CN103778276B - Composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION - Google Patents

Abstract

The invention discloses a kind of composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION, first pass through Mittag Leffler distribution and portray the FATIGUE LIFE DISTRIBUTION of composite, then in conjunction with the Miner equivalent damage principle revised, determine the progressive damage statistical distribution of composite, finally use Monte Carlo method prediction composite Fatigue Reliability under different loads cycle-index.Mittag Leffler distribution is fewer than the parameter of existing frequently-used three-parameter weibull distribution, and the explicit physical meaning of parameter；The Miner equivalent damage principle improved, it is contemplated that act on the impact of load sequence on composite, and technical characterstic is simple, facilitates the use of engineers and technicians.The present invention can be used for predicting the Fatigue Reliability of composite, is the reference formulating composite element rehabilitation plan, has important theory and engineering significance.

Description

Composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION

Technical field

The invention belongs to Composites Fatigue reliability field, be specifically related to a kind of based on FATIGUE LIFE DISTRIBUTION answer Condensation material Predicting Reliability method.

Background technology

At present, composite is widely used to Aero-Space, biomedicine, construction material, electronic equipments etc. Field.Composite belongs to fragile material, and mechanical property dispersibility is very big, generally uses material in engineering design Statistical value corresponding to certain RELIABILITY INDEX, it is therefore desirable to the statistical law of clear and definite material property.

In the method for existing prediction composite reliability, generally using progressive damage as stochastic variable, utilize multiple The FATIGUE LIFE DISTRIBUTION of condensation material determines the random distribution of progressive damage.A large number of experiments show that the Composites Fatigue longevity The dispersibility of life is very big, and the variance of corresponding distribution is infinitely great, and the afterbody so causing distribution is power law.So And conventional distribution, the afterbody form including logarithm normal distribution, Weibull distribution etc. is index, it is impossible to accurate Really describing the dispersibility of fatigue life, design and reparation to composite element bring the biggest trouble.

At home and abroad, existing multinomial patented technology is applied to Fatigue Reliability prediction, such as patent CN102567622A " bank bridge construction wind-induced fatigue Predicting Reliability method based on accumulated damage of probability ", utilizes FEM (finite element) model meter Calculate stress response time-histories, and then the fatigue statisic obtaining luffing stress spectra is distributed, and predicts that wind-induced fatigue is reliable with this Degree；Patent US2012/8285522Materials-based failure analysis in design of electronic Devices, by stress response, the numerical model of join probability method, the fatigue reliability of assessment electronics； Patent CN103018063A " bridge Stochastic Fatigue Life Forecasting Methodology based on Mittag-Leffler distribution ", The probability density function using Mittag-Leffler distribution portrays stress amplitude distribution, in conjunction with the system of Miner criterion The dynamic fatigue lifetime of meter form prediction bridge；CN102331343A " booster turbine fatigue life prediction and Its method for evaluating reliability ", by calculating its average and the standard deviation of turbine material fatigue life, determine function Function, evaluates the reliability of turbine.Above-mentioned first two method is computationally intensive, is unfavorable for that Practical Project operates；The Three kinds of methods use the probability density distribution of Mittag-Leffler distribution, it was predicted that Fatigue Reliability needs to adopt Be Mittag-Leffler distribution Cumulative Distribution Function；4th kind of method tests number only with turbine material According to average and standard deviation, be not suitable for describe composite materials testing data dispersibility；

Accordingly, it would be desirable to a kind of new compound material Fatigue Reliability Forecasting Methodology is to solve the problems referred to above.

Summary of the invention

The present invention is directed to Composites Fatigue Predicting Reliability method in prior art defect, it is provided that Yi Zhongji Composite Predicting Reliability method in FATIGUE LIFE DISTRIBUTION.

