P-s-n curve solving method for small sample fatigue test data

Abstract

In order to overcome the problems that the existing P-S-N curve solving method has small applicable range, low accuracy, and is limited in engineering application due to dependence on a large number of fatigue test data, the present application provides a P-S-N curve solving method for small sample fatigue test data. The present application first classifies and standardizes the fatigue life test data according to the stress level; then detects and eliminates the outliers in the data, and performs resampling judgment; then performs distribution test on the effective fatigue life data of each stress level to determine the distribution type of the overall test data; finally, the S-N curve of P at different survival rates is solved in combination with the distribution type, and the monotonicity is verified, and when the curve exists, the cross is eliminated by using the equal slope constraint to eliminate the cross phenomenon of the curve. The present application solves the problems of small sample fatigue test data, optimal distribution type of data, and P-S-N curve solving difficulty of violating the physical monotonicity rule of the prediction life.

Description

Technical Field

[0001] This invention belongs to the field of fatigue life prediction, specifically involving a method for solving PSN curves for small sample fatigue test data. Background Technology

[0002] Fatigue life testing is a test method used to evaluate the ability of metal components to resist fatigue failure under actual service conditions. It simulates periodic loads under actual working conditions, observes and measures the fatigue damage of components during long-term operation, thereby verifying the rationality of structural design and providing basic data for fatigue life prediction.

[0003] Typical fatigue tests primarily acquire conventional data such as stress and fatigue life, and strain and fatigue life. While the data types are clearly defined, fatigue test data often exhibit significant dispersion and potential invalidity due to random defects in the material itself, manufacturing errors, and clamping methods during the test. This is especially true for small sample data obtained at high costs; without scientific screening and removal of these abnormal and invalid data, the accuracy of the final fatigue life prediction will be severely affected. Furthermore, traditional life assessment methods mostly only provide an average level or a single life estimate of "50% survival rate." However, in practical engineering applications, components of different importance have drastically different reliability requirements. This necessitates data processing that not only avoids the interference of outliers but also provides corresponding levels of life estimates based on different reliability indices, thus offering more valuable reference for engineering design decisions. Therefore, a precise, efficient, and comprehensive method for processing test data that fully reflects the dispersion of fatigue life is essential.

[0004] The PSN curve, or SN curve corresponding to the survival rate P, describes the relationship between the stress level and fatigue life of a component under different survival rates. When analyzing and processing experimental data, the PSN curve can reflect the dispersion of the fatigue characteristics of the component, providing more comprehensive and accurate information for engineering design and fatigue analysis. The literature "Bai Enjun, Yang Chaojie. PSN curve fitting method for regression fatigue life distribution parameters [J]. Internal Combustion Engine and Parts, 2022, (22): 102-104" describes the solution method for the PSN curve. It is based on the weighted least squares method to fit the distribution parameters of fatigue life at each stress level, obtains the mean and standard deviation of fatigue life at each stress level after regression, and fits the SN curve under different survival rates P. Although the above method can complete the fitting of fatigue test data and solve the PSN curve, the above method assumes that the test data is log-normally distributed, which is difficult to handle the case where the test data is not log-normally distributed, and the scope of application is small; at the same time, the above method does not consider identifying and eliminating abnormal data in the test process, which leads to the abnormal data of individual tests seriously affecting the accuracy of the overall results. In addition, the above methods rely on a large amount of fatigue test data, which is difficult to obtain, thus limiting their engineering applications. Summary of the Invention

[0005] To overcome the technical problems of existing PSN curve solving methods having a limited scope of application, low accuracy, and reliance on a large amount of fatigue test data, which restricts their engineering applications, this invention proposes a PSN curve solving method for small sample fatigue test data.

[0006] The technical solution adopted by this invention to solve its technical problem is as follows:

[0007] The method for solving PSN curves for small sample fatigue test data is unique in that it includes the following steps:

[0008] Step 1: Obtain experimental data;

[0009] Fatigue life test data were obtained from specimens of the same batch with the same structure, size, material and process, including the test stress amplitude and the corresponding fatigue life value.

[0010] Step 2, data normalization processing;

[0011] After arranging the test stress amplitudes in sequence, they are defined as different stress levels, and the fatigue life values ​​under each stress level are arranged in sequence.

[0012] Step 3, outlier removal;

[0013] For the sample set of fatigue life values ​​under each stress level, outliers are iteratively removed based on the dynamic threshold Schauvinay criterion. If the current iteration number is greater than 1, after removing outliers in the current round, a resampling judgment is made based on the rate of change of the subsample standard deviation of the sample set before and after removing outliers in the current round. If the rate of change of the standard deviation is greater than the preset safety threshold, fatigue life data under the current stress level needs to be supplemented and merged with the fatigue life data under the stress level obtained in step 1, and then return to step 2.

[0014] Step 4: Determine the distribution type;

[0015] For the effective fatigue life values ​​after removing outliers under each stress level, the single-sample KS test is used to preliminarily determine the assumed distribution of the effective fatigue life values ​​under each stress level. The assumed distribution is at least one of the normal distribution, log-normal distribution and Weibull distribution.

[0016] Secondary determination of distribution type: Frequency statistics are performed from a global perspective, and the distribution type with the highest frequency of fatigue life data of all stress levels is taken as the unified distribution type of the overall test data; if the frequency of fatigue life data of each stress level following multiple assumed distributions is consistent, the assumed distribution with the smallest sum of squared residuals is selected as the unified distribution type of the overall test data.

[0017] The formula for calculating the sum of squared residuals is:

[0018]

[0019] In the formula, For a certain stress level The total number of valid fatigue life data after removing outliers; Let be the empirical cumulative distribution function value of the p-th effective fatigue life data; This is the theoretical cumulative distribution function value assumed to be distributed at the p-th effective fatigue life data point;

[0020] Step 5: Solve for the PSN curve;

[0021] Based on the uniform distribution type determined in step 4, different survival rates P were selected, and the stress-life model was used to solve the preliminary PSN curve equation.

[0022] Verify the monotonicity of the initial PSN curve. If it fails, it indicates that the curves intersect. Then, use the equal slope constraint method to refit and obtain the corrected PSN curve equation. The corrected PSN curve equation is the final PSN curve obtained by solving. If it passes, the initial PSN curve equation is the final PSN curve obtained by solving.

[0023] Furthermore, the method for outlier removal in step 3 for fatigue life data under a single stress level is as follows:

[0024] Step 3.1, Algorithm initialization;

[0025] Set the number of iterations The initial sample set is defined as the set of fatigue life data under a certain stress level. Its initial sample size is ;

[0026] Step 3.2, Calculation of statistics and threshold values;

[0027] In the In the next iteration, based on the current sample set Calculate the sample mean and sample standard deviation ;

[0028] Based on the current sample set sample size The threshold value of the Chauvinistic criterion is obtained by querying the threshold value test table or by performing linear interpolation based on the data in the Chauvinistic criterion threshold value test table. The dynamic threshold value corresponding to the next iteration ;

[0029] Step 3.3, extreme deviation test;

[0030] Find the current sample set Mean deviation of sample Largest data point Calculate its value compared to the sample average. The absolute value of the deviation, if the absolute value of the deviation is greater than the dynamic threshold value Then determine the data point If it is an outlier, remove it from the current sample set. If the sample is removed from the current set, proceed to step 3.4; otherwise, it indicates that the current sample set is not in its final form. If no outliers are found, the iteration terminates and the final valid dataset is output.

