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Biaxial fatigue crack propagation behavior of perfluorosulfonic-acid membranes
Journal of Power Sources 384 (2018) 58–65
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Biaxial fatigue crack propagation behavior of perfluorosulfonic-acid membranes
T
Qiang Lin, Shouwen Shi∗, Lei Wang, Xu Chen, Gang Chen∗∗ School of Chemical Engineering and Technology, Tianjin University, Tianjin, 300072, China
H I G H L I G H T S
G RA P H I C A L AB S T R A C T
biaxial effect on fatigue crack • In-plane propagation behavior is investigated. introduction of transverse stress • The retards fatigue crack growth. and curved cracks are ob• Branched served under biaxial loading condition.
crack growth paths are dis• Fatigue cussed from a mechanical perspective. after ex-situ fatigue test is si• Crack milar in geometry to that after in-situ test.
A R T I C L E I N F O
A B S T R A C T
Keywords: Nafion Biaxial Fatigue crack propagation PEM fuel cells Crack path
Perfluorosulfonic-acid membranes have long been used as the typical electrolyte for polymer-electrolyte fuel cells, which not only transport proton and water but also serve as barriers to prevent reactants mixing. However, too often the structural integrity of perfluorosulfonic-acid membranes is impaired by membrane thinning or cracks/pinholes formation induced by mechanical and chemical degradations. Despite the increasing number of studies that report crack formation, such as crack size and shape, the underlying mechanism and driving forces have not been well explored. In this paper, the fatigue crack propagation behaviors of Nafion membranes subjected to biaxial loading conditions have been investigated. In particular, the fatigue crack growth rates of flat cracks in responses to different loading conditions are compared, and the impact of transverse stress on fatigue crack growth rate is clarified. In addition, the crack paths for slant cracks under both uniaxial and biaxial loading conditions are discussed, which are similar in geometry to those found after accelerated stress testing of fuel cells. The directions of initial crack propagation are calculated theoretically and compared with experimental observations, which are in good agreement. The findings reported here lays the foundation for understanding of mechanical failure of membranes.
1. Introduction Perfluorosulfonic-acid (PFSA) membranes are the most widely used ion-conductive membranes in polymer electrolyte fuel cells (PEFCs) for their superior mechanical and electrochemical properties [1,2]. In PEFCs, electrodes are separated by the membrane that is used as a solid-
∗
state electrolyte for proton transport and an electrical insulator for electron insulation. In addition, the membrane also functions as a separator to prevent reactants mixing and a structural framework to support catalysts [3]. Therefore, the structural integrity of the membrane is of vital importance for maintaining high performance and durability of PEFCs. During fuel cell practical operation, the structural
Corresponding author. Corresponding author. E-mail addresses: [email protected] (S. Shi), [email protected] (G. Chen).
∗∗
https://doi.org/10.1016/j.jpowsour.2018.02.002 Received 14 November 2017; Received in revised form 31 January 2018; Accepted 2 February 2018 0378-7753/ © 2018 Elsevier B.V. All rights reserved.
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Fig. 1. Different biaxial loading paths: (a) represents uniaxial loading; (b)–(d) represent biaxial loading at different stress ratios where stresses of x and y directions are always the same; (e) represents 180° out-of-phase biaxial loading; (f)–(g) represent biaxial loading at biaxial stress ratios of 1.5 and 0.5, respectively; (h) represents constant stress of x-direction during biaxial loading; (i)–(j) represent in-phase and 180° out-of-phase biaxial loading of slant cracks.