For solving above-mentioned technical problem, the composite Predicting Reliability side based on FATIGUE LIFE DISTRIBUTION of the present invention The technical scheme that method is used is:

A kind of composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION, comprises the following steps:

(1) data of fatigue life of composite, is obtained；

(2) Mittag-Leffler distribution, is utilized to determine the statistics of the fatigue life of composite in step (1) Distribution, the expression formula of the cumulative distribution function of described Mittag-Leffler distribution is:

$F_N\left(n\right)=1-\frac{\sin \mathrm{\π\α}}{\mathrm{\π}}{\∫}_0^8\frac{y^{\mathrm{\α}-1}\exp (-\mathrm{ny}/\mathrm{\σ})}{y^{2\mathrm{\α}}+1+{2y}^{\mathrm{\α}}\cos \mathrm{\π\α}}\mathrm{dy}$

Wherein, α is index of stability, and σ is scale parameter, and n is fatigue life；

(3), utilize the data of fatigue life in step (1), determine that the Miner equivalent damage of improvement is former Composite parameter in then, the Miner equivalent damage of described improvement passes following formula in principle and represents:

$\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{M_i^{{\mathrm{\λ}}_i}}{N_i}=\frac{M^{\mathrm{\λ}}}{N}$

Wherein, N_iFor the fatigue life that i-stage load is corresponding, M_iFor the cycle-index that i-stage load is corresponding, λ_iFor The composite parameter that i-stage load is corresponding, N is equivalent fatigue life, and M is equivalent cycle number of times, and λ is The composite parameter of equivalence, m is the number of times that load loads；

(4), improvement that the statistical distribution of the fatigue life obtained according to step (2) and step (3) obtain Miner equivalent damage principle, determines the statistical distribution that accumulated damage is corresponding, the statistical that described accumulated damage is corresponding The expression formula of the cumulative distribution function of cloth is:

$F_D\left(d\right)=\frac{\sin \mathrm{\π\α}}{\mathrm{\π}}{\∫}_0^8\frac{y^{\mathrm{\α}-1}\exp (-M^{\mathrm{\λ}}y/\left(\mathrm{\σd}\right))}{y^{2\mathrm{\α}}+1+{2y}^{\mathrm{\α}}\cos \mathrm{\π\α}}\mathrm{dy}$

Wherein, d is accumulated damage, and M is equivalent cycle number of times, and α is index of stability, and σ is scale parameter, and λ is The composite parameter of equivalence；

(5), utilize the statistical distribution of the accumulated damage that step (4) obtains, use Monte Carlo method prediction multiple Condensation material Fatigue Reliability under different loads cycle-index.

Further, step (5) use Monte Carlo method prediction composite in different loads circulation time Fatigue Reliability under several comprises the following steps:

1), use following formula generate fatigue life corresponding Mittag-Leffler distribution random number τ,

τ=-σ ln u (sin (α π)/tan (α π v)-cos (α π))^1/α

Wherein, α is index of stability, and σ is scale parameter, u and v is being uniformly distributed on (0,1) interval；

2), utilize random number τ to calculate the value of accumulated damage, number h of statistics d ＜ 1, and utilize following formula to calculate The Fatigue Reliability R of composite,

$R=\frac{h}{H}$

Wherein, H is the number of the random number generated.

Further, by making axially loaded test on fatigue machine in step (1), obtain to be tested The data of fatigue life of composite.

Further, step (2) utilize fractional order square to determine the scale parameter that Mittag-Leffler is distributed The value of σ, utilizes method of least square to determine the value of index of stability α.

Beneficial effect: the composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION of the present invention is passed through The FATIGUE LIFE DISTRIBUTION of composite is portrayed in Mittag-Leffler distribution, then in conjunction with the Miner equivalent damage revised Principle, determines the progressive damage statistical distribution of composite, finally uses Monte Carlo method prediction composite to exist Fatigue Reliability under different loads cycle-index.Mittag-Leffler is distributed than three existing frequently-used parameters The parameter of Weibull distribution is few, and the explicit physical meaning of parameter；The Miner equivalent damage principle improved, examines Consider and acted on the impact of load sequence on composite, and technical characterstic is simple, facilitates engineers and technicians' Use.The present invention can be used for predicting the Fatigue Reliability of composite, is to formulate composite element rehabilitation plan Reference, there is important theory and engineering significance.

