Comparison of Threshold Stress Intensity Factor and Fatigue Limit for Micro-crack of Offshore Structural Steel F690

Abstract

In this paper, the evaluation equations proposed by Tange et al. and Ando et al. were used to evaluate the threshold stress intensity factor ∆K^R_th(s) and fatigue limit ∆𝜎^R_wc, according to the small crack of offshore structural steel F690. Despite the differences in concept and shape of the two equations, the ∆K^R_th(s) and ∆𝜎^R_wc proved completely consistent. It is possible to use these equations to evaluate the dependence of the crack length on the ∆K^R_th(s) and ∆𝜎^R_wc of structures made of all steel grades. With these equations, the characteristics of microcracks can be quantitatively evaluated, and the safety and reliability of the structure can be secured.

1. Introduction

Generally, the safety of fracture is ensured when the applied stress on the structure satisfies a certain limit. The limit value is determined by the material properties. Material properties are obtained from the material strength test without cracks or defects. However, most of the actual structures contain, or are likely to cause, cracks. As a result, fracture occurs, even under stresses less than the yield strength or tensile strength. Fracture mechanics predicts the fracture behavior and safety based on the dynamic relationship of the crack tip using continuum mechanics, under the premise that cracks exist in the structure. Fracture mechanics quantitatively determines how dangerous a crack is, and how it will grow, dependent on the existing cracks or the environment, and the loads in which the cracks are probabilistically expected to exist [1-9].

In particular, linear fracture mechanics requires that the plastic or nonlinear region occurring at the crack tip be very small, compared to the crack length. The stress intensity factor (\(K\)) [10-12] of linear fracture mechanics depends on the fracture toughness of metals and ceramics, and the threshold stress intensity factor of fatigue crack propagation [13]. However in small cracks, as the crack length increases, these values increase; but above a certain crack length, the values become constant, regardless of the crack length. Therefore, the fracture of the structure will depend on how the microcrack problem is controlled. El Haddad et al [14]. proposed a threshold stress intensity factor that depends on the microcrack length, while Tange et al.[15] proposed a modification of the El Haddad equation. Ando et al.[16] proposed an equation for evaluating the threshold stress intensity factor in consideration of the plastic region size by the Dugdale model [17]. Using this equation, the reliability improvement of F690[18] and the fatigue limit of SCM435[19] for microcracks were studied.

In this study, we evaluated the threshold stress intensity factor (\(\Delta K _ { t h } ^ { R }\)) and the fatigue limit (\(\sigma _ { w c } ^ { R } \)) according to the microcrack length in offshore structural steel F690 using the equations proposed by Tange[15] and Ando[16], which can be unified to evaluate the microcrack problems.

2. Evaluation Method

The threshold stress intensity factor range (\(\Delta K _ { t h ( s ) } ^ { R } \)) of the fatigue crack growth of the small crack in the material can be obtained from Eq. (1):

\(\Delta K _ { t h ( s ) } ^ { R } = \Delta \sigma _ { w c } ^ { R } \sqrt { \pi l }\)       (1)

Equation (2) can be used to evaluate the crack length dependence of the threshold stress intensity factor range (\(\Delta K _ { t h ( s ) } ^ { R } \)) of a small crack as a function of the crack length (\(l\)), the fatigue limit of a smooth specimen (\(\Delta \sigma _ { w } ^ { R } =740MPa\)), and the threshold stress intensity factor range (\(\Delta K _ { th(l)} ^ { R } = 6.51MPa \sqrt { m }\)) of a large crack[16]:

\(\)       (2)

Meanwhile, Tange et al. obtained Eq. (3) by removing from the El Haddad equation[15]:

\(\Delta K _ { th ( s ) } ^ { R } = \Delta \sigma _ { wl } ^ { R } \sqrt { \pi l } = \frac { \Delta K _ { th ( I ) } ^ { R } \sqrt { l } } { \{ l + \frac { 1 } { \pi } ( \frac { \Delta K _ { th ( l ) } ^ { R } } { \Delta \sigma _ { w } ^ { R } } ) ^ { 2 } \} ^ { 0.5 } }\)       (3)

The dependence of \(\Delta \sigma_{w}^{R}\) on the stress ratio was obtained from modified Goodman Eq. (4).

\(\sigma _ { a } = \sigma _ { w } ( 1 - \frac { \sigma _ { m } } { \sigma _ { B } } )\)       (4)

Where \(\sigma _ { a }\), \(\sigma _ { w }\), \(\sigma _ { m }\), and \(\sigma _ { B }\)is stress amplitude, fatigue limit of R=-1, mean stress and tensile strength, respectively.

The dependence of \(\Delta K _ { t h ( l ) } ^ { R }\) on the stress ratio was obtained from Eq. (5).

\(\Delta K _ { t h ( l ) } ^ { R } = \Delta K _ { t h ( l ) } ^ { R } \sqrt { ( 1 - R }\)       (5)

Table 1. Chemical compositions (wt.%) and mechanical properties of F690

3. Specimen and Properties of Material

The material used in this study is the offshore structural steel F690. Table 1 shows the chemical compositions and mechanical properties of F690. Fatigue tests were conducted with four-point bending at the stress ratio (R = 0.1). Fig. 1 shows the shape of the specimen, and a schematic of the four-point bending load, with lower span of 150 mm, and upper span of 20 mm. The fatigue limit (\( \sigma_{w}^{R}\) ) obtained at the stress ratio (R = 0.1) is 740 MPa, and the threshold stress intensity factor (\(K_{th(l)}^{R}\)) is 6.51\(MPa \sqrt{m}\) obtained at 1TCT with a thickness of 12.5 mm.