mechanical behaviors under biaxial loading conditions before small cracks or defects were formed, i.e. the crack initiation period. However, such a period could not represent the whole lifetime of the proton exchange membrane in that more time was needed for cracks to propagate till failure. The fuel cells were found to run for additional 50% of the whole lifetime after the diagnostic of pinholes [27]. Such a phenomenon was ascribed to the presence of water that covered the surface of pinhole or cracks, preventing further crossover of reactants [19,27,28]. Till now, only the fatigue crack propagation behaviors of PFSA membrane under uniaxial loading conditions have been experimentally investigated [29]. In most cases, the fatigue crack propagation behaviors were studied through numerical simulations [30,31]. To have a better understanding of the fatigue crack propagation behaviors of PFSA membrane during fuel cell practical operation conditions, it is instructive to study the fatigue crack propagation behaviors under biaxial loading conditions. Hence, the objective of this paper is to investigate the impact of biaxial stress state on fatigue crack propagation behaviors, with the aim of finding the underlying mechanism of crack growth under practical operation conditions of fuel cells. For this reason, the fatigue crack growth under different biaxial loading conditions will be compared, and the fatigue crack growth paths will be discussed. The findings in this paper are expected to provide insight into the understanding of failure mechanism of fuel cell membranes, which will potentially help optimize and develop ion-conductive membranes of higher mechanical durability.
integrity of membrane is frequently impaired by combined chemical and mechanical degradation which manifest themselves as membrane thinning, creep, delamination, cracks, pinhole etc. [4]. Among them, cracks formation and development in the membrane are considered as the lifetime-limiting failure mode for fuel fell applications. Failure analysis is a common method to explore the possible failure mechanism of fuel cell during fuel cell durability research, where different techniques have been applied before, during and after fuel cell applications. Prior to fuel cell operation, cracks were found on the surfaces of membrane electrode assemblies which were thought to arise from manufacturing process and identified as the potential source for pinhole formation [5]. Defects in fuel cell catalyst layers could be detected during manufacturing process with the assist of infrared thermography as a quality control tool [6]. The same technique could also be applied on membrane defects detection during fuel cell operation [7–14]. During post-mortem analysis of failure membrane after fuel cell operation, scanning electron microscopy (SEM) might be the most widely used technique. Cracks of different lengths were found, ranging from several microns to hundreds of microns and even larger than 1 cm [15–19]. Additionally, cracks of different geometries and features were also found where both throughout and partially throughout cracks were observed in through-plane directions [8,16,20], and both branched and curved cracks were found in in-plane directions [8,17]. Regardless of the increasing number of studies concerning membrane failure modes and features, the underlying driving force and mechanism for pre-exist defect/crack to propagate is still unclear. During fuel cell operation, the catalyst coated membrane is sandwiched between two gas diffusion layers where the planar constraint keeps the membrane in a nearly in-plane strain state. Upon water sorption or desorption, the constraint prevents membrane from swelling or shrinking, thus generating in-plane biaxial compressive or tensile stresses [21]. Hence, compared to uniaxial loading conditions, the biaxial loading conditions represent more closely to the actual stress state of membrane during fuel cell practical operation. The equibiaxial stress state had been produced by pressurized blister method where the strength, fatigue, creep behaviors of proton exchange membranes [22–24] had been investigated. To bridge the gap between stress-strain responses and fatigue behaviors, the in-plane biaxial cyclic mechanical behavior was also studied using cruciform specimens [25], and an elastic-viscoplastic model was established to describe the biaxial behavior [26]. Such investigations were mainly concerned with the
2. Experiment 2.1. Materials and preparation In this study, Nafion® 212 membranes in protonated (H+) form from Dupont were used with the nominal thickness of 50 μm. Specimens were cut into cruciform shape using a sophisticated molding cutting die with the central gauge area dimension of 30 mm × 30 mm. Detailed description of cruciform specimen could be found in our previous work [25]. During biaxial fatigue crack propagation tests, center-cracked tension (CCT) specimens were used. The specimens were made by cutting notches at the middle of cruciform specimens using fresh razor blade to serve as stress concentration sites. Two kinds of notches were made: one is the notch that was either parallel or perpendicular to 59
Journal of Power Sources 384 (2018) 58–65
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J = Je + Jp
loading directions, while the other one was the slant notch that was 45° degree relative to each loading direction, as shown in Fig. 1 (a) and (i), respectively. Here, the stress paralleling to crack length was termed as σx and that being perpendicular to crack length was termed as σy, as shown in Fig. 1 (a). Biaxial experiments were conducted with biaxial cyclic testing system (IPBF-300, CARE Measurement & Control Co., Ltd.) [25,32]. The specimens were firstly cycled at low stress level to make pre-cracks from notches. The stress level was selected to make sure that the plastic zone ahead of the crack tip was not too large. In this study, the maximum stress at this stage was selected to be 4 MPa, and the stress ratio which represents the ratio between minimum and maximum stress in one loading direction (R=σmin/σmax) was chosen to be 0.1. During this process, the biaxial stress ratio λ, which represents the ratio between σx and σy at the same time (λ=σx/σy), was chosen to be 1. Crack length was calculated as the sum of notch length and pre-crack length, and was measured by a travelling microscope with a resolution of 2 μm. Afterward, the specimens were subjected to cyclic biaxial stress loading, and the number of cycles as well as corresponding crack length were recorded.
