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A fatigue damage accumulation model based on stiffness degradation of composite materials
Materials and Design 88 (2015) 1290–1295
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Materials and Design journal homepage: www.elsevier.com/locate/jmad
A fatigue damage accumulation model based on stiffness degradation of composite materials Saeed Shiri, Mojtaba Yazdani, Mohammad Pourgol-Mohammad ⁎ Department of Mechanical Engineering, Sahand University of Technology, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 1 July 2015 Received in revised form 17 September 2015 Accepted 18 September 2015 Available online 21 September 2015 Keywords: Polymeric composites Phenomenological models Fatigue damage Life prediction
a b s t r a c t Many of the stiffness-based fatigue damage models contain four main limitations for composite materials life determination: (i) lack of capability in simulating the whole periods of damage evolution, (ii) applicability for a specific type of composite and in a limited range of loading levels, (iii) existence of many parameters in the models requiring substantial experimental data, (iv) application for constant amplitude loading only. In this paper, a stiffness-based fatigue damage model is proposed aiming to address the posed shortcomings. Nine sets of experimental data are utilized for this purpose. The results show that the model succeeds in dealing with these limitations. Then, a life prediction model is developed based on the damage model for two-stage loading. The model is assessed by three sets of experimental data and compared with some existing models. The comparison demonstrates that the model predictions are promising for remaining fatigue cycles in comparison with the studied models. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Fatigue is the dominant failure mechanism for structures under cyclic loading causing damage and material property degradation in a cumulative manner [1]. Composite fatigue leads to several different damage mechanisms, such as matrix cracking, delamination and fiber breakage [2]. Hence, fatigue damage is a complex process in composite materials. Proper modeling of the damage evolution is the foundation for predicting the fatigue life of composite structures for engineering applications [3]. Macroscopic models are capable of simulating the damage evolution and useful for practical design of composite structures. They track the changes in material properties, such as strength and stiffness [4]. Several residual strength models have been developed describing the degradation of the initial strength during fatigue life. The most popular ones are available in [5,6]. The main disadvantage of these models is that the remaining life cannot be assessed by non-destructive testing techniques. However, the residual stiffness can be measured non-destructively [7]. Therefore, a fatigue damage model based on the residual strength may not be suitable for predicting and tracking the fatigue damage. Stiffness can be used as a potential non-destructive parameter to monitor frequently or even continuously the damage development in a component during service life [8,9]. Several stiffness-based models [10–19] have been presented to describe the fatigue damage evolution or life prediction of composites in ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (M. Pourgol-Mohammad).
http://dx.doi.org/10.1016/j.matdes.2015.09.114 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
recent decades. They quantify the extent of damage by measuring the Young's modulus of the material. Most of these models are not capable of simulating the damage progress in all stages explained in the following section, especially if the stage of final failure is concerned [20]. High number of parameters is another deficiency in many of these models requiring extensive experimental data for their definition. Wu and Yao [21] presented a model capable of describing all the stages of damage evolution with the minimum required parameters. However, they evaluated their model in a limited range of loading level and for common fibers. This is true about most of the similar models. That is, most of the stiffness-based models are validated for a specific type of composite and are not evaluated in a wide range of loading levels. Recently, authors [22] proposed a modified model. They used more various fibers for model validation, but their model is assessed in a very limited range of loading as well. Besides, Peng et al. [23] have proposed a rate-based stiffness degradation model overcoming most of the existent shortcomings, but the model was not examined for life prediction of composites under non-constant amplitude loading. Fatigue life prediction is of vital importance in composite structures. Evaluating the performance of the structure in early life stages prevents from catastrophic failures. Some studies have been conducted on this subject based on residual strength or stiffness. Yao and Himmel [24] assumed that the cumulative damage was associated with the strength decrease. They proposed an analysis to predict residual strength caused by fatigue damage in glass and carbon fiber reinforced plastics. Aghazadeh Mohandesi and Majidi [3] applied a residual strength model for life prediction and investigated the effect of high-stress peaks on fatigue life of carbon fiber reinforced plastics. Wu and Yao [21] extended their stiffness-based model for life prediction and
S. Shiri et al. / Materials and Design 88 (2015) 1290–1295