[0031] Step 3.4, Dynamic Update and Resampling Determination;

[0032] Step 3.4.1: Determine if the current iteration count is greater than 1. If yes, proceed to step 3.4.2; otherwise, proceed to step 3.4.3.

[0033] Step 3.4.2, Resampling determination;

[0034] Calculate the rate of change of the subsample standard deviation of the sample set before and after removing outliers in the current round, and compare it with the set safety threshold. If it exceeds the safety threshold, the algorithm stops iterating and needs to supplement the fatigue life data under the current stress level. The supplemented fatigue life data is then merged with the original fatigue life data under the current stress level to form a new sample set, and the algorithm returns to the normalization process in step 2. If it does not exceed the safety threshold, proceed to step 3.4.3.

[0035] Step 3.4.3, Dynamic Update;

[0036] renew Update the sample set after removing outliers as follows: Update the subsample size to Then, return to step 3.2 to proceed to the next round of elimination iteration.

[0037] Furthermore, the safety threshold mentioned in step 3 is 50%-70%.

[0038] Furthermore, in step 4, the method for initially determining the assumed distribution of the effective fatigue life value under a single stress level using the single-sample KS test is as follows:

[0039] Step 4.1: For the effective fatigue life data under a certain stress level after outlier removal, propose initial distribution assumptions that it follows a normal distribution, a log-normal distribution, and a Weibull distribution, respectively, and construct parallel three types of known distribution models in this way;

[0040] Step 4.2: Determine the empirical cumulative distribution functions of the effective fatigue life data under this stress level, which are subject to three initial distribution assumptions, and calculate their function values.

[0041] Step 4.3: Determine the theoretical cumulative distribution function of the effective fatigue life data under this stress level, which follows three initial distribution assumptions, and calculate its function value;

[0042] Step 4.4: Calculate the KS test statistics for the three initial distribution hypotheses corresponding to this stress level. , and ;

[0043] Step 4.5, determine the critical value D for the KS test. α ;

[0044] Step 4.6, based on the KS test statistic , and and critical value D α To determine the assumed distribution of effective fatigue life data under this stress level.

[0045] Furthermore, the survival rates P selected in step 5 are 50%, 84.1%, 90%, 95%, 99%, 99.9%, and 99.99%.

[0046] Furthermore, the stress-life model used in step 5 is the Basquin formula.

[0047] Furthermore, the method for refitting using the equal slope constraint in step 5 is as follows:

[0048] The logarithmic linear equations of each survival rate P Modified to an equal slope constraint model ; For survival rate The slope of the logarithmic linear equation at time ;

[0049] Substitute the fatigue life data under each survival rate P into the modified isoclinic constraint model. In this paper, the intercept parameter is solved using the least squares method. ;

[0050] Based on the intercept parameter Combined with slope Reconstruct the PSN curve equation.

[0051] Furthermore, in step 5, the least squares method is used to solve for the intercept parameter. The process is as follows:

[0052] Suppose there are q stress levels in the experiment, and let the logarithm of the i-th stress level be... The logarithmic value of fatigue life corresponding to a specific survival rate P is: ,in ;

[0053] In the model with equal slope constraint Next, construct the objective function Q for the sum of squared residuals:

[0054]

[0055] Let Q be the parameter to be calculated. The first-order partial derivative is equal to zero:

[0056]

[0057] Expand and simplify the above equation:

[0058]

[0059] This allows us to obtain a result specific to the intercept. The analytical solution expression:

[0060]

[0061] In the formula, This is the arithmetic mean of the logarithmic values ​​of fatigue life corresponding to all experimental stress levels under this specific survival rate P; This is the arithmetic mean of the logarithmic values ​​of the stress levels in each test.

[0062] Substitute the known data into the intercept. The corrected intercept parameter can be obtained by solving the analytical expression. .

[0063] Furthermore, step 5 also includes the step of plotting the PSN curve based on the final PSN curve obtained from the solution.

[0064] The beneficial effects of this invention are:

[0065] 1. This invention first classifies and standardizes fatigue life test data according to stress level; secondly, it detects and removes outliers in the fatigue life test data and performs resampling; thirdly, it performs distribution testing on the effective fatigue life data for each stress level to determine the distribution type of the overall test data; then, based on the distribution type, it solves the SN curves for different survival rates P and verifies monotonicity. When curves intersect, it uses equal slope constraints to refit to eliminate curve intersection. This invention effectively solves the problems of outlier interference in small sample fatigue test data, difficulty in determining the optimal data distribution type, and the problem of solving PSN curves where predicted life violates the physical monotonicity law, laying the foundation for predicting the fatigue life of components under different reliability levels.

[0066] 2. This invention introduces a resampling judgment in the outlier iterative elimination process based on the Chauvinistic criterion. When the number of iterations exceeds 1, after each iteration, the standard deviation change rate is calculated and compared with a set safety threshold. When the standard deviation change rate exceeds the safety threshold, it indicates that the basic physical distribution of the current sample set may have been destroyed, or that the fatigue life test data itself has gross errors introduced by abnormal factors, or that the specimen has experienced unexpected mixed failure under the current stress level. At this time, the algorithm will forcibly terminate the iterative elimination process and give a warning, thus avoiding excessive elimination that may lead to data distortion or obscure the true fatigue dispersion law of the material.

[0067] 3. When performing distribution testing on valid fatigue test data, this invention introduces a mechanism for secondary distribution determination based on frequency statistics and residual sum of squares, in addition to the traditional single-sample KS test, thus solving the problem of selecting multiple assumed distributions that simultaneously pass the KS test.

[0068] 4. Considering that PSN curves may intersect in practical engineering with small sample sizes, this invention introduces monotonicity verification and equal slope constraint refitting. If monotonicity verification reveals curve intersection, a reference slope is selected. Constrain the intercept of the modified PSN curve and base it on the reference slope. The PSN curve equation is reconstructed using the corrected intercept. This invention employs an analytical method of finding extrema through first-order partial derivatives, and re-optimizes using the least squares method with actual fatigue data. This ensures that the optimized slope is within the range of minimum error, the PSN curves do not intersect, and the life degradation law is satisfied.

[0069] 5. This invention is a PSN curve solving method designed for small sample fatigue test data, and the entire process has been adapted and optimized for the characteristics of small samples:

[0070] 5.1 Outlier removal adopts the Chauvinistic criterion, which can dynamically adjust the threshold according to the sample size, and can accurately identify outlier data without a large sample size;

[0071] 5.2 The distribution type determination system, which uses the KS test and the global frequency and residual sum of squares as the best approach, solves the blind spot in determining the simultaneous passing of multiple distributions under small sample conditions.

[0072] 5.3 An equal slope constraint mechanism is introduced in the curve fitting process to compensate for the random fluctuations of small samples with the inherent physical laws of material fatigue damage and avoid curve intersection.

[0073] The entire method replaces the traditional approach of relying on large sample sizes to improve the reliability of results with the above-mentioned approach. It can output accurate PSN curves that conform to physical laws with only a small amount of experimental data. Therefore, it does not need to rely on a large amount of fatigue test data and can effectively overcome the engineering application limitations caused by the difficulty of data acquisition.

