- Email: [email protected]
Fatigue of polymer composites
Fatigue of polymer composites
10
Ramesh Talreja Department of Aerospace Engineering, Department of Materials Science and Engineering, Texas A&M University, College Station, TX, United States
10.1
Introduction
The recent success of polymer composites in civil aviation is attributed not only to fuel savings by lightweight properties of these materials but also to their high fatigue resistance. The higher fatigue limit of unidirectional composites in cyclic axial tension compared to that of Al alloys, when measured in strain, has allowed higher design loads in composite structures. However, the fatigue performance of polymer composites in different loading combinations and different laminate configurations differ significantly. Optimal designs cannot, therefore, be achieved by arbitrarily setting the allowable strain levels to low values. The alternative of testing to find safe design values is not cost-effective. The mechanisms underlying fatigue failure of the constituents and their interfaces must, therefore, be understood, and this knowledge incorporated in predictive models. Addressing this problem is exacerbated by new developments in fibers and polymers and by a variety of manufacturing methods to construct composite structures. Beginning with the paper in 1981 [1], this author introduced a conceptual framework for the interpretation of the fatigue behavior of composites called a fatigue life diagram (FLD). In its baseline form, the FLD separates the fatigue failure in three regions based on the underlying dominant mechanisms. One region denoted as Region I, characterizes the fiber-failure dominant mechanism, where failure is heavily governed by the statistical nature of the fiber failure, and therefore, the resulting composite fatigue shows negligible dependence on load cycles. Region II is controlled by progressive mechanisms, such as irreversible deformation of the matrix and matrix crack growth, leading to damage accumulation culminating in a critical condition of ultimate failure. Finally, Region III is where failure mechanisms do not progress to criticality within a prespecified large number of load cycles because of local obstacles to growth imparted by fiber-architectural features and interfaces. The demarcation between Region II and Region III forms the fatigue limit. The FLD clarifies the relative significance of these regions for a given combination of fibers and matrix.
10.2
Construction of FLDs
Let us begin with Region I, which, as stated above, signifies the fiber failure dominated mechanism. Fig. 10.1 depicts three failure scenarios under a tensile load P. Fig.10.1(a) Polymer Composites in the Aerospace Industry. https://doi.org/10.1016/B978-0-08-102679-3.00010-1 Copyright © 2020 Elsevier Ltd. All rights reserved.
290
Polymer Composites in the Aerospace Industry
(a)
(b)
(c)
P
P
P
P
P
P
Figure 10.1 Fiber failure scenarios under a tensile load P. (a) A dry fiber bundle with the weakest fiber failure; (b) A fiber composite with the weakest fiber failure and debonding over a short length; (c) Composite failure from the linkage of failed fiber regions.
shows a dry fiber bundle in which the first fiber failure is indicated at the weakest point. At the instant of this fiber failure, the load is shared equally by the surviving fibers, and on increasing the load, the next failure occurs at the weakest point in these fibers. If, however, the load is not increased but is reduced to some value, say zero, and reapplied, no further fiber failures occur. Thus, there is no fatigue failure process. This assumes that fibers do not have a fatigue degradation mechanism. Consider now the scenario depicted in Fig. 10.1(b) where a unidirectional composite is loaded in axial tension and the first fiber has failed at a certain value of P. The failed fiber will likely also debond over a length depending on the interfacial characteristics. If the load P is reduced to zero and reapplied to this value, the question arises: will another fiber fail? To answer this question, we note first that on the failure of the first fiber, not all other fibers but a few fibers in close vicinity of the failed fiber are affected. This leads to what is known as local load sharing, the extent of which depends on the debonded length of the failed fiber. Thus, in the local load sharing, an affected fiber can fail if its strength is exceeded at some point over its affected length. If this occurs, it will occur instantaneously, if no time dependency of deformation exists in the fibers, matrix, or interfaces. To avoid unnecessary complication at this point, we assume that all deformation is time-independent. If the load P is reduced to zero and reapplied to the previous value, no further failure is possible unless an irreversible deformation has occurred. Such deformation will redistribute the stresses in the local load sharing region during the unloading and reloading process and create a new stress field, which can result in new fiber failure. Due to the statistical nature of fiber failures, however, there is no certainty that the next fiber failure will occur in the second or a
Fatigue of polymer composites
291
subsequent load cycle. Thus, a random process of fiber failure ensues. More significantly, the critical condition in which the failed fibers in the local load sharing region connect through the matrix and interface cracks, leading to an unstable growth of the connecting failure plane, occurs much more randomly. In fact, this condition of composite failure depicted in Fig. 10.1(c), is of such random nature that it may be rightly described as a chaotic process because of its high sensitivity to the initial condition, i.e., the first fiber failure. This also means that a well-defined progression in the fiber failure process that can be described by a rate equation does not exist. Therefore, the final failure event is for all purposes cycle-independent. Region I of an FLD corresponding to the fiber failure dominant mechanism is sketched in Fig. 10.2. The vertical axis of the FLD is the maximum strain εmax applied to the composite in the fiber direction. Although the load P is applied and reapplied, the first applied maximum strain makes a proper reference for the load level since it is the same irrespective of the fiber volume fraction, while fiber stress is not. Fig. 10.2 shows Region I as a scatter band centered around the mean strain to static (first load application) failure εc of the composite. The scatter band shown in the figure is for 5% and 95% probabilities of failure, as an example. The scatter band is shown flat (horizontal) to indicate that the initial (static) statistical variability does not change with the number of load cycles applied. As described and argued above, if on the first application of the tensile load the composite does not fail, but at least one fiber within the composite fails, then the composite can fail at any number of subsequent cycles. This will then lead to the scatter band of composite failure becoming independent of the number of cycles. It is possible, however, that the width of the scatter band changes (possibly reduces) with the number of cycles applied, but an analysis to determine that change is currently not feasible. Consider now the case where the first application of load (εmax) is below the scatter band depicted in Region I. The composite failure then will be less likely to be dominated by fiber failures. Instead, the irreversible deformation and crack formation mechanisms in the polymer matrix and at the fiber/matrix interfaces will gain more εmax
95% failure probability εC
Region I 5% failure probability
N
Figure 10.2 Region I of a fatigue life diagram.
