An Advanced Fatigue Life Predicting Model for High-Strength Steel Sucker Rods Integrating Material Strength Parameters

Abstract

Conventional fatigue life prediction models for sucker rods, such as the Normal Distribution Basquin model and Miner’s Rule, have notable limitations. This research introduces a new fatigue life prediction model for ultra-high-strength sucker rods by employing a three-parameter Weibull distribution and incorporating material tensile strength. Tensile and fatigue tests were conducted on sucker rods to gather life data at various stress levels. Subsequent P-S-N curve fitting and life prediction analyses showed that the fatigue life data conformed to both normal and Weibull distributions, with the Weibull distribution providing a superior fit. A comparison of S-N curves at a 50% failure probability demonstrated that the three-parameter Weibull model more accurately matches experimental data across both high-cycle and low-cycle fatigue regimes. Fatigue life estimates indicated average prediction errors of 12.50% for the normal distribution model and 5.39% for the Weibull distribution model, confirming the enhanced accuracy of the proposed method. “The new model more precisely captures the actual fatigue behavior of ultra-high-strength sucker rods by considering fatigue life under varying tensile forces and ultimate tensile strength, offering valuable support for assessing fatigue durability and ensuring operational safety in oilfield operations while reducing maintenance costs”.

1. Introduction

Essential for the rod pumping system used in the oilfield, the sucker rod can convert the reciprocating motion of the surface pumping unit into lifting crude oil with a downhole sucker rod pump. It must also withstand cyclic tensile and compressive loads during pumping [1], which necessitates selecting appropriate materials and specifications to meet various oil production needs based on factors such as well depth, crude oil viscosity, downhole pump size, and dogleg severity. The performance of the sucker rod string directly affects the operational efficiency and stability of the artificial lift system, thereby influencing overall productivity and reliability in oil production [2]. Additionally, it impacts production costs, since its durability and reliability determine the frequency of pump maintenance and associated expenses. The lifespan of the sucker rod string is also critical for preventing environmental risks and avoiding equipment damage. Therefore, this study examines the fatigue life of sucker rods to predict their service life under ideal conditions, providing a theoretical foundation for fatigue life prediction in practical applications. Moreover, the novel model developed herein enhances the accuracy of fatigue life predictions for sucker rods, reducing unplanned downtime and maintenance costs, ensuring safe oilfield operations, and preventing safety accidents caused by rod failures.

Many scholars have studied the fatigue life of sucker rods. Chen Guoming et al. [3] determined the fatigue life of an improved ultra-high-strength sucker rod made from chromium–molybdenum alloy structural steel (20CrMo) through fatigue tests and analyzed whether the fatigue life follows a normal distribution. Song Kaili et al. [4] summarized the features of ultra-high-strength sucker rods and proposed a prediction method based on fitting the P-S-N curve with a normal distribution and fatigue life. Li Dajian et al. [5] developed the P-S-N curve for ultra-high-strength sucker rods using statistical processing of test data and determined the ultimate fatigue strength of the rod. Fan Song, Liang Yi et al. [6] assume that sucker rod fatigue data follow a normal distribution, and they fit the Basquin formula, a power function, to create a fatigue life prediction model. These studies assume that sucker rod fatigue data follow a normal distribution and that fatigue life is linearly related to the logarithm of cyclic stress at a certain stress amplitude.

Due to the tensile stress on the sucker rod during operation, the accuracy of fatigue life prediction from the original Basquin model diminishes, making it challenging for the normal distribution model to forecast fatigue life accurately [7,8,9,10,11]. The more precise the fatigue life prediction for sucker rods, the greater the potential to extend the pump inspection cycle, thereby reducing maintenance costs during oil extraction. To maximize economic benefits, developing a new, more accurate prediction model is essential. Apart from the normal distribution, the Weibull distribution is also widely used in predicting fatigue life. Ma, Junming et al. [12] employed generalized hierarchical Bayesian inference for fatigue life prediction based on the general multidimensional Weibull model. Chuanxi Jin et al. [13] described the hard failure process using the Weibull distribution. Ł. Blacha et al. [14] validated a probabilistic model based on the concept of the weakest link by comparing it with fatigue test results. Xu Jiajin [15] studied the fatigue life confidence interval within the Weibull distribution in fatigue statistical analysis, offering an intelligent solution based on the high-town method. Han Qinghua et al. [16] established a corrosion fatigue life assessment method for cast steel and butt welds based on the Weibull distribution. Strzelecki Przemysław [17] determined low-probability fatigue life at different stress levels using a three-parameter Weibull distribution, while Toasa Caiza et al. [18] applied the Weibull distribution P-S-N curve fitting to S355J2+N steel. Lan, Chengming et al. [19] investigated the effect of corrosion on the fatigue life of steel bars in RC structures, utilizing the Weibull distribution. Wen Debing [20] found that the flexibility and accuracy of the three-parameter Weibull distribution surpass those of the two-parameter Weibull and lognormal distributions through a comparative study on the fatigue life probability distributions of 2024-T3 Alclad aluminum alloys. Wu Huazhi et al. [21] examined the application of Weibull and normal distributions in the probabilistic and statistical analysis of structural fatigue failure life. Cai Wenbin et al. [22] concluded that, when tensile strength is not considered, the fatigue life prediction model based on the three-parameter Weibull distribution exhibits higher accuracy.

