In this paper, fatigue master curves are adopted to predict fatigue life of composite laminates made by unidirectional and braided fiber reinforced composite. Fatigue failure is then determined by Puck’s criterion. Uniaxial and multiaxial S-N curve could be obtained from the fatigue master curves. Fatigue lives of composite laminates made by unidirectional carbon and glass fiber reinforced composite are predicted by the proposed method. To extend the fatigue master curve to predict the braided composite laminate, tensile-tensile, compressive-compressive and tensile-compressive fatigue experiments are carried out in this paper. The predictions from the proposed fatigue model are compared with various experimental results and reasonably good agreement is observed.

Due to the excellent mechanical properties, fiber reinforced composites are widely used in industry [1]. Many fatigue models for fiber reinforced composite materials have been developed, and could be classed into three categories [2]: fatigue life models based on the S-N curves and fatigue failure criteria; residual stiffness/strength models; and progressive fatigue damage models. In all these three models, S-N curves are the basement of fatigue life assessment of fiber reinforced composite structure. Draw of traditional S-N curves demand so many fatigue experiments, which is time consuming and expensive [3]. Meanwhile, multiaxial fatigue life prediction is very necessary due to the composite anisotropy and contact between the different composite structures [4]. So compared with traditional S-N curves, a new simple and effective fatigue S-N curve which need less fatigue experiments and take multiaxial fatigue loading into consideration is quite important and necessary for the fatigue life prediction of composite structures.

Ellyin [5] has developed a fatigue failure criterion for unidirectional fiber reinforced laminae under plane stress, and in the failure criterion strain energy density was used. The fatigue life was related to the total energy through a power law type. Fawaz [6] proposed a semi-log linear relationship between applied cyclic stress and the number of cycles to failure, the model succeeded in predicting the fatigue life of laminate under uniaxial and biaxial fatigue loading, and this S-N curve is widely used in the fatigue life prediction. Harris [7,8] proposed the normalized constant-life model, in his model mean and peak stress amplitudes give rise to the same number of cycles to failure, and the normalized constant-life model is quite useful for the progressive fatigue modelling, but the cost and time are still quite high. The fatigue life of off-axis unidirectional fibre reinforced composite materials was investigated by Plumtree [9], the fatigue damage parameter based on that of Smith Watson Topper, which took into account the maximum shear and normal stresses, and shear and normal alternating strains on the fracture plane parallel to the fibres. Bond [10] adopted a semi-empirical fatigue life prediction methodology to predict fatigue life of glass fibre reinforced composite subjected to variable amplitude fatigue loading. S-N curve wiht a four-parameter power law relation was developed by Xiao [11] to investigate the load frequency effect for thermoplastic carbon/PEEK composites. Epaarachchi [12] proposed an empirical fatigue law, which derived from assumptions on fatigue crack propagation rate and often used in the fatigue crack propagation modelling.

Compared with uniaxial stress fatigue, multiaxial stress fatigue is quite important, due to composite anisotropy. Many researchers have investigated this problem. Lessard [13,14] presents a generalized residual material property degradation model to study fatigue of composite under multiaxial stress state, in his model the multiaxial is taken into consideration by Hashin polynomial failure criterion. Diao [15] extended the generalized residual material property degradation model to the statistical model for the prediction of the fatigue life of unidirectional composite laminae subjected to multiaxial fatigue loading. The distribution function of fatigue life is determined in terms of the distribution function of static strength of the composite laminae in different loading modes. Kawai [16,17] introduced effective stress to study fatigue of composite under and room and high temperature. In his model, effective stress is treated as plane stress state and experimental results match the numerical results well. Zhou [18] extended the effective stress to the equivalent stress for the composite bolted joints fatigue life prediction. Mahadevan [19] introduces characteristic plane to investigate multiaxial fatigue of isotropic and anisotropic materials. The effect of the mean normal stress is also included in his model and calculation results of fatigue life of unidirectional and multidirectional composite laminate show excellent agreement with experiments. Kawai [20] modified Tsai-Hill static failure criterion as fatigue cirterion and combined with fatigue life (CFL) diagrams to calculate fatigue life of unidirectional carbon/epoxy laminates, fatigue life of composite with any stress ratio and any ply orientation was predicted. Naik [21] presented an analytical model based on minimum strength model and fiber failure criterion to investigated fatigue of composite laminates with a circular hole. Analytical predictions are compared with the experimental results for uniaxial and multiaxial fatigue loading cases. Schön [22,23] used a fatigue prediction model based on the kinetic theory of fracture for polymer matrices to model fatigue failure of composite bolted joints under biaxial variable amplitude loading at elevated temperature.

