LOAD-RATIO DEPENDENCE ON FATIGUE LIFE OF COMPOSITES Joakim Shön 1 and Anders F. Blom 1, 1 Strutures Department, The Aeronautial Researh Institute of Sweden Box 1101, SE-161 11 Bromma, Sweden Department of Aeronautis, Royal Institute of Tehnology SE-100 44 Stokholm, Sweden SUMMARY: A model for prediting the fatigue life of omposites at different load ratios have been developed. The basi assumption of the model is that the fatigue life of a speimen is governed by delamination growth. The model is based on alulation of delamination growth by integration of Pari s law. It is assumed that frature for different load ratios, but with the same peak-load magnitude, will our at the same delamination length. In addition, it is assumed that the energy release rate does not vary with delamination length. The model is able to predit the fatigue life for tension-tension, ompression-ompression, and tensionompression fatigue. In omparison with experimental results for unnothed speimens the model shows good agreement. Due to the large exponent in Pari s law for omposites the model is sensitive to errors in the parameters used. KEYWORDS: fatigue life, load ratio, delamination growth, noth, life-predition model INTRODUCTION Different omposite strutures in appliations suh as airraft are often subjeted to spetrum fatigue loading with different average load ratios, R = σ min σ. When haraterizing the max material for suh strutures onstant amplitude fatigue data of oupons at the speifi R- values are often used. This results in a large number of fatigue test having to be run whih is expensive. Hene, there is a need to develop models for alulating the fatigue life of omposites at different R-values. There are a large number of different models available for prediting fatigue life of omposites and some of them will be mentioned here. The model whih probably is most well known is Miner s rule. It has been found that the model usually overestimates the fatigue life of omposites.[1][] However, for some simple spetra it gives reasonable results.[3] In the strength degradation model fatigue failure ours when the residual strength has dropped to
the maximum stress of the fatigue loading.[4][5] By using the fatigue failure envelope method to desribe the fatigue behavior of omposite laminates Rotem was able predit the fatigue life at any load ratio.[6] The fatigue life of a omposite an be estimated with fatigue failure funtions established on a laminate level.[7] Different R-values are then handled with the Goodman orretion approah. The model by Ratwani and Kan[8] models delamination growth in a speimen. The delamination growth is predited from interlaminar shear stresses. They found that if the interlaminar shear stress is plotted against fatigue life data for different R-values and speimens the results will ollapse into a single satter band. The objetive of this paper is to develop a model for prediting the fatigue life at different R- values and then ompare the model with experimental results. The model should be based on material data whih an be measured with simple test speimens. MODEL The basi assumption of the present model is that the fatigue life of a speimen is governed by delamination growth. When the delaminations have grown to a ritial size the speimen will fail. This ritial size is dependent on the peak load at failure. All other damage mehanisms, for example matrix raking and fiber failure, are assumed to be of seondary importane and will only have a minor influene on the total life of the speimen. A onsequene of this assumption is that the fatigue life of a speimen an be estimated by alulating the delamination growth rate in the speimen whih is what the model in this paper will do. The model will be appliable to all situations whih fulfill the basi assumption above. Extensive delamination growth has been observed for unnothed speimens fatigue loaded in tensiontension and tension-ompression.[9] Delamination growth of the ritial delamination whih will ause failure is alulated by integrating Paris law. In fatigue of omposites several delaminations in thikness diretion are often growing at the same time. But, it is reasonable to assume that the delamination growth of the ritial delamination will still be governed by Paris law when the energy release rate of that delamination is used. From the global quantity ompliane of nothed speimens it has been possible to obtain Paris law.