(1)
where Je and Jp are the elastic and plastic portion of J J-integral, respectively. For power-law materials, the J expression could be expanded in the general form [34]. n+1
J=
K2 ⎧ F 2 ⎛ n − 1 ⎞ ⎡ (σ∞/ σ0)2 ⎤ ⎫ σ + ασ0 ε0 ah1 ⎛ ∞ ⎞ 1+ 2 ⎥⎬ ⎢ + + E′ ⎨ C n 1 1 ( σ / σ ) ⎝ ⎠ 2 0 ∞ ⎝ σ0 ⎠ ⎣ ⎦⎭ ⎩ ⎜
⎟
(2)
where C2=2 and E’ = E for plane stress conditions. F represents the geometry factor in elastic K expression. The strain-hardening exponent n and the coefficient α, ε0, and σ0 are determined from uniaxial stressstrain curve with Ramberg-Osgood (power-law) constitutive relationship. The fully plastic factor, h1, is obtained from the Electric Power Research Institute (EPRI) handbooks. For in-plane biaxial tension loading conditions, the effects of multiaxiality are accounted for by changing h1 expression, where the biaxial solution h1′ is given by Refs. [34,35].
h1′ = h1 ( 1 − λ + λ2 )
n−1
(3)
For cyclic loading conditions, the estimate of ΔJ expression is similar to that of monotonic J expression but with single value of σ and ε replaced by their ranges, Δσ and Δε. Combining Eq. (3), then Eq. (2) is transformed into
2.2. Biaxial fatigue crack propagation tests The biaxial fatigue crack propagation tests consist of several loading paths as illustrated in Fig. 1. Fatigue crack propagation under uniaxial loading was firstly performed to set as the baseline for subsequent comparison, as shown in Fig. 1 (a). To investigate the effect of stress ratio, biaxial fatigue crack propagation tests were carried out at three different stress ratios, e.g. 0.1, 0.3 and 0.5, at the biaxial stress ratio of 1, as shown in Fig. 1 (b)-(d). Considering the fact that stresses in two directions does not always function at the same time, the effect of phase difference was studied by performing 180° out-of-phase biaxial fatigue crack propagation experiments where 180° phase difference exists between stresses of two directions (Fig. 1 (e)). In the above experiments, the maximum stress experienced at two loading directions are the same. However, there are cases where the maximum stresses at two loading directions are not the same. Hence, biaxial fatigue crack propagation experiments were carried out at different biaxial stress ratios, e.g. 0.5 and 1.5, as shown in Fig. 1 (f)-(g). As an extreme condition where the stresses paralleling to crack length might keep constant during fuel cell operation, the impact of constant stress on biaxial fatigue crack propagation was thus investigated by keeping σx as 6 MPa during experiments (Fig. 1 (h)). The flat cracks aforementioned are ideal assumptions where stress in both directions are either parallel or perpendicular to crack length. More common conditions are slant cracks where an angle (not 0°) exists between crack length and loading condition, as shown in Fig. 1 (i)-(j). In such cases, the biaxial fatigue crack propagation behaviors are different with flat cracks as the slant cracks are subjected to a mixed loading mode. Therefore, the biaxial fatigue crack propagation behaviors of slant cracks subjected to biaxial loadings were studied at different loading phase differences (0° and 180°). For comparison purpose, the fatigue crack propagation behavior under uniaxial loading was also investigated which is not shown in Fig. 1. For slant cracks subjected to biaxial loading, crack lengths were recorded as the crack projection lengths on two loading directions. Triangular waveform was used during all tests with the time period of 2.8 s for each cycle. All experiments were conducted at ambient room condition (25 °C/60% RH). The crack length and fatigue crack propagation rate were estimated using the 7-point incremental polynomial method as recommended by ASTM E647-11 [33].
