16 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS STRENGTH DEGRADATION MODEL FOR FATIGUE LIFE PREDICTION Hiroshige Kikukawa: kikukawa@neptune.kanazawa-it.ac.jp Department of Aeronautics, Kanazawa Institute of Technology Keywords: Probability of failure, Weibull parameters, Residual strength, Fatigue life, Scatter factor, Fudge factor Abstract As application of composites in aerospace structures extends into primary structures, the needs for fatigue life prediction is critical for the reliability and safety of composite structures. Composite structures used in high cycle fatigue applications are often over designed and are somewhat heavier and more costly than necessary. Improved life prediction methodologies may results in more efficient use of these materials and may result in lower weight and lower cost structures. Many models have been proposed to predict fatigue life of composite structures subjected to a variety of cyclic loading. Since composite laminates exhibit very complex failure processes, statistical life prediction methodologies have been prevailed. In this paper we study the following statistical life prediction methodologies, tension(fl), 90-degree tension(ft), 0-degree compression(fl ), 90-degree compression(ft ), ± 45-degree shear(flt), laminate strengths of tension(ftu), compression(fcu), open hole tension(oht) and crippling strength of flat plate(fcc-flat) or T-type(Fcc-T-type) shape.. The fatigue data were obtained for the specimens of open hole, single shear joint and double shear joint under constant amplitude loading of stress ratio R=-1.0. The configurations of fatigue test specimen are shown in fig.1. (1) Scatter factor in the recognition of structural configuration (2) Consistency of estimated life to tested life (Introduction of Fudge factor) (3) Simple fatigue life estimation method The concerns in this study are (1) Weibull parameter conduction and its characteristics, (ii) Parameters necessary in the fatigue life estimations and (iii) )Possibility of simple fatigue life estimation 1 Conduction of Weibull parameters and its characteristics The Weibull parameters, i.e. the shape parameter and the scale parameter, are conducted from test data for static strength distribution and fatigue life distribution of carbon/epoxy laminates. The static strength data presented in this paper were obtained for various specimens of 0-degree Fig.1 Configuration of fatigue test specimens The Weibull parameters of static strength are shown in fig.2. (i)ft αo=7.1 βo=59 (l)flt αo=105 βo=80 (k)ft αo=9.1 βo=211 T300/3601 0 500 1000 1500 2000 Static Strength (MPa) (j)fl αo=14.5 βo=1376 (h)fl αo=8.2 βo=1650 1
Hiroshige KIKUKAWA (a) Unidirectional properties (b) Laminate properties (c) Element properties Fig.2 Weibull parameters of static strength The Weibull parameters of fatigue life are shown in fig.3. (a) &(b): [+-45/02/90/03/-+45] (c )&(d): [(+-45)/0/(0,90)/02/(+-45)] 0 200 400 600 800 1000 1200 [+-452/05/90]s Open Hole [+-45/03/+-45/02/90]s R=-1.0 2-15Hz RoomTemp., Dry Static Static Strength (MPa) (g)fcc-t-type αo=21.0 βo=263 (a)ftu (c )Ftu αo=13.5 αo=12.4 βo=670 βo=800 0 100 200 300 400 500 600 700 Static Strength (MPa) (f)fcc-flat αo=13.1 βo=422 (d)fcu αo=8.4 βo=856 Max,Fatigue Stress 354 (MPa) 337 310 287 271 245 αf=3.2 αf=1.5 αf=0.75 (e)oht αo=14.5 βo=499 (b)fcu αo=12.4 βo=1005 Probability of ailure E+03 E+04 E+05 E+06 Single Shear Joint [+-45/03/+-45/02/90]s R=-1.0 3-4Hz RoomTemp., Dry (b) Single Shear Joint Double Shear Joint Max,Fatigue Stress (MPa) 426(1-3Hz) 589(5-15Hz) αf=0.65 [+-45/03/+-45/02/90]s R=-1.0 RoomTemp., Dry Max,Fatigue Stress (MPa) 296 224 150 α f=2.7 Cycle to Failure E+02 E+03 E+04 E+05 E+06 Cycle to Failure (c) Double Shear Joint Fig.3 Weibull parameters of fatigue life Fig.3(a) indicates shape parameter of fatigue life does not always have the same value for stress change. The conducted Weibull parameters of static strength are listed in table 1 and fig.4 with probability density function and also those of fatigue life are listed in table 2 and fig.5 with cumulative distribution function. E+03 E+04 E+05 E+06 E+07 Cycle to Failure (a) Open hole Table 1 Weibull parameters of static strength Shape Parameter: Static Strength Strength Shape Parameter, αo Bending Specimen 28.0 Ftu 13.5 12.4 Fcu 12.4 8.4 OHT 14.5 Fcc (Flat Plate) 13.1 Fcc (T-type) 21.0 FL (0 tension) 8.2 FT (90 tension) 7.1 FL' (0 compression) 14.5 FT' (90 compression) 9.1 FLT (±45 shear) 105.0 2