concluded that their model can predict residual fatigue life of composites quite well. However, these models mostly yield a high percent of error in fatigue life prediction. Although, their predicted remaining cycles are acceptable by taking to account the significant scatter of fatigue life in composite materials, however there is a noticeable difference between experimental and predicted values in most of cases. A proper way to decrease the model and experiment discrepancy is to develop a life prediction model based on a suitable damage model. Besides, considering the load sequence effect and proper definition of model parameters is of high importance which has usually been ignored. Recently, the authors [25] introduced a life prediction model considering the simultaneous degradation of stiffness and strength. They succeed in dealing with the neglected points. However, the main disadvantage of their model is utilization of two constants to consider the effect of load sequence. In this paper, fatigue damage evolution is briefly discussed in composites. Based on that analysis, a model is proposed for simulation of the damage progress. Validation of the model shows that the proposed damage model resolves the discussed limitations of stiffness-based models. Then, a life prediction model is developed based on the first model. A logarithmic expression is used to consider the load sequence effect. Besides, weighted parameters are introduced for the model to enhance the extent of accuracy in parameter estimation. Numerical application of the cumulative model demonstrates that most of the predicted residual lives lead to quantitatively better errors estimations. 2. Fatigue damage 2.1. A stiffness-based descriptive model Damage is accumulated nonlinearly in composite materials. According to Fig. 1, this process is divided into three stages. The damage accumulates rapidly during the first few cycles. During this stage, micro cracks initiate in multiple locations in the matrix. Some fibers with low strength may break during this stage. The next stage shows a slow and steady damage growth rate. This occurs when the crack density is saturated in the matrix. Finally, the damage again grows rapidly. In brief, it is commonly known that matrix cracking, interfacial debonding, delamination and fiber breakage are the four basic dominant mechanisms in the explained stages [12,16,26]. Based on the illustrated trend of damage development in Fig. 1, trigonometric terms are chosen to simulate the explained stages of damage progress. Besides, trigonometric terms can provide a better life prediction due to their non-constant trend. Although it is possible to do the simulation with some other functions, but they mostly require more than two parameters requiring extensive amount of experimental data. In addition, they cannot present the capabilities of trigonometric expression in life prediction of composites. It is also commonly accepted that the measured stiffness just before complete failure of the specimen
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is not zero. Hence, a new fatigue damage model is proposed based on stiffness degradation of composites in the loading direction as follows: DðnÞ ¼
E0 −EðnÞ sinqx cosðq−pÞ ¼ E0 −E f sinq cosðqx−pÞ
ð1Þ
where, E0, E(n) and Ef are the magnitudes of stiffness corresponding to the initial cycle, the nth cycle, and the final stable cycle (or the failure Young modulus) respectively, x = n / N, n is the number of applied cycles, N is the fatigue life, D(n) is the fatigue damage index, p and q are material dependent parameters. In fact, trigonometric expression at the right side of Eq. (1) is a derived relationship for the simulation of damage evolution based on stiffness degradation in composites. The above model is the basis of this study and is used for the development of life prediction model. 2.2. Model validation and model parameters estimation The published data in [27–29] are utilized for evaluation of the proposed damage model. Two main factors are considered for selecting of experimental data: (i) A wide range of loading levels and (ii) different types of composites. By utilization of the chosen data, parameters of Eq. (1) are obtained through curve fitting. These values are listed in Table 1 along with correlative coefficient, R2. Then, fatigue damage evolution curves are plotted and shown in Fig. 2. Fig. 2 shows that the proposed model is capable of simulating the three stages of damage evolution in hemp fibers with good agreement in addition to common fibers. The model can adapt itself with a wide range of loadings, i.e. 40–90% ultimate strength of the material. The values of correlative coefficients R2 show the excellent accuracy of the model in curve fitting. As noted in Table 1, the values of parameters are in a limited range. A range of 0.383 and 0.871 for p and q values were observed. This short range of parameters shows that the model is capable to adjust itself with various datasets by keeping a logical range for parameter values. Due to existence of the trigonometric terms, development of certain proportionality seems unsuitable between damage index and model variables. However, it is obvious that model parameters are nondimensional and fatigue life, N is inversely proportional to the loading level. Based on these two points and the parameter values in Table 1, the following expressions are defined for the model parameters: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðσ max =σ u Þ p¼C logN
ð2Þ
q ¼ 2:5p−0:85
ð3Þ
where, σmax and σu are the maximum and ultimate stress respectively. C is a proportional constant. The linear relationship between parameters is a good approximation. This can be inferred by curve fitting of the obtained experimental values of parameters. The presented approximation leads to an error of less than 4% in 6 out of 9 cases. In overall, it is
Table 1 Results of curve fitting for the proposed model. Type of composite
Stacking sequence
Loading
p
q
R2
AS4/PR500 [27]
[0/902w]s
T650/polyimide [28]
[0/±60]
Hemp/epoxy [29]
[0/90]7
Unaged sample Aged sample %65 σu %80 σu %40 σu %60 σu %80 σu %75 σu %90 σu
1.2812 1.3122 1.5415 1.5323 1.1593 1.4848 1.5001 1.2830 1.4472
2.3689 2.4217 2.9469 2.8958 2.5986 2.9129 2.7902 2.1028 2.0743
0.9798 0.9906 0.9521 0.9524 0.9936 0.9837 0.9730 0.9936 0.9966
[±45]7 Fig. 1. Schematic of fatigue damage evolution in composites.