While existing studies have provided valuable insights, they generally suffer from two key limitations: either they rely on the conventional Basquin model, which does not accurately capture the physical boundaries of fatigue behavior, or they employ standard Weibull distributions without integrating key material properties. Consequently, a gap exists for a more physically realistic and robust probabilistic model.

The primary originality of this study lies in addressing this gap by proposing a novel fatigue modeling framework that achieves the following for the first time for sucker rods:

(1) Integrates Material Physics with Probabilistic Analysis: Weibull-distributed fatigue life data processing is well known for requiring small sample sizes and delivering high prediction accuracy. We uniquely combine the physically grounded Stüssi function, which explicitly incorporates the material’s ultimate tensile strength R_m and fatigue limit S∞, with the statistical power of a three-parameter Weibull distribution. This synergy creates a model that is both probabilistic and physically realistic.

(2) Employs a More Flexible Distribution: Unlike previous works that may use a two-parameter Weibull, our use of the three-parameter version allows for a non-zero minimum life to be modeled, providing greater flexibility to fit the experimental data more accurately.

Furthermore, the practical relevance of this study to the journal Coatings is direct and substantial. High-strength steel components like sucker rods are rarely used in their bare state in corrosive service environments; they critically rely on protective coatings (e.g., metallic, ceramic, or polymer) to prevent corrosion fatigue, a primary failure mechanism. However, the performance of any coating cannot be evaluated in a vacuum.

This research addresses a fundamental prerequisite for the entire field of protective coatings for fatigue applications: it establishes a high-fidelity, predictive baseline of the substrate’s intrinsic fatigue life. This baseline is indispensable for two critical reasons. First, it provides a quantitative benchmark against which the performance enhancement of any coating can be rigorously measured. The true value of a coating is its ability to make the component’s fatigue life in a corrosive environment approach the ideal, uncoated life in a benign environment, as predicted by our model. Second, the coating process itself (e.g., thermal spraying, electroplating, PVD: Physical Vapor Deposition/CVD: Chemical Vapor Deposition) can induce significant changes in the substrate’s surface, such as residual stresses or microstructural alterations, which directly impact fatigue initiation. Our model provides the essential “before” state, allowing researchers to isolate and quantify the precise “after” effects of the coating process itself on the component’s fatigue durability.

Therefore, this work is not merely about steel fatigue; it provides an essential and foundational tool for the design, validation, and quantitative assessment of protective coating systems intended for demanding cyclic loading applications.

2. Experiment

The experimental content mainly includes two tests: the ultra-high-strength sucker rod fatigue test and the tensile test of the sucker rod, which determine the fatigue life and ultimate tensile strength under different stress amplitudes, respectively.

2.1. Fatigue Experiments

The experiment was carried out following the People’s Republic of China’s oil and gas industry standard ultra-high-strength sucker rod SY/T 6272-1997 (equivalent to API 11B) [23]. The experimental sample was a Type I specimen, produced through a consistent manufacturing process and mass production. We selected fifteen 500 mm long sections from ultra-high-strength sucker rods, with tensile strengths between 900 MPa and 1100 MPa. The samples were oriented with the white side facing up, and a PLG-300C electromagnetic resonance high-frequency fatigue testing machine (Changchun Research Institute for Mechanical Science Co., Ltd., Changchun, China) was used. Specimens were divided into three groups based on stress levels, with five samples per group, and subjected to fully reversed axial loading at 150 Hz frequency (R = 0.1). Figure 1 shows the sample after fatigue failure, and the experimental results are displayed in

Table 1. Experimental fatigue life data at different stress amplitudes.

Stress Amplitude S/MPa	Fatigue Life N/Times
500	833,120, 767,320, 853,620, 722,682, 753,018
540	420,356, 419,652, 433,111, 413,256, 408,352
600	195,968, 251,597, 265,439, 197,851, 156,850

.

2.2. Tensile Experiments

Common materials for high-strength sucker rods include AISI 4130, AISI 4140, and 35CrMo. In this experiment, 35CrMo, known for its ultra-high strength, was chosen as the test material. Its tensile strength usually ranges from 900 MPa to 1100 MPa. Three 500 mm segments were cut from the middle portion of the ultra-high-strength sucker rod. After cutting, any burrs or flashing on the surfaces were carefully polished to ensure smoothness.