[10] This would suggest that Paris law also is valid on a global sale. Reently, a model for prediting fatigue delamination growth rate has been developed.[11] The model uses Paris law and predits the growth rate for different R-values and mixed mode ratios. It is based on the observation that the peak energy release rate an not be larger than the ritial quasi-stati energy release rate and that the hange in energy release rate G r at the threshold is independent on mixed mode ratio and load ratio. This gives two known values from whih the onstants in Paris law an be alulated. This model will be the basis for alulating the onstants in Paris law. Model for tension-tension and ompression-ompression fatigue When the delamination growth rate is measured experimentally with test speimens, for example DCB and ENF speimens, the R-value is defined suh that -1 R 1. For other test speimens suh as nothed and unnothed oupons there is a lear differene in loading between, for example, R=0.1 and R=10. But, when energy release rates of delaminations in those speimens are ompared with those from simple test speimens there is no differene between R=0.1 and R=10. Therefore, a new parameter Q is introdued and defined as Q = R if -1 R 1 (1)
or Q = 1 R if R<-1 or R>1 () To alulate the delamination length a( N) as funtion of number of yles in a speimen Paris law is integrated as an ( )= N da dn = D G dn r N n [ ] dn (3) where N is number of yles, is number of yles at whih a delamination is initiated, D and n are onstants in Paris law, and G r is the range of hange in energy release rate. The diffiulty is to determine G r. It is assumed that the energy release rate as a funtion of rak length Ga ()for the ritial delamination of the speimen has been alulated at an applied load. The energy release rate at an applied load P is then G P ()= a Ga () P (4) For the ase Q 0 the range of hange in energy release rate an now be given as G r = G P max () a G P min a ()= Ga () [ P min ] (5) P P max ref where P max and P min are maximum and minimum applied fore magnitude. Fore magnitude is used sine it will make it possible to treat tension-tension and ompression-ompression loading in the same way in the following derivation. The Q-value is related to the applied fore magnitude as Substitution of Eqn 6 into Eqn 5 gives G r = Ga Q = P min P max (6) () P max [ ] (7) 1 Q Substitution of Eqn 7 into Eqn 3 gives the delamination length as an ( )= D P max 1 Q [ ] P ref n N n [ Ga ()] dn (8) Consider now two speimens subjeted to onstant amplitude fatigue loading at two different Q-values, Q 1 and Q. The fatigue loading is either ompression-ompression on both speimens or tension-tension on both speimens with P max equal for both speimens. It is then reasonable to assume that both speimens will break when the delamination has grown to a
ritial size a, whih is equal for both speimens, sine P max is equal for both speimens. This gives a,q1 ( N f 1 )= a,q ( N f ) (9) where N f1 and N f is number of yles at failure for the speimens loaded at Q 1 and Q, respetively. Substitution of Eqn 8 into Eqn 9 gives D 1 P max 1 Q P 1 ref [ ] n1nf 1 n1 [ Ga ()] dn = D P max 1 Q P ref [ ] n Nf [ Ga ()] n dn (10) Assume that N i is zero. That means a delamination will begin to grow from the first yle or after only a few yles whih an be negleted. Considering that some damage might be present at the interfae from speimen preparation this is not unrealisti. Also, assume that the energy release rate of the ritial delamination G is independent of length and do not hange mode ratio. For unnothed speimens with a delamination growing from the edge it has been found that the energy release rate inreases for short delamination lengths and then remains onstant for delamination lengths above 4 ply thiknesses.[1][13][14] The energy release rate mode ratio also remains onstant. For a nothed speimens a FEM solution found that the total energy release rate dereases slightly with inreasing delamination length and the mode ratio was fairly onstant.