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Fig. 2. Damage evolution curves of various composites under different loadings. (a, b) [0/902w]s laminates [27], (c, d) [0/±60] laminates [28], (e–g) [0/90]7 laminates and (h, i) [±45]7 laminates [29].
concluded that the presented damage model has four main capabilities simultaneously: (i) more accurate fitting to various composites, (ii) better adaptability with a wide range of loading levels, (iii) containing the minimum number of required parameters, (iv) capable of extension for variable amplitude loading, discussed in the next section. Above capabilities are mainly related to the introduction of trigonometric terms in Eq. (1). The investigation done in this research shows that any similar expression cannot provide all the above capabilities at the same time.
inapplicability of Eq. (1) for non-constant amplitude loading necessitates introduction of a damage accumulation model expressed as Eqs. (4a)-(4b). According to Eq. (1), the accumulated damage will be in the range between 0 and 1. Therefore, the critical damage is defined as Eq. (5) which stands for damage index of failure. 0
logðσ i Þ
1qw pð1−RÞ
log σ Bni þ ni;i−1 ð i−1 Þ C Dðni Þ ¼ @ A Ni
w
ð4aÞ
3. Fatigue life prediction 3.1. Proposed cumulative damage model Constant amplitude loadings are rarely observed in industrial applications and structural behaviors; usually, a loading whose amplitude change intricately is applied [30]. Therefore, it is important to assess the fatigue life of composite structures subject to variable amplitude cyclic loadings. Under this type of loading, the produced damage in the earlier stage of loading will affect the damage of the next stage. Hence,
ni;i−1 ¼ N i Dðn F Þ ¼ 1
pw sinqi−1 xi−1 cosðqi−1 −pi−1 Þ qw ð1−RÞ sinqi−1 cosðqi−1 xi−1 −pi−1 Þ
ð4bÞ ð5Þ
where, R is stress ratio (σmin/σmax), xi -1 = ni -1/Ni -1, σi and σi -1 are the stress levels under ith and (i - 1)th cyclic loadings, ni and ni - 1 are the number of applied cycles under ith and (i − 1)th cyclic loadings and
S. Shiri et al. / Materials and Design 88 (2015) 1290–1295 Table 2 Predicted remaining fatigue cycles by the proposed model for E-glass/epoxy laminates [31]. Experimental
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Table 4 Predicted remaining fatigue cycles by the proposed model for carbon/epoxy laminates [33].
Current study (PREa)
Experimental
Current study (PREa)
Case
σ1 (MPa)
σ1 (MPa)
n1
n2
n2
Case
σ1 (MPa)
σ2 (MPa)
n1
n2
n2
1 2 3 4
539 577.5 654.5 654.5
654.5 654.5 539 577.5
10,000 5000 500 500
2031 3240 33,049 21,450
1128(44.48) 1451(55.22) 25,738(22.12) 13,676(36.24)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
315 315 315 315 315 315 315 315 315 315 340 340 340 340 340 340 340 340 340
340 340 340 340 340 340 340 340 340 340 315 315 315 315 315 315 315 315 315
87,200 87,000 86,300 57,700 57,550 40,300 28,700 26,500 25,300 17,650 8500 7480 6800 6500 4600 4400 2500 1500 1350
520 150 1408 1750 2280 2027 3320 2640 2464 6170 15,250 17,060 29,939 48,760 73,910 89,350 90,150 111,120 99,520
592(13.84) 602(301.05) 635(54.89) 1928(10.16) 1935(15.15) 2816(38.92) 3584(7.94) 3758(42.34) 3858(56.58) 4613(25.23) 20,665(35.51) 30,025(76.00) 34,063(13.77) 35,572(27.05) 43,574(41.04) 44,382(50.33) 53,642(40.50) 62,035(44.17) 63,834(35.86)
a
Percent of relative error.
Ni, Ni -1 are their corresponding fatigue lives, ni ,i -1 is the relation factor between the two stages of loading and logarithmic expression is introduced for consideration of load sequence effect in this factor, D(nF) is the critical damage causing fatigue failure under Fth cyclic loading, pi 1 and qi - 1 are the parameters under (i - 1)th cyclic loadings, pw and qw are the weighted values of parameters. They are defined based on the physical trend of fatigue failure. Values of p and q for each stage of two-stress level loading refer to the failure at the end of each stage. However, the failure occurs at the end of second stage in this type of loading. Also, experimental compatibility is considered in the definition of weighted parameters. Hence, weighted values for parameters are introduced as Eqs. (6)–(7). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ pw ¼ min pi−1 þ pi σ max qw ¼
ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qi−1 þ qi
ð7Þ
where, pi and qi are the parameters under ith cyclic loadings, σmin and σmax are selected from the two applied stress of each stage. Therefore, a fatigue life prediction model is developed for two-stage loading. 3.2. Numerical implementation In order to validate the proposed model, published data in [31–33] are used. Proportional constant, C is set to the value of 1.75 to make a good compatibility between the experimental values of model parameters and those for two-stage tension-tension data. Residual lives are Table 3 Predicted remaining fatigue cycles by the proposed model for E-glass/epoxy laminates [32]. Current study (PREa)
Experimental Case
σ1 (MPa)
σ2 (MPa)
n1
n2
n2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
241 241 241 241 241 289 289 289 337 337 386 386 386 386 386 386 337 337 337 337 289 289
289 289 337 337 386 337 337 386 386 386 241 241 289 289 337 337 241 241 289 289 241 241
49,950 19,975 49,950 19,975 19,975 10,000 2000 2000 1000 250 250 100 250 100 250 100 1000 250 1000 250 10,000 2000
3730 9490 391 804 124 293 1290 355 297 503 192,000 193,000 5940 11,970 1250 1635 86,000 162,500 8670 8000 96,500 110,800
3531(5.35) 7601(19.90) 84(78.49) 858(6.67) 113(8.97) 40(86.52) 1077(16.53) 153(56.77) 84(71.85) 187(62.92) 129,792(32.40) 127,864(33.75) 9372(60.48) 9074(24.19) 1391(11.30) 1319(19.31) 105,879(23.11) 119,371(26.54) 6391(26.29) 8407(5.08) 77,910(19.26) 116,526(5.17)
a
Percent of relative error.
