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discrete loading
Composites: Part A 36 (2005) 1236–1245 www.elsevier.com/locate/compositesa
A new cumulative fatigue damage model for glass fibre reinforced plastic composites under step/discrete loading Jayantha A. Epaarachchia,*, Philip D. Clausenb a
James Goldston Faculty of Engineering and Physical Systems, Central Queensland University, Bruce Highway, North Rockhampton, QLD 4701, Australia b Mechanical Engineering, School of Engineering, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia Received 4 December 2003; revised 3 December 2004; accepted 31 January 2005
Abstract This paper presents and discusses a new fatigue damage accumulation model for Glass Fibre Reinforced Plastics under step loading. This model was formulated from a single-step constant amplitude fatigue model previously developed by the authors. Fatigue damage is defined and quantified in this paper. The model was tested using independent experimental data and predictions were found to be in excellent agreement with this data. The model was found to give a damage index closer to unity than Palmgren–Miner sum and the models proposed by Broutman and Sahu and Hashin and Rotem. The model predictions were also in excellent agreement with the two-stage fatigue data for carbon/epoxy composites [0/90, G452, 0/90], suggesting that this model may be valid for composite systems other than GFRP. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Glass fibres; B. Fatigue; C. Computational modelling; Cumulative damage
1. Introduction Components and structures manufactured from glass fibre reinforced plastics (GFRP) are used increasingly in situations, where high fatigue loading is present, for example for wind turbine blades and aircraft components. It is therefore, important to have a fatigue damage accumulation model that accurately predicts the life span of a composite component. The behaviour of GFRP composites under fatigue loading is more complex than that of metallic materials because there are many more factors that influence the fatigue crack growth including the matrix material, fibre material, volume fractions, fibre orientation, moisture content, porosity, applied stress and strain rate. The fatigue behaviour of composites has also been shown to be highly dependent on the stress ratio, R and the frequency of applied cyclic load, f. Consequently, it is vital that a comprehensive fatigue model be developed to suit a range of composite materials, which can be extended to include damage accumulation mechanisms, and be * Corresponding author. Fax: C61 7 493 06984. E-mail address: [email protected] (J.A. Epaarachchi).
1359-835X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2005.01.021
reliable in addressing experimental data scatter, Epaarachchi and Clausen [1]. A fatigue model has been proposed by the authors [1], that includes the non-linear effect of stress ratio and load frequency on the fatigue life of a composite material. This model is capable of predicting the total fatigue behaviour under constant amplitude loading of a composite using the results from only a few key experiments. Predictions from this, model were found to be in good agreement with experimental data obtained from the literature and selected research laboratories. A logical extension of this model is to accurately predict the fatigue life of a composite under the action of spectrum or step loading, i.e. loading experienced by a large range of components and structures. This paper describes a model, which can predict fatigue life under step loading, and presents result validating its accuracy. Many cumulative damage theories have been developed for metallic materials, the most widely used was proposed by Palmgren–Miner [2]. Palmgren–Miner’s theory assumes that the damage under repeated or cyclic loads can be related to the net work absorbed by the specimen. From this, they proposed that the ratio of the number of load cycles applied at a particular stress level to the total number of cycles for fatigue failure to occur at the same stress level is proportion to the life spent under the particular loading
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245
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List of symbol relative damage at the stress level of sn frequency (Hz) fatigue life (cycles) of the material at smax fatigue life (cycles) of the material at sn stress level number of fatigue cycles stress ratio (smin/smax) work done by the ith fatigue loading step total work done on the material by fatigue loading
Dsn f N Nn n R wi W
cycles, i.e. wi n X Z i wi Z W W Ni where wi is the work absorbed during the ni cycles, W is the total work absorbed at failure, ni is the number of cycles applied at given stress level si, and Ni is the number of cycles to failure at stress level si. From this, it is easy to derive the well-known Palmgren– Miner’s rule: X ni Ni
Z1
It is well known that Palmgren–Miner’s rule is adequate for predicting the cumulative damage in metallic materials but is inadequate for GFRP composite due to their dominant viscoelastic characteristics. When composites are subjected to spectrum loading or stepped loading, an important and critical effect called ‘load sequence effect’ influences the fatigue life and strength degradation of composite material, Yang and Jones [4,5]. This ‘load sequence’ is the time-based arrangement of the load application sequence or the sequential order of the load applied onto the component. The ‘load sequence effect’ is due to contributions from the ‘boundary effect’, i.e. the difference between residual strength levels, and the ‘memory effect’, where the strength/fatigue properties of the composite are influenced by an accumulation of the previous load history, since the matrix is viscoelastic. The viscoelastic properties of the composites, in combination with the load levels, load duration, temperature and moisture content can lead to substantial amounts of viscoelastic creep, relaxation, damping and damage which can accumulate to induce delayed failure. Many researchers have shown, for example [4–7,21,22], that when the Palmgren–Miner’s rule is applied to composites under a ‘low–high’ load sequence, i.e. low amplitude loading followed by high amplitude loading, the results are generally smaller than unity and for a ‘high–low’ load sequence, i.e. high amplitude loading followed by low amplitude loading, the results are greater than unity. However, it has been shown that ‘high–low’ sequence is
a, b q sres(n) sn smax su
material constants the smallest angle of fibres between the loading direction and the fibre direction, (see Fig. 1) residual strength in the loading direction after nth loading step maximum applied stress in nth loading step maximum applied stress in loading direction ultimate stress of the virgin material in the loading direction
not always less damaging than ‘low–high’ sequence [20,21]. This disparity is primarily due to memory effects and boundary effects and Palmgren–Miner rule does not address these effects. The discrete fatigue processes such as two or multi-steps loading and spectrum loading are significantly different from a constant amplitude fatigue process. The state of the fatigue process has many complex controlling parameters such as loading rate, crack growth rate, stress state and temperature. As such when spectrum loading is applied, the material’s state changes with every load changeover. Dillard [3] argues that the total fatigue process of a composite is a combination of damage accumulation of the fibre and matrix and viscoelastic deformation of the matrix. The latter may be reversible or partially recoverable but the fibre damage is irreversible. The changed level of stress during spectrum loading may lead to a longer fatigue life depending on the applied stress level. Dillard [3] and Sendeckyj [9] show that when some composite materials are subject to severe fatigue conditions, their final strength is much higher than predicted. Furthermore, it has been shown that if a specimen is subject to a single-step load, for example a proof load, its final strength can be up to 30% higher than the strength of the virgin material, Dillard [3]. Dillard suggests that a single-step loading may relieve stress raisers and improve fibre alignment. Several models have been developed for damage accumulation and predictions of residual strength after applying a spectrum/step load on GFRP composites. Sendeckyj [9] briefly reviewed the available fatigue accumulation models and compares the applicability and accuracy of each model. In Sendeckyj’s review, he discusses the life prediction models proposed by Hashin and Rotem [18], Wang et al. [8] and Broutman and Sahu [6]. Also he raised concerns about the danger of formulating P
θ
P
Fig. 1. q the smallest angle between the fibre direction and the loading direction (Note if there are fibres in loading direction. such as [0/90/G45]s then qZ0, in the case of [G45/90]s then qZ458).
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elaborate fatigue theories that are not founded on experimental observation. A model proposed by Hwang and Han [11] based on stiffness degradation and coined ‘fatigue modulus’ has recently been modified by the Hwang et al. [12] by introducing another parameter to account for load sequencing. More recently, Otani and Song [10] have proposed a model for a composite under two-stage loading that was based on the propagation of micro-cracks within the composite. An interesting damage model with five material constants has been proposed by Paepegem and Degrieck [20] based on stiffness degradation at each of the damage stages. Their model considered tension–tension and compression–compression cases separately and used both experimental data and the results of finite element modelling from a flexural sample. Unfortunately, all these models depend on extensive experimental data to determine the value of key control parameters and generally cannot be extended to other operational conditions. Schaff and Davidson [13], Aboussaleh and Boukhili [15] and Fu et al. [16] have proposed models based on a statistical approach. These models do not directly interpret any of the material’s fatigue controlling properties but can be extended to predict fatigue life at the same conditions used to obtain their parameters. Wahl et al. [14] has recently undertaken experimental work on repeated block loading and spectrum loading of fibreglass laminates. They concluded that the load sequencing effect is not significant if there are more than two blocks in the loading sequence and spectrum loading situations. They also concluded that better agreements for lifetime calculations could be found using rules based on strength degradation. In a fatigue process the mechanical properties of composite materials, like its strength, are under continuous degradation. Strength degradation fatigue models assume a material fails due to fatigue when the applied stress level is equal to its strength in the loading direction. Therefore, the damage done on a material due to fatigue loading can be expressed as the amount of strength reduced from the original level to the present strength under the action of fatigue loading. Extensive studies have been carried out on the model developed by the authors [1] for its suitability for use with spectrum and step loading processes. The results of this work suggest that the model can be used to calculate damage in discrete loading situations such as multi-step loading.