In our paper, the fatigue master curve [24] is adopted to predict the fatigue life unidirectional, multidirectional and braided composite laminates. The fatigue master curves are established based on the accelerated testing methodology, which is used to determine the uniaxial ply fatigue strengths and the multi-axial fatigue failure is then determined by Puck’s criterion, and multi-axiality ratio is introduced to predict the multiaxial fatigue life of composite laminate. Fatigue lives of GFRP and CFRP laminates are predicted by our model. To extend this method to the braided laminate, fatigue tests of T700 braided composite laminate under tensile-tensile and compressive-compressive loading are carried out to obtain the fatigue parameters in the fatigue master curves, and the tensile-compressive fatigue tests are used to verify the calculation results of fatigue master curve.

2Fatigue master curve theoryIn this paper, in order to solve the problem that the fatigue life prediction model of the fiber reinforced composite laminate is suitable for any stress ratios and lay-up configurations, it is necessary to establish a uniaxial fatigue model for the UD laminates accurately first which is based on the accelerated test method (ATM) with Time-temperature superposition principle (TTSP). In addition, it is also needed to simplify the stress state of the laminates by multi-axiality ratios, so that the multi-axial fatigue master curve of the MD laminates could be determined.

2.1Fatigue master curve of UD laminates2.1.1Time-temperature superposition principle (TTSP)A same mechanical creep phenomenon can be observed at a higher temperature in a short period of time, or at a lower temperature in a longer period of time, so increasing the temperature and prolonging the observation time are equivalent to molecular motion, this equivalent can be achieved by a time-temperature shift factor. By using the time-temperature shift factor, the mechanical data obtained at one temperature can be changed to the mechanical data at another temperature but in different time ranges

Through the time-temperature shift factor aT, the data at the actual temperature can be linked with the data under the reference data, and the complex test which takes a long time can be converted into a simple test which can be completed in a short time at different temperatures, as shown in Fig. 1.

The basic expression formula of the temperature shift factor is as follows:

where T0 is the reference temperature and c1 and c2 are empirical constants.2.1.2Fatigue master curves and S-N curvesThe accelerated test method based on the principle of time-temperature superposition in this article is based on the following three conditions:(a) the same TTSP is applicable for both non-destructive viscoelastic behavior and destructive strength properties of matrix resin and their composites, (b) the linear cumulative damage law is applicable to the strength by the monotonic loading, and (c) the fatigue strengths exhibit linear dependence on the stress ratio of the cyclic loadings [25]. For (a), many scholars have proved its applicability to various composites and their structures, and given a large amount of experimental data. Through (b), the creep curve main curve can be calculated from the constant strain rate strength master curve, and the constant strain rate strength master curve can be measured by a constant strain rate test at different temperatures at a relatively easy fixed strain rate. For (c), the fatigue strength at any stress ratio can be interpolated from the fatigue strength of the material at the other two stress ratios, and these strengths can be obtained from the creep and fatigue strength master curves, respectively. Therefore, for the fatigue strength under any conditions (frequency, temperature and stress ratio), the fatigue strain strength can be obtained at different temperatures at different temperatures and the fatigue strength at different temperatures at a different stress rate. Therefore, for the fatigue strength under any conditions (frequency, temperature and stress ratio), it can be obtained from the constant strain rate strength at different temperatures at a determining strain rate and the fatigue strength at different temperatures under a determining stress ratio.