a
Percent of relative error.
calculated from Eqs. (4a)-(7). The results are presented in Tables 2-4. The numbers in parentheses indicate the percent of relative error for each prediction. According to results in Tables 2-4, the proposed model's error is less than 50% in more than about 75% of the cases. 3.3. Discussion In a recent research by authors [25], some life prediction models are selected and compared for two-stage loadings. The reader is referred to [25] for observing the predicted values and corresponding error of each model. The three worst predictions of each model are chosen. Table 5 shows these values for E-glass/epoxy laminates presented by Broutman and Sahu [32]. As shown in Table 5, the proposed model's worst prediction is 50.4% better than the associated value of the next most accurate model. Also, the model's error is less than 35% in 16 out of 22 cases. The developed model yields the maximum error of about 55% for E-glass/epoxy laminates presented by Hwang and Han [31]. Table 6 is related to carbon/ epoxy laminates [33]. Table 6 shows that the model is more reliable than other examined models, i.e. there is a noticeable difference between the error values of studied models. Based on Tables 5–6, the developed life prediction model is capable of estimating the remaining life better than studied models. If we examine the results of presented model more deeply, an important capability is observed. Under the same stress levels in the first and second stage of loading, it is expected that increasing the number of applied cycles in the first stage leads to decreasing the number of applied cycles in the second stage. However, experimental results do not always follow it (such as cases 19–20 of Table 3 or case 2 of Table 4.). Investigation of authors shows that the studied models cannot adapt themselves with this situation and provide a high percent of Table 5 Percent of relative error in fatigue life Prediction by different models for E-glass/epoxy laminates [32]. Model
First worst prediction
Second worst prediction
Third worst prediction
Yao & Himmel [24] Aghazadeh & Majidi [3] Wu & Yao [21] Current Study
136.92 262.15 442.2 86.52
82.81 208.87 295.16 78.49
75.68 138.56 211.5 71.85
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S. Shiri et al. / Materials and Design 88 (2015) 1290–1295
Table 6 Percent of relative error in fatigue life prediction by different models for carbon/epoxy laminates [33]. Model
First worst prediction
Second worst prediction
Third worst prediction
Wu & Yao [21] Shiri et al. [1] Current study
1388 253.67 301.05
325.77 105.29 76
209.13 100.08 56.58
Fig. 3. Percent of relative error for proposed life prediction model in different datasets.
relative error in such cases. However, the proposed model is able to predict the remaining cycles better than its similar counterparts in such cases. This is related to the consideration of trigonometric terms. Another good characteristic of the proposed model is uniformity of the predictions, i.e. the dispersion of errors is of low degree in different data sets. Fig. 3 provides a scheme for better understanding of relative error values. According to Fig. 3, the model is consistent with different data sets. If we ignore just one case (the worst case), the maximum error of life prediction is about 87% in three data sets. Totally, it is concluded that the model predicts better than similar models due to the consideration of three main factors which had usually been ignored: (i) consideration of the load sequence effect by introducing a logarithmic expression in equivalent cycle term, (ii) definition of weighted parameters and (iii) development of the model based on a proper damage model. Hence, the model may be reliable for other datasets as well. 4. Conclusion A fatigue damage model is proposed tracking the degradation of stiffness in composite materials. It aimed to resolve the common explained shortcomings of similar models. The model has the minimum number of required parameters. Experimental validation shows that the model is able to accurately simulate the whole life cycle of damage evolution in several different sets of experimental data. Adaptability with a wide range of loading levels is another key feature of the
proposed model. Then, the model is extended for non-constant amplitude loading and a fatigue life prediction model is developed. Considering the load sequence effect, definition of weighted parameters and development of the model based on a proper damage model are the three main contributions of the presented cumulative model. Hence, the model leads to much less error than other examined models in many cases. The authors intend to add statistical analysis to their current research; experimental work is inevitable and is on progress. The aim is to examine and validate the model with the research's own data as well.