[15] This makes the assumption reasonable for unnothed and nothed speimens. In the future it might be possible to obtain G numerially whih would make it possible to numerially integrate the delamination growth. The onstants D are given from[11] D = da dn th ( ) n (11) G r, th da where is the threshold delamination growth rate when da/dn versus G r makes a th da sharp orner, G r,th is the threshold hange in energy release rate at this point. Both th and G r,th are independent of Q-value and mixed mode ratio and as a result D 1 D = [ G r,th ] n n1 (1) Equation 10 now beomes [ G r,th ] n n1 P max 1 Q [ 1 ]Ga [ ()] n1 N f 1 = P max 1 Q [ P ]Ga ref [ ()] n N f (13) It has been observed for unnothed speimens that a fatigue threshold exists.[9] It is reasonable to assume that the fatigue threshold is related to G r,th. Therefore, it is possible to write
G r,th = Ga () P th,1 [ ] (14) 1 Q 1 where P th,1 is the maximum load magnitude at threshold for the speimen loaded at Q 1. Substitution into Eqn 13 gives N f, = P th,1 P max ( n n1) 1 Q 1 1 Q n N f 1 (15) This equation makes it possible to alulate the fatigue life at the load ration Q and maximum load magnitude P max if the onstant n in Paris law is known at the two load ratios. Model for tension-ompression fatigue Now, it is possible to estimate the fatigue life for tension-tension and ompressionompression fatigue. But, if Q<0, tension-ompression fatigue, the situation is more ompliated. Assume that the delamination rak is open during the ompressive part of the load yle. This means that G I >0. During the tension part of the load yle the rak will be losed whih means that G I =0. Therefore, the energy release rate of the delamination rak during ompression will not be equal the orresponding G t for the tension part of the load yle. The objetive of the following derivation is to find an expression similar to Eqn 15 for the fatigue life when Q<0 whih should be based on fatigue lives for Q>0. Following the previous derivation Eqn 5 need to be hanged to G r = G Pt ()+ a G P ()= a G t P t + a () P (16) where subsript and t stands for ompressive and tensile loading. Assume that the speimen will fail in ompression with a delamination length a ( N f 3 ) after N f3 yles. The speimens fails at the same delamination length as a speimen loaded in ompressionompression, Q 1 0, with the peak load magnitude P whih fails after N f1 yles. This gives a ( N f 3 )= a,q1 ( N f 1 ) (17) If Eqn 16 is substituted into Eqn 3 and part of Eqn 10 is used then Eqn 17 an be written as D 3 P t n3 Nf 3 [ G t ] n3 dn + P n3 Nf 3 [ ] n3 dn = D P 1 [ 1 Q 1 ] P ref n1nf 1 n1 [ ] dn (18) where n 3 and D 3 are onstants in Paris law for the Q<0 ase studied. Now, assume that N i is zero and that both and G t are independent of delamination length and that the energy release rate ratio is independent of delamination length. Substitution of Eqns 11, 1 and 14 into 18 gives
N f 3 = P P th,1 n1 n3 [ 1 Q 1 ] n3 N f 1 n3 P t G t P +1 (19) where P th,1 is the threshold load magnitude at load ratio Q 1. In order to ontinue it is neessary to find an expression for G t() a. Consider a speimen loaded in ompression-ompression at load ratio Q 4 and one loaded in tension-tension at load ratio Q 5 at the same load magnitude P max,4. Following the tehnique used above it is assumed that failure in the speimens our at the same delamination length. This might not be orret in all situations sine is not equal to G t and loal bukling might our in ompression. Equations 9 and 10 now beomes D 4 P max,4 [ ] 1 Q 4 n4 Nf 4 Substitution of Eqns 1 and 14 into Eqn 0 gives Substitution into Eqn 19 gives n 4 P [ ] dn = max, 4 D 5 1 Q [ 5 ] P ref G t = P 1 n 4 th,4 n5 1 Q 4 P max,4 1 Q 5 N f 4 N f 5 n5 Nf 5 n 5 [ G t ] dn (0) 1 n 5 (1) N f 3 = P P th,1 n1 n3 P t P P th,4 P max,4 [ 1 Q 1 ] n3 N f 1 1 n4 n5 1 Q 4 1 Q 5 N f 4 N f 5 1 n 5 n3 +1 () An alternative to using Eqn would be to run one test at Q<0 and then use Eqn 19 to alulate G t() a. It would then be possible to alulate N f3 for other negative Q-values. If failure ours in tension instead of in ompression, whih was assumed above, subsript should be hanged to t and vie versa in equations above. Comparison with experimental results To ompare the model with experimental results data from the work by Gatherole et al.[16] are used. The data are for unnothed T800/545 omposites tested in fatigue at different R- values, see Fig. 1. Before it is possible to use the model it is neessary to find the n-value in Paris law for the different R-values. This an be done with the delamination growth model by da Shön[11] whih require that the mix mode ratio and the parameters, da th, G r,th, and G max, are known together with a quasi-stati failure riteria for mixed mode delamination da growth. The parameter G r,th is the hange in energy release rate at whih the versus