[0074] 6. This invention has excellent effectiveness and accuracy:

[0075] 6.1 The PSN curves obtained by the method of this invention strictly conform to the inherent law of high-cycle fatigue of materials that "the higher the stress amplitude, the shorter the fatigue life, and the higher the survival rate, the lower the fatigue life." The survival rate curves do not intersect throughout the entire process, which meets the physical monotonicity requirement for fatigue life prediction. At the same time, the deviation between the fitted life values ​​at each stress level and the corresponding effective experimental data is small, the overall fitting residual is at a low level, and the numerical accuracy can meet the needs of engineering applications.

[0076] 6.2 The outlier iteration elimination and resampling judgment mechanism built into the method of this invention can effectively resist the interference of gross errors during the experiment: even if abnormal data caused by equipment failure or improper clamping is mixed in the original data, the algorithm can accurately identify and eliminate it, avoiding outliers from skewing the overall fitting results.

[0077] 6.3 In the research and development practice project, the inventors of this invention cross-verified the stress level life prediction conclusions output by this invention with the results of bench physical tests on actual components. The experimental data showed that the error between the predicted life and the actual service life was within the allowable range of engineering. Attached Figure Description

[0078] Figure 1 This is a flowchart of the method of the present invention.

[0079] Figure 2 This is a flowchart of the iterative removal process for small sample outliers based on the dynamic threshold Schauvinay criterion in this invention.

[0080] Figure 3 This is a flowchart of the distribution verification and quantification optimization process for experimental data in this invention.

[0081] Figure 4 This is the PSN curve. Detailed Implementation

[0082] The present invention will be further described in detail below with reference to the accompanying drawings.

[0083] Reference Figure 1 The method for solving PSN curves for small sample fatigue test data proposed in this invention includes the following steps:

[0084] Step 1: Obtain experimental data;

[0085] The fatigue life test data obtained by this invention only includes the test stress amplitude and the corresponding fatigue life value under each test stress. To clearly describe the distribution of the test data, fatigue life test data are obtained based on specimens from the same batch with the same structure, dimensions, material, and manufacturing process.

[0086] Step 2, data normalization processing;

[0087] First, the test stress amplitudes are sorted in ascending order (or descending order) and defined as different stress levels. Then, within each defined test stress level, all corresponding fatigue life values ​​are rearranged in ascending order (or descending order). For example, the test stress amplitudes can be sorted in ascending order, and the corresponding stress levels can be defined as follows: There are a total of q levels, among which ; to a certain stress level The fatigue life values ​​obtained under (i=1, 2,…, q) Arranged in ascending order as follows: .

[0088] Step 3: Iterative removal of small sample outliers based on the dynamic threshold Chauvinistic criterion;

[0089] Fatigue testing is often limited by high costs and time constraints, resulting in life data at the same stress level that are typically obtained from small samples (usually small sample size). In engineering data processing, the commonly used Laida criterion (i.e., The criteria assume a normal distribution and usually require a sufficiently large sample size, such as... For fatigue life test data with high dispersion in small samples, the Raida criterion's exclusion interval is too broad, easily leading to the omission of true outliers. In contrast, the Chauvinistic criterion, based on probability distribution theory, has a discrimination threshold that can dynamically adjust with changes in sample size, exhibiting higher sensitivity and scientific rigor under small sample conditions. Therefore, this invention, specifically addressing the characteristics of small samples, employs the Chauvinistic criterion for outlier identification and processing.

[0090] To prevent statistical distortion caused by a single elimination, this invention designs a dynamic iterative elimination and resampling judgment algorithm for small samples based on the Chauvinistic criterion, which eliminates outliers in the fatigue life data under each stress level divided in step 1.

[0091] The specific algorithm flowchart is as follows: Figure 2 As shown, it includes the following steps:

[0092] Step 3.1, Algorithm initialization;

[0093] Set the number of iterations The sample set consisting of fatigue life data at a certain stress level as defined in step 2 is denoted as the initial sample set. Its initial subsample size (sample size) is .

[0094] Step 3.2, Calculation of statistics and threshold values;

[0095] In the In the next iteration, based on the current sample set Calculate the sample mean Sample Standard Deviation The calculation formula is as follows:

[0096] Sample average The calculation formula is:

[0097]

[0098] Sample standard deviation The calculation formula is:

[0099]

[0100] In the formula, For the current sample set The fatigue life of the i-th specimen (i.e., the value of the i-th subsample). , i=1, 2,…, ; For the current sample set The sample size (sample size).

[0101] At the same time, based on the current sample set sample size Retrieve the first table by querying Table 1 The dynamic threshold value corresponding to the next iteration If the actual sample size If it is not in Table 1, it is assumed to be between the sizes of its adjacent subsamples in Table 1. and Between, and the corresponding threshold value is and Then its corresponding threshold value It can be obtained through the following linear interpolation formula:

[0102]

[0103] Table 1. Test table for the threshold value k of Chauvinistic criterion.

[0104]

[0105] Step 3.3, extreme deviation test;

[0106] Find the current sample set Mean deviation of sample Largest data point Calculate its value compared to the sample average. The absolute value of the deviation, if it satisfies:

[0107]

[0108] Then determine the data point If it is an outlier, remove it from the current sample set. Remove from the list and proceed to step 3.4.

[0109] If this condition is not met, it indicates that the current sample set... If no outliers are found, the iteration terminates, and the final valid dataset is output.

[0110] Step 3.4, Dynamic Update and Resampling Determination;

[0111] Step 3.4.1: Determine if the current iteration count is greater than 1. If yes, proceed to step 3.4.2; otherwise, proceed to step 3.4.3.

[0112] Step 3.4.2, Resampling determination;

[0113] Considering the extreme sensitivity of small sample data to changes in a single data point, excessive filtering may misjudge the inherently high discreteness of the material as outliers and remove them, leading to data distortion or masking the true fatigue dispersion patterns of the material. Therefore, this invention introduces the standard deviation variation rate. As a monitoring indicator:

[0114]

[0115] After the current iteration of elimination, the rate of change of standard deviation is calculated using the above formula. The results are then compared with a set safety threshold. In general engineering problems, the reasonable range of the safety threshold is usually 50% to 70%. If the safety threshold is exceeded, it indicates that the basic physical distribution of the current sample set may have been severely disrupted, or that the fatigue life test data has been mixed with gross errors introduced by abnormal factors such as test equipment failure, improper specimen clamping, or macroscopic defects in the material, or that the batch of specimens has experienced unexpected mixed failures at the current stress level. At this time, the algorithm will forcibly stop the iterative elimination process, prompting that the number of test samples needs to be increased at the current stress level. After the sample size is increased, the newly added test data needs to be merged with the original test data to form a new sample set, and the normalization process in step 2 is returned to be executed again. This is equivalent to starting a new round of fatigue test data processing and PSN curve solving process to help re-determine the true distribution of fatigue life.

[0116] If the safety threshold is not exceeded, proceed to step 3.4.3;

[0117] Step 3.4.3, Dynamic Update;

[0118] renew Update the sample set after removing outliers as follows: Update the subsample size (sample size) to Then, return to step 3.2 to proceed to the next round of elimination iteration.

[0119] Step 4: Determine the optimal distribution function by performing distribution tests on the experimental data;

[0120] Fatigue life data typically follow one of the following distributions: normal, log-normal, exponential, or Weibull. The exponential distribution can be considered as a shape parameter... The exponential distribution follows the Weibull distribution, therefore it can be considered a special case of the Weibull distribution.

[0121] This invention employs the traditional single-sample KS test method, which examines the degree of agreement between the empirical cumulative distribution and the known theoretical cumulative distribution of the effective fatigue life data sample set corresponding to a single stress level. This method is used to perform distribution tests on the fatigue life data corresponding to each stress level after outlier removal in step 3, and to determine the distribution function.