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Materials and Design journal homepage: www.elsevier.com/locate/jmad
A fatigue damage accumulation model based on stiffness degradation of composite materials Saeed Shiri, Mojtaba Yazdani, Mohammad Pourgol-Mohammad ⁎ Department of Mechanical Engineering, Sahand University of Technology, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 1 July 2015 Received in revised form 17 September 2015 Accepted 18 September 2015 Available online 21 September 2015 Keywords: Polymeric composites Phenomenological models Fatigue damage Life prediction
a b s t r a c t Many of the stiffness-based fatigue damage models contain four main limitations for composite materials life determination: (i) lack of capability in simulating the whole periods of damage evolution, (ii) applicability for a specific type of composite and in a limited range of loading levels, (iii) existence of many parameters in the models requiring substantial experimental data, (iv) application for constant amplitude loading only. In this paper, a stiffness-based fatigue damage model is proposed aiming to address the posed shortcomings. Nine sets of experimental data are utilized for this purpose. The results show that the model succeeds in dealing with these limitations. Then, a life prediction model is developed based on the damage model for two-stage loading. The model is assessed by three sets of experimental data and compared with some existing models. The comparison demonstrates that the model predictions are promising for remaining fatigue cycles in comparison with the studied models. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Fatigue is the dominant failure mechanism for structures under cyclic loading causing damage and material property degradation in a cumulative manner [1]. Composite fatigue leads to several different damage mechanisms, such as matrix cracking, delamination and fiber breakage [2]. Hence, fatigue damage is a complex process in composite materials. Proper modeling of the damage evolution is the foundation for predicting the fatigue life of composite structures for engineering applications [3]. Macroscopic models are capable of simulating the damage evolution and useful for practical design of composite structures. They track the changes in material properties, such as strength and stiffness [4]. Several residual strength models have been developed describing the degradation of the initial strength during fatigue life. The most popular ones are available in [5,6]. The main disadvantage of these models is that the remaining life cannot be assessed by non-destructive testing techniques. However, the residual stiffness can be measured non-destructively [7]. Therefore, a fatigue damage model based on the residual strength may not be suitable for predicting and tracking the fatigue damage. Stiffness can be used as a potential non-destructive parameter to monitor frequently or even continuously the damage development in a component during service life [8,9]. Several stiffness-based models [10–19] have been presented to describe the fatigue damage evolution or life prediction of composites in ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (M. Pourgol-Mohammad).
http://dx.doi.org/10.1016/j.matdes.2015.09.114 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
recent decades. They quantify the extent of damage by measuring the Young's modulus of the material. Most of these models are not capable of simulating the damage progress in all stages explained in the following section, especially if the stage of final failure is concerned [20]. High number of parameters is another deficiency in many of these models requiring extensive experimental data for their definition. Wu and Yao [21] presented a model capable of describing all the stages of damage evolution with the minimum required parameters. However, they evaluated their model in a limited range of loading level and for common fibers. This is true about most of the similar models. That is, most of the stiffness-based models are validated for a specific type of composite and are not evaluated in a wide range of loading levels. Recently, authors [22] proposed a modified model. They used more various fibers for model validation, but their model is assessed in a very limited range of loading as well. Besides, Peng et al. [23] have proposed a rate-based stiffness degradation model overcoming most of the existent shortcomings, but the model was not examined for life prediction of composites under non-constant amplitude loading. Fatigue life prediction is of vital importance in composite structures. Evaluating the performance of the structure in early life stages prevents from catastrophic failures. Some studies have been conducted on this subject based on residual strength or stiffness. Yao and Himmel [24] assumed that the cumulative damage was associated with the strength decrease. They proposed an analysis to predict residual strength caused by fatigue damage in glass and carbon fiber reinforced plastics. Aghazadeh Mohandesi and Majidi [3] applied a residual strength model for life prediction and investigated the effect of high-stress peaks on fatigue life of carbon fiber reinforced plastics. Wu and Yao [21] extended their stiffness-based model for life prediction and