2. Analysis There are many definitions for fatigue damage, D, depending on the degradation property chosen as an indicator. Hwang and Han [11] have used fatigue modulus and failure strain to define the damage parameter. They quantified the damage as the ratio of degradation of strain under fatigue cycles (from its static value) to the degradation of strain at failure for the same stress level.
A similar definition was given for fatigue modulus—the ratio of the stiffness at failure to the original stiffness. Broutman and Sahu [6] used degradation of strength to define their damage parameter. Sendeckyj [9] provides a comprehensive summary of many fatigue accumulation models and of the damage parameters used in those models. In all models, the damage is defined relative to the final loading step that cause fatigue failure. In other words, the damage parameter should represent the residual life or residual strength relative to the final or the concerned stress level. Following this we define the relative damage as Dsl Z ðsu K sresðnÞ Þ=ðsu K sl Þ which is the damage after n cycles at sn loading, relative to load level sl, where sl!su. Here sl denotes the final loading level and su is the ultimate stress of the virgin material. The relative damage, Dsl has a value 0!Dsl!1. DslR1 suggests that there is no usable strength relative to the stress level considered, sl. If a composite component is subject to a spectrum or step loading situation which has the final stress level sl, and under that stress level the component failed by fatigue, then DslZ1. More generally, Pk rZ1 ðsresðrK1Þ K sresðrÞ Þ D sk Z ðsk ! su Þ (1) ðsu K sk Þ Let us consider a situation of step loading in which fatigue failure is caused by stress level sk. So, say for a particular sample, the calculated value for DskO1 when using experimental data from the sample population, and would suggest that this sample is stronger than ‘average’. Conversely, if Dsk!1 then the sample is weaker than ‘average’. Therefore, in a spectrum or multi-step loading situations the applied load that causes the final failure may be one of the major factors which will determine the life usage of the material. The model developed by the authors’ for a continuous single-step (i.e. constant amplitude loading) fatigue process is given by i smax 0:6Kjjsin qj h su K smax Z a smax ð1 K jÞ1:6Kjjsin qj su 1 b ! b N K1 ð2Þ f where smax is the applied stress, f is the frequency, N is the fatigue life at applied stress level, a and b are material parameters which can be obtained using traditional S–N fatigue data of the material. A brief explanation of the method to determine a and b is given in Appendix B. Here q is the smallest ply angle of the laminate to the loading direction, see Fig. 1, and R is the stress ratio [1]. The significance of the indices in Eq. (2) is explained in Ref. [1]. Also jZR for KN%R!1 (tension–tension and reverse loading) jZ1/R for 1!R%N (compression–compression).
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245
To avoid confusion the right side of Eq. (2) will be replaced with ðDsÞsn ;n Z a
sn
sresðnK1Þ 1 ! b ðnb K 1Þ fn
ð3Þ
where (Ds)sn, n is the strength degraded after n cycles at stress level sn. This model was derived for the constant amplitude fatigue loading of (s, N with loading frequency f and stress ratio R) a specimen subject to tension–tension, reversal, and compression–compression or flexural fatigue. The model in Eq. (2) must be modified to deal with discontinuous loading situations, where many state changes occur in the specimen’s fatigue life during the stress level changeovers. The time dependent nature of matrix material and glass fibre properties will immediately change the damage accumulation process with any change of stress levels, stress ratios and the frequency. During a step/discrete fatigue process, any change of stress level will results in a comparable amount of viscoelastic deformations being reversed and a part of the damage in the material’s memory erased [3]. As such, the reduction of strength of the composite calculated by the model in Eq. (2) is not accurate for a discontinuous process. Many researchers have shown that the degradation of strength during a discontinuous fatigue process is a function of (n/N) [1,2,4,6,11,14,18,23]. Let F(n/N) be the factor needed to correct the strength degradation in Eq. (2) which was proposed for the continuous fatigue process under a constant amplitude fatigue load. As such the amount of strength remaining after n fatigue cycles at a particular stress level during a discontinuous (i.e. variable amplitude loading) fatigue process can be explained as; Remaining strength after n cycles Z ðDsÞsn ;n ðFðn=NÞÞ
(4)
The fraction of strength degraded after n cycles is therefore Z(1KF(n/N)). Taking the general form of the model [1], Eq. (2) can be approximated for a discontinuous fatigue process by introducing fn a ‘factor’ that accounts for the fraction of strength degraded in a discontinuous fatigue process (i.e. variable amplitude loading) as the material has not used all it fatigue life under the previous loading level. So fn Z ð1 K Fðn=NÞÞ
Then "
n fn Z 1 K 1 K Nsn
0:6Kjjsin qj 1:6Kjjsin qj sn 1 K j
(5)
Assuming the damage accumulation process is nonlinear with respect to the number of loading cycles, we propose function F(n/N) has the form: " #b n
n b F (6) Z 1K N Nsn
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b #b (7)
where Ns,n is the residual life at sn after the loading at the (nK1)th step. Incorporating this into Eq. (2) yields 0:6Kjjsin qj smax sresðnK1Þ KsresðnÞ Za sresðnK1Þ 1:6Kjjsin qj 1 ! smax ð1KjÞ ðnb K1Þfn ZðDsÞsn ;n fn b fn ð8Þ where sres(n) is the residual strength after n cycles at sn, and sres(nK1) is the residual strength after n cycles at snK1. Now let us consider a multiple step loading spectrum of {(s1,n1@f1,R1).(sn,nn@fn,Rn)} with fatigue failure at (sn,nn@fn,Rn), where ni, si, fi and Ri are the number of cycles, applied stress, load frequency and stress ratio at the ith load step, respectively. The residual stress, sres1, at the end of first load step (s1,n1@f1,R1) is given by sult K sres1 Z ðDsÞs1n1 f1
(9)
Similarly for the final load step, (sn,nn@fn,Rn), where fatigue failure occurs, sresðnK1Þ K sn Z ðDsÞsn nn fn :
(10)
Note that fnZ1, where the nth step is the last step. Summing for n steps yields Pn 1Z
sk kZ1 a sresðkK1Þ
0:6Kjjsin qj
½sk ð1Kjk Þ1:6Kjk jsin qj f1b ðnbk K1Þfk k
su Ksn ð11Þ
which is the relative damage, Dsn at sn.
If a specimen is subject to a constant amplitude test under stress level sn at any arbitrary value of f and R, until it is fatigue failure, then 0:6Kjjsin qj sn su K sn Z a ½sn ð1 K jÞ1:6Kjjsin qj sult 1 !
fnb
ðNnb K 1Þ
(12)
Substituting Eq. (12) into Eq. (11) yields Pn sk 0:6Kjjsin qj ½sk ð1Kjk Þ1:6Kjk jsin qj f1b ðnbk K1Þfk kZ1 sresðkK1Þ k 1Z 0:6Kjjsin qj sn 1:6Kjjsin qj 1 b ½sn ð1KjÞ f b ðN K1Þ su n
(13)
Using Eq. (11) or (13) the residual strength or the indicator for damage Dsn , can be calculated for step loading situations.
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Table 1 Damage calculated by proposed algorithm and experimental results published by Broutman and Sahu [6] for E-Glass/Epoxy [08/908] Experiment
Calculated
Test no.
s1 (MPa)
s2 (MPa)
n1
n2 (mean)
s(res) after n1 (MPa)
Sum (Ds2)
Palmgren– Miner sum
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
386 386 386 386 386 386 337 337 337 337 289 289 241 241 241 241 241 289 289 289 337 337
241 241 289 289 337 337 241 241 289 289 241 241 289 289 337 337 386 337 337 386 386 386
250 100 250 100 250 100 1000 249 1000 249 9996 1999 49,938 19,975 49,938 19,975 19,975 9996 1999 1999 1000 249
192,000 193,000 5840 11,970 1250 1635 86,000 162,500 8670 8000 96,500 110,800 3730 9490 391 804 124 293 1290 355 297 503
437 463 437 463 437 463 448 469 448 469 425 463 431 454 431 454 454 425 463 463 448 469
1.15 1.09 0.83 0.92 0.94 0.93 0.93 1.03 0.89 0.82 1.01 0.96 0.75 0.89 0.70 0.79 0.84 0.66 0.88 1.12 1.12 1.22
1.62 1.33 0.9 1.02 1.01 0.87 0.91 1.05 0.99 0.64 1.24 0.78 0.64 0.76 0.45 0.44 0.37 0.8 0.66 0.86 1.01 1.12
Broutman and Sahu [6] sum
Hashin and Rotem sum
1.36 1.14 1.31 1.12 1.22 1.09 1.19 1.05 1.13 1.03 1.16 1.03 0.91 0.96 0.77 0.89 0.73 0.7 0.94 0.79 0.69 0.92
1.30 1.14 0.65 0.86 0.87 0.75 0.72 0.97 0.89 0.59 1.18 0.73 0.63 0.82 0.63 0.60 0.67 0.87 0.75 1.09 1.15 1.24
Calculated values for aZ0.197 and bZ0.2265 (sultZ482 MPa, RZ0.05 and fZ10 Hz).
