Fatigue Life Prediction of High Strength Steel with Pitting Corrosion under Three-Point Bending Load

Abstract

Offshore structures often suffer from pitting corrosion, which leads to local stress concentrations, a decrease in the cross-sectional area, the subsequent initiation and gradual propagation of cracks, and a shorter service life as a result. This study aims to investigate the impact of pitting corrosion on the fatigue life endurance of high-strength steel used for offshore structures. To this end, a three-point bending fatigue test was first performed on the specimens to obtain the fatigue test data. Then, a fatigue life prediction model consisting of two terms is proposed based on fracture mechanics, and the fatigue test data are used to verify the reliability of the model. Finally, the experimental results are discussed, and conclusions are drawn. The first term was designed for crack initiation. Combining with the energy theory and slip band dislocation theory, a novel equivalent surface defect model was proposed and used to predict the fatigue life of pitted corroded specimens before crack initiation. The second term is designed for crack propagation. The generalized Paris model is adopted for fatigue life prediction during the crack propagation process after the crack angle is taken as a variable. The mathematical model for predicting three-point bending fatigue life was finally obtained, and the average relative error of the data validation results did not exceed 16%, which proved the reliability of the prediction model.

1. Introduction

Corrosion is a common issue in mechanical structures, causing significant damage to their structural integrity [1]. Pitting corrosion is a kind of metal corrosion that usually occurs under harsh working conditions, and localized pitting corrosion accounts for approximately 70% of overall destruction caused by corrosion. It was reported that under identical mass loss rates, pitting corrosion is typically more harmful than uniform corrosion [2]. Under the influence of alternating load, dramatically increased local stress fields are generated around corrosion pits, which consequently decrease the bearing capacity of the structure and shorten its service life as well [3]. Therefore, it is necessary to deeply understand the process of fatigue crack initiation and propagation and explore approaches to fatigue life prediction of pitted corroded metal structures under alternating load.

Currently, studies on the prediction of fatigue life of corroded metal structures are mainly divided into two categories. One is corrosion fatigue, which takes into account the synchronization and coupling of corrosion and alternating load and believes that corrosion and fatigue damage always exist and interact with each other all the time. The other is pre-corrosion fatigue, which applies mechanical theories to corroded metal structures and focuses on the influence of pitting geometry and fatigue load on the damage process and fatigue life. This method can improve the accuracy and efficiency of fatigue life prediction [4]. For example, Zhang adopted the damage mechanics evolution equation and the finite element method to predict fatigue life. Although the prediction was basically well in agreement with the experimental result, the initial damage caused by pre-corrosion was not taken into consideration [5]. Zhan regarded the initial damage of pre-corrosion as a constant and calculated the additional damage caused by alternating load using the same theory. But, they found that it was difficult to generalize the damage mechanics to predict fatigue life by the damage evolution method [6]. At present, methods based on fracture mechanics are commonly used for fatigue life prediction of pre-corroded structures, and good accuracy has been achieved [7]. Barter proposed an equivalent surface defect size method, where the critical pits on the specimen were assumed to be initial cracks, and the service life of the specimen was defined as the propagation time of the cracks [8]. This method effectively predicted the crack propagation life, but the service life before the crack initiation was not taken into consideration. Alvarez clearly pointed out that the crack propagation curve is conservative and can only provide a conservative prediction [9]. Fernandez proposed a cross-section fiber model to evaluate the fatigue performance of metal structures suffering from pitting corrosion, in which local effects caused by pitting corrosion, such as uneven distribution of material properties, stress concentration, and local deformation, were considered, and both generalized and local pitting effects were taken into consideration except the fatigue crack propagation behavior [10]. Huang treated pits as initial cracks and proposed an equivalent crack size model to predict fatigue life using linear elastic fracture mechanics [11]. However, the model only focused on the crack propagation life and ignored the period before crack initiation. The whole fatigue life of a metal structure should consist of both crack initiation life and crack propagation life. In order to foretell exactly the fatigue life of pitted corroded materials, both periods should be taken into consideration [12,13].

