CN106126856A - It is applicable to the crack growth rate Forecasting Methodology based on low cycle fatigue property parameter of negative stress ratio - Google Patents

Abstract

The invention discloses a kind of crack growth rate Forecasting Methodology based on low cycle fatigue property parameter being applicable to negative stress ratio, low cycle fatigue test data are used to simulate low-cycle fatigue parameter, stress ratio threshold values different from material substitute into model formation, measurable crack growth rate jointly.The present invention overcomes existing based on fatigue crack growth rate test, the crack growth rate Forecasting Methodology of threshold value test, expensive, to expend time length shortcoming；In addition the result of what is more important this method prediction is more consistent with experimental data, accurately；Even be difficult to effectively, the situation of the negative stress ratio of Accurate Prediction, the prediction effect of this model is the most accurate.Fatigue crack growth rate prediction, predicting residual useful life that present invention Important Project crucial for Aero-Space, nuclear power, pipeline, high-speed railway etc. is widely present have great importance.

Description

Crack Growth Rate Prediction Based on Low Cycle Fatigue Performance Parameters for Negative Stress Ratio test method

technical field

The invention relates to the prediction of fatigue crack growth speed in fracture mechanics, especially the prediction of fatigue crack growth speed and fatigue crack life prediction of advanced and important engineering structures and components in service.

Background technique

Most engineering fractures are caused by fatigue, so the low-cycle fatigue and crack growth rate performance of metal materials has always been concerned by the safety design department. For a long time, scholars at home and abroad have conducted multi-material and multi-angle research on the establishment of the relationship between the low-cycle fatigue behavior of metal materials and the crack growth rate performance. Based on the small-scale yield stress-strain field at the tip of the plane stress crack and the failure criterion of fatigue crack growth, a prediction model for mode I fatigue crack growth rate is proposed.

Fatigue has always been one of the main causes of engineering failure, and has always attracted the attention of safety design departments ^[1] . Fatigue damage of metal materials has to go through three stages: crack initiation, crack stable growth and crack instability growth, so fatigue analysis should study both crack initiation and crack stable growth ^[2] . Considering that fatigue crack growth is a process of local plastic damage accumulation, the mechanism made of materials will suffer fatigue failure under reciprocating load ^[3] . According to linear elastic fracture mechanics, the stress field near the crack tip in fatigue failure analysis is controlled by the stress intensity factor K, so the crack propagation behavior under fatigue load can be qualitatively and quantitatively described by the stress intensity factor. Based on the low-cycle fatigue behavior of materials to predict the growth behavior of type I fatigue cracks, a variety of models have been proposed abroad. From the perspective of research objects, the models are divided into crack growth models described by material microscopic parameters ^[4-5] and macroscopic parameters characterized by materials. Crack growth models, such as GLINKA et al. ^[6-7] based on the assumption of constant strain in the crack tip growth zone, KUJAWSKI et al. ^[8-12] introduced passivation assumptions at the crack tip: research on various theoretical models for predicting fatigue crack growth, etc. ^{[13 -16]} based on the crack tip plastic strain energy assumption and CHEN et al. ^[17] based on the crack tip average plastic damage. The crack growth model described by the material microscopic parameters is combined with the crack growth rate test and the material microscopic parameters comparison and analysis to obtain the crack growth zone size of the material, and then the crack growth rate. It is worth noting that the microscopic parameters obtained by comparing the crack growth rate test and the material microscopic parameter model can only describe the microstructure and size of the material, but cannot be easily applied to the fatigue crack growth rate analysis of other materials. However, since the crack growth model based on material macroscopic parameter characterization does not involve the material microscopic size or its microscopic parameters are evaluated by macroscopic mechanical parameters, the crack growth model based on material macroscopic parameter characterization reflects the average rate of crack growth, and the model is convenient for application. In plane stress analysis, there are two stress-strain fields at the tip of a mode I crack under cyclic loading: the Rice-Kujawski-Ellyin (RKE) field and the Hutchinson-Rice-Rosengren (HRR) field. Relevant empirical results show that the stress-strain field at the crack tip based on RKE theory is closer to the real situation, so this patent model selects the RKE field based on the crack tip and the strain energy failure criterion in the cyclic plastic zone of the crack tip, and compares the plastic cumulative energy With the linear damage accumulation criterion, a plastic accumulation energy-based process model is adopted, and a fatigue crack velocity prediction model for predicting the mode I crack tip under plane stress is proposed.

