Research ArticleLife Prediction on a T700 Carbon Fiber Reinforced Cylinderwith Limited Accelerated Life Testing Data
Ma Xiaobing1 and Zhang Yongbo2
1School of Reliability and Systems Engineering Beihang University Beijing 100191 China2Research Center of Small Sample Technology Beihang University Beijing 100191 China
Correspondence should be addressed to Zhang Yongbo zhang19840504163com
Received 14 July 2014 Revised 10 September 2014 Accepted 10 September 2014
Academic Editor Shaofan Li
Copyright copy 2015 M Xiaobing and Z Yongbo This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
An accelerated life testing investigation was conducted on a composite cylinder that consists of aluminum alloy and T700 carbonfiberThe ultimate failure stress predictions of cylinders were obtained by themixing rule and verified by the blasting static pressuremethod Based on the stress prediction of cylinder under working conditions the constant stress accelerated life test of the cylinderwas designed However the failure data cannot be sufficiently obtained by the accelerated life test due to the time limitationTherefore most of the data presented to be high censored in high stress level and zero-failure data in low stress level When usingthe traditional method for rupture life prediction the results showed to be of lower confidence In this study the consistency offailure mechanism for carbon fiber and cylinder was analyzed firstly According to the analysis result the statistical test informationof carbon fiber could be utilized for the accelerated model constitution Then rupture life prediction method for cylinder wasproposed based on the accelerated life test data and carbon fiber test data In this way the life prediction accuracy of cylinder couldbe improved obviously and the results showed that the accuracy of this method increased by 35
1 Introduction
In recent years T700 carbon fiber replacing T300 becomesthe new general carbon fiber since densification gives theT700 higher tensile strength [1ndash6] The cylinder of newspecial equipment consists of T700 carbon fiber compositematerials glass fiber composites and aluminum alloy Themain role of aluminum alloy is to improve the axial modulusand corrosion resistance of cylinder the glass fiber compositematerial is to guarantee the aluminum alloy being adaptedto higher working because of large prestress and the carbonfiber composite material is to improve the strength andmodulus of the cylinder and it is also the main load-bearingmaterial between the three-layer materials
The introduction of carbon fiber into the new generationcylinder improves the failure stress of the cylinder but italso brings new problem to the reliability evaluation of thecylinder The cylinder wound by T700 carbon fiber exhibitslong life and high reliability in high-speed rotation modeand the sufficient data cannot be obtained by the traditional
life tests Therefore the accelerated test must be introducedand the reliability index in the normal stress could beextrapolated by the statistical analysis using the high stresslevel data [7] The maximum likelihood estimation method[8 9] is very suitable for the censored data analysis but thismethod only has good properties for the large sample And itneeds to iterate for the transcendental equations sometimesthe computation is difficulties and not convergence Theintegral best linear unbiased estimationmethod [10 11] solvesproblems by regression analysis based on linear transforma-tion acceleration model The information between differentstresses is used comprehensively and the estimation accuracycan be improved But the evaluation accuracy is still unableto meet the actual engineering requirement of the cylinderstructure with great life dispersion
This study found that the failure mode of the cylinderwound by T700 carbon fiber was carbon fiber breakagein the working mode with high-speed rotation Thereforethere would be some relationship between the rupture life offiber carbon and the cylinder In this study we analyzed the
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 902157 9 pageshttpdxdoiorg1011552015902157
2 Mathematical Problems in Engineering
mechanical properties of the cylinder compositematerial andobtained the ultimate failure stress by mixing rules whichwere verified by the static pressure burst test method Thenwe designed accelerated life tests on the carbon fiber andthe cylinder According to the test results we proposed theintegral best unbiased estimate method for the cylinder lifeprediction The evaluation accuracy was greatly improvedby comprehensive evaluation using the information of T700carbon fiber based on the test results of the acceleratedequivalence and failure mechanisms
2 Mechanical Analysis
21 Mechanical Analysis of the Composite Material The newgeneration cylinder consists of aluminium alloy glass fiberand T700 carbon fiber and its structure is illustrated inFigure 1
The main role of the aluminium alloy is to improve theaxial modulus and anticorrosion of the cylinder The glassfiber is to impose prestress to the aluminium alloy accordingto winding and so it could improve the maximum capacityThe carbon fiber is the main bearing materials and for theirhigh strength and modulus the ultimate failure stress ofcylinder is greatly improved The mechanical properties ofthe glass fiber and the carbon fiber can be obtained bymixingrules according to tensile strength modulus and the volumecontent of each fiber Thus the overall mechanical propertiesof the cylinder could also be estimated by the mixing rulebasing on the material properties and structures
According to the property test of T700 carbon fiber thetensile strength is 4900Mpa themodulus is 230GPa and thestrain is approximately 2 Also the matrix strength of thecomposite material is 85MPa the modulus is 28 Gpa andthe strain is 4 The mix rule is
119883119905= 120590119891119881119891+ 120590119898(1 minus 119881
119891) (1)
where the119883119905is tensile strength of composite 120590
119891is the tensile
strength of fiber 120590119898is the tensile strength ofmatrix and119881
119891is
the fiber volumeTherefore for the fiber volume of compositematerial used in this study is 72 the tensile strength of thiscomposite would be calculated as 35518MPa
For the glass fiber the tensile strength is 3400MPa themodulus is 91 GPa and the strain is approximately 37 Andthe matrix is the same with the T700 carbon fiber reinforcedcomposite While the percentage of fiber volume is 76according to the mix rule the tensile strength of this glassfiber reinforced composite would be calculated as 26044
The lining of the cylinder is aluminium alloy its elasticmodulus is 70GPa yield strength is 610Mpa tensile strengthis 640Mpa and strain is greater than 4 [12]
22 Failure Stress Analysis of the Cylinder The tensile modu-lus of the glass fiber and the aluminium alloy is significantlylower than the outer carbon fiber composite material by theprevious analysis And according to the thickness of eachlaminate listed in Table 1 it is obvious that the carbon fiberis the main load-bearing material in the entire cylinderWhen the outer T700 carbon fiber reinforced composite
Aluminium alloyGlass fiberT700 carbon fiber
XC
YC
ZC
Figure 1 Structure of the cylinder
is destructed the glass fiber and aluminum alloy wouldbe instantaneously destroyed due to the large load Thestress-strain curves of glass fiber and carbon fiber reinforcedcomposite materials presented to be linear like the tensilestress-strain curves of glass fiber and carbon fiber that areshown in Figures 2 and 3
The bearing stress of glass fiber is 380Mpa that is thereverse stress to impose on the aluminum alloy therefore itsbearing stress is 1777Mpa when the conditions are the samewith the outer carbon fiberThe tensile strength of aluminumalloy is 640MPa we consider that it imposed the reverseprestress of 380MPa its bearing stress is 260Mpa when thebreaking elongation is 2 and its thickness is 12mm Thethickness of glass fiber and the carbon fiber is 085mm and15mm respectively Therefore according to the mixing rulethe ultimate failure stress of the cylinder is theoretical as
The strip tensile testmethod [13] for ultimate failure stressof the cylinder cannot meet the requirements of test accuracywith a few samples The NOL [13] ring stretching methodis sensitive to the boundary effects and sample processing isvery difficult Therefore we chose the blasting static pressuremethod [14] for the ultimate failure stress testing which dealtwith the data as a whole and thus could reflect themechanicalproperties better
The test samples were produced according to the nationalstandardGBT15560 and the processing technical rawmate-rials were the same with the cylinder Figure 4 shows the sizeof this test sample As shown two ends of the cylinder werestrengthened by carbon fiber layers of 30mm width Table 2shows the parameters of this static pressure burst test Table 3shows the test result
It can be seen from Table 3 that the mean of the ultimatetest failure stress is 20676MPa which is slightly higher thanthe theoretical prediction of 2004Mpa It may be the reasonthat the overall performance of the cylinder material wouldbe slightly higher than fiber samples The parameters usedfor theoretical prediction were the mean value of the tensiletest results of fiber sample and thus its volatility is relativelylarge Another main reason may be that the length of the testsample in the static pressure burst test was required to be
Mathematical Problems in Engineering 3
Table 1 Geometry parameters and properties of each laminate
Table 2 Parameters of the static pressure burst test
Aluminium alloy(mm)
Process conditions Resin formula Curing systemGlass fiber Carbon fiber
Thickness 12Length 350 5 layers 10 layers Proprietary formula 75∘C4 h + 80∘C12 h
0
1500
3000
4500
0 1 2 3 4
Stre
ss (M
Pa)
Strain ()
Figure 2 Tensile stress-strain curves of the glass fiber
0
2000
4000
6000
0 05 1 15 2 25
Stre
ss (M
Pa)
Strain ()
Figure 3 Tensile stress-strain curves of the carbon fiber
at least 5 times greater than the diameter in test standard ofGBT15560 But this long diameter ratio of our test sampleis only 27 As the shorter sample is sensitive to the effectsof the end portion in the testing process this may be themain reason that the test results are slightly higher than theprediction result In all the test results of the static pressureburst test made a good agreement with the prediction resultand it is feasible to verify the overall performance of thecylinder This test result would be taken as the importantreference for the constant stress accelerated life test in thefollowing research
3 Accelerated Life Test
31 Experiment Preparation In the working conditions thebearing stress of cylinder is about 9138Mpa which is equiv-alent to 455 of the ultimate failure stress We calculated
Table 3 Test result of the static pressure burst test
140Figure 4 Size of the static pressure burst test
the stress level of the cylinder by the finite element methodand the result shows that the bearing stress of the liningaluminum alloy is 240MPa the bearing stress of the glassfiber is 1060Mpa and the bearing of carbon fiber is 1350MPaThe related parameters are shown in Table 1
The cylinder used in the accelerated life test is the sameas that used in the static pressure burst test as shown inFigure 5 Then the cylinder is filled with hydraulic oil up tocertain pressure and maintains this pressure for a long timeso that the cylinder bears a uniform inner pressure in everydirectionThe test device is placed in one ovenwhereworkingtemperature could be maintained The pressure can be testedby the pressure sensor If the pressure of the cylinder dropssignificantly the cylinder was considered to be failure and thefailure time is automatically recorded
4 Mathematical Problems in Engineering
Compression nut
Exhaust port
Gland
Seal ring
Mandrel
Hydraulic shaft
Test sample
Oil-in
P
P
P
P
P
P
Figure 5 Diagram of the reliability test unit of the cylinder
32 The Accelerated Test Plan of the Cylinder We knowthe ultimate failure stress is 20676Mpa given by the staticpressure burst test and the working stress is 9138Mpa whichis equivalent to 455 of the ultimate failure stress of thecylinderThe stress-strain curve is close to linear in the scopeof the ultimate failure stress of the cylinder
According to GB 26891-81 we divided the cylinders intofive groups for constant stress accelerated life test And weset the minimum stress level to be 1315MPa (almost 636of the ultimate failure stress) and the maximum stress levelto be 1861MPa (almost 90 of the ultimate failure stress)Also other stress levels were set to be 727 80 and 85 ofthe ultimate failure stress respectively According to the teststandard of GB 26891-81 this test level setting could ensurethat the failure mechanism of cylinders was the same Theworking temperature of the cylinder will not exceed 40∘C sothe test temperature is controlled at 40 plusmn 2∘C
4 Results and Discussion
41 The Method on Rupture Life Evaluation of the CylinderWe assume that there are 119899
119895samples prepared for type-I
censored test in the V119895(119895 = 1 2 119904) stress level and the
censored time is 120585lowast119895 There are 119902
119895(1 le 119902
119895le 119899119895) failures and
the censored data is 1205851198951le sdot sdot sdot le 120585
119895119902119895
Hypothesis 1 The product life follows Weibull distribution119882(119898119895 120578119895) in the V
119895stress level
Hypothesis 2 The failure mechanisms of the carbon fiber andthe cylinder are the same in each stress level
Hypothesis 3 The relationship between characteristic life 120578119895
and stress level V119895follows the inverse power law model 120578
119895=
119860Vminus119888119895 119895 = 1 2 119904 And 119860 119888 are parameters that should be
estimated
The linear expression can be obtained by logarithmictransformation on the accelerated model in Hypothesis 3
ln 120578119895= 119886 + 119887119909
119895 (3)
Then the logarithmic life follows the extreme value distribu-tion and the relationship between the logarithmic life and thelogarithmic stress can be written as
119895 119886 = ln119860 and 119887 = minus119888 Parameter
119886 reflects the characteristic of test product parameter 119887
reflects the acceleration characteristic of the test 120590 is the scaleparameter of the extreme value distribution and the measureparameter for the consistency of the failure mechanism
In the condition of Hypothesis 1 type-I censored data1199101198951
le sdot sdot sdot le 119910119895119902119895
can be taken as the value of the former 119902119895
order statistics1198841198951le sdot sdot sdot le 119884
119895119902119895for the extreme value distribu-
tion with size 119899119895 119910119895(119902119895+1)
= 119910lowast
119895can be taken as the value of the
119902119895th interval statistics 119884
119895(119902119895+1)with the same sample
