Damage index for crack initiation of structural steel under cyclic loading

Journal of Constructional Steel Research 114 (2015) 1–7

Contents lists available at ScienceDirect

Journal of Constructional Steel Research

Damage index for crack initiation of structural steel under cyclic loading Z.G. Zhou ⁎, L.J. Jia Research Institute of Structural Engineering and Disaster Reduction, Tongji University, Shanghai 200092, China

a r t i c l e

i n f o

Article history: Received 5 January 2013 Received in revised form 21 June 2015 Accepted 3 July 2015 Available online xxxx Keywords: Cyclic loading Crack initiation Ductile fracture Fatigue Finite element analysis (FEA) Notch

a b s t r a c t A damage index for crack initiation (DICI) with damage accumulation effects considered was introduced in this study. Cyclic tension and compression tests of notched bars were used to verify the proposed DICI. The local cyclic elasto-plastic stress–strain responses, used to calculate DICI, were analyzed using the incremental plasticity procedures of ABAQUS finite element code for various strain amplitudes. Two crack initiation criteria with DICI used, single point criterion and characteristic length criterion, were employed to investigate the crack initiation behavior of the specimens. The single point criterion for crack initiation is met when DICI exceeds 1 at any point of the continuum. The characteristic length criterion for crack initiation is met when DICI exceeds 1 over a critical length equal to the characteristic length. It was found that the single point criterion is applicable to fatigue crack initiation and ductile crack initiation separately, while the characteristic length criterion is feasible to simulate fatigue crack initiation and ductile crack initiation simultaneously. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Brittle fracture in steel buildings occurred during the 1994 Northridge earthquake and the 1995 Hyogoken Nanbu earthquake can actually take place only after several or dozens of cyclic loadings [1,2]. The characteristic of this type of fracture is that ductile crack occurs first before sudden propagation of the crack in a brittle manner. The size of the ductile crack can be small or relatively large. To understand the law of ductile crack propagation, it is necessary to find a method to predict ductile crack initiation. There are several approaches to predict ductile fracture of metals under cyclic large strain loading, e.g., critical plastic strain criterion model, void growth model, Gurson–Tvergaard–Needleman (GTN) model and cohesive zone model. Ductile fracture under monotonic loading has been investigated by a number of researchers, e.g., [3–8], while studies on ductile fracture under cyclic loading are still limited [9–12], especially for the transition between fatigue fracture and ductile fracture. In this paper, ductile crack initiation models were summarized first. Then damage index for crack initiation (DICI) based on the critical plastic strain criterion model was introduced in incremental form to predict crack initiation under cyclic large strain loading. Two types of criteria, i.e. single point criterion and characteristic length criterion, were introduced to investigate crack initiation behavior of steels under various fracture modes, i.e., fatigue fracture and ductile fracture. Based on the experiments of Kuwamura [13], finite element analysis ⁎ Corresponding author at: Research Institute of Structural Engineering and Disaster Reduction, Tongji University, 1239 Siping Road, Shanghai 200092 China. E-mail address: [email protected] (Z.G. Zhou).

http://dx.doi.org/10.1016/j.jcsr.2015.07.005 0143-974X/© 2015 Elsevier Ltd. All rights reserved.

was conducted to validate the applicability of the DICI and the two related crack initiation criteria to the two fracture modes. 2. Crack initiation for ductile metal 2.1. Summary of crack initiation models In general, ductile cracking in metal occurs through a process of void nucleation, growth, and coalescence. Generally, computational theories of fracture mechanics of metal include three categories of ductile crack initiation models: model with void implicitly considered, model with void explicitly considered, and model without void considered. Three representative models corresponding to each category, respectively, are introduced in the following sections. 2.1.1. Critical plastic strain criterion model Longitudinal and equivalent plastic strains are used for uniaxial and multiaxial stress states, respectively, to evaluate ductile crack by the critical plastic strain criterion. Although this kind of criterion can predict ductile crack roughly, it cannot take into account some important factors that have effects on the occurrence of ductile crack. That is, plastic strain only is not appropriate to predict ductile crack, as the necessity of considering the effect of stress state is pointed out. Especially, it is necessary to consider the effect of stress triaxiality. Formulae proposed by Hancock and MacKenzie [14], Kuwamura and Yamamoto [15], and the stress modified critical strain (SMCS) [16] are representative models for the critical plastic strain criterion. Although plastic constitutive theories without considering void are used in these models, the effect of void is considered implicitly by introducing the stress triaxiality.