Accompanying drawing explanation

Fig. 1 is the flow chart of the composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION of the present invention；

Fig. 2 is the cumulative distribution table of Mittag-Leffler distribution simulation carbon/epoxy composite material fatigue life；

Fig. 3 is the tired of the accumulated damage statistical distribution that determines of Mittag-Leffler based on fatigue life distribution Integration Butut；

Fig. 4 is under different loads cycle-index, the decay pattern of carbon/epoxy composite material Fatigue Reliability.

Detailed description of the invention

Below in conjunction with the accompanying drawings and specific embodiment, it is further elucidated with the present invention, it should be understood that these embodiments are only used for The present invention is described rather than limits the scope of the present invention, after having read the present invention, those skilled in the art Amendment to the various equivalent form of values of the present invention all falls within the application claims limited range.

Mittag-Leffler distribution includes that a big class is distributed, and exponential is that its specific form is (see document 1 Kozubowski,T.J.(1999)Geometric stable laws:Estimation and Applications[J]. Math.Comput.Model29(10-12),241-253.).The cumulative distribution function of this distribution is:

$F\left(x\right)=1-\frac{\sin \mathrm{\π\α}}{\mathrm{\π}}{\∫}_0^8\frac{y^{\mathrm{\α}-1}\exp (-\mathrm{xy}/\mathrm{\σ})}{y^{2\mathrm{\α}}+1+{2y}^{\mathrm{\α}}\cos \mathrm{\π\α}}\mathrm{dy}$

Wherein, index of stability α ∈ (0,1), scale parameter σ ＞ 0.When index of stability is 1, above-mentioned distribution is degenerated For exponential.The numerical computations of this integrated form is simple, and precision is high.

The statistical property of a large amount of composite structures fatigue lifes shows exponential form and power-law distribution form. Mittag-Leffler distribution is as the popularization of exponential, and its parameter has clear and definite physical significance, be one more Add modeling method flexibly.This distribution is used can relatively accurately to determine the distribution rule of composite structures fatigue life Rule, and predict composite Fatigue Reliability under different loads effect based on this.

Refer to shown in Fig. 1, present invention composite based on FATIGUE LIFE DISTRIBUTION Predicting Reliability method, bag Include following steps:

(1) data of fatigue life of composite is obtained；

Generally, by making axially loaded test on fatigue machine, obtain the fatigue of institute's test compound material Lifetime data.

(2) Mittag-Leffler distribution is utilized to determine the statistical distribution of fatigue life；

The expression formula of the Mittag-Leffler profile accumulation distribution function used is:

$F_N\left(n\right)=1-\frac{\sin \mathrm{\π\α}}{\mathrm{\π}}{\∫}_0^8\frac{y^{\mathrm{\α}-1}\exp (-\mathrm{ny}/\mathrm{\σ})}{y^{2\mathrm{\α}}+1+{2y}^{\mathrm{\α}}\cos \mathrm{\π\α}}\mathrm{dy}$

Wherein α is index of stability, and σ is scale parameter, and n is fatigue life.Use fractional order moments estimation and Little square law determines the parameter of Mittag-Leffler distributed model.

(3) the composite parameter in the Miner equivalent damage principle of improvement is determined；

The Miner equivalent damage principle improved is:

$\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{M_i^{{\mathrm{\λ}}_i}}{N_i}=\frac{M^{\mathrm{\λ}}}{N}$

Wherein, N_iFor the fatigue life that i-stage load is corresponding, M_iFor the cycle-index that i-stage load is corresponding, λ_iFor The composite parameter that i-stage load is corresponding, N is equivalent fatigue life, and M is equivalent cycle number of times, and λ is The composite parameter of equivalence, m is the number of times that load loads.With the load acted on the composite and load The most relevant.According to the data of fatigue life obtained, determine answering in the Miner equivalent damage principle of improvement Condensation material parameter；

(4) according to the cumulative distribution of fatigue life, the statistical distribution that accumulated damage is corresponding is determined；

$\begin{array}{c}{F}_D\left(d\right)=P(D\left(M\right)\≤d)=P(M^{\mathrm{\λ}}/N\≤d)\\ =\frac{\sin \mathrm{\π\α}}{\mathrm{\π}}{\∫}_0^8\frac{y^{\mathrm{\α}-1}\exp ({-M}^{\mathrm{\λ}}y/\left(\mathrm{\σd}\right))}{y^{2\mathrm{\α}}+1+{2y}^{\mathrm{\α}}\cos \mathrm{\π\α}}\end{array}$

Wherein, d is accumulated damage, and M is equivalent cycle number of times, and α is index of stability, and σ is scale parameter, and λ is The composite parameter of equivalence.