From literature [9 10] we can obtain the estimations of 119886119887 and 120590 by partial derivative for 119876
119876 =
119904
sum
119895=1
119902119895+1
sum
119896119897=1
(119910119895119896minus 119886 minus 119887119909
119895minus 120590119906119895119896)
times 119892119895119896119897(119910119895119897minus 119886 minus 119887119909
The Failure Mechanism Consistency Analysis This analysisfocused on the consistency of failure mechanism and acceler-ated model parameter between the carbon fiber and cylinderin the accelerated life test Denote the model parameters ofthe carbon fiber and the cylinder as 119886
1 1198871 and 120590
1and 1198862 1198872
and 1205902 respectively We assume the following
(1) The distribution parameters 1205901 1205902are two indepen-
dent normal populations If the failure mechanism ofthe carbon fiber is the same with the cylinder themean and variance of the two normal populations arethe same [16 17]
(2) Parameters of 1198861and 119886
2reflect the life characteristic
of carbon fiber and cylinder and then 1198861and 1198862have
no relation
(3) The model parameters 1198871 1198872are two independent
normal populations If the acceleration of the carbonfiber is the same with the cylinder the mean andvariance of the two normal populations are the same
Based on the above assumptions (1 1) and (
2 2) can
be taken as two bivariate normal populations We can judgethe consistency of the mean vector and covariance matrix ofthe two bivariate normal populations by hypothesis test
(1) The Consistency Judgment of the Mean Vector The twoindependent normal populations are denoted by (
1 1) sim
1198732(1205831 Σ1) and (
2 2) sim 119873
2(1205832 Σ2) We sample 119899 119898 gt
2 specimens from them respectively and denote the meanvectors by 119883 119884 respectively and the variance matrix by119878119894(119894 = 1 2) The hypothesis is
1198670 1205831= 1205832
1198671 1205831
= 1205832 (10)
When Σ1= Σ2and they were unknown the test statistic
1198792=
119899119898
119899 + 119898(119883 minus 119884)
119879
119878minus1(119883 minus 119884) (11)
where
119878 =(119899 minus 1) 119878
1+ (119898 minus 1) 119878
2
119899 + 119898 minus 2 (12)
And 119865 = (((119899 +119898minus2) minus 1)2(119899 +119898minus2))1198792sim 119865(2 119899 +119898minus3)
Then the rejection region with the significance level 120572 is
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) (13)
(2) The Consistency Judgment of the Variance Matrix Thehypothesis is
1198670 Σ1= Σ2
1198671 Σ1
= Σ2 (14)
The amendatory likelihood ratio statistic is
120582lowast=
(119899 + 119898 minus 2)(119899+119898minus2)10038161003816100381610038161198781
(119898 minus 1)(119898minus1)210038161003816100381610038161198781 + 1198782
1003816100381610038161003816
(119899+119898minus2)2 (15)
Then
minus2 ln 120582lowast sim 1205942(
119891
1 minus 119889) (16)
6 Mathematical Problems in Engineering
where
119891 =1
2119901 (119901 + 1) (119896 minus 1)
119889 =21199012+ 3119901 minus 1
6 (119901 + 1) (119896 minus 1)(
1
119899 minus 1+
1
119898 minus 1minus
1
119899 + 119898 minus 119896)
(17)
and 119896 = 119901 = 2Then the rejection region with the significance level 120572 is
1205942lt 1205942
120572(
119891
1 minus 119889) (18)
The information fusion methodBased on the result of the integral best linear unbiased
estimation and the consistency analysis of failuremechanismwe proposed an information fusion method for the covari-ance matrixes of the T700 carbon fiber and the cylinderThe fusion process can be accomplished by the two stepsfirstly fusing the information only related to parameters 119887and 120590 in the covariance matrix and then secondly to furtherimprove the evaluation accuracy of the covariance matrix formodel parameter of the cylinder fusing the other informationrelated to 119887 and 120590
Denote the point estimations of T700 carbon fiber and thecylinder by 119886
1 1 and
1and 1198862 2 and
2 respectively and
their covariance matrix is
cov (119886119894 119894 119894) =
[[
[
var (119886119894) cov (119886
119894 119894) cov (119886
119894 119894)
cov (119886119894 119894) var (
119894) cov (
119894 119894)
cov (119886119894 119894) cov (
119894 119894) var (
119894)
]]
]
(119894 = 1 2)
(19)
Denote the covariance matrix of the bivariate normaldistribution of 119887
1 1205901and 1198872 1205902 respectively by
119894= [
var (119894) cov (
119894 119894)
cov (119894 119894) var (
119894)] (119894 = 1 2) (20)
The elements of the above matrixes are one part of (19)The first step is fusing the information only related to
parameters 119887 and 120590 in the covariance matrixIf the covariance matrixes 119881
1and 119881
2are certified to be
the same by the consistency analysis of the bivariate normaldistribution (119887
1 1205901) and (119887
2 1205902) the unbiased estimation of
the covariancematrix of the bivariate normal distribution canbe obtained by the following equation
=(1198991minus 1)
1+ (1198992minus 1)
2
1198991+ 1198992minus 2
(21)
Compared to the small ratio of failure and high censoredtime of the cylinder life test the data and failures of the carbonfiber are much greater and thus the evaluation result is moreaccurateTherefore when the matrix
2is replaced by with
fusing the information of parameters 119887 and 120590 according to theabove method the prediction result of the cylinder is more
accurate The covariance matrix of 119887 and 120590 based on fusioncould be written as
= [var (
12) cov (
12 12)
cov (12 12) var (
12)
] (22)
The covariancematrix cov(1198862 2 2) for the cylinder with
bottom right four elements replaced by could be written as
cov (1198862 2 2)1015840
=[[
[
var (1198862) cov (119886
2 2) cov (119886
2 2)
cov (1198862 2) var (
12) cov (
12 12)
cov (1198862 2) cov (
12 12) var (
12)
]]
]
(23)
The second step is fusing the other elements related to 119887 120590of the covariance matrix
Based on the same correlation coefficient between theparameters the value of cov(119886
2 12) cov(119886
2 12) with fusing
the information of the T700 carbon fiber can be calculated as
cov (1198862 12) = radic
var (12)
var (2)
sdot cov (1198862 2)
cov (1198862 12) = radic
var (12)
var (2)sdot cov (119886
2 2)
(24)
According to the above two steps the covariance matrixcov(1198862 12 12) can be written as
cov (1198862 12 12)
=[[
[
var (1198862) cov (119886
2 12) cov (119886
2 12)
cov (1198862 12) var (
12) cov (
12 12)
cov (1198862 12) cov (
12 12) var (
12)
]]
]
(25)
Comparing with (25) and (19) (119894 = 2) it can be found thatthe elements of parameter covariance matrix of the cylinderhave changed in addition to the variance var(119886
2) by fusing
the carbon fiber test information thus making the evaluationresult more reasonable It should be noted that the parametercovariance matrix can be obtained by the integrated bestlinear unbiased estimation as follows
cov (119886 ) = 1205902119862 (26)
In the calculation of the upper and lower limits for thereliability rupture life thematrix119862would be usedThereforein fusion process of the covariance matrix with T700 carbonfiber information and cylinder information we can introducethe matrix 119862 directly in the above method to make thecalculation easier
42 Examples To verify the design level of the rupture life ofa certain type of cylinder a unit made accelerated life testsfor T700 carbon fiber composite material and the cylinder
Mathematical Problems in Engineering 7
Table 4 Accelerated life test information for T700 carbon fiber andthe cylinder
structure respectively The test temperature is 40∘C and thetest load conditions are shown in Table 4
The diagramof rupture life data for T700 carbon fiber andthe cylinder structure from the test is as in Figures 6 and 7
(1) The Result of the Integral Best Unbiased Estimation Theresults of model parameters and the matrix 119862 obtained bythe integrated best linear unbiased estimation are shown inTable 5
(2) The Test of Mean Vector and Covariance Matrix Equalto ( ) of the Carbon Fiber and the Cylinder We considerparameters ( ) as a bivariate normal population and testwhether the mean vectors and covariance matrixes of ( )for the carbon fiber and the cylinder are equal or not and theresults are shown in Table 6
From Table 6 the observed values of the test statistic are
119865 = 809725 lt 1198650975
(2 5) = 843
1205942= 19468 gt 120594
2
0025(5)
(27)
Then we can receive the null hypothesis The mean vectorsand the covariance matrixes of ( ) for the carbon fiber andthe cylinder are equal
(3) The Information Fusion of the T700 Carbon Fiber andthe Cylinder By using the method in this paper the matrixinformation of carbon fiber in Table 5 can be fused into thecylinder and the upper and lower limits of the reliabilityrupture life at confidence level 120574 = 095 can be calculatedThe comparison results are shown in Table 7
From Table 7 when the reliability is 09 the evaluationaccuracy of the reliability life of the cylinder is increased by35
The curves of upper and lower limits of the logarithmicreliability rupture life 119910
119877changed with the reliability 119877 are
shown in Figure 7Figure 8 shows that the upper and lower limits of the
reliability rupture lifewill bemore accuratewith the reliabilitychanged after fusion of the information of T700 carbon fiberand the interval length is much shorter
5 Conclusion
(1) Since the carbon fiber bears themain load at work theacceleration and failure mechanism of T700 carbonfiber and the cylinder are the same And it can beproved by the structure analysis and the statistic testof the test data
1700 1750 1800 1850 1900 1950 2000
0
2000
4000
6000
8000
10000
12000
14000
Failure timeCensored time
17 failure2 censored
25 failure6 censored
20 failure13 censored
Stress level (N)
Rupt
ure l
ife120585
(h)
Figure 6 Rupture life of T700 carbon fiber
60 65 70 75 80 85 90
0
5000
10000
15000
20000
25000
30000
35000
Stress level (N)
0 failure8 censored
0 failure9 censored
0 failure9 censored
3 failure3 censored
3 failure3 censored
Failure timeCensored time
Rupt
ure l
ife120585
(h)
Figure 7 Ruptures life of the cylinder
(2) When the acceleration and failure mechanism are thesame the evaluation accuracy of the reliability lifefor the cylinder can be improved by fusion of theinformation of the carbon fiber
(3) The method in this paper is based on Weibull distri-bution and the inverse power law model for struc-tured products It can be applied to other location-scale family distribution and acceleration models
8 Mathematical Problems in Engineering
Table 5 Contrast result between T700 carbon fiber and the cylinder with the integrated best linear unbiased estimation
Table 6 The test results of mean vector and covariance matrix for a bivariate normal population ( ) of carbon fiber and cylinder
Null hypothesis Rejection region Observation value Critical value 120572 = 0025
1205831= 1205832
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) 809725 843Σ1= Σ2
1205942lt 1205942
120572(119891(1 minus 119889)) 19468 0831
Table 7 Contrast result of cylinder after fusion information of T700 carbon fiber
Result The original data Fusion of the information of T700
119862[[
[
053 270 minus006
270 2149 minus105
minus006 minus105 013
]]
]
[[
[
053 208 minus0037
208 1276 minus037
minus0037 minus037 0049
]]
]
The interval estimation of 119910119877
[1379 2735] [1429 2315]Interval length 1356 886
080 085 090 095 10010
15
20
25
30
35
40
Reliability (R)
Loga
rithm
ic re
liabi
lity
lifey
R(h
)
After fusion yRLAfter fusion yRUyR
Before fusion yRLBefore fusion yRU
Figure 8 Change curves for 119910119877and upper and lower limits
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work described in this paper was supported by theNational Natural Science Foundation of China (Grants nos61104133 and 61473014) The authors fully appreciate thefinancial support The authors would like to thank thereviewers and the editor for many suggestions that helpedimprove this paper
References
[1] D F Li H JWang and F He ldquoStructure and properties of T300and T700 carbon fiberrdquoNew CarbonMaterials vol 22 no 1 pp59ndash64 2007
[2] G T Zhang W G Chen and B Yang ldquoTesting research onmechanical properties of T700 carbon fiberepoxy compositesrdquoFiber Composites vol 2 pp 49ndash52 2009
[3] Y X Zhou M A Baseer H Mahfuz and S Jeelani ldquoStatisticalanalysis on the fatigue strength distribution of T700 carbonfiberrdquo Composites Science and Technology vol 66 no 13 pp2100ndash2106 2006
[4] Z G Yu S C Yang and B F Song ldquoComparison of wet and hotaging resistance of T700 and T300 carbon fiber strengthenedepoxy resin compositesrdquoMaterials for Mechanical Engineeringvol 33 no 6 pp 48ndash51 2009
[5] X Gu and X Xu ldquoNumerical simulation of damage in fiberreinforced composite laminates under high velocity impactrdquoActaMateriae Compositae Sinica vol 29 no 1 pp 150ndash161 2012
[6] J Liu Y X Bai Y L Tian X Y Huang C H Wang and JY Liang ldquoEffect of the process of electrochemical modificationon the surface structure and properties of PAN-based carbonfirbersrdquo Acta Materiae Compositae Sinica vol 29 no 2 pp 16ndash25 2012
Mathematical Problems in Engineering 9
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
mechanical properties of the cylinder compositematerial andobtained the ultimate failure stress by mixing rules whichwere verified by the static pressure burst test method Thenwe designed accelerated life tests on the carbon fiber andthe cylinder According to the test results we proposed theintegral best unbiased estimate method for the cylinder lifeprediction The evaluation accuracy was greatly improvedby comprehensive evaluation using the information of T700carbon fiber based on the test results of the acceleratedequivalence and failure mechanisms
2 Mechanical Analysis
21 Mechanical Analysis of the Composite Material The newgeneration cylinder consists of aluminium alloy glass fiberand T700 carbon fiber and its structure is illustrated inFigure 1
The main role of the aluminium alloy is to improve theaxial modulus and anticorrosion of the cylinder The glassfiber is to impose prestress to the aluminium alloy accordingto winding and so it could improve the maximum capacityThe carbon fiber is the main bearing materials and for theirhigh strength and modulus the ultimate failure stress ofcylinder is greatly improved The mechanical properties ofthe glass fiber and the carbon fiber can be obtained bymixingrules according to tensile strength modulus and the volumecontent of each fiber Thus the overall mechanical propertiesof the cylinder could also be estimated by the mixing rulebasing on the material properties and structures
According to the property test of T700 carbon fiber thetensile strength is 4900Mpa themodulus is 230GPa and thestrain is approximately 2 Also the matrix strength of thecomposite material is 85MPa the modulus is 28 Gpa andthe strain is 4 The mix rule is
119883119905= 120590119891119881119891+ 120590119898(1 minus 119881
119891) (1)
where the119883119905is tensile strength of composite 120590