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For cyclic loadings, the Manson–Coffin law is often used as an empirical method to determine crack initiation. 2.1.2. GTN model A representative model with void explicitly considered is the Gurson–Tvergaard–Needleman (GTN) model [17]. There are microvoids originated from inclusions or impurities in ductile metal. When stress increases, voids grow up and coalesce. When stress increases more, ductile failure happens. A yield function for porous metal was derived from a spherical void model by Gurson [18]. Tvergaard modified the yield function by introducing a factor. Needleman and Tvergaard modified the development law of the volume fraction of the void. The GTN model treats a porous metal as a homogeneous continuum, where the effect of voids is considered through the continuum material averagely. The main difference between a traditional plastic model and the GTN model is that the former does not take hydrostatic pressure into account while the latter does. The above GTN model considered isotropic hardening only and cannot be applicable to cyclic loadings. Leblond considered both the isotropic hardening and kinematic hardening, and extended the GTN model to cyclic loadings [19]. 2.1.3. Cohesive zone model (CZM) In the cohesive zone model, crack path is assumed previously. Crack path is treated as a thin layer of material with its own material properties [20]. The traction–separation method is used to decide whether a crack develops. For cyclic loadings, cyclic cohesive zone model (CCZM) is introduced for simulating crack propagation [21]. 2.2. Proposed damage index for crack initiation (DICI) From Rice and Tracey's derivation [22], it has been shown that void growth rate is proportional to exponent of the stress triaxiality. The stress triaxiality is defined as follows: T r ¼ σ m =σ e

ð1Þ

where σm is mean or hydrostatic stress, and σe is von Mises' equivalent stress. Based on this derivation, Hancock and Mackenzie expressed the crack initiation plastic strain by the following relationship ε pd ¼ α expð−1:5T r Þ

ð2Þ

where α is a material dependent constant, and the coefficient of 1.5 in the exponent is theoretically derived. Eq. (2) is mainly applicable to monotonic loading, and it cannot be employed directly for the cases where the stress triaxiality covers both positive and negative values. In this paper, a state variable, ωd, namely DICI, is introduced to evaluate the cumulative damage leading to the crack initiation, as shown in Eq. (3): Z ωd ¼

dε p ¼ εpd

Z

1 expð1:5T r Þdεp α

ð3Þ

2.2.1. Single point criterion The single point criterion for crack initiation is met when the following condition is satisfied at any point of the continuum. ωd ¼ 1:

ð5Þ

2.2.2. Characteristic length criterion The characteristic length criterion for crack initiation is met when the following condition is satisfied. ωd ≥1

for

 r ≥l :

ð6Þ

Here r is the maximum distance between any two points of the area over which ωd exceeds 1. The parameter, l⁎, termed characteristic length, is determined as average size of the dimple plateaus and valleys which are commonly observed at the fracture surface [14]. This criterion includes a length scale to describe the critical volume of the continuum over which ductile crack initiation index is exceeded. The inequality r ≥ l* implies that crack initiation will occur when ωd exceeds 1 over a critical length of r equaling to the characteristic length. Thus, this criterion has two parameters, α and l⁎, which has to be calibrated first. 3. Verification by FEA 3.1. Summary of experiment used in FEA The experimental results from Kuwamura [13] were used to verify the proposed DICI. The configuration of the test specimens is shown in Fig. 1. The specimens have an hourglass-type shape with a circumferential notch at the mid-length. The depth of the notch is 1 mm; the radius of the notch root is 0.5 mm, and the diameter of the minimum cross section is 12 mm. The material is SM490 of Japanese Industrial Standard. Deformation controlled cyclic loading tests were performed with a gauge length of 30 mm spanning the center notch. Deformation amplitude, namely δ, is between 0.06 mm and 1.00 mm. The number of cycles until crack initiation at the notch root was recorded. 3.2. Finite element analysis Since the section of the specimens is circular and the loadings are axisymmetric, it can be simplified as an axisymmetric model. FEA model is illustrated in Fig. 2. The edge length of the mesh near the notch root is 0.03 mm. ABAQUS 6.9 was used to conduct the analysis [23]. An axisymmetric solid element type, CAX4R, is employed as it is computational efficient for plastic deformation problems. Material properties used in the analysis are illustrated in Fig. 3, which are based on the true stress–true strain data and obtained by coupon test results. As for the plasticity model, a combined nonlinear isotropic/kinematic hardening model termed Chaboche model was used, with three backstresses to describe the nonlinear kinematic hardening effect, and

where εp is the equivalent plastic strain. The ratio of incremental equivalent plastic strain to the crack initiation plastic strain expressed in Eq. (2), i.e., dεp/εpd is supposed as incremental damage herein. The integration of dεp/εpd is defined as ωd, and the physical implication of ωd is cumulative damage. ωd increases monotonically with εp. At each increment of an FE analysis, incremental increase in ωd is computed as follows: Δωd ¼

1 expð1:5T r ÞΔεp ≥0: α

ð4Þ Notch Details

Through this definition, crack initiation under cyclic loading can be evaluated by the monotonic increasing index, ωd.

