Stiffness degradation of composite laminates due to matrix cracking and induced delamination during tension-tension fatigue

Stiffness degradation of composite laminates due to matrix cracking and induced delamination during tension-tension fatigue

Engineering Fracture Mechanics 216 (2019) 106489 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.else...

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Engineering Fracture Mechanics 216 (2019) 106489

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Stiffness degradation of composite laminates due to matrix cracking and induced delamination during tension-tension fatigue Hamed Pakdel, Bijan Mohammadi

T



School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

A R T IC LE I N F O

ABS TRA CT

Keywords: Fatigue Stiffness reduction Induced delamination Matrix cracking

Degradation of mechanical properties of laminates during uniaxial tension-tension fatigue is analytically and experimentally investigated. A unit-cell based stress analysis is developed accompanied by a two-stage energy-based power law evolution rule to predict sequential propagation of matrix cracks and induced delaminations. Two power law damage evolutions are proposed to relate the rate of crack density accumulation or induced delamination propagation to the corresponding energy release rates. Analytically predicted stiffness reductions of carbonepocy specimens are shown to be in agreement with experimental observations. The procedure is argued to be capable of predicting fatigue behavior of laminates with limited empirical parameters.

1. Introduction Cyclic loading of laminated composites leads to several different damage mechanisms such as matrix cracking, induced delamination and fiber breakage [1,2]. Damage accumulation, as the common source of fatigue in composites, degrades the mechanical properties of the laminate. Hence, proper modeling of the damage evolution is the basis for predicting the fatigue life and behavior of composite structures. Despite the variety and complexity of damage mechanisms, fatigue behavior of composites has been investigated and modeled through various approaches. Inspired by the developed approaches for predicting fatigue life of metals, some researchers [3–7] have established fatigue life models upon S-N curves. The proposed fatigue life models try to relate the fatigue life of laminates to loading conditions such as mean applied stress, stress ratio and loading frequency. However, other affective parameters such as laminate layup have been disregarded in the majority of proposed models. Thus, most of the fatigue life models are unable to predict the fatigue life of an arbitrary layup based on limited experiments. However, they provide rapid and uncomplicated procedures for industrial demands where fatigue life of a certain laminate layup is of interest. Many other researchers established their fatigue models on phenomenological approaches. In phenomenological fatigue models, results of experimentally observed phenomena, mainly residual strength or stiffness, are expressed mathematically. Several phenomenological residual strength models [8–10] have been proposed based on wear-out and sudden death approaches which deal with strength degradation during low-cycle and high-cycle fatigue, respectively. Residual strength models necessitate extensive destructive experiments which result in high data scatter. Therefore, residual strength models are often accompanied by probability models such as Weibull distribution models. In comparison with residual strength, residual stiffness models are established upon non-destructive experiments. Numerous



Corresponding author. E-mail address: [email protected] (B. Mohammadi).

https://doi.org/10.1016/j.engfracmech.2019.106489 Received 24 March 2019; Received in revised form 11 May 2019; Accepted 20 May 2019 Available online 25 May 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 216 (2019) 106489

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Fig. 1. Three regions of damage accumulation in laminates due to fatigue.