Based on the above three conditions, the establishment of the fatigue master curves of the UD laminates can be achieved, as shown in Fig. 2.

Flowchart for Establishing of the fatigue master curve [25].

Fig. 2a–c represents the three conditions in the accelerated test method, and they are also the theoretical basis for the establishment of the fatigue master curve in this paper.

From the constant strain rate test and fatigue test data, the fatigue master curve at the same frequency can be obtained,

where σf is the failure fatigue stress of the UD laminates, f is frequency, t is time to failure, σs,0 is the constant strain rate (CSR) strength at the initial reduced time; t1 is the transient reduced time at the reference temperature, nc, nr, nf are exponent parameters determined by fitting test data [24].By introducing coefficient gf, the fatigue master curve for different stress ratio can be measured:

where:Here, -1≤r＜1 is a load amplitude ratio and ‘sgn’ denotes the sign function, andwhere R and R0 are the stress ratios at the actual temperature and the reference temperature, and R0 ≥ 0.To establish the relationship between the fatigue failure strength σf and the initial static strength σ0, assume that σf can be simplified to σ0 when Nf=f⋅t=1/2,so that:

whereIt can be seen that the fatigue master curve in Eq. (2) is determined by σ0, t1, R0 and fitting parameters nc, nr, nf. In this paper, the reference stress ratio R0 in the fatigue master curve can be selected in a finite UD laminates fatigue tests. t1, nc, nr, nf are obtained by fitting the corresponding fatigue test data.

Correspondingly, the S-N curve can also be derived from the fatigue master curve [24]:

where:For composite materials, the static failure stress σ0 is usually different in the tensile and compressive test, therefore, the static failure stress σ0 in Eq. (8) has to be distinguished between the tensile or compressive loading state, that is, the tensile strength, the compressive strength, and their combination determined by the stress ratio R, as follows:

where:

Eq. (12) shows that for R<0, there are two fatigue master curves, as the tensile fatigue master curve and the compressive fatigue master curve. It should be noted that material fatigue failure is determined by the curve which is lower in the current cycle, which indicates that the fatigue master curve in the tension-compressive loading is caused both by tensile fatigue and compressive fatigue shown in Fig. 3.

Another situation not shown in the figure is that if the tensile and compression master curves are not crossed, then the tensile fatigue main curve can be considered to be always higher or lower than the compression fatigue master curve, and the fatigue failure is completely caused by tension or compression, the fatigue master curve is also completely consistent with the tensile or compression master curve, that is, the one with a lower stress level.

Up to now, the uniaxial fatigue master curve of the UD laminates has been established. For the fatigue master curve and S-N curve under any frequency or loading, it can be always obtained through a relatively simple constant strain rate test and a tensile and compression fatigue test under a determined stress ratio.

2.2Multiaxial fatigue model2.2.1Multi-axiality ratioThe S-N curve shown in Eq. (3) can only represent the failure condition of the UD laminates. In order to correctly characterize the off-axis and multi-axis laminate, it is introduced the concept of multi-axiality ratios. First, a scalar equivalent fatigue stress σeq is used to represent the stress components in the stress matrix:

where:Here, the superscript ‘T’ denotes transpose of an array; σN is the first non-zero stress component of σ, a group of multi-axiality ratios [λ] comprises the multi-axiality ratios λi covering all stress components in a given stress space. For off-axis laminates, the multi-axiality ratios depend on the fiber laying angle θ, that is:By introducing a multi-axiality ratio, the equivalent fatigue stress σeq in the multi-axial fatigue loading σ=σf(R,f,Nf) can be determined with Eq. (14), as σeq=σeq,f(R,f,Nf). So the multi-axial fatigue stress can be obtained from the fatigue master curve in Eq. (14) and a given multi-axiality ratio λ, thereby the multi-axial fatigue master curve modeling can be completed:

where Fp represents the Puck criterion function under fatigue loading.2.2.2Fatigue failure criteriaIn order to determine the laminates failure, the improved Puck failure criterion is selected by the fatigue model based on the fatigue master curve as the fatigue failure criterion, which contains fiber and matrix fatigue failure modes.