The electro-hydraulic servo universal testing machine, model SHT4106 (Shenzhen Sansi Vertical and Horizontal Testing Technology Co., Ltd., Shenzhen, China), was used in this experiment. The procedure followed the standard “SY/T 6272-1997 (equivalent to API 11B)—Tensile Test of Metal Materials, Part 1: Room Temperature Tensile Test Method”. The tensile strength was calculated by averaging the results of three specimens.

Figure 2 shows the fractured tensile test specimen, including the tensile test result of a sucker rod with a ferrule. However, the specimen containing the ferrule did not meet industry standards and was therefore excluded from the final data. This exclusion was necessary because the goal of the test is to measure the intrinsic strength of the base material. The ferrule introduces a stress concentration that caused a premature, non-representative failure at the ferrule–rod interface, rather than in the uniform gauge section of the rod. This invalid failure mode yields a strength value for the assembly, not the material, and including it would have introduced a significant negative bias to the average ultimate tensile strength Rm, thereby compromising the foundation of our fatigue model. The results of the valid tensile tests are presented in

Table 2. Tensile test results and average ultimate tensile strength Rm.

	Ultimate Tensile Strength Rm/MPa	Average Value MPa
Sample 1	1028	1060.67
Sample 2	1099
Sample 3	1055

.

3. Fatigue Life Modeling Based on the Three-Parameter Weibull Distribution

The following is the derivation process of the new model of the three-parameter Weibull distribution: Firstly, a basic fatigue life prediction model is obtained by using the cumulative failure distribution function of the three-parameter Weibull distribution. This model has five unknowns: λ, η, α, β, and γ. The parameters λ and η, which are shape parameters, are estimated using linear regression. Then, the three parameters α, β, and γ of the three-parameter Weibull distribution are calculated using the probability-weighted matrix method.

3.1. Establishment of the Basic Model

The cumulative failure distribution function of the three-parameter Weibull distribution is [11]

$$F\left(x\mid \alpha ,\beta ,\gamma \right)=1-\exp \left[-{\left({\displaystyle \frac{x-\alpha }{\beta }}\right)}^{\gamma }\right]$$

(1)

where x is the fatigue life span function formula, x ≥ α; α∈R is the location parameter; β > 0 is the scale parameter; γ > 0 is the form parameter.

To describe the S-N curve of the material, Stüssi proposed a nonlinear function [24] that considers both the fatigue limit and the ultimate tensile strength, making it more realistic, as shown in Equation (2):

$$S={\displaystyle \frac{R_m+\lambda N^{\eta }{S}_{\mathrm{\infty }}}{1+\lambda N^{\eta }}}$$

(2)

where S is the stress amplitude of the fatigue experiment, MPa; R_m is the limit tensile strength of the material, MPa; N is the fatigue life, times; S_∞ is the theoretical fatigue limit, MPa; λ, η is the form parameter, zero dimension.

The fatigue life function can be defined by Equation (2) as

$$x=S-{\displaystyle \frac{R_m+\lambda N^{\eta }{S}_{\mathrm{\infty }}}{1+\lambda N^{\eta }}}$$

(3)

If x follows a three-parameter Weibull distribution, bring Equation (3) into Equation (1) to obtain the ultra-high-strength sucker rod P-S-N curve model equation, the fatigue life prediction model is as follows:

$$p=1-\exp \left\{-{\left({\displaystyle \frac{S-\frac{R_m+\lambda N^{\eta }{S}_{\mathrm{\infty }}}{1+\lambda N^{\eta }}-\alpha }{\beta }}\right)}^{\gamma }\right\}$$

(4)

where p is the probability of failure, %.

Equation (4) can be used to fit the fatigue data of the ultra-high-strength sucker rod, and the Weibull distribution can be obtained with three parameters, α, β, γ, and two shape parameters η, λ.

3.2. Parameter Estimation Method

3.2.1. Estimation of the Shape Parameters λ and H of the Stüssi Function

Estimation of the two shape parameters λ and η, using linear regression according to Equation (2), obtains the following:

$$\lambda N^n={\displaystyle \frac{R_m-S}{S-S_{\mathrm{\infty }}}}$$

(5)

Taking the logarithm on both sides of Equation (5),

$$\log \left(N\right)={\displaystyle \frac{1}{\eta }}\log \left({\displaystyle \frac{R_m-S}{S-S_{\mathrm{\infty }}}}\right)-{\displaystyle \frac{1}{\eta }}\log \left(\lambda \right)$$

(6)

If $Y=\log \left(N\right), X=\mathrm{l}\mathrm{o}\mathrm{g}\left({\displaystyle \frac{R_m-S}{S-S_{\mathrm{\infty }}}}\right)$, $A={\displaystyle \frac{1}{\eta }}$, $B=-{\displaystyle \frac{1}{\eta }}\log \left(\lambda \right),$ then Equation (6) can be written as

$$Y=AX+B$$

(7)