residual stiffness models [11–17] have been presented during recent decades. In residual stiffness models, a mathematical curve is fitted to the stiffness degradation obtained from fatigue experiments of composite laminates. The proposed mathematical formulation is meant to be capable of predicting stiffness degradation at each load cycle based on numerous loading, material and layup parameters. However, high number of parameters affecting fatigue behavior of laminates (such as ply thickness, layup, material characteristics, loading type, magnitude, ratio and frequency) requires extensive experiments to propose a mathematical model capable of predicting stiffness degradation of an arbitrary laminate layup. Besides, different fatigue behavior of laminates during different stages of fatigue, as depicted in Fig. 1, makes it difficult to fit a particular mathematical model to all regions. Hence, many researchers have validated their proposed models for limited range of parameters at specific fatigue regions. The multiplicity of the residual stiffness models has made it the most developed approach for predicting fatigue behavior of laminates. Shokrieh and Lessard [18–23] implemented the developed approach of stiffness degradation accompanied by fatigue life and classical failure criteria for different loading conditions and proposed a progressive model to predict stiffness reduction of laminates in a cycle by cycle manner. Progressive approach was also proposed in the framework of continuum damage mechanics in sporadic studies [24–26]. Despite fundamental differences between various fatigue modeling approaches, all fatigue life, phenomenological and progressive fatigue models have a common point. All above models disregard the physics of damage as the source of fatigue and focus on the consequences in the form of fatigue life or degradation of mechanical properties. In the present paper, the damage accumulation as the source of mechanical degradation during fatigue of laminates is taken into account. Multiplication of matrix cracks at the first stage of fatigue and propagation of induced delaminations as the most common mode of damage at the second stage of fatigue are studied experimentally and analytically. A power law damage evolution containing two empirical parameters is proposed to relate the rate of crack density accumulation to the energy release rate. A similar power law model is also proposed for induced delamination growth rate which possesses a different pair of empirical parameters. All empirical parameters are obtained from experimental observations. An energy-based damage competition criterion is implemented to predict the characteristic damage state (CDS), as the boundary between first and second fatigue stages. A unit cell based stress analysis based upon variational principles is implemented to derive the stiffness degradation of [θn /0]s and [0/ θn]s laminates with different damage states. Fatigue experiments are performed on [θn /0]s and [0/ θn]s carbon-epoxy laminates with θ = 60, 75, 90 and experimental results of longitudinal stiffness degradation are compared to analytical predictions. The whole procedure developed through numerous papers by the authors [27–31] is believed to be capable of predicting stiffness reduction of laminates due to fatigue implementing limited empirical parameters independent from layup configuration.

2. Damage propagation criterion As depicted schematically in Fig. 1, stiffness degradation of off-axis dominated composite laminates under fatigue loading consists of three regions through the whole fatigue life. Extensive experimental observations [32–34] confirm an alteration in governing damage mode at each region. During the initial region, which includes 10–20 percent of the fatigue life, intra-laminar matrix cracks initiate and multiply along the laminate. Despite the short duration of regions I and III, the major proportion of the stiffness reduction of the laminate due to fatigue is experienced through these regions. At the characteristic damage state (CDS) of an off-axis dominated laminate, matrix cracks saturate to a specific crack density and inter-laminar induced delaminations initiate from the tips of existing cracks. In comparison with the rather rapid multiplication of matrix cracks, induced delaminations propagate slowly through the interface between adjacent plies. Therefore, the major portion of the fatigue life is assigned to region II which includes propagation and coalescence of induced delaminations. After the gradual degradation of mechanical properties due to slow propagation of induced delaminations in region II, fiber breakage becomes the governing damage mode at the final region III. 2

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Fig. 2. Unit cell of laminate containing mid or outer-ply matrix cracks with induced delaminations.

Predicting stiffness reduction of a laminate through both regions I and II requires: 1. 2. 3. 4.

Deriving stiffness of the damaged laminate containing matrix cracks with or without induced delaminations. Prediction of crack density at each load cycle through region I, Prediction of the crack density at saturation or CDS, Prediction of induced delamination length at each load cycle through region II,