[0122] Because the traditional single-sample KS test often has a blind spot when dealing with small sample data, where multiple distributions pass the test simultaneously, this invention introduces a quantitative selection mechanism based on frequency statistics and residual sum of squares to scientifically determine the unique optimal distribution type of fatigue life data.

[0123] Reference Figure 3 This invention addresses a specific stress level. The specific steps for performing distribution testing and quantitative selection of the corresponding effective fatigue life data are as follows:

[0124] Step 4.1 proposes initial distribution assumptions for the effective fatigue life data after outlier removal, suggesting they follow a normal, log-normal, and Weibull distribution, respectively, to construct parallel models of three known distribution types. The formulas for each distribution function are as follows:

[0125] Normal distribution function: ;

[0126] Log-normal distribution function: ;

[0127] Weibull distribution function: .

[0128] Step 4.2, determine the stress level The effective fatigue life data are subjected to three different initial distribution assumptions (or assumed distributions), and their empirical cumulative distribution functions are calculated. These functions are the cumulative distribution functions of random variables formed based on the effective fatigue life data corresponding to the stress level, and are defined as follows:

[0129]

[0130] In the formula, m is the current stress level. The total number of effective working life data samples after removing outliers; p is the current stress level. The effective working life data samples after removing outliers are sorted in ascending order and their ranks (i.e., serial numbers), p=1, 2,…,m; For the current stress level After removing outliers, the effective working life data samples are sorted in ascending order, and the fatigue life value corresponding to the sample with rank p is obtained. For the current stress level The smallest fatigue life value in the effective fatigue life data sample after removing outliers, i.e. the first observation value after ascending order; For the current stress level The maximum fatigue life value in the effective working life data sample after removing outliers, i.e., the last observation value after ascending order.

[0131] Step 4.3, Determine the stress level The effective fatigue life data are respectively subjected to the theoretical cumulative distribution functions of three initial distribution assumptions (or assumed distributions), and their function values ​​are calculated;

[0132] First, based on the effective fatigue life data after removing outliers, it is necessary to estimate the parameters of the initial hypothesis distribution function to obtain the characteristic parameters required for the corresponding distribution. The specific steps and results of parameter estimation are as follows:

[0133] (1) If the stress level is assumed in step 4.1 The effective fatigue life data follows a normal distribution, and the parameter to be estimated is the mean of the normal distribution function. and standard deviation Calculate using the following formula:

[0134]

[0135]

[0136] (2) If the stress level is assumed in step 4.1 The effective fatigue life data follows a log-normal distribution, and the parameter to be estimated is the log-normal distribution function's log-mean. and logarithmic standard deviation Calculate using the following formula:

[0137]

[0138]

[0139] In the formula, Indicates stress level Below, the i-th effective fatigue life observation value, where ; Indicates stress level The total number of valid fatigue data samples after removing outliers; Let represent the natural logarithm of the i-th effective fatigue life.

[0140] (3) If in step 4.1 it is assumed that the data follows a Weibull distribution, the parameter to be estimated is the shape parameter. Scale parameters and position parameters Typically, the maximum likelihood estimation (MLE) method or the least squares method combined with correlation coefficient optimization is used to obtain the specific values ​​of these three parameters through numerical iteration.

[0141] After completing the parameter estimation above, substitute the respective parameter values ​​into the three types of distribution function formulas corresponding to step 4.1 to obtain the current stress level. The effective fatigue life data corresponds to three theoretical cumulative distribution functions: normal distribution, log-normal distribution, and Weibull distribution.

[0142] Step 4.4, calculate the KS statistic D. n ;

[0143] KS statistic D n This is used to measure the difference between the empirical cumulative distribution function and the theoretical cumulative distribution function of effective fatigue life data. Based on the distribution assumptions in step 4.1, the empirical cumulative distribution function obtained in step 4.2 needs to be substituted and compared with the theoretical cumulative distribution functions of the normal, log-normal, and Weibull distributions obtained in step 4.3, respectively, to calculate the stress level. The effective fatigue life data correspond to the three KS statistics of these three distributions. , and Its definition is:

[0144] ,

[0145] in, Stress level The empirical cumulative distribution function of effective fatigue life data; Assuming stress level The theoretical cumulative distribution function when the effective fatigue life data follows a normal distribution; Assuming stress level The theoretical cumulative distribution function when the effective fatigue life data follows a log-normal distribution; Assuming stress level The effective fatigue life data follows a theoretical cumulative distribution function that conforms to a Weibull distribution.

[0146] Step 4.5, determine the critical value D for the KS test. α ;

[0147] First, the significance level is set based on fatigue life engineering statistical standards and reliability requirements. Significance level In hypothesis testing, the significance level refers to the probability that a null hypothesis, which is actually true, is incorrectly rejected (i.e., the probability of a Type I error or "false positive" error in statistics). In the distribution test of fatigue life data, the significance level... The larger the value of , the lower the threshold for rejecting the null hypothesis. This means a more stringent requirement for the goodness of fit of the data to the theoretical distribution. The KS critical value D obtained at this point... α The smaller the value, the easier it is to conclude that the effective fatigue life data does not follow a specific distribution; conversely, the larger the value, the lower the significance level. The smaller the value, the higher the tolerance, and the more likely it is to accept the null hypothesis. To effectively control the probability of misjudgment during the distribution test and ensure the accuracy of subsequent fatigue life calculations, during routine assessments... A value of 0.05 is acceptable; for critical components with high reliability requirements and where strict control of evaluation errors is necessary, A value of 0.01 is acceptable. Furthermore, given the high dispersion of fatigue life data, when conducting theoretical research on novel materials, or when it is necessary to rigorously eliminate distribution models with poor fit, in order to improve the sensitivity and rejection rate of the algorithm's verification, The value can also be expanded to 0.15 or 0.2.

[0148] In practice, the KS critical value D α The specific value depends only on the effective fatigue life data sample size N and the set significance level. Closely correlated, and does not change with the type of theoretical distribution being tested. D α The results were obtained by referring to Table 2, which shows the test conditions.

[0149] Table 2 Critical values ​​D for the KS test α

[0150]

[0151] Step 4.6: Make a preliminary determination of the distribution based on the test criteria;

[0152] The KS test statistics obtained in step 4.4 for the corresponding normal distribution, log-normal distribution, and Weibull distribution are used. (j=1,2,3), respectively compared with the critical value D determined in step 4.5 α Perform numerical comparisons. If the KS test statistic calculated for a certain assumed distribution satisfies... If the null hypothesis is rejected at the set significance level, then the stress level is determined. The effective fatigue life data do not follow this type of assumed distribution; conversely, if the following conditions are met... If so, then the null hypothesis is accepted, meaning the stress level is determined in a statistically significant sense. The effective fatigue life data follows this type of assumed distribution.

[0153] Step 4.7: Perform a secondary determination of the distribution based on the joint quantitative index system of frequency statistics and residual sum of squares (SSE);

[0154] Based on the fatigue failure mechanism of materials, different stress states can lead to changes in the failure mode or mechanism. However, under similar stress levels, fatigue damage in the same batch of materials is often dominated by similar physical mechanisms. Considering that the stress levels selected in conventional PSN curve testing are usually within similar ranges and have similar failure mechanisms, it is reasonable to infer that the effective test data at each stress level should conform to the same probability distribution model. However, the traditional KS test can only provide a qualitative conclusion on whether to reject the null hypothesis. In actual engineering operations, due to the limited sample size of fatigue tests and the inherent dispersion of data, data at a single stress level can easily accept the null hypothesis of multiple distributions simultaneously in a statistical sense, thus leading to the dilemma of choosing which of the assumed distributions pass the KS test at the same time.