STRENGTH DEGRADATION MODEL FOR FATIGUE LIFE PRDICTION P ro b a b ility D e n sity F u n c tio n, f(x /β ) Table 2 Weibull parameters of fatigue life Shape Parameter: Fatigue Life Configuration Shape Parameter, αf Bending Specimn 0.63 1.18 Open Hole 3.2 1.5 0.75 Single Shear Joint 2.7 Double Shear Join 0.65 8.0 6.0 4.0 2.0 f(x/β)=α(x/β) α-1 exp[- (x/β) α ] Fatigue Life α=0.65 α=1.5 α=3 Static Strength α=105 α=21 α=13 α=7 Fatigue life distribution, F N (n) =1-exp[-(n/N) αf ] (n/n) αf = -ln[1-f N (n)] Scatter factor SF, SF = (n/n) = [-ln[1-f N (n)]] 1/αf (1) MIL-HDBK-17 provides the two kinds of probability of failure F N (n) in confidence level 95%. A value = Probability of failure F N (n) = 1% B value = Probability of failure F N (n) = 10% Mean value = Probability of failure F N (n) = 63.2% for Weibull distribution The scatter factors were calculated by (1) and are shown in fig.6 and table 3. 0.0 0.0 0.5 1.0 1.5 2.0 2.5 X/β 10,000 A value Cumulative Distribution Function, F(x/β ) 1.0 0.8 0.6 0.4 0.2 Fig.4 Probability density Function Fatigue Life α=0.65 α=1.5 α=3 F(x/β)=1-exp[- (x/β) α ] Static Strength α=105 α=21 α=13 α=7 Scatter Factor 1,000 100 10 1 B value 0 2 4 Weibull Fatigue Life Shape Parameter αf 0.0 0.0 0.5 1.0 1.5 2.0 2.5 Fig.5 Cumulative distribution function 2 Scatter factor in the recognition of structural configuration 2.1 Conduction scatter factor As shown in table 2, the shape parameters of fatigue life vary with every configuration of structural element. The scatter band is very wide, Therefore scatter factor should be established for the each structural configuration. Scatter factor depends on he shape parameter as follows, X/β Fig.6 Scatter factor vs shape parameter Table 3 Scatter factor for A value and B value Configuration αf Scatter Factor A value B value Bending Specimen 0.63 1,483 36 1.18 49 6.7 3.2 4.2 2.0 Open Hole 1.5 21 4.5 0.75 461 20 Single Shear Joint 2.7 5.5 2.3 Double Shear Joint 0.65 1,184 32 2.2 Sample of safe life estimation It is so surprised that the scatter factor of double shear joint is very large in comparison 3
Hiroshige KIKUKAWA with one of single shear joint. However we have a good information that the tested lives of double shear joint is superior to single shear joint as shown in fig.7. Max. bearing stress s (MPa) 1000 800 600 400 200 Fig.7 Tested life of double shear joint and single shear joint Safe lives were calculated for the maximum bearing stress 800MPa and shown in table 4. Safe Life = Tested Life / Scatter Factor Table 4 Safe life comparison between single shear joint and double shear joint Sample of Safe Life Calculation at max.fbr 800MPa Scatter Joint Tested Life Safe Life Factor Configulation (Flight hours) (FLT hours) (B value) Single Shear Joint 2.00E+02 2.3 87 Double Shear 0 E+02 E+03 E+04 E+05 Flight hours to failure Joint E+05 32 3,125 The safe life of double shear joint is much longer than that of single shear joint in spite of the large scatter factor. Safe life depends deeply both on mean fatigue life (tested life in above) and scatter factor (shape parameter in table 3). 3 Comparison between Prediction and Test 3.1 Fatigue life estimation method Double Shear Joint Single Shear Joint Variable Amplitude Fatigue Life In this paper we study the following four estimation models, (1) Miner s Rule (2) Linear Reduction Fatigue Model [1] (3) Degradation Parameter Model [2] (4) Degradation Rate Model [3][4] 3.2 Parameters necessary to estimate fatigue life The necessary parameters of each model are listed in a table 5. Residual strength degradation parameter ν, cycle mix constant Cm and load sequence parameter γ are introduced supplementarily into the life estimation to improve the accuracy. Table 5 Parameters necessary in fatigue life estimation Prediction Items study Probability of Methodology Fatigue Life model Failure Miner's Rule NA ni/ni NA Linear Residual model 1 Strength Reduction ni/ni, βo >Spectrum Stress Fatigue Model NA model 2 Degradation ni, Ni, βo, ν, Cm ni, Ni, αo, αf,, >Spectrum Stress Parameter ν ν model 3 ni, Ni, βo, αo, ni, Ni, βo, αo, Degradation Rate >Spectrum Stress αf,γ αf, γ ni = number of cycle in segment i Ni = fatigue life scale parameter for segment i βo = static strength scale parameter αo = static strength shape parameter αf = fatigue life shape parameter ν = residual strength degradation parameter Cm = cycle mix constant γ = load sequence parameter The procedure needs new disinclined test data to fit in to the degradation rate or the load sequence effects. Therefore the estimation becomes complicated and troublesome. 3.3 Consistency of estimated life to tested life For the joint test articles as illustrated in fig.8, the ratios of calculated lives to the tested lives are 2.16 for Linear Reduction Fatigue Model (LRSRFM) and 2.12 for Degradation Parameter Model (ν=1.0). However, the ratio is 4.91 for Miner s Rule. The applicability of strength degradation models is better than Miner s Rule. In the case of ν=1.0, Degradation Parameter Model gives almost the same life as Linear Reduction Fatigue Model (LRSRFM). 4