According to the size of the plastic zone at the crack tip that exists at the crack front, we can divide the problems of fracture mechanics into linear elastic fracture mechanics and elastoplastic fracture mechanics. The propagation and fracture judgment of linear elastic cracks are mainly based on the stress intensity factor K; the propagation and fracture judgment of elastic-plastic cracks are mainly based on J integral. Since the research on linear elastic fracture mechanics is relatively mature, and linear elastic cracks actually account for a large proportion in engineering, the calculation of the stress intensity factor K of various cracks and the study of the fracture toughness K _C of the material itself are the main tasks in fracture mechanics. content.

In general, crack growth can be divided into three stages: low-speed growth, medium-speed growth, and high-speed growth, which correspond to the three regions on the crack growth rate curve, as shown in Figure 1.

The crack growth rate formula is usually used to calculate and fit the crack growth rate curve. According to the factors considered from simple to complex, the commonly used ones are Walker formula, Forman formula, etc. These formulas have their own characteristics and shortcomings.

Paris formula:

da/dN=C(ΔK) ^m (1)

Among them, m and C are constants related to material properties. Equation (1) is simple in form, few parameters and easy to use, but the Paris formula has serious limitations. The formula does not involve crack growth threshold ΔK _th , fracture toughness K _C , and stress ratio R, the three stages of crack initiation in crack growth , Stable crack growth, accelerated crack growth, formula (1) is only applicable to the second stage of crack growth (stable growth stage), without considering the other two stages, but we know that the crack life is mainly concentrated in the crack initiation stage, and Paris The formula does not take this factor into account, which makes the result error larger.

Forman formula:

where m and C are material constants. Equation (2) can be used for variable amplitude loading under various stress ratios R, but still does not consider the influence of the threshold value of the stress intensity factor ΔK _th .

Walker formula:

da/dN＝C{(1-R) ^(n-1) (ΔK)} ^m (3)

where m, n and C are material constants. Equation (3) takes into account the influence of R, but does not consider the influence of the stress intensity factor threshold value ΔK _th and the fracture toughness K _C. This model is also only applicable to the stable growth stage of fatigue cracks.

NASGRO formula:

The NASGRO formula includes the complete life interval of crack initiation, propagation and instantaneous break, while considering the influence of different stress ratios (-2<R<0.7), namely:

In the formula: K _max is the maximum stress intensity factor in the loading cycle, and p and q are empirical constants.

ΔK _th is not a constant value, but is related to the crack morphology and the stress state of the crack tip. Its calculation formula is as follows:

In the formula: α ₀ is E1-Haddad constant (=0.0381), and C _th is the constant (=0.1) that is used to distinguish C _th+ and C _th- , is respectively used in the situation of +R and-R, simultaneously R is neither constant -1.

In order to consider the effect of plastic crack closure, the equivalent stress intensity factor amplitude ΔK _eff =ΔK·(1-f)/(1-R) is introduced, where:

In the formula: f is called the crack opening function (=σ _op /σ _max ), the constraint factor α=2.5 is introduced, and other unknown constants are calculated as follows:

It can be seen that although the nasgro formula has considered various factors, it takes a long time to obtain experimental data based on the material crack growth experiment, and the disadvantages of high cost are very significant; in addition, many fitting parameters such as C, m, p, q, and C _th need to be calculated, which is It has a huge impact on the calculation accuracy, is time-consuming and laborious, and is inconvenient to use.