[0155] To overcome the evaluation blind spots of the traditional KS test, this invention constructs a joint quantitative index system based on frequency statistics and sum of squared residuals (SSE):

[0156] First, frequency statistics are performed from a global perspective to determine which assumed distribution has the highest frequency for fatigue life data of all stress levels. The assumed distribution type with the highest frequency is then used as the unified distribution type for the entire batch of test data (test data of all stress levels).

[0157] If the fatigue life data for each stress level follow a consistent frequency of multiple assumed distributions, then the sum of squared residuals (SSE) is further introduced as a depth quantification index, with the formula:

[0158]

[0159] In the formula: For a certain stress level The total number of valid fatigue life data after removing outliers; Let be the empirical cumulative distribution function value of the p-th effective fatigue life data; Let be the theoretical cumulative distribution function value assumed to be distributed at the p-th effective fatigue life data point.

[0160] After calculating the sum of squared residuals (SSE) for each distribution using the above formulas, the goodness of fit of each assumed distribution function can be evaluated. The assumed distribution function with the smallest SSE is selected as the unified distribution type for all stress levels. The sum of squared residuals (SSE) essentially reflects the degree of deviation between the theoretical cumulative probability predicted by the assumed distribution and the actual empirical cumulative frequency shown by real experimental data. Therefore, this method overcomes the limitation of the traditional KS test, which relies solely on critical values, by directly utilizing the minimum deviation between theory and reality (i.e., the corresponding minimum SSE value) to achieve quantitative comparison of each qualified distribution, thereby determining and establishing the assumed distribution function with the best fit, which is then used as the stress level. The optimal distribution function for effective fatigue life data.

[0161] To facilitate understanding, examples are given below. First scenario: Assume a set of test data has 5 stress levels. The fatigue life data for one stress level follows a normal distribution, while the fatigue life data for the other 4 stress levels follows both a normal and a log-normal distribution. The normal distribution has the highest frequency, therefore, it is determined to be the unified distribution type for all 5 stress levels. Second scenario: Assume a set of test data has 5 stress levels, and the fatigue life data for all 5 stress levels follows both a normal and a log-normal distribution. The frequencies of following both the normal and log-normal distributions are the same. In this case, the stress-weighted average fatigue life (SSE) needs to be calculated, and the distribution type with the smallest SSE is selected as the unified distribution type for all 5 stress levels.

[0162] Step 5: Solve and plot the PSN curve;

[0163] Step 5.1: Select the survival rate P for the PSN curve. Typically, P can be set to 50%, 84.1%, 90%, 95%, 99%, 99.9%, or 99.99%.

[0164] Step 5.2, based on stress level Based on the effective fatigue life data, the fatigue life under different survival rates P corresponding to each stress level was calculated.

[0165] (1) If the stress level The effective fatigue life data below follows a normal distribution, and the fatigue life N P Calculate using the following formula:

[0166]

[0167] in, σ is the mean fatigue life; s is the standard deviation of fatigue life. This represents the standard normal skewness corresponding to the survival rate P.

[0168] (2) If the stress level The effective fatigue life data follows a log-normal distribution, and the fatigue life N P Calculate using the following formula:

[0169]

[0170] in, This represents the logarithmic mean fatigue life. The standard deviation of the logarithmic fatigue life; This represents the standard normal skewness corresponding to the survival rate P.

[0171] (3) If the stress level The effective fatigue life data below follows a Weibull distribution, and the fatigue life N P Calculate using the following formula:

[0172]

[0173] in, These are dimensional parameters; For shape parameters; Survival rate; For position parameters.

[0174] Step 5.3: Solve for the preliminary SN curve corresponding to the survival rate P;

[0175] In high-cycle fatigue, the logarithm of the fatigue life of metallic materials typically exhibits a good linear relationship with the logarithm of the stress amplitude. Although other life models, such as the Coffin-Manson model, exist for low-cycle fatigue, the most commonly used stress-life model in engineering for high-cycle fatigue is represented by the Basquin formula. This invention uses the Basquin formula as the theoretical basis to solve for the preliminary SN curves under different survival rates.

[0176] Let N be the fatigue life under the survival rate P. P The corresponding stress level (i.e., stress amplitude) is denoted as The basic expression for the SN curve under the survival rate P is:

[0177]

[0178] Taking the logarithm of both sides of the above equation, we can transform it into a logarithmic linear equation:

[0179]

[0180] In the formula, a and C are constants to be determined that are related to the fatigue characteristics of the metallic material itself.

[0181] Using the least squares method, the stress levels obtained in step 4.2 are... and the corresponding fatigue life under the survival rate P. Substitute these values ​​into the logarithmic linear equation above to perform linear regression fitting. The corresponding material constants a and C are then calculated. At this point, the specific equation for the SN curve under a specific survival rate P can be obtained.

[0182] By summarizing all the fitting results under the set survival rate P, the preliminary PSN curve equation is obtained.

[0183] Step 5.4, Monotonicity verification and refitting of the initial PSN curve equation;

[0184] In high-cycle fatigue service, the fatigue life of metallic materials inevitably decreases with increasing survival rate P. However, due to the random fluctuations of small sample data, independently solving the logarithmic linear equations for each survival rate in step 5.3 can easily lead to deviations in the slope of the obtained curves. This can cause the PSN curves for different survival rates to intersect in certain stress ranges, meaning that the predicted life corresponding to a high survival rate is actually greater than that of a low survival rate, violating the physical monotonicity of fatigue failure. Therefore, this invention designs the following monotonicity verification and iterative correction:

[0185] Step 5.4.1, Monotonicity verification;

[0186] Define the engineering stress range for component service. The PSN curves for each survival rate obtained in step 5.3 are iterated and calculated to verify that for any survival rate... Throughout the stress range Does the internal condition always satisfy? If satisfied, the verification passes, proceed to step 5.5; if it appears in a certain interval... This indicates that the curves intersect, and we proceed to step 5.4.2 for refitting.

[0187] Step 5.4.2, refit with equal slope constraints;

[0188] To eliminate curve intersections, the equal slope constraint method is used for parameter correction. Under high-cycle fatigue mechanisms, the fatigue damage evolution rate of the same material exhibits consistency; therefore, the survival rate is extracted. The parameters of the logarithmic linear equation at time t are used as a benchmark, and its slope constant is denoted as the reference slope. This invention focuses on survival rate. The slope of the logarithmic linear equation at time t is used as the reference slope. Because of the survival rate The SN curve is the median curve, which reflects the essential law of macroscopic fatigue damage of materials under the statistical average of a large number of micro-defects. Compared with high survival rate or lower survival rate curves controlled by a few extreme micro-defects, the median curve contains the largest amount of effective sample information, and its slope can best reflect the inherent fatigue damage evolution rate of the same material.

[0189] In subsequent fitting, the log-linear equation is fixed. slope constant Keeping it unchanged, the equation Modified to an equal slope constraint model Subsequently, the fatigue life data for each survival rate P obtained in step 4.2 are substituted back into the corrected equation. In this context, the least squares method is applied only to the intercept parameter. The solution is re-solved, and the process is as follows:

[0190] Suppose there are q stress levels in the experiment. Let the logarithm of the i-th stress level be... The logarithmic value of fatigue life corresponding to a specific survival rate P is: ,in .