[22]. The results of this work are presented in Table 1–4 inclusive and Fig. 3 in Section 4.
3. Model validation The comprehensive data published by Broutman and Sahu [6], Bonnee [7] and Hwang and Han [11] has been used to validate our model. Also test results for residual strength published in DOE/MSU database [17] has been used to check the accuracy of the model for residual strength calculations. The model proposed here was an extension of a previous fatigue life prediction model [1], which was defined for glass-fibre reinforced composites and not for carbon/epoxy composites. However, since the performance of this model will be compared against other available damage accumulation models, such as Palmgren–Miner, Broutman and Sahu and Hasin and Rotem, for completeness we decided to check the accuracy of the model’s predictions for the carbon/epoxy composites published in reference
3.1. Statistical investigation of model predictions The statistical method of addressing scatter in fatigue data has been described in detail by many researchers [19,24]. As such the characteristic values of fatigue data can be estimated in desired confidence intervals. It has been warranted that the parameters a and b of Eq. (2) should be obtained using statistically analysed fatigue data, for better performance of the model [1]. In this paper, the model parameters a and b for all cases were determined using statistically analysed fatigue data. However, it is useful to evaluate the confidence level of the predictions made by each cumulative damage model. Consequently, the predictions from the proposed model,
Table 2 Damage calculated by proposed algorithm and experimental results published by NRL Investigation-[7] on E-Glass/Polyester 08/G458, a Z0.1938 and bZ0.238 (3ultZ2.57% and RZK1, fZ5 Hz) Experiment
Calculated
31 (%)
32 (%)
n1
n2
3(res) % after n1
Sum (D32)
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
1 1 1 1 0.5 0.5 0.5 0.5
1.00!104 1.00!104 1.00!104 1.00!104 1.00!104 1.00!104 1.00!104 1.00!104
5850 5430 3280 9490 1635,210 925,240 1509,550 830,050
2.3814 2.3814 2.3814 2.3814 2.3814 2.3814 2.3814 2.3814
1.16 1.14 1.01 1.31 1.20 1.05 1.17 1.03
Palmgren– Miner sum
Broutman and Sahu
Hasin and Rotem
0.99 0.92 0.58 1.57 0.64 0.39 0.60 0.35
1.05 0.98 0.62 1.67 1.29 0.75 1.20 0.68
1.02 0.95 0.61 1.60 0.63 0.37 0.58 0.34
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245
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Table 3 Damage calculated by proposed algorithm and experimental results published by Hwang and Han-[11] on E-Glass/Epoxy Uni-directional composite Type 1002, aZ0.0182 and bZ0.34 (sultZ770 MPa and RZ0.05, fZ2 Hz) Experiment
Calculated
s1 (MPa)
s2 (MPa)
n1
539 654.5 577.5 654.5
654.5 539 654.5 577.5
10,000 500 5000 500
Mean
Std. dev.
s(res) after n1
2031 33,049 3240 21,450
2983 3012 2986 14,742
739.4 757.1 746.1 757.1
n2
Palmgren–Miner rule, Broutman and Sahu and Hashin and Rotem, have all been statistically analysed. It was found that the damage index from all three models approximately follows a Weibull distribution. As such the Weibull probability function was fitted into the damage index
Sum Ds2 1.1 1.0 1.1 1.0
Palmgren– Miner sum
Broutman and Sahu
Hashin and Rotem
0.8 0.8 1.1 0.9
0.97 0.76 1.18 0.86
1.01 0.71 1.24 0.81
The Weibull parameters were determined in 95% confident levels using ‘MINITAB’ statistical software. The Weibull PDF graphs for the damage index D for each case are shown in Figs. 4–6 inclusive.
Probability density of index D pdfðDÞ a a Z DaK1 eKðD=bÞ for DO 0 (14) b where a and b are shape and scale parameters of Weibull distribution. As an example, the probability of damage D within the range ð 1:1 a aK1 KðD=bÞa D Prð0:9! D! 1:1Þ Z e dD 0:9 b
and illustrated by the shaded area in the Fig. (2).
4. Results and discussion The proposed model has been tested against several twostep fatigue data sets for tension–tension and reverse loading, published in the literature and found to be in good agreement. These results are presented in Tables 1–4 inclusive and Fig. 3. All fatigue data exhibits significant scatter, which is normal for glass fibre composites. This scatter reflects the non-homogeneity presented in material batches even if the specimens are sourced from the same stock. This suggests a comprehensive statistical analysis to
Table 4 Damage calculated by proposed algorithm and experimental results published by Found and Quaresimin-[22] on carbon-fibre/Epoxy angle-ply composite [0/90, G452, 0/90], aZ0.25106 and bZ0.094 (sultZ422 MPa and RZ0.05, fZ3 Hz) Experiment s1
s2
n1
n2
315 315 315 315 315 315 315 315 315 315 315 315 315 340 340 340 340 340 340 340 340 340 340 340 340
340 340 340 340 340 340 340 340 340 340 340 340 340 315 315 315 315 315 315 315 315 315 315 315 315
87,200 87,000 86,300 57,700 57,550 40,300 28,700 26,500 25,300 17,650 17,000 13,000 12,500 8500 7480 7480 6800 6500 4600 4400 4400 2500 1500 1500 1350
520 150 1408 1750 2280 2027 3320 2640 2464 6170 38,140 14,300 24,030 15,250 17,060 79,496 29,939 48,760 73,910 89,350 80,605 90,150 41,840 111,120 99,520
Model sum (D32)
Palmgren– Miner sum
Broutman and Sahu
Hashin and Rotem
1.10 0.94 1.23 1.10 1.14 1.06 1.10 1.05 1.04 1.15 1.48 1.27 1.37 1.17 1.17 1.38 1.14 1.19 1.17 1.19 1.17 1.13 1.00 0.83 1.11
0.82 0.77 0.91 0.70 0.76 0.58 0.63 0.53 0.50 0.85 4.48 1.74 2.84 1.10 1.00 1.54 1.03 1.16 1.16 1.28 1.20 1.07 0.53 0.97 1.02
1.05 1.00 1.14 0.85 0.91 0.69 0.70 0.60 0.57 0.90 4.53 1.77 2.87 0.87 0.80 1.34 0.85 0.99 1.04 1.16 1.08 1.00 0.49 0.23 0.98
0.87 0.82 0.96 0.79 0.85 0.68 0.72 0.62 0.59 0.94 4.56 1.81 2.91 1.09 0.96 1.50 0.97 1.10 1.07 1.18 1.10 0.98 0.46 0.20 0.95
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J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245 3 Model Hashin & Rotem Palmgren-Miner Broutman & Sahu
Probability Density
2.5
2
1.5
1
Fig. 2. Weibull probability distribution of damage index D.
Predicted
Experimental
800
Res. Strength (MPa)
0.5
0 0
700 600
0.5
1
1.5
2
Damage Index Fig. 4. Comparison of probability of prediction of damage index by proposed model, P–M, Broutman and Sahu and Hashin and Rotem, for the two-step loading results published by Broutman and Sahu [6].
The specimen has been subject to 249 cycles at 341 MPa followed by 503 cycles at a higher stress level of 389 MPa. The virgin material, Table A1, can only withstand 493 cycles at 389 MPa. Therefore, these specimens are stronger than average as indicated by D being greater than unity. Consider now the results in Table 2, for the step loading data set of reverse loading RZK1. It can be seen that the proposed model and the damage index Ds2 are consistently closer to unity than the other models. As shown in Fig. 5, the probability of the predicted damage D being within the arbitrary chosen 0.9!D!1.1 is higher for the proposed model than from all other models. It is clear, from 4 Model Hashin & Rotem Palmgren-Miner Broutman & Sahu
3.5 3
Probability Density
obtain the most probable limits of the material properties and their fatigue life from scattered data sets, [19,24]. For the proposed model, it is vital to accurately calculate material parameters a and b. To overcome the effects of the scatter in the data, we used the statistically analysed fatigue life data presented in each set, to obtain a value for parameters a and b. The results shown in Table 1 are for E-Glass/Epoxy [08/908] [6]. The index proposed by Broutman and Sahu shows the load sequencing effect, i.e. the summation is less than unity for low–high loading and greater than unity for high–low loading. The proposed damage index Ds2, Palmgren–Miner and Hashin and Rotem models, do not show any significant load sequencing effect. However, the probability of damage D for each model to be within the arbitrary chosen limits 0.9!D!1.1 is considerably higher for the proposed model than for Palmgren - Miner and Hashin and Rotem models, see Fig. (4) for details. The damage, D, depends on the final loading conditions, such as applied stress, stress ratio and load frequency. D can be used to compare strong or weak material batches. For example consider test 1 in Table 1. Here the specimen has been subject to a few cycles at a relatively high stress level followed by 192,000 cycles at a stress of 244 MPa. The virgin material, Table A1 (in Appendix A), can only withstand 172,000 cycles at 244 MPa. Also consider test 22 in Table 1.