Shi-Cai model ^[18] :

By considering the cyclic stress-strain relationship at the crack tip and the Manson-Coffin strain-life relationship, the following crack growth rate prediction formula can be obtained:

The model predicts the crack growth rate based on the low-cycle fatigue parameters of the material, and the obtained results fit well in the I and II areas of the growth rate curve, but it does not take into account the crack closure effect and cannot be applied to negative stress ratios. Load conditions become a major bottleneck limiting its application.

references:

[1] Du Yannan, Zhu Mingliang, Xuan Fuzhen. Influence of Stress Ratio on Turning Point of Fatigue Crack Growth Curve in Near Threshold Area of Welded Joint of 25Cr2Ni2MoV Steel [J]. Chinese Journal of Mechanical Engineering, 2015, 51(8): 44-49.

[2] Liu Chunxiao, Wang Haiyan. A review of research on fatigue crack growth [J]. Journal of Jiaozuo University, 2008, 2(4): 91-93.

[3] Gao Wenzhu, Wu Huan, Zhao Yongqing. A review of metal materials fatigue crack growth research [J]. Titanium Industry Progress, 2007,24(6):33-37.

[4] Antolovich, S.D., Saxena, A., & Chanani, G.R. (1975). A model for fatigue crack propagation. Engineering fracture mechanics, 7(4), 649-652.

[6] Glinka, G.(1982). A cumulative model of fatigue crackgrowth. International journal of fatigue, 4(2), 59-67.

[7] Glinka, G., Robin, C., Pluvinage, G., & Chehimi, C. (1984). A cumulative model of fatigue crack growth and the crack closure effect. International journal of fatigue, 6(1), 37-47 .

[8] Kujawski, D., & Ellyin, F. (1984). A fatigue crack propagation model. Engineering Fracture Mechanics, 20 (5-6), 695-704.

[9]Ellyin,F.(1986).Crack growth rate under cyclic loading and effect of different singularity fields.Engineering fracture mechanics,25(4),463-473.

[10]Ellyin,F.(1986).Stochastic modeling of crack growth based ondamage accumulation.Theoretical and applied fracture mechanics,6(2),95-101.

[11] Wu, S.X., Mai, Y.W., & Cotterell, B.(1992). A model of fatigue crackgrowth based on Dugdale model and damage accumulation. International journal of fracture, 57(3), 253-267.

[12] Li, D.M., Nam, W.J., & Lee, C.S. (1998). An improvement on prediction offatigue crack growth from low cycle fatigue properties. Engineering fracture mechanics, 60(4), 397-406.

[13] Skelton, R.P., Vilhelmsen, T., & Webster, G.A. (1998). Energy criteria and cumulative damage during fatigue crack growth. International journal offatigue, 20(9), 641-649.

[14]Pandey,K.N.,&Chand,S.(2003).An energy based fatigue crack growth model.International journal of fatigue,25(8),771-778.

[15]Pandey,K.N.,&Chand,S.(2004).Fatigue crack growth model for constant amplitude loading.Fatigue&Fracture of Engineering Materials&Structures,27(6),459-472.

[16] Noroozi, A.H., Glinka, G., & Lambert, S. (2005). A two parameter drivingforce for fatigue crack growth analysis. International Journal of Fatigue, 27(10), 1277-1296.

[17]Chen,L.,Cai,L.,&Yao,D.(2013).A new method to predict fatiguecrack growth rate of materials based on average cyclic plasticity straindamage accumulation.Chinese Journal of Aeronautics,26(1),130 -135.

[18]Shi K K,Cai L X,Chen L,et al.Prediction of fatigue crack growth based on low cycle fatigue properties[J].International Journal of Fatigue,2014,61(61):220-225.

