Acceleration stress lower limit-considered composite material load-bearing structure storage life evaluation method

Abstract

The invention provides a composite material bearing structure storage life evaluation method considering an acceleration stress lower limit, and the method comprises the steps: 1, carrying out the initialization modeling: enabling the service life of a product to obey the logarithmic normal distribution, and building a three-parameter inverse power law acceleration model of the service life feature theta and the stress level S; 2, estimating parameters: estimating unknown parameters by applying a maximum likelihood estimation method for the three-parameter inverse power law acceleration model; 3, model selection: comparing the calculated likelihood function values, wherein the larger the likelihood function value is, the more stable and more accurate the model is; meanwhile, the variation coefficients of the acceleration factors are calculated in combination with test data for comparison, and the model with the small variation coefficient is more accurate; and 4, accelerated life evaluation: giving reliability, and evaluating the storage life of the three-parameter inverse power law model. The method is suitable for accelerated test evaluation of a product with a threshold effect; the three-parameter inverse power law model provided by the invention can show a better model fitting effect when the acceleration effect does not meet the linear characteristic.

Description

Technical Field

[0001] The present invention provides a method for evaluating the storage life of a composite load-bearing structure considering the lower limit of the acceleration stress, which relates to a method for evaluating the accelerated storage life of a composite load-bearing structure based on a three-parameter inverse power law acceleration model; aiming at the accelerated storage test data of the composite load-bearing structure, by introducing a location parameter to characterize the lower limit of the acceleration stress, a three-parameter inverse power law acceleration model is established. Taking the lognormal distribution as an example, model parameter estimation and accelerated life evaluation are carried out, which is applicable to the field of accelerated storage life evaluation of composite load-bearing structures. Background Art

[0002] For high-reliability and long-life products, the accelerated life test and evaluation method is an effective means to evaluate their life. According to the failure mechanism and sensitive stress of the product, selecting a matching acceleration model is an important step in accelerated life evaluation. The inverse power law acceleration model is applicable to the stress model failure mechanism occasions, so it is often widely used in the accelerated life test design and data evaluation of mechanical products and electrical products. Physically, many experimental data prove that under the action of mechanical stress, the relationship between the life characteristics of the product and the stress level generally conforms to the inverse power law model.

[0003] The traditional two-parameter inverse power law model is widely used in the accelerated life evaluation of products. However, its limitations have gradually emerged in engineering practice:

[0004] Threshold effect under complex failure mechanisms: In actual application scenarios, the failure mechanism of the product affected by environmental stress may be very complex, and there may be a threshold effect, that is, when exceeding a certain threshold, the corresponding environmental stress will produce an acceleration effect. In this case, the traditional two-parameter inverse power law model is not applicable;

[0005] Insufficient model prediction accuracy under a wide acceleration stress range: If the acceleration stress range is too large, when the acceleration effect generated by the environmental stress shows a non-linear characteristic and the number of accelerated samples is small, the acceleration model fitted based on the traditional two-parameter inverse power law model may lead to large prediction errors;

[0006] Therefore, the present invention aims at products subject to lognormal life distribution, and by introducing a location parameter to characterize the lower limit of the acceleration stress, a method for evaluating the storage life of a composite load-bearing structure considering the lower limit of the acceleration stress is proposed, which can effectively improve the evaluation accuracy. Summary of the Invention

[0007] The object of the present invention is to provide a method for evaluating the storage life of a composite load-bearing structure considering the lower limit of the acceleration stress for products subject to a lognormal life distribution. By introducing a location parameter to characterize the lower limit of the acceleration stress, a three-parameter inverse power law acceleration model is established, and a parameter estimation method for the acceleration model and the life model is provided to evaluate the reliable life of the product under given storage conditions.

[0008] To this end, the present invention needs to establish the following basic settings:

[0009] Setting 1 The data type is all accelerated life type censored data, and the acceleration stress of the product is mechanical stress. There are k stress levels, namely S1, S2, …, S k , and at the stress level S i there are r i failure time data, that is the censoring time is τ i ;

[0010] It is assumed that the relationship between the life characteristic θ and the stress level S basically conforms to the inverse power law, and follows a three-parameter inverse power law acceleration model, that is, the life characteristic parameter θ and the stress S satisfy:

[0011]

[0012] where: A and B are constants; S is the mechanical stress; S0 is the initial mechanical stress.

[0013] Setting 2 The product life follows a lognormal distribution, and its probability density function at the stress level S is:

[0014]

[0015] where: t is the product life; σ(S) is the standard deviation of the lognormal distribution; μ(S) is the mean of the lognormal distribution.

