a slower debonding scenario provides predicted results in better agreement with the experimental observations. However, slow debonding requires significant computation time to simulate a certain range of crack length. For the current investigation, debonding one node for every five loading cycles provides satisfactory predictions of crack growth for the material under investigation with realistic computation time. 5. Conclusions An approach developed earlier by Jiang et al. [49-51] was applied to predict the fatigue crack growth of Q345R pressure vessel steel under constant amplitude loading and a single tensile overload during constant amplitude loading. The crack growth of the material under investigation is characterized by the insensitivity of the crack growth to the R -ratio and an instantaneous acceleration of crack growth rate right after the application of an overload with a clear check mark in the da / dN K curve.
The approach can satisfactorily describe these crack growth
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[43] Zhang J. Z., Zhang J. Z., Du S. Y. Elastic-plastic finite element analysis and experimental study of short and long fatigue crack growth. Engng Fract Mech 2001;68:1591-605. [44] Shi J., Chopp D., Lua J., Sukumar N., Belytchko T. Abaqus implementation of extended finite element method using a level set representation for three-dimensional fatigue crack growth and life predictions. Engng Fract Mech 2010;77:2840-63. [45] Sander M., Richard H. A. Fatigue crack growth under variable amplitude loading Part I: experimental investigations. Fatigue Fract Engng Mater Struct 2006;29:291-301. [46] Sander M., Richard H. A. Fatigue crack growth under variable amplitude loading Part II: analytical and numerical investigations. Fatigue Fract Engng Mater Struct 2006;29:303-19. [47] Ghidini T., Dalle Donne C. Fatigue crack propagation assessment based on residual stresses obtained through cut-compliance technique. Fatigue Fract Engng Mater Struct 2007;30:214-22. [48] Sun B., Li Z. A multi-scale damage model for fatigue accumulation due to short cracks nucleation and growth. Engng Fract Mech 2014;127:280-95. [49] Jiang Y., Ding F., Feng M. An approach for fatigue life prediction. ASME J Engng Mater Tech 2007;129:182-89. [50] Jiang Y. A fatigue criterion for general multiaxial loading. Fatigue Fract Engng Mater Struct 2000;23:19-32. [51] Jiang Y., Feng M. Modeling of fatigue crack propagation. ASME J Eneng Mater Tech 2004;126:77-86. [52] Ding F,. Feng M., Jiang Y. Modeling of fatigue crack growth from a notch. Int J Plasticity 2007;23:1167-88. [53] Fan F., Kalnaus S., Jiang Y. Modelling of fatigue crack growth of stainless steel 304L. Mech Mater 2008;40:961-73. [54] Gao Z., Zhao T., Wang X., Jiang Y. Multiaxial fatigue of 16MnR steel. J Pressure Vessel Technol 2009;131(2):021403-1-9. [55] Jiang Y., Sehitoglu H. Modeling of cyclic ratchetting plasticity, Part I: Development of constitutive relations. J Appl Mech 1996;63:720-25. [56] Jiang Y., Sehitoglu H. Modeling of cyclic ratchetting plasticity, Part II: Comparison of model 18