The presence of notch and size effects should also be considered. Considering the cost of full-size testing of components and the limitations of test conditions, it is difficult to carry out fatigue tests on them, and local fatigue life is often measured by the fatigue test data of small-size specimens [14]. Factors to be considered are stress concentration due to notched structures (notch effect) [15] and fatigue life changes due to dimensional changes (dimensional effect) [16]. The presence of notch and size effects has a significant impact on the experimental results. For the size effect and notch effect, it can be simply understood that the larger the size of the structural notch, the lower its strength [15]. Similarly, the fatigue life of a material typically decreases as its defect size rises; i.e., the larger the influence of the specimen’s zone, the greater the probability that it will develop more severe structural defects or lead to failure [17,18]. Therefore, it is necessary to establish an equivalent surface defect model for different structures and defect sizes, through which notch and size are transformed into one variable to facilitate the establishment and validation of subsequent fatigue life prediction models.

In this paper, firstly, the influence of the notch effect and size effect on the experimental results is taken into account to establish an equivalent surface defect model with one variable to represent the influence of specimens’ notch size and structure on fatigue performance. Then, the three-point bending fatigue life prediction of pitted metal structures is divided into two parts: crack initiation life and crack extension life. For the crack initiation life, a new equivalent surface defect model is proposed in this paper, which utilizes the volume loss rate and the pitting depth-to-diameter ratio to quantitatively assess the effect of pitting on fatigue life. For crack extension life, the crack angle is used to represent the crack extension process, and the Paris model is employed to predict the fatigue life of pitted specimens. Finally, the reliability of the fatigue life prediction model is verified by a three-point bending fatigue test, and the prediction results are in good agreement with the experimental results.

2. Materials and Methods

2.1. Equivalent Surface Defect Model

An equivalent surface defect model can be used to evaluate the influence of pitting damage on the fatigue life of structural members. Although the size of an equivalent surface defect can be represented by the roughness of the material, it does not take the influence of pitting characteristics into consideration, which includes pitting size and shape [19]. In the methods based on fracture mechanics, pits are regarded as surface defects, and the size of initial defect can be obtained by establishing the relationship between the size of initial defect and pit geometrical parameters [20]. Following this idea, the relationship between equivalent surface defect size, volume loss rate, and pitting hole size can be defined as follows:

lna₀ = AV₀ + B(h/d) + C,

(1)

where $a_0$ is the size of equivalent surface defect, $V_0$ means the volume loss rate of specimens, $h/d$ is pitting depth to diameter ratio, and $A$, $B,$ and $C$ are model parameters.

2.2. Fatigue Life Prediction Model

The total fatigue life $N_f$ is composed of crack initial life $N_i$ and crack propagation life $N_p$:

N_f = N_i + Np,

(2)

2.2.1. Prediction of Crack Initiation Life

Crack initiation life is defined as the number of load cycles before the generation of the crack. In order to foretell the crack initiation life, Tanaka and Mura assumed that the surface crack propagated forward along the grain boundary and proposed a method based on energy theory and slip band dislocation theory [21,22]:

$$N_i=\frac{9∆K_{th}^{2}G}{2E{\left(S-S_e\right)}^2{a}_0},$$

(3)

where $G$ is the shear modulus, $E$ is the elastic modulus, $S$ is the stress amplitude, and $a_0$ is the size of equivalent surface defect. The shear modulus and modulus of elasticity of DH36 steel are 80 GPa and 210 GPa, respectively.

$S$ is given as follows:

S = 0.5Qβ(1 − R),

where Q is the destructive load, β is the stress level, and $R$ is the stress ratio. In this paper, Q is shown in the experimental results section, β = 0.7, 0.8, 0.9. R = 0.2.

${∆K}_{th}$ is threshold of the range of stress intensity factors at which cracks do not extend [23] and $S_e$ is fatigue strength, MPa·m½.

${∆K}_{th}$ is given as follows:

∆K_th = 7(1 − 0.85R),

(4)

$S_e$ is defined as follows:

S_e = (0.65f_u + 14.48)/K_t,

(5)

where $f_u$ is the ultimate tensile strength of the material, which is 630 Mpa, and $K_t$ is the stress concentration factor. The stress concentration factor is related to the size of the pit and is defined as follows:

K_t = (1 + 6.6h/d)/(1 + 2h/d),

(6)