Fig. 1. Configuration of the test specimen [13].

Z.G. Zhou, L.J. Jia / Journal of Constructional Steel Research 114 (2015) 1–7

3

Uniform deformation (mm) 15

dx=0 10

R=50

5

Y

R=0.5

0 X

0

1

2

3 4 5 dy=0

6 7

Fig. 2. FEA model.

3.3. Verification of averaged true stress–true strain curves Averaged true stress–true strain curves calculated from FEA and experimental results [13] are plotted in Fig. 4. The horizontal coordinate is the averaged true strain calculated from variation of the diameter of the minimum cross section. The vertical coordinate means the averaged true stress of the minimum cross section. Results of three cases of deformation amplitude, i.e., 0.08, 0.30, and 1.00 mm, are illustrated here. It can be found that the FEA curves can compare well with the experimental results. These FEA results were used to evaluate the DICI. 3.4. Distribution of stress triaxiality and equivalent plastic strain Distribution of the stress triaxiality and equivalent plastic strain along the radial direction of the minimum cross section at the first cycle is illustrated in Fig. 5, where +δ means positive peak deformation amplitude, and −δ means negative peak deformation amplitude. The horizontal coordinate 0 and 6 correspond to the center of the minimum cross section and the notch root, respectively. The vertical coordinates

True stress (MPa)

1000 800 600 400

Young’s modulus=205 GPa Poisson’s ratio=0.3

200

of the left and the right side indicate the stress triaxiality and equivalent plastic strain, respectively. It can be seen from the figure that the maximum stress triaxiality occurs at the point apart slightly from the notch root, and that maximum equivalent plastic strain occurs at the notch root. The curves are flat near the center and sharp near the notch root. Comparing the results at +δ and −δ, it can be found that the node with the maximum stress triaxiality at −δ locates closer to the notch root than that at +δ, and this tendency becomes more apparent as the amplitude increases. The absolute value of the maximum stress triaxiality at +δ and that of the minimum stress triaxiality at −δ have no difference when amplitude is small, while have difference when the amplitude is large. For example, the values are approximately 1.2 and 0.9 respectively when δ = 1.00 mm. Fig. 6 shows the stress triaxiality versus equivalent plastic strain curves at different locations of the minimum cross section. The horizontal coordinate means equivalent plastic strain, the vertical coordinate means stress triaxiality, and each point of the curves corresponds to one state of stress triaxiality-equivalent plastic strain. The curves are history states of the first two cycles at the notch root and the center 0.08 mm, Test

0.3 mm, Test

1mm, Test

0.08 mm, FEA

0.3 mm, FEA

1 mm, FEA

1000 800 Averaged true stress (MPa)

the isotropic hardening effect was also simulated using the method proposed in the literature [24]. Vertical displacement is applied to the top of the model, as shown in Fig. 2. The deformation amplitudes have 10 different magnitudes, i.e., 0.06, 0.07, 0.08, 0.1, 0.2, 0.3, 0.4, 0.5, 0.8, and 1.0 mm, which are the same as those of the experiments.

600 400 200 0 -200 -400 -600 -800

0

-1000

0

0.2

0.4 0.6 True strain

Fig. 3. Half cycle of stress–strain data.

0.8

1

-6

-4

-2

0 2 4 6 Averaged true strain (%)

8

Fig. 4. Averaged true stress–true strain correlations.

10

12

4

Z.G. Zhou, L.J. Jia / Journal of Constructional Steel Research 114 (2015) 1–7

Stress triaxiality Equivalent plastic strain 1.2

+δ +δ

1.5

-δ -δ

Notch root 0.16

δ =0.08 mm

0.9

0.12

0.3

0.10 Stress Triaxiality

0

0

0.08

-0.3

0.06

-0.6

0.04 Equivalent plastic strain

-0.9

-1.5

0.02

-1.2

0

1

2

3

0.00

-1.5

-0.02 2

3

4

5

1.2

1

2

3

4

5

0

1

2

3

1.5

1.35 1.20 1.05

0.3

0.90

0

0.75

-0.3

0.60

-0.6

0.45

-0.9

0.30 0.15

-1.2

0.00

-1.5 1

-1.5

δ =1.0 mm

0.6

0

0

6

δ =1.0 mm

0.9

δ =0.30 mm 0.54 0.48 0.42 0.36 0.30 0.24 0.18 0.12 0.06 0.00

δ =0.30 mm

0

1.5

6

Stress triaxiality

1.2 0.9 0.6 0.3 0 -0.3 -0.6 -0.9 -1.2 -1.5

1

Equivalent plastic strain

0

Stress triaxiality

Center

0.14

0.6

δ =0.08 mm

2

3

4

5

6

0

-1.5 0

1

2

3

Equivalent plastic strain Fig. 6. Triaxiality-equivalent plastic strain state for the first two cycles.