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19
Table1 Chemical composition of Q345R (wt %) Element
C
Al
Ni
Cr
Mn
Co
Nb
Cu
Ti
Si
P
V
Fe
Weight
0.203
0.035
0.003
0.008
1.535
0.004
0.039
0.010
0.002
0.236
0.009
0.003
Balance
Table 2 Jiang-Sehitoglu cyclic plasticity model used in the finite element simulations ~ S deviatoric stress tensor ~ ~ f (S ~) : (S ~) 2k 2 0
Yield Function
~ backstress k yield stress in shear
n~ normal of yield surface
d~ p
Flow law
1 ~ ~ ~ dS : n n h
h plastic modulus
~ p plastic strain
~
M
~
(i )
~ i ith backstress part
i 1
Hardening Rule
d~ (i )
~ (i ) (i ) (i ) ~ c r n ( i ) r
( i ) 1
~ (i ) dp ~ (i )
M number of backstress parts dp equivalent plastic strain increment
c (i ) , r (i ) and (i ) material constants
(i 1,2,...,M )
Table 3 Material constants of Q345R in Jiang-Sehitoglu cyclic plasticity model E 212.5GPa c1 221.0 r 1 83.0
0.31 c2 46.0
r 2 91.0
k 14.5GPa c3 14.5 r 3 89.0
c4 5.8
c5 2.721
r 4 88.0
r 5 212.0
1 2 3 4 5 1
20
63.5 W=50.8 F 30.5 12.7
r0 =0.1,1.0,2.0
14 0.2
a
an
14 30.5 F
Fig. 1 Standard compact tension specimen (mm) (a)
y
x a Crack growth direction (b)
Notch Root
Fig. 2 Mesh model for CT specimen: (a) overall model; (b) Fine mesh near crack tip
21
D, Fatigue Damage, MJ/m
3
8 Damaging Zone
6 4 A
Specimen CP17S R=0.1, Pmax=3.5kN a=9.48mm
2 0 0.0
0.2
0.4
0.6
0.8
1.0
r, Distance From Crack Tip, mm
Fig. 3 Distribution of fatigue damage per loading cycle along the x-direction
22
10
10
-3 -3
da/dN, Crack Growth Rate, mm/cycle
da/dN, Crack Growth Rate, mm/cycle
10
-4
-5
R=-1.0, Pmax=2.7kN Prediction
10
-6
10
-4
10
-5
10
R=0.1, Pmax=4.0kN Prediction -6
2
3
4
5
10
6 7 8 9
10 100 1/2 K, Stress Intensity Factor Range, MPa m
2
3
4
5
6 7 8 9
10 100 1/2 K, Stress Intensity Factor Range, MPa m 2
10
da/dN, Crack Growth Rate, mm/cycle
da/dN, Crack Growth Rate, mm/cycle
2
-3 8 6 4
2
10
-4 8 6
R=0.2, Pmax=8.0kN Prediction
4
10
-3 8 6 4
2
10
-4 8 6 4
2
2
10
10
-5 1
4
5
6
R=0.3, Pmax=6.9kN Prediction
-5 2
3
4
5
6 7 8 9
10
7
3x10 1/2 K, Stress Intensity Factor Range, MPa m
100
K, Stress Intensity Factor Range, MPa m
1/2
da/dN, Crack Growth Rate, mm/cycle
2
da/dN, Crack Growth Rate, mm/cycle
-3
10
-4
10
-5
10
R=0.5, Pmax=6.0kN Prediction
8 6 4
2
-5
10
8 6 4
2
R=0.75, Pmax=10.0kN Prediction
-6
10
-6
10
-4
10
2
3
4
5
6 7 8 9
10 100 1/2 K, Stress Intensity Factor Range, MPa m
1
3
4
5
2x10 10 1/2 K, Stress Intensity Factor Range, MPa m
Fig. 4 R -ratio effect on crack growth at constant amplitude loading (experimental data taken from [34,59])
23
mean, Mean Axial Stress, MPa
300
250
200
150 Experiment 100
(i)
(i=1,2,3,4,5)=1 (i)
(i=1,2,3,4,5)=5 (i)
50
/2 = 0.16%, mean = 0.84%
(i=1,2,3,4,5)=10 (i)
(i=1,2,3,4,5)=20 0
2
3
4
5 6 7 8 9
1
2
3
4