Stress state and the corresponding mechanical properties of a damaged [θm(o) / θn(i) ]s laminate containing either mid or outer-ply matrix cracks with or without induced delaminations have been derived using a developed unit cell based analysis based upon variational principles by the authors [27–29]. In the developed variational approach, matrix cracks are assumed to distribute uniformly through the length of the laminate. This is in contrast with experimental observations specially at low crack densities. However, as crack density increases, the matrix cracks form a fairly uniform distribution. Thus, assuming uniform distribution of cracks may deviate the predicted values at lower crack densities, however it gives acceptable results at higher cracks densities. The stress state of the damaged laminate can be derived using any unit cell approach such as variational [35–37] shear-lag [38,39] or finite element [40] analysis. In unit cell stress analysis, the whole laminate is considered as a repetition of unit cells bounded by adjacent matrix cracks. Matrix cracks are assumed to distribute uniformly through the length of the specimen and induced delaminations are considered to grow with identical length from all matrix cracks tips as depicted schematically in Fig. 2. Outer-ply matrix cracks are considered to distribute with a staggered pattern in agreement with experimental observations as investigated by the authors [30]. As far as the whole laminate is made up of identical unit cells, the stress state and mechanical properties of the unit cell can be generalized to the laminate. Having the stress state, complementary energy of the unit cell and thereupon energy release rate during any finite amount of damage accumulation can be calculated. The onset of matrix crack saturation and induced delamination initiation has been regarded as a characteristic damage state (CDS) independent of loading conditions by many researchers [41–44]. It should be noted that as discussed thoroughly in a prior study by the authors [28,29] An energy based criterion for predicting CDS in laminates has been proposed and validated in comparison with experimental observations by the authors [28,29]. In the proposed criterion, induced delaminations start when matrix crack density reaches the saturation state. It should be noted that as discussed thoroughly in the prior study by the authors [28,29] premature induced delaminations may occur at regions with local high crack density before the CDS is achieved. However, investigating the overal length of the laminate, the majority of delaminations start growing after saturation of matrix cracks. Whereas, the assumption may induce a deviation in the prediction of CDS from experiments, however it causes an insignificant deviation in the prediction of stiffness degradation. In the proposed criterion, energy release rates for matrix crack accumulation Gm , and induced delamination initiation Gd are calculated and compared at different crack densities as depicted schematically in Fig. 3. The crack density at which energy release rate of induced delamination initiation surpasses the one for matrix crack multiplication is known as CDS:

Fig. 3. Energy release rate as the criterion for predicting CDS. 3

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CDS if:

Gd ⩾ Gm

(1)

The energy release rate for matrix crack accumulation and induced delamination initiation and propagation is calculated introducing the stress state of the unit cell of damage into thermodynamics rules. The first law of thermodynamics states that during development of any finite amount of damage area ΔA , total energy of the system must be conserved as:

Δq ΔW ΔU ΔK − = − +γ ΔA ΔA ΔA ΔA

(2)

where ΔU and ΔK are the change in internal and kinetic energy of the system during development of crack surfaces, respectively, γ is the surface energy, Δq is the heat added to the system and ΔW is the work done on the system. The left hand side of Eq. (2) is the driving energy needed for formation of unit crack surface area identified as energy release rate (G), whereas the right hand side represents all dissipated energies of the system per crack area well-known as “effective critical energy release rate” denoted by Gc . It should be noted that if the formation of matrix cracks is under load-control condition, as for the case of most fatigue loading tests, the energy release rate can be written in the form:

G=

ΔW ΔU ΔUc load − control − = ΔA ΔA ΔA

(3)

where ΔUc is the change in the complementary energy of the unit cell during any finite accumulation of damage area. Prediction of matrix crack accumulation due to static loading has also been investigated by the authors [31].Besides, an energy based evolution criterion is proposed in the present study for predicting crack density at each load cycle. In the following a summary of the proposed criterion is presented. A similar energy based criterion is also proposed capable of predicting induced delamination length at each load cycle which fulfills all of the 4 requirements for prediction of stiffness reduction of a laminate through both regions I and II. 2.1. Matrix crack multiplication-Region I According to experimental data achieved during tension-tension fatigue of laminated composites, a power law relation is proposed between crack density growth per cycle (dρ / dN ) and energy release rate calculated at the maximum applied stress (Gm ) as:

dρ = c·[Gm (1 − R2)]b dN

(4)

where ρ = 1/2a is the crack density, R = σmin/ σmax is the stress ratio, c and b are effective material properties characterized based on experimental data, and Gm is the energy release rate due to formation of new matrix cracks. The energy release rate during accumulation of mid-ply matrix cracks under uniaxial tensile loading is derived using variational stress analysis of the unit cell of damage as:

Gm(mid − ply) =

2 σ¯ XX a (ti + to) 1 1 − d [ d ] ti EXX (2ρ) EXX (ρ)

(5)

d EXX

is the effective stiffness of the damaged laminate in the X direction and σ¯XX is the total average longitudinal stress applied where to the laminate. Stress analysis of the unit cell of staggered outer-ply matrix cracks gives the energy release rate for multiplication of outer-ply matrix cracks as:

Gm(outer − ply) =

2 σ¯ XX a (ti + to) 1 1 [ d ] − d 2to EXX (3ρ) EXX (ρ)

(6)

Having the energy release rate Gm for any crack density ρ from variational approach, and the material properties c and b from experiments, Eq. (4) can be integrated over an incremental increase in load cycles which results in a recursive equation as:

ρ2 = c·ΔN ·[Gm (1 − R2)]b + ρ1

(7)

where ρ1 and ρ2 are crack densities at load cycles N1 and N2 , respectively, and ΔN = N2 − N1. Using the recursive relation of Eq. (7), crack density at each load cycle can be predicted based on the crack density at the previous load cycle all through region I. 2.2. Induced delamination propagation-Region II A similar damage evolution criterion is proposed for the growth of induced delaminations in region II as:

d (d ) = c′·[Gd (1 − R2)]b′ dN

(8)

where (d ) is the induced delamination length, and c′ and b′ are two empirical parameters derived from fatigue experiments. The energy release rate Gd due to an incremental increase in delamination length is derived introducing the stress state of the unit cell into Eq. (3) which gives: 4

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Table 1 Material properties of carbon-epoxy. E11 = 90000 (MPa) E22 = 6700 (MPa) Ply thickness = 0.3 (mm)

Gd = lim

d2 → d1

G12 = 3500 (MPa) G23 = 2500 (MPa)

ν12 = 0.3 ν23 = 0.5

Uc (ρs , d2) − Uc (ρs , d1) 1 ∂Uc (ρs , d ) = |d = d1 ∂d 4(d2 − d1) 4

(9)

Having the energy release rate Gd for any induced delamination length d1 at saturated crack density ρs from Eq. (9), and the material properties c′ and b′ from experiments, Eq. (8) can be integrated over an incremental increase in load cycles which results in a recursive equation as:

d2 = c′·ΔN . [Gd (1 − R2)]b′ + d1

(10)

where d1 and d2 are delamination lengths at load cycles N1 and N2 , respectively, and ΔN = N2 − N1. Using the recursive relation of Eq. (10), induced delamination length and the corresponding damage state at each load cycle can be predicted based on the damage state of the previous load cycle all through region II. 3. Experimental procedure In order to check the validity of the proposed damage evolution criteria, tensile fatigue tests were performed on carbon-epoxy test specimens with the mechanical properties in Table 1. Hand layup technique was implemented to prepare [0/ θn]s and [θn /0]s laminates with θ = 60, 75, 90 . Test specimens nominally 20 mm by 200 mm, were cut out of the laminated plates after cure at room temperature under vacuum bag pressure. Carbon-epoxy end tabs were prepared and bonded to the specimens using an epoxy-based adhesive. All specimens and end tabs were cut using a water cooled circular diamond saw to ensure heat removal. A hand-held digital microscope was utilized to detect matrix cracks and induced delaminations. The digital microscope was assembled on a linear mechanism to travel along the specimen edge during fatigue tests as depicted in Fig. 4. In order to provide sufficient finishing surface condition for damage detection by optical microscopy, the specimen edges were polished using wet sanding with medium to extra fine grit sizes. All load-control fatigue tests were performed at a frequency of 5 Hz with zero or positive stress ratios. Residual stiffness of the specimens were calculated at certain load cycle intervals based on the hysteresis loops. Besides, in order to provide sufficient data for characterization of the empirical parameters, the experiment was periodically paused during fatigue test of each specimen, and the damage state was detected using the digital microscope. Despite the scatter in distribution of matrix cracks, the detected crack density at each interval was averaged over the length of the tested specimen. 4. Results and discussion In order to characterize the empirical parameters for matrix crack accumulation and induced delamination propagation, both damage evolution criteria in Eqs. (4) and (8) are rewritten in a logarithmic form which gives:

log(

dρ ) = log(c ) + b·log[Gm (1 − R2)] dN

(11)

and

Fig. 4. Optical microscope setup on Dartec-9600 for detection of damage during Fatigue. 5