Distance from the center of the section (mm) Fig. 5. Distribution of stress triaxiality and equivalent plastic strain of the first cycle.

respectively. The upside of the curves corresponds to tensile state and the underside compressive state. It can be seen that the stress triaxiality varies sharply at the unloading stage.

The variation of DICI with the increase of loading cycles is illustrated in Fig. 8, taking the cases of δ = 0.30 mm and α = 5 as an example. It can be seen that the curves are flat in the compressive part of each cycle, resulting from the fact that Δωd is relatively small in Eq. (4) when the stress triaxiality is negative. So an increase of DICI is mainly due to the tensile loading half cycles. It can also be seen that the increase of DICI at the center is much slower than that at the notch root.

3.5. Analysis of damage index for crack initiation For most of structural steels, typical values of the parameter α in Eq. (3) are in the range of 1 to 10. A large value of α indicates a good plastic deformation capacity of the steel. Chi et al. gave a way to determine α [16]. In this study, although the cases of α equaling 1, 2, 3, 4, 5, and 8 were calculated, the cases of α equaling 3, 5, and 8 represent the range of experimental results, so the results of the three cases were used to analysis. 3.5.1. Distribution of DICI Distribution of DICI along the radial direction of the minimum cross section versus loading cycles when α equals 5 is illustrated in Fig. 7. Each curve shows the distribution of DICI at the end of each cycle with the bottom curve corresponding to the first cycle and subsequent cycle upwards one by one. The curve with mark corresponds to the cycle at which experimental crack is initiated. It can be observed that the variation of DICI concentrated on an area 1 mm beneath the notch root, meaning that ductile crack will initiate at this area.

3.5.2. DICI corresponding to crack initiation cycle This section discusses distribution of DICI at the experimental crack initiation cycle. In the experiment, Kuwamura used a microscope to monitor the state of crack initiation [13]. Distribution of DICI along the radial direction of the minimum section at crack initiation cycle is illustrated in Fig. 9. It can be seen that DICI decreases with the growth of α. Fig. 10 shows the contour plots of DICI near the notch for the case when α = 5. From Fig. 10 it can be concluded that DICI concentrates on a band through the notch root and about 45° to the radial direction. In the case of α = 5, the maximum DICI is 1.63 for δ = 0.08 mm, 1.12 for δ = 0.30 mm, and 0.88 for δ = 1.00 mm. 3.5.3. Evaluation by single point criterion As for the single point criterion, DICI = 1 means ductile crack initiation. The maximum value of DICI occurs at the notch root, so crack initiates at the notch root first. The cycle at which DICI of the notch root grows up to 1 is considered as the crack initiation cycle.

Z.G. Zhou, L.J. Jia / Journal of Constructional Steel Research 114 (2015) 1–7

2.0

3

40

When α=5

δ =0.08 mm

Experimental crack cycle

35

1.5

5

δ =0.08 mm

2.5

30

Cycle number

2

25

1.0

20

1.5

15

0.5

10

1

5

0.0

0.5

1

0

1

2

3

4

5

6

0 0

1

2

3

4

5

6

1

2

3

4

5

6

δ =0.30 mm

1.5

9

1.0

7

5 3

0.5

1

0.0 0

2.0

1

2

3

4

5

6

δ =1.0 mm

DICI, ω d

DICI, ω d

2.0

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

3

1.5

δ =0.3 mm

1.6 1.0

2

0.5

1

0.0 0

1

2 3 4 5 Distance from the center of the section (mm)

6

δ =1.0 mm

1.4

α=3

1.2

α=5

1

α=8

0.8 0.6 0.4

Fig. 7. DICI along the radial direction at cycles.