292
Polymer Composites in the Aerospace Industry
prominence. These mechanisms were studied extensively and reported in Ref. [2] for a carbon/epoxy (AS4/8552) unidirectional composite loaded in cyclic tension at the load ratio (Pmin/Pmax) of R ¼ 0.1 and cyclic frequency of 10 Hz. Surface replicas were collected periodically at different load levels using cellulose acetate films. After replication, the films were hardened and coated with a layer of carbon by sputtering. They were then observed under a microscope. Fig. 10.3 shows a series of micrographs showing the progression of a crack on the specimen surface and Fig. 10.4 shows more clearly how a crack is bridged by fibers. The crack opening profile evident in Fig. 10.3 has the crack-tip region squeezed, which can be attributed to the closing pressure on crack surfaces induced by the bridging fibers. The study [2] also indicated the effect of fiber/matrix debonding on the fiberbridged matrix cracking as the operating progressive failure mechanism in Region II. The progression of this cracking mechanism is governed by fiber breakage, which is accelerated when a greater length of fibers is exposed to an enhanced stress by debonding, thereby increasing their propensity to failure. With sufficient debonding,
Fibre direction
Pristine
1 cycle
100 cycles
1,000 cycles 10 μ m
Figure 10.3 Micrographs of surface replicas showing a matrix crack at different applied load cycles [2].
Fatigue of polymer composites
293
Figure 10.4 Micrograph showing a matrix crack bridged by fibers [2].
the progressive failure mechanism is then essentially fiber breakage, whose unstable progression leads to the composite failure. Fig. 10.5 illustrates the appearance of Region II in the FLD. The random rate of fiber breakage leads to the scatter band in Region II, as indicated. Finally, Region III represents the cracking mechanisms that have such low driving forces that the energy barriers placed by fibers in the crack paths cannot be overcome to reach the critical state of unstable growth. Fig. 10.6 shows this region of the FLD with the insert in the figure illustrating a matrix crack arresting mechanism. The fatigue limit, below which Region III lies, is the limiting state of the progressive mechanisms in Region II. Fig. 10.7 is now the entire FLD for a unidirectional composite under axial fatigue where the three regions of the FLD (Figs. 10.2, 10.5 and 10.6) have been put together. The effect of increasing the stiffness of fibers is indicated in the FLD. This effect is a consequence of the bridging action of fibers running across a growing matrix crack (insert in Fig. 10.5) and the constraining effect on the matrix cracks by the surrounding fibers (insert in Fig. 10.6). Fig. 10.8 shows the actual fatigue life data with superimposed FLDs for a lower fiber stiffness case (glass/epoxy) and a higher fiber stiffness case (carbon/PEEK). As can εmax εC Region II
5% failure probability
95% failure probability
N
Figure 10.5 Region II of the fatigue life diagram.
294
Polymer Composites in the Aerospace Industry
max
C
Cracks Fibers
fl
Region III N 107
Figure 10.6 Region III of fatigue life diagram indicating the mechanism of matrix crack arrest by fibers. εmax
εC
Region I
Stiffer fibers Region II εfl Region III N
Figure 10.7 The baseline fatigue life diagram of unidirectional composites under cyclic tension.
be seen, the trends predicted by the conceptual FLD (Fig. 10.7) are confirmed by these data. It is also noteworthy that the glass/epoxy data for three different fiber volume fractions (Vf ¼ 0.16, 0.33 and 0.5) fall together within the same Region II scatter band. It is recalled that the fatigue testing was done conventionally, i.e., with load control, and the data when plotted on the stress basis (as SeN plots) fall as three separate curves, each corresponding to one fiber volume fraction. The fatigue limit, as shown in Fig. 10.8(a) is the fatigue limit reported by the authors [3] obtained by testing neat epoxy in strain control. The runouts (with arrows) shown in Fig. 10.8(a) suggest that the glass/epoxy composite fatigue limit is slightly higher, in accordance with the FLD in Fig. 10.7. For carbon/PEEK data in
Fatigue of polymer composites
295
Vf 0.50 0.33 0.16
(a)
εmax
0.024 εc 0.016
0.008 εm
2
0
6
4 log Nf
(b)
8
εmax (%) 1.5 εc 1.2 0.9 εm 0.6 0
1
2
3
4
5
6
7 log Nf
Figure 10.8 Fatigue life data and superimposed FLDs for a glass/epoxy composite (a) at three fiber volume fractions [3] and for a carbon-PEEK composite (b) [4].
Fig. 10.8(b) the same fatigue limit of epoxy (εm ¼ 0.6%) is also indicated for reference, as the PEEK fatigue limit has not been reported. It is expected that PEEK being tougher than epoxy will have a higher fatigue limit. In any case, the carbon/PEEK data shown in Fig. 10.8(b) show a fatigue limit higher than 0.6% strain. A comparison of carbon/PEEK and carbon/epoxy in Ref. [2] showed the fatigue limit values of 0.75% and 0.85%, respectively. The lower value of the fatigue limit for carbon/PEEK is attributed to more extensive debonding observed in this composite [2]. Again, this is in accordance with the expected trend predicted by the FLD (Fig. 10.7).