2.2.2. Prediction of Crack Propagation Life

The crack propagation life is defined as the number of load cycles from crack initiation to structural failure. The generalized Paris model is adopted to predict the crack propagation life, in which two parameters, namely fatigue coefficient C and fatigue index m, are used to represent the influence of the environment on crack propagation [24]. Although C and m can be computed by using experimental results, after referring to the fatigue coefficient and fatigue index of high-strength steel, C = 1 × 10⁻¹⁰ and m = 2.0 are used in this study. The prediction model is given as follows:

$$N_p={\int }_{a_1}^{a_f}\frac{1}{C{\left(∆K\right)}^m}da,$$

(7)

where a is crack length. The crack angle is used to represent the crack propagation process. $a_1$ is the initial angle during crack generation and $a_f$ is the final angle after specimen failure. $∆K$ is the range of stress intensity factors, which is related to pit size and fatigue load. The expression of $∆K$ is given below:

$$∆K=(2.2K_t∆\sigma \sqrt{\pi a}/\pi ),$$

(8)

where $K_t$ is the stress concentration factor, and $∆\sigma$ is the stress amplitude.

The crack propagation life of pitted corroded metal structures can be obtained by taking crack angles into the model, and the accuracy of prediction can be verified by comparing it to the experimental results.

2.3. Materials and Specimens

The material used in the study is DH36 high-strength steel, and its chemical composition was obtained from the factory test report of the material. It is shown in

Table 1. Chemical composition of DH36 (wt%).

Element	%	Element	%	Element	%
C	0.14	Als	0.039	Ceq	0.36
Si	0.21	Nb	0.012	Mo	0.007
Mn	1.19	Cr	0.05	V	0.003
S	0.003	Ni	0.02	Ti	0.011
P	0.014	Cu	0.05

.

According to GB/T 232-2010 [25], standard specimens are designed as plates. There are two approaches for producing pits through pre-corrosion treatment, namely electrochemical corrosion and mechanical processing. It has been reported that the shape of pits obtained by electrochemical etching is often arbitrary, leading to suboptimal outcomes. On the contrary, mechanical processing methods can bring high accuracy to cylindrical pits. Moreover, the shape of the corrosion spot has minimal impact on the mechanical properties of a component or structure, provided that the corrosion volume loss rate remains equivalent [26]. Therefore, the mechanical processing method is adopted to manufacture specimens with pitting corrosion. The volume loss rate $V_0$ can be defined in terms of the volume of the pitting hole and the volume of the specimen scale section. Figure 1 shows the design and machining process of a specimen.

2.4. Three-Point Bending Static Test

Specimens with different pits were manufactured by machining. In order to simultaneously consider the effects of volume loss rate and diameter-to-depth ratio on the experimental results, it is necessary to design the depth and diameter of the pits as a difference in the presence of a certain gradient. This gradient should also be expressed in the volume loss rate and the diameter-to-depth ratio [27]. The details of the specimens are shown in

Table 2. Specimens for three-point bending static tests.

Specimen Index	Pitting Hole
	Diameter d (mm)	Depth h (mm)	${\mathit{V}}_0$ (%)
${\mathrm{P}}_0$	0	0	0
${\mathrm{P}}_1$	2	4	0.168
${\mathrm{P}}_2$	4	4	0.670
${\mathrm{P}}_3$	6	2	0.754
${\mathrm{P}}_4$	6	3	1.131
${\mathrm{P}}_5$	6	4	1.508
${\mathrm{P}}_6$	6	5	1.885
${\mathrm{P}}_7$	8	4	2.681

.

Three-point bending static tests were carried out on an electronic universal testing machine, as shown in Figure 2. The loading speed is 1 mm/min, and the span of supports is 50 mm. The pitting hole is on the downside of the specimen, as shown in Figure 2b.

2.5. Three-Point Bending Fatigue Test

The three-point bending fatigue tests were carried out using the SD100 electro-hydraulic servo fatigue testing machine, shown in Figure 3. The supports had a span of 50 mm, loading frequency was 5 Hz, and stress ratio was 0.2. Referring to the fatigue test standard ASTM D7615 [28], loads of 70%, 80%, and 90% of static strength were applied in the fatigue test. The applied loads were defined as three different stress levels of 0.7, 0.8, and 0.9, respectively. Considering the high divergence of the fatigue test, each group of tests was repeated 5 times, and mean values were used as experimental results.

3. Results and Discussion

A discussion of the three-point bending static test data provides relevant results.

Each type of specimen was conducted five times, and averaged values were used to fit a curve. A load–displacement curve was obtained and shown in Figure 4. Due to the influence of pitting holes, the load–displacement curves can be divided into four stages for the sake of analysis and discussion. In the graph the curve is divided into four parts by red dashed lines.