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Fig. 5. Matrix cracking in (a) [0/753]s and (b) [753 /0]s laminates.

log(

d (d ) ) = log(c′) + b′·log[Gd (1 − R2)] dN dρ dN

(12) d (d) , dN

or are derived based on experimental observations. Experimental observations The damage accumulation rate, namely confirmed sequential accumulation of matrix cracks and induced delaminations in regions I and II. Initial highly scattered matrix cracks formed in the vicinity of material deficiencies such as voids or resin rich regions. As load cycles increased, matrix cracks accumulated rapidly and distributed in a more uniform pattern through the length of the specimen. Outer-ply cracks formed a staggered distribution pattern (Fig. 5) as considered in the unit cell based stress analysis. Matrix crack accumulation slowed down at high crack densities and induced delaminations initiated from crack tips and propagated at a low rate through the interface with adjacent plies as illustrated in Fig. 6. Both crack density and induced delamination length were recoded at different load cycles and damage accumulation rates were derived from recorded data. The energy release rates for matrix crack accumulation Gm or induced delamination propagation Gd are derived introducing the stress state of unit cell based variational analysis into Eqs. (5), (6) and (9). Therefore, a graph indicating logarithmic form of matrix crack accumulation rate against crack accumulation energy release rate can be plotted including data from all tested specimens as depicted in Fig. 7. The empirical parameters can also be characterized based on more limited experiments. As far as experiments are more convenient to be performed on cross-ply laminates, master plot for cross-ply specimens is also depicted in Fig. 8. It can be seen that the empirical parameters derived based on two cross-ply laminates has a slight deviation from the parameters derived based on all tested specimens. Hence, the material characterization can be performed based on convenient experiments on limited cross-ply specimens. The empirical parameters are then used to predict crack density at each load cycle during fatigue with different positive stress ratios as depicted for some specific specimens in Fig. 9. As it can be seen in the figure, stress ratio affects matrix crack development during fatigue. Matrix cracks multiplicate more rapidly at lower stress ratios. Besides, the crack density is affected by the off-axis ply thickness. Laminates with thinner off-axis ply experience higher crack densities. Having the crack density at each load cycle, unit cell based variational approach can be used to derive the residual stiffness of the damaged laminate through region I. A similar logarithmic graph is plotted for induced delamination length versus delamination growth energy release rate as depicted in Fig. 10. According to Eqs. (11) and (12), all empirical properties can be derived from the inclination and vertical intercept of the linear curves fitted to the data points of Figs. 7 and 10. Using the characterized empirical parameters, stiffness reduction of laminates subject to tension-tension fatigue can be predicted through regions I and II implementing Eqs. (7) and (10), respectively. Experimental observations and analytical predictions for longitudinal stiffness reduction of laminates containing mid and outer off-axis plies are depicted in Figs. 11 and 12, respectively. Experimental data are derived at load cycles from at least 2 tested specimens for each layup. As depicted in both graphs, the initial deviation between experiments and predictions at region I, decreases through region II and brings about more compatible results at higher load cycles. In order to investigate the effects of off-axis ply orientation on stiffness degradation of laminates subject to tensile fatigue, analytical results for longitudinal stiffness reduction of [0/ θ2]s laminates during identical fatigue loading are depicted in Fig. 13. As depicted in the graph, as the off-axis ply orientation increases, the laminate experiences more stiffness degradation. This can be argued to be a consequence of the orientation of the matrix cracks. As mentioned before, matrix cracks are aligned parallel to fiber orientations. In laminate with transverse ply, the crack surface is perpendicular to the applied stress direction. This results in more decrease of load carrying capability of the cracked layer which causes more stiffness degradation. Besides, crack density reaches higher values in layers with higher off-axis orientationθ This can be argued to be an other source of more stiffness degradation in laminates with higher off-axis ply orientation.

Fig. 6. Induced delaminations in (a) [0/752]s and (b) [752 /0]s laminates. 6

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Fig. 7. Mater plot for characterizing empirical parameters of matrix crack accumulation criterion.

Fig. 8. Mater plot of cross-ply laminates for characterizing empirical parameters of matrix crack accumulation criterion.