10.3
Modeling of FLD trends
Some of the qualitative trends in the fatigue behavior of unidirectional composites in cyclic tension emerge logically from the FLD, e.g., the direct dependence of Region I on the fiber strain to failure and the effect of fiber stiffness on Region II and Region III as indicated in Fig. 10.7. There have, however, been efforts to pin down the trends by quantitative modeling. Experimental evidence gathered by surface replicas
296
Polymer Composites in the Aerospace Industry
(Figs. 10.2 and 10.3) [2] indicated that the fiber-bridged cracks attributed to Region II (Fig. 10.2) tended to stop growing unless, as described above, a progressive fiber breakage occurred. It was found that the fiber/matrix debond length grew with fatigue cycles, exposing the increasing length of the debonded fiber to enhanced stress, as well as affecting the neighboring fibers. A shear-lag model was used to determine the stress transfer from the broken fiber to the neighboring fibers, and the Weibull distribution was assumed for the probability of fiber failure [5]. Although quantitative fatigue life prediction could not be made, the trends regarding how Region II would change with the cyclic rate of fiber/matrix debonding and the Weibull shape parameter were inferred [5]. An approach to fiber failure progression in fatigue was put forth recently [6] in which the fiber/matrix interface failure is represented by a cohesive zone model for static strength and the Paris relationship (power-law) for mode-II fatigue crack growth is assumed for the debond crack growth. The statistical fiber failure is given a Weibull distribution, and the fiber bundle failure is modeled by a hierarchical scaling method developed earlier [7]. While evaluating various material parameters entering the models is a challenge, with reasonably assumed values, the prediction of fatigue life in tension-tension loading by the approach seems to confirm the proposed FLD (Fig. 10.9). Strictly speaking, it is noted that Region I of the FLD is not the same as that in Fig. 10.7, where the idealized behavior does not assume cycle dependency. The authors [6] predict low dependency on cycles in this region because of low debond growth rate and high rate of fiber failures. The debond crack growth dictates the slope of Region II in their model. Finally, the mode-II fatigue crack growth threshold of debond cracks determines the fatigue limit. The uncertainty in model inputs and their effects on fatigue life prediction was discussed by the authors [6]. Assuming a cohesive zone for the static shear failure of the
1.6
Peak strain (%)
1.4
Model prediction I
1.2 1.0
II
0.8 III 0.6 100
101
102
103
104
105
106
Number of cycles
Figure 10.9 Model prediction by Ref. [6] compared to experimental data reported in Ref. [2]. The three regions of FLD in Fig. 10.6 have been superimposed on the data.
Fatigue of polymer composites
297
fiber/matrix interface has inherent uncertainty, as the material constants entering this model cannot be obtained independently. For debond growth, assuming a mode-II crack growth rate also induces uncertainty since the driving forces for a debond crack on the surface of a fiber embedded in a composite cannot be accurately reproduced in a test specimen devised for measuring crack growth rates. Finally, applying Weibull parameters found in a single fiber test, where the fiber is under uniform stress, to debonded fibers in a cluster of other debonded fibers introduces difficulties, which necessitate simplifying assumptions, leading to inaccuracies. At best, therefore, the type of modeling in Ref. [6] provides a useful trend analysis such as the effect of the mode-II crack growth threshold on the fatigue limit, the effect of the shear strength of the fiber/matrix interface and the crack growth resistance on the slope of the fatigue life scatter band. Another recent work [8] also conducts modeling of the fatigue failure process in UD composites under cyclic axial tension starting with a broken fiber surrounded by matrix and accompanied by fiber/matrix debond of a certain length. The debond crack is assumed to grow as a mode-II crack following the Paris relationship. Additionally, in the debonding process, the role of interfacial frictional shear stress is considered. The fibers affected by the broken and debonded fiber are placed around this fiber in a hexagonal pattern, and the enhanced stress in these fibers is evaluated from the singular stress field at the debond crack tip. Failure of these fibers is then assessed assuming Weibull distribution for fiber strength. A novel feature of [8] is that the fiber breakage process is assumed to cease when none of the six fibers surrounding the broken and debonded fibers reaches its failure condition expressed as the probability of failure (1/6, i.e., one out of six failing). This provides a way to estimate the fatigue limit. With this model of fatigue limit, the effects of parameters, such as the fiber volume fraction (based on a hexagonal fiber pattern), as well as the fiber/matrix interfacial characteristics, expressed as frictional shear stress and fracture energy can be evaluated. These effects are shown in Fig. 10.10, reproduced from Ref. [8]. It is noted that there is no unique way to estimate the interfacial characteristics, and this introduces a source of the inherent uncertainty in modeling fatigue behavior, where the irreversibility underlying fatigue is attributed to the interface. In both models cited above, the role of the matrix in the process of stress transfer from the broken and debonded fiber to the surrounding fibers is not explicitly considered. It is known from observations and analysis that a debond crack can deviate into the matrix if favorable conditions exist for that to occur. A study [9] of failure in UD composites from a local fracture plane under monotonic axial tension looked at two scenarios: one, where a broken fiber exists from manufacturing and the other where a fiber fails at a weak point during loading. While in the first case the matrix crack formed at the broken fiber end was found to grow normal to the fiber axis, in the second case, the initial (assumed) debond crack kinks out in the matrix toward the neighboring fibers. The consequence of fiber failure progression, evaluated in probabilistic terms, was found to be higher when matrix cracking was considered than when it was ignored. It is expected that this will also be the case under cyclic loading.
298
Polymer Composites in the Aerospace Industry
Figure 10.10 Variation of the fatigue limit with (a) fiber volume fraction, (b) interfacial frictional shear stress, and (c) interfacial fracture energy [8]. The function A1 shown in the figures is the frictional prefactor [8] that depends on the R-factor, the frictional shear stress, and its critical value, as indicated.
Fatigue of polymer composites
299
Figure 10.10 cont’d.