Stage I: The displacement falls in the range of 0.0–0.7 mm, and the test is in its initial loading stage. During this stage, the specimen does not show any apparent deformation, and the load–displacement curves are increased linearly.

Stage II: The displacement falls in the range of 0.7–3.8 mm, and the specimen begins to deform with the increase in load. The load–displacement curves are still linearly increasing, but their slopes decrease a lot when compared to the first stage.

Stage III: The displacement falls in the range of 3.8–5.0 mm, and the slope of the load–displacement curve of the pitting-free specimen shows a sudden increase, as shown in the green dotted box in Figure 3, which may be due to the tensile strengthening occurred in the materials on the tensile side of the specimen. On the other hand, the load–displacement curves of pitted corroded specimens are nearly unchanged because of the appearance of pitting holes.

Stage IV: The displacement falls in the range of 5.0–15.0 mm, and the tensile strengthening effect has the greatest influence on the mechanical performance of the specimen. When displacement reaches 10.0 mm, the difference among load–displacement curves is the largest. After that, the changes in the load–displacement curves tend to be consistent.

Based on the analysis above, the displacement of 10.0 mm is regarded as the failure point of the specimen in the subsequent fatigue test. Thus, the corresponding load is defined as the failure load of the specimen. The failure loads of specimens with different pitting holes are obtained via the three-point bending static test and given in

Table 3. Failure loads of pitted corroded specimens.

Specimen	${\mathbf{P}}_0$	${\mathbf{P}}_1$	${\mathbf{P}}_2$	${\mathbf{P}}_3$
Load (kN)	17.1	15.5	14.2	15.0
	${\mathbf{P}}_4$	${\mathbf{P}}_5$	${\mathbf{P}}_6$	${\mathbf{P}}_7$
Load (kN)	14.0	13.0	12.6	11.5

.

From the point of view of fatigue damage, this study focuses on the relationship between the generation and propagation of cracks and the number of load cycles under the influence of pitting corrosion, and then the prediction of fatigue life. As shown in Figure 5, during the loading process, materials under the neutral plane of the specimen bear a tensile load, and those above the neutral plane carry a compressive load. Cracks always appear in the tensile region, generate at the bottom edge of the pitting hole, and simultaneously evolve along the wall of the pitting hole and the bottom surface of the specimen until the failure of the specimen. So, in the process of crack initiation and propagation, crack length is not a suitable variable that can be used to establish the relationship between the crack and the mechanical performance of the specimen.

However, based on the observation of the failure process of the specimen, it is clear that the crack opening angle continuously increases after the crack initiation, as shown in Figure 6. Thus, the crack angle is chosen as an indicator for measuring the level of fatigue damage in this study.

It is difficult to accurately capture the number of load cycles $N$ when a crack is generated during an actual test. As an alternative, $N$ is defined as follows:

N = 0.5(N₁ + N₂),

(9)

where $N_1$ is the maximum number of load cycles when the crack is not observed on the specimen, and $N_2$ is the number of load cycles when the crack is first observed during the test. So, the accuracy of N absolutely depends on the time interval of observation during the fatigue test. However, considering the divergence of fatigue tests, the error caused by the observation interval is always acceptable.

A discussion of the three-point bending fatigue test data provides relevant results.

The results of three-point bending fatigue tests are shown in

Table 4. Results of fatigue tests.

Specimen Index	Maximum Load (kN)	Fatigue Life N (Cycle)	Crack Initiation Life (Cycle)
${\mathrm{P}}_{1-1}$	10.9	91,035	68,276
${\mathrm{P}}_{1-2}$	12.4	41,218	34,072
${\mathrm{P}}_{1-3}$	14.0	25,291	20,175
${\mathrm{P}}_{2-1}$	9.9	77,021	58,536
${\mathrm{P}}_{2-2}$	11.4	35,987	29,116
${\mathrm{P}}_{2-3}$	12.8	19,211	15,177
${\mathrm{P}}_{3-1}$	10.5	79,196	62,565
${\mathrm{P}}_{3-2}$	12.0	38,401	31,265
${\mathrm{P}}_{3-3}$	13.5	21,132	16,694
${\mathrm{P}}_{4-1}$	9.8	68,781	53,649
${\mathrm{P}}_{4-2}$	11.2	31,025	24,197
${\mathrm{P}}_{4-3}$	12.6	16,445	12,745
${\mathrm{P}}_{5-1}$	9.1	68,218	52,528
${\mathrm{P}}_{5-2}$	10.4	25,169	19,124
${\mathrm{P}}_{5-3}$	11.7	13,978	10,484
${\mathrm{P}}_{6-1}$	8.8	59,005	43,075
${\mathrm{P}}_{6-2}$	10.1	19,256	14,311
${\mathrm{P}}_{6-3}$	11.3	12,391	8922
${\mathrm{P}}_{7-1}$	8.1	43,836	31,225
${\mathrm{P}}_{7-2}$	9.2	20,125	15,036
${\mathrm{P}}_{7-3}$	10.4	10,018	7013