119891is the tensile
strength of fiber 120590119898is the tensile strength ofmatrix and119881
119891is
the fiber volumeTherefore for the fiber volume of compositematerial used in this study is 72 the tensile strength of thiscomposite would be calculated as 35518MPa
For the glass fiber the tensile strength is 3400MPa themodulus is 91 GPa and the strain is approximately 37 Andthe matrix is the same with the T700 carbon fiber reinforcedcomposite While the percentage of fiber volume is 76according to the mix rule the tensile strength of this glassfiber reinforced composite would be calculated as 26044
The lining of the cylinder is aluminium alloy its elasticmodulus is 70GPa yield strength is 610Mpa tensile strengthis 640Mpa and strain is greater than 4 [12]
22 Failure Stress Analysis of the Cylinder The tensile modu-lus of the glass fiber and the aluminium alloy is significantlylower than the outer carbon fiber composite material by theprevious analysis And according to the thickness of eachlaminate listed in Table 1 it is obvious that the carbon fiberis the main load-bearing material in the entire cylinderWhen the outer T700 carbon fiber reinforced composite
Aluminium alloyGlass fiberT700 carbon fiber
XC
YC
ZC
Figure 1 Structure of the cylinder
is destructed the glass fiber and aluminum alloy wouldbe instantaneously destroyed due to the large load Thestress-strain curves of glass fiber and carbon fiber reinforcedcomposite materials presented to be linear like the tensilestress-strain curves of glass fiber and carbon fiber that areshown in Figures 2 and 3
The bearing stress of glass fiber is 380Mpa that is thereverse stress to impose on the aluminum alloy therefore itsbearing stress is 1777Mpa when the conditions are the samewith the outer carbon fiberThe tensile strength of aluminumalloy is 640MPa we consider that it imposed the reverseprestress of 380MPa its bearing stress is 260Mpa when thebreaking elongation is 2 and its thickness is 12mm Thethickness of glass fiber and the carbon fiber is 085mm and15mm respectively Therefore according to the mixing rulethe ultimate failure stress of the cylinder is theoretical as
The strip tensile testmethod [13] for ultimate failure stressof the cylinder cannot meet the requirements of test accuracywith a few samples The NOL [13] ring stretching methodis sensitive to the boundary effects and sample processing isvery difficult Therefore we chose the blasting static pressuremethod [14] for the ultimate failure stress testing which dealtwith the data as a whole and thus could reflect themechanicalproperties better
The test samples were produced according to the nationalstandardGBT15560 and the processing technical rawmate-rials were the same with the cylinder Figure 4 shows the sizeof this test sample As shown two ends of the cylinder werestrengthened by carbon fiber layers of 30mm width Table 2shows the parameters of this static pressure burst test Table 3shows the test result
It can be seen from Table 3 that the mean of the ultimatetest failure stress is 20676MPa which is slightly higher thanthe theoretical prediction of 2004Mpa It may be the reasonthat the overall performance of the cylinder material wouldbe slightly higher than fiber samples The parameters usedfor theoretical prediction were the mean value of the tensiletest results of fiber sample and thus its volatility is relativelylarge Another main reason may be that the length of the testsample in the static pressure burst test was required to be
Mathematical Problems in Engineering 3
Table 1 Geometry parameters and properties of each laminate
Table 2 Parameters of the static pressure burst test
Aluminium alloy(mm)
Process conditions Resin formula Curing systemGlass fiber Carbon fiber
Thickness 12Length 350 5 layers 10 layers Proprietary formula 75∘C4 h + 80∘C12 h
0
1500
3000
4500
0 1 2 3 4
Stre
ss (M
Pa)
Strain ()
Figure 2 Tensile stress-strain curves of the glass fiber
0
2000
4000
6000
0 05 1 15 2 25
Stre
ss (M
Pa)
Strain ()
Figure 3 Tensile stress-strain curves of the carbon fiber
at least 5 times greater than the diameter in test standard ofGBT15560 But this long diameter ratio of our test sampleis only 27 As the shorter sample is sensitive to the effectsof the end portion in the testing process this may be themain reason that the test results are slightly higher than theprediction result In all the test results of the static pressureburst test made a good agreement with the prediction resultand it is feasible to verify the overall performance of thecylinder This test result would be taken as the importantreference for the constant stress accelerated life test in thefollowing research
3 Accelerated Life Test
31 Experiment Preparation In the working conditions thebearing stress of cylinder is about 9138Mpa which is equiv-alent to 455 of the ultimate failure stress We calculated
Table 3 Test result of the static pressure burst test
140Figure 4 Size of the static pressure burst test
the stress level of the cylinder by the finite element methodand the result shows that the bearing stress of the liningaluminum alloy is 240MPa the bearing stress of the glassfiber is 1060Mpa and the bearing of carbon fiber is 1350MPaThe related parameters are shown in Table 1
The cylinder used in the accelerated life test is the sameas that used in the static pressure burst test as shown inFigure 5 Then the cylinder is filled with hydraulic oil up tocertain pressure and maintains this pressure for a long timeso that the cylinder bears a uniform inner pressure in everydirectionThe test device is placed in one ovenwhereworkingtemperature could be maintained The pressure can be testedby the pressure sensor If the pressure of the cylinder dropssignificantly the cylinder was considered to be failure and thefailure time is automatically recorded
4 Mathematical Problems in Engineering
Compression nut
Exhaust port
Gland
Seal ring
Mandrel
Hydraulic shaft
Test sample
Oil-in
P
P
P
P
P
P
Figure 5 Diagram of the reliability test unit of the cylinder
32 The Accelerated Test Plan of the Cylinder We knowthe ultimate failure stress is 20676Mpa given by the staticpressure burst test and the working stress is 9138Mpa whichis equivalent to 455 of the ultimate failure stress of thecylinderThe stress-strain curve is close to linear in the scopeof the ultimate failure stress of the cylinder
According to GB 26891-81 we divided the cylinders intofive groups for constant stress accelerated life test And weset the minimum stress level to be 1315MPa (almost 636of the ultimate failure stress) and the maximum stress levelto be 1861MPa (almost 90 of the ultimate failure stress)Also other stress levels were set to be 727 80 and 85 ofthe ultimate failure stress respectively According to the teststandard of GB 26891-81 this test level setting could ensurethat the failure mechanism of cylinders was the same Theworking temperature of the cylinder will not exceed 40∘C sothe test temperature is controlled at 40 plusmn 2∘C
4 Results and Discussion
41 The Method on Rupture Life Evaluation of the CylinderWe assume that there are 119899
119895samples prepared for type-I
censored test in the V119895(119895 = 1 2 119904) stress level and the
censored time is 120585lowast119895 There are 119902
119895(1 le 119902
119895le 119899119895) failures and
the censored data is 1205851198951le sdot sdot sdot le 120585
119895119902119895
Hypothesis 1 The product life follows Weibull distribution119882(119898119895 120578119895) in the V
119895stress level
Hypothesis 2 The failure mechanisms of the carbon fiber andthe cylinder are the same in each stress level
Hypothesis 3 The relationship between characteristic life 120578119895
and stress level V119895follows the inverse power law model 120578
119895=
119860Vminus119888119895 119895 = 1 2 119904 And 119860 119888 are parameters that should be
estimated
The linear expression can be obtained by logarithmictransformation on the accelerated model in Hypothesis 3
ln 120578119895= 119886 + 119887119909
119895 (3)
Then the logarithmic life follows the extreme value distribu-tion and the relationship between the logarithmic life and thelogarithmic stress can be written as
119895 119886 = ln119860 and 119887 = minus119888 Parameter
119886 reflects the characteristic of test product parameter 119887
reflects the acceleration characteristic of the test 120590 is the scaleparameter of the extreme value distribution and the measureparameter for the consistency of the failure mechanism
In the condition of Hypothesis 1 type-I censored data1199101198951
le sdot sdot sdot le 119910119895119902119895
can be taken as the value of the former 119902119895
order statistics1198841198951le sdot sdot sdot le 119884
119895119902119895for the extreme value distribu-
tion with size 119899119895 119910119895(119902119895+1)
= 119910lowast
119895can be taken as the value of the
119902119895th interval statistics 119884
119895(119902119895+1)with the same sample
From literature [9 10] we can obtain the estimations of 119886119887 and 120590 by partial derivative for 119876
119876 =
119904
sum
119895=1
119902119895+1
sum
119896119897=1
(119910119895119896minus 119886 minus 119887119909
119895minus 120590119906119895119896)
times 119892119895119896119897(119910119895119897minus 119886 minus 119887119909
The Failure Mechanism Consistency Analysis This analysisfocused on the consistency of failure mechanism and acceler-ated model parameter between the carbon fiber and cylinderin the accelerated life test Denote the model parameters ofthe carbon fiber and the cylinder as 119886
1 1198871 and 120590
1and 1198862 1198872
and 1205902 respectively We assume the following
(1) The distribution parameters 1205901 1205902are two indepen-
dent normal populations If the failure mechanism ofthe carbon fiber is the same with the cylinder themean and variance of the two normal populations arethe same [16 17]
(2) Parameters of 1198861and 119886
2reflect the life characteristic
of carbon fiber and cylinder and then 1198861and 1198862have
no relation
(3) The model parameters 1198871 1198872are two independent
normal populations If the acceleration of the carbonfiber is the same with the cylinder the mean andvariance of the two normal populations are the same
Based on the above assumptions (1 1) and (
2 2) can
be taken as two bivariate normal populations We can judgethe consistency of the mean vector and covariance matrix ofthe two bivariate normal populations by hypothesis test
(1) The Consistency Judgment of the Mean Vector The twoindependent normal populations are denoted by (
1 1) sim
1198732(1205831 Σ1) and (
2 2) sim 119873
2(1205832 Σ2) We sample 119899 119898 gt
2 specimens from them respectively and denote the meanvectors by 119883 119884 respectively and the variance matrix by119878119894(119894 = 1 2) The hypothesis is
1198670 1205831= 1205832
1198671 1205831
= 1205832 (10)
When Σ1= Σ2and they were unknown the test statistic
1198792=
119899119898
119899 + 119898(119883 minus 119884)
119879
119878minus1(119883 minus 119884) (11)
where
119878 =(119899 minus 1) 119878
1+ (119898 minus 1) 119878
2
119899 + 119898 minus 2 (12)
And 119865 = (((119899 +119898minus2) minus 1)2(119899 +119898minus2))1198792sim 119865(2 119899 +119898minus3)
Then the rejection region with the significance level 120572 is
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) (13)
(2) The Consistency Judgment of the Variance Matrix Thehypothesis is
1198670 Σ1= Σ2
1198671 Σ1
= Σ2 (14)
The amendatory likelihood ratio statistic is
120582lowast=
(119899 + 119898 minus 2)(119899+119898minus2)10038161003816100381610038161198781
(119898 minus 1)(119898minus1)210038161003816100381610038161198781 + 1198782
1003816100381610038161003816
(119899+119898minus2)2 (15)
Then
minus2 ln 120582lowast sim 1205942(
119891
1 minus 119889) (16)
6 Mathematical Problems in Engineering
where
119891 =1
2119901 (119901 + 1) (119896 minus 1)
119889 =21199012+ 3119901 minus 1
6 (119901 + 1) (119896 minus 1)(
1
119899 minus 1+
1
119898 minus 1minus
1
119899 + 119898 minus 119896)
(17)
and 119896 = 119901 = 2Then the rejection region with the significance level 120572 is
1205942lt 1205942
120572(
119891
1 minus 119889) (18)
The information fusion methodBased on the result of the integral best linear unbiased
estimation and the consistency analysis of failuremechanismwe proposed an information fusion method for the covari-ance matrixes of the T700 carbon fiber and the cylinderThe fusion process can be accomplished by the two stepsfirstly fusing the information only related to parameters 119887and 120590 in the covariance matrix and then secondly to furtherimprove the evaluation accuracy of the covariance matrix formodel parameter of the cylinder fusing the other informationrelated to 119887 and 120590
Denote the point estimations of T700 carbon fiber and thecylinder by 119886
1 1 and
1and 1198862 2 and
2 respectively and
their covariance matrix is
cov (119886119894 119894 119894) =
[[
[
var (119886119894) cov (119886
119894 119894) cov (119886
119894 119894)
cov (119886119894 119894) var (
119894) cov (
119894 119894)
cov (119886119894 119894) cov (
119894 119894) var (
119894)
]]
]
(119894 = 1 2)
(19)
Denote the covariance matrix of the bivariate normaldistribution of 119887
1 1205901and 1198872 1205902 respectively by
119894= [
var (119894) cov (
119894 119894)
cov (119894 119894) var (
119894)] (119894 = 1 2) (20)
The elements of the above matrixes are one part of (19)The first step is fusing the information only related to
parameters 119887 and 120590 in the covariance matrixIf the covariance matrixes 119881
1and 119881
2are certified to be
the same by the consistency analysis of the bivariate normaldistribution (119887
1 1205901) and (119887
2 1205902) the unbiased estimation of
the covariancematrix of the bivariate normal distribution canbe obtained by the following equation
=(1198991minus 1)
1+ (1198992minus 1)
2
1198991+ 1198992minus 2
(21)
Compared to the small ratio of failure and high censoredtime of the cylinder life test the data and failures of the carbonfiber are much greater and thus the evaluation result is moreaccurateTherefore when the matrix
2is replaced by with
fusing the information of parameters 119887 and 120590 according to theabove method the prediction result of the cylinder is more
accurate The covariance matrix of 119887 and 120590 based on fusioncould be written as
= [var (
12) cov (
12 12)
cov (12 12) var (
12)
] (22)
The covariancematrix cov(1198862 2 2) for the cylinder with
bottom right four elements replaced by could be written as
cov (1198862 2 2)1015840
=[[
[
var (1198862) cov (119886
2 2) cov (119886
2 2)
cov (1198862 2) var (
12) cov (
12 12)
cov (1198862 2) cov (
12 12) var (
12)
]]
]
(23)
The second step is fusing the other elements related to 119887 120590of the covariance matrix
Based on the same correlation coefficient between theparameters the value of cov(119886
2 12) cov(119886
2 12) with fusing
the information of the T700 carbon fiber can be calculated as