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concluded that their model can predict residual fatigue life of composites quite well. However, these models mostly yield a high percent of error in fatigue life prediction. Although, their predicted remaining cycles are acceptable by taking to account the significant scatter of fatigue life in composite materials, however there is a noticeable difference between experimental and predicted values in most of cases. A proper way to decrease the model and experiment discrepancy is to develop a life prediction model based on a suitable damage model. Besides, considering the load sequence effect and proper definition of model parameters is of high importance which has usually been ignored. Recently, the authors [25] introduced a life prediction model considering the simultaneous degradation of stiffness and strength. They succeed in dealing with the neglected points. However, the main disadvantage of their model is utilization of two constants to consider the effect of load sequence. In this paper, fatigue damage evolution is briefly discussed in composites. Based on that analysis, a model is proposed for simulation of the damage progress. Validation of the model shows that the proposed damage model resolves the discussed limitations of stiffness-based models. Then, a life prediction model is developed based on the first model. A logarithmic expression is used to consider the load sequence effect. Besides, weighted parameters are introduced for the model to enhance the extent of accuracy in parameter estimation. Numerical application of the cumulative model demonstrates that most of the predicted residual lives lead to quantitatively better errors estimations. 2. Fatigue damage 2.1. A stiffness-based descriptive model Damage is accumulated nonlinearly in composite materials. According to Fig. 1, this process is divided into three stages. The damage accumulates rapidly during the first few cycles. During this stage, micro cracks initiate in multiple locations in the matrix. Some fibers with low strength may break during this stage. The next stage shows a slow and steady damage growth rate. This occurs when the crack density is saturated in the matrix. Finally, the damage again grows rapidly. In brief, it is commonly known that matrix cracking, interfacial debonding, delamination and fiber breakage are the four basic dominant mechanisms in the explained stages [12,16,26]. Based on the illustrated trend of damage development in Fig. 1, trigonometric terms are chosen to simulate the explained stages of damage progress. Besides, trigonometric terms can provide a better life prediction due to their non-constant trend. Although it is possible to do the simulation with some other functions, but they mostly require more than two parameters requiring extensive amount of experimental data. In addition, they cannot present the capabilities of trigonometric expression in life prediction of composites. It is also commonly accepted that the measured stiffness just before complete failure of the specimen
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is not zero. Hence, a new fatigue damage model is proposed based on stiffness degradation of composites in the loading direction as follows: DðnÞ ¼
E0 −EðnÞ sinqx cosðq−pÞ ¼ E0 −E f sinq cosðqx−pÞ
ð1Þ
where, E0, E(n) and Ef are the magnitudes of stiffness corresponding to the initial cycle, the nth cycle, and the final stable cycle (or the failure Young modulus) respectively, x = n / N, n is the number of applied cycles, N is the fatigue life, D(n) is the fatigue damage index, p and q are material dependent parameters. In fact, trigonometric expression at the right side of Eq. (1) is a derived relationship for the simulation of damage evolution based on stiffness degradation in composites. The above model is the basis of this study and is used for the development of life prediction model. 2.2. Model validation and model parameters estimation The published data in [27–29] are utilized for evaluation of the proposed damage model. Two main factors are considered for selecting of experimental data: (i) A wide range of loading levels and (ii) different types of composites. By utilization of the chosen data, parameters of Eq. (1) are obtained through curve fitting. These values are listed in Table 1 along with correlative coefficient, R2. Then, fatigue damage evolution curves are plotted and shown in Fig. 2. Fig. 2 shows that the proposed model is capable of simulating the three stages of damage evolution in hemp fibers with good agreement in addition to common fibers. The model can adapt itself with a wide range of loadings, i.e. 40–90% ultimate strength of the material. The values of correlative coefficients R2 show the excellent accuracy of the model in curve fitting. As noted in Table 1, the values of parameters are in a limited range. A range of 0.383 and 0.871 for p and q values were observed. This short range of parameters shows that the model is capable to adjust itself with various datasets by keeping a logical range for parameter values. Due to existence of the trigonometric terms, development of certain proportionality seems unsuitable between damage index and model variables. However, it is obvious that model parameters are nondimensional and fatigue life, N is inversely proportional to the loading level. Based on these two points and the parameter values in Table 1, the following expressions are defined for the model parameters: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðσ max =σ u Þ p¼C logN
ð2Þ
q ¼ 2:5p−0:85
ð3Þ
where, σmax and σu are the maximum and ultimate stress respectively. C is a proportional constant. The linear relationship between parameters is a good approximation. This can be inferred by curve fitting of the obtained experimental values of parameters. The presented approximation leads to an error of less than 4% in 6 out of 9 cases. In overall, it is
Table 1 Results of curve fitting for the proposed model. Type of composite
Stacking sequence
Loading
p
q
R2
AS4/PR500 [27]
[0/902w]s
T650/polyimide [28]
[0/±60]
Hemp/epoxy [29]
[0/90]7
Unaged sample Aged sample %65 σu %80 σu %40 σu %60 σu %80 σu %75 σu %90 σu
1.2812 1.3122 1.5415 1.5323 1.1593 1.4848 1.5001 1.2830 1.4472
2.3689 2.4217 2.9469 2.8958 2.5986 2.9129 2.7902 2.1028 2.0743
0.9798 0.9906 0.9521 0.9524 0.9936 0.9837 0.9730 0.9936 0.9966
[±45]7 Fig. 1. Schematic of fatigue damage evolution in composites.