, where the subscript i in P_i–j is the same type of specimen as that shown in Table 2 and Table 3, i = 1, 2, … 7, j denotes the stress level, and where j = 1 corresponds to 0.7, j = 2 to 0.8, and j = 3 to 0.9. In addition, specimens are regarded as failures when the loading displacement reaches 10 mm, as explained above.

The relationships between fatigue life and volume loss rate V₀ at different stress levels are shown in Figure 7. With the increase in V₀, the fatigue life of the specimen decreases monotonically and significantly. When V₀ is 2.68%, the fatigue life of the specimen decreases by more than 50%. It can be concluded that a small increase in the volume loss rate results in a substantial reduction in fatigue life. This experimental result proves that the value of the volume loss rate has a great influence on the fatigue life.

The changes in crack angle were recorded by a high-resolution camera after the first observation of a crack, and a photo was taken every 500 load cycles. The crack angles were measured by image processing software, and the changes in crack angles at 0.8 stress level are shown in

Table 5. Changes in crack angles.

Specimen No.	Observation Times	Extension Angle (°)	Life Percentage (%)
${\mathrm{P}}_{1-2}$	1	6.77	82.9
	2	7.21	85.4
	3	7.98	87.8
	4	9.15	90.2
	5	12.11	92.7
	6	17.82	95.1
	7	25.41	98.6
	8	79.23	100
${\mathrm{P}}_{2-2}$	1	6.96	80.7
	2	7.38	83.5
	3	8.91	86.3
	4	10.77	89.0
	5	13.32	91.8
	6	18.49	94.6
	7	25.93	97.9
	8	76.83	100
${\mathrm{P}}_{3-2}$	1	5.59	78.2
	2	5.98	80.8
	3	6.81	84.6
	4	7.93	88.5
	5	9.15	92.3
	6	13.38	96.2
	7	52.12	100
${\mathrm{P}}_{4-2}$	1	6.34	76.9
	2	6.65	80.1
	3	7.75	83.3
	4	8.91	86.5
	5	11.01	89.7
	6	13.55	92.9
	7	18.11	96.1
	8	64.10	100
${\mathrm{P}}_{5-2}$	1	8.04	75.3
	2	8.61	79.2
	3	9.45	83.1
	4	11.37	87.1
	5	14.29	91.1
	6	19.92	95.1
	7	27.96	97.0
	8	70.25	100
${\mathrm{P}}_{6-2}$	1	8.06	73.4
	2	8.64	77.0
	3	10.70	81.6
	4	13.74	85.3
	5	18.27	89.9
	6	25.54	93.5
	7	36.45	97.1
	8	73.78	100
${\mathrm{P}}_{7-2}$	1	8.99	72.9
	2	9.98	77.5
	3	11.55	82.0
	4	14.52	86.6
	5	18.74	91.1
	6	23.93	95.9
	7	62.44	100

. It can be seen that with the increase in load cycles, the crack angles increase.

3.1. Validation of Equivalent Surface Defect Model

Parameter regression analysis was adopted to validate the proposed equivalent surface defect model. The main steps are as follows: (1) bring one of the three sets of experimental data into the equivalent surface defect model expression to obtain the model parameters. (2) Verify the reliability of the equivalent surface defect model with the remaining two sets of the three sets of experimental data. (3) Further determine the feasibility of the model through the Bayesian approach. Firstly, substitute the results of P_i−2 in Table 4 and related parameters in Table 2 into Equation (1), and the following equation is obtained:

[lna₀]^T = Σ[A B C]^T,

where

[lna₀]^T = [−17.66 − 17.24 − 16.93 − 16.76 − 16.53 − 16.41 − 15.84],

The coefficient matrix is as follows:

$$\Sigma =\left[\begin{array}{cc}\begin{array}{c}0.168\\ 0.67\\ 0.754\\ \begin{array}{c}1.131\\ 1.508\\ \begin{array}{c}1.885\\ 2.681\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}2\\ 1\\ 0.333\\ \begin{array}{c}0.5\\ 0.667\\ \begin{array}{c}0.833\\ 0.5\end{array}\end{array}\end{array}& \begin{array}{c}1\\ 1\\ 1\\ \begin{array}{c}1\\ 1\\ \begin{array}{c}1\\ 1\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$

The parameter matrix of the model is then given as follows:

[A B C]^T = [0.598 − 0.229 − 17.331]

So, the proposed equivalent surface defect model is obtained:

lna₀ = 0.598V₀ − 0.229(h/d) − 17.331,

In order to verify the proposed method, any set of experimental results can theoretically be used for verification. However, it is well known that different sizes of data sets and components will result in different values of model parameters. Uncertainties between different specimens and tests arise for a variety of reasons, such as variability in tests and test procedures, which affect the validation result. The uncertainty of model parameters can be fully evaluated and quantified by the Bayesian method [29]. To understand the influence of volume loss rate and size of the pitting hole on the equivalent surface defect model, a new regression model is introduced based on the Bayesian method:

lna₀ = AV₀ + B(h/d) + C(h/d)V₀ + D,

Using the same calculation process as mentioned above, the following equation is obtained:

lna₀ = 0.653V₀ − 0.203 (h/d) − 0.088(h/d)V₀ − 17.349,

The computed $a_0$ is compared with the regression analysis value, and the relationship between them is shown in Figure 8. It can be seen that the calculated $a_0$ is in good agreement with the regression analysis one. The validation results show that the equivalent surface defect model can measure the influence of the notch effect and size effect on the fatigue life of structural members well. The equivalent surface defect size can be applied to the fatigue life prediction model.

3.2. Validation of Fatigue Life Prediction Model

3.2.1. Life Crack Initiation Life N_i

To verify the prediction of crack initiation life, P_i−1 and P_i−3 are used. The verification process includes the following steps: first, the equivalent surface defect model is obtained by using Equation (1) and $a_0$ is obtained under the condition of given V₀ and (h/d). Parameters are then substituted to Equations (3)–(6) to calculate the crack initiation life. The specific values of all practical significance parameters in Equations (3)–(6) have been given. By bringing the test data into the formula and connecting the equations, the fatigue crack emergence life can be solved. Results are shown in

Table 6. Comparison of experimental results and predicted crack initiation life.

Specimen	${\mathbf{a}}_0$ (×10−8)	Predicted Results (Cycles)	Experimental Results (Cycles)	Relative Error (%)
${\mathrm{P}}_{1-1}$	1.724	76,398	68,276	11.90
${\mathrm{P}}_{2-1}$	2.842	64,160	58,536	9.61
${\mathrm{P}}_{3-1}$	3.448	67,388	62,565	7.71
${\mathrm{P}}_{4-1}$	4.099	52,443	53,649	2.25
${\mathrm{P}}_{5-1}$	4.878	54,195	52,528	3.17
${\mathrm{P}}_{6-1}$	5.816	36,192	43,075	15.91
${\mathrm{P}}_{7-1}$	9.754	34,403	31,225	10.18
${\mathrm{P}}_{1-3}$	1.724	24,347	20,175	21.84
${\mathrm{P}}_{2-3}$	2.842	13,693	15,177	9.76
${\mathrm{P}}_{3-3}$	3.448	13,968	16,694	16.33
${\mathrm{P}}_{4-3}$	4.099	11,512	12,745	9.67
${\mathrm{P}}_{5-3}$	4.878	11,162	10,484	6.47
${\mathrm{P}}_{6-3}$	5.816	8538	8922	4.3
${\mathrm{P}}_{7-3}$	9.754	8144	7013	16.13

. It can be seen that the average relative error is 10.37%. The maximum relative error of P_i−1 is 15.91%, and that of P_i−3 is 21.84%.

Figure 9 shows the predicted crack initiation life versus the experimental results. In the figure, the test results at 0.7 stress level and 0.9 stress level refer to the test results at 70% and 90% static loading conditions for fatigue loading, respectively. Without loss of generality, the correlation lines with a scattering limit factor of 1.3 are used to evaluate the performance of the prediction of crack initiation life [30]. It can be seen that all the correlation points fall within the correlation lines, which means a good prediction performance. The validation results show that the prediction results obtained from the fatigue life prediction model at the crack initiation stage are reliable and accurate.