cov (1198862 12) = radic
var (12)
var (2)
sdot cov (1198862 2)
cov (1198862 12) = radic
var (12)
var (2)sdot cov (119886
2 2)
(24)
According to the above two steps the covariance matrixcov(1198862 12 12) can be written as
cov (1198862 12 12)
=[[
[
var (1198862) cov (119886
2 12) cov (119886
2 12)
cov (1198862 12) var (
12) cov (
12 12)
cov (1198862 12) cov (
12 12) var (
12)
]]
]
(25)
Comparing with (25) and (19) (119894 = 2) it can be found thatthe elements of parameter covariance matrix of the cylinderhave changed in addition to the variance var(119886
2) by fusing
the carbon fiber test information thus making the evaluationresult more reasonable It should be noted that the parametercovariance matrix can be obtained by the integrated bestlinear unbiased estimation as follows
cov (119886 ) = 1205902119862 (26)
In the calculation of the upper and lower limits for thereliability rupture life thematrix119862would be usedThereforein fusion process of the covariance matrix with T700 carbonfiber information and cylinder information we can introducethe matrix 119862 directly in the above method to make thecalculation easier
42 Examples To verify the design level of the rupture life ofa certain type of cylinder a unit made accelerated life testsfor T700 carbon fiber composite material and the cylinder
Mathematical Problems in Engineering 7
Table 4 Accelerated life test information for T700 carbon fiber andthe cylinder
structure respectively The test temperature is 40∘C and thetest load conditions are shown in Table 4
The diagramof rupture life data for T700 carbon fiber andthe cylinder structure from the test is as in Figures 6 and 7
(1) The Result of the Integral Best Unbiased Estimation Theresults of model parameters and the matrix 119862 obtained bythe integrated best linear unbiased estimation are shown inTable 5
(2) The Test of Mean Vector and Covariance Matrix Equalto ( ) of the Carbon Fiber and the Cylinder We considerparameters ( ) as a bivariate normal population and testwhether the mean vectors and covariance matrixes of ( )for the carbon fiber and the cylinder are equal or not and theresults are shown in Table 6
From Table 6 the observed values of the test statistic are
119865 = 809725 lt 1198650975
(2 5) = 843
1205942= 19468 gt 120594
2
0025(5)
(27)
Then we can receive the null hypothesis The mean vectorsand the covariance matrixes of ( ) for the carbon fiber andthe cylinder are equal
(3) The Information Fusion of the T700 Carbon Fiber andthe Cylinder By using the method in this paper the matrixinformation of carbon fiber in Table 5 can be fused into thecylinder and the upper and lower limits of the reliabilityrupture life at confidence level 120574 = 095 can be calculatedThe comparison results are shown in Table 7
From Table 7 when the reliability is 09 the evaluationaccuracy of the reliability life of the cylinder is increased by35
The curves of upper and lower limits of the logarithmicreliability rupture life 119910
119877changed with the reliability 119877 are
shown in Figure 7Figure 8 shows that the upper and lower limits of the
reliability rupture lifewill bemore accuratewith the reliabilitychanged after fusion of the information of T700 carbon fiberand the interval length is much shorter
5 Conclusion
(1) Since the carbon fiber bears themain load at work theacceleration and failure mechanism of T700 carbonfiber and the cylinder are the same And it can beproved by the structure analysis and the statistic testof the test data
1700 1750 1800 1850 1900 1950 2000
0
2000
4000
6000
8000
10000
12000
14000
Failure timeCensored time
17 failure2 censored
25 failure6 censored
20 failure13 censored
Stress level (N)
Rupt
ure l
ife120585
(h)
Figure 6 Rupture life of T700 carbon fiber
60 65 70 75 80 85 90
0
5000
10000
15000
20000
25000
30000
35000
Stress level (N)
0 failure8 censored
0 failure9 censored
0 failure9 censored
3 failure3 censored
3 failure3 censored
Failure timeCensored time
Rupt
ure l
ife120585
(h)
Figure 7 Ruptures life of the cylinder
(2) When the acceleration and failure mechanism are thesame the evaluation accuracy of the reliability lifefor the cylinder can be improved by fusion of theinformation of the carbon fiber
(3) The method in this paper is based on Weibull distri-bution and the inverse power law model for struc-tured products It can be applied to other location-scale family distribution and acceleration models
8 Mathematical Problems in Engineering
Table 5 Contrast result between T700 carbon fiber and the cylinder with the integrated best linear unbiased estimation
Table 6 The test results of mean vector and covariance matrix for a bivariate normal population ( ) of carbon fiber and cylinder
Null hypothesis Rejection region Observation value Critical value 120572 = 0025
1205831= 1205832
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) 809725 843Σ1= Σ2
1205942lt 1205942
120572(119891(1 minus 119889)) 19468 0831
Table 7 Contrast result of cylinder after fusion information of T700 carbon fiber
Result The original data Fusion of the information of T700
119862[[
[
053 270 minus006
270 2149 minus105
minus006 minus105 013
]]
]
[[
[
053 208 minus0037
208 1276 minus037
minus0037 minus037 0049
]]
]
The interval estimation of 119910119877
[1379 2735] [1429 2315]Interval length 1356 886
080 085 090 095 10010
15
20
25
30
35
40
Reliability (R)
Loga
rithm
ic re
liabi
lity
lifey
R(h
)
After fusion yRLAfter fusion yRUyR
Before fusion yRLBefore fusion yRU
Figure 8 Change curves for 119910119877and upper and lower limits
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work described in this paper was supported by theNational Natural Science Foundation of China (Grants nos61104133 and 61473014) The authors fully appreciate thefinancial support The authors would like to thank thereviewers and the editor for many suggestions that helpedimprove this paper
References
[1] D F Li H JWang and F He ldquoStructure and properties of T300and T700 carbon fiberrdquoNew CarbonMaterials vol 22 no 1 pp59ndash64 2007
[2] G T Zhang W G Chen and B Yang ldquoTesting research onmechanical properties of T700 carbon fiberepoxy compositesrdquoFiber Composites vol 2 pp 49ndash52 2009
[3] Y X Zhou M A Baseer H Mahfuz and S Jeelani ldquoStatisticalanalysis on the fatigue strength distribution of T700 carbonfiberrdquo Composites Science and Technology vol 66 no 13 pp2100ndash2106 2006
[4] Z G Yu S C Yang and B F Song ldquoComparison of wet and hotaging resistance of T700 and T300 carbon fiber strengthenedepoxy resin compositesrdquoMaterials for Mechanical Engineeringvol 33 no 6 pp 48ndash51 2009
[5] X Gu and X Xu ldquoNumerical simulation of damage in fiberreinforced composite laminates under high velocity impactrdquoActaMateriae Compositae Sinica vol 29 no 1 pp 150ndash161 2012
[6] J Liu Y X Bai Y L Tian X Y Huang C H Wang and JY Liang ldquoEffect of the process of electrochemical modificationon the surface structure and properties of PAN-based carbonfirbersrdquo Acta Materiae Compositae Sinica vol 29 no 2 pp 16ndash25 2012
Mathematical Problems in Engineering 9
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
Table 2 Parameters of the static pressure burst test
Aluminium alloy(mm)
Process conditions Resin formula Curing systemGlass fiber Carbon fiber
Thickness 12Length 350 5 layers 10 layers Proprietary formula 75∘C4 h + 80∘C12 h
0
1500
3000
4500
0 1 2 3 4
Stre
ss (M
Pa)
Strain ()
Figure 2 Tensile stress-strain curves of the glass fiber
0
2000
4000
6000
0 05 1 15 2 25
Stre
ss (M
Pa)
Strain ()
Figure 3 Tensile stress-strain curves of the carbon fiber
at least 5 times greater than the diameter in test standard ofGBT15560 But this long diameter ratio of our test sampleis only 27 As the shorter sample is sensitive to the effectsof the end portion in the testing process this may be themain reason that the test results are slightly higher than theprediction result In all the test results of the static pressureburst test made a good agreement with the prediction resultand it is feasible to verify the overall performance of thecylinder This test result would be taken as the importantreference for the constant stress accelerated life test in thefollowing research
3 Accelerated Life Test
31 Experiment Preparation In the working conditions thebearing stress of cylinder is about 9138Mpa which is equiv-alent to 455 of the ultimate failure stress We calculated
Table 3 Test result of the static pressure burst test
140Figure 4 Size of the static pressure burst test
the stress level of the cylinder by the finite element methodand the result shows that the bearing stress of the liningaluminum alloy is 240MPa the bearing stress of the glassfiber is 1060Mpa and the bearing of carbon fiber is 1350MPaThe related parameters are shown in Table 1
The cylinder used in the accelerated life test is the sameas that used in the static pressure burst test as shown inFigure 5 Then the cylinder is filled with hydraulic oil up tocertain pressure and maintains this pressure for a long timeso that the cylinder bears a uniform inner pressure in everydirectionThe test device is placed in one ovenwhereworkingtemperature could be maintained The pressure can be testedby the pressure sensor If the pressure of the cylinder dropssignificantly the cylinder was considered to be failure and thefailure time is automatically recorded
4 Mathematical Problems in Engineering
Compression nut
Exhaust port
Gland
Seal ring
Mandrel
Hydraulic shaft
Test sample
Oil-in
P
P
P
P
P
P
Figure 5 Diagram of the reliability test unit of the cylinder
32 The Accelerated Test Plan of the Cylinder We knowthe ultimate failure stress is 20676Mpa given by the staticpressure burst test and the working stress is 9138Mpa whichis equivalent to 455 of the ultimate failure stress of thecylinderThe stress-strain curve is close to linear in the scopeof the ultimate failure stress of the cylinder
According to GB 26891-81 we divided the cylinders intofive groups for constant stress accelerated life test And weset the minimum stress level to be 1315MPa (almost 636of the ultimate failure stress) and the maximum stress levelto be 1861MPa (almost 90 of the ultimate failure stress)Also other stress levels were set to be 727 80 and 85 ofthe ultimate failure stress respectively According to the teststandard of GB 26891-81 this test level setting could ensurethat the failure mechanism of cylinders was the same Theworking temperature of the cylinder will not exceed 40∘C sothe test temperature is controlled at 40 plusmn 2∘C
4 Results and Discussion
41 The Method on Rupture Life Evaluation of the CylinderWe assume that there are 119899
119895samples prepared for type-I
censored test in the V119895(119895 = 1 2 119904) stress level and the
censored time is 120585lowast119895 There are 119902
119895(1 le 119902
119895le 119899119895) failures and
the censored data is 1205851198951le sdot sdot sdot le 120585
119895119902119895
Hypothesis 1 The product life follows Weibull distribution119882(119898119895 120578119895) in the V
119895stress level
Hypothesis 2 The failure mechanisms of the carbon fiber andthe cylinder are the same in each stress level
Hypothesis 3 The relationship between characteristic life 120578119895
and stress level V119895follows the inverse power law model 120578
119895=
119860Vminus119888119895 119895 = 1 2 119904 And 119860 119888 are parameters that should be
estimated
The linear expression can be obtained by logarithmictransformation on the accelerated model in Hypothesis 3
ln 120578119895= 119886 + 119887119909
119895 (3)
Then the logarithmic life follows the extreme value distribu-tion and the relationship between the logarithmic life and thelogarithmic stress can be written as
119895 119886 = ln119860 and 119887 = minus119888 Parameter
119886 reflects the characteristic of test product parameter 119887
reflects the acceleration characteristic of the test 120590 is the scaleparameter of the extreme value distribution and the measureparameter for the consistency of the failure mechanism
In the condition of Hypothesis 1 type-I censored data1199101198951
le sdot sdot sdot le 119910119895119902119895
can be taken as the value of the former 119902119895
order statistics1198841198951le sdot sdot sdot le 119884
119895119902119895for the extreme value distribu-
tion with size 119899119895 119910119895(119902119895+1)
= 119910lowast
119895can be taken as the value of the
119902119895th interval statistics 119884
119895(119902119895+1)with the same sample
From literature [9 10] we can obtain the estimations of 119886119887 and 120590 by partial derivative for 119876
119876 =
119904
sum
119895=1
119902119895+1
sum
119896119897=1
(119910119895119896minus 119886 minus 119887119909
119895minus 120590119906119895119896)
times 119892119895119896119897(119910119895119897minus 119886 minus 119887119909
The Failure Mechanism Consistency Analysis This analysisfocused on the consistency of failure mechanism and acceler-ated model parameter between the carbon fiber and cylinderin the accelerated life test Denote the model parameters ofthe carbon fiber and the cylinder as 119886
1 1198871 and 120590
1and 1198862 1198872
and 1205902 respectively We assume the following
(1) The distribution parameters 1205901 1205902are two indepen-
dent normal populations If the failure mechanism ofthe carbon fiber is the same with the cylinder themean and variance of the two normal populations arethe same [16 17]
(2) Parameters of 1198861and 119886
2reflect the life characteristic
of carbon fiber and cylinder and then 1198861and 1198862have
no relation
(3) The model parameters 1198871 1198872are two independent
normal populations If the acceleration of the carbonfiber is the same with the cylinder the mean andvariance of the two normal populations are the same
Based on the above assumptions (1 1) and (
2 2) can
be taken as two bivariate normal populations We can judgethe consistency of the mean vector and covariance matrix ofthe two bivariate normal populations by hypothesis test
(1) The Consistency Judgment of the Mean Vector The twoindependent normal populations are denoted by (
1 1) sim
1198732(1205831 Σ1) and (
2 2) sim 119873
2(1205832 Σ2) We sample 119899 119898 gt
2 specimens from them respectively and denote the meanvectors by 119883 119884 respectively and the variance matrix by119878119894(119894 = 1 2) The hypothesis is
1198670 1205831= 1205832
1198671 1205831
= 1205832 (10)
When Σ1= Σ2and they were unknown the test statistic
1198792=
119899119898
119899 + 119898(119883 minus 119884)
119879
119878minus1(119883 minus 119884) (11)
where
119878 =(119899 minus 1) 119878
1+ (119898 minus 1) 119878
2
119899 + 119898 minus 2 (12)
And 119865 = (((119899 +119898minus2) minus 1)2(119899 +119898minus2))1198792sim 119865(2 119899 +119898minus3)
Then the rejection region with the significance level 120572 is
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) (13)
(2) The Consistency Judgment of the Variance Matrix Thehypothesis is
