1. Introduction

2. Materials and Methods

3. Results and discussion

3.3. Low-temperature fatigue test results and analysis

3.3.1. Low-temperature fatigue test results

After the low-temperature fatigue tests were completed, a total of 30 sets of valid data were obtained, and the results were shown in Table 6.

The table showed that the number of cycles to failure of the specimens under maximum stress of 480, 500, 520, 540 and 560 MPa were all higher than 10⁵ cycles, while the number of cycles to failure of the specimen under the maximum stress of 590 MPa was less than 10⁵ cycles. It can be concluded that high-cycle fatigue occurred when the maximum stress was 560 MPa and below, while low-cycle fatigue occurred when the maximum stress was 590 MPa and above, as shown in Table 7.

According to the literature [38], the expression for the three-parameter model was:

$${(S_{\max }-S_f)}^{m}N=C$$

(2)

Where, S was the maximum cyclic stress, N was fatigue life, m and C were parameters related to the material properties and the specific loading conditions, and S_f was the fatigue limit of the material.

Taking the logarithm of equation (2), we obtained:

$$\lg N=\lg C-m\lg (S-S_f)$$

(3)

Let x = lg(S-S_f), y = lgN, a = lgC, and b = -m. Then we had:

$$y=a+bx$$

(4)

For a series of Si and fatigue life N_i (i=1,2,3,...30), using the method of least squares, we obtained:

$$\left\{\begin{array}{l}a=\overline{y}-\overline{x}b\\ b=L_{xy}/L_{xx}\end{array}\right.$$

(5)

where, L_xx was total sum of squares, which represents the sum of squares of the differences between the x and $\overline{x}$, L_xy was regression sum of squares, which represents the sum of squares of the differences between the predicted values from the regression model and $\overline{y}$.

The square of the linear correlation coefficient R was:

$$R^2=\frac{L_{xy}^2}{L_{xx}\cdot L_{yy}}$$

(6)

where, L_yy was residual sum of squares, which represents the sum of squares of the differences between the observed values and the predicted values from the regression model.

To obtain the best linear correlation, it was necessary to maximize the absolute value of R, thus we had:

$$\frac{d(R^2)}{d{S}_f}=0$$

(7)

That was:

$$\frac{d(R^2)}{d{S}_f}=\frac{L_{xy}^2}{L_{xx}\cdot L_{yy}}(\frac{2}{L_{xy}}\cdot \frac{d{L}_{xy}}{d{S}_f}-\frac{1}{L_{xx}}\cdot \frac{d{L}_{xx}}{d{S}_f})$$

(8)

Where,

$$\frac{d{L}_{xy}}{d{S}_f}=-\frac{1}{\ln 10}\left[{\displaystyle \sum _{i=1}^n\frac{y_i}{S_i-S_f}}-\frac{1}{n}\cdot \left({\displaystyle \sum _{i=1}^n{y}_i}\right)\cdot \left({\displaystyle \sum _{i=1}^n\frac{1}{S_i-S_f}}\right)\right]=-\frac{L_{y0}}{\ln 10}$$

(9)

$$\frac{d{L}_{xx}}{d{S}_f}=-\frac{2}{\ln 10}\left[{\displaystyle \sum _{i=1}^n\frac{x_i}{S_i-S_f}}-\frac{1}{n}\cdot \left({\displaystyle \sum _{i=1}^n{x}_i}\right)\cdot \left({\displaystyle \sum _{i=1}^n\frac{1}{S_i-S_f}}\right)\right]=-\frac{2L_{x0}}{\ln 10}$$

(10)

Where, S_i represents the applied maximum stress for the i-th fatigue test. L_y0 is the intercept parameter in the regression equation, which represents the predicted value of the dependent variable y when the independent variable x is zero. L_x0 is the slope parameter in the regression equation, which represents the effect of a unit change in the independent variable x on the predicted value of the y.