The following function can be constructed:

$$Q=\sum _{i=1}^n{\left(Y_i-{\widehat{Y}}_i\right)}^2=\sum _{i=1}^n{\left(Y_i-\left(\widehat{A}{X}_i+\widehat{B}\right)\right)}^2$$

(8)

Q reaches a minimum when the first-order partial derivatives of Q with respect to $\widehat{A}$ and $\widehat{B}$ are 0:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial Q}{\partial \widehat{A}}}=0\\ {\displaystyle \frac{\partial Q}{\partial \widehat{B}}}=0\end{array}\right.\Rightarrow \left\{\begin{array}{c}\sum \left(Y_i-\widehat{A}{X}_i-\widehat{B}\right)=0\\ \sum \left(Y_i-\widehat{A}{X}_i-\widehat{B}\right)X_i=0\end{array}\right.\Rightarrow \left\{\begin{array}{c}\sum Y_i=n\widehat{B}+\widehat{A}\sum X_i\\ \sum Y_i{X}_i=\widehat{B}\sum X_i+\widehat{A}\sum X_i^2\end{array}\right.$$

(9)

Solving the above equation yields

$$\left\{\begin{array}{c}\widehat{A}={\displaystyle \frac{n\sum Y_i{X}_i-\sum Y_i\sum X_i}{n\sum X_i^2-{\left(\sum X_i\right)}^2}}\\ \widehat{B}=\stackrel{-}{Y}-\widehat{A}\stackrel{-}{X}\end{array}\right.$$

(10)

Let

$$\left\{\begin{array}{c}\stackrel{-}{X}={\displaystyle \frac{1}{n}}\sum X_i\\ \stackrel{-}{Y}={\displaystyle \frac{1}{n}}\sum Y_i\\ x_i=X_i-\stackrel{-}{X}\\ y_i=Y_i-\stackrel{-}{Y}\end{array}\right.$$

(11)

Then, Equation (11) can be written as

$$\left\{\begin{array}{c}\widehat{A}={\displaystyle \frac{\sum x_i{y}_i}{\sum x_i^2}}\\ \widehat{B}=\stackrel{-}{Y}-\widehat{A}\stackrel{-}{X}\end{array}\right.$$

(12)

The estimates of A and B, $\widehat{A}, \widehat{B}$, are obtained by Equation (12), and then the estimates of the geometric parameters η and λ are obtained by Equation (7).

3.2.2. Estimation of the Three Parameters α, β, and γ

The probability-weighted moment method is used to estimate the parameters of the three-parameter Weibull distribution, and the probability-weighted moment function formula is [25]

$$M_{\mathrm{p},\mathrm{r},\mathrm{s}}={\int }_0^1{\left[x\left(F\right)\right]}^p{F}^r{\left(1-F\right)}^{s}dF$$

(13)

where p, r, s ∈ N.

Let p = 1, r = 0 to obtain the probability-weighted moment M_1,0,s of the Weibull distribution, expressed as

$$M_{\mathrm{1,0},s}={\displaystyle \frac{\alpha }{s+1}}+{\displaystyle \frac{\beta }{{\left(s+1\right)}^{1+\frac{1}{\gamma }}}}\Gamma \left(1+{\displaystyle \frac{1}{\gamma }}\right)\gamma >0$$

(14)

Estimating the values of the three parameters requires three equations, so s = 0,1,2 can be substituted into Equation (14) to obtain the following set of equations:

$$\left\{\begin{array}{c}{M}_{\mathrm{1,0},0}=\alpha +\beta \cdot \Gamma \left(1+{\displaystyle \frac{1}{\gamma }}\right)\\ M_{\mathrm{1,0},1}={\displaystyle \frac{\alpha }{2}}+{\displaystyle \frac{\beta }{2^{1+\frac{1}{\gamma }}}}\cdot \Gamma \left(1+{\displaystyle \frac{1}{\gamma }}\right)\\ M_{\mathrm{1,0},2}={\displaystyle \frac{\alpha }{3}}+{\displaystyle \frac{\beta }{3^{1+\frac{1}{\gamma }}}}\cdot \Gamma \left(1+{\displaystyle \frac{1}{\gamma }}\right)\end{array}\right.$$

(15)

The expressions for the three parameters of the Weibull distribution derived from the Equation (15) are

$$\alpha =M_{\mathrm{1,0},0}-\beta \cdot \Gamma \left(1+{\displaystyle \frac{1}{\gamma }}\right)$$

(16)

$$\beta ={\displaystyle \frac{2M_{\mathrm{1,0},1}-M_{\mathrm{1,0},0}}{\left(2^{\frac{-1}{\gamma }}-1\right)\Gamma \left(1+\frac{1}{\gamma }\right)}}$$