Contents of the invention

In view of the above deficiencies in the prior art, the purpose of the present invention is to provide a method for predicting crack growth rate based on low cycle fatigue performance parameters suitable for negative stress ratios, so as to overcome the deficiencies in the prior art.

The purpose of the invention of the present invention is achieved by the following means.

A crack growth rate prediction method based on low-cycle fatigue performance parameters suitable for negative stress ratios. The stress and strain data and the cycle life of the corresponding samples are obtained through the low-cycle fatigue test of the material, and the low-cycle fatigue performance parameters are fitted. The calculation of the expansion speed, the brief process includes:

(1) The tested sample is subjected to low-cycle fatigue test according to the general test method, and seven low-cycle fatigue performance parameters can be obtained: cyclic strength coefficient K', cyclic strain hardening exponent n', fatigue strength coefficient _σ'f , fatigue strength index b, fatigue ductility coefficient ε' _f , fatigue ductility index c, cyclic yield strength σ _yc ; use the above parameters to substitute into the formula

The relationship between the low-cycle cycle life ΔN of the tested sample and the crack tip stress intensity factor ΔK can be obtained; where: σ _m is the average stress, which can be obtained by σ _m = (σ _min + σ _max )/2; ΔK _th is the crack growth threshold, which can be obtained by the following formula under different stress ratios

In the above formula, ΔK _th,0 is the crack growth threshold value when the stress ratio is 0, which is obtained through the crack growth threshold measurement test of the material; a is the crack length, a _H is the crack length increment, R is the stress ratio, ΔK _{th, R} is the crack tip stress intensity factor threshold value of the material when the stress ratio is R, which are all obtained from the data records during the test; C _th is the material constant used in the Pairs formula, α _H is the small crack correction constant, P is the material fitting constant, P is different for different materials, and the stress ratio P for the same material is the same, the three coefficients are empirical values; f is a closed function, which can be obtained by the following two formulas:

In the formula, Y is a geometric factor related to the shape of the sample, α is the constraint factor, and σ _max is the maximum stress of the sample during the low cycle fatigue test.

(2) Substitute the closed function f obtained in (1) into the following formula

Obtain the relationship between the equivalent stress intensity factor ΔK _eff and the stress intensity factor ΔK;

(3) Substituting the ΔN, ΔK _th , and ΔK _eff obtained in (1) and (2) into the following formula to calculate the crack growth rate da/dN:

Compared with prior art, the advantage of this method is obvious:

(1) This method can predict the negative stress ratio that cannot be predicted by other methods, and the results are in good agreement with the experimental data, especially for the load situation of the negative stress ratio of trains in complex working conditions during high-speed rail operation. Predict crack growth and give appropriate maintenance recommendations;

(2) The results predicted by this method are more consistent and accurate than the previously proposed model and experimental data;

(3) This method is based on the cheap and fast low-cycle fatigue test, which replaces the expensive and lengthy crack growth rate test, which greatly saves the time and expense of researchers.

The invention has important significance for the prediction of fatigue crack growth speed and crack life prediction widely existing in key major projects such as aerospace, nuclear power, pipelines, high-speed railways and the like.

Description of drawings:

Figure 1. A typical crack growth rate curve.

Fig. 2. Comparison of crack propagation results of alloy steel 25CrMo4 used for high-speed rail axles predicted by the method of the present invention with experimental data and predicted by Paris formula.

Fig. 3. Comparison of the results predicted by the method of the present invention with the experimental data and the paris formula prediction under the negative stress ratio of Ti-6Al-4V titanium alloy according to the embodiment of the present invention.

Fig. 4.E36 steel crack growth The comparison situation of the result predicted by the inventive method and the experimental data and the paris formula prediction.