[0016] Its life distribution function at the stress level S is:

[0017]

[0018] Based on the above basic settings, the present invention proposes a method for evaluating the storage life of a composite load-bearing structure considering the lower limit of the acceleration stress, which mainly includes initialization modeling, parameter estimation, model selection, and accelerated life evaluation; the steps are as follows:

[0019] Step 1: Initialization modeling

[0020] First, for the k stress levels S1, S2, …, S k , since the product life follows a lognormal distribution, a three-parameter inverse power law acceleration model of its life characteristic θ and the stress level S can be established.

[0021] Taking the logarithm of both sides of Equation (1), a linearized three-parameter inverse power law model is obtained, that is, the characteristic life θ of the product i and the applied stress level S i satisfy the following acceleration equation:

[0022]

[0023] where: a, b, S0 are the parameters to be estimated in the acceleration model, is a known function, and here

[0024] Applying the lognormal distribution as the life distribution, the mechanism consistency condition under the lognormal distribution is: the logarithmic standard deviation σ(S) is a constant independent of the stress level S, that is, σ(S) = σ. Selecting the median life t 0.5 as a parameter of the acceleration equation, for the three-parameter inverse power law model, there is:

[0025]

[0026] where: F -1 is the inverse function of the lognormal life distribution function, and μ(S i ) is the mean of the lognormal distribution at the stress level S i .

[0027] Step 2: Parameter estimation

[0028] For the three-parameter inverse power law acceleration model, the maximum likelihood estimation method is used to estimate the unknown parameters a, b, and S0

[0029] If it is complete data, its log-likelihood function is:

[0030]

[0031] where: n i is the sample size corresponding to the stress level S i , t ij is the life of the jth product at the stress level S i , i = 0, 1,..., k, j = 0, 1,..., n i , μ i is the mean of the lognormal distribution at the stress level S i , and σ is the standard deviation of the lognormal distribution to be estimated

[0032] For the three-parameter inverse power law model, there is

[0033] μ i = a + bln (S i - S0) (7)

[0034] If it is time-truncated data, its likelihood function is as follows:

[0035]

[0036]

[0037] where: n i is the sample size corresponding to the stress level S i and r i is the number of failures at the stress level S i and τ i is the truncation time, Φ(·) is the failure distribution function of the standard normal distribution, and (·)! is the factorial of a natural number.

[0038] The model parameter values and the log-likelihood function values can be obtained by maximizing the log-likelihood function:

[0039]

[0040] where: are the estimated values of the unknown model parameters a, b, σ, S0, and is the corresponding array of parameter estimated values.

[0041] Step 3: Model selection

[0042] To determine the model with the best fitting effect under different parameter estimation methods, the calculated likelihood function values can be compared. Generally, it is considered that the model with a larger likelihood function value is more stable and accurate.

[0043] At the same time, the coefficient of variation of the acceleration factor can also be calculated and compared in combination with the experimental data. The model with a smaller coefficient of variation is more accurate. The coefficient of variation of the acceleration factor can be calculated through the following steps. First, select the acceleration factor K M of the maximum stress S 1,M and the minimum stress S1, and its expression is:

[0044]

[0045] where: θ1 is the life characteristic parameter under the minimum stress S1, and θ M is the life characteristic parameter under the minimum stress S M and b, S0 are the unknown model parameters.

[0046] The calculation formula for the coefficient of variation is:

[0047]

[0048] where: σ K is the standard deviation, and μ Kis the average value. Here, the mean can be directly calculated by the formula, that is

[0049]

[0050] Standard deviation σ K can be calculated by the Fisher information matrix. When performing data statistical analysis, if the coefficient of variation is greater than 15%, it is necessary to consider that the data may be abnormal and should be excluded.

[0051] Step Four: Accelerated life assessment

[0052] Given the reliability R, for the three-parameter inverse power law model, its storage life can be evaluated as:

[0053]

[0054] where S N is the normal stress level.

[0055] where the "Fisher information matrix" mentioned in Step Three refers to the second-order partial derivative of the negative log-likelihood, that is

[0056]

[0057] where: I is the Fisher information matrix, and E(·) is the expectation of the second derivative at the true value of the parameter.

[0058] Inverting the above Fisher information matrix can obtain the covariance matrix of the parameter vector, that is

[0059]

[0060] where: are the estimated values of the unknown parameters a, b, σ, S0 in Step Two, and Var(·) is the variance at the true value of the parameter.