n+1
ΔJ =
ΔK 2 ⎧ F 2 ⎛ n − 1 ⎞ ⎡ (σ∞/ σ0)2 ⎤ ⎫ Δσ + 4ασ0 ε0 ah1′ ⎛ ∞ ⎞ 1+ 2 ⎥⎬ ⎢ + + E′ ⎨ C n 1 1 ( σ / σ ) 2σ0 ⎠ ⎝ ⎠ 2 0 ∞ ⎝ ⎣ ⎦ ⎭ ⎩ ⎜
⎟
(4) 3. Results and discussion 3.1. Biaxial fatigue crack growth rate The effect of stress ratio on biaxial fatigue crack grow rate is investigated as shown in Fig. 2. At the same ΔJ, the biaxial fatigue crack growth rate is the smallest when R value equals 0.1 and increases with
2.3. Determination of ΔJ-integral The J-integral is used as a parameter to describe elastic-plastic fatigue crack growth, which could be estimated by general engineering approach as:
Fig. 2. Biaxial fatigue crack growth rate at different stress ratios. The fatigue crack growth rate under uniaxial loading condition (R = 0.3) is shown here for comparison.
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the increase of R, similar with the stress ratio dependence during uniaxial loading conditions [36]. It is interesting to note that after increasing R value to 0.3, further increasing R value to 0.5 does not cause any increase in biaxial fatigue crack growth rate, implying that the biaxial fatigue crack growth rate is not sensitive to stress ratio at this stage. A power law relationship exists between fatigue crack growth rate da/dN and ΔJ, as follows:
da/ dN = C (ΔJ )m
(5)
where C and m are material constants. The parameter C obtained for membranes tested at R = 0.1, 0.3 and 0.5 are 3.45, 4.05 and 5.26, respectively. To illustrate the impact of stress of the transverse direction (σx) on fatigue crack growth rate, the fatigue crack growth rate under uniaxial loading condition at the stress ratio of 0.3 is also shown for comparison. In general, the fatigue crack growth rate under uniaxial loading conditions is larger than that under biaxial loading conditions, indicating that the presence of stress paralleling to crack length serves to hinder crack growth. Such a finding implies that the fatigue crack growth rate reported before under uniaxial loading conditions might be non-conservative, especially when considering the actual stress state during practical operation of fuel cells. Due to the inhomogeneous water distribution and water transportation kinetics, different locations in the membrane might experience different humidity cyclings, and it is much likely that a phase difference of resultant stress between different locations exists, particularly between different loading directions. The influence of phase difference between two loading directions is also studied as shown in Fig. 3. The phase differences of 0° (in-phase) and 180° (out-of-phase) are investigated and compared. It is found that fatigue crack growth rate of 180° phase difference is larger than that of 0° difference, and this discrepancy decreases with the increase of ΔJ. Such an observation is consistent with the observation for metals where the fatigue crack growth rate is found to be the largest for 180° phase difference and decreases with the decrease of phase difference [37]. It is important to note that for biaxial loading with 180° phase difference, the biaxial stress ratio is not maintained constant through one cycle even though the maximum stress at two directions are the same, which might
Fig. 4. Comparison of biaxial fatigue crack growth rates at constant and cycling transverse stress. Inset shows the enlarged view where the line in the inset figure is to guide for eyes.