(5) Fatigue Reliability under prediction different loads cycle-index.

Following formula is used to generate the random number of correspondence Mittag-Leffler distribution fatigue life.

τ=-σ lnu (sin (α π)/tan (α π v)-cos (α π))^1/α

Wherein u and v is being uniformly distributed on (0,1) interval.

Utilize random number τ to calculate the value of accumulated damage, number h (d ＜ 1) of statistics d ＜ 1, and utilize following formula to calculate The Fatigue Reliability R of composite.

$R=\frac{h(d<1)}{H}$

Wherein, H is the number generating random number.

Embodiment 1

(1) present invention with Distribution of Unidirectional Graphite/epoxy Composite at Swiss AMSLER10HFP1-478 high-cycle fatigue As a example by making axial La-draw loading on machine, fatigue life the sample size of data be 50 (see: Data Source is in document 2 Xu's people's equalitys. composite structures fatigue life data research [J]. developing material and application, 9,18-21,1994).

(2) refer to shown in Fig. 2, the cumulative distribution function simulation Unidirectional of employing Mittag-Leffler distribution/ The cumulative distribution of epoxy composite material fatigue life.Determine that Mittag-Leffler divides first with fractional order moments estimation The value of cloth scale parameter σ, then utilizes the value of Least Square Method index of stability α, is shown in Table 1.

The index of stability α and scale parameter σ of table 1Mittag-Leffler distribution correspondence

FATIGUE LIFE DISTRIBUTION	α	σ
			Mittag-Leffler	0.74	2008400

(3) composite parameter lambda is relevant with the load acted on the composite and load sequence.Due to examination Test the restriction of data, when at this, test λ is 1.15, the impact on Fatigue Reliability.

(4) statistical distribution that accumulated damage is corresponding is

$F_D\left(d\right)=\frac{\sin \mathrm{\&pi;\&alpha;}}{\mathrm{\&pi;}}{\&Integral;}_0^8\frac{y^{\mathrm{\&alpha;}-1}\exp (-M^{\mathrm{\&lambda;}}y/\left(\mathrm{\&sigma;d}\right))}{y^{2\mathrm{\&alpha;}}+1+{2y}^{\mathrm{\&alpha;}}\cos \mathrm{\&pi;\&alpha;}}\mathrm{dy},$

Wherein α=0.74, σ=2008400.Referring to shown in Fig. 3, without loss of generality, M is taken as examination fatigue life Test the median of data, calculate the value of accumulated damage, and with being distributed F_DThe distribution of (d) simulation accumulated damage.

(5) following formula is used to generate 10⁴The random number of individual correspondence Mittag-Leffler distribution fatigue life.

τ=-σ lnu (sin (α π)/tan (α π v)-cos (α π))^1/α

Utilize the random number generated to calculate the value of accumulated damage, add up under number h (d ＜ 1) of d ＜ 1, and utilization Formula assumed (specified) load cycle-index is 10³To 10⁸Time, the Fatigue Reliability R of composite, is shown in Fig. 4.

$R=\frac{h(d<1)}{{10}^4}$

From fig. 4, it can be seen that the Fatigue Reliability of starting stage composite is higher, then along with load cycle number of times Continuous increase, Fatigue Reliability drastically declines, the two stage respectively with crack initiation stage and extension phase Unanimously.

The present invention first passes through Mittag-Leffler distribution and portrays the FATIGUE LIFE DISTRIBUTION of composite, then ties Close the Miner equivalent damage principle revised, determine the progressive damage statistical distribution of composite, finally use and cover spy Caro method prediction composite Fatigue Reliability under different loads cycle-index.Mittag-Leffler is distributed ratio The parameter of existing frequently-used three-parameter weibull distribution is few, and the explicit physical meaning of parameter；The Miner improved Equivalent damage principle, it is contemplated that act on the impact of load sequence on composite, and technical characterstic is simple, convenient The use of engineers and technicians.