2.5 2 1.5
500 400
1
300 200
0.5
100 0
0 0
50000
100000
150000
200000
Cycles
Fig. 3. Residual strength of material DD16A [17] under applied stress level of 241 MPa, RZ0.1 and frequency 8 Hz. Glass/Polyester [90/0/G45/0]s., aZ0.61 and bZ0.2 (sultZ673 MPa and RZ0.1, fZ13 Hz).
0
0.5
1
1.5
2
Damage Index Fig. 5. Comparison of probability of prediction of damage index by proposed model, P–M, Broutman and Sahu and Hashin and Rotem, for the two-step loading results published by NRL [7].
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245
and spectrum fatigue data. Whal et al. [14] concluded that for repeated step loading and spectrum loading situations, the Palmgren–Miner’s sum will always be less than unity. The results in Tables 1–4 inclusive, show that Palmgren– Miner’s, Broutman and Sahu and Hashin and Rotem models did not give consistent results in all cases considered here. However, the proposed model is consistent in it’s damage index and probability of prediction is very much higher than to the other models.
3
Model Hashin & Rotem Palmgren-Miner Broutman & Sahu
Probability Density
2.5
1243
2
1.5
1
5. Conclusions 0.5
0 0
0.5
1
1.5
2
Damage Index Fig. 6. Comparison of probability of prediction of damage index by proposed model, P–M, Broutman and Sahu and Hashin and Rotem, for the two-step loading results published by Found and Quaresimin [22].
a statistical point of view that the proposed model appears to be superior to Palmgren–Miner’s Broutman and Sahu and Hashin and Rotem models. Results shown in the Table 3, for E-Glass/Epoxy UD Type 1002 [11], clearly show the predictions from the proposed model are much closer to unity than the predictions from the other three models. However, as the size of this data set was too small, a statistical analysis for the damage index was not performed. The results presented in Table 4 for Carbon/Epoxy system of [22] show that the proposed model yields more accurate results than the other three models. Also Fig. 6 confirmed that the probability of damage D within the arbitrary chosen 0.9!D!1.1 range is much higher for the proposed model than the other models with the probability density of Palmgren–Miner’s Broutman and Sahu and Hashin and Rotem models collapsing approximately into a single curve. Let us consider the residual strength calculations for the material [90/0G45/0]s [17] as shown in Fig. 3. Here it clearly shows that the predicted values closely follow the experimental values. The model indicates sudden failure after 176,877 cycles at stress level of 241 MPa, which is close to the experimental life of 169,974 cycles as shown in Table A4. According to the model, the loading sequence, s1, n1Ks2, n2 and the loading sequence, s2, n2Ks1, n1, (where s1Os2) should give different residual strength values that are dependent on the values of s1, n1, s2 and n2. Whal et al. [14] concluded that the load sequencing effect is insignificant at multi-stress level or repeated step applications. The damage index by the proposed model does not shows any significant load sequencing effect, the damage index fluctuates about unity irrespective of loading sequence. However, the information given by Whal et al. [14] is insufficient to be reliably apply to the model for their step
This paper describes a fatigue damage accumulation model for GFRP in discrete loading situations. This model was based on a fatigue model previously developed by the authors’ [1]. Here we define the damage to the composite using strength degradation concept, which have been shown to be dependent on the final failure conditions. The predictions from the model for two-step loading in tension–tension and reverse loading were found to be in good agreement with the experimental data obtained from the literature. Unfortunately, the model could not be tested for compression–compression two-step loading condition as well as multi-block and spectrum loading due to lack of reliable experimental data. The model does indicate that load-sequencing effects in two-step-loading conditions have been sufficiently addressed in the proposed model. The proposed model also shows excellent agreement with twostep loading data for a carbon/epoxy composite, but further testing will need to be undertaken before confirming its accuracy for all carbon/epoxy composite systems.
Acknowledgements The work presented in this paper was supported by an Australian Research Council SPIRT grant. JE was financially supported by an Australian Postgraduate Award (industry) scholarship.
Appendix A Table A1 Life time calculated by model [1] and experimental results published by Broutman and Sahu [6] for E-Glass/Epoxy [08/908] Experiment
Calculated N
smax(ksi)
N (mean)
386 337 289 241
493 2470 14,700 172,000
304 2583 19,861 158,994
Calculated values for aZ0.197 and bZ0.2265 (sultZ483 MPa, RZ0.05 and fZ10 Hz).
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Table A2 Life time calculated by model [1] and experimental results published by NRL investigation-[7] on E-Glass/Polyester 08/G458, aZ 0.1938 and bZ0.238 (3ultZ2.57% and RZK1, fZ5 Hz) Experimental
D
Calculated N
3max (%)
N (Weibull mean)
0.5 0.7 1.0
2767,904 185,361 6260
1316,058 101,812 5928
Table A3 Life time calculated by model [1] and experimental results published by Hwang and Han-[11] on E-Glass/Epoxy Uni-directional composite type 1002, aZ0.0182 and bZ0.34 (sultZ770 MPa and RZ0.05, fZ1–3 Hz) Experiment
Calculated N
smax (MPa)
N
693 654.50 616 577.5 539
Mean
Std. dev.
513 3694 10,922 27,577 47,739
346 4322 4194 21,382 24,642
463 3590 11,863 29,949 55,699
Table A4 Life time calculated by model [1] and experimental results published for DD16A [17] Experiment
(N β-1) Fig. B1. Illustration of determining parameters a and b.
!
1 fb ½ð1 K jÞ1:6Kjjsin qj
su K1 Where D Z smax
su smax
0:6Kjjsin qj
(B3)
The complexity of Eq. (B1) suggests a trial-and-error method to determine appropriate values of a and b. For each experimental data set, D from Eq. (B‘3) was determined for a trial value of b. D was then plotted against (NbK1) as shown in Fig. B1. This procedure is repeated by changing value of b, until the linear regression line passes through the origin. The values of a and b are noted.
Calculated N
smax (MPa)
N Mean
Std. dev.
241
169,974
98,418
References
176,877
Glass/Polyester [90/0/G45/0]s., aZ0.61 and bZ0.2 (sultZ673 MPa and RZ0.1, fZ13 Hz).
Table A5 Calculated lifetime and experimental results published by Found and Quaresimin-[22] on carbon-fibre/Epoxy angle-ply composite [0/90, G 452, 0/90], aZ0.25106 and bZ0.094 (sultZ422 MPa and RZ0.05, fZ 3 Hz) Experiment
Calculated N
smax (MPa)
N
315 340
115,150 8800
176,877
Appendix B
smax 0:6Kjjsin qj smax ð1 K jÞ1:6Kjjsin qj au 1 b ! B ðN K 1Þ (B1) f
su K smax Z a
rearranging Eq. (B2) D Z aðN b K 1Þ
(B2)
[1] Epaarachchi JA, Clausen PD. An empirical model for fatigue behavior prediction of glass fibre-reinforced plastics composites for various stress ratios and test frequencies, Composite A. Appl Sci Manuf 2003; 34:313–26. [2] Miner MA. Cumulative damage in fatigue. J Appl Mech 1945;12: A159–A64. [3] Dillard DA. Viscoelastic behavior of laminated composite materials. In: Reifsnider KL, editor. Fatigue of composite materials. Amsterdam: Elsevier; 1990. p. 339–84. [4] Yang JN, Jones DL. Load sequence effects on graphite/epoxy [G 35]2s. In: O’Brien TK, editor. Long term behavior of composites, ASTM STP, vol. 813, 1983. p. 246–62. [5] Yang JN, Jones DL. Effect of load sequence on the statistical fatigue of composites. AIAA J 1980;18(12):1525–31. [6] Broutman LJ, Sahu SA. New theory to predict cumulative fatigue damage. In: Fiberglass reinforced plastics, composite materials: testing and design (second conference), ASTM STP, vol. 497;1972. p. 170–88. [7] Bonnee WJA. NLR investigation of polyester composite materials. In: Kensche CW, editor. Fatigue of materials and components for wind turbine rotor blades. German Aerospace Establishment, European Commission-EUR 16684 p. 39–70. [8] Wang ASD, Chou PC, Alper J. Effect of proof test on the strength and fatigue life of a unidirectional composite. Fatigue of Fibrous Composite Materials ASTM STP. vol. 723; 1981 p. 116–132. [9] Sendeckyj GP. Life prediction for resin_matrix composite materials. In: Reifsnider KL, editor. Fatigue composite materials. Amsterdam: Elsevier; 1990.