[0191] In such slope constraint models Next, construct the objective function Q for the sum of squared residuals:

[0192]

[0193] To minimize the fitting error, i.e., the objective function Q, let Q be the parameter to be solved. The first-order partial derivative is equal to zero:

[0194]

[0195] Expand and simplify the above equation:

[0196]

[0197] This allows us to obtain a result specific to the intercept. The analytical solution expression:

[0198]

[0199] In the formula, This is the arithmetic mean of the logarithmic values ​​of fatigue life corresponding to all experimental stress levels under this specific survival rate P; This is the arithmetic mean of the logarithmic values ​​of the stress levels in each test.

[0200] Substitute the known data into the intercept. The corrected intercept parameter can be directly obtained from the analytical solution expression. .

[0201] Based on the corrected intercept parameter Combined with the reference slope Reconstruct the SN curve equations for each survival rate P, and then proceed to step 5.5.

[0202] Step 5.5 Drawing curve

[0203] Based on the monotonicity verification results of the PSN curve in step 5.4, the final curve equation is constructed and plotted for each case:

[0204] (1) Curve plotting when verification is successful: The logarithm of fatigue life is used as the horizontal axis and the stress amplitude is used as the vertical axis. The preliminary SN curve equations under each survival rate P obtained by least squares linear regression fitting in step 5.3 are used to plot these equations in the coordinate system to obtain the complete PSN curve.

[0205] (2) Correction and plotting when verification fails: Using the logarithm of fatigue life as the abscissa and the stress amplitude as the ordinate, plot the data based on the corrected intercept parameter. Combined with reference slope By reconstructing the SN curve equations for different survival rates P and plotting them uniformly on the same coordinate system, a complete, non-intersecting PSN curve can be obtained.

[0206] Example

[0207] Step 1: Obtain experimental data;

[0208] This embodiment calculates the PSN curve based on the raw test data recorded by the fatigue testing machine at various test stress amplitudes of a certain component. The raw test data of this component includes the actual fatigue life observation values ​​recorded by each specimen in the order of testing at different stress levels, as shown in Table 3.

[0209] Table 3 Original experimental data

[0210]

[0211] The method for solving the PSN curve based on the original experimental data in Table 3 in this embodiment is as follows:

[0212] Step 2: Standardization of experimental data;

[0213] The test data were categorized according to the magnitude of the test stress level and sorted from smallest to largest based on the lifespan. The standardized test data are shown in Table 4.

[0214] Table 4. Standardized experimental data

[0215]

[0216] Step 3: Iterative removal of small sample outliers based on the dynamic threshold Chauvinistic criterion;

[0217] For the experimental data in Table 4, an iterative algorithm based on the dynamic threshold Chauvinistic criterion was used for outlier identification and processing. Now, taking the first stress level as an example... Taking this as an example, the specific execution and iteration process of the algorithm will be explained in detail:

[0218] First iteration ( ):

[0219] Step 3.1, Algorithm initialization;

[0220] Set the current iteration number Extract the first stress level from Table 4. Based on the fatigue life data below, construct an initial sample set:

[0221]

[0222] Its initial sample size .

[0223] Step 3.2, Calculation of statistics and dynamic thresholds;

[0224] Based on the current sample set Calculate the sample average of its 6 data points. Sample Standard Deviation ;

[0225] At the same time, according to the sample size Refer to Table 1 above to obtain the dynamic threshold value corresponding to the current iteration. .

[0226] Step 3.3, extreme deviation test;

[0227] Find the current set Mean deviation of sample The largest data point, i.e. Calculate its value compared to the sample mean. The absolute value of the deviation is taken, compared with the standard deviation, and substituted into the formula to calculate the result:

[0228]

[0229] Because the calculation result is greater than the current threshold value According to the Chauvinistic criterion, the data point (425800) is determined to be an outlier and is removed from the current sample set. Remove from the list.

[0230] Step 3.4, Dynamic Update and Resampling Determination;

[0231] Since only a single exclusion has been performed so far, and the standard deviation change rate condition for two consecutive exclusions has not been triggered, the algorithm will be updated. Update the set of remaining valid samples to The sample size was updated to Then, return to step 3.2 to continue the iteration.

[0232] Second iteration ( ):

[0233] Step 3.2, Calculation of statistics and dynamic thresholds;

[0234] Based on the current sample set Calculate the sample average of the 5 data points within it. Sample Standard Deviation ;

[0235] At the same time, according to the sample size Referring to Table 1 above, the dynamic threshold value corresponding to the current iteration is obtained. .

[0236] Step 3.3, extreme deviation test;

[0237] Find the current sample set Mean deviation of sample The largest data point at this time is Substituting this into the test formula and calculating, the result is:

[0238]

[0239] Because the calculation result is less than the threshold value This indicates that, based on the current threshold, the data point is considered valid and should be retained; it also proves that the current set... There are no abnormal values ​​in the middle, first stress level. The iterative decision-making process has ended.

[0240] Step 3.4, Dynamic Update and Resampling Determination;

[0241] The iteration terminates normally. At this point, it is necessary to determine the rate of change of the standard deviation and the current set. Is it effective? A reasonable range for the safety threshold is typically 50% to 70%, but this invention uses 60%. The rate of change of the standard deviation at this point... for:

[0242]

[0243] Current set The experimental data is valid and can be used as the first stress level. The final valid dataset.

[0244] Repeat the iterative logic of initialization, calculation of statistics and dynamic thresholds, extreme deviation test and resampling determination to complete the outlier processing of fatigue life data under the remaining three stress levels (839MPa, 810MPa, 783MPa) in sequence.

[0245] The effective fatigue life data after processing are shown in Table 5.

[0246] Table 5. Experimental data after outlier handling

[0247]

[0248] Step 4: Determine the optimal distribution function by performing a distribution test on the effective fatigue life data;

[0249] The distribution function is determined by performing a one-sample KS test. Now, taking the first stress level as an example... Taking this as an example, the process is as follows:

[0250] Step 4.1, propose the null hypothesis;

[0251] Hypotheses are proposed for this distribution test, assuming the first stress level. The fatigue life data follow a normal distribution, with the distribution function being:

[0252] ;

[0253] Step 4.2, calculate the empirical cumulative distribution function;

[0254] First stress level The empirical cumulative distribution function is:

[0255]

[0256] Step 4.3: Calculate the theoretical cumulative distribution function;

[0257] Based on the effective fatigue life data after removing outliers in Table 5, the characteristic parameters required for a normal distribution are calculated. , Substituting into the normal distribution function, we obtain the theoretical normal distribution function for the first stress level as follows:

[0258] ;

[0259] Step 4.4, calculate the KS statistic D. n ;

[0260] when hour, ,

[0261] when hour, ,

[0262] when hour, ,

[0263] when hour, ,

[0264] when hour, ,

[0265] but,

[0266]

[0267] Step 4.5: Find the KS critical value D α ;

[0268] Set significance level ;

[0269] When the sample size N=5, the significance level At that time, referring to Table 2, we can obtain:

[0270]

[0271] Step 4.6: Make a preliminary determination of the distribution based on the test criteria;

[0272] The calculated KS statistic The critical value of KS obtained from the table lookup Compare them. If Then we accept the null hypothesis and consider the first stress level to be... The experimental data at the significance level If the distribution follows a normal distribution, then the null hypothesis is rejected.