3.2.2. Life Crack Initiation Life Np

Based on the measured crack propagation data, as shown in Table 5, the relationships between the crack angle and normalized fatigue life are shown in Figure 10. It can be seen that when the normalized fatigue life reaches 95%, the crack angle increases suddenly. The crack initiation life accounts for more than 70% of the fatigue life. In addition, fatigue cracks often first appear on the bottom edge of the pitting hole in this study and gradually evolve mainly along the wall of pits toward the bottom in the axial direction, as shown in Figure 11. At this period, crack angles remain almost constant, and specimens do not show any failure phenomenon. After the crack reaches the bottom of the pit, it begins to evolve transversely, and the crack angle begins to increase rapidly.

Therefore, the final crack angle is defined as the angle when normalized fatigue life reaches 95%. The proposed model for crack propagation life is verified by using the data given in Table 5. The calculated fatigue life values can be obtained by substituting the crack initiation angle $a_1$ and crack extension angle $a_f$ of each specimen in Table 5 into the crack extension stage Expression (7) of the fatigue life prediction model. The calculated crack extension fatigue life values are compared with the experimental values. Results are shown in

Table 7. Comparison of experimental results and predicted crack propagation life.

Specimen	Predicted Results (Cycles)	Experimental Results (Cycles)	Relative Error (%)
${\mathrm{P}}_{1-2}$	4191	5029	16.67
${\mathrm{P}}_{2-2}$	5766	6190	6.85
${\mathrm{P}}_{3-2}$	8443	6912	22.16
${\mathrm{P}}_{4-2}$	6753	5957	13.36
${\mathrm{P}}_{5-2}$	5878	4983	17.96
${\mathrm{P}}_{6-2}$	5433	4564	19.04
${\mathrm{P}}_{7-2}$	4171	4629	9.89

. It can be seen that the average relative error is 15.13%, and the maximum relative error is 22.16%.

Figure 12 shows the predicted fatigue life versus the actual fatigue life. Without loss of generality, correlation lines with a scattering limit factor of 1.2 are used to evaluate the prediction performance of crack propagation life. It can be seen that almost all the correlation points are within the correlation lines, which indicates a good prediction performance. The validation results show that the prediction results obtained from the fatigue life prediction model at the crack extension stage are reliable and accurate.

4. Conclusions

Considering the influence of notch effect and size effect on the fatigue life of specimens subjected to a three-point bending load, an equivalent surface defect model related to the volume loss rate and the depth-to-diameter ratio of the pitting hole is proposed, and the model is utilized to define the pitting hole notch as the equivalent surface defect size. The fatigue life of pit corrosion specimens subjected to three-point bending loads was predicted using fracture mechanics theory. The prediction consists of two stages: crack initiation and extension. In the crack initiation stage, the equivalent surface defect size was introduced into the prediction model in order to quantify the effect of pitting corrosion hole defects on the fatigue life at this stage. During the crack extension stage, a Paris model with crack angle as a variable was utilized to predict fatigue life. The model is verified by experimental results, and some findings are shown:

(1)

The variation of volume loss rate has a significant effect on fatigue life. A small increment in the volume loss rate can reduce the fatigue life substantially. Under the same stress level, the three-point bending fatigue life of DH36 steel decreases with the increase in volume loss rate. When the volume loss rate reaches 2.68%, the fatigue life decreases by more than 50%.

(2)

The influence of the pitting hole notch effect and size effect on fatigue life was quantified by volume loss rate and the pitting depth-to-diameter ratio, and an equivalent surface defect model was proposed. The model was validated using the Bayesian method, and the results were in good agreement that the equivalent surface defect model can be used to measure the influence of the notch effect and size effect on fatigue life.

(3)

Based on energy theory and slip band dislocation theory, an equivalent surface defect model is combined to predict crack initiation life. Compared with the experimental results, the average relative error is 10.37%, and the maximum relative errors are 15.91% and 21.84%, respectively. The reliability of the fatigue life prediction model at the crack initiation stage is verified.

(4)

Based on the Paris fatigue life prediction model, the crack propagation life is predicted, where the crack angle is adopted as a variable. Compared with the experimental results, the average error is 15.13%, and the maximum relative error is 22.16%. The reliability of the fatigue life prediction model in the crack extension stage is verified.