(17)

$${\displaystyle \frac{3M_{\mathrm{1,0},2}-M_{\mathrm{1,0},0}}{2M_{\mathrm{1,0},1}-M_{\mathrm{1,0},0}}}={\displaystyle \frac{3^{\frac{-1}{\gamma }}-1}{2^{\frac{-1}{\gamma }}-1}}$$

(18)

M_1,0,0, M_1,0,1, and M_1,0,2 can be calculated by Equations (19)–(21).

$${\widehat{M}}_{\mathrm{1,0},0}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{x}_i$$

(19)

$${\widehat{M}}_{\mathrm{1,0},1}={\displaystyle \frac{1}{n\left(n-1\right)}}\sum _{i=1}^{n-1}\left(n-i\right)x_i$$

(20)

$${\widehat{M}}_{\mathrm{1,0},2}={\displaystyle \frac{1}{n\left(n-1\right)\left(n-2\right)}}\sum _{i=1}^{n-2}\left(n-i\right)\left(n-i-1\right)x_i$$

(21)

where n is the number of fatigue experimental samples; x_i is the sample data (I = 1, 2 … n) calculated from Equation (3).

4. Model Validation

The Basquin model based on the normal distribution and the new model based on the three-parameter Weibull distribution were derived, and the final models from both verification methods were obtained.

4.1. Basquin Model Based on Normal Distribution

Basquin model S-N equation based on logarithmic Equations (8):

$$\mathit{log}S=\mathit{log}C+m\mathit{log}N$$

(22)

where S represents the stress amplitude, MPa; N stands for fatigue life, times; C and m are parameters related to the loading mode of the experimental material and can be obtained by linear regression.

Suppose the logarithmic fatigue life $\mathit{log}N$ obeys normal distribution, and the expression of fatigue life under certain reliability obeys standard normal distribution is

$$\mathit{log}{N}_p=\mu +{\mu }_p\sigma$$

(23)

where $\mathit{log}{N}_p$ represents the logarithmic fatigue life that obeys the standard normal distribution; $\mu$ represents the logarithmic average fatigue life; ${\mu }_p$ represents the standard normal bias corresponding to the reliability P (check the standard normal bias obtained); $\sigma$ represents the logarithmic fatigue life standard deviation.

Let $B=\mathit{log}C,A=m$ construct the fitting equation; then, Equation (24) is obtained:

$$\mathit{log}S=A\mathit{log}{N}_p+B$$

(24)

Based on fatigue experiment data, A and B are solved using the least squares method:

$$\begin{array}{l}\widehat{A}={\displaystyle \frac{n\sum \mathrm{l}\mathrm{o}\mathrm{g}{N}_{pi}\mathrm{l}\mathrm{o}\mathrm{g}{S}_i-\sum \mathrm{l}\mathrm{o}\mathrm{g}{N}_{pi}\sum \mathrm{l}\mathrm{o}\mathrm{g}{S}_i}{n\sum {\left(\log S_i\right)}^2-{\left(\mathrm{l}\mathrm{o}\mathrm{g}\sum S_i\right)}^2}}\\ \widehat{B}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}\log N_{pi}-\widehat{A}{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}\log S_i\end{array}$$

(25)

The fatigue test data is substituted (Equation (25)) to obtain an estimate of A and B, and the results are shown in

Table 3. Estimated parameters for the Basquin model at different reliabilities.

Ultra-High-Strength Sucker Rod
Reliability	99.99%		95%		50%
Geometric parameters	A	B	A	B	A	B
	−0.091	3.238	−0.104	3.313	−0.129	3.461

.

According to Equation (24), the P-S-N curve equation with different reliability degrees in a normal distribution can be obtained.

4.2. A New Model Based on a Three-Parameter Weibull Distribution

Using the experimental data from Table 1 and Table 2, linear regression of A and B is performed according to Equation (7), and the results are shown in Figure 3. Further estimates of the shape parameters λ and η of the Stüssi function were obtained, and the results are shown in

Table 4. Estimated shape parameters for the Stüssi function.

Parameter Estimates
A	B	η	λ
2.1484	4.418	0.4655	0.0057

.

The fatigue test data and the data in Table 4 are substituted into Equation (3) to calculate the value of formula x of the fatigue life function, combined with Equations (16)–(21). The three parameters of the Weibull distribution are estimated, and the results are shown in

Table 5. Estimated parameters (α, β, γ) for the Weibull distribution.

Parameter Estimates
$\alpha$	$\beta$	$\gamma$
−17.5733	19.1316	1.8038

.