G r urve turns down. In this ase this value is assumed to be 80 J/m. The parameter da is the delamination growth rate at whih G r,th is reahed. In this ase this is assumed th to be at 1E-6 mm/yle. The parameter G max, is the energy release rate at stati failure. In this ase a linear failure riteria for the different mixed-mode ratios was used whih some experiments would suggest.[17] The parameter da is the delamination growth rate at whih stati failure is reahed. In this ase it is assumed to our at 0.1 mm/yle. It is also neessary to know the mixed mode ratio for the ritial delamination in the speimens during tensile and ompressive loading. One way to estimate this is by onsidering the stati frature. In tensile loading frature ourred at 1.67 GPa and in ompressive loading at -0.88 GPa. It is assumed that frature in both ases will be due to delamination growth. It is reasonable to assume that during tensile loading the delaminations will be losed, G I =0, sine the tensile strength is higher than the ompressive one. It is assumed that quasi-stati delamination growth will our when G II +G III =750 J/m. During ompressive loading there will be a mode I frature omponent together with mode II and III. It is assumed that the mode II and III frature omponents will be equal to that during tensile loading at a given load magnitude. A linear mixed mode failure riteria is used together with G IC =15 J/m.[17][18] This makes it G possible to alulate the mode I to mode II+III ratio as I = 1.0. It is now possible to G II + G III alulate the n-value in Paris law for the different R-values used, see Table 1. For R=-0.3 and - 0.6 frature is assumed to our in tension when G I =0 and for R=-1.0 and -1.5 frature is assumed to our in ompression when all three frature modes are present. That is the reason for the n-value to be larger for R=-1.0 and -1.5 than for R=-0.3 and -0.6. These values are similar to those in the literature.[19] Table 1: Calulated n-values in Paris law. R-value +0.5 +0.1-0.3-0.6-1.0-1.5 +10 n-value 5.90 5.17 4.76 4.08 5.43 5.97 6.64 It is now possible to ompare the theoretial preditions with the experimental results. Using Eqn. 15 and the fatigue results for R=0.1 (Q=0.1) at 1.3 and 1.4 GPa the fatigue lives are predited for R=0.5 (Q=0.5). As P th,1 was 0.55 GPa used whih was alulated from the stati tensile strength and G II +G III =750 J/m together with G r,th =80 J/m. The results an be seen in Fig. 1. For the ase of Q<0 two methods are possible for determining G t() a. Either it an be determined from tension-tension and ompression-ompression fatigue results at the same P max -value or from Q<0 and Q>0 fatigue data at the same P max -value and failure mode, tension or ompression failure. Sine there is suh a large differene between tensile and ompressive strength it is not possible to use the first method. There are no fatigue data at the same P max - value. The seond method is then used. Based on the fatigue urves it is reasonable to assume
R=0.5 (Q=0.5) R=-0.6 (Q=-0.6) R=10 (Q=0.1) R=-1.5 (predited) R=0,1 (Q=0.1) R=-1.0 (Q=-1.0) Quasi-stati R=-0.3 (predited) R=-0.3 (Q=-0.3) R=-1.5 (Q=0.667) R=0.5 (predited) 1,5 1 Peak stress (GPa) 0,5 0-0,5-1 1E+00 1E+01 1E+0 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 Number of yles Fig. 1 Experimental [16] and predited fatigue results that for R=-1 and -1.5 the failure is ompressive. The parameter G t a was determined from the R=-1 and R=10 urves at P max =0.7 GPa and Eqn. 19 with a P th,1 of -0.368 whih was determined from quasi-stati ompressive strength. It was found that G t( a) =1.77. The fatigue life for R=-1.5 (Q=0.67) was then alulated using Eqn. 19 and the results an be seen in Fig. 1. For the R-values -0.3 and -0.6 it is reasonable to assume that the failure mode is tensile. ()