1198670 Σ1= Σ2
1198671 Σ1
= Σ2 (14)
The amendatory likelihood ratio statistic is
120582lowast=
(119899 + 119898 minus 2)(119899+119898minus2)10038161003816100381610038161198781
(119898 minus 1)(119898minus1)210038161003816100381610038161198781 + 1198782
1003816100381610038161003816
(119899+119898minus2)2 (15)
Then
minus2 ln 120582lowast sim 1205942(
119891
1 minus 119889) (16)
6 Mathematical Problems in Engineering
where
119891 =1
2119901 (119901 + 1) (119896 minus 1)
119889 =21199012+ 3119901 minus 1
6 (119901 + 1) (119896 minus 1)(
1
119899 minus 1+
1
119898 minus 1minus
1
119899 + 119898 minus 119896)
(17)
and 119896 = 119901 = 2Then the rejection region with the significance level 120572 is
1205942lt 1205942
120572(
119891
1 minus 119889) (18)
The information fusion methodBased on the result of the integral best linear unbiased
estimation and the consistency analysis of failuremechanismwe proposed an information fusion method for the covari-ance matrixes of the T700 carbon fiber and the cylinderThe fusion process can be accomplished by the two stepsfirstly fusing the information only related to parameters 119887and 120590 in the covariance matrix and then secondly to furtherimprove the evaluation accuracy of the covariance matrix formodel parameter of the cylinder fusing the other informationrelated to 119887 and 120590
Denote the point estimations of T700 carbon fiber and thecylinder by 119886
1 1 and
1and 1198862 2 and
2 respectively and
their covariance matrix is
cov (119886119894 119894 119894) =
[[
[
var (119886119894) cov (119886
119894 119894) cov (119886
119894 119894)
cov (119886119894 119894) var (
119894) cov (
119894 119894)
cov (119886119894 119894) cov (
119894 119894) var (
119894)
]]
]
(119894 = 1 2)
(19)
Denote the covariance matrix of the bivariate normaldistribution of 119887
1 1205901and 1198872 1205902 respectively by
119894= [
var (119894) cov (
119894 119894)
cov (119894 119894) var (
119894)] (119894 = 1 2) (20)
The elements of the above matrixes are one part of (19)The first step is fusing the information only related to
parameters 119887 and 120590 in the covariance matrixIf the covariance matrixes 119881
1and 119881
2are certified to be
the same by the consistency analysis of the bivariate normaldistribution (119887
1 1205901) and (119887
2 1205902) the unbiased estimation of
the covariancematrix of the bivariate normal distribution canbe obtained by the following equation
=(1198991minus 1)
1+ (1198992minus 1)
2
1198991+ 1198992minus 2
(21)
Compared to the small ratio of failure and high censoredtime of the cylinder life test the data and failures of the carbonfiber are much greater and thus the evaluation result is moreaccurateTherefore when the matrix
2is replaced by with
fusing the information of parameters 119887 and 120590 according to theabove method the prediction result of the cylinder is more
accurate The covariance matrix of 119887 and 120590 based on fusioncould be written as
= [var (
12) cov (
12 12)
cov (12 12) var (
12)
] (22)
The covariancematrix cov(1198862 2 2) for the cylinder with
bottom right four elements replaced by could be written as
cov (1198862 2 2)1015840
=[[
[
var (1198862) cov (119886
2 2) cov (119886
2 2)
cov (1198862 2) var (
12) cov (
12 12)
cov (1198862 2) cov (
12 12) var (
12)
]]
]
(23)
The second step is fusing the other elements related to 119887 120590of the covariance matrix
Based on the same correlation coefficient between theparameters the value of cov(119886
2 12) cov(119886
2 12) with fusing
the information of the T700 carbon fiber can be calculated as
cov (1198862 12) = radic
var (12)
var (2)
sdot cov (1198862 2)
cov (1198862 12) = radic
var (12)
var (2)sdot cov (119886
2 2)
(24)
According to the above two steps the covariance matrixcov(1198862 12 12) can be written as
cov (1198862 12 12)
=[[
[
var (1198862) cov (119886
2 12) cov (119886
2 12)
cov (1198862 12) var (
12) cov (
12 12)
cov (1198862 12) cov (
12 12) var (
12)
]]
]
(25)
Comparing with (25) and (19) (119894 = 2) it can be found thatthe elements of parameter covariance matrix of the cylinderhave changed in addition to the variance var(119886
2) by fusing
the carbon fiber test information thus making the evaluationresult more reasonable It should be noted that the parametercovariance matrix can be obtained by the integrated bestlinear unbiased estimation as follows
cov (119886 ) = 1205902119862 (26)
In the calculation of the upper and lower limits for thereliability rupture life thematrix119862would be usedThereforein fusion process of the covariance matrix with T700 carbonfiber information and cylinder information we can introducethe matrix 119862 directly in the above method to make thecalculation easier
42 Examples To verify the design level of the rupture life ofa certain type of cylinder a unit made accelerated life testsfor T700 carbon fiber composite material and the cylinder
Mathematical Problems in Engineering 7
Table 4 Accelerated life test information for T700 carbon fiber andthe cylinder
structure respectively The test temperature is 40∘C and thetest load conditions are shown in Table 4
The diagramof rupture life data for T700 carbon fiber andthe cylinder structure from the test is as in Figures 6 and 7
(1) The Result of the Integral Best Unbiased Estimation Theresults of model parameters and the matrix 119862 obtained bythe integrated best linear unbiased estimation are shown inTable 5
(2) The Test of Mean Vector and Covariance Matrix Equalto ( ) of the Carbon Fiber and the Cylinder We considerparameters ( ) as a bivariate normal population and testwhether the mean vectors and covariance matrixes of ( )for the carbon fiber and the cylinder are equal or not and theresults are shown in Table 6
From Table 6 the observed values of the test statistic are
119865 = 809725 lt 1198650975
(2 5) = 843
1205942= 19468 gt 120594
2
0025(5)
(27)
Then we can receive the null hypothesis The mean vectorsand the covariance matrixes of ( ) for the carbon fiber andthe cylinder are equal
(3) The Information Fusion of the T700 Carbon Fiber andthe Cylinder By using the method in this paper the matrixinformation of carbon fiber in Table 5 can be fused into thecylinder and the upper and lower limits of the reliabilityrupture life at confidence level 120574 = 095 can be calculatedThe comparison results are shown in Table 7
From Table 7 when the reliability is 09 the evaluationaccuracy of the reliability life of the cylinder is increased by35
The curves of upper and lower limits of the logarithmicreliability rupture life 119910
119877changed with the reliability 119877 are
shown in Figure 7Figure 8 shows that the upper and lower limits of the
reliability rupture lifewill bemore accuratewith the reliabilitychanged after fusion of the information of T700 carbon fiberand the interval length is much shorter
5 Conclusion
(1) Since the carbon fiber bears themain load at work theacceleration and failure mechanism of T700 carbonfiber and the cylinder are the same And it can beproved by the structure analysis and the statistic testof the test data
1700 1750 1800 1850 1900 1950 2000
0
2000
4000
6000
8000
10000
12000
14000
Failure timeCensored time
17 failure2 censored
25 failure6 censored
20 failure13 censored
Stress level (N)
Rupt
ure l
ife120585
(h)
Figure 6 Rupture life of T700 carbon fiber
60 65 70 75 80 85 90
0
5000
10000
15000
20000
25000
30000
35000
Stress level (N)
0 failure8 censored
0 failure9 censored
0 failure9 censored
3 failure3 censored
3 failure3 censored
Failure timeCensored time
Rupt
ure l
ife120585
(h)
Figure 7 Ruptures life of the cylinder
(2) When the acceleration and failure mechanism are thesame the evaluation accuracy of the reliability lifefor the cylinder can be improved by fusion of theinformation of the carbon fiber
(3) The method in this paper is based on Weibull distri-bution and the inverse power law model for struc-tured products It can be applied to other location-scale family distribution and acceleration models
8 Mathematical Problems in Engineering
Table 5 Contrast result between T700 carbon fiber and the cylinder with the integrated best linear unbiased estimation
Table 6 The test results of mean vector and covariance matrix for a bivariate normal population ( ) of carbon fiber and cylinder
Null hypothesis Rejection region Observation value Critical value 120572 = 0025
1205831= 1205832
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) 809725 843Σ1= Σ2
1205942lt 1205942
120572(119891(1 minus 119889)) 19468 0831
Table 7 Contrast result of cylinder after fusion information of T700 carbon fiber
Result The original data Fusion of the information of T700
119862[[
[
053 270 minus006
270 2149 minus105
minus006 minus105 013
]]
]
[[
[
053 208 minus0037
208 1276 minus037
minus0037 minus037 0049
]]
]
The interval estimation of 119910119877
[1379 2735] [1429 2315]Interval length 1356 886
080 085 090 095 10010
15
20
25
30
35
40
Reliability (R)
Loga
rithm
ic re
liabi
lity
lifey
R(h
)
After fusion yRLAfter fusion yRUyR
Before fusion yRLBefore fusion yRU
Figure 8 Change curves for 119910119877and upper and lower limits
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work described in this paper was supported by theNational Natural Science Foundation of China (Grants nos61104133 and 61473014) The authors fully appreciate thefinancial support The authors would like to thank thereviewers and the editor for many suggestions that helpedimprove this paper
References
[1] D F Li H JWang and F He ldquoStructure and properties of T300and T700 carbon fiberrdquoNew CarbonMaterials vol 22 no 1 pp59ndash64 2007
[2] G T Zhang W G Chen and B Yang ldquoTesting research onmechanical properties of T700 carbon fiberepoxy compositesrdquoFiber Composites vol 2 pp 49ndash52 2009
[3] Y X Zhou M A Baseer H Mahfuz and S Jeelani ldquoStatisticalanalysis on the fatigue strength distribution of T700 carbonfiberrdquo Composites Science and Technology vol 66 no 13 pp2100ndash2106 2006
[4] Z G Yu S C Yang and B F Song ldquoComparison of wet and hotaging resistance of T700 and T300 carbon fiber strengthenedepoxy resin compositesrdquoMaterials for Mechanical Engineeringvol 33 no 6 pp 48ndash51 2009
[5] X Gu and X Xu ldquoNumerical simulation of damage in fiberreinforced composite laminates under high velocity impactrdquoActaMateriae Compositae Sinica vol 29 no 1 pp 150ndash161 2012
[6] J Liu Y X Bai Y L Tian X Y Huang C H Wang and JY Liang ldquoEffect of the process of electrochemical modificationon the surface structure and properties of PAN-based carbonfirbersrdquo Acta Materiae Compositae Sinica vol 29 no 2 pp 16ndash25 2012
Mathematical Problems in Engineering 9
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
Figure 5 Diagram of the reliability test unit of the cylinder
32 The Accelerated Test Plan of the Cylinder We knowthe ultimate failure stress is 20676Mpa given by the staticpressure burst test and the working stress is 9138Mpa whichis equivalent to 455 of the ultimate failure stress of thecylinderThe stress-strain curve is close to linear in the scopeof the ultimate failure stress of the cylinder
According to GB 26891-81 we divided the cylinders intofive groups for constant stress accelerated life test And weset the minimum stress level to be 1315MPa (almost 636of the ultimate failure stress) and the maximum stress levelto be 1861MPa (almost 90 of the ultimate failure stress)Also other stress levels were set to be 727 80 and 85 ofthe ultimate failure stress respectively According to the teststandard of GB 26891-81 this test level setting could ensurethat the failure mechanism of cylinders was the same Theworking temperature of the cylinder will not exceed 40∘C sothe test temperature is controlled at 40 plusmn 2∘C
4 Results and Discussion
41 The Method on Rupture Life Evaluation of the CylinderWe assume that there are 119899
119895samples prepared for type-I
censored test in the V119895(119895 = 1 2 119904) stress level and the
censored time is 120585lowast119895 There are 119902
119895(1 le 119902
119895le 119899119895) failures and
the censored data is 1205851198951le sdot sdot sdot le 120585
119895119902119895
Hypothesis 1 The product life follows Weibull distribution119882(119898119895 120578119895) in the V
119895stress level
Hypothesis 2 The failure mechanisms of the carbon fiber andthe cylinder are the same in each stress level
Hypothesis 3 The relationship between characteristic life 120578119895
and stress level V119895follows the inverse power law model 120578
119895=
119860Vminus119888119895 119895 = 1 2 119904 And 119860 119888 are parameters that should be
estimated
The linear expression can be obtained by logarithmictransformation on the accelerated model in Hypothesis 3
ln 120578119895= 119886 + 119887119909
119895 (3)
Then the logarithmic life follows the extreme value distribu-tion and the relationship between the logarithmic life and thelogarithmic stress can be written as
119895 119886 = ln119860 and 119887 = minus119888 Parameter
119886 reflects the characteristic of test product parameter 119887
reflects the acceleration characteristic of the test 120590 is the scaleparameter of the extreme value distribution and the measureparameter for the consistency of the failure mechanism
In the condition of Hypothesis 1 type-I censored data1199101198951
le sdot sdot sdot le 119910119895119902119895
can be taken as the value of the former 119902119895
order statistics1198841198951le sdot sdot sdot le 119884
119895119902119895for the extreme value distribu-
tion with size 119899119895 119910119895(119902119895+1)
= 119910lowast
119895can be taken as the value of the
119902119895th interval statistics 119884
119895(119902119895+1)with the same sample
From literature [9 10] we can obtain the estimations of 119886119887 and 120590 by partial derivative for 119876
119876 =
119904
sum
119895=1
119902119895+1
sum
119896119897=1
(119910119895119896minus 119886 minus 119887119909
119895minus 120590119906119895119896)
times 119892119895119896119897(119910119895119897minus 119886 minus 119887119909
The Failure Mechanism Consistency Analysis This analysisfocused on the consistency of failure mechanism and acceler-ated model parameter between the carbon fiber and cylinderin the accelerated life test Denote the model parameters ofthe carbon fiber and the cylinder as 119886
1 1198871 and 120590
1and 1198862 1198872
and 1205902 respectively We assume the following
(1) The distribution parameters 1205901 1205902are two indepen-
dent normal populations If the failure mechanism ofthe carbon fiber is the same with the cylinder themean and variance of the two normal populations arethe same [16 17]
(2) Parameters of 1198861and 119886
2reflect the life characteristic
of carbon fiber and cylinder and then 1198861and 1198862have
no relation
(3) The model parameters 1198871 1198872are two independent
normal populations If the acceleration of the carbonfiber is the same with the cylinder the mean andvariance of the two normal populations are the same