The P-S-N curve equation based on the three-parameter Weibull distribution, which is the fatigue life prediction model function formula, is as follows:

$$p=1-\exp \left\{-{\left({\displaystyle \frac{S-\frac{1060.67+0.0057N^{0.4655}\times 327.49}{1+0.0057N^{0.4655}}+17.5733}{19.1316}}\right)}^{1.8038}\right\}$$

(26)

4.3. Goodness-of-Fit K-S Test

The K-S test [26] was used to evaluate the goodness of fit of the normal and Weibull distributions. $F_0\left(x\right)$ is a distribution function, $F_n\left(x\right)$ is the sample cumulative frequency function, experimental data follow a hypothetical distribution denoted as H₀, and the insubordination assumed distribution is denoted as H₁. Formula (27) is as follows:

$${\mathrm{F}}_{\mathrm{n}}\left(\mathrm{x}\right)\left\{\begin{array}{cc}0& \mathrm{x}<{\mathrm{X}}_{\left(1\right)}\\ {\displaystyle \frac{\mathrm{k}}{\mathrm{n}}}& {\mathrm{X}}_{\left(\mathrm{k}\right)}\le \mathrm{x}\le {\mathrm{X}}_{\left(\mathrm{k}+1\right)},\mathrm{k}=1,\cdot \cdot \cdot ,\mathrm{n}-1\\ 1& \mathrm{x}\ge {\mathrm{X}}_{\left(\mathrm{n}\right)}\end{array}\right.$$

(27)

where $X_{\left(1\right)}\le X_{\left(2\right)}\le \cdots X_{\left(n\right)}$ is called the $X_{\left(1\right)},\cdots ,X_{\left(n\right)}$-ascending order statistic.

Let D be the maximum value of the gap between $F_0\left(x\right)$ and $F_n\left(x\right)$—that is, $D=\mathrm{m}\mathrm{a}\mathrm{x}\left|F_n\left(x\right)\right.-\left.F_0\left(x\right)\right|$—and reject the H₀ hypothesis when $D>D\left(\mathrm{n},\mathsf{\alpha }\right)$ is actually observed. Conversely, accept the H₀ hypothesis: the smaller the D value, the higher the goodness-of-fit. The value of $D\left(\mathrm{n},\mathsf{\alpha }\right)$ can be found in the cut-off table of the K-S test.

Taking the ultra-high-strength sucker rod stress amplitude of 500 MPa as an example, there are five experimental samples, with the significance level $\mathsf{\alpha }=0.05$, and the cut-off value $D\left(5,0.05\right)$ can be obtained by looking at the table as 0.563. According to the K-S test method, the K-S test results of the three-parameter Weibull distribution goodness of fit are displayed in

Table 6. K-S goodness-of-fit test results for the Weibull distribution.

k	1	2	3	4	5
$X_{\left(k\right)}$	−0.753	0.022	1.511	1.846	8.141
$F_0\left(X_{\left(k\right)}\right)$	0.427	0.577	0.63	0.642	0.818
$F_n\left(x\right)$	0.2	0.4	0.6	0.8	1
$\left|F_n\left(x\right){-F}_0\right|$	0.227	0.178	0.03	0.258	0.181

.

The maximum value according to the parameters in the last row of Table 6 can be used to obtain the D value of 0.258, which is less than the critical value $D\left(5,0.05\right)=0.563$; therefore, assume H₀, that is, the sucker rod fatigue data when the stress amplitude is 500 MPa, follows the Weibull distribution. Similarly, the D values of the stress amplitudes of 540 MPa and 600 MPa are 0.249 and 0.169, respectively, and both values are less than the critical value of 0.563. Therefore, the fatigue life data under the three stress amplitudes all follow the three-parameter Weibull distribution.

The K-S test for goodness-of-fit of normal distribution at a stress amplitude of 500 MPa was also performed, and the results are shown in

Table 7. K-S goodness-of-fit test results for the normal distribution.

k	1	2	3	4	5
$X_{\left(k\right)}$	−0.919	−0.677	−0.209	−0.104	1.909
$F_0\left(X_{\left(k\right)}\right)$	0.46	0.179	0.462	0.251	0.972
$F_n\left(x\right)$	0.2	0.4	0.6	0.8	1
$\left|F_n\left(x\right){-F}_0\right|$	0.26	0.221	0.138	0.549	0.028

.

The D value in Table 7 is 0.549, less than the critical value $D\left(5,0.05\right)=0.563$, so assumption H₀ is accepted; that is, the sucker rod fatigue data at a stress amplitude of 500 MPa is considered to follow a normal distribution. Similarly, it can be calculated that the stress amplitudes of 540 MPa and 600 MPa yield D values of 0.349 and 0.281, respectively. Both values are less than the critical value, and the fatigue life data at all three stress amplitudes follow a normal distribution.

Compared with the D values under the three stress amplitudes of the three-parameter Weibull distribution and the normal distribution, the D value of the three-parameter Weibull distribution is smaller, and the three-parameter Weibull distribution fits better.