Fiber tensile failure:

Fiber compressive failure:where ε1T and ε1C is the longitudinal tensile and compressive failure strain respectively, ε1 is axial strain, νf12 is fiber longitudinal Poisson's ratio, Ef1 is fiber longitudinal tensile modulus, σ22 is transverse stress, γ21 is longitudinal shear strain, mσf characterized the difference in transverse stress between fiber and matrix. For carbon fiber, mσf≈1.1, mσf≈1.3 for glass fiber. In order to apply the criterion to the fatigue failure determination, it is necessary to convert the failure strain under the static failure load into the fatigue failure strain.The matrix failure modes can be divided into three categories, named A, B, and C:

Failure mode A corresponds to a fracture angle of 0°

Failure mode BFailure mode Cwhere τ21 is longitudinal shear stress, σ11 is longitudinal tensile stress, σ22 is transverse tensile stress, YT is transverse compressive strength, p⊥//+ and p⊥//- is the slope of failure envelope of (σ2, τ21), σ1D is degradation stress. To meet the fatigue failure determination requirements, the strength parameter can be converted into a function related to the stress ratio, the stress amplitude and the number of cyclic loads.The fatigue failure envelope of laminates can be determined by the given number of cycles and S-N curve with given multi-axiality ratio, as shown in Fig. 4. The fatigue master curve of the multi-axial laminates can also be plotted by the fatigue test data of UD laminates and fitting parameters associated with it.

For UD laminates, the multi-axial fatigue strength can be determined by the uniaxial fatigue strength derived from the fatigue master curve with the Puck’s criterion. For multi-axial laminates, the fatigue strength can be determined by multi-axiality ratios in the global coordinate system, which is converted from the off-axis ratio in the local coordinate system of UD laminates. Meanwhile, the fatigue life prediction is based on the last-ply-failure (LPF) theory. Under the assumption that the laminate is under average stress, the stress of the multi-axial laminate is determined by the Puck’s criterion and last-ply-failure theory.

3Fatigue life prediction of composite laminatesIn order to verify the accuracy of the model, fatigue test data for UD and MD laminates with different materials were selected, including glass fiber reinforced composites, T300, T700 and T800 carbon fiber reinforced composites. After inputting the material properties and fitting parameters, test results were compared with the fatigue data calculated by the multiaxial fatigue model.

3.1Fatigue life prediction of GFRPIn this section, different glass/epoxy UD laminates and MD laminates were tested. The material properties and fitting parameters are shown in Table 1:

From the fatigue life data of 0° UD laminates under R=0.1 and R=10, the fatigue master curve of glass/epoxy UD laminates and MD laminates can be established by Eq. (3), the fitting parameters are shown in Table 2. When the off-axis UD laminates were characterized with the fatigue master curve, the off-axis ratio λ is used to indicate their stress states. From Eq. (17), the off-axis ratio λ of 0°, 19°, 45°, 71°,90° UD laminates are 1:0:0, 1:0.12:0.34, 1:1:1, 1:8.43:2.9, 0:1:0. Some scholars have compared the prediction results with the test data, and the results have a higher fitting degree [24].

Meanwhile, the multiaxial fatigue model can also establish an accurate multi-axial fatigue model for MD laminates. Figs. 5 and 6 plot multi-axial S-N curves for different lay-up configurations of [45/02/-45]s, [0/902/0]s, [45/0/-45/90]s, [45/-452/45]s and [0/90]4s with different stress ratios. In order to simplify the stress state under fatigue damage, the multi-axiality ratio λ is 1:0:0 in this paper.