Fig. 9. Analytical prediction and experimental observation of crack density during fatigue.

An identical fatigue behavior is observed in laminates containing outer off-axis plies. As depicted in Fig. 14, the cross-ply laminates [902/0]s experiences more stiffness reduction during fatigue in comparison with [752 /0]s and [602/0]s laminates with identical loading conditions. 7

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Fig. 10. Master plot for characterizing empirical parameters of induced delamination propagation criterion.

Fig. 11. Stiffness reduction of laminates containing mid off-axis plies.

Fig. 12. Stiffness reduction of laminates containing outer off-axis plies.

The characteristic damage state as the distinguishing point between regions I and II differs with off-axis ply orientation. As depicted in Fig. 13, in laminates containing mid off-axis ply, as the off-axis orientation θ decreases, CDS is reached at higher load cycles. Thus, region I possesses higher proportions of fatigue life as the mid off-axis ply orientation increases. On the other hand, increasing the outer off-axis ply orientation advances occurrence of CDS as depicted in Fig. 14. Besides, it should be noted that midply matrix crack density reaches higher values at saturation in comparison with outer-plies. Staggered distribution of the outer-ply 8

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Fig. 13. Effect of off-axis angle on stiffness reduction of laminates containing mid off-axis plies.

Fig. 14. Effect of off-axis angle on stiffness reduction of laminates containing outer off-axis plies.

cracks results in a non-symmetric redistribution of stress state. The non-symmetric stress state causes local bending in the vicinity of matrix crack tips. The local bending of the laminate results in rapid formation of induced delaminations and saturation of matrix cracks at lower densities. Sooner initiation of induced delamination results in shorter region I during fatigue of laminates with outer off-axis plies.

5. Conclusion A two stage damage evolution criterion was proposed for prediction of damage accumulation during first and second stages of mechanical degradation of laminated composites subject to tension-tension fatigue. The rate of matrix crack accumulation during first stage of fatigue was related to the energy release rate through a power law criterion containing two empirical parameters. A similar power law equation with a different pair of empirical parameters was proposed to predict propagation of induced delaminations at the second stage of fatigue. A unit cell based stress analysis was implemented to derive degradation of mechanical properties of the laminate due to presence of mid or outer-ply matrix cracks with or without induced delaminations. In order to characterize the empirical parameters and validate the proposed damage evolution criterion, experimental tests were performed on [0/ θn]s and [θn /0]s carbon-epoxy laminates. Analytical predictions of the longitudinal stiffness reduction of laminates were shown to be in agreement with experimental observations. Off-axis ply orientation in both [0/ θn]s and [θn /0]s laminates was shown to affect CDS and consequently the duration of first region. Increasing the mid off-axis ply orientation in [0/ θn]s laminates extended the duration of the first stage. Whereas, increasing outer offaxis ply orientation reduced the number of cycles to CDS and thereupon shortened region I. Besides, off-axis ply orientation affected the stiffness degradation of laminates subject to fatigue. Laminates with higher off-axis ply orientation experienced more stiffness reduction at identical load cycles. As discussed previously, a proper fatigue model must be capable of predicting degraded mechanical properties such as stiffness reduction at each load cycle during 1st and 2nd regions of fatigue with limited empirical parameters. A procedure was developed through numerous papers by the authors [27–31] and concluded in the present paper to predict the damage evolution and the corresponding stiffness degradation due to fatigue of laminates. All geometrical parameters such as layup configuration, damage 9

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mode and propagation pattern was regarded through unit-cell based analysis and the parameters were shown to be only affected by material properties. All empirical parameters can be characterized through limited experiments of a cross-ply laminate and implemented to predict the damage evolution in arbitrary layup configurations. The proposed procedure can be developed for further modes of damage in general materials and layups. However, it should be noted that the validity of the developed procedure was investigated through limited experiments of selective material and layups. Hence, further extensive experiments must be performed to investigate the validity of the proposed model. References [1] Degrieck J, Paepegem WV. Fatigue damage modelling of fibre-reinforced composite materials: review. Appl Mech Rev 2001;54:279–300. https://doi.org/10. 1115/1.1381395. [2] Berthelot J. 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