5 6 7 8 9
10
100
2
3
4
5 6 7 8 9
1000
N, Loading Cycles, cycle
Fig. 5 Mean stress relaxation under strain-controlled uniaxial loading with a mean strain (experimental data taken from [34])
24
-3
10 -3 9 8 7 6 5
da/dN, Crack Growth Rate, mm/cycle
4
Specimen CP17S R=0.1 Pmax=3.5kN POL=6.0kN
Specimen CP18S R=0.1 Pmax=4.0kN POL=6.0kN
4
3
2
-4
10
9 8 7 6 5
da/dN, Crack Growth Rate, mm/cycle
10
9 8 7 6 5 4
3
2
-4
10
9 8 7 6 5 4
3
3
2
2
Experiment Prediction
Experiment Prediction
-5
-5
10
1
10
3
2x10
4
K, Stress Intensity Factor Range, MPa m
10
5
1
3
4
2x10 1/2 K, Stress Intensity Factor Range, MPa m
1/2
5
-3
10
8 7 6 5
Specimen CP08N R=0.1 Pmax=2.5kN POL=4.5kN
4
da/dN, Crack Growth Rate, mm/cycle
3 2
-4
10
8 7 6 5 4 3 2
-5
10
8 7 6 5 4 3
Experiment Prediction
2
-6
10
1
3
2x10
10
K, Stress Intensity Factor Range, MPa m
1/2
Fig. 6 Overload effect on crack growth (experimental data taken from [34]): (a) CP17S ( R =0.1, P =3.15 kN, POL =6.0 kN, a=9.48mm); (b) CP18S ( R =0.1, P =3.15 kN, POL =6.0 kN, a=10.32
mm); (c) CP08N ( R =0.1, P =2.25 kN, POL =4.5 kN, a=9.42mm).
25
1000
Specimen CP17S R=0.1 Pmax=3.5kN POL=6.0kN
a=13.5mm a=9.54mm
yy, y-Direction Normal Stress, MPa
a=10.68mm
500
0 B
B
B B
-500 A
A
A
A
a=9.48mm (After overload) a=9.48mm (Befor overload)
-1000 0.00
0.02
0.04 yy, y-Direction Normal Strain
0.06
0.08
Fig. 7 yy yy hysteresis loops of a material point near the crack tip
0.04
Specimen CP17S R=0.1 Pmax=3.5kN POL=6.0kN
3 5
y,y-Direction Displacement,mm
1
0.03 2
4
6
3
0.02
5
1
0.01
4 6 2
0.00 symmetrical plane
Crack tip -0.01 0.0
0.5
1.0 1.5 r, Distance to Crack Tip,mm
2.0
Fig. 8 y-direction displacements of nodes in the crack wake zone
26
2.5
Fig. 9 Influence of parameters (i ) on the predicted crack growth rate
27
-3
10
9 8 7 6
Specimen DJ07 R=0.1 Pmax=4.0kN
5
da/dN, Crack Growth Rate, mm/cycle
4
Overload/Underload PUL=-6.0kN POL=6.0kN
3
2
-4
10
9 8 7 6 5 4
Underload PUL=-6.0kN
3
2
Experiment Prediction -5
10
1
2x10
3
4
K, Stress Intensity Factor Range, MPa m
5 1/2
Fig. 10 Crack growth with the underload effect
28
20 Constant amplitude loading CP02N Experiment Prediction CP04N Experiment Overload influence zone Prediction
Crack Length, mm
15
10
Overload CP17S Pmax=3.5kN POL=6.0kN Experiment Prediction CP18S Pmax=4.0kN POL=6.0kN Experiment Prediction
5
0 2
3
4
5 6 7 8 9
4
10
2
3
4
5 6 7 8 9
5
10 Number of cycle, cycle
2
3
4
5 6 7 8 9
6
2
10
Fig. 11 Crack length versus number of loading cycles for four specimens (experimental data taken from [34])
Fig. 12 Influence of debonding method on predicted crack growth rate 29
Highlights 1. The crack growth behavior with a single tensile overload is properly predicted. 2. The low sensitivity of R-ratio is attributed to the fast mean stress relaxation. 3. The retarding effect is the competing result of different influence factors. 4. Different material constants in plasticity model used in modeling are discussed.
30