[0273] Similarly, following steps 4.1 to 4.6 above, the first stress level is respectively... The experimental data were subjected to hypothesis tests for log-normal and Weibull distributions. Furthermore, the fatigue life data for the remaining three stress levels (839 MPa, 810 MPa, and 783 MPa) were subjected to the same one-sample KS tests for these three distribution types. To determine the uniform distribution type for all fatigue life data, the test results for each stress level under the three distribution hypotheses are summarized in Table 6 below.

[0274] Table 6 Summary of Test Data Distribution and Verification Results for Each Stress Level

[0275]

[0276] As shown in Table 6, at the significance level... The KS statistic for fatigue life data at four stress levels (877MPa, 839MPa, 810MPa, and 783MPa) under the assumptions of normal, log-normal, and Weibull distributions. All are less than the corresponding critical value This indicates that, outside the rejection region, the null hypothesis is accepted, namely that the fatigue life data set simultaneously follows a normal distribution, a log-normal distribution, and a Weibull distribution at each stress level.

[0277] Step 4.7: Perform secondary distribution determination based on the joint quantitative index system of frequency statistics and residual sum of squares (SSE);

[0278] Given each stress level The fatigue life data simultaneously follows multiple distributions, and the frequencies of each assumed distribution are consistent. In order to determine the optimal distribution model for this batch of fatigue life data, it is necessary to further calculate the sum of squared residuals (SSE) of each distribution function, and use the minimum error between the estimated values ​​and the actual values ​​of the three distribution functions to determine the optimal distribution function.

[0279] The formula for calculating the sum of squared residuals (SSE) is:

[0280]

[0281] The sum of squared residuals for the three distribution types at each stress level was calculated separately, and the results are summarized in Table 7.

[0282] Table 7 Summary of Residual Sum of Squares (SSE) Calculation Results for the Three Distribution Functions

[0283]

[0284] Comparing the total sum of squared residuals of the three distribution functions in Table 7, we can see that the log-normal distribution has the smallest sum of squared residuals (0.764), followed by the Weibull distribution (0.0914), and the normal distribution has the largest sum of squared residuals (0.1071).

[0285] Based on the principle of minimum error, the log-normal distribution is determined to be the best-fit distribution for this set of fatigue life data (test data under all stress levels). Therefore, in subsequent PSN curve calculations, the log-normal distribution will be uniformly used as the life distribution model for solution.

[0286] Step 5: Solving for the PSN curve and correcting for self-consistency;

[0287] Based on the analysis results in step 3, it was determined that the fatigue life data in this set follows a log-normal distribution. Based on this distribution model, the fatigue life under different survival rates was calculated and curves were fitted according to the following steps.

[0288] Step 5.1: Select the survival rate P;

[0289] To comprehensively evaluate the fatigue reliability of the components, this embodiment selects seven typical survival rates for calculation, namely 50%, 84.1%, 90%, 95%, 99%, 99.9%, and 99.99%.

[0290] Step 5.2: Calculate the fatigue life corresponding to the survival rate P;

[0291] Take the logarithm of the effective fatigue life data for each stress level (877MPa, 839MPa, 810MPa, 783MPa). And calculate its logarithmic mean. and standard deviation The calculation results are summarized in Table 8.

[0292] Table 8. Calculation results of logarithmic fatigue life statistical parameters for each stress level.

[0293]

[0294] The fatigue life calculation formula based on the log-normal distribution The logarithmic fatigue life values ​​for each stress level under different survival rates were calculated. Among these, the standard normal skewness was... Based on survival rate Determining by referring to a table (as is known in the art):

[0295] when hour, ;

[0296] when hour, ;

[0297] when hour, ;

[0298] when hour, ;

[0299] when hour, ;

[0300] when hour, ;

[0301] when hour, .

[0302] Substituting the parameters from Table 8 into the formula, the fatigue life under each survival rate can be calculated. The results are shown in Table 9.

[0303] Table 9. Calculation results of fatigue life under different survival rates.

[0304]

[0305] Step 5.3: Solve for the preliminary SN curve corresponding to the survival rate P;

[0306] Based on the fatigue life data at each survival rate calculated above, the Basquin model is used. describe Curve. The fatigue life at survival rate P is denoted as N. P The corresponding stress level is denoted as The basic expression for the SN curve under the survival rate P is:

[0307]

[0308] To facilitate parameter fitting, taking the logarithm of both sides of the formula yields a linear equation:

[0309]

[0310] In the formula, For fatigue life, For stress values, a and Let be the material constant to be determined.

[0311] Using the least squares method, the stress levels corresponding to each survival rate in Table 9 are... and the obtained fatigue life Substitute the values ​​into the above equation and perform linear regression fitting. For each survival rate... A set of corresponding material constants a and can be obtained from each. The fitting calculation results are summarized in Table 10.

[0312] Table 10. Survival rates at different levels Curve fitting parameters

[0313]

[0314] Step 5.4: Monotonicity verification and refitting of the PSN curve;

[0315] According to the monotonicity verification specifications in the technical solution, the obtained survival rate (SN) curves need to be verified for physical laws.

[0316] (1) Setting the service stress range of the component and performing traversal calculations: First, set the service stress range of the metal component, i.e., 550MPa-1200MPa. Then, use a computer or numerical analysis software to perform traversal calculations on the preliminary SN curve fitting equations obtained in Table 10 for each survival rate;

[0317] (2) The verification results show that, within the entire set engineering stress range, for any survival rate All of them strictly meet their corresponding predicted fatigue life. .

[0318] (3) Since the survival rate curves do not intersect in any stress interval, the predicted lifetime corresponding to the high survival rate is always less than that of the low survival rate, which is consistent with the physical monotonicity law of high-cycle fatigue failure of metallic materials. Therefore, the monotonicity verification is passed, confirming that no law has been violated, and there is no need to use the equal slope constraint method to refit and correct the intercept parameter. The formulas obtained in step 5.3 are the final SN curve equations.

[0319] Step 5.5: Drawing curve

[0320] Based on the fitting formulas for each survival rate obtained in step 5.3, the logarithm of fatigue life is used. As the horizontal axis, maximum stress Using the vertical axis as the ordinate, the SN curves corresponding to each survival rate (50%, 84.1%, 90%, 95%, 99%, 99.9%, 99.99%) are plotted on the same coordinate system.

[0321] Finally obtained Figure 4 The PSN curve shown visually illustrates the stress-life relationship under different reliability requirements. From... Figure 4 As can be seen, under the same stress level, the higher the survival rate requirement (e.g., 99.99%), the shorter the allowable fatigue life; conversely, under the same predetermined life, the higher the survival rate requirement, the lower the allowable stress that the component can withstand.

[0322] Thus, this embodiment, through standardized processing of experimental data, iterative removal of outliers from small samples using the dynamic threshold Chauvinistic criterion, and selection of the optimal distribution model through single-sample KS test combined with residual sum of squares quantification, and by introducing monotonicity verification and self-consistency correction mechanisms in the curve fitting stage, finally completes the adaptive distribution test and PSN curve solution method for fatigue test data with small samples. The results verify the engineering feasibility and process integrity of the method in processing fatigue test data with small samples, and can provide reasonable data reference for strength verification in engineering structural design.