10.4
Fatigue of laminates
In straight-fiber laminates subjected to in-plane cyclic loading, the individual plies will be subjected to biaxial cyclic stressing. In the material coordinates of a ply, the three stress components viz., the normal stress in the fiber direction s1, the normal stress transverse to the fiber direction s2, and the in-plane shear stress s12, will generally vary cyclically in various in-phase and out-of-phase combinations. It is obvious that testing and analysis of fatigue in all such combinations is a tremendous task. The approaches taken often in the literature are to resort to phenomenological criteria, commonly used for quasi-static failure, and predict fatigue life by simply generalizing to cyclic loading. A literature survey of such approaches was given in Ref. [10]. In Ref. [11], a critical assessment of common phenomenological models for fatigue life prediction was made, and using a large set of data from the literature, it was shown that such models are not reliable. Since then, unfortunately, more phenomenological models have appeared in the literature, and the reliability of life prediction by such models remains uncertain. In Ref. [11], a way forward to address fatigue behavior by the mechanisms path was laid out. Resorting to the FLD concept, the progressive mechanisms of fiber-bridged matrix cracking and interfiber matrix cracking (including fiber/matrix debonding) were identified as the dominant ply fatigue mechanisms. It was emphasized that through systematic studies of these mechanisms by experiments and observations, modeling strategies could develop. Several such studies have been conducted and models proposed for the interfiber matrix cracking under cyclic loading [12e16]. As the ply cracks approach the ply/ply interfaces, the diversion and deflection of these cracks take place, leading to delamination. The delamination cracks then grow under cyclic loading. In the initial stage of the delamination fatigue, the interaction with the
300
Polymer Composites in the Aerospace Industry
ply cracks is expected. This interaction needs to be studied before proper models for it can be developed. After sufficient delamination has occurred, the separated plies behave without the constraint from the previously bonded plies. In this phase of the fatigue process, the focus should shift to the most load-bearing ply (or set of plies), whose failure will cause the final failure of the laminate. Here, the FLD for UD composites, discussed above, will be a useful guide to modeling the final failure stage. The issue of the fatigue limit of a given laminate has not been addressed sufficiently so far. Generally speaking, there are two ways to approach the fatigue limit, one by asking at what applied load the progressive fatigue mechanism slows down to not fail the composite in a prescribed large number of cycles (106, 107, etc.), and second, to identify conditions for not initiating a progressive mechanism. The two approaches will produce upper and lower bounds to the fatigue limit, respectively. As an example, if transverse cracking in 90-degree plies is the first failure mechanism under axial tension of a cross-ply laminate, then its progression with load cycles is the key to determining the fatigue limit. Thus, if a model is set up for describing the cyclic rate of transverse crack density, then the fatigue limit will fall naturally from the model. This is then the first approach stated above, providing the upper bound of the fatigue limit. To find the lower limit, one would need to determine the conditions for the formation of a transverse crack. The upper bound approach was demonstrated in Ref. [17] for cross-ply laminates under cyclic axial tension.
10.5
Concluding remarks
This chapter has focused on the fundamentals of the fatigue failure process in composite materials. Without reviewing all works in fatigue, the emphasis has been placed on clarifying the roles of failure mechanisms, and for this purpose, a conceptual framework proposed by the author in 1981 called a fatigue life diagram (FLD) has been revisited. It is shown how easily the FLD reveals the effects of fiber stiffness and fiber failure strain on the fatigue behavior of UD composites. The trends induced in the fatigue life and the fatigue limit by the progressive fiber failure mechanism have been modeled recently, and these have been discussed to identify further improvements needed in the models. The implications of the FLD concept on understanding fatigue of laminates has then been discussed. An example of how this was utilized for axial tension fatigue of cross-ply laminates suggests a direction for addressing other laminates by mechanisms-based approaches.
References [1] R. Talreja, Fatigue of composite materials: damage mechanisms and fatigue life diagrams, Proc. R. Soc. Lond. A378 (1981) 461e475.
Fatigue of polymer composites
301
[2] E.K. Gamstedt, R. Talreja, Fatigue damage mechanisms in unidirectional carbon-fibrereinforced plastics, J. Mater. Sci. 34 (1999) 2535e2546. [3] C.K.H. Dharan, Fatigue failure in graphite fibre and glass fibre-polymer composites, J. Mater. Sci. 10 (1975) 1665e1670. [4] P.T. Curtis, An investigation of the tensile fatigue behaviour of improved carbon fibre composite materials, in: Proceedings of the Sixth International Conference on Composite Materials, ICCM-6, Imperial College, London, 1987, pp. 4.54e64. [5] E.K. Gamstedt, Effects of debonding and fiber strength distribution on fatigue-damage propagation in carbon fiber-reinforced epoxy, J. Appl. Polym. Sci. 76 (2000) 457e474. [6] M. Alves, S. Pimenta, A computationally-efficient micromechanical model for the fatigue life of unidirectional composites under tension-tension loading, Int. J. Fatigue 116 (2018) 677e690. [7] S. Pimenta, S.T. Pinho, Hierarchical scaling law for the strength of composite fibre bundles, J. Mech. Phys. Solids 61 (2013) 1337e1356. [8] B.F. Sørensen, S. Goutianos, Micromechanical model for prediction of the fatigue limit for unidirectional fibre composites, Mech. Mater. 131 (2019) 169e187. [9] L. Zhuang, R. Talreja, J. Varna, Tensile failure of unidirectional composites from a local fracture plane, Compos. Sci. Technol. 133 (2016) 119e127. [10] J. Degrieck, W. Van Paepegem, Fatigue damage modelling of fibre-reinforced composite materials: a review, Appl. Mech. Rev. 54 (2001) 279e300. [11] M. Quaresimin, L. Susmel, R. Talreja, Fatigue behaviour and life assessment of composite laminates under multiaxial loadings, Int. J. Fatigue 32 (2010) 2e16. [12] M. Quaresimin, P.A. Carraro, On the investigation of the biaxial fatigue behaviour of unidirectional composites, Compos. B Eng. 54 (2013) 200e208. [13] M. Quaresimin, P. Carraro, Damage initiation and evolution in glass/epoxy tubes subjected to combined tension-torsion fatigue loading, Int. J. Fatigue 63 (2014) 22e35. [14] M. Quaresimin, P.A. Carraro, L.P. Mikkelsen, N. Lucato, L. Vivian, P. Brønsted, B.F. Sørensen, J. Varna, R. Talreja, Damage evolution under cyclic multiaxial stress state: a comparative analysis between glass/epoxy laminates and tubes, Compos. B Eng. 61 (2014) 282e290. [15] (a) M. Quaresimin, P.A. Carraro, A damage based model for crack initiation in unidirectional composites under multiaxial cyclic loading, Compos. Sci. Technol. 99 (2014) 154e163; (b) M. Quaresimin, P.A. Carraro, L. Maragoni, Early stage damage in off-axis plies under fatigue loading, Compos. Sci. Technol. 128 (2016) 147e154. [16] J.A. Glud, P.A. Carraro, M. Quaresimin, J.M. Dulieu-Barton, O.T. Thomsen, L.C.T. Overgaard, A damage-based model for mixed-mode crack propagation in composite laminates, Compos. Appl. Sci. Manuf. 107 (2018) 421e431. [17] N.V. Akshantala, R. Talreja, A micromechanics based model for predicting fatigue life of composite laminates, Mater. Sci. Eng. A285 (2000) 303e313.
