复合材料的疲劳损伤模型---英文

A fatigue damage model of composite materials
Fuqiang Wu *,WeiXing Yao
Key Laboratory of Fundamental Science for National Defense-Advanced Design Technology of Flight Vehicle,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China
a r t i c l e i n f o Article history:
Available online 20February 2009Keywords:Composite Fatigue
Accumulative damage Predicted life
a b s t r a c t
The mechanical properties of composite materials degrade progressively with the increasing of the num-ber of cyclic loadings.Based on the stiffness degradation rule of composites,a phenomenological fatigue damage model is presented in this paper,which contains two material parameters.They are proportional to the fatigue life of materials and inversely proportional to the fatigue loading level.Thirteen sets of experimental data of composite stiffness degradation were employed to verify the presented model,and the statistical results showed that this model is capable of describing the damage evolution of com-posite materials.The characteristics of damage development and accumulation of composite materials subjected to variable loading were studied in this paper.Four sets of two-level loading experimental data were cited to verify the damage model,and the results showed that the predicted life is in good agree-ment with the experimental ones.
Ó2009Elsevier Ltd.All rights reserved.
1.Introduction
The damage evolution mechanism is one of the important fo-cuses of fatigue behavior investigation of composite materials and also is the foundation to predict fatigue life of composite struc-tures for engineering applications.As known,the fatigue damage and failure mechanism of composites is more complex than that of metals and four basic failure types will occurr in composites un-der cyclic loading,which are matrix cracking,interfacial debond-ing,delamination and fiber breakage.Based on a great deal of experimental investigations,many damage models [1–8],which have been,respectively,defined by strength degradation,stiffness degradation and energy dissipation of composites,have been em-ployed to describe the damage development of materials in the re-cent decades.The cognition to damage evolution mechanism had been developed from linear model to nonlinear model.However,most models are just suited to a special composite and are not capable of fitting others.To obtain the parameters of the models,a mass of fatigue experimental data is necessary.The fatigue dam-age mechanism of composites has not yet been recognized wholly.In this paper,the factors related to fatigue damage development of composites were analyzed and a phenomenological fatigue damage model defined by material stiffness degradation is de-scribed.Thirteen sets of experimental data were employed to ver-ify the model,and the results show that the model can describe the damage evolution of composite laminates under the different fati-gue loadings.And it is also verified that the model can predict
residual fatigue life of composite laminates quite well by four sets of two-level experimental data.2.Damage model
Under cyclic stress or strain,the non-inverse structural change will occur in micro local field in composite materials and these changes lead to fatigue damage of composites.With an increase in the number of loading cycles,the quantity of this change will in-crease and the damage will cumulate synchronously.The accumu-lation of damage leads to a change in the macroscopic mechanical properties of the composites,such as the degradation of strength or stiffness of the material.Based on the experimental investigation,Reifsnider [1]concluded that fatigue damage evolution is nonlin-ear in composite materials.During the initial period of fatigue life,many non-interactive cracks occur in the matrix.When the matrix crack density reaches saturation,the fiber failure,interfacial deb-onding and delamination occur in the composites.Damage will rapidly develop and the material causes ‘‘sudden death”in the end period of fatigue life,as shown in Fig.1.
To test the change in Young’s modulus of materials,the damage development of composite materials can be described by stiffness degradation of materials in fatigue behavior investigation.Based on this technique that spends less experimental time and cost,many nonlinear damage evolution models [8]were presented.And the models defined by stiffness degradation of composite lam-inates are widely investigated theoretically and experimentally and they fairly described the damage progress in the initial or/and middle period of the fatigue life.However,they are not capable of fitting the damage progress in the whole period,as shown in
0142-1123/$-see front matter Ó2009Elsevier Ltd.All rights reserved.doi:10.1016/j.ijfatigue.2009.02.027
*Corresponding author.Tel.:+862584892576.E-mail address:stonefuq@ (F.Wu).
International Journal of Fatigue 32(2010)
134–138
Contents lists available at ScienceDirect
International Journal of Fatigue
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e
Fig.1.According to the fatigue mechanisms of composites,a versa-tile new fatigue damage model is presented to describe the stiff-ness degradation rule of composite materials in the loading direction.The proposed model of the damage is that
DðnÞ¼E0ÀEðnÞ
E0ÀE f
¼1À1À
n
N
B
A
ð1Þ
where E0is initial Young’s modulus,E f is the failure Young’s modu-lus,E(n)is Young’s modulus of the material subjected to the n th cy-cling loading,n is the cycle,N is the fatigue life,A and B are model parameters,D(n)is the fatigue damage,which equals0when n=0 and equals1when n=N.
3.Statistical analysis
According to the stiffness degradation experimental data of composite materials,the material fatigue damage values are gotten under different cycles.Then,the curve of Eq.(1)can be gotten by the least squaresfitting.The comprehensive data published in Refs. [3,9,10]were used to validate the proposed damage model.The values of A and B in Eq.(1)and the correlative coefficient R2are listed out in Table1and are shown in Fig.2.