STRENGTH DEGRADATION MODEL FOR FATIGUE LIFE PRDICTION Calculated Life (Flight Hour) Miner s Rule-Single Shear Joint * E+06 LRSRFM-Single Shear Joint * Miner s Rule-Double Shear Joint* LRSRFM-Double Shear Joint* Miner s Rule-Bending Specimen** LRSRFM-Bending Specimen** * 6 step-low High loading, R= -1 E+05 ** 2 step loading, R=0.05 Miner s rule gives good calculation results for simple specimen, but not for joint configurations. E+04 [model 2] Degradation Parameter (ν=1.0) gives almost the same calculated lives as [model 1] LRSRFM. E+03 E+02 E+03 E+04 E+05 Tested Life (Flight Hour) Fig.8 Consistency with test results 3.4 Introduction of fudge factor The consistency of estimation model to test result is defined as the fudge factor, Fudge Factor=Calculated Life / Tested Life For the joints, Fudge Factor = 5 to 10 for Miner s rule Fudge Factor = 2 for LRSRM Fudge Factor = 2 for Degradation Parameter Model (ν=1.0) When we can get the Weibull parameters for static strength distribution and fatigue life distribution of constant amplitude loading, the residual strength distribution under variable amplitude loading can be predicted using the strength degradation models. The Linear Reduction Fatigue Model (LRSRFM) is easier to calculate the fatigue life and can give the about twice longer life, but can not give the probability of failure. The Degradation Parameter Model and the Degradation Rate Model can give both the fatigue life and the probability of failure. By introducing the parameters ν, Cm, γ, it is expected to improve the accuracy of life estimation. However, its calculation is rather complicated and troublesome. An extensive program needs to evaluate more the applicability of the residual strength degradation models under the varying amplitude fatigue loadings. And also it is desired for simple calculations not using the fitting parameters ν, Cm, γ, because it needs hard testing and difficult interpretation of test results. 4 Proposal of simple fatigue life estimation method We can estimate safe lives simply using the scatter factor (paragraph 2.1) and the fudge factor (paragraph 3.4). The schematic illustration is shown in fig.9. That is, Safe Life = Calculated Life/(Scatter Factor x Fudge Factor) Probability of failure 100% Mean life 63.2% Safe life Failure rate 10% or 1% 3 Tested life Scatter Factor Fudge Factor Cycle to failure N Estima life Fig.9 Schematic illustration of simple fatigue life estimation method 5 Conclusion (1) Shape factors of fatigue life vary widely with every configuration of structural element. Therefore scatter should be established for each structural configuration. (2) Safe life depends deeply both on mean life (tested life) and scatter factor (shape parameter). As shown in the sample estimation, the safe life of double shear joint is much longer than that of single shear joint in spite of the very large scatter factor (32 vs. single shear joint 2.3). (3) Fudge factor depends on the statistical life prediction methodologies. The more tested parameters are used, the better consistencies to test life (fudge factor ~1.0) can be got. However it needs hard testing and difficult interpretation of test results. (4) Safe life can be estimated simply by using the scatter factor and the fudge factor. Safe life=calculated Life/(Scatter Factor x Fudge Factor) 2 1 5
Hiroshige KIKUKAWA References [1]Badaliance R., Hill H.D. and Potter J.M, Effects of spectrum variations on fatigue life of composites, ASTM STP 787, 1982. [2]Schaff J.R. and Davidson B.D. Life prediction methodology for composite structures, Journal of composite materials, 1995. [3]Yang J.N. Fatigue and residual strength degradation for graphite/epoxy composites under tension-compression cyclic loadings, Journal of composite materials, 1978. [4]Kim R.Y. Fatigue life prediction methodology, 9 th Japan SAMPE, 2005. 6