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245 [10] Otani N, Song DY. Fatigue life prediction of composites under twostage loading. J Mater Sci 1997;32(3):755–60. [11] Hwang WB, Han KS. Fatigue of composite materials-damage model and life prediction. In: Composite materials fatigue and fracture, ASTM STP, vol. 1012;1989. p. 87–102. [12] Hwang H, Lee CS, Park HC, Han KS. Single- and multi-stress level fatigue life prediction of glass/epoxy composites. J Adv Mater 1995; 26(4):3–9. [13] Schaff JR, Davidson BD. Life prediction methodology for composite structures. Part II—spectrum Fatigue. J Composite Mater 1977;31(2): 158–81. [14] Wahl N, Samborsky D, Mandell J, Cairns D. Spectrum fatigue lifetime and residual strength for fiberglass laminates, AIAA-20010025;2001. [15] Aboussaleh M, Boukhili R. Life prediction for composite laminates submitted to service loading spectra. Polym Composites 1998;19(3): 241–5. [16] Fu KE, Lee LJ, Yang JN. Statistical fatigue of composite laminates subject to service loading spectrum. AIAA structural dynamics and material conference; 1996 p. 2148–58.
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[17] DOE-MSU wind turbine blade composite material fatigue database 1998–1999, Department of Chemical Engineering, Montana State University at Bozman, Montan 59717, USA. [18] Hashin Z, Rotem A. A cumulative damage theory of fatigue failure. J Mater Sci Eng 1978;(34):147–60. [19] Sendeckyj GP. Fitting models to composite materials fatigue data. In: Chaims CC, editor. Test methods and design allowable for fibrous composites, ASTM SIP, vol. 743; 1981. p. 245–60. [20] Van Paepegem W, Degrieck J. Effects of Load Sequence and Block Loading on the Fatigue Response of Fibre-reinforced Composites, Mechanics of Advanced Materials and Structures;9(1):19–35. [21] Gamstedt EK, Sjo¨gren BA. An experimental investigation of the sequence effect in block amplitude loading of cross-ply composite laminates. Int J Fatigue 2002;24:437–46. [22] Found MS, Quaresimin M. Two-stage loading of woven carbon fibre reinforced laminates. Fatigue Fracture Eng Mater 2003;26:17–26. [23] Scharpery RA. Deformation and Failure Analysis of Viscoelastic Composite Materials. AMD, ASME. vol. 13; 1975. [24] Ronold Knut O, Echtermeyer Andreas T. Estimation of fatigue curves for design of composite laminates. Composite: part A 1996;27A: 485–91.
A new cumulative fatigue damage model for glass fibre reinforced plastic composites under step/discrete loading Jayantha A. Epaarachchia,*, Philip D. Clausenb a
James Goldston Faculty of Engineering and Physical Systems, Central Queensland University, Bruce Highway, North Rockhampton, QLD 4701, Australia b Mechanical Engineering, School of Engineering, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia Received 4 December 2003; revised 3 December 2004; accepted 31 January 2005
Abstract This paper presents and discusses a new fatigue damage accumulation model for Glass Fibre Reinforced Plastics under step loading. This model was formulated from a single-step constant amplitude fatigue model previously developed by the authors. Fatigue damage is defined and quantified in this paper. The model was tested using independent experimental data and predictions were found to be in excellent agreement with this data. The model was found to give a damage index closer to unity than Palmgren–Miner sum and the models proposed by Broutman and Sahu and Hashin and Rotem. The model predictions were also in excellent agreement with the two-stage fatigue data for carbon/epoxy composites [0/90, G452, 0/90], suggesting that this model may be valid for composite systems other than GFRP. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Glass fibres; B. Fatigue; C. Computational modelling; Cumulative damage
1. Introduction Components and structures manufactured from glass fibre reinforced plastics (GFRP) are used increasingly in situations, where high fatigue loading is present, for example for wind turbine blades and aircraft components. It is therefore, important to have a fatigue damage accumulation model that accurately predicts the life span of a composite component. The behaviour of GFRP composites under fatigue loading is more complex than that of metallic materials because there are many more factors that influence the fatigue crack growth including the matrix material, fibre material, volume fractions, fibre orientation, moisture content, porosity, applied stress and strain rate. The fatigue behaviour of composites has also been shown to be highly dependent on the stress ratio, R and the frequency of applied cyclic load, f. Consequently, it is vital that a comprehensive fatigue model be developed to suit a range of composite materials, which can be extended to include damage accumulation mechanisms, and be * Corresponding author. Fax: C61 7 493 06984. E-mail address: [email protected] (J.A. Epaarachchi).
1359-835X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2005.01.021
reliable in addressing experimental data scatter, Epaarachchi and Clausen [1]. A fatigue model has been proposed by the authors [1], that includes the non-linear effect of stress ratio and load frequency on the fatigue life of a composite material. This model is capable of predicting the total fatigue behaviour under constant amplitude loading of a composite using the results from only a few key experiments. Predictions from this, model were found to be in good agreement with experimental data obtained from the literature and selected research laboratories. A logical extension of this model is to accurately predict the fatigue life of a composite under the action of spectrum or step loading, i.e. loading experienced by a large range of components and structures. This paper describes a model, which can predict fatigue life under step loading, and presents result validating its accuracy. Many cumulative damage theories have been developed for metallic materials, the most widely used was proposed by Palmgren–Miner [2]. Palmgren–Miner’s theory assumes that the damage under repeated or cyclic loads can be related to the net work absorbed by the specimen. From this, they proposed that the ratio of the number of load cycles applied at a particular stress level to the total number of cycles for fatigue failure to occur at the same stress level is proportion to the life spent under the particular loading
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245
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List of symbol relative damage at the stress level of sn frequency (Hz) fatigue life (cycles) of the material at smax fatigue life (cycles) of the material at sn stress level number of fatigue cycles stress ratio (smin/smax) work done by the ith fatigue loading step total work done on the material by fatigue loading
Dsn f N Nn n R wi W
cycles, i.e. wi n X Z i wi Z W W Ni where wi is the work absorbed during the ni cycles, W is the total work absorbed at failure, ni is the number of cycles applied at given stress level si, and Ni is the number of cycles to failure at stress level si. From this, it is easy to derive the well-known Palmgren– Miner’s rule: X ni Ni
Z1
It is well known that Palmgren–Miner’s rule is adequate for predicting the cumulative damage in metallic materials but is inadequate for GFRP composite due to their dominant viscoelastic characteristics. When composites are subjected to spectrum loading or stepped loading, an important and critical effect called ‘load sequence effect’ influences the fatigue life and strength degradation of composite material, Yang and Jones [4,5]. This ‘load sequence’ is the time-based arrangement of the load application sequence or the sequential order of the load applied onto the component. The ‘load sequence effect’ is due to contributions from the ‘boundary effect’, i.e. the difference between residual strength levels, and the ‘memory effect’, where the strength/fatigue properties of the composite are influenced by an accumulation of the previous load history, since the matrix is viscoelastic. The viscoelastic properties of the composites, in combination with the load levels, load duration, temperature and moisture content can lead to substantial amounts of viscoelastic creep, relaxation, damping and damage which can accumulate to induce delayed failure. Many researchers have shown, for example [4–7,21,22], that when the Palmgren–Miner’s rule is applied to composites under a ‘low–high’ load sequence, i.e. low amplitude loading followed by high amplitude loading, the results are generally smaller than unity and for a ‘high–low’ load sequence, i.e. high amplitude loading followed by low amplitude loading, the results are greater than unity. However, it has been shown that ‘high–low’ sequence is
a, b q sres(n) sn smax su
material constants the smallest angle of fibres between the loading direction and the fibre direction, (see Fig. 1) residual strength in the loading direction after nth loading step maximum applied stress in nth loading step maximum applied stress in loading direction ultimate stress of the virgin material in the loading direction
not always less damaging than ‘low–high’ sequence [20,21]. This disparity is primarily due to memory effects and boundary effects and Palmgren–Miner rule does not address these effects. The discrete fatigue processes such as two or multi-steps loading and spectrum loading are significantly different from a constant amplitude fatigue process. The state of the fatigue process has many complex controlling parameters such as loading rate, crack growth rate, stress state and temperature. As such when spectrum loading is applied, the material’s state changes with every load changeover. Dillard [3] argues that the total fatigue process of a composite is a combination of damage accumulation of the fibre and matrix and viscoelastic deformation of the matrix. The latter may be reversible or partially recoverable but the fibre damage is irreversible. The changed level of stress during spectrum loading may lead to a longer fatigue life depending on the applied stress level. Dillard [3] and Sendeckyj [9] show that when some composite materials are subject to severe fatigue conditions, their final strength is much higher than predicted. Furthermore, it has been shown that if a specimen is subject to a single-step load, for example a proof load, its final strength can be up to 30% higher than the strength of the virgin material, Dillard [3]. Dillard suggests that a single-step loading may relieve stress raisers and improve fibre alignment. Several models have been developed for damage accumulation and predictions of residual strength after applying a spectrum/step load on GFRP composites. Sendeckyj [9] briefly reviewed the available fatigue accumulation models and compares the applicability and accuracy of each model. In Sendeckyj’s review, he discusses the life prediction models proposed by Hashin and Rotem [18], Wang et al. [8] and Broutman and Sahu [6]. Also he raised concerns about the danger of formulating P
θ
P
Fig. 1. q the smallest angle between the fibre direction and the loading direction (Note if there are fibres in loading direction. such as [0/90/G45]s then qZ0, in the case of [G45/90]s then qZ458).