Eq.(1)is capable of describing the nonlinear damage evolution macro-mechanically in all periods of the fatigue life of composite materials subjected to different fatigue loadings,as shown in Fig.2.During the initial period of the fatigue life,the main damage type is matrix cracking in the composite.The bigger the applied loading is or the less the ratio R of stress or strain(R=r min/r max or R=e min/e max)is,the faster the damage development is.When the crack density is saturated in the matrix,the rate of damage development of the material is steady and slow.During thefinal period of fatigue life,fiber breaking controls the composite failure. The faster the fractured rate of thefiber is,the shorter the fatigue life is.With the increase in the numbers of fracturedfibers,the rate of damage development of material increases quickly and again. Therefore,the change rule of the damage development rate in com-posites is from quick to slow and to quick again in the whole period of the fatigue life.
In Eq.(1),the normalization fatigue life n/N is rewritten as x=n/ N.Then,the rate of damage development of laminate is
d D
d x
¼ABx BÀ1ð1Àx BÞAÀ1ð2ÞAccording to the values of parameters A and B in Table1,the dam-age development rates of the laminates can be gotten,as shown in Fig.3.
Based on the characteristics of damage evolution of composite materials,the rates of damage development between the initial period and thefinal period of fatigue life are same,as shown in Fig.3.Then,an assumption is proposed,which is that the rates at any normalization life x1and x2(0<x1<x2<1)are same.From the Eq.(2),the parameters A and B can be expressed as
A¼1þðBÀ1Þ
lg x1
2
lg1Àx2
B
1
ð3Þ
It can be verified mathematically that the relation between A and B in Eq.(3)approximates the linear relation,when x1and x2are discretionarily given.Therefore,Eq.(3)can be approximately ex-pressed as
A¼pBþqð4Þwhere p and q are constants.Tofit the values of parameters A and B, as shown in Table1,a quantitative relationship between the param-eters is proposed
A¼0:67Bþ0:44ð5ÞWhen the laminates are subjected to the fatigue loading,the less the ultimate strength,the ratio R of stress or strain,the fatigue life under given loading is or the bigger the loading is,the bigger the fatigue damage in the initial period of the fatigue life is.In Eq.(1),the parameter B describes the characteristics of laminate damage in the initial period of the fatigue life.The less B is,the big-ger the laminate damage is.Therefore,the parameter B is propor-tional to the fatigue life N and is inversely proportional to the loading level r max=r ult.
B¼k
lg N
ð1ÀRÞðr max=r ultÞð6Þwhere r max is the maximum stress,r ult is the ultimate strength,and k is a proportional
constant.
Fig.1.Fatigue damage evolution in composite laminates[1].
Table1
The values of the parameters of the presented model.
Materials Loading/sequence A B R2 Glass/HC9106-3[0/903]S[3]75%r ult0.3140.0250.949
80%r ult0.4190.0550.9805 T300/QY8911[9][45/90/À45/02/À45/90/45]S509.7MPa0.6150.2960.9997
441.7MPa0.7030.4450.9991
424.7MPa0.7420.5110.9995
[À45/0/45/902/45/0/À45]S462.1MPa0.6640.2920.9527
431.3MPa0.7530.4010.9809
400.5MPa0.8420.5140.9888
[02/45/02/À45/0/90]S946.2MPa0.5030.0250.8855
917.5MPa0.5710.0570.8668
888.8MPa0.5810.1090.9804 AS4/PR500[0/90W2]S[10]Unaged specimen0.7150.4750.9842
Aged specimen0.6790.3840.9992
F.Wu,W.Yao/International Journal of Fatigue32(2010)134–138135
4.Damage accumulation
When the composite materials are subjected to the constant amplitude fatigue loading,the damage development of materials can be described by Eq.(1).Under the variable amplitude fatigue loading,the damage that is produced in the former stage loading will affect the damage that is produced in the next stage loading.Then,the cumulative fatigue damage D (n )in the composite mate-rials subjected to the i th loading is calculated as
D ðn i Þ¼1À1À
n i þn i ;i À1
i
B i
!A i ð7a Þ
n i ;i À1¼N i 1À1Àn i À1þn i À1;i À2i À1
B i À1 !A i À1
i 0B @1C
A
1=B i
ð7b Þ
where A i ,A i À1,B i ,B i À1are the parameters under the i th and (i À1)th fatigue loadings,respectively,n i and n i À1are the
cycles
Fig.2.Damage development of composite materials under different loadings.