5. Comparative Analysis

The K-S test confirms that both distributions provide a statistically acceptable fit to the experimental data within the tested range. However, a statistical test of fit is not the sole criterion for selecting the most appropriate predictive model, especially when considering physical realism and theoretical foundations. The superiority of the three-parameter Weibull distribution in this context is based on several key aspects.

First, in terms of theoretical appropriateness, the Weibull distribution is mathematically derived from the “weakest link” theory, which posits that failure is governed by the most critical flaw. This is highly representative of the fatigue failure mechanism, which initiates at material defects. The normal (or log-normal) distribution, based on the central limit theorem, is less physically representative of this type of failure initiation.

Second, regarding model flexibility and physical realism, the three-parameter Weibull model, particularly when integrated with the Stüssi function, has the flexibility to accurately model the entire S-N curve. It correctly captures the physically realistic asymptotic behavior as the stress approaches the ultimate tensile strength R_m at low life and converges to a fatigue limit S∞ at high life. The conventional Basquin model, being linear on a log–log scale, fails to represent these critical physical boundaries.

Third, concerning robustness with small samples, it is widely established in reliability engineering that the Weibull distribution provides more robust and reliable estimates from small sample sizes, which are characteristic of destructive and costly fatigue tests.

Therefore, while the normal distribution serves as a statistically plausible approximation and a vital benchmark, the proposed Weibull-based framework provides a more robust, flexible, and physically meaningful foundation for fatigue life prediction.

According to Equation (26), the P-S-N curve based on the three-parameter Weibull distribution is shown in Figure 4. According to Equation (24), the P-S-N curve based on normal distribution is shown in Figure 5. The results of the fatigue life predictions with different stress amplitudes are paired with experimental data, as shown in

Table 8. Comparison of prediction errors between the proposed Weibull model and the conventional Basquin model at 50% reliability.

Stress Amplitude S/MPa	Experimental Values	Three-Parameter Weibull Model			Normally Distributed Basquin Model
		Fatigue Life N/Times	Error	Average Error	Fatigue Life N/Times	Error	Average Error
500	864,152	792,213	8.32%	5.39%	959,431	11.03%	-
540	418,945	416,919	0.48%		485,548	15.9%	12.50%
600	213,541	197,808	7.37%		190,956	10.58%	-

.

As shown in Figure 4 and Figure 5, the P-S-N curve derived from the three-parameter Weibull distribution model exhibits a more physically realistic nonlinear shape that reflects the material’s asymptotic behavior near its strength and endurance limits and aligns more closely with the experimental data, which is quantitatively demonstrated by its lower average prediction error (5.39%) compared to the Basquin model (12.50%), as detailed in Table 8.

The fatigue limit value S∞ and the ultimate tensile strength R_m can be visually identified in Figure 4. It is important to note that the portion of the curves extending beyond approximately 2 × 10⁶ cycles is a mathematical extrapolation intended only to illustrate the model’s asymptotic convergence towards the fatigue limit. This region is beyond the range of our experimental data, and the model is not validated for making predictive claims in the very-high-cycle fatigue (VHCF) regime, where failure mechanisms may differ. The S-N curves with a 50% failure probability from Figure 4 and Figure 5 are plotted within the same coordinate system, as depicted in Figure 6. In the low-cycle fatigue life region (N < 6 × 10⁴), the Weibull model’s S-N curve gradually approaches the ultimate tensile strength. The predicted fatigue life value drops to zero when the stress amplitude equals the ultra-high-strength sucker rod’s ultimate tensile strength, causing the rod to rupture under tension. Conversely, for the normal distribution, as the stress amplitude increases, the S-N curve extends upward without converging to the ultimate tensile strength. When the stress amplitude matches the ultimate tensile strength, the predicted fatigue life for the normal distribution is 2405 cycles, which significantly deviates from the actual behavior. In the middle- and high-cycle fatigue life regions (6 × 10⁴ < N < 6 × 10⁶), the prediction accuracy of both models is comparable. In the high-cycle fatigue region (N > 6 × 10⁶), the Weibull model’s S-N curve slowly converges towards the theoretical fatigue limit. It is important to reiterate that the portion of the curves extending far beyond the experimental data is for illustrative purposes only, intended to highlight the different asymptotic behaviors of the two models. These extrapolations into the very-high-cycle fatigue (VHCF) regime are not scientifically validated predictions. For the normal distribution, as the stress amplitude decreases, the S-N curve continues downward without approaching the ultimate tensile strength. A similar pattern is observed in the low-cycle fatigue region. As illustrated in Figure 5, the normal distribution-based P-S-N curve intersects in the high-cycle fatigue region. Beyond this point, for the same stress amplitude, predicted fatigue life diminishes as reliability decreases, which contradicts actual fatigue behavior. This unphysical behavior is a known limitation of the purely statistical Basquin approach. In contrast, the proposed Weibull-based model avoids this discrepancy entirely. This is because our model is physically constrained by the need to incorporate both the ultimate tensile strength R_m at the high-stress limit and the fatigue limit S∞ at the low-stress limit. These physical anchors ensure the P-S-N curves maintain their logical order across the entire fatigue spectrum, thus providing a more realistic and robust representation of the sucker rod’s fatigue characteristics.