Claims

1. A method for solving PSN curves for small sample fatigue test data, characterized in that, Including the following steps: Step 1: Obtain experimental data; Fatigue life test data were obtained from specimens of the same batch with the same structure, size, material and process, including the test stress amplitude and the corresponding fatigue life value. Step 2, data normalization processing; After arranging the test stress amplitudes in sequence, they are defined as different stress levels, and the fatigue life values ​​under each stress level are arranged in sequence. Step 3, outlier removal; For the sample set of fatigue life values ​​under each stress level, outliers are iteratively removed based on the dynamic threshold Schauvinay criterion. If the current iteration number is greater than 1, then after removing outliers in the current round, a resampling judgment is made based on the rate of change of the subsample standard deviation of the sample set before and after removing outliers in the current round. If the rate of change of the standard deviation is greater than the preset safety threshold, then fatigue life data under the current stress level needs to be supplemented and merged with the fatigue life data under the stress level obtained in step 1, and then return to step 2. Step 4: Determine the distribution type; For the effective fatigue life values ​​after removing outliers under each stress level, the single-sample KS test is used to preliminarily determine the assumed distribution of the effective fatigue life values ​​under each stress level. The assumed distribution is at least one of the normal distribution, log-normal distribution and Weibull distribution. Secondary determination of distribution type: Frequency statistics are performed from a global perspective, and the distribution type with the highest frequency of fatigue life data of all stress levels is taken as the unified distribution type of the overall test data; if the frequency of fatigue life data of each stress level following multiple assumed distributions is consistent, the assumed distribution with the smallest sum of squared residuals is selected as the unified distribution type of the overall test data. The formula for calculating the sum of squared residuals is: In the formula, For a certain stress level The total number of valid fatigue life data after removing outliers; Let be the empirical cumulative distribution function value of the p-th effective fatigue life data; This is the theoretical cumulative distribution function value assumed to be distributed at the p-th effective fatigue life data point; Step 5: Solve for the PSN curve; Based on the uniform distribution type determined in step 4, different survival rates P were selected, and the stress-life model was used to solve the preliminary PSN curve equation. Verify the monotonicity of the initial PSN curve. If it fails, it indicates that the curves intersect. Then, use the equal slope constraint method to refit and obtain the corrected PSN curve equation. The corrected PSN curve equation is the final PSN curve obtained by solving. If it passes, the initial PSN curve equation is the final PSN curve obtained by solving.

2. The method for solving PSN curves for small sample fatigue test data according to claim 1, characterized in that, The method for outlier removal in step 3 for fatigue life data under a single stress level is as follows: Step 3.1, Algorithm initialization; Set the number of iterations The initial sample set is defined as the set of fatigue life data under a certain stress level. Its initial sample size is ; Step 3.2, Calculation of statistics and threshold values; In the In the next iteration, based on the current sample set Calculate the sample mean Sample Standard Deviation ; Based on the current sample set sample size The threshold value of the Chauvinistic criterion is obtained by querying the threshold value test table or by performing linear interpolation based on the data in the Chauvinistic criterion threshold value test table. The dynamic threshold value corresponding to the next iteration ; Step 3.3, extreme deviation test; Find the current sample set Mean deviation of sample Largest data point Calculate its value compared to the sample average. The absolute value of the deviation, if the absolute value of the deviation is greater than the dynamic threshold value Then determine the data point If it is an outlier, remove it from the current sample set. If the sample is removed from the current set, proceed to step 3.4; otherwise, it indicates that the current sample set is not in its final form. If no outliers are found, the iteration terminates and the final valid dataset is output. Step 3.4, Dynamic Update and Resampling Determination; Step 3.4.1: Determine if the current iteration count is greater than 1. If yes, proceed to step 3.4.2; otherwise, proceed to step 3.4.

3. Step 3.4.2, Resampling determination; Calculate the rate of change of the subsample standard deviation of the sample set before and after the removal of outliers in the current round, and compare it with the set safety threshold. If the safety threshold is exceeded, the algorithm stops iterating. It is necessary to supplement the fatigue life data under the current stress level, merge the supplemented fatigue life data with the original fatigue life data under the current stress level to form a new sample set, and return to the normalization process in step 2. If the safety threshold is not exceeded, proceed to step 3.4.3; Step 3.4.3, Dynamic Update; renew Update the sample set after removing outliers as follows: Update the subsample size to Then, return to step 3.2 to proceed to the next round of elimination iteration.

3. The method for solving PSN curves for small sample fatigue test data according to claim 1 or 2, characterized in that, The safety threshold mentioned in step 3 is 50%-70%.

4. The method for solving PSN curves for small sample fatigue test data according to claim 3, characterized in that, Step 4 uses the single-sample KS test to preliminarily determine the assumed distribution of the effective fatigue life value under a single stress level. Step 4.1: For the effective fatigue life data under a certain stress level after outlier removal, propose initial distribution assumptions that it follows a normal distribution, a log-normal distribution, and a Weibull distribution, respectively, and construct parallel three types of known distribution models in this way; Step 4.2: Determine the empirical cumulative distribution functions of the effective fatigue life data under this stress level, which are subject to three initial distribution assumptions, and calculate their function values. Step 4.3: Determine the theoretical cumulative distribution function of the effective fatigue life data under this stress level, which follows three initial distribution assumptions, and calculate its function value; Step 4.4: Calculate the KS test statistics for the three initial distribution hypotheses corresponding to this stress level. , and ; Step 4.5, determine the critical value D for the KS test. α ; Step 4.6, based on the KS test statistic , and and critical value D α To determine the assumed distribution of the effective fatigue life data under this stress level.

5. The method for solving PSN curves for small sample fatigue test data according to claim 4, characterized in that, The survival rates P selected in step 5 were 50%, 84.1%, 90%, 95%, 99%, 99.9%, and 99.99%.

6. The method for solving PSN curves for small sample fatigue test data according to claim 5, characterized in that, The stress-life model used in step 5 is the Basquin formula.

7. The method for solving PSN curves for small sample fatigue test data according to claim 6, characterized in that, The method for refitting using the equal slope constraint in step 5 is as follows: The logarithmic linear equations of each survival rate P Modified to an equal slope constraint model ; For survival rate The slope of the logarithmic linear equation at time ; Substitute the fatigue life data under each survival rate P into the modified isoclinic constraint model. In this paper, the intercept parameter is solved using the least squares method. ; Based on the intercept parameter Combined with slope Reconstruct the PSN curve equation.

8. The method for solving PSN curves for small sample fatigue test data according to claim 7, characterized in that, In step 5, the least squares method is used to solve for the intercept parameter. The process is as follows: Suppose there are q stress levels in the experiment, and let the logarithm of the i-th stress level be... The logarithmic value of fatigue life corresponding to a specific survival rate P is: ,in ; In the model with equal slope constraint Next, construct the objective function Q for the sum of squared residuals: Let Q be the parameter to be calculated. The first-order partial derivative is equal to zero: Expand and simplify the above equation: This allows us to obtain a result specific to the intercept. The analytical solution expression: In the formula, This is the arithmetic mean of the logarithmic values ​​of fatigue life corresponding to all experimental stress levels under this specific survival rate P; This is the arithmetic mean of the logarithmic values ​​of the stress levels in each test. Substitute the known data into the intercept. The corrected intercept parameter can be obtained by solving the analytical expression. .

9. The method for solving PSN curves for small sample fatigue test data according to claim 8, characterized in that, Step 5 also includes the step of plotting the PSN curve based on the final PSN curve obtained from the solution.

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