explain the different biaxial fatigue growth rates. The dependence of fatigue crack growth rate on phase difference might change with the applied stress level, where the fatigue crack growth rate was found to increase with phase difference at low applied stress level while the opposite was true at higher applied stress level [36]. Nevertheless, the fatigue crack growth rate of biaxial 180° phase difference is still lower than that of uniaxial loading. To investigate how biaxial fatigue crack growth rate changes at constant transverse stress, the fatigue crack growth rates at constant and cycling transverse stress are compared, as shown in Fig. 4. The difference of fatigue crack growth rate between these two conditions is not significant, and the inset figure shows the enlarged view. Even though the difference is minor, the fatigue crack growth rate at the constant transverse stress is still larger than that at cycling transverse stress. The constant transverse stress could be seen as an extreme of the cycling stress where the amplitude is zero and the mean stress is the maximum stress. Hence, the slight increase in fatigue crack growth rate is consistent with our observation in Fig. 2 where the biaxial fatigue crack growth rate does not increase with stress ratio after the stress ratio reaches 0.3. For constant transverse stress, the stress ratio is not constant but rather larger than 0.3. As the fatigue crack growth rate is not sensitive to stress ratio at this stage, the fatigue crack growth rate at constant maximum transverse stress shows no big difference with that at cycling transverse stress. Aside from the phase difference induced by inhomogeneous water distribution, stress difference also exists as a result of water content difference. Therefore, the effect of biaxial stress ratio on biaxial fatigue crack propagation rate is investigated, as shown in Fig. 5. Fatigue crack growth rates at biaxial stress ratio of 0.5, 1 and 1.5 are compared. To serve as a baseline, the fatigue crack growth rate at uniaxial loading which corresponding to biaxial stress ratio of 0 is also compared. Increasing biaxial stress ratio from 0 to 0.5 significantly reduces the fatigue crack growth rate, which is in agreement with the finding in Fig. 2 that the presence of transverse stress acts to retard fatigue crack growth. Further increasing biaxial stress ratio from 0.5 to 1 and 1.5
Fig. 3. Comparison of biaxial fatigue crack growth rate at different phase differences.
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Fig. 6. Comparison of biaxial fatigue crack growth rates of slant crack at different phases.
KI = KII =
σ πa {(1 + λ ) + (1 2 σ πa {(1 − λ )sin 2 2
− λ )cos 2 β } β}
(6)
For flat cracks where β is 0, KII reduces to 0, the cracks are only subjected to mode I growth and the resultant crack paths are straight, which is not discussed here. However, for slant cracks, KI and KII are no longer 0 except for λ = 1, and the cracks are subjected to a mixed mode growth. At this time, the fatigue crack propagation paths are no longer straight any more. The fatigue crack propagation paths for slant cracks under biaxial and uniaxial loading conditions are shown in Fig. 7. As the precrack is generated by subjecting notch to in-plane biaxial loading condition where λ = 1, the precrack path is thus straight. As soon as the biaxial loading with 180° phase difference is applied, the crack starts to deviate from straight path. More importantly, the original single crack splits into two cracks when the phase difference biaxial stress is applied and grows without further branching. The direction of initial crack propagation path with respect to precrack is measured to be 40°. It should be noted that this angle only represents the angle between the initial stage of crack and precrack, and the region of the initial stage is relatively small in comparison with the overall crack length due to the continuous changing of crack path directions during fatigue crack growth. However, the region is still visible with closer inspection. The theoretical value of initial crack propagation θ in mixed fracture mode (I and II) has been derived by Erdogan and Sih [38]:
Fig. 5. Comparison of biaxial fatigue crack growth rates at different biaxial stress ratios.