As can be seen from Table 8, the average fatigue life and experimental error of ultra-high-strength sucker rods, calculated based on the three-parameter Weibull distribution model, are 5.39%, and the fatigue life and experimental error calculated based on the normal distribution Basquin model are 12.50%. The fatigue life prediction accuracy of the model based on the three-parameter Weibull distribution proposed in this paper is higher.

6. Future Prospects

The advanced fatigue model established in this work provides a robust foundation for the analysis of sucker rod reliability. Several promising avenues exist for future research to enhance its practical applicability and scientific scope:

(1) Incorporating Environmental and Operational Effects: The most critical next step is to extend the model to include the effects of real-world service conditions. This requires conducting new fatigue experiments under various corrosive environments (e.g., H₂S, brine) and temperatures to develop a database of environment-specific model parameters.

(2) Generalization and Database Development for API Steels: The methodology should be applied to a wider range of API-standard sucker rod steels (e.g., Grade K, D, KD). This would lead to the creation of a comprehensive parameter library, which is an essential prerequisite for integrating the model into industrial design standards and software.

(3) Advanced Model Validation: Further experimental work, particularly with more samples and additional stress levels in the very-high cycle fatigue (VHCF) regime, would help to improve statistical confidence and more accurately validate the model’s long-life asymptotic behavior.

(4) Integration with Bayesian Methods for Predictive Maintenance: The model serves as an ideal “prior” for advanced reliability frameworks. Coupling it with Bayesian updating techniques using real-time, in-service data could enable the development of “digital twins” for individual sucker rod strings, paving the way for a shift from preventative to truly predictive maintenance.

7. Discussion

(1) The reason the new model based on the three-parameter Weibull distribution provides a more realistic depiction is that the Weibull-based model, by integrating material strength parameters via the Stüssi function, correctly captures the asymptotic behavior of the S-N curve at its physical limits.

In the low-cycle (high-stress) region, the curve correctly converges towards the ultimate tensile strength R_m. This is physically realistic, as fatigue life must approach zero when the applied stress equals the material’s ultimate strength. In contrast, the Basquin model unrealistically extends to predict a finite life at stresses far above this physical limit.

In the high-cycle (low-stress) region, the curve correctly flattens and converges towards a theoretical fatigue limit, S∞. This accurately represents the endurance behavior of many ferrous alloys, whereas the Basquin model, being linear on a log–log scale, incorrectly predicts that failure will eventually occur at any stress level.

(2) The primary quantitative comparison in this study was made against the conventional Basquin model. However, it is valuable to position our proposed model’s advantages in relation to other advanced models in the literature. Many probabilistic studies employ a two-parameter Weibull distribution, which is less flexible than the three-parameter version used here, particularly in modeling a non-zero minimum life. More advanced physics-based models, such as those derived from fracture mechanics (e.g., Paris’ Law), offer a rigorous description of crack growth but require knowledge of an initial flaw size, which is often unknown in design and can be computationally intensive. Our model offers a practical and powerful compromise: it retains the accessible framework of S-N analysis while significantly enhancing it with a robust probabilistic foundation and, crucially, grounding it in key physical parameters R_m and S∞. This integration of material strength makes it more physically realistic than purely statistical fitting models, without requiring the complex inputs of fracture mechanics.

8. Conclusions

Through the analysis and comparison of the two models presented above, the following conclusions are drawn:

(1) For the first time, a three-parameter Weibull distribution was employed, considering the fatigue limit and ultimate tensile strength, leading to the development of a new, ultra-high-strength, sucker rod fatigue life prediction model. The results of the K-S goodness-of-fit test indicate that the fatigue life data under the three stress amplitudes follow both the three-parameter Weibull distribution and the normal distribution, with the Weibull distribution exhibiting a higher goodness-of-fit.

(2) Comparing the S-N curves at a 50% failure probability of the two models, the fatigue life prediction accuracy in the middle- to high-cycle fatigue region is similarly consistent. However, in the low-cycle and high-cycle fatigue regions, the S-N curve based on the three-parameter Weibull distribution offers a more realistic representation.

(3) The fatigue life prediction results demonstrate that the errors in fatigue life estimation are 5.39% for the three-parameter Weibull distribution model and 12.50% for the normal distribution model. Overall, the fatigue life prediction model grounded in the three-parameter Weibull distribution is demonstrably more accurate than the conventional normal distribution model.