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elaborate fatigue theories that are not founded on experimental observation. A model proposed by Hwang and Han [11] based on stiffness degradation and coined ‘fatigue modulus’ has recently been modified by the Hwang et al. [12] by introducing another parameter to account for load sequencing. More recently, Otani and Song [10] have proposed a model for a composite under two-stage loading that was based on the propagation of micro-cracks within the composite. An interesting damage model with five material constants has been proposed by Paepegem and Degrieck [20] based on stiffness degradation at each of the damage stages. Their model considered tension–tension and compression–compression cases separately and used both experimental data and the results of finite element modelling from a flexural sample. Unfortunately, all these models depend on extensive experimental data to determine the value of key control parameters and generally cannot be extended to other operational conditions. Schaff and Davidson [13], Aboussaleh and Boukhili [15] and Fu et al. [16] have proposed models based on a statistical approach. These models do not directly interpret any of the material’s fatigue controlling properties but can be extended to predict fatigue life at the same conditions used to obtain their parameters. Wahl et al. [14] has recently undertaken experimental work on repeated block loading and spectrum loading of fibreglass laminates. They concluded that the load sequencing effect is not significant if there are more than two blocks in the loading sequence and spectrum loading situations. They also concluded that better agreements for lifetime calculations could be found using rules based on strength degradation. In a fatigue process the mechanical properties of composite materials, like its strength, are under continuous degradation. Strength degradation fatigue models assume a material fails due to fatigue when the applied stress level is equal to its strength in the loading direction. Therefore, the damage done on a material due to fatigue loading can be expressed as the amount of strength reduced from the original level to the present strength under the action of fatigue loading. Extensive studies have been carried out on the model developed by the authors [1] for its suitability for use with spectrum and step loading processes. The results of this work suggest that the model can be used to calculate damage in discrete loading situations such as multi-step loading.
2. Analysis There are many definitions for fatigue damage, D, depending on the degradation property chosen as an indicator. Hwang and Han [11] have used fatigue modulus and failure strain to define the damage parameter. They quantified the damage as the ratio of degradation of strain under fatigue cycles (from its static value) to the degradation of strain at failure for the same stress level.
A similar definition was given for fatigue modulus—the ratio of the stiffness at failure to the original stiffness. Broutman and Sahu [6] used degradation of strength to define their damage parameter. Sendeckyj [9] provides a comprehensive summary of many fatigue accumulation models and of the damage parameters used in those models. In all models, the damage is defined relative to the final loading step that cause fatigue failure. In other words, the damage parameter should represent the residual life or residual strength relative to the final or the concerned stress level. Following this we define the relative damage as Dsl Z ðsu K sresðnÞ Þ=ðsu K sl Þ which is the damage after n cycles at sn loading, relative to load level sl, where sl!su. Here sl denotes the final loading level and su is the ultimate stress of the virgin material. The relative damage, Dsl has a value 0!Dsl!1. DslR1 suggests that there is no usable strength relative to the stress level considered, sl. If a composite component is subject to a spectrum or step loading situation which has the final stress level sl, and under that stress level the component failed by fatigue, then DslZ1. More generally, Pk rZ1 ðsresðrK1Þ K sresðrÞ Þ D sk Z ðsk ! su Þ (1) ðsu K sk Þ Let us consider a situation of step loading in which fatigue failure is caused by stress level sk. So, say for a particular sample, the calculated value for DskO1 when using experimental data from the sample population, and would suggest that this sample is stronger than ‘average’. Conversely, if Dsk!1 then the sample is weaker than ‘average’. Therefore, in a spectrum or multi-step loading situations the applied load that causes the final failure may be one of the major factors which will determine the life usage of the material. The model developed by the authors’ for a continuous single-step (i.e. constant amplitude loading) fatigue process is given by i smax 0:6Kjjsin qj h su K smax Z a smax ð1 K jÞ1:6Kjjsin qj su 1 b ! b N K1 ð2Þ f where smax is the applied stress, f is the frequency, N is the fatigue life at applied stress level, a and b are material parameters which can be obtained using traditional S–N fatigue data of the material. A brief explanation of the method to determine a and b is given in Appendix B. Here q is the smallest ply angle of the laminate to the loading direction, see Fig. 1, and R is the stress ratio [1]. The significance of the indices in Eq. (2) is explained in Ref. [1]. Also jZR for KN%R!1 (tension–tension and reverse loading) jZ1/R for 1!R%N (compression–compression).
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245
To avoid confusion the right side of Eq. (2) will be replaced with ðDsÞsn ;n Z a
sn
sresðnK1Þ 1 ! b ðnb K 1Þ fn
ð3Þ
where (Ds)sn, n is the strength degraded after n cycles at stress level sn. This model was derived for the constant amplitude fatigue loading of (s, N with loading frequency f and stress ratio R) a specimen subject to tension–tension, reversal, and compression–compression or flexural fatigue. The model in Eq. (2) must be modified to deal with discontinuous loading situations, where many state changes occur in the specimen’s fatigue life during the stress level changeovers. The time dependent nature of matrix material and glass fibre properties will immediately change the damage accumulation process with any change of stress levels, stress ratios and the frequency. During a step/discrete fatigue process, any change of stress level will results in a comparable amount of viscoelastic deformations being reversed and a part of the damage in the material’s memory erased [3]. As such, the reduction of strength of the composite calculated by the model in Eq. (2) is not accurate for a discontinuous process. Many researchers have shown that the degradation of strength during a discontinuous fatigue process is a function of (n/N) [1,2,4,6,11,14,18,23]. Let F(n/N) be the factor needed to correct the strength degradation in Eq. (2) which was proposed for the continuous fatigue process under a constant amplitude fatigue load. As such the amount of strength remaining after n fatigue cycles at a particular stress level during a discontinuous (i.e. variable amplitude loading) fatigue process can be explained as; Remaining strength after n cycles Z ðDsÞsn ;n ðFðn=NÞÞ
(4)
The fraction of strength degraded after n cycles is therefore Z(1KF(n/N)). Taking the general form of the model [1], Eq. (2) can be approximated for a discontinuous fatigue process by introducing fn a ‘factor’ that accounts for the fraction of strength degraded in a discontinuous fatigue process (i.e. variable amplitude loading) as the material has not used all it fatigue life under the previous loading level. So fn Z ð1 K Fðn=NÞÞ
Then "
n fn Z 1 K 1 K Nsn
0:6Kjjsin qj 1:6Kjjsin qj sn 1 K j
(5)
Assuming the damage accumulation process is nonlinear with respect to the number of loading cycles, we propose function F(n/N) has the form: " #b n
n b F (6) Z 1K N Nsn
1239
b #b (7)
where Ns,n is the residual life at sn after the loading at the (nK1)th step. Incorporating this into Eq. (2) yields 0:6Kjjsin qj smax sresðnK1Þ KsresðnÞ Za sresðnK1Þ 1:6Kjjsin qj 1 ! smax ð1KjÞ ðnb K1Þfn ZðDsÞsn ;n fn b fn ð8Þ where sres(n) is the residual strength after n cycles at sn, and sres(nK1) is the residual strength after n cycles at snK1. Now let us consider a multiple step loading spectrum of {(s1,n1@f1,R1).(sn,nn@fn,Rn)} with fatigue failure at (sn,nn@fn,Rn), where ni, si, fi and Ri are the number of cycles, applied stress, load frequency and stress ratio at the ith load step, respectively. The residual stress, sres1, at the end of first load step (s1,n1@f1,R1) is given by sult K sres1 Z ðDsÞs1n1 f1
(9)
Similarly for the final load step, (sn,nn@fn,Rn), where fatigue failure occurs, sresðnK1Þ K sn Z ðDsÞsn nn fn :
(10)
Note that fnZ1, where the nth step is the last step. Summing for n steps yields Pn 1Z
sk kZ1 a sresðkK1Þ
0:6Kjjsin qj
½sk ð1Kjk Þ1:6Kjk jsin qj f1b ðnbk K1Þfk k
su Ksn ð11Þ
which is the relative damage, Dsn at sn.
If a specimen is subject to a constant amplitude test under stress level sn at any arbitrary value of f and R, until it is fatigue failure, then 0:6Kjjsin qj sn su K sn Z a ½sn ð1 K jÞ1:6Kjjsin qj sult 1 !
fnb
ðNnb K 1Þ
(12)
Substituting Eq. (12) into Eq. (11) yields Pn sk 0:6Kjjsin qj ½sk ð1Kjk Þ1:6Kjk jsin qj f1b ðnbk K1Þfk kZ1 sresðkK1Þ k 1Z 0:6Kjjsin qj sn 1:6Kjjsin qj 1 b ½sn ð1KjÞ f b ðN K1Þ su n
(13)
Using Eq. (11) or (13) the residual strength or the indicator for damage Dsn , can be calculated for step loading situations.