Substituting equation (8), (9), and (10) into equation (7), we get:

$$H(S_f)=\frac{L_{y0}}{L_{xy}}-\frac{L_{x0}}{L_{xx}}$$

(11)

By solving the nonlinear equation system H(S_f), we can obtain S_f. Then substituting Sf into equation (5) and (6), we can obtain a and b. Therefore, we had:

$$\left\{\begin{array}{l}C={10}^a\\ m=-b\end{array}\right.$$

(12)

The equation (2) can be rearranged as:

$$S=S_f+\frac{a^{*}}{N^{b*}}$$

(13)

Where:

$$\left\{\begin{array}{l}{b}^{*}=1/m\\ a^{*}=C^{b*}\end{array}\right.$$

(14)

According to a series of S_i and N_i, a nonlinear fitting of equation (13) was performed using MATLAB programming language, and the related parameters were calculated as follows: S_f=499.87, m=1.2566, C=2.66×10⁷. Based on the three-parameter model, the S-N curve expression was:

$${(S_{\max }-499.87)}^{1.2566}N=2.66\times {10}^7$$

(15)

Therefore, it can be concluded that the fatigue limit S_f(10⁷) of the steel plate at -60℃ was approximately the maximum stress of 500 MPa. Shiozawa and Lu [39,40] had summarized the characteristics of S-N curves as follows: in the high-stress range below 10⁶ cycles, the S-N curve was attributed to the initiation and propagation of surface cracks, and the critical stress that prevented surface crack growth was referred to as the fatigue limit. According to Table 8, it can be observed that when the maximum stress was greater than or equal to 500 MPa, the number of cycles to failure was less than 10⁶ cycles and cracks would initiate and propagate on the surface.

Figure 19 showed the predicted S-N curve obtained using the expression above. In comparison to the curve, fatigue data exhibited significant scatter. In general, even when using the same batch of specimens and identical test conditions, variations in test results can still occur. In this study, several factors influencing the scatter of fatigue test results were considered, including the inconsistency of test materials, variations in specimen processing and rolling processing, and accidental changes in the test environment [41]. Compared with room temperature, S_f(10⁷) at -60℃ was increased to some extent, indicating that the fatigue performance of the steel plate was better at low temperatures. This was because the crack propagation mechanism within the temperature range of -60℃ to room temperature was based on the dislocation slip mechanism. When the temperature decreased, the yield strength of the steel plate increased. Dislocations can only move when they were subjected to stress greater than the yield strength. Therefore, the stress required for driving dislocation motion increased, and dislocations were less likely to slip, which made it difficult for cracks to propagate and thus increased the fatigue limit [42,43]. Furthermore, as the temperature decreased, the grain boundaries were strengthened and less prone to cracking. This was also an important reason why the fatigue limit increased when the temperature decreased.

3.3.2. Observation of fracture surface in low-temperature fatigue tests

Figure 20 showed the photographs of the fatigue fracture specimens under different maximum stress at room temperature. From Figure 20(a), (b), (c) and (d), it can be seen that the fractures of these specimens were all oblique shear fractures, without obvious plastic deformation. The specimen in Figure 20(d) had a crack perpendicular to the direction of the applied stress. According to Figure 20(e), it can be seen that the fracture surface of the specimen was a cup-cone shape, with a shrunken cross-section and obvious plastic deformation.

Figure 21 showed the SEM image of the crack on the side of the specimen in Figure 20(d), where the direction of the applied cyclic tensile stress was perpendicular to the crack surface. From the figure, it can be seen that under tensile stress, the two surfaces were pulled apart, and the crack propagation mode was opening type. While the crack propagated inwardly along the specimen, it also propagated upward along the machining scratch on the right side of the specimen. Based on the local magnified image in Figure 21(b), it can be seen that there was an crack source that had not yet propagated on the machining scratch, indicating that cracks were prone to initiate at surface defects such as these machining scratches [44].