Fig. 5. The comparison of the results predicted by the method of the present invention with the experimental data and the paris formula prediction of 8630 steel crack growth.

detailed description

The object of the present invention is to provide a method for predicting crack growth rate based on low cycle fatigue test, material stress intensity factor threshold value test, and low cycle fatigue performance parameters: obtain stress and strain data and corresponding sample data through low cycle fatigue test of material The cycle life N _{f is} used to fit the low-cycle fatigue performance parameters, and the threshold value of the stress intensity factor under the corresponding stress ratio of the material is obtained through the threshold value test, which is substituted into this model to calculate the crack growth rate; the detailed process includes:

(1) Material low-cycle fatigue test, the stress and strain data and the cycle life N _f of the corresponding sample are obtained in the following table:

Table 1 Raw data of low cycle fatigue test

(2) Using the low-cycle fatigue test data in Table 1, the least squares method is used for data fitting, and the low-cycle fatigue

Labor performance parameters:

The original data include the maximum stress σ _max , the minimum stress σ _min , the maximum strain ε _max , the minimum strain ε _min and the number of cycles N _f , and the corresponding parameters are obtained from the original data according to the following formulas:

Stress amplitude σ _a :

Total strain amplitude ε _a : In this process, since the original strain is given in the form of a percentage, it is necessary to divide the strain by 100 and substitute it into the formula;

Reverse times: 2N _f ;

Elastic strain range ε _a ^e and plastic strain range ε _a ^p :

The specific process uses σ _a /E to obtain the elastic strain range ε _a ^e , and then obtains the plastic strain range ε _a ^p from (ε _a -ε _a ^e ). The relevant data information obtained from the above formula is shown in Table 2:

Table 2 Summary of useful fatigue data

Fatigue strength coefficient σ _f , fatigue strength index b, fatigue toughness coefficient ε _f , fatigue toughness index c, cyclic strength coefficient K and cyclic strain hardening exponent n:

The specific process uses the least squares method to carry out linear regression, according to

Y=A+BX

Where: Y=lg(2N _f )

X=lg(ε _a ) or X=lg(ε _a ^e ) or X=lg(ε _a ^p )

(where A and B are the plastic strain-life stages, ε _f is the fatigue toughness coefficient, and c is the fatigue toughness index)

(where A and B are the elastic strain-life stages, σ _f is the fatigue strength coefficient, and b is the fatigue strength index)

(K is the cyclic strength coefficient, n is the cyclic strain hardening exponent, both of which are used to draw the subsequent hysteresis stress-strain curve)

The relevant parameters obtained from the above formula are shown in Table 3:

Table 3 Summary of fatigue parameters

(1) Through the material threshold value test, the threshold value of the stress intensity factor under the corresponding stress ratio of the material is obtained.

The test records the stress σ, stress ratio R, crack depth a and other data during the test, and the calculation method of the threshold value of the stress intensity factor is as follows:

The calculation method of fatigue crack growth threshold is as follows, according to the formula:

In the formula: a represents the calculated crack length of the sample, that is, crack length a = notch depth a ₀ + actual crack length a ₁ ;

ε=a/W, W represents the width before opening the notch of the sample;

Δσ = σ _max (1-R);

When 0.2<ε≦0.6, F(ε)＝1.12-0.23ε+10.55ε ² -21.72ε ³ +30.39ε ⁴ ;

When ε<0.2, F(ε)=0.265(1−ε) ⁴ +(0.857+0.265ε)/(1−ε) ^3/2 .

In summary, the threshold value of stress intensity factor under each stress ratio R is obtained. Several groups of data with large deviations were discarded, and then the average value of other groups was taken.