10
Ramesh Talreja Department of Aerospace Engineering, Department of Materials Science and Engineering, Texas A&M University, College Station, TX, United States
10.1
Introduction
The recent success of polymer composites in civil aviation is attributed not only to fuel savings by lightweight properties of these materials but also to their high fatigue resistance. The higher fatigue limit of unidirectional composites in cyclic axial tension compared to that of Al alloys, when measured in strain, has allowed higher design loads in composite structures. However, the fatigue performance of polymer composites in different loading combinations and different laminate configurations differ significantly. Optimal designs cannot, therefore, be achieved by arbitrarily setting the allowable strain levels to low values. The alternative of testing to find safe design values is not cost-effective. The mechanisms underlying fatigue failure of the constituents and their interfaces must, therefore, be understood, and this knowledge incorporated in predictive models. Addressing this problem is exacerbated by new developments in fibers and polymers and by a variety of manufacturing methods to construct composite structures. Beginning with the paper in 1981 [1], this author introduced a conceptual framework for the interpretation of the fatigue behavior of composites called a fatigue life diagram (FLD). In its baseline form, the FLD separates the fatigue failure in three regions based on the underlying dominant mechanisms. One region denoted as Region I, characterizes the fiber-failure dominant mechanism, where failure is heavily governed by the statistical nature of the fiber failure, and therefore, the resulting composite fatigue shows negligible dependence on load cycles. Region II is controlled by progressive mechanisms, such as irreversible deformation of the matrix and matrix crack growth, leading to damage accumulation culminating in a critical condition of ultimate failure. Finally, Region III is where failure mechanisms do not progress to criticality within a prespecified large number of load cycles because of local obstacles to growth imparted by fiber-architectural features and interfaces. The demarcation between Region II and Region III forms the fatigue limit. The FLD clarifies the relative significance of these regions for a given combination of fibers and matrix.
10.2
Construction of FLDs
Let us begin with Region I, which, as stated above, signifies the fiber failure dominated mechanism. Fig. 10.1 depicts three failure scenarios under a tensile load P. Fig.10.1(a) Polymer Composites in the Aerospace Industry. https://doi.org/10.1016/B978-0-08-102679-3.00010-1 Copyright © 2020 Elsevier Ltd. All rights reserved.
290
Polymer Composites in the Aerospace Industry
(a)
(b)
(c)
P
P
P
P
P
P
Figure 10.1 Fiber failure scenarios under a tensile load P. (a) A dry fiber bundle with the weakest fiber failure; (b) A fiber composite with the weakest fiber failure and debonding over a short length; (c) Composite failure from the linkage of failed fiber regions.
shows a dry fiber bundle in which the first fiber failure is indicated at the weakest point. At the instant of this fiber failure, the load is shared equally by the surviving fibers, and on increasing the load, the next failure occurs at the weakest point in these fibers. If, however, the load is not increased but is reduced to some value, say zero, and reapplied, no further fiber failures occur. Thus, there is no fatigue failure process. This assumes that fibers do not have a fatigue degradation mechanism. Consider now the scenario depicted in Fig. 10.1(b) where a unidirectional composite is loaded in axial tension and the first fiber has failed at a certain value of P. The failed fiber will likely also debond over a length depending on the interfacial characteristics. If the load P is reduced to zero and reapplied to this value, the question arises: will another fiber fail? To answer this question, we note first that on the failure of the first fiber, not all other fibers but a few fibers in close vicinity of the failed fiber are affected. This leads to what is known as local load sharing, the extent of which depends on the debonded length of the failed fiber. Thus, in the local load sharing, an affected fiber can fail if its strength is exceeded at some point over its affected length. If this occurs, it will occur instantaneously, if no time dependency of deformation exists in the fibers, matrix, or interfaces. To avoid unnecessary complication at this point, we assume that all deformation is time-independent. If the load P is reduced to zero and reapplied to the previous value, no further failure is possible unless an irreversible deformation has occurred. Such deformation will redistribute the stresses in the local load sharing region during the unloading and reloading process and create a new stress field, which can result in new fiber failure. Due to the statistical nature of fiber failures, however, there is no certainty that the next fiber failure will occur in the second or a
Fatigue of polymer composites
291
subsequent load cycle. Thus, a random process of fiber failure ensues. More significantly, the critical condition in which the failed fibers in the local load sharing region connect through the matrix and interface cracks, leading to an unstable growth of the connecting failure plane, occurs much more randomly. In fact, this condition of composite failure depicted in Fig. 10.1(c), is of such random nature that it may be rightly described as a chaotic process because of its high sensitivity to the initial condition, i.e., the first fiber failure. This also means that a well-defined progression in the fiber failure process that can be described by a rate equation does not exist. Therefore, the final failure event is for all purposes cycle-independent. Region I of an FLD corresponding to the fiber failure dominant mechanism is sketched in Fig. 10.2. The vertical axis of the FLD is the maximum strain εmax applied to the composite in the fiber direction. Although the load P is applied and reapplied, the first applied maximum strain makes a proper reference for the load level since it is the same irrespective of the fiber volume fraction, while fiber stress is not. Fig. 10.2 shows Region I as a scatter band centered around the mean strain to static (first load application) failure εc of the composite. The scatter band shown in the figure is for 5% and 95% probabilities of failure, as an example. The scatter band is shown flat (horizontal) to indicate that the initial (static) statistical variability does not change with the number of load cycles applied. As described and argued above, if on the first application of the tensile load the composite does not fail, but at least one fiber within the composite fails, then the composite can fail at any number of subsequent cycles. This will then lead to the scatter band of composite failure becoming independent of the number of cycles. It is possible, however, that the width of the scatter band changes (possibly reduces) with the number of cycles applied, but an analysis to determine that change is currently not feasible. Consider now the case where the first application of load (εmax) is below the scatter band depicted in Region I. The composite failure then will be less likely to be dominated by fiber failures. Instead, the irreversible deformation and crack formation mechanisms in the polymer matrix and at the fiber/matrix interfaces will gain more εmax
95% failure probability εC
Region I 5% failure probability
N
Figure 10.2 Region I of a fatigue life diagram.