136 F.Wu,W.Yao /International Journal of Fatigue 32(2010)134–138
under the i th and (i À1)th cyclic loadings,N i and N i À1are the fatigue lives corresponding to the i th and (i À1)th applied loadings,n i,i À1is equivalent cycles.According to the same produced damage,the equivalent cycles n i,i À1under the i th cyclic loading are equaled to the sum of cycles (n i À1+n i À1,i À2)under the (i À1)th cyclic loading,and i !2;n 1;0¼0.When the fatigue failure occurs under M th cyclic
loading that is the last step cyclic loading,the critical damage is de-fined as
D ðn M Þ¼1
ð8Þ
5.Verification
In order to calculate the damage of materials by the presented model,it is necessary to know the fatigue life under constant amplitude loading.The fatigue life can be gotten from the experi-mental data or the material S–N curve.The following S–N curve model [13]has been applied in this paper
S ¼1þm exp Àlg N
b
a À1
ð9Þ
where S =r max /r ult or S =e max /e ult ,e max is the maximum strain,e ult is the ultimate strain.a ,b and m are the experimental parameters.According to the experimental data of composite materials,the material damage evolution curve under the constant amplitude fa-tigue loading that is Eq.(1)was gotten.Then,the fatigue damage of materials can be calculated by Eq.(1)under different constant amplitude loadings.Based on the accumulative rules of Eq.(7)and (8),the residual fatigue life of laminates can be predicted final-ly.The comprehensive data published in Refs.[5,6,11,12]were used to validate the proposed cumulative damage model.According to the proposed algorithm,the residual fatigue life of composites is calculated and the results of this work are presented in Tables 2–5with experimental data.
According to the experimental data,it is a good regression that the proportional constant k equals 0.06.The results show that
the
Fig.3.The rate of damage development of laminates.
Table 2
Life estimated by proposed model and experimental results for carbon/epoxy [12].Experimental
Predicted
r 1(MPa)
r 2(MPa)
n 1n 2
n 231534087,200520221431534087,000150223231534086,3001408229331534057,7001750480331534057,5502280481631534040,3002027626631534028,7003320716931534026,5002640733131534025,3002464741831534017,6506170794231534017,00038,140798431534013,00014,300822931534012,50024,0308259340315850015,2504665340315748017,06017,517340315748079,49617,517340315680029,93925,482340315650048,76028,931340315460073,91050,523340315440089,35052,809340315440080,60552,809340315250090,15075,271340315150041,84088,2223403151500111,12088,222340
315
1350
99,520
90,296
Table 3
Life estimated by proposed model and experimental results for E-glass/epoxy [11].Experimental
Predicted
e 1(%)
e 2(%)
n 1n 2
n 2
0.7 1.010,000585062180.7 1.010,000543062180.7 1.010,000328062180.7 1.010,000949062180.70.510,0001,635,2102,382,1660.70.510,000925,2402,382,1660.70.510,000150,95502,382,1660.7
0.5
10,000
830,050
2,382,166
Table 4
Life estimated by proposed model and experimental results for E-glass/epoxy [5].Experimental
Predicted
r 1(MPa)
r 2(MPa)
n 1n 2
n 2
386241250192,00064,267386241100193,000100,3203862892505840594438628910011,97096053863372501250109038633710016351784337241100086,00083,293337241249162,500130,098337289100086707905337289249800012,298289241999696,50052,2922892411999110,800135,150********,938373011,61924128919,975949013,87724133749,938391212024133719,975804241424138619,975124490289337999629384628933719991290227628938619993554763373861000297328337
386
249
503
468
Table 5
Life estimated by proposed model and experimental results for E-glass/epoxy [6].Experimental
Predicted
r 1(MPa)
r 2(MPa)
n 1n 2n 2539654.510,00020313241654.553950033,04937046577.5654.5500032403256654.5
577.5
500
21,450
22,054
F.Wu,W.Yao /International Journal of Fatigue 32(2010)134–138137
difference between the predicted residual fatigue life and the experimental data is acceptable because of the big scatter of fati-gue life and most points are within three times range as shown in Fig.4.Experimental fatigue life among composite samples sub-jected to the same fatigue loading is very different and the differ-ence of experimental data is even bigger than 10times.The predicted residual fatigue life by the proposed algorithm is in good agreement with the experiment ones,considering the bigger scat-ter of composites.6.Conclusion
Based on the stiffness degradation rule of composite materials under fatigue loading,a phenomenological fatigue damage model is presented in this paper.The numerical examples show that the fatigue damage model is capable of describing the nonlinear dam-age evolution in the whole fatigue life period of the materials.The characteristics of damage development of composite materials are
analyzed and the quantitative relation is gotten,that the parame-ters of the model are proportional to the fatigue life and are inver-sely proportional to the fatigue loading level.The two-level loading examples show that the model can predict residual fatigue life of composite materials quite well.References
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Fig.4.The experimental and predicted residual fatigue lives of laminates.
138 F.Wu,W.Yao /International Journal of Fatigue 32(2010)134–138。