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Table 1 Damage calculated by proposed algorithm and experimental results published by Broutman and Sahu [6] for E-Glass/Epoxy [08/908] Experiment
Calculated
Test no.
s1 (MPa)
s2 (MPa)
n1
n2 (mean)
s(res) after n1 (MPa)
Sum (Ds2)
Palmgren– Miner sum
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
386 386 386 386 386 386 337 337 337 337 289 289 241 241 241 241 241 289 289 289 337 337
241 241 289 289 337 337 241 241 289 289 241 241 289 289 337 337 386 337 337 386 386 386
250 100 250 100 250 100 1000 249 1000 249 9996 1999 49,938 19,975 49,938 19,975 19,975 9996 1999 1999 1000 249
192,000 193,000 5840 11,970 1250 1635 86,000 162,500 8670 8000 96,500 110,800 3730 9490 391 804 124 293 1290 355 297 503
437 463 437 463 437 463 448 469 448 469 425 463 431 454 431 454 454 425 463 463 448 469
1.15 1.09 0.83 0.92 0.94 0.93 0.93 1.03 0.89 0.82 1.01 0.96 0.75 0.89 0.70 0.79 0.84 0.66 0.88 1.12 1.12 1.22
1.62 1.33 0.9 1.02 1.01 0.87 0.91 1.05 0.99 0.64 1.24 0.78 0.64 0.76 0.45 0.44 0.37 0.8 0.66 0.86 1.01 1.12
Broutman and Sahu [6] sum
Hashin and Rotem sum
1.36 1.14 1.31 1.12 1.22 1.09 1.19 1.05 1.13 1.03 1.16 1.03 0.91 0.96 0.77 0.89 0.73 0.7 0.94 0.79 0.69 0.92
1.30 1.14 0.65 0.86 0.87 0.75 0.72 0.97 0.89 0.59 1.18 0.73 0.63 0.82 0.63 0.60 0.67 0.87 0.75 1.09 1.15 1.24
Calculated values for aZ0.197 and bZ0.2265 (sultZ482 MPa, RZ0.05 and fZ10 Hz).
[22]. The results of this work are presented in Table 1–4 inclusive and Fig. 3 in Section 4.
3. Model validation The comprehensive data published by Broutman and Sahu [6], Bonnee [7] and Hwang and Han [11] has been used to validate our model. Also test results for residual strength published in DOE/MSU database [17] has been used to check the accuracy of the model for residual strength calculations. The model proposed here was an extension of a previous fatigue life prediction model [1], which was defined for glass-fibre reinforced composites and not for carbon/epoxy composites. However, since the performance of this model will be compared against other available damage accumulation models, such as Palmgren–Miner, Broutman and Sahu and Hasin and Rotem, for completeness we decided to check the accuracy of the model’s predictions for the carbon/epoxy composites published in reference
3.1. Statistical investigation of model predictions The statistical method of addressing scatter in fatigue data has been described in detail by many researchers [19,24]. As such the characteristic values of fatigue data can be estimated in desired confidence intervals. It has been warranted that the parameters a and b of Eq. (2) should be obtained using statistically analysed fatigue data, for better performance of the model [1]. In this paper, the model parameters a and b for all cases were determined using statistically analysed fatigue data. However, it is useful to evaluate the confidence level of the predictions made by each cumulative damage model. Consequently, the predictions from the proposed model,
Table 2 Damage calculated by proposed algorithm and experimental results published by NRL Investigation-[7] on E-Glass/Polyester 08/G458, a Z0.1938 and bZ0.238 (3ultZ2.57% and RZK1, fZ5 Hz) Experiment
Calculated
31 (%)
32 (%)
n1
n2
3(res) % after n1
Sum (D32)
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
1 1 1 1 0.5 0.5 0.5 0.5
1.00!104 1.00!104 1.00!104 1.00!104 1.00!104 1.00!104 1.00!104 1.00!104
5850 5430 3280 9490 1635,210 925,240 1509,550 830,050
2.3814 2.3814 2.3814 2.3814 2.3814 2.3814 2.3814 2.3814
1.16 1.14 1.01 1.31 1.20 1.05 1.17 1.03
Palmgren– Miner sum
Broutman and Sahu
Hasin and Rotem
0.99 0.92 0.58 1.57 0.64 0.39 0.60 0.35
1.05 0.98 0.62 1.67 1.29 0.75 1.20 0.68
1.02 0.95 0.61 1.60 0.63 0.37 0.58 0.34
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245
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Table 3 Damage calculated by proposed algorithm and experimental results published by Hwang and Han-[11] on E-Glass/Epoxy Uni-directional composite Type 1002, aZ0.0182 and bZ0.34 (sultZ770 MPa and RZ0.05, fZ2 Hz) Experiment
Calculated
s1 (MPa)
s2 (MPa)
n1
539 654.5 577.5 654.5
654.5 539 654.5 577.5
10,000 500 5000 500
Mean
Std. dev.
s(res) after n1
2031 33,049 3240 21,450
2983 3012 2986 14,742
739.4 757.1 746.1 757.1
n2
Palmgren–Miner rule, Broutman and Sahu and Hashin and Rotem, have all been statistically analysed. It was found that the damage index from all three models approximately follows a Weibull distribution. As such the Weibull probability function was fitted into the damage index
Sum Ds2 1.1 1.0 1.1 1.0
Palmgren– Miner sum
Broutman and Sahu
Hashin and Rotem
0.8 0.8 1.1 0.9
0.97 0.76 1.18 0.86
1.01 0.71 1.24 0.81
The Weibull parameters were determined in 95% confident levels using ‘MINITAB’ statistical software. The Weibull PDF graphs for the damage index D for each case are shown in Figs. 4–6 inclusive.
Probability density of index D pdfðDÞ a a Z DaK1 eKðD=bÞ for DO 0 (14) b where a and b are shape and scale parameters of Weibull distribution. As an example, the probability of damage D within the range ð 1:1 a aK1 KðD=bÞa D Prð0:9! D! 1:1Þ Z e dD 0:9 b
and illustrated by the shaded area in the Fig. (2).
4. Results and discussion The proposed model has been tested against several twostep fatigue data sets for tension–tension and reverse loading, published in the literature and found to be in good agreement. These results are presented in Tables 1–4 inclusive and Fig. 3. All fatigue data exhibits significant scatter, which is normal for glass fibre composites. This scatter reflects the non-homogeneity presented in material batches even if the specimens are sourced from the same stock. This suggests a comprehensive statistical analysis to
Table 4 Damage calculated by proposed algorithm and experimental results published by Found and Quaresimin-[22] on carbon-fibre/Epoxy angle-ply composite [0/90, G452, 0/90], aZ0.25106 and bZ0.094 (sultZ422 MPa and RZ0.05, fZ3 Hz) Experiment s1
s2
n1
n2
315 315 315 315 315 315 315 315 315 315 315 315 315 340 340 340 340 340 340 340 340 340 340 340 340
340 340 340 340 340 340 340 340 340 340 340 340 340 315 315 315 315 315 315 315 315 315 315 315 315
87,200 87,000 86,300 57,700 57,550 40,300 28,700 26,500 25,300 17,650 17,000 13,000 12,500 8500 7480 7480 6800 6500 4600 4400 4400 2500 1500 1500 1350
520 150 1408 1750 2280 2027 3320 2640 2464 6170 38,140 14,300 24,030 15,250 17,060 79,496 29,939 48,760 73,910 89,350 80,605 90,150 41,840 111,120 99,520
Model sum (D32)
Palmgren– Miner sum
Broutman and Sahu
Hashin and Rotem
1.10 0.94 1.23 1.10 1.14 1.06 1.10 1.05 1.04 1.15 1.48 1.27 1.37 1.17 1.17 1.38 1.14 1.19 1.17 1.19 1.17 1.13 1.00 0.83 1.11
0.82 0.77 0.91 0.70 0.76 0.58 0.63 0.53 0.50 0.85 4.48 1.74 2.84 1.10 1.00 1.54 1.03 1.16 1.16 1.28 1.20 1.07 0.53 0.97 1.02
1.05 1.00 1.14 0.85 0.91 0.69 0.70 0.60 0.57 0.90 4.53 1.77 2.87 0.87 0.80 1.34 0.85 0.99 1.04 1.16 1.08 1.00 0.49 0.23 0.98
0.87 0.82 0.96 0.79 0.85 0.68 0.72 0.62 0.59 0.94 4.56 1.81 2.91 1.09 0.96 1.50 0.97 1.10 1.07 1.18 1.10 0.98 0.46 0.20 0.95
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J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245 3 Model Hashin & Rotem Palmgren-Miner Broutman & Sahu
Probability Density
2.5
2
1.5
1
Fig. 2. Weibull probability distribution of damage index D.