does not cause significant increase in biaxial fatigue crack growth, indicating that the biaxial fatigue crack growth rate is not sensitive to biaxial stress ratio at this regime. It seems that the retardation effect of transverse stress on fatigue crack growth rate reaches a plateau after the biaxial stress ratio reaches 0.5. The insensitivity of fatigue crack growth rate on stress ratio (Fig. 2) accounts for the insensitivity of fatigue crack growth rate on mean stress, as the amplitude difference at different stress ratios are already accounted for by ΔJ. The insensitive of fatigue crack growth rate in this case also reflects the insensitivity of fatigue crack growth rate on mean stress. Hence, combining the results from Figs. 2–5, it is not hard to find out that even though discussing about different factors, the underlying controlling mechanism is the same. The investigations mentioned above are all dealing with ideal conditions where the crack is either parallel or normal to loading directions. However, in most cases during fuel cell practical operation, the orientation of crack relative to biaxial loading directions is random. It is most likely that an angle exists between crack and two loading directions. Thus, the biaxial fatigue growth rates of slant cracks might characterize more accurately the fatigue crack behavior during fuel cell practical operation conditions. The biaxial fatigue crack growth rates for slant cracks at different phases are compared in Fig. 6. As it is complicated and almost impossible to calculate ΔJ in this case, only the half crack length is compared as a function of the number of cycles. Different from the phase dependence of biaxial fatigue crack growth rate in flat cracks, almost no difference in fatigue crack growth rate is observed at different phase differences. Repeated experiments were also conducted (not shown here) to verify the reproducibility of experimental results. In addition, the fatigue crack growth rate under biaxial loading conditions is lower than that under uniaxial loading, similar with the observation for flat cracks.
⎛1 − θ = 2 arctan ⎜ ⎜ ⎜ ⎝
1+8 K 4 KII I
KII 2 KI
( )
⎞ ⎟ ⎟ ⎟ ⎠
(7)
The ratio between KII and KI is given by Ref. [39].
σy − σx KII = KI σy + σx
(8)
The theoretical initial crack propagation values during the loading history calculated using the above equations are shown in Fig. 7 (b). The maximum value in a cycle equals to +41.8° and the minimum value equals to −41.8°, which means that the crack could either propagate in the direction of +41.8° with respect to the direction of precrack, or in the direction of −41.8°. Such an explanation well illustrates the branching behavior of precrack and the symmetry of two branching cracks. With regard to direction of initial crack propagation, the measured value is close to the theoretical value. Following initial crack branching and propagation, the stress states of the two branching crack are different from the initial ones. It was reported in the literature that the angle of crack propagation during subsequent cycles had only one
3.2. Crack paths of slant cracks under biaxial loading For a sheet with an inclined crack subjected to in-plane biaxial stress fields, the crack tip elastic stress intensity factors KI and KII are given by
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Fig. 7. Crack propagation paths for slant cracks under biaxial loading and uniaxial loading conditions: (a) shows the crack path of slant crack under biaxial loading with 180° phase difference and (b) is the corresponding theoretical value of direction of initial crack propagation during a cycle; (c) shows the crack path for slant crack under uniaxial loading conditions and (d) is the corresponding theoretical value of direction of initial crack propagation during a cycle.
precracks. The slant crack subjected to biaxial loading with 180° phase difference exhibits branched crack, while that subjected to uniaxial loading exhibits bent crack. Singh et al. [17] examined the crack geometry of crack in membranes after accelerated stress testing of fuel cells through non-invasive and non-destructive 3D X-ray computed tomography. The obtained results is shown in Fig. 8 (c) where I-shaped and Y-shaped cracks were found after subjecting membranes to combined mechanical and chemical degradation. By comparison, it is interesting to note that the branched crack (Fig. 8 (a)) formed in this study is similar in geometry to the Y-shaped crack reported by Singh et al., while the bent crack (Fig. 8 (b)) is similar to the I-shaped crack. In this sense, our findings provides a mechanical foundation which serves as an explanation for crack formation and growth mechanism. This would be helpful for further understanding of crack formation in membranes in
maximum absolute value [39], explaining the no further branching behavior of the cracks. For slant crack subjected to uniaxial loading, the fatigue crack path is shown in Fig. 7 (c) where the direction of initial crack propagation is found to be 54°. Different from the branching behavior of precrack under biaxial loading, the precrack in this case shows no branching behavior. Using the above equation, the theoretical value of initial crack propagation equals to −53.1° and keeps constant during one cycle, in well agreement with the observation results, which also explains the no branching behavior of precrack. 3.3. Comparison between in-situ and ex-situ fatigue crack growth To have a full picture of fatigue crack propagation paths, Fig. 8 schematically illustrates the overall paths including initial notch and
Fig. 8. Schematic illustration of crack propagation path under (a) biaxial loading with 180° phase difference conditions and (b) uniaxial loading conditions, inspired by Kitagawa et al. [40]; (c) shows the two type of cracks (Y-shaped and I-shaped) found in membranes after accelerated stress testing of fuel cells, reprinted from Ref. [17], Copyright (2017), with permission from Elsevier.