Based on the above assumptions (1 1) and (
2 2) can
be taken as two bivariate normal populations We can judgethe consistency of the mean vector and covariance matrix ofthe two bivariate normal populations by hypothesis test
(1) The Consistency Judgment of the Mean Vector The twoindependent normal populations are denoted by (
1 1) sim
1198732(1205831 Σ1) and (
2 2) sim 119873
2(1205832 Σ2) We sample 119899 119898 gt
2 specimens from them respectively and denote the meanvectors by 119883 119884 respectively and the variance matrix by119878119894(119894 = 1 2) The hypothesis is
1198670 1205831= 1205832
1198671 1205831
= 1205832 (10)
When Σ1= Σ2and they were unknown the test statistic
1198792=
119899119898
119899 + 119898(119883 minus 119884)
119879
119878minus1(119883 minus 119884) (11)
where
119878 =(119899 minus 1) 119878
1+ (119898 minus 1) 119878
2
119899 + 119898 minus 2 (12)
And 119865 = (((119899 +119898minus2) minus 1)2(119899 +119898minus2))1198792sim 119865(2 119899 +119898minus3)
Then the rejection region with the significance level 120572 is
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) (13)
(2) The Consistency Judgment of the Variance Matrix Thehypothesis is
1198670 Σ1= Σ2
1198671 Σ1
= Σ2 (14)
The amendatory likelihood ratio statistic is
120582lowast=
(119899 + 119898 minus 2)(119899+119898minus2)10038161003816100381610038161198781
(119898 minus 1)(119898minus1)210038161003816100381610038161198781 + 1198782
1003816100381610038161003816
(119899+119898minus2)2 (15)
Then
minus2 ln 120582lowast sim 1205942(
119891
1 minus 119889) (16)
6 Mathematical Problems in Engineering
where
119891 =1
2119901 (119901 + 1) (119896 minus 1)
119889 =21199012+ 3119901 minus 1
6 (119901 + 1) (119896 minus 1)(
1
119899 minus 1+
1
119898 minus 1minus
1
119899 + 119898 minus 119896)
(17)
and 119896 = 119901 = 2Then the rejection region with the significance level 120572 is
1205942lt 1205942
120572(
119891
1 minus 119889) (18)
The information fusion methodBased on the result of the integral best linear unbiased
estimation and the consistency analysis of failuremechanismwe proposed an information fusion method for the covari-ance matrixes of the T700 carbon fiber and the cylinderThe fusion process can be accomplished by the two stepsfirstly fusing the information only related to parameters 119887and 120590 in the covariance matrix and then secondly to furtherimprove the evaluation accuracy of the covariance matrix formodel parameter of the cylinder fusing the other informationrelated to 119887 and 120590
Denote the point estimations of T700 carbon fiber and thecylinder by 119886
1 1 and
1and 1198862 2 and
2 respectively and
their covariance matrix is
cov (119886119894 119894 119894) =
[[
[
var (119886119894) cov (119886
119894 119894) cov (119886
119894 119894)
cov (119886119894 119894) var (
119894) cov (
119894 119894)
cov (119886119894 119894) cov (
119894 119894) var (
119894)
]]
]
(119894 = 1 2)
(19)
Denote the covariance matrix of the bivariate normaldistribution of 119887
1 1205901and 1198872 1205902 respectively by
119894= [
var (119894) cov (
119894 119894)
cov (119894 119894) var (
119894)] (119894 = 1 2) (20)
The elements of the above matrixes are one part of (19)The first step is fusing the information only related to
parameters 119887 and 120590 in the covariance matrixIf the covariance matrixes 119881
1and 119881
2are certified to be
the same by the consistency analysis of the bivariate normaldistribution (119887
1 1205901) and (119887
2 1205902) the unbiased estimation of
the covariancematrix of the bivariate normal distribution canbe obtained by the following equation
=(1198991minus 1)
1+ (1198992minus 1)
2
1198991+ 1198992minus 2
(21)
Compared to the small ratio of failure and high censoredtime of the cylinder life test the data and failures of the carbonfiber are much greater and thus the evaluation result is moreaccurateTherefore when the matrix
2is replaced by with
fusing the information of parameters 119887 and 120590 according to theabove method the prediction result of the cylinder is more
accurate The covariance matrix of 119887 and 120590 based on fusioncould be written as
= [var (
12) cov (
12 12)
cov (12 12) var (
12)
] (22)
The covariancematrix cov(1198862 2 2) for the cylinder with
bottom right four elements replaced by could be written as
cov (1198862 2 2)1015840
=[[
[
var (1198862) cov (119886
2 2) cov (119886
2 2)
cov (1198862 2) var (
12) cov (
12 12)
cov (1198862 2) cov (
12 12) var (
12)
]]
]
(23)
The second step is fusing the other elements related to 119887 120590of the covariance matrix
Based on the same correlation coefficient between theparameters the value of cov(119886
2 12) cov(119886
2 12) with fusing
the information of the T700 carbon fiber can be calculated as
cov (1198862 12) = radic
var (12)
var (2)
sdot cov (1198862 2)
cov (1198862 12) = radic
var (12)
var (2)sdot cov (119886
2 2)
(24)
According to the above two steps the covariance matrixcov(1198862 12 12) can be written as
cov (1198862 12 12)
=[[
[
var (1198862) cov (119886
2 12) cov (119886
2 12)
cov (1198862 12) var (
12) cov (
12 12)
cov (1198862 12) cov (
12 12) var (
12)
]]
]
(25)
Comparing with (25) and (19) (119894 = 2) it can be found thatthe elements of parameter covariance matrix of the cylinderhave changed in addition to the variance var(119886
2) by fusing
the carbon fiber test information thus making the evaluationresult more reasonable It should be noted that the parametercovariance matrix can be obtained by the integrated bestlinear unbiased estimation as follows
cov (119886 ) = 1205902119862 (26)
In the calculation of the upper and lower limits for thereliability rupture life thematrix119862would be usedThereforein fusion process of the covariance matrix with T700 carbonfiber information and cylinder information we can introducethe matrix 119862 directly in the above method to make thecalculation easier
42 Examples To verify the design level of the rupture life ofa certain type of cylinder a unit made accelerated life testsfor T700 carbon fiber composite material and the cylinder
Mathematical Problems in Engineering 7
Table 4 Accelerated life test information for T700 carbon fiber andthe cylinder
structure respectively The test temperature is 40∘C and thetest load conditions are shown in Table 4
The diagramof rupture life data for T700 carbon fiber andthe cylinder structure from the test is as in Figures 6 and 7
(1) The Result of the Integral Best Unbiased Estimation Theresults of model parameters and the matrix 119862 obtained bythe integrated best linear unbiased estimation are shown inTable 5
(2) The Test of Mean Vector and Covariance Matrix Equalto ( ) of the Carbon Fiber and the Cylinder We considerparameters ( ) as a bivariate normal population and testwhether the mean vectors and covariance matrixes of ( )for the carbon fiber and the cylinder are equal or not and theresults are shown in Table 6
From Table 6 the observed values of the test statistic are
119865 = 809725 lt 1198650975
(2 5) = 843
1205942= 19468 gt 120594
2
0025(5)
(27)
Then we can receive the null hypothesis The mean vectorsand the covariance matrixes of ( ) for the carbon fiber andthe cylinder are equal
(3) The Information Fusion of the T700 Carbon Fiber andthe Cylinder By using the method in this paper the matrixinformation of carbon fiber in Table 5 can be fused into thecylinder and the upper and lower limits of the reliabilityrupture life at confidence level 120574 = 095 can be calculatedThe comparison results are shown in Table 7
From Table 7 when the reliability is 09 the evaluationaccuracy of the reliability life of the cylinder is increased by35
The curves of upper and lower limits of the logarithmicreliability rupture life 119910
119877changed with the reliability 119877 are
shown in Figure 7Figure 8 shows that the upper and lower limits of the
reliability rupture lifewill bemore accuratewith the reliabilitychanged after fusion of the information of T700 carbon fiberand the interval length is much shorter
5 Conclusion
(1) Since the carbon fiber bears themain load at work theacceleration and failure mechanism of T700 carbonfiber and the cylinder are the same And it can beproved by the structure analysis and the statistic testof the test data
1700 1750 1800 1850 1900 1950 2000
0
2000
4000
6000
8000
10000
12000
14000
Failure timeCensored time
17 failure2 censored
25 failure6 censored
20 failure13 censored
Stress level (N)
Rupt
ure l
ife120585
(h)
Figure 6 Rupture life of T700 carbon fiber
60 65 70 75 80 85 90
0
5000
10000
15000
20000
25000
30000
35000
Stress level (N)
0 failure8 censored
0 failure9 censored
0 failure9 censored
3 failure3 censored
3 failure3 censored
Failure timeCensored time
Rupt
ure l
ife120585
(h)
Figure 7 Ruptures life of the cylinder
(2) When the acceleration and failure mechanism are thesame the evaluation accuracy of the reliability lifefor the cylinder can be improved by fusion of theinformation of the carbon fiber
(3) The method in this paper is based on Weibull distri-bution and the inverse power law model for struc-tured products It can be applied to other location-scale family distribution and acceleration models
8 Mathematical Problems in Engineering
Table 5 Contrast result between T700 carbon fiber and the cylinder with the integrated best linear unbiased estimation
Table 6 The test results of mean vector and covariance matrix for a bivariate normal population ( ) of carbon fiber and cylinder
Null hypothesis Rejection region Observation value Critical value 120572 = 0025
1205831= 1205832
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) 809725 843Σ1= Σ2
1205942lt 1205942
120572(119891(1 minus 119889)) 19468 0831
Table 7 Contrast result of cylinder after fusion information of T700 carbon fiber
Result The original data Fusion of the information of T700
119862[[
[
053 270 minus006
270 2149 minus105
minus006 minus105 013
]]
]
[[
[
053 208 minus0037
208 1276 minus037
minus0037 minus037 0049
]]
]
The interval estimation of 119910119877
[1379 2735] [1429 2315]Interval length 1356 886
080 085 090 095 10010
15
20
25
30
35
40
Reliability (R)
Loga
rithm
ic re
liabi
lity
lifey
R(h
)
After fusion yRLAfter fusion yRUyR
Before fusion yRLBefore fusion yRU
Figure 8 Change curves for 119910119877and upper and lower limits
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work described in this paper was supported by theNational Natural Science Foundation of China (Grants nos61104133 and 61473014) The authors fully appreciate thefinancial support The authors would like to thank thereviewers and the editor for many suggestions that helpedimprove this paper
References
[1] D F Li H JWang and F He ldquoStructure and properties of T300and T700 carbon fiberrdquoNew CarbonMaterials vol 22 no 1 pp59ndash64 2007
[2] G T Zhang W G Chen and B Yang ldquoTesting research onmechanical properties of T700 carbon fiberepoxy compositesrdquoFiber Composites vol 2 pp 49ndash52 2009
[3] Y X Zhou M A Baseer H Mahfuz and S Jeelani ldquoStatisticalanalysis on the fatigue strength distribution of T700 carbonfiberrdquo Composites Science and Technology vol 66 no 13 pp2100ndash2106 2006
[4] Z G Yu S C Yang and B F Song ldquoComparison of wet and hotaging resistance of T700 and T300 carbon fiber strengthenedepoxy resin compositesrdquoMaterials for Mechanical Engineeringvol 33 no 6 pp 48ndash51 2009
[5] X Gu and X Xu ldquoNumerical simulation of damage in fiberreinforced composite laminates under high velocity impactrdquoActaMateriae Compositae Sinica vol 29 no 1 pp 150ndash161 2012
[6] J Liu Y X Bai Y L Tian X Y Huang C H Wang and JY Liang ldquoEffect of the process of electrochemical modificationon the surface structure and properties of PAN-based carbonfirbersrdquo Acta Materiae Compositae Sinica vol 29 no 2 pp 16ndash25 2012
Mathematical Problems in Engineering 9
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
The Failure Mechanism Consistency Analysis This analysisfocused on the consistency of failure mechanism and acceler-ated model parameter between the carbon fiber and cylinderin the accelerated life test Denote the model parameters ofthe carbon fiber and the cylinder as 119886
1 1198871 and 120590
1and 1198862 1198872
and 1205902 respectively We assume the following
(1) The distribution parameters 1205901 1205902are two indepen-
dent normal populations If the failure mechanism ofthe carbon fiber is the same with the cylinder themean and variance of the two normal populations arethe same [16 17]
(2) Parameters of 1198861and 119886
2reflect the life characteristic
of carbon fiber and cylinder and then 1198861and 1198862have
no relation
(3) The model parameters 1198871 1198872are two independent
normal populations If the acceleration of the carbonfiber is the same with the cylinder the mean andvariance of the two normal populations are the same
Based on the above assumptions (1 1) and (
2 2) can
be taken as two bivariate normal populations We can judgethe consistency of the mean vector and covariance matrix ofthe two bivariate normal populations by hypothesis test
(1) The Consistency Judgment of the Mean Vector The twoindependent normal populations are denoted by (
1 1) sim
1198732(1205831 Σ1) and (
2 2) sim 119873
2(1205832 Σ2) We sample 119899 119898 gt
2 specimens from them respectively and denote the meanvectors by 119883 119884 respectively and the variance matrix by119878119894(119894 = 1 2) The hypothesis is
1198670 1205831= 1205832
1198671 1205831
= 1205832 (10)
When Σ1= Σ2and they were unknown the test statistic
1198792=
119899119898
119899 + 119898(119883 minus 119884)
119879
119878minus1(119883 minus 119884) (11)
where
119878 =(119899 minus 1) 119878
1+ (119898 minus 1) 119878
2
119899 + 119898 minus 2 (12)
And 119865 = (((119899 +119898minus2) minus 1)2(119899 +119898minus2))1198792sim 119865(2 119899 +119898minus3)
Then the rejection region with the significance level 120572 is
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) (13)
(2) The Consistency Judgment of the Variance Matrix Thehypothesis is
1198670 Σ1= Σ2
1198671 Σ1
= Σ2 (14)
The amendatory likelihood ratio statistic is
120582lowast=
(119899 + 119898 minus 2)(119899+119898minus2)10038161003816100381610038161198781
(119898 minus 1)(119898minus1)210038161003816100381610038161198781 + 1198782
1003816100381610038161003816
(119899+119898minus2)2 (15)
Then
minus2 ln 120582lowast sim 1205942(
119891
1 minus 119889) (16)
6 Mathematical Problems in Engineering
where
119891 =1
2119901 (119901 + 1) (119896 minus 1)
119889 =21199012+ 3119901 minus 1
6 (119901 + 1) (119896 minus 1)(
1
119899 minus 1+
1
119898 minus 1minus
1
119899 + 119898 minus 119896)
(17)
and 119896 = 119901 = 2Then the rejection region with the significance level 120572 is
1205942lt 1205942
120572(
119891
1 minus 119889) (18)
The information fusion methodBased on the result of the integral best linear unbiased