Comparison of predictions of multi-axial S-N curve for MD laminates for different lay-up configurations with test data [26].

Comparison of predictions of multi-axial S-N curve for MD laminates for different stress ratio with test data [27].

From the test data, it can be concluded that the higher proportion of 0° ply contained in four laminates, the longer fatigue life the laminates have. Compared [45/02/-45]s with [0/902/0]s laminates which have the same proportion of 0° ply, the fatigue life of [45/02/-45]s laminates is longer than [45/02/-45]s, because the 0°ply of [45/02/-45]s laminates is continuous layer and cross-ply for [0/902/0]s laminates. For [45/02/-45]s and[45/-452/45]s laminates, the fatigue life of [45/02/-45]s laminates is longer than [45/-452/45]s laminates which is continuous layer with 0°ply and 45°ply, this is because the fatigue strength of 0° UD laminates is stronger than -45° laminates.

From the comparison of theoretical S-N curve and test data, the prediction results match test data accurately.

3.2Fatigue life prediction of T300 CFRPIn this section, different T300 UD laminates and MD laminates were tested. The material properties and fitting parameters are shown in Table 3:

In the model of the fatigue master curve, by inputting material parameters and fitting parameters (Tables 3 and 4), loading frequency and maximum stress, the tension S-N curve of 90°UD laminates can be calculated with the fatigue test data of 0°UD laminates under R=0.1, and the compressive S-N curve of 90°UD laminates can be calculated with 0°UD laminates under R=10 similarly shown in Fig. 7, because the longitudinal fitting parameters and the transverse fitting parameters are assumed to be consistent.

Comparison of predictions of multi-axial S-N curve for UD laminates with test data [28].

In addition, in order to promote the application scope of the model, different MD laminates were test with different lay-up configurations as [±45]4s, [0/904]s, [02/902]s, [0/902]s and [45/-45/0/0/-45/90/0/90/45/0]s. Figs. 8 and 9 show their S-N curve under stress ratio of R=0.1.

Comparison of predictions of multi-axial S-N curve for MD laminates with test data [29,30].

Comparison of predictions of multi-axial S-N curve for cross-ply laminates with test data [31].

From Figs. 13 and 14, it can be seen that the number of 0°ply determines the fatigue life of laminates. For the laminates with same materials, the trend of S-N curve is roughly the same for different lay-up configurations because they have same multi-axiality ratios and same S-N curve for R=0.1 and R=10.

Comparison of predictions of multi-axial S-N curve for UD laminates for R=−1 with test data [32].

Comparison of predictions of multi-axial S-N curve for UD laminates for R=0.1 with test data [32,33].

Comparison of predictions of multi-axial S-N curve for UD laminates for R=0.5 with test data [32,33].

Comparison of predictions of multi-axial S-N curve for UD laminates for R=10 with test data [32,33].

Comparison of predictions of multi-axial S-N curve for MD laminates for different lay-up configurations with test data [34–36].

In this section, different T700 UD laminates and MD laminates were tested. The material properties and fitting parameters are shown in Table 5 and 6.

From the fatigue master curve established by the fatigue life data of the 0° UD laminates under R=0.1 and R=10, the S-N curve of 0°, 10°, 15°, 30°, 45°, 90° UD laminates under different stress ratios could be calculated respectively. Their multi-axiality ratios are1:0.03:0.18, 1:0.07:0.26, 1:0.33:0.58, 1:1:1and 0:1:0. The comparison of test data and the calculated results were shown in Figs. 10–13.

From Eq. (17), it can be concluded that the fatigue life of off-axis laminates is related with the off angle θ, the UD laminates has longer fatigue life with smaller off angle θ.