(2) Substitute the data obtained from the experiment into the formula of this model:

(3) Substituting the data obtained from the test into the formula of this model to predict the crack growth rate:

(4) The calculation formula for predicting the crack growth rate between r _c -ρ _p (crack tip plastic zone) is as follows:

(5) There are multiple parameters involved, which are calculated according to the following formulas:

1), ΔN is the low-cycle cycle life for the crack to grow between r _c -ρ _p :

2), ΔK _eff and ΔK _th are the equivalent stress intensity factor and the crack growth threshold under different stress ratios, respectively:

In the formula, C _th is the material constant used in the Pairs formula, ΔK _th is the crack tip stress intensity factor threshold of the material, and α _H is the small crack correction constant. ΔK _th,0 is the crack tip stress intensity factor threshold value of the material when the stress ratio is 0. P is the material fitting constant, P is different for different materials, and the stress ratio P for the same material is the same. ΔK _th,0 is the crack tip stress intensity factor threshold value of the material when the stress ratio is R. f is a closed function, calculated as follows:

Y is a geometric factor related to the sample. α is the constraint factor.

Claims (1)

1. Crack growth rate prediction method based on low-cycle fatigue performance parameters suitable for negative stress ratio: obtain stress and strain data and cycle life of corresponding samples through low-cycle fatigue tests of materials, and fit low-cycle fatigue performance parameters, and finally To calculate the crack growth rate, the brief process includes: (1) The tested sample is subjected to low-cycle fatigue test according to the general test method, and seven low-cycle fatigue performance parameters can be obtained: cyclic strength coefficient K', cyclic strain hardening exponent n', fatigue strength coefficient _σ'f , fatigue strength index b, fatigue ductility coefficient ε' _f , fatigue ductility index c, cyclic yield strength σ _yc ; use the above parameters to substitute into the formula $\mathrm{<span\; class="notranslate">\; \&Delta;</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; \&lsqb;</span>\frac{{\mathrm{<span\; class="notranslate">\; K</span>}}^{<span\; class="notranslate">\; \&prime;</span>}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; c</span>}}^{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\frac{{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>}^{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ The relationship between the low-cycle cycle life ΔN of the tested sample and the crack tip stress intensity factor ΔK can be obtained; where: σ _m is the average stress, which can be obtained by σ _m = (σ _min + σ _max )/2; ΔK _th is the crack growth threshold, which can be obtained by the following formula under different stress ratios ${\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}}\sqrt{\frac{\mathrm{<span\; class="notranslate">\; a</span>}}{\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}}{<span\; class="notranslate">\; \&lsqb;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; f</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; \&rsqb;</span>}^{<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; )</span>}& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 0</span>}\\ {<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; R</span>}}{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; R</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ In the above formula, ΔK _th,0 is the crack growth threshold value when the stress ratio is 0, which is obtained through the crack growth threshold measurement test of the material; a is the crack length, a _H is the crack length increment, R is the stress ratio, ΔK _{th, R} is the crack tip stress intensity factor threshold value of the material when the stress ratio is R, which are all obtained from the data records during the test; C _th is the material constant used in the Pairs formula, α _H is the small crack correction constant, P is the material fitting constant, P is different for different materials, and the stress ratio P for the same material is the same, the three coefficients are empirical values; f is a closed function, which can be obtained by the following two formulas: $\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; R</span>}& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ $\begin{array}{c}{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0.825</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.34</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 0.05</span>}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{<span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&pi;Y\&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>}^{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}\\ {\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0.415</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.071</span>}\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; Y\&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}\\ {\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ In the formula, Y is a geometric factor related to the shape of the sample, α is a constraint factor, and _σmax is the maximum stress of the sample during the low cycle fatigue test; (2) Substitute the closed function f obtained in (1) into the following formula ${\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; f</span>}\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; R</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}\mathrm{<span\; class="notranslate">\; K</span>}& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 0</span>}\\ {<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 0.5</span>}\mathrm{<span\; class="notranslate">\; f</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; R</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ Obtain the relationship between the equivalent stress intensity factor ΔK _eff and the stress intensity factor ΔK; (3) Substituting the ΔN, ΔK _th , and ΔK _eff obtained in (1) and (2) into the following formula to calculate the crack growth rate da/dN: $\frac{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; a</span>}}{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; p</span>}}}{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}_{\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; f</span>}\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&Delta;K</span>}}_{\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; |</span>}{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; no</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$