estimation and the consistency analysis of failuremechanismwe proposed an information fusion method for the covari-ance matrixes of the T700 carbon fiber and the cylinderThe fusion process can be accomplished by the two stepsfirstly fusing the information only related to parameters 119887and 120590 in the covariance matrix and then secondly to furtherimprove the evaluation accuracy of the covariance matrix formodel parameter of the cylinder fusing the other informationrelated to 119887 and 120590
Denote the point estimations of T700 carbon fiber and thecylinder by 119886
1 1 and
1and 1198862 2 and
2 respectively and
their covariance matrix is
cov (119886119894 119894 119894) =
[[
[
var (119886119894) cov (119886
119894 119894) cov (119886
119894 119894)
cov (119886119894 119894) var (
119894) cov (
119894 119894)
cov (119886119894 119894) cov (
119894 119894) var (
119894)
]]
]
(119894 = 1 2)
(19)
Denote the covariance matrix of the bivariate normaldistribution of 119887
1 1205901and 1198872 1205902 respectively by
119894= [
var (119894) cov (
119894 119894)
cov (119894 119894) var (
119894)] (119894 = 1 2) (20)
The elements of the above matrixes are one part of (19)The first step is fusing the information only related to
parameters 119887 and 120590 in the covariance matrixIf the covariance matrixes 119881
1and 119881
2are certified to be
the same by the consistency analysis of the bivariate normaldistribution (119887
1 1205901) and (119887
2 1205902) the unbiased estimation of
the covariancematrix of the bivariate normal distribution canbe obtained by the following equation
=(1198991minus 1)
1+ (1198992minus 1)
2
1198991+ 1198992minus 2
(21)
Compared to the small ratio of failure and high censoredtime of the cylinder life test the data and failures of the carbonfiber are much greater and thus the evaluation result is moreaccurateTherefore when the matrix
2is replaced by with
fusing the information of parameters 119887 and 120590 according to theabove method the prediction result of the cylinder is more
accurate The covariance matrix of 119887 and 120590 based on fusioncould be written as
= [var (
12) cov (
12 12)
cov (12 12) var (
12)
] (22)
The covariancematrix cov(1198862 2 2) for the cylinder with
bottom right four elements replaced by could be written as
cov (1198862 2 2)1015840
=[[
[
var (1198862) cov (119886
2 2) cov (119886
2 2)
cov (1198862 2) var (
12) cov (
12 12)
cov (1198862 2) cov (
12 12) var (
12)
]]
]
(23)
The second step is fusing the other elements related to 119887 120590of the covariance matrix
Based on the same correlation coefficient between theparameters the value of cov(119886
2 12) cov(119886
2 12) with fusing
the information of the T700 carbon fiber can be calculated as
cov (1198862 12) = radic
var (12)
var (2)
sdot cov (1198862 2)
cov (1198862 12) = radic
var (12)
var (2)sdot cov (119886
2 2)
(24)
According to the above two steps the covariance matrixcov(1198862 12 12) can be written as
cov (1198862 12 12)
=[[
[
var (1198862) cov (119886
2 12) cov (119886
2 12)
cov (1198862 12) var (
12) cov (
12 12)
cov (1198862 12) cov (
12 12) var (
12)
]]
]
(25)
Comparing with (25) and (19) (119894 = 2) it can be found thatthe elements of parameter covariance matrix of the cylinderhave changed in addition to the variance var(119886
2) by fusing
the carbon fiber test information thus making the evaluationresult more reasonable It should be noted that the parametercovariance matrix can be obtained by the integrated bestlinear unbiased estimation as follows
cov (119886 ) = 1205902119862 (26)
In the calculation of the upper and lower limits for thereliability rupture life thematrix119862would be usedThereforein fusion process of the covariance matrix with T700 carbonfiber information and cylinder information we can introducethe matrix 119862 directly in the above method to make thecalculation easier
42 Examples To verify the design level of the rupture life ofa certain type of cylinder a unit made accelerated life testsfor T700 carbon fiber composite material and the cylinder
Mathematical Problems in Engineering 7
Table 4 Accelerated life test information for T700 carbon fiber andthe cylinder
structure respectively The test temperature is 40∘C and thetest load conditions are shown in Table 4
The diagramof rupture life data for T700 carbon fiber andthe cylinder structure from the test is as in Figures 6 and 7
(1) The Result of the Integral Best Unbiased Estimation Theresults of model parameters and the matrix 119862 obtained bythe integrated best linear unbiased estimation are shown inTable 5
(2) The Test of Mean Vector and Covariance Matrix Equalto ( ) of the Carbon Fiber and the Cylinder We considerparameters ( ) as a bivariate normal population and testwhether the mean vectors and covariance matrixes of ( )for the carbon fiber and the cylinder are equal or not and theresults are shown in Table 6
From Table 6 the observed values of the test statistic are
119865 = 809725 lt 1198650975
(2 5) = 843
1205942= 19468 gt 120594
2
0025(5)
(27)
Then we can receive the null hypothesis The mean vectorsand the covariance matrixes of ( ) for the carbon fiber andthe cylinder are equal
(3) The Information Fusion of the T700 Carbon Fiber andthe Cylinder By using the method in this paper the matrixinformation of carbon fiber in Table 5 can be fused into thecylinder and the upper and lower limits of the reliabilityrupture life at confidence level 120574 = 095 can be calculatedThe comparison results are shown in Table 7
From Table 7 when the reliability is 09 the evaluationaccuracy of the reliability life of the cylinder is increased by35
The curves of upper and lower limits of the logarithmicreliability rupture life 119910
119877changed with the reliability 119877 are
shown in Figure 7Figure 8 shows that the upper and lower limits of the
reliability rupture lifewill bemore accuratewith the reliabilitychanged after fusion of the information of T700 carbon fiberand the interval length is much shorter
5 Conclusion
(1) Since the carbon fiber bears themain load at work theacceleration and failure mechanism of T700 carbonfiber and the cylinder are the same And it can beproved by the structure analysis and the statistic testof the test data
1700 1750 1800 1850 1900 1950 2000
0
2000
4000
6000
8000
10000
12000
14000
Failure timeCensored time
17 failure2 censored
25 failure6 censored
20 failure13 censored
Stress level (N)
Rupt
ure l
ife120585
(h)
Figure 6 Rupture life of T700 carbon fiber
60 65 70 75 80 85 90
0
5000
10000
15000
20000
25000
30000
35000
Stress level (N)
0 failure8 censored
0 failure9 censored
0 failure9 censored
3 failure3 censored
3 failure3 censored
Failure timeCensored time
Rupt
ure l
ife120585
(h)
Figure 7 Ruptures life of the cylinder
(2) When the acceleration and failure mechanism are thesame the evaluation accuracy of the reliability lifefor the cylinder can be improved by fusion of theinformation of the carbon fiber
(3) The method in this paper is based on Weibull distri-bution and the inverse power law model for struc-tured products It can be applied to other location-scale family distribution and acceleration models
8 Mathematical Problems in Engineering
Table 5 Contrast result between T700 carbon fiber and the cylinder with the integrated best linear unbiased estimation
Table 6 The test results of mean vector and covariance matrix for a bivariate normal population ( ) of carbon fiber and cylinder
Null hypothesis Rejection region Observation value Critical value 120572 = 0025
1205831= 1205832
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) 809725 843Σ1= Σ2
1205942lt 1205942
120572(119891(1 minus 119889)) 19468 0831
Table 7 Contrast result of cylinder after fusion information of T700 carbon fiber
Result The original data Fusion of the information of T700
119862[[
[
053 270 minus006
270 2149 minus105
minus006 minus105 013
]]
]
[[
[
053 208 minus0037
208 1276 minus037
minus0037 minus037 0049
]]
]
The interval estimation of 119910119877
[1379 2735] [1429 2315]Interval length 1356 886
080 085 090 095 10010
15
20
25
30
35
40
Reliability (R)
Loga
rithm
ic re
liabi
lity
lifey
R(h
)
After fusion yRLAfter fusion yRUyR
Before fusion yRLBefore fusion yRU
Figure 8 Change curves for 119910119877and upper and lower limits
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work described in this paper was supported by theNational Natural Science Foundation of China (Grants nos61104133 and 61473014) The authors fully appreciate thefinancial support The authors would like to thank thereviewers and the editor for many suggestions that helpedimprove this paper
References
[1] D F Li H JWang and F He ldquoStructure and properties of T300and T700 carbon fiberrdquoNew CarbonMaterials vol 22 no 1 pp59ndash64 2007
[2] G T Zhang W G Chen and B Yang ldquoTesting research onmechanical properties of T700 carbon fiberepoxy compositesrdquoFiber Composites vol 2 pp 49ndash52 2009
[3] Y X Zhou M A Baseer H Mahfuz and S Jeelani ldquoStatisticalanalysis on the fatigue strength distribution of T700 carbonfiberrdquo Composites Science and Technology vol 66 no 13 pp2100ndash2106 2006
[4] Z G Yu S C Yang and B F Song ldquoComparison of wet and hotaging resistance of T700 and T300 carbon fiber strengthenedepoxy resin compositesrdquoMaterials for Mechanical Engineeringvol 33 no 6 pp 48ndash51 2009
[5] X Gu and X Xu ldquoNumerical simulation of damage in fiberreinforced composite laminates under high velocity impactrdquoActaMateriae Compositae Sinica vol 29 no 1 pp 150ndash161 2012
[6] J Liu Y X Bai Y L Tian X Y Huang C H Wang and JY Liang ldquoEffect of the process of electrochemical modificationon the surface structure and properties of PAN-based carbonfirbersrdquo Acta Materiae Compositae Sinica vol 29 no 2 pp 16ndash25 2012
Mathematical Problems in Engineering 9
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
and 119896 = 119901 = 2Then the rejection region with the significance level 120572 is
1205942lt 1205942
120572(
119891
1 minus 119889) (18)
The information fusion methodBased on the result of the integral best linear unbiased
estimation and the consistency analysis of failuremechanismwe proposed an information fusion method for the covari-ance matrixes of the T700 carbon fiber and the cylinderThe fusion process can be accomplished by the two stepsfirstly fusing the information only related to parameters 119887and 120590 in the covariance matrix and then secondly to furtherimprove the evaluation accuracy of the covariance matrix formodel parameter of the cylinder fusing the other informationrelated to 119887 and 120590
Denote the point estimations of T700 carbon fiber and thecylinder by 119886
1 1 and
1and 1198862 2 and
2 respectively and
their covariance matrix is
cov (119886119894 119894 119894) =
[[
[
var (119886119894) cov (119886
119894 119894) cov (119886
119894 119894)
cov (119886119894 119894) var (
119894) cov (
119894 119894)
cov (119886119894 119894) cov (
119894 119894) var (
119894)
]]
]
(119894 = 1 2)
(19)
Denote the covariance matrix of the bivariate normaldistribution of 119887
1 1205901and 1198872 1205902 respectively by
119894= [
var (119894) cov (
119894 119894)
cov (119894 119894) var (
119894)] (119894 = 1 2) (20)
The elements of the above matrixes are one part of (19)The first step is fusing the information only related to
parameters 119887 and 120590 in the covariance matrixIf the covariance matrixes 119881
1and 119881
2are certified to be
the same by the consistency analysis of the bivariate normaldistribution (119887
1 1205901) and (119887
2 1205902) the unbiased estimation of
the covariancematrix of the bivariate normal distribution canbe obtained by the following equation
=(1198991minus 1)
1+ (1198992minus 1)
2
1198991+ 1198992minus 2
(21)
Compared to the small ratio of failure and high censoredtime of the cylinder life test the data and failures of the carbonfiber are much greater and thus the evaluation result is moreaccurateTherefore when the matrix
2is replaced by with
fusing the information of parameters 119887 and 120590 according to theabove method the prediction result of the cylinder is more
accurate The covariance matrix of 119887 and 120590 based on fusioncould be written as
= [var (
12) cov (
12 12)
cov (12 12) var (
12)
] (22)
The covariancematrix cov(1198862 2 2) for the cylinder with
bottom right four elements replaced by could be written as
cov (1198862 2 2)1015840
=[[
[
var (1198862) cov (119886
2 2) cov (119886
2 2)
cov (1198862 2) var (
12) cov (
12 12)
cov (1198862 2) cov (
12 12) var (
12)
]]
]
(23)
The second step is fusing the other elements related to 119887 120590of the covariance matrix
Based on the same correlation coefficient between theparameters the value of cov(119886
2 12) cov(119886
2 12) with fusing
the information of the T700 carbon fiber can be calculated as
cov (1198862 12) = radic
var (12)
var (2)
sdot cov (1198862 2)
cov (1198862 12) = radic
var (12)
var (2)sdot cov (119886
2 2)
(24)
According to the above two steps the covariance matrixcov(1198862 12 12) can be written as
cov (1198862 12 12)
=[[
[
var (1198862) cov (119886
2 12) cov (119886
2 12)
cov (1198862 12) var (
12) cov (
12 12)
cov (1198862 12) cov (
12 12) var (
12)
]]
]
(25)
Comparing with (25) and (19) (119894 = 2) it can be found thatthe elements of parameter covariance matrix of the cylinderhave changed in addition to the variance var(119886
2) by fusing
the carbon fiber test information thus making the evaluationresult more reasonable It should be noted that the parametercovariance matrix can be obtained by the integrated bestlinear unbiased estimation as follows
cov (119886 ) = 1205902119862 (26)
In the calculation of the upper and lower limits for thereliability rupture life thematrix119862would be usedThereforein fusion process of the covariance matrix with T700 carbonfiber information and cylinder information we can introducethe matrix 119862 directly in the above method to make thecalculation easier
42 Examples To verify the design level of the rupture life ofa certain type of cylinder a unit made accelerated life testsfor T700 carbon fiber composite material and the cylinder
Mathematical Problems in Engineering 7
Table 4 Accelerated life test information for T700 carbon fiber andthe cylinder
structure respectively The test temperature is 40∘C and thetest load conditions are shown in Table 4
The diagramof rupture life data for T700 carbon fiber andthe cylinder structure from the test is as in Figures 6 and 7
(1) The Result of the Integral Best Unbiased Estimation Theresults of model parameters and the matrix 119862 obtained bythe integrated best linear unbiased estimation are shown inTable 5
(2) The Test of Mean Vector and Covariance Matrix Equalto ( ) of the Carbon Fiber and the Cylinder We considerparameters ( ) as a bivariate normal population and testwhether the mean vectors and covariance matrixes of ( )for the carbon fiber and the cylinder are equal or not and theresults are shown in Table 6
From Table 6 the observed values of the test statistic are
119865 = 809725 lt 1198650975
(2 5) = 843