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and I-shaped cracks reported by other researchers. The mechanism of crack branching was revealed through theoretical calculation, and the theoretical values of direction of initial crack propagation were in good agreement with experimental observations. In conclusion, this study demonstrated how crack paths evolved in response to different external loading conditions and the underlying mechanisms, which could potentially benefit the understanding of membrane failure mechanism and optimization of fuel cell.
that most reports in the literature only discussed the crack geometry [4,15,17,19,20], such as size and shape, while the underlying mechanism on how the crack propagated is left unsolved. If the membrane is loaded only under uniaxial loading conditions, it is impossible to form branched cracks, whether for flat or slant initial cracks. However, the above explanation might be one of the possible reasons. In a subsequent study, Singh et al. [8] analyzed cracks on membranes subjected to pure mechanical and pure chemical degradations so as to uncouple the factors that affect membrane failures. It was found that the Y-shaped cracks did not appear after pure mechanical degradation, but instead only present under the combination of chemical and mechanical degradation. After pure mechanical degradation, only the I-shaped cracks were found on membrane surfaces. They attributed such findings to the embrittlement induced by chemical stressors. Indeed, chemical degradation attacks the main chain and side chains, causing loss in macromolecular entanglements and resulting in ductile to brittle transition [41,42]. The embrittlement of membranes makes it easier for crack propagation due to the reduction in toughness. From our findings, the phenomena mentioned above could be explained from another perspective. As seen from Fig. 8, the fatigue crack growth rate under uniaxial loading conditions is much higher than that under biaxial loading conditions. Thus, under the same stress level, it is harder for cracks to develop under biaxial loadings. As a result, cracks under uniaxial loading condition grow into larger and discernable cracks, while those under biaxial loading condition might scarcely to grow. This condition could be alleviated with the embrittlement of membrane where it is easier for cracks to grow under both uniaxial and biaxial loading conditions. The presence of branching cracks or Y-shaped cracks in our study might be ascribed to the large crack size or high stress level used, which imposes stronger impetus for cracks to propagate. Even though the potential factors affecting crack paths have been well explored by Singh et al. [8,17], the underlying driving forces for cracks branching and growing are identified in this paper. The questions as to why crack branching is observed for cracks, how flat or slant cracks respond to external uniaxial or biaxial loading conditions, and what are the fatigue crack propagation rates under those conditions have been clearly answered and clarified. Thus the findings in our paper could be helpful for understanding crack growth mechanism and even membrane mechanical failure mechanism, which could aid the design of new membrane materials and optimization of fuel cells. For instance, anisotropic materials with preferential mechanical orientation, such as Nafion XL membrane [43], could be used and oriented at a certain direction during fuel cell assembling according to the possible crack path obtained in this study, thus increasing the resistance for cracks to develop and propagate at certain directions.