[0061] The variance of the coefficient of variation σ AF can be obtained through the following formula:

[0062]

[0063] where Ω i represents the i-th term of Ω.

[0064] The advantages and beneficial effects of the present invention are as follows:

[0065] ① The present invention is applicable to the accelerated test evaluation of products with threshold effects;

[0066] ② Compared with the traditional two-parameter inverse power law acceleration model, the three-parameter inverse power law model proposed by the present invention can show a better model fitting effect when the acceleration effect does not satisfy the linear characteristic. Description of the Drawings

[0067] Figure 1 Flow chart of the method according to the present invention Detailed Description of the Invention

[0068] The present invention will be further described in detail below with reference to examples

[0069] A time-truncated life test was carried out on a composite fiber cylinder. Its ultimate failure stress is 2067.6 MPa, and the bearing stress during operation is about 913.8 MPa, which is about 45.5% of the ultimate failure stress. The accelerated stress levels are set as: 90%, 85%, 80%, 72.7%, 63.6%. A total of 39 samples were put into this test. The number of samples and test data at each stress level are shown in Table 1

[0070] Table 1 Time-truncated life data of a composite fiber cylinder (unit: hour)

[0071]

[0072] Among them, "*" indicates a censored non-failed sample

[0073] Step 1: First, for the 5 stress levels S1, S2, …, S5, select the median life t 0.5 as the parameter of the acceleration equation, and establish its three-parameter inverse power law acceleration model, that is

[0074]

[0075] Step 2: Use the maximum likelihood estimation method to estimate the unknown parameters of the model. For the time-truncated data in Table 1, its log-likelihood function is

[0076]

[0077] To highlight the accuracy of the three-parameter inverse power law acceleration model, a two-parameter inverse power law model can be used for comparison. For the two-parameter inverse power law model, there is

[0078] μ i = a + bln(S i ) (20)

[0079] where: a, b are the parameters to be estimated in the acceleration model

[0080] By maximizing the log-likelihood function, the parameter estimation values and log-likelihood function values of the two-parameter inverse power law model can be obtained

[0081]

[0082] lnL = -183.2737

[0083] If the three-parameter inverse power law model is adopted, there is

[0084] μ i = a + bln(S i - S0) (21)

[0085] where: a, b, and S0 are the parameters to be estimated in the acceleration model.

[0086] By maximizing the log-likelihood function, the estimated values of the model parameters and the log-likelihood function value can be obtained:

[0087]

[0088] lnL = -183.2513

[0089] Step 3: According to the log-likelihood function value lnL, calculate the likelihood function values L under the two-parameter inverse power law model and the three-parameter inverse power law model respectively, that is

[0090] Table 2 Likelihood function values under two acceleration models

[0091]

[0092] It is generally considered that the model with a larger likelihood function value is more stable and accurate. According to the comparison of the likelihood function values of the models, the three-parameter inverse power law model has higher accuracy, and the three-parameter inverse power law model should be selected for the life assessment of the composite fiber cylinder.

[0093] Step 4: When the reliability is given as 0.8, for the three-parameter inverse power law model, its storage life is:

[0094]

[0095] When the reliability is given as 0.9891, its storage life is:

[0096]

Claims

1. A storage life assessment method for composite material load-bearing structures considering the lower limit of accelerated stress requires the following settings: Setting 1: The data types are all accelerated life truncated data, the accelerated stress of the product is mechanical stress, and there are k stress levels, namely S1, S2, …, S k , stress level S i There are a total of r i The failure time data is The censoring time is τ i ; Assume that the relationship between the life characteristic θ and the stress level S basically conforms to the inverse power law and obeys the three-parameter inverse power law acceleration model, that is, the life characteristic parameter θ and the stress S satisfy: Where: A, B are constants; S is the mechanical stress; S0 is the initial mechanical stress; Setting 2: The product life follows a log-normal distribution, and its probability density function at stress level S is: Where: t is the product life; σ(S) is the standard deviation of the lognormal distribution; μ(S) is the mean of the lognormal distribution; Its life distribution function under stress level S is: Based on the above configuration, it is characterized in that: The steps include: Step 1: Initialize modeling For k stress levels S1, S2, ..., S k , the product life follows a log-normal distribution, and a three-parameter inverse power-law acceleration model of its life characteristic θ and stress level S is established; Step 2: Parameter estimation For the three-parameter inverse power law acceleration model, the maximum likelihood estimation method is used to estimate the unknown parameters a, b, and S0; Step 3: Model selection In order to determine the model with the best fitting effect under different parameter estimation methods, the calculated likelihood function values ​​are compared. The model with a larger likelihood function value is more stable and accurate. At the same time, the coefficient of variation of the acceleration factor is calculated based on the test data for comparison. The model with a smaller coefficient of variation is more accurate. Step 4: Accelerated Life Assessment Given the reliability R, the storage life of the three-parameter inverse power law model is evaluated.