292
Polymer Composites in the Aerospace Industry
prominence. These mechanisms were studied extensively and reported in Ref. [2] for a carbon/epoxy (AS4/8552) unidirectional composite loaded in cyclic tension at the load ratio (Pmin/Pmax) of R ¼ 0.1 and cyclic frequency of 10 Hz. Surface replicas were collected periodically at different load levels using cellulose acetate films. After replication, the films were hardened and coated with a layer of carbon by sputtering. They were then observed under a microscope. Fig. 10.3 shows a series of micrographs showing the progression of a crack on the specimen surface and Fig. 10.4 shows more clearly how a crack is bridged by fibers. The crack opening profile evident in Fig. 10.3 has the crack-tip region squeezed, which can be attributed to the closing pressure on crack surfaces induced by the bridging fibers. The study [2] also indicated the effect of fiber/matrix debonding on the fiberbridged matrix cracking as the operating progressive failure mechanism in Region II. The progression of this cracking mechanism is governed by fiber breakage, which is accelerated when a greater length of fibers is exposed to an enhanced stress by debonding, thereby increasing their propensity to failure. With sufficient debonding,
Fibre direction
Pristine
1 cycle
100 cycles
1,000 cycles 10 μ m
Figure 10.3 Micrographs of surface replicas showing a matrix crack at different applied load cycles [2].
Fatigue of polymer composites
293
Figure 10.4 Micrograph showing a matrix crack bridged by fibers [2].
the progressive failure mechanism is then essentially fiber breakage, whose unstable progression leads to the composite failure. Fig. 10.5 illustrates the appearance of Region II in the FLD. The random rate of fiber breakage leads to the scatter band in Region II, as indicated. Finally, Region III represents the cracking mechanisms that have such low driving forces that the energy barriers placed by fibers in the crack paths cannot be overcome to reach the critical state of unstable growth. Fig. 10.6 shows this region of the FLD with the insert in the figure illustrating a matrix crack arresting mechanism. The fatigue limit, below which Region III lies, is the limiting state of the progressive mechanisms in Region II. Fig. 10.7 is now the entire FLD for a unidirectional composite under axial fatigue where the three regions of the FLD (Figs. 10.2, 10.5 and 10.6) have been put together. The effect of increasing the stiffness of fibers is indicated in the FLD. This effect is a consequence of the bridging action of fibers running across a growing matrix crack (insert in Fig. 10.5) and the constraining effect on the matrix cracks by the surrounding fibers (insert in Fig. 10.6). Fig. 10.8 shows the actual fatigue life data with superimposed FLDs for a lower fiber stiffness case (glass/epoxy) and a higher fiber stiffness case (carbon/PEEK). As can εmax εC Region II
5% failure probability
95% failure probability
N
Figure 10.5 Region II of the fatigue life diagram.
294
Polymer Composites in the Aerospace Industry
max
C
Cracks Fibers
fl
Region III N 107
Figure 10.6 Region III of fatigue life diagram indicating the mechanism of matrix crack arrest by fibers. εmax
εC
Region I
Stiffer fibers Region II εfl Region III N
Figure 10.7 The baseline fatigue life diagram of unidirectional composites under cyclic tension.
be seen, the trends predicted by the conceptual FLD (Fig. 10.7) are confirmed by these data. It is also noteworthy that the glass/epoxy data for three different fiber volume fractions (Vf ¼ 0.16, 0.33 and 0.5) fall together within the same Region II scatter band. It is recalled that the fatigue testing was done conventionally, i.e., with load control, and the data when plotted on the stress basis (as SeN plots) fall as three separate curves, each corresponding to one fiber volume fraction. The fatigue limit, as shown in Fig. 10.8(a) is the fatigue limit reported by the authors [3] obtained by testing neat epoxy in strain control. The runouts (with arrows) shown in Fig. 10.8(a) suggest that the glass/epoxy composite fatigue limit is slightly higher, in accordance with the FLD in Fig. 10.7. For carbon/PEEK data in
Fatigue of polymer composites
295
Vf 0.50 0.33 0.16
(a)
εmax
0.024 εc 0.016
0.008 εm
2
0
6
4 log Nf
(b)
8
εmax (%) 1.5 εc 1.2 0.9 εm 0.6 0
1
2
3
4
5
6
7 log Nf
Figure 10.8 Fatigue life data and superimposed FLDs for a glass/epoxy composite (a) at three fiber volume fractions [3] and for a carbon-PEEK composite (b) [4].
Fig. 10.8(b) the same fatigue limit of epoxy (εm ¼ 0.6%) is also indicated for reference, as the PEEK fatigue limit has not been reported. It is expected that PEEK being tougher than epoxy will have a higher fatigue limit. In any case, the carbon/PEEK data shown in Fig. 10.8(b) show a fatigue limit higher than 0.6% strain. A comparison of carbon/PEEK and carbon/epoxy in Ref. [2] showed the fatigue limit values of 0.75% and 0.85%, respectively. The lower value of the fatigue limit for carbon/PEEK is attributed to more extensive debonding observed in this composite [2]. Again, this is in accordance with the expected trend predicted by the FLD (Fig. 10.7).