Acknowledgements The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No.11602164, No.11172202). References [1] A. Kusoglu, A.Z. Weber, New insights into perfluorinated sulfonic-acid ionomers, Chem. Rev. 117 (2017) 987–1104. [2] K.A. Mauritz, R.B. Moore, State of understanding of nafion, Chem. Rev. 104 (2004) 4535–4585. [3] H. Zhang, P.K. Shen, Recent development of polymer electrolyte membranes for fuel cells, Chem. Rev. 112 (2012) 2780–2832. [4] C.S. Gittleman, F.D. Coms, Y.-H. Lai, Membrane durability: physical and chemical degradation, in: M. Mench, E.C. Kumbur, T.N. Veziroglu (Eds.), Polymer Electrolyte Fuel Cell Degradation, Elsevier Inc., 2012, pp. 15–88. [5] S. Kundu, M.W. Fowler, L.C. Simon, S. Grot, Morphological features (defects) in fuel cell membrane electrode assemblies, J. Power Sources 157 (2006) 650–656. [6] N.V. Aieta, P.K. Das, A. Perdue, G. Bender, A.M. Herring, A.Z. Weber, M.J. Ulsh, Applying infrared thermography as a quality-control tool for the rapid detection of polymer-electrolyte-membrane-fuel-cell catalyst-layer-thickness variations, J. Power Sources 211 (2012) 4–11. [7] G. Bender, W. Felt, M. Ulsh, Detecting and localizing failure points in proton exchange membrane fuel cells using IR thermography, J. Power Sources 253 (2014) 224–229. [8] Y. Singh, F.P. Orfino, M. Dutta, E. Kjeang, 3D failure analysis of pure mechanical and pure chemical degradation in fuel cell membranes, J. Electrochem. Soc. 164 (2017) F1331–F1341. [9] F. Nandjou, J.P. Poirot-Crouvezier, M. Chandesris, J.F. Blachot, C. Bonnaud, Y. Bultel, Impact of heat and water management on proton exchange membrane fuel cells degradation in automotive application, J. Power Sources 326 (2016) 182–192. [10] G. De Moor, C. Bas, N. Charvin, J. Dillet, G. Maranzana, O. Lottin, N. Caqué, E. Rossinot, L. Flandin, Perfluorosulfonic acid membrane degradation in the hydrogen inlet region: a macroscopic approach, Int. J. Hydrogen Energy 41 (2016) 483–496. [11] Y.-H. Lai, G.W. Fly, In-situ diagnostics and degradation mapping of a mixed-mode accelerated stress test for proton exchange membranes, J. Power Sources 274 (2015) 1162–1172. [12] A.S. Alavijeh, R.M.H. Khorasany, Z. Nunn, A. Habisch, M. Lauritzen, E. Rogers, G.G. Wang, E. Kjeang, Microstructural and mechanical characterization of catalyst coated membranes subjected to in situ hygrothermal fatigue, J. Electrochem. Soc. 162 (2015) F1461–F1469. [13] X.-Z. Yuan, S. Zhang, S. Ban, C. Huang, H. Wang, V. Singara, M. Fowler, M. Schulze, A. Haug, K. Andreas Friedrich, R. Hiesgen, Degradation of a PEM fuel cell stack with Nafion® membranes of different thicknesses. Part II: ex situ diagnosis, J. Power Sources 205 (2012) 324–334. [14] S. Vengatesan, K. Panha, M.W. Fowler, X.-Z. Yuan, H. Wang, Membrane electrode assembly degradation under idle conditions via unsymmetrical reactant relative humidity cycling, J. Power Sources 207 (2012) 101–110. [15] G. De Moor, C. Bas, N. Charvin, E. Moukheiber, F. Niepceron, N. Breilly, J. André, E. Rossinot, E. Claude, N.D. Albérola, L. Flandin, Understanding membrane failure in PEMFC: comparison of diagnostic tools at different observation scales, Fuel Cell. 12 (2012) 356–364. [16] K. Patankar, D.A. Dillard, S.W. Case, M.W. Ellis, Y. Li, Y.-H. Lai, M.K. Budinski, C.S. Gittleman, Characterizing fracture energy of proton exchange membranes using a knife slit test, J. Polym. Sci., Part B: Polym. Phys. 48 (2010) 333–343. [17] Y. Singh, F.P. Orfino, M. Dutta, E. Kjeang, 3D visualization of membrane failures in fuel cells, J. Power Sources 345 (2017) 1–11. [18] S. Kreitmeier, P. Lerch, A. Wokaun, F.N. Büchi, Local degradation at membrane defects in polymer electrolyte fuel cells, J. Electrochem. Soc. 160 (2013) F456–F463. [19] S. Kreitmeier, M. Michiardi, A. Wokaun, F.N. Büchi, Factors determining the gas crossover