After the subsripts and t in Eqns 19 and had been shifted the parameter G a G t was determined from the R=-0.6 and R=0.1 urves at P t =1.1 GPa. It was found that G G t =17. whih is quite different to what was found for R=-1. The predited fatigue life for R=-0.3 an be seen in Fig. 1. There are several possible soures of errors for the omparison with experiments. One is the parameters used for alulating the n-value in Paris law for the different R-values. Sine the n-values are large a small error in n-value or in the other parameters used to alulate the fatigue life will introdue a large error in predited fatigue life. This is a basi problem of delamination growth in omposites. As tougher matries are developed the n-value will derease and this problem might be less ritial. CONCLUSIONS A model for prediting the fatigue life of omposites at different R-values has been developed. It an be used for tension-tension, ompression-ompression, and tension-ompression fatigue. In a omparison with experimental results the model provides good preditions. REFERENCES 1. Rosenfeld, M. S. and Huang, S. L., Fatigue harateristis of graphite-epoxy laminates under ompression loading, AIAA/ASME 18th Strutures, Strutural Dynamis and Materials Conferene, Marh 1977, Paper No. 473.. Phillips, E. P., Effets of trunation of a predominantly ompression load spetrum on the life of a nothed graphite/epoxy laminate, ASTM STP 73, 1981, pp. 197-1. 3. Adam, T., Gatherole, Reiter, H., and Harris, B., Life predition for fatigue of T800/545 arbon-fibre omposites: II. Variable-amplitude loading, Fatigue, 1994, Vol. 16, pp. 533-547. 4. Yang, J. N., and Liu, M. D., Residual strength degradation model and theory of periodi proof tests for graphite/epoxy laminates, J. Comp. Mater., Vol. 11, 1977, pp. 176-04. 5. Yang, J. N., and Jones, D. L., Load sequene effets on the fatigue of unnothed omposite materials, ASTM STP 674, 1981, pp. 13-3. 6. Rotem, A., The fatigue behavior of omposite laminates under various mean stresses, Composite strutures, Vol. 17, 1991, pp. 113-16. 7 Nyman, T., Composite fatigue design methodology: a simplified approah, Composite strutures, Vol. 35, 1996, pp. 183-194. 8. Ratwani, M. M. and Kan, H. P., Compression fatigue analysis of fiber omposites, AIAA/ASME/ASCE/AHS 1st Strutures, Strutural dynamis & Materials Conferene, Part 1, May 1980, Seattle, Washington, USA, pp. 79-84. 9. Ryder, J. T., and Walker, E. K., Effet of ompression on fatigue properties of a quasi-isotropi graphite/epoxy omposite, ASTM STP 636, 1977, pp. 3-6. 10. Blom, A. F., Frature mehanis analysis and predition of delamination growth in omposite strutures, Fatigue 84, Proeedings of the nd International Conferene on Fatigue and Fatigue Thresholds held at the University of Birmingham, UK, 3-7 Sep. 1984, Vol. III, pp. 1881-1891 11. Shön, J., Fatigue delamination growth model for omposites, Submitted for publiation in Comp. Si. Teh. 1. O Brien, T. K., Mixed-mode strain-energy-release rate effets on edge delamination of omposites, ASTM STP 836, 1984, pp. 15-14. ()
13. O Brien, T. K. and Raju, I. S., Strain-energy-release rate analysis of delamination around an open hole in omposites laminates, AIAA 5th strutures, strutural dynamis & materials onferene, Part I, 1984, California, pp. 56-536. 14. Armanios, E. A, Rehfield, L. W., Raju, I. S., and O Brien, T. K., Sublaminate analysis of interlaminar frature in omposites: part II-appliations, J. Comp. Teh. & Res., Vol. 11, (1989), pp. 147-153. 15. Shön, J. and Andersson, B., Interation between a delamination rak and a matrix rak, Proeedings of the Amerian Soiety for Composites, 13 th annual tehnial onferene, Sept. 1-3, 1998, USA, 676-710. 16. Gatherole, N., Reiter, H., Adam, T., and Harris, B., Life predition for fatigue of T800/545 arbon-fibre omposites: I. Constant-amplitude loading, Fatigue, 1994, Vol. 16, pp. 53-53. 17. Greenhalgh, E. S., and Matthews, F. L., Charaterisation of mexed-mode frature in unidiretional laminates, ECCM-7: Seventh European Conferene on Composite Materials, Vol. 1, London, UK, May 1996, pp. 135-140. 18. Beland, S., Komorowski, J. P., and Roy, C., Hygrothermal influene on the interlaminar frature energy of graphite/bismaleimide modified epoxy omposite (IM6/545C), ICCM 6 and ECCM, 1987, London UK, Vol. 3, pp. 3.305-3.316. 19. Neft, J. F., Interlaminar fatigue rak growth in arbon fiber reinfored plastis, SECTAM XIX: 19th Southeastern Conferene on Theoretial and Applied Mehanis, FL, USA, 1998, pp. -10.