10.3
Modeling of FLD trends
Some of the qualitative trends in the fatigue behavior of unidirectional composites in cyclic tension emerge logically from the FLD, e.g., the direct dependence of Region I on the fiber strain to failure and the effect of fiber stiffness on Region II and Region III as indicated in Fig. 10.7. There have, however, been efforts to pin down the trends by quantitative modeling. Experimental evidence gathered by surface replicas
296
Polymer Composites in the Aerospace Industry
(Figs. 10.2 and 10.3) [2] indicated that the fiber-bridged cracks attributed to Region II (Fig. 10.2) tended to stop growing unless, as described above, a progressive fiber breakage occurred. It was found that the fiber/matrix debond length grew with fatigue cycles, exposing the increasing length of the debonded fiber to enhanced stress, as well as affecting the neighboring fibers. A shear-lag model was used to determine the stress transfer from the broken fiber to the neighboring fibers, and the Weibull distribution was assumed for the probability of fiber failure [5]. Although quantitative fatigue life prediction could not be made, the trends regarding how Region II would change with the cyclic rate of fiber/matrix debonding and the Weibull shape parameter were inferred [5]. An approach to fiber failure progression in fatigue was put forth recently [6] in which the fiber/matrix interface failure is represented by a cohesive zone model for static strength and the Paris relationship (power-law) for mode-II fatigue crack growth is assumed for the debond crack growth. The statistical fiber failure is given a Weibull distribution, and the fiber bundle failure is modeled by a hierarchical scaling method developed earlier [7]. While evaluating various material parameters entering the models is a challenge, with reasonably assumed values, the prediction of fatigue life in tension-tension loading by the approach seems to confirm the proposed FLD (Fig. 10.9). Strictly speaking, it is noted that Region I of the FLD is not the same as that in Fig. 10.7, where the idealized behavior does not assume cycle dependency. The authors [6] predict low dependency on cycles in this region because of low debond growth rate and high rate of fiber failures. The debond crack growth dictates the slope of Region II in their model. Finally, the mode-II fatigue crack growth threshold of debond cracks determines the fatigue limit. The uncertainty in model inputs and their effects on fatigue life prediction was discussed by the authors [6]. Assuming a cohesive zone for the static shear failure of the
1.6
Peak strain (%)
1.4
Model prediction I
1.2 1.0
II
0.8 III 0.6 100
101
102
103
104
105
106
Number of cycles
Figure 10.9 Model prediction by Ref. [6] compared to experimental data reported in Ref. [2]. The three regions of FLD in Fig. 10.6 have been superimposed on the data.
Fatigue of polymer composites
297
fiber/matrix interface has inherent uncertainty, as the material constants entering this model cannot be obtained independently. For debond growth, assuming a mode-II crack growth rate also induces uncertainty since the driving forces for a debond crack on the surface of a fiber embedded in a composite cannot be accurately reproduced in a test specimen devised for measuring crack growth rates. Finally, applying Weibull parameters found in a single fiber test, where the fiber is under uniform stress, to debonded fibers in a cluster of other debonded fibers introduces difficulties, which necessitate simplifying assumptions, leading to inaccuracies. At best, therefore, the type of modeling in Ref. [6] provides a useful trend analysis such as the effect of the mode-II crack growth threshold on the fatigue limit, the effect of the shear strength of the fiber/matrix interface and the crack growth resistance on the slope of the fatigue life scatter band. Another recent work [8] also conducts modeling of the fatigue failure process in UD composites under cyclic axial tension starting with a broken fiber surrounded by matrix and accompanied by fiber/matrix debond of a certain length. The debond crack is assumed to grow as a mode-II crack following the Paris relationship. Additionally, in the debonding process, the role of interfacial frictional shear stress is considered. The fibers affected by the broken and debonded fiber are placed around this fiber in a hexagonal pattern, and the enhanced stress in these fibers is evaluated from the singular stress field at the debond crack tip. Failure of these fibers is then assessed assuming Weibull distribution for fiber strength. A novel feature of [8] is that the fiber breakage process is assumed to cease when none of the six fibers surrounding the broken and debonded fibers reaches its failure condition expressed as the probability of failure (1/6, i.e., one out of six failing). This provides a way to estimate the fatigue limit. With this model of fatigue limit, the effects of parameters, such as the fiber volume fraction (based on a hexagonal fiber pattern), as well as the fiber/matrix interfacial characteristics, expressed as frictional shear stress and fracture energy can be evaluated. These effects are shown in Fig. 10.10, reproduced from Ref. [8]. It is noted that there is no unique way to estimate the interfacial characteristics, and this introduces a source of the inherent uncertainty in modeling fatigue behavior, where the irreversibility underlying fatigue is attributed to the interface. In both models cited above, the role of the matrix in the process of stress transfer from the broken and debonded fiber to the surrounding fibers is not explicitly considered. It is known from observations and analysis that a debond crack can deviate into the matrix if favorable conditions exist for that to occur. A study [9] of failure in UD composites from a local fracture plane under monotonic axial tension looked at two scenarios: one, where a broken fiber exists from manufacturing and the other where a fiber fails at a weak point during loading. While in the first case the matrix crack formed at the broken fiber end was found to grow normal to the fiber axis, in the second case, the initial (assumed) debond crack kinks out in the matrix toward the neighboring fibers. The consequence of fiber failure progression, evaluated in probabilistic terms, was found to be higher when matrix cracking was considered than when it was ignored. It is expected that this will also be the case under cyclic loading.
298
Polymer Composites in the Aerospace Industry
Figure 10.10 Variation of the fatigue limit with (a) fiber volume fraction, (b) interfacial frictional shear stress, and (c) interfacial fracture energy [8]. The function A1 shown in the figures is the frictional prefactor [8] that depends on the R-factor, the frictional shear stress, and its critical value, as indicated.
Fatigue of polymer composites
299
Figure 10.10 cont’d.