Figure 22 showed the fatigue fracture morphology of the specimen under different maximum stress. It can be seen from the figure that the fracture morphology was composed of the I zone, II zone, and III zone, and the I zone was located at the surface. Upon comparing the Figure 22 (a), (d), (g), and (j), it can be seen that the fracture surface was relatively flat under maximum stress of 500 MPa, 520 MPa, and 540 MPa, and the boundary between the II zone and III zone was clear. However, under the maximum stress of 560 MPa, the boundary was less clear than other maximum stress. By comparing the Figure 22(c), (f), (i), and (j), it can be seen that the III zone under the maximum stress of 500 MPa, 520 MPa, and 540 MPa contained a certain number of D_el and larger D_eq. By comparison, under the maximum stress of 560 MPa, the III zone contained more D_el and larger D_eq. The value of ac in Figure 22(a) was about 2.2 mm, which was higher than the value of 1.9mm at room temperature as shown in Figure 10(a), indicating that the N value at low temperature was relatively higher than that at room temperature [45].

Figure 23 showed the fatigue fracture morphology of the specimen under the maximum stress of 590 MPa. From Figure 23(a), it can be seen that the fracture surface showed plastic deformation and shrinkage, and there were multiple fatigue sources on the surface. The zone where the crack propagated inwards was the II zone, and the center was the fibrous III zone. From Figure 23(b), it can be seen that the fatigue source zone was relatively smooth without the formation of dimples. At the boundary between the I source and II zones, there were D_eq, while D_el were present in the II zone. From Figure 23(c), it can be seen that the III zone also mainly consisted of D_el.

Figure 24 showed the morphology of inclusions at location 1, 2, 3, 4 and 5 in the I and II zones. From the figure, it can be seen that inclusions at location 1, 2 and 3 were irregular block-shaped with sizes between 25-40 μm, while inclusions at location 4 and 5 were ellipsoidal and spherical with diameters between 6-10 μm, with the former being much larger than the latter. Table 8 showed the energy spectrum analysis results of the inclusions at different location in Figure 24. Based on the results in the table, it was inferred that the inclusions at locations 1, 2, and 3 were composite inclusions consisting of CaO, SiO₂, TiO₂, and FeS [29,46]. The location and content of these inclusions were relatively close, which may be due to a large block inclusion undergoing deformation and splitting into three smaller inclusions because of stress concentration during the fatigue test, becoming the source of fatigue failure in the steel [47]. The inclusion at location 4 was speculated to be primarily composed of SiO₂, Al₂O₃, and a small amount of TiO₂. It was suggested that this inclusion was mainly formed due to secondary oxidation of silicon and the precipitation of dissolved oxygen during the steel refining process, resulting in the formation of glassy silicate inclusions rich in SiO₂. The inclusion at location 5 was inferred to have a core composition of Al₂O₃·CaO·MgO·TiO₂, with the deposition of composite inclusions such as MnS on its surface [48].

Table 8. Energy spectrum analysis results of inclusions in Figure 24.

Table 8. Energy spectrum analysis results of inclusions in Figure 24.

Location	Element ( at%)
	Fe	C	O	Ti	S	Ca	Si	K	Cl	Al	Mg	Mn
1	1.01	79.45	17.86	0.08	0.48	0.12	0.42	0.16	0.27	0.13	-	-
2	0.32	72.70	25.30	0.09	0.33	0.24	0.27	0.10	0.65	-	-	-
3	1.40	82.51	14.08	0.07	0.53	0.13	0.39	0.25	0.64	-	-	-
4	1.42	23.82	58.55	0.040	-	-	13.82	0.53	-	1.82	-	-
5	1.24	28.74	45.25	0.230	0.19	4.69	-	-	-	18.95	0.59	0.110

Figure 24. Inclusions in different zones of fatigue fracture under the maximum stress of 590 MPa (a) I zone; (b) II zone.

Figure 24. Inclusions in different zones of fatigue fracture under the maximum stress of 590 MPa (a) I zone; (b) II zone.

4. Conclusions

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