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Fig. 2. Damage evolution curves of various composites under different loadings. (a, b) [0/902w]s laminates [27], (c, d) [0/±60] laminates [28], (e–g) [0/90]7 laminates and (h, i) [±45]7 laminates [29].
concluded that the presented damage model has four main capabilities simultaneously: (i) more accurate fitting to various composites, (ii) better adaptability with a wide range of loading levels, (iii) containing the minimum number of required parameters, (iv) capable of extension for variable amplitude loading, discussed in the next section. Above capabilities are mainly related to the introduction of trigonometric terms in Eq. (1). The investigation done in this research shows that any similar expression cannot provide all the above capabilities at the same time.
inapplicability of Eq. (1) for non-constant amplitude loading necessitates introduction of a damage accumulation model expressed as Eqs. (4a)-(4b). According to Eq. (1), the accumulated damage will be in the range between 0 and 1. Therefore, the critical damage is defined as Eq. (5) which stands for damage index of failure. 0
logðσ i Þ
1qw pð1−RÞ
log σ Bni þ ni;i−1 ð i−1 Þ C Dðni Þ ¼ @ A Ni
w
ð4aÞ
3. Fatigue life prediction 3.1. Proposed cumulative damage model Constant amplitude loadings are rarely observed in industrial applications and structural behaviors; usually, a loading whose amplitude change intricately is applied [30]. Therefore, it is important to assess the fatigue life of composite structures subject to variable amplitude cyclic loadings. Under this type of loading, the produced damage in the earlier stage of loading will affect the damage of the next stage. Hence,
ni;i−1 ¼ N i Dðn F Þ ¼ 1
pw sinqi−1 xi−1 cosðqi−1 −pi−1 Þ qw ð1−RÞ sinqi−1 cosðqi−1 xi−1 −pi−1 Þ
ð4bÞ ð5Þ
where, R is stress ratio (σmin/σmax), xi -1 = ni -1/Ni -1, σi and σi -1 are the stress levels under ith and (i - 1)th cyclic loadings, ni and ni - 1 are the number of applied cycles under ith and (i − 1)th cyclic loadings and
S. Shiri et al. / Materials and Design 88 (2015) 1290–1295 Table 2 Predicted remaining fatigue cycles by the proposed model for E-glass/epoxy laminates [31]. Experimental
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Table 4 Predicted remaining fatigue cycles by the proposed model for carbon/epoxy laminates [33].
Current study (PREa)
Experimental
Current study (PREa)
Case
σ1 (MPa)
σ1 (MPa)
n1
n2
n2
Case
σ1 (MPa)
σ2 (MPa)
n1
n2
n2
1 2 3 4
539 577.5 654.5 654.5
654.5 654.5 539 577.5
10,000 5000 500 500
2031 3240 33,049 21,450
1128(44.48) 1451(55.22) 25,738(22.12) 13,676(36.24)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
315 315 315 315 315 315 315 315 315 315 340 340 340 340 340 340 340 340 340
340 340 340 340 340 340 340 340 340 340 315 315 315 315 315 315 315 315 315
87,200 87,000 86,300 57,700 57,550 40,300 28,700 26,500 25,300 17,650 8500 7480 6800 6500 4600 4400 2500 1500 1350
520 150 1408 1750 2280 2027 3320 2640 2464 6170 15,250 17,060 29,939 48,760 73,910 89,350 90,150 111,120 99,520
592(13.84) 602(301.05) 635(54.89) 1928(10.16) 1935(15.15) 2816(38.92) 3584(7.94) 3758(42.34) 3858(56.58) 4613(25.23) 20,665(35.51) 30,025(76.00) 34,063(13.77) 35,572(27.05) 43,574(41.04) 44,382(50.33) 53,642(40.50) 62,035(44.17) 63,834(35.86)
a
Percent of relative error.
Ni, Ni -1 are their corresponding fatigue lives, ni ,i -1 is the relation factor between the two stages of loading and logarithmic expression is introduced for consideration of load sequence effect in this factor, D(nF) is the critical damage causing fatigue failure under Fth cyclic loading, pi 1 and qi - 1 are the parameters under (i - 1)th cyclic loadings, pw and qw are the weighted values of parameters. They are defined based on the physical trend of fatigue failure. Values of p and q for each stage of two-stress level loading refer to the failure at the end of each stage. However, the failure occurs at the end of second stage in this type of loading. Also, experimental compatibility is considered in the definition of weighted parameters. Hence, weighted values for parameters are introduced as Eqs. (6)–(7). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ pw ¼ min pi−1 þ pi σ max qw ¼
ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qi−1 þ qi
ð7Þ
where, pi and qi are the parameters under ith cyclic loadings, σmin and σmax are selected from the two applied stress of each stage. Therefore, a fatigue life prediction model is developed for two-stage loading. 3.2. Numerical implementation In order to validate the proposed model, published data in [31–33] are used. Proportional constant, C is set to the value of 1.75 to make a good compatibility between the experimental values of model parameters and those for two-stage tension-tension data. Residual lives are Table 3 Predicted remaining fatigue cycles by the proposed model for E-glass/epoxy laminates [32]. Current study (PREa)
Experimental Case
σ1 (MPa)
σ2 (MPa)
n1
n2
n2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
241 241 241 241 241 289 289 289 337 337 386 386 386 386 386 386 337 337 337 337 289 289
289 289 337 337 386 337 337 386 386 386 241 241 289 289 337 337 241 241 289 289 241 241
49,950 19,975 49,950 19,975 19,975 10,000 2000 2000 1000 250 250 100 250 100 250 100 1000 250 1000 250 10,000 2000
3730 9490 391 804 124 293 1290 355 297 503 192,000 193,000 5940 11,970 1250 1635 86,000 162,500 8670 8000 96,500 110,800
3531(5.35) 7601(19.90) 84(78.49) 858(6.67) 113(8.97) 40(86.52) 1077(16.53) 153(56.77) 84(71.85) 187(62.92) 129,792(32.40) 127,864(33.75) 9372(60.48) 9074(24.19) 1391(11.30) 1319(19.31) 105,879(23.11) 119,371(26.54) 6391(26.29) 8407(5.08) 77,910(19.26) 116,526(5.17)
a
Percent of relative error.