10.4
Fatigue of laminates
In straight-fiber laminates subjected to in-plane cyclic loading, the individual plies will be subjected to biaxial cyclic stressing. In the material coordinates of a ply, the three stress components viz., the normal stress in the fiber direction s1, the normal stress transverse to the fiber direction s2, and the in-plane shear stress s12, will generally vary cyclically in various in-phase and out-of-phase combinations. It is obvious that testing and analysis of fatigue in all such combinations is a tremendous task. The approaches taken often in the literature are to resort to phenomenological criteria, commonly used for quasi-static failure, and predict fatigue life by simply generalizing to cyclic loading. A literature survey of such approaches was given in Ref. [10]. In Ref. [11], a critical assessment of common phenomenological models for fatigue life prediction was made, and using a large set of data from the literature, it was shown that such models are not reliable. Since then, unfortunately, more phenomenological models have appeared in the literature, and the reliability of life prediction by such models remains uncertain. In Ref. [11], a way forward to address fatigue behavior by the mechanisms path was laid out. Resorting to the FLD concept, the progressive mechanisms of fiber-bridged matrix cracking and interfiber matrix cracking (including fiber/matrix debonding) were identified as the dominant ply fatigue mechanisms. It was emphasized that through systematic studies of these mechanisms by experiments and observations, modeling strategies could develop. Several such studies have been conducted and models proposed for the interfiber matrix cracking under cyclic loading [12e16]. As the ply cracks approach the ply/ply interfaces, the diversion and deflection of these cracks take place, leading to delamination. The delamination cracks then grow under cyclic loading. In the initial stage of the delamination fatigue, the interaction with the
300
Polymer Composites in the Aerospace Industry
ply cracks is expected. This interaction needs to be studied before proper models for it can be developed. After sufficient delamination has occurred, the separated plies behave without the constraint from the previously bonded plies. In this phase of the fatigue process, the focus should shift to the most load-bearing ply (or set of plies), whose failure will cause the final failure of the laminate. Here, the FLD for UD composites, discussed above, will be a useful guide to modeling the final failure stage. The issue of the fatigue limit of a given laminate has not been addressed sufficiently so far. Generally speaking, there are two ways to approach the fatigue limit, one by asking at what applied load the progressive fatigue mechanism slows down to not fail the composite in a prescribed large number of cycles (106, 107, etc.), and second, to identify conditions for not initiating a progressive mechanism. The two approaches will produce upper and lower bounds to the fatigue limit, respectively. As an example, if transverse cracking in 90-degree plies is the first failure mechanism under axial tension of a cross-ply laminate, then its progression with load cycles is the key to determining the fatigue limit. Thus, if a model is set up for describing the cyclic rate of transverse crack density, then the fatigue limit will fall naturally from the model. This is then the first approach stated above, providing the upper bound of the fatigue limit. To find the lower limit, one would need to determine the conditions for the formation of a transverse crack. The upper bound approach was demonstrated in Ref. [17] for cross-ply laminates under cyclic axial tension.
10.5
Concluding remarks
This chapter has focused on the fundamentals of the fatigue failure process in composite materials. Without reviewing all works in fatigue, the emphasis has been placed on clarifying the roles of failure mechanisms, and for this purpose, a conceptual framework proposed by the author in 1981 called a fatigue life diagram (FLD) has been revisited. It is shown how easily the FLD reveals the effects of fiber stiffness and fiber failure strain on the fatigue behavior of UD composites. The trends induced in the fatigue life and the fatigue limit by the progressive fiber failure mechanism have been modeled recently, and these have been discussed to identify further improvements needed in the models. The implications of the FLD concept on understanding fatigue of laminates has then been discussed. An example of how this was utilized for axial tension fatigue of cross-ply laminates suggests a direction for addressing other laminates by mechanisms-based approaches.
References [1] R. Talreja, Fatigue of composite materials: damage mechanisms and fatigue life diagrams, Proc. R. Soc. Lond. A378 (1981) 461e475.
Fatigue of polymer composites
301
[2] E.K. Gamstedt, R. Talreja, Fatigue damage mechanisms in unidirectional carbon-fibrereinforced plastics, J. Mater. Sci. 34 (1999) 2535e2546. [3] C.K.H. Dharan, Fatigue failure in graphite fibre and glass fibre-polymer composites, J. Mater. Sci. 10 (1975) 1665e1670. [4] P.T. Curtis, An investigation of the tensile fatigue behaviour of improved carbon fibre composite materials, in: Proceedings of the Sixth International Conference on Composite Materials, ICCM-6, Imperial College, London, 1987, pp. 4.54e64. [5] E.K. Gamstedt, Effects of debonding and fiber strength distribution on fatigue-damage propagation in carbon fiber-reinforced epoxy, J. Appl. Polym. Sci. 76 (2000) 457e474. [6] M. Alves, S. Pimenta, A computationally-efficient micromechanical model for the fatigue life of unidirectional composites under tension-tension loading, Int. J. Fatigue 116 (2018) 677e690. [7] S. Pimenta, S.T. Pinho, Hierarchical scaling law for the strength of composite fibre bundles, J. Mech. Phys. Solids 61 (2013) 1337e1356. [8] B.F. Sørensen, S. Goutianos, Micromechanical model for prediction of the fatigue limit for unidirectional fibre composites, Mech. Mater. 131 (2019) 169e187. [9] L. Zhuang, R. Talreja, J. Varna, Tensile failure of unidirectional composites from a local fracture plane, Compos. Sci. Technol. 133 (2016) 119e127. [10] J. Degrieck, W. Van Paepegem, Fatigue damage modelling of fibre-reinforced composite materials: a review, Appl. Mech. Rev. 54 (2001) 279e300. [11] M. Quaresimin, L. Susmel, R. Talreja, Fatigue behaviour and life assessment of composite laminates under multiaxial loadings, Int. J. Fatigue 32 (2010) 2e16. [12] M. Quaresimin, P.A. Carraro, On the investigation of the biaxial fatigue behaviour of unidirectional composites, Compos. B Eng. 54 (2013) 200e208. [13] M. Quaresimin, P. Carraro, Damage initiation and evolution in glass/epoxy tubes subjected to combined tension-torsion fatigue loading, Int. J. Fatigue 63 (2014) 22e35. [14] M. Quaresimin, P.A. Carraro, L.P. Mikkelsen, N. Lucato, L. Vivian, P. Brønsted, B.F. Sørensen, J. Varna, R. Talreja, Damage evolution under cyclic multiaxial stress state: a comparative analysis between glass/epoxy laminates and tubes, Compos. B Eng. 61 (2014) 282e290. [15] (a) M. Quaresimin, P.A. Carraro, A damage based model for crack initiation in unidirectional composites under multiaxial cyclic loading, Compos. Sci. Technol. 99 (2014) 154e163; (b) M. Quaresimin, P.A. Carraro, L. Maragoni, Early stage damage in off-axis plies under fatigue loading, Compos. Sci. Technol. 128 (2016) 147e154. [16] J.A. Glud, P.A. Carraro, M. Quaresimin, J.M. Dulieu-Barton, O.T. Thomsen, L.C.T. Overgaard, A damage-based model for mixed-mode crack propagation in composite laminates, Compos. Appl. Sci. Manuf. 107 (2018) 421e431. [17] N.V. Akshantala, R. Talreja, A micromechanics based model for predicting fatigue life of composite laminates, Mater. Sci. Eng. A285 (2000) 303e313.