At the same time, different MD laminates were tested with different lay-up configurations as [±45]4s, [0/±45/90]s, [45/90/-45/0/45/0/-45/90/0]s, [-45/90/45/0/90/90/-45/90]s, Fig. 14 shows their S-N curve under stress ratio of R=0.1.

It can be seen that the fitting results are in good agreement between test data and prediction results.

3.4Fatigue life prediction of T800 CFRPIn this section, different T800 UD laminates were tested. The material properties and fitting parameters are shown in Table 7.

From the fatigue master curve established by the fatigue life data of the 0° UD laminates under R=0.1 and R=10, the S-N curve of 10°, 15°, 30° UD laminates under different stress ratios could be calculated, respectively. The comparation of test data and the calculated results were shown in Fig. 15.

Comparison of predictions of multi-axiality S-N curve for UD laminates with test data [37].

From the comparison of the prediction results of the UD laminates with different materials, it can be seen that the trend of S-N curve depends on master fatigue curve. The predicted S-N curves of the off-axis UD laminates are roughly the same, which is caused by using the same reference fatigue master curves and the different multi-axiality ratios cause the different fatigue strength. However the MD laminates use the same multi-axiality ratio, the different lay-up configurations lead to the difference of different fatigue strength.

4Fatigue of braided composite laminate2D braid composites are widely used in the practice, then this fatigue master curve model could be expanded to 2D braid composite laminates which is treated as cross ply composite laminates as its stress is evenly distributed.

4.1Fatigue experiments of braided composite laminateIn order to expand the application scope of the model further, a fatigue test of braided composite laminates was carried out on the MTS fatigue machine under stress ratio R=0.1, R=10, R=−1 and R=−0.8 shown in Fig. 16. The fitting parameters and material properties are shown in Table 8 and Table 9.

The final fatigue failure is shown in Fig. 17.

From the above figures of failure specimens, it can be seen that the fatigue failure mode of tension-tension specimen is mostly brittle broken, the fiber fracture is relatively neat without delamination and other gap. For compression-compression specimen, there is micro-buckling and stacking failure in the failure area and delamination between each plies. As the same time, the tension-compression specimens have both fatigue failure mode of tension-tension specimen and compression-compression specimen. The outer layer is brittle fracture and the inner layer has buckling, kinking and stacking which leads to delamination between plies.

The fatigue test data of braided composite laminates under different stress ratio is shown in Fig. 18.

4.2Fatigue life prediction of braided composite laminatesTo expand the scope of the model and verify the accuracy of the model, the test results of braided laminates are compared with the prediction results. From the two fatigue master curves of R=0.1 and R=10 obtained from the test data, and then the S-N curve could be determined from the fatigue master curve of R=−1 and R=−0.8 as the loading frequency and cycle were known shown in Fig. 19. The fitting parameters are shown in Table 10. Under the assumption of stress in the laminate is distributed equally, each layer of the braided composite laminates can be approximated as a 0° UD laminate, and its life could be predicted by classical laminate theory(CLT theory). Fig. 20 shows the test results and fitting curve of braided composite laminates under R=0.1 and R=10.

From the test data and prediction results, it can be seen that the prediction results are well fitted to the test data. In the above tension-compression fatigue test, the fatigue life for R=−0.8 is shorter than R=−1 as the S-N curve shown, this is because the longitude strength of braided composite laminates is less than transverse strength.

5ConclusionsIn this paper, fatigue master curve model is adopted to predict the fatigue lives of UD and MD glass and carbon fibre reinforeced composite lamminates. The calculation results of fatigue master curve model show good agreements with exiperiments.

The fatigue master curve model could predict the fatigue life of 2D carbon fibre braid composite laminates by asuming the 2D braid composite laminates as cross ply composite laminates. The caluculation results of 2D braid composite laminates under fatgiue loading with R=−1 and −0.8 are consistent with experimental results.

This study was supported by the National Natural Science Foundation of China (NO. 11402064).