2. The method for evaluating the storage life of a composite material load-bearing structure considering the lower limit of accelerated stress according to claim 1, characterized in that: In step 1, the logarithm of both sides of equation (1) is taken to obtain a linearized three-parameter inverse power law model, that is, the characteristic life θ of the product i With the applied stress level S i The following acceleration equation is satisfied: Among them: a, b, S0 are the parameters to be estimated of the acceleration model, is a known function, here 3. A method for evaluating the storage life of a composite material load-bearing structure taking into account the lower limit of accelerated stress according to claim 1 or 2, characterized in that: The lognormal distribution is used as the life distribution. The mechanism consistency condition under the lognormal distribution is: the logarithmic standard deviation σ(S) is a constant that does not vary with the stress level S, that is, σ(S) = σ; the median life t 0.5 As the parameters of the acceleration equation, for the three-parameter inverse power law model, we have: Among them: F -1 is the inverse function of the log-normal life distribution function, μ(S i ) is at stress level S i The mean of the lower lognormal distribution.

4. The method for evaluating the storage life of a composite material load-bearing structure considering the lower limit of accelerated stress according to claim 1, characterized in that: In step 2, if the data is complete, the log-likelihood function is: Where: n i is the stress level S i The corresponding sample size, t ij is the stress level S i The life of the next j-th product, i = 0, 1, ..., k, j = 0, 1, ..., n i , μ i At stress level S i is the mean of the lognormal distribution, and σ is the standard deviation of the lognormal distribution to be estimated.

5. The method for evaluating the storage life of a composite material load-bearing structure considering the lower limit of accelerated stress according to claim 4, characterized in that: For the three-parameter inverse power law model, we have m i =a+bln (S i -S0) (7) If the data is time-censored, the likelihood function is: Where: n i is the stress level S i The corresponding sample size is r i is the stress level S i The number of failures under i is the censoring time, Φ(·) is the failure distribution function of the standard normal distribution, and (·)! is the factorial of a natural number.

6. The method for evaluating the storage life of a composite material load-bearing structure considering the lower limit of accelerated stress according to claim 4, characterized in that: The model parameter values ​​and log-likelihood function values ​​are obtained by maximizing the log-likelihood function: in: are the estimated values ​​of the unknown parameters a, b, σ, and S0 of the model, is the corresponding array of parameter estimates.

7. The method for evaluating the storage life of a composite material load-bearing structure considering the lower limit of accelerated stress according to claim 1, characterized in that: In step 3, select the maximum stress S M and the acceleration factor K of the minimum stress S1 1,M , whose expression is: Where: θ1 is the life characteristic parameter under the minimum stress S1, θ M is the minimum stress S m The life characteristic parameters under this condition, b, S0 are unknown parameters of the model.

8. The method for evaluating the storage life of a composite material load-bearing structure considering the lower limit of accelerated stress according to claim 7, characterized in that: The formula for calculating the coefficient of variation is: Where: K is the standard deviation, μ K is the average value; here, the mean is calculated directly by the formula, that is, Standard deviation K Calculated by Fisher information matrix; when performing statistical analysis of data, if the coefficient of variation is greater than 15%, the data should be considered abnormal and should be eliminated.

9. The method for evaluating the storage life of a composite material load-bearing structure considering the lower limit of accelerated stress according to claim 1, characterized in that: In step 3, for the three-parameter inverse power law model, the storage life is estimated as: Among them, S N is the normal stress level; Among them, the Fisher information matrix described in step 3 refers to the second-order partial derivative of the negative log-likelihood, that is, Where: I is the Fisher information matrix, E(·) is the expectation of the second-order derivative at the true value of the parameter.

10. The method for evaluating the storage life of a composite material load-bearing structure considering the lower limit of accelerated stress according to claim 9, characterized in that: Inverting the Fisher information matrix obtains the covariance matrix of the parameter vector, that is, in: is the estimated value of the unknown parameters a, b, σ, S0 of the model in step 2, Var(·) is the variance at the true value of the parameter; Variance of coefficient of variation σ AF That is, it is obtained by the following formula: Among them, Ω i represents the i-th term of Ω.

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