Fig. 1 Shape of specimen, and schematic of four-point bending

4. Evaluation Results and Discussion

The aspect ratio (As) was made into three types of (1.0, 0.6, and 0.4). The stress ratio (R) was set to (0.0, 0.1, and 0.5). \( \sigma_{w}^{R}\) and \(K_{th(l)}^{R}\) according to the stress ratio were obtained from Eqs. (4) and (5) using the result of R = 0.1 obtained in the experiment. Table 2 shows \( \sigma_{w}^{R}\) and \(K_{th(l)}^{R}\) according to the stress ratio:

Table 2. Data of F690 for the evaluation of

The results of Table 1 were used to evaluate \(K_{th(s)}^{R}\) and \( \sigma_{wc}^{R}\) according to the crack size. The Ando and Tange equations (Eqs. (2) and (3), respectively), were used for evaluation.

Fig. 2 shows the crack depth dependence of \(K_{th(s)}^{R}\). Fig. 2 (a), (b), and (c) evaluate according to the stress ratio (R) of As = (1.0, 0.6, and 0.4), respectively. depends on the crack depth, and as the crack grows, increases. The convergence value is . The crack depth satisfying is highly dependent on the crack shape. The larger the and of the material, the shorter the crack depth at which converges to . Table 3 shows five crack depths by the Tange and Ando equations. The table shows that was almost similar, regardless of the stress ratio (R) and aspect ratio (As). Therefore, hereinafter, by the Ando equation will be described. In the case of As = (1.0, 0.6, and 0.4), it can be seen that as the crack depth increases, the stress ratio R = (0, 0.1, and 0.5) converges to each . For , the smaller As was found to converge faster at = 6.86 . That is, when compared at the stress ratio (R = 0) and the crack depth a = 3.0 mm, As = 1.0 is = 6.79 , As = 0.6 is = 6.83 , and As = 0.4 is = 6.84 . In addition, the of the crack depth a = 0.04 mm is 4.39 (As = 1.0), 4.98 (As = 0.6), and 5.26 (As = 0.4), respectively, indicating that the for the smaller As was larger. This is because when the surface length is larger than the depth, the crack propagation is faster in the depth direction.

Fig. 2 Threshold stress intensity factor (\(\Delta K_{th(s)}^{R}\)) depending on crack depth. (a) As = 1.0, (b) As = 0.6, and (c) As = 0.4

Table 3. Threshold stress intensity factor (\(\Delta K_{th(s)}^{R}\)) for each crack depth depending on the stress ratio and aspect ratio (unit: \(MPa \sqrt {m}\))

Fig. 3 shows the dependence of on the crack depth. Fig. 3 (a), (b) and (c) evaluate according to the stress ratio (R) of As = (1.0, 0.6 and 0.4), respectively. depends on the crack size, and as the crack grows, decreases. Table 4 shows the by the Tange and Ando equations for the five crack depths. The values are almost similar, regardless of the stress ratio (R) and the aspect ratio (As). Therefore, hereinafter, by the Ando equation will be described. The of the cracked material decreases by approximately 8 % (As = 1.0), 12 % (As = 0.6), and 15 % (As = 0.4) at the crack depth a = 0.01 mm, regardless of the stress ratio (R), and then rapidly decreases. The abruptly decreasing slope corresponds to the threshold stress intensity factor of a large crack. The crack depth a = 3 mm decreases approximately 86 % (As = 1.0), 92 % (As = 0.6), and 93 % (As = 0.4). At the same aspect ratio (As), as the stress ratio (R) increase, gets smaller. This is because the increase of the stress ratio (R) increases the average stress, so that the crack propagation rate increases. In addition, the smaller the As, the smaller the fatigue limit ( ). This is because the crack propagation of the depth direction increases. Therefore, since the fatigue limit depends on the stress ratio (R) and aspect ratio (As), it is necessary to pay attention to the crack control of the structure by nondestructive inspection.

Fig. 3 Fatigue limit ( ) depending on crack depth (a) As = 1.0, (b) As = 0.6, and (c) As = 0.4

Table 4. Fatigue limit ( ) for each crack depth depending on the stress ratio (R) and aspect ratio (As) (unit: )

The crack dimension dependencies of and obtained from Eqs. (1) and (2) agree very well. The two equations agree very well, regardless of the aspect ratio (As) and the stress ratio (R). From this result, the fatigue limit of the El Haddad research can be evaluated by the Ando Eq. (2). El Haddad et al. induced an evaluation equation of depending on the crack length, by summing the initial crack length and the microcrack length grown by cyclic stress. On the other hand, Ando et al. induced the evaluation equation for the size of the Dugdale model plastic zone by cyclic stress, which does not depend on the crack size. Although the basic concepts and the types of equation differ, the values by the Ando Eq. (2) and Tange Eq. (3) are almost the same. They are judged to have similar results, because the microcrack size due to cyclic stress or the plastic zone size of the Dugdale model satisfies the small-scale yield range.

5. Conclusions

The threshold stress intensity factor and fatigue limit were evaluated according to the small crack of the offshore structural steel F690 using the Tange equation that modified the El Haddad equation, and the Ando equation that considered the plasticity area of the fatigue crack tip. Although the two equations differ in the induced concept and the shape of equation, the and results of the small crack are completely consistent. These equations will therefore be able to evaluate the small crack dimensional dependence of the lower limit stress intensity factor and fatigue limit of all steel structures.