The present invention can be used for predicting the Fatigue Reliability of composite, is to formulate composite element rehabilitation plan Reference, has important theory and engineering significance.

Claims (3)

1. a composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION, it is characterised in that include following Step:

(1) data of fatigue life of composite, is obtained；

(2) Mittag-Leffler distribution, is utilized to determine the statistical of the fatigue life of composite in step (1) Cloth, the expression formula of the cumulative distribution function of described Mittag-Leffler distribution is:

$F_N\left(n\right)=1-\frac{sin\mathrm{\&pi;}\mathrm{\&alpha;}}{\mathrm{\&pi;}}{\&Integral;}_0^{\mathrm{\&infin;}}\frac{y^{\mathrm{\&alpha;}-1}\exp (-ny/\mathrm{\&sigma;})}{y^{2\mathrm{\&alpha;}}+1+2y^{\mathrm{\&alpha;}}cos\mathrm{\&pi;}\mathrm{\&alpha;}}dy$

Wherein, α is index of stability, and σ is scale parameter, and n is fatigue life；

(3), utilize the data of fatigue life in step (1), determine in the Miner equivalent damage principle of improvement Composite parameter, the Miner equivalent damage of described improvement passes following formula in principle and represents:

$\underset{i=1}{\overset{m}{\&Sigma;}}\frac{M_i^{{\mathrm{\&lambda;}}_i}}{N_i}=\frac{M^{\mathrm{\&lambda;}}}{N}$

Wherein, N_iFor the fatigue life that i-stage load is corresponding, M_iFor the cycle-index that i-stage load is corresponding, λ_iFor The composite parameter that i-stage load is corresponding, N is equivalent fatigue life, and M is equivalent cycle number of times, and λ is The composite parameter of equivalence, m is the number of times that load loads；

(4), improvement that the statistical distribution of the fatigue life obtained according to step (2) and step (3) obtain Miner equivalent damage principle, determines the statistical distribution that accumulated damage is corresponding, the statistical that described accumulated damage is corresponding The expression formula of the cumulative distribution function of cloth is:

$F_D\left(d\right)=\frac{sin\mathrm{\&pi;}\mathrm{\&alpha;}}{\mathrm{\&pi;}}{\&Integral;}_0^{\mathrm{\&infin;}}\frac{y^{\mathrm{\&alpha;}-1}\exp (-M^{\mathrm{\&lambda;}}y/(\mathrm{\&sigma;}d\left)\right)}{y^{2\mathrm{\&alpha;}}+1+2y^{\mathrm{\&alpha;}}cos\mathrm{\&pi;}\mathrm{\&alpha;}}dy$

Wherein, d is accumulated damage, and M is equivalent cycle number of times, and α is index of stability, and σ is scale parameter, and λ is The composite parameter of equivalence；

(5), utilize the statistical distribution of the accumulated damage that step (4) obtains, use Monte Carlo method prediction composite wood Material Fatigue Reliability under different loads cycle-index:

1), use following formula generate fatigue life corresponding Mittag-Leffler distribution random number τ,

τ=-σ lnu (sin (α π)/tan (α π v)-cos (α π))^1/α

Wherein, α is index of stability, and σ is scale parameter, u and v is being uniformly distributed on (0,1) interval；

2), utilize random number τ to calculate the value of accumulated damage, number h of statistics d ＜ 1, and it is compound to utilize following formula to calculate The Fatigue Reliability R of material,

$R=\frac{h}{H}$

Wherein, H is the number of the random number generated.

Composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION the most according to claim 1, it is special Levy and be, by making axially loaded test on fatigue machine in step (1), obtain composite to be tested Data of fatigue life.

Composite Predicting Reliability method based on FATIGUE LIFE DISTRIBUTION the most according to claim 1, it is special Levy and be, the value of utilize fractional order square to determine in step (2) scale parameter σ that Mittag-Leffler is distributed, Method of least square is utilized to determine the value of index of stability α.