1205942= 19468 gt 120594
2
0025(5)
(27)
Then we can receive the null hypothesis The mean vectorsand the covariance matrixes of ( ) for the carbon fiber andthe cylinder are equal
(3) The Information Fusion of the T700 Carbon Fiber andthe Cylinder By using the method in this paper the matrixinformation of carbon fiber in Table 5 can be fused into thecylinder and the upper and lower limits of the reliabilityrupture life at confidence level 120574 = 095 can be calculatedThe comparison results are shown in Table 7
From Table 7 when the reliability is 09 the evaluationaccuracy of the reliability life of the cylinder is increased by35
The curves of upper and lower limits of the logarithmicreliability rupture life 119910
119877changed with the reliability 119877 are
shown in Figure 7Figure 8 shows that the upper and lower limits of the
reliability rupture lifewill bemore accuratewith the reliabilitychanged after fusion of the information of T700 carbon fiberand the interval length is much shorter
5 Conclusion
(1) Since the carbon fiber bears themain load at work theacceleration and failure mechanism of T700 carbonfiber and the cylinder are the same And it can beproved by the structure analysis and the statistic testof the test data
1700 1750 1800 1850 1900 1950 2000
0
2000
4000
6000
8000
10000
12000
14000
Failure timeCensored time
17 failure2 censored
25 failure6 censored
20 failure13 censored
Stress level (N)
Rupt
ure l
ife120585
(h)
Figure 6 Rupture life of T700 carbon fiber
60 65 70 75 80 85 90
0
5000
10000
15000
20000
25000
30000
35000
Stress level (N)
0 failure8 censored
0 failure9 censored
0 failure9 censored
3 failure3 censored
3 failure3 censored
Failure timeCensored time
Rupt
ure l
ife120585
(h)
Figure 7 Ruptures life of the cylinder
(2) When the acceleration and failure mechanism are thesame the evaluation accuracy of the reliability lifefor the cylinder can be improved by fusion of theinformation of the carbon fiber
(3) The method in this paper is based on Weibull distri-bution and the inverse power law model for struc-tured products It can be applied to other location-scale family distribution and acceleration models
8 Mathematical Problems in Engineering
Table 5 Contrast result between T700 carbon fiber and the cylinder with the integrated best linear unbiased estimation
Table 6 The test results of mean vector and covariance matrix for a bivariate normal population ( ) of carbon fiber and cylinder
Null hypothesis Rejection region Observation value Critical value 120572 = 0025
1205831= 1205832
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) 809725 843Σ1= Σ2
1205942lt 1205942
120572(119891(1 minus 119889)) 19468 0831
Table 7 Contrast result of cylinder after fusion information of T700 carbon fiber
Result The original data Fusion of the information of T700
119862[[
[
053 270 minus006
270 2149 minus105
minus006 minus105 013
]]
]
[[
[
053 208 minus0037
208 1276 minus037
minus0037 minus037 0049
]]
]
The interval estimation of 119910119877
[1379 2735] [1429 2315]Interval length 1356 886
080 085 090 095 10010
15
20
25
30
35
40
Reliability (R)
Loga
rithm
ic re
liabi
lity
lifey
R(h
)
After fusion yRLAfter fusion yRUyR
Before fusion yRLBefore fusion yRU
Figure 8 Change curves for 119910119877and upper and lower limits
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work described in this paper was supported by theNational Natural Science Foundation of China (Grants nos61104133 and 61473014) The authors fully appreciate thefinancial support The authors would like to thank thereviewers and the editor for many suggestions that helpedimprove this paper
References
[1] D F Li H JWang and F He ldquoStructure and properties of T300and T700 carbon fiberrdquoNew CarbonMaterials vol 22 no 1 pp59ndash64 2007
[2] G T Zhang W G Chen and B Yang ldquoTesting research onmechanical properties of T700 carbon fiberepoxy compositesrdquoFiber Composites vol 2 pp 49ndash52 2009
[3] Y X Zhou M A Baseer H Mahfuz and S Jeelani ldquoStatisticalanalysis on the fatigue strength distribution of T700 carbonfiberrdquo Composites Science and Technology vol 66 no 13 pp2100ndash2106 2006
[4] Z G Yu S C Yang and B F Song ldquoComparison of wet and hotaging resistance of T700 and T300 carbon fiber strengthenedepoxy resin compositesrdquoMaterials for Mechanical Engineeringvol 33 no 6 pp 48ndash51 2009
[5] X Gu and X Xu ldquoNumerical simulation of damage in fiberreinforced composite laminates under high velocity impactrdquoActaMateriae Compositae Sinica vol 29 no 1 pp 150ndash161 2012
[6] J Liu Y X Bai Y L Tian X Y Huang C H Wang and JY Liang ldquoEffect of the process of electrochemical modificationon the surface structure and properties of PAN-based carbonfirbersrdquo Acta Materiae Compositae Sinica vol 29 no 2 pp 16ndash25 2012
Mathematical Problems in Engineering 9
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
structure respectively The test temperature is 40∘C and thetest load conditions are shown in Table 4
The diagramof rupture life data for T700 carbon fiber andthe cylinder structure from the test is as in Figures 6 and 7
(1) The Result of the Integral Best Unbiased Estimation Theresults of model parameters and the matrix 119862 obtained bythe integrated best linear unbiased estimation are shown inTable 5
(2) The Test of Mean Vector and Covariance Matrix Equalto ( ) of the Carbon Fiber and the Cylinder We considerparameters ( ) as a bivariate normal population and testwhether the mean vectors and covariance matrixes of ( )for the carbon fiber and the cylinder are equal or not and theresults are shown in Table 6
From Table 6 the observed values of the test statistic are
119865 = 809725 lt 1198650975
(2 5) = 843
1205942= 19468 gt 120594
2
0025(5)
(27)
Then we can receive the null hypothesis The mean vectorsand the covariance matrixes of ( ) for the carbon fiber andthe cylinder are equal
(3) The Information Fusion of the T700 Carbon Fiber andthe Cylinder By using the method in this paper the matrixinformation of carbon fiber in Table 5 can be fused into thecylinder and the upper and lower limits of the reliabilityrupture life at confidence level 120574 = 095 can be calculatedThe comparison results are shown in Table 7
From Table 7 when the reliability is 09 the evaluationaccuracy of the reliability life of the cylinder is increased by35
The curves of upper and lower limits of the logarithmicreliability rupture life 119910
119877changed with the reliability 119877 are
shown in Figure 7Figure 8 shows that the upper and lower limits of the
reliability rupture lifewill bemore accuratewith the reliabilitychanged after fusion of the information of T700 carbon fiberand the interval length is much shorter
5 Conclusion
(1) Since the carbon fiber bears themain load at work theacceleration and failure mechanism of T700 carbonfiber and the cylinder are the same And it can beproved by the structure analysis and the statistic testof the test data
1700 1750 1800 1850 1900 1950 2000
0
2000
4000
6000
8000
10000
12000
14000
Failure timeCensored time
17 failure2 censored
25 failure6 censored
20 failure13 censored
Stress level (N)
Rupt
ure l
ife120585
(h)
Figure 6 Rupture life of T700 carbon fiber
60 65 70 75 80 85 90
0
5000
10000
15000
20000
25000
30000
35000
Stress level (N)
0 failure8 censored
0 failure9 censored
0 failure9 censored
3 failure3 censored
3 failure3 censored
Failure timeCensored time
Rupt
ure l
ife120585
(h)
Figure 7 Ruptures life of the cylinder
(2) When the acceleration and failure mechanism are thesame the evaluation accuracy of the reliability lifefor the cylinder can be improved by fusion of theinformation of the carbon fiber
(3) The method in this paper is based on Weibull distri-bution and the inverse power law model for struc-tured products It can be applied to other location-scale family distribution and acceleration models
8 Mathematical Problems in Engineering
Table 5 Contrast result between T700 carbon fiber and the cylinder with the integrated best linear unbiased estimation
Table 6 The test results of mean vector and covariance matrix for a bivariate normal population ( ) of carbon fiber and cylinder
Null hypothesis Rejection region Observation value Critical value 120572 = 0025
1205831= 1205832
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) 809725 843Σ1= Σ2
1205942lt 1205942
120572(119891(1 minus 119889)) 19468 0831
Table 7 Contrast result of cylinder after fusion information of T700 carbon fiber
Result The original data Fusion of the information of T700
119862[[
[
053 270 minus006
270 2149 minus105
minus006 minus105 013
]]
]
[[
[
053 208 minus0037
208 1276 minus037
minus0037 minus037 0049
]]
]
The interval estimation of 119910119877
[1379 2735] [1429 2315]Interval length 1356 886
080 085 090 095 10010
15
20
25
30
35
40
Reliability (R)
Loga
rithm
ic re
liabi
lity
lifey
R(h
)
After fusion yRLAfter fusion yRUyR
Before fusion yRLBefore fusion yRU
Figure 8 Change curves for 119910119877and upper and lower limits
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work described in this paper was supported by theNational Natural Science Foundation of China (Grants nos61104133 and 61473014) The authors fully appreciate thefinancial support The authors would like to thank thereviewers and the editor for many suggestions that helpedimprove this paper
References
[1] D F Li H JWang and F He ldquoStructure and properties of T300and T700 carbon fiberrdquoNew CarbonMaterials vol 22 no 1 pp59ndash64 2007
[2] G T Zhang W G Chen and B Yang ldquoTesting research onmechanical properties of T700 carbon fiberepoxy compositesrdquoFiber Composites vol 2 pp 49ndash52 2009
[3] Y X Zhou M A Baseer H Mahfuz and S Jeelani ldquoStatisticalanalysis on the fatigue strength distribution of T700 carbonfiberrdquo Composites Science and Technology vol 66 no 13 pp2100ndash2106 2006
[4] Z G Yu S C Yang and B F Song ldquoComparison of wet and hotaging resistance of T700 and T300 carbon fiber strengthenedepoxy resin compositesrdquoMaterials for Mechanical Engineeringvol 33 no 6 pp 48ndash51 2009
[5] X Gu and X Xu ldquoNumerical simulation of damage in fiberreinforced composite laminates under high velocity impactrdquoActaMateriae Compositae Sinica vol 29 no 1 pp 150ndash161 2012
[6] J Liu Y X Bai Y L Tian X Y Huang C H Wang and JY Liang ldquoEffect of the process of electrochemical modificationon the surface structure and properties of PAN-based carbonfirbersrdquo Acta Materiae Compositae Sinica vol 29 no 2 pp 16ndash25 2012
Mathematical Problems in Engineering 9
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
Table 6 The test results of mean vector and covariance matrix for a bivariate normal population ( ) of carbon fiber and cylinder
Null hypothesis Rejection region Observation value Critical value 120572 = 0025
1205831= 1205832
119865 gt 1198651minus120572
(2 119899 + 119898 minus 3) 809725 843Σ1= Σ2
1205942lt 1205942
120572(119891(1 minus 119889)) 19468 0831
Table 7 Contrast result of cylinder after fusion information of T700 carbon fiber
Result The original data Fusion of the information of T700
119862[[
[
053 270 minus006
270 2149 minus105
minus006 minus105 013
]]
]
[[
[
053 208 minus0037
208 1276 minus037
minus0037 minus037 0049
]]
]
The interval estimation of 119910119877
[1379 2735] [1429 2315]Interval length 1356 886
080 085 090 095 10010
15
20
25
30
35
40
Reliability (R)
Loga
rithm
ic re
liabi
lity
lifey
R(h
)
After fusion yRLAfter fusion yRUyR
Before fusion yRLBefore fusion yRU
Figure 8 Change curves for 119910119877and upper and lower limits
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work described in this paper was supported by theNational Natural Science Foundation of China (Grants nos61104133 and 61473014) The authors fully appreciate thefinancial support The authors would like to thank thereviewers and the editor for many suggestions that helpedimprove this paper
References
[1] D F Li H JWang and F He ldquoStructure and properties of T300and T700 carbon fiberrdquoNew CarbonMaterials vol 22 no 1 pp59ndash64 2007
[2] G T Zhang W G Chen and B Yang ldquoTesting research onmechanical properties of T700 carbon fiberepoxy compositesrdquoFiber Composites vol 2 pp 49ndash52 2009
[3] Y X Zhou M A Baseer H Mahfuz and S Jeelani ldquoStatisticalanalysis on the fatigue strength distribution of T700 carbonfiberrdquo Composites Science and Technology vol 66 no 13 pp2100ndash2106 2006
[4] Z G Yu S C Yang and B F Song ldquoComparison of wet and hotaging resistance of T700 and T300 carbon fiber strengthenedepoxy resin compositesrdquoMaterials for Mechanical Engineeringvol 33 no 6 pp 48ndash51 2009
[5] X Gu and X Xu ldquoNumerical simulation of damage in fiberreinforced composite laminates under high velocity impactrdquoActaMateriae Compositae Sinica vol 29 no 1 pp 150ndash161 2012
[6] J Liu Y X Bai Y L Tian X Y Huang C H Wang and JY Liang ldquoEffect of the process of electrochemical modificationon the surface structure and properties of PAN-based carbonfirbersrdquo Acta Materiae Compositae Sinica vol 29 no 2 pp 16ndash25 2012
Mathematical Problems in Engineering 9
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
[7] W Yurkosky R E Schafter and J M Finkerlstein ldquoAcceleratedtesting technologyrdquo Tech Rep NORADC-TR-67-420 1-2 1967
[8] J S Zhao G M Zhuang and Z G Wang ldquoThe introductionof the maximum likelihood estimation methodrdquo Journal ofChangchun University of Science and Technology vol 5 no 6pp 53ndash54 2010
[9] H-M Fu and X-R Yue ldquoRegression analysis method for type-Icensored datardquo Journal of Aerospace Power vol 25 no 1 pp142ndash147 2010
[10] X Ma T Wang J Wang and Z Liu ldquoMethod on acceleratedrupture life evaluation for composite material based on type-IIcensored datardquo Advanced Materials Research vol 284ndash286 pp439ndash443 2011
[11] A J Watkins ldquoReview Likelihood method for fitting Weibulllog-linear models to accelerated life-test datardquo IEEE Transac-tions on Reliability vol 43 no 3 pp 361ndash365 1994
[12] G Q Jiao and P R JiaMechanics of Composites NorthwesternPolytechnical University Press Xian China 2008
[13] L F Gui and Y T Cao Handbook of Mechanical EngineeringMaterials Testing Liaoning Science and Technology PressLiaoning China 2001
[14] J H Sun and Y HWang ldquoStandard test method for short-timehydraulic failure and resistance to constant internal pressure ofthe plastics pipes for the transport of fluidsrdquo GBT15560 China1995
[15] P S B Chan ldquoOrder statistics from extreme value distributionI-table of mean variances and covariancesrdquoCommunications inStatistics-Simulation and Computation vol 21 no 4 pp 1199ndash1217 1992
[16] Y Q Zhou Z X Weng and X T Ye ldquoStudy on acceleratedfactor and condition for constant failure mechanism (I)mdashthecase for lifetime is a random variablerdquo Journal of SystemsEngineering and Electronics vol 1 pp 55ndash67 1996
[17] Z L Sun ldquoA condition for constant failure mechanismrdquoElectronic Product Reliability and Environmental Testing vol 26no 4 pp 6ndash8 2008