Predicted
Experimental
800
Res. Strength (MPa)
0.5
0 0
700 600
0.5
1
1.5
2
Damage Index Fig. 4. Comparison of probability of prediction of damage index by proposed model, P–M, Broutman and Sahu and Hashin and Rotem, for the two-step loading results published by Broutman and Sahu [6].
The specimen has been subject to 249 cycles at 341 MPa followed by 503 cycles at a higher stress level of 389 MPa. The virgin material, Table A1, can only withstand 493 cycles at 389 MPa. Therefore, these specimens are stronger than average as indicated by D being greater than unity. Consider now the results in Table 2, for the step loading data set of reverse loading RZK1. It can be seen that the proposed model and the damage index Ds2 are consistently closer to unity than the other models. As shown in Fig. 5, the probability of the predicted damage D being within the arbitrary chosen 0.9!D!1.1 is higher for the proposed model than from all other models. It is clear, from 4 Model Hashin & Rotem Palmgren-Miner Broutman & Sahu
3.5 3
Probability Density
obtain the most probable limits of the material properties and their fatigue life from scattered data sets, [19,24]. For the proposed model, it is vital to accurately calculate material parameters a and b. To overcome the effects of the scatter in the data, we used the statistically analysed fatigue life data presented in each set, to obtain a value for parameters a and b. The results shown in Table 1 are for E-Glass/Epoxy [08/908] [6]. The index proposed by Broutman and Sahu shows the load sequencing effect, i.e. the summation is less than unity for low–high loading and greater than unity for high–low loading. The proposed damage index Ds2, Palmgren–Miner and Hashin and Rotem models, do not show any significant load sequencing effect. However, the probability of damage D for each model to be within the arbitrary chosen limits 0.9!D!1.1 is considerably higher for the proposed model than for Palmgren - Miner and Hashin and Rotem models, see Fig. (4) for details. The damage, D, depends on the final loading conditions, such as applied stress, stress ratio and load frequency. D can be used to compare strong or weak material batches. For example consider test 1 in Table 1. Here the specimen has been subject to a few cycles at a relatively high stress level followed by 192,000 cycles at a stress of 244 MPa. The virgin material, Table A1 (in Appendix A), can only withstand 172,000 cycles at 244 MPa. Also consider test 22 in Table 1.
2.5 2 1.5
500 400
1
300 200
0.5
100 0
0 0
50000
100000
150000
200000
Cycles
Fig. 3. Residual strength of material DD16A [17] under applied stress level of 241 MPa, RZ0.1 and frequency 8 Hz. Glass/Polyester [90/0/G45/0]s., aZ0.61 and bZ0.2 (sultZ673 MPa and RZ0.1, fZ13 Hz).
0
0.5
1
1.5
2
Damage Index Fig. 5. Comparison of probability of prediction of damage index by proposed model, P–M, Broutman and Sahu and Hashin and Rotem, for the two-step loading results published by NRL [7].
J.A. Epaarachchi, P.D. Clausen / Composites: Part A 36 (2005) 1236–1245
and spectrum fatigue data. Whal et al. [14] concluded that for repeated step loading and spectrum loading situations, the Palmgren–Miner’s sum will always be less than unity. The results in Tables 1–4 inclusive, show that Palmgren– Miner’s, Broutman and Sahu and Hashin and Rotem models did not give consistent results in all cases considered here. However, the proposed model is consistent in it’s damage index and probability of prediction is very much higher than to the other models.
3
Model Hashin & Rotem Palmgren-Miner Broutman & Sahu
Probability Density
2.5
1243
2
1.5
1
5. Conclusions 0.5
0 0
0.5
1
1.5
2
Damage Index Fig. 6. Comparison of probability of prediction of damage index by proposed model, P–M, Broutman and Sahu and Hashin and Rotem, for the two-step loading results published by Found and Quaresimin [22].
a statistical point of view that the proposed model appears to be superior to Palmgren–Miner’s Broutman and Sahu and Hashin and Rotem models. Results shown in the Table 3, for E-Glass/Epoxy UD Type 1002 [11], clearly show the predictions from the proposed model are much closer to unity than the predictions from the other three models. However, as the size of this data set was too small, a statistical analysis for the damage index was not performed. The results presented in Table 4 for Carbon/Epoxy system of [22] show that the proposed model yields more accurate results than the other three models. Also Fig. 6 confirmed that the probability of damage D within the arbitrary chosen 0.9!D!1.1 range is much higher for the proposed model than the other models with the probability density of Palmgren–Miner’s Broutman and Sahu and Hashin and Rotem models collapsing approximately into a single curve. Let us consider the residual strength calculations for the material [90/0G45/0]s [17] as shown in Fig. 3. Here it clearly shows that the predicted values closely follow the experimental values. The model indicates sudden failure after 176,877 cycles at stress level of 241 MPa, which is close to the experimental life of 169,974 cycles as shown in Table A4. According to the model, the loading sequence, s1, n1Ks2, n2 and the loading sequence, s2, n2Ks1, n1, (where s1Os2) should give different residual strength values that are dependent on the values of s1, n1, s2 and n2. Whal et al. [14] concluded that the load sequencing effect is insignificant at multi-stress level or repeated step applications. The damage index by the proposed model does not shows any significant load sequencing effect, the damage index fluctuates about unity irrespective of loading sequence. However, the information given by Whal et al. [14] is insufficient to be reliably apply to the model for their step
This paper describes a fatigue damage accumulation model for GFRP in discrete loading situations. This model was based on a fatigue model previously developed by the authors’ [1]. Here we define the damage to the composite using strength degradation concept, which have been shown to be dependent on the final failure conditions. The predictions from the model for two-step loading in tension–tension and reverse loading were found to be in good agreement with the experimental data obtained from the literature. Unfortunately, the model could not be tested for compression–compression two-step loading condition as well as multi-block and spectrum loading due to lack of reliable experimental data. The model does indicate that load-sequencing effects in two-step-loading conditions have been sufficiently addressed in the proposed model. The proposed model also shows excellent agreement with twostep loading data for a carbon/epoxy composite, but further testing will need to be undertaken before confirming its accuracy for all carbon/epoxy composite systems.
Acknowledgements The work presented in this paper was supported by an Australian Research Council SPIRT grant. JE was financially supported by an Australian Postgraduate Award (industry) scholarship.
Appendix A Table A1 Life time calculated by model [1] and experimental results published by Broutman and Sahu [6] for E-Glass/Epoxy [08/908] Experiment
Calculated N
smax(ksi)
N (mean)
386 337 289 241
493 2470 14,700 172,000
304 2583 19,861 158,994
Calculated values for aZ0.197 and bZ0.2265 (sultZ483 MPa, RZ0.05 and fZ10 Hz).
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Table A2 Life time calculated by model [1] and experimental results published by NRL investigation-[7] on E-Glass/Polyester 08/G458, aZ 0.1938 and bZ0.238 (3ultZ2.57% and RZK1, fZ5 Hz) Experimental
D
Calculated N
3max (%)
N (Weibull mean)
0.5 0.7 1.0
2767,904 185,361 6260
1316,058 101,812 5928
Table A3 Life time calculated by model [1] and experimental results published by Hwang and Han-[11] on E-Glass/Epoxy Uni-directional composite type 1002, aZ0.0182 and bZ0.34 (sultZ770 MPa and RZ0.05, fZ1–3 Hz) Experiment
Calculated N
smax (MPa)
N
693 654.50 616 577.5 539
Mean
Std. dev.
513 3694 10,922 27,577 47,739
346 4322 4194 21,382 24,642
463 3590 11,863 29,949 55,699
Table A4 Life time calculated by model [1] and experimental results published for DD16A [17] Experiment
(N β-1) Fig. B1. Illustration of determining parameters a and b.
!
1 fb ½ð1 K jÞ1:6Kjjsin qj
su K1 Where D Z smax
su smax
0:6Kjjsin qj
(B3)
The complexity of Eq. (B1) suggests a trial-and-error method to determine appropriate values of a and b. For each experimental data set, D from Eq. (B‘3) was determined for a trial value of b. D was then plotted against (NbK1) as shown in Fig. B1. This procedure is repeated by changing value of b, until the linear regression line passes through the origin. The values of a and b are noted.
Calculated N
smax (MPa)
N Mean
Std. dev.
241
169,974
98,418
References
176,877
Glass/Polyester [90/0/G45/0]s., aZ0.61 and bZ0.2 (sultZ673 MPa and RZ0.1, fZ13 Hz).
Table A5 Calculated lifetime and experimental results published by Found and Quaresimin-[22] on carbon-fibre/Epoxy angle-ply composite [0/90, G 452, 0/90], aZ0.25106 and bZ0.094 (sultZ422 MPa and RZ0.05, fZ 3 Hz) Experiment
Calculated N
smax (MPa)
N
315 340
115,150 8800
176,877
Appendix B
smax 0:6Kjjsin qj smax ð1 K jÞ1:6Kjjsin qj au 1 b ! B ðN K 1Þ (B1) f
su K smax Z a
rearranging Eq. (B2) D Z aðN b K 1Þ
(B2)
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