0.2 0 0

Fig. 11 shows the correlations of the crack initiation life and the averaged true strain amplitude ranging from 0.41%–7.47%, which corresponds to the deformation amplitude of 0.06–1 mm. The three curves in Fig. 11 correspond to the cases of α equaling 3, 5, and 8, respectively, where the separate dot points are the experimental results. Comparing the FEA results and the experimental results, it can be found that they are close to each other for large amplitudes of 0.3 to 1.00 mm if α = 5, and for small amplitudes of 0.06 to 0.20 mm if α = 8. Kuwamura has pointed out that the small amplitudes of 0.06 to 0.1 mm correspond to fatigue crack, the medium amplitudes of 0.1 to 0.2 mm correspond to transition from fatigue crack to ductile crack, and the large amplitudes of 0.2 to 1.00 mm correspond to ductile crack [13]. Combining these results it can be seen that the value of α is not the same for the two cracking modes, and ductile crack corresponds to a smaller value of α. 1.8 Notch root

1.6

DICI, ω d

2 3 4 5 Distance from the center of the section (mm)

DICI values at the notch root of the experimental crack initiation cycle corresponding to different averaged true strain amplitudes are illustrated in Fig. 12. The horizontal coordinate means the averaged true strain amplitude, and the vertical coordinate means DICI value at the notch root. For large amplitudes of 0.3 mm to 1.00 mm, DICI fluctuates near 1 when α = 5; and for small amplitudes of 0.06 mm to 0.20 mm, DICI fluctuates near 1 when α = 8. 3.5.4. Evaluation by characteristic length criterion As for the characteristic length criterion, crack initiates when Eq. (6) is satisfied. The critical length r over which DICI exceeds 1 is illustrated in Fig. 13. The critical length is along the radial direction in this study. It can be seen that the critical length is the most even when α = 3, in Radial direction

δ =0.30 mm

1.2 1

δ =0.08 mm

0.8 0.6 0.4

δ =0.30 mm

0.2 0 0

1

2

3

4 5 6 Cycle number

Fig. 8. DICI versus cycles.

7

8

9

10

6

Fig. 9. DICI along the radial direction at the experimental crack cycle.

Center

1.4

1

When α =5

δ =1.0 mm Fig. 10. DICI plots at the experimental crack cycle.

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Z.G. Zhou, L.J. Jia / Journal of Constructional Steel Research 114 (2015) 1–7

100

0.5 0.45

Critical length (mm)

Number of cycles to crack initiation

0.4

10

α=3

0.35 Avarage

0.3 0.25

α=5

0.2 Avarage

0.15 0.1

α=3

0.05

α=5

0

α=8

α=8 Avarage 0.1

1 Averaged true strain amplitude (%)

Test results

10

Fig. 13. Critical length at the experimental crack cycle.

1 0.1

1 Averaged true strain amplitude (%)

10

2) The characteristic length criterion is feasible to simulate fatigue crack initiation and ductile crack initiation simultaneously with the same value of the material parameter, α.

Fig. 11. Cycles to crack initiation-deformation amplitude correlation.

which case the average length is 0.318 mm, and the standard deviation value is 0.104 mm. It means that this criterion will be feasible if the characteristic length l⁎ takes the value of 0.318 mm. 4. Conclusions Elasto–plastic finite element simulations of cyclic loadings of notched bars were performed in this study. A damage index for crack initiation (DICI) is proposed in incremental form to evaluate crack initiation under cyclic loading. It is also used for establishing two types of criteria, i.e., single point criterion and characteristic length criterion, to investigate crack initiation behavior of steel under different cracking modes, i.e., fatigue fracture and ductile fracture. It can be concluded that: 1) For the single point criterion, the material parameter, α, should be determined for fatigue fracture and ductile fracture, respectively, where fatigue fracture corresponds to a larger value of α.

Nomenclature mean or hydrostatic stress σm σ von Mises' equivalent stress stress triaxiality Tr damage initiation plastic strain εpd α material dependent constant damage index for crack initiation (DICI) ωd equivalent plastic strain εp r the maximum distance between any two points of the area over which ωd exceeds 1 l⁎ characteristic length δ deformation amplitude Acknowledgment

3.5

The authors wish to gratefully acknowledge the support of this work by Research Fund for the Doctoral Program of Higher Education of China under grant no. 20130072120009 and National Natural Science Foundation of China under grant no. 51478357. It is also gratefully acknowledged that Prof. Hitoshi Kuwamura provided the detailed experimental data.

3

2.5 α=3 DICI, ω d

The 1.5 coefficient in Eq. (2) is theoretically derived [22]. Some coefficients other than 1.5 are reported to work better for Eq. (2). So it is worthy to study the effects of the coefficient in future work. More experiments under mixed-mode loading, especially for the case of presence of compressive-shear stresses, are also expected to validate the proposed criteria.

2

References 1.5

α=5

1 α=8 0.5

0 0.1

1 Averaged true strain amplitude (%)

Fig. 12. DICI values at the notch root at the experimental crack cycle.

10

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