a
Percent of relative error.
calculated from Eqs. (4a)-(7). The results are presented in Tables 2-4. The numbers in parentheses indicate the percent of relative error for each prediction. According to results in Tables 2-4, the proposed model's error is less than 50% in more than about 75% of the cases. 3.3. Discussion In a recent research by authors [25], some life prediction models are selected and compared for two-stage loadings. The reader is referred to [25] for observing the predicted values and corresponding error of each model. The three worst predictions of each model are chosen. Table 5 shows these values for E-glass/epoxy laminates presented by Broutman and Sahu [32]. As shown in Table 5, the proposed model's worst prediction is 50.4% better than the associated value of the next most accurate model. Also, the model's error is less than 35% in 16 out of 22 cases. The developed model yields the maximum error of about 55% for E-glass/epoxy laminates presented by Hwang and Han [31]. Table 6 is related to carbon/ epoxy laminates [33]. Table 6 shows that the model is more reliable than other examined models, i.e. there is a noticeable difference between the error values of studied models. Based on Tables 5–6, the developed life prediction model is capable of estimating the remaining life better than studied models. If we examine the results of presented model more deeply, an important capability is observed. Under the same stress levels in the first and second stage of loading, it is expected that increasing the number of applied cycles in the first stage leads to decreasing the number of applied cycles in the second stage. However, experimental results do not always follow it (such as cases 19–20 of Table 3 or case 2 of Table 4.). Investigation of authors shows that the studied models cannot adapt themselves with this situation and provide a high percent of Table 5 Percent of relative error in fatigue life Prediction by different models for E-glass/epoxy laminates [32]. Model
First worst prediction
Second worst prediction
Third worst prediction
Yao & Himmel [24] Aghazadeh & Majidi [3] Wu & Yao [21] Current Study
136.92 262.15 442.2 86.52
82.81 208.87 295.16 78.49
75.68 138.56 211.5 71.85
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Table 6 Percent of relative error in fatigue life prediction by different models for carbon/epoxy laminates [33]. Model
First worst prediction
Second worst prediction
Third worst prediction
Wu & Yao [21] Shiri et al. [1] Current study
1388 253.67 301.05
325.77 105.29 76
209.13 100.08 56.58
Fig. 3. Percent of relative error for proposed life prediction model in different datasets.
relative error in such cases. However, the proposed model is able to predict the remaining cycles better than its similar counterparts in such cases. This is related to the consideration of trigonometric terms. Another good characteristic of the proposed model is uniformity of the predictions, i.e. the dispersion of errors is of low degree in different data sets. Fig. 3 provides a scheme for better understanding of relative error values. According to Fig. 3, the model is consistent with different data sets. If we ignore just one case (the worst case), the maximum error of life prediction is about 87% in three data sets. Totally, it is concluded that the model predicts better than similar models due to the consideration of three main factors which had usually been ignored: (i) consideration of the load sequence effect by introducing a logarithmic expression in equivalent cycle term, (ii) definition of weighted parameters and (iii) development of the model based on a proper damage model. Hence, the model may be reliable for other datasets as well. 4. Conclusion A fatigue damage model is proposed tracking the degradation of stiffness in composite materials. It aimed to resolve the common explained shortcomings of similar models. The model has the minimum number of required parameters. Experimental validation shows that the model is able to accurately simulate the whole life cycle of damage evolution in several different sets of experimental data. Adaptability with a wide range of loading levels is another key feature of the
proposed model. Then, the model is extended for non-constant amplitude loading and a fatigue life prediction model is developed. Considering the load sequence effect, definition of weighted parameters and development of the model based on a proper damage model are the three main contributions of the presented cumulative model. Hence, the model leads to much less error than other examined models in many cases. The authors intend to add statistical analysis to their current research; experimental work is inevitable and is on progress. The aim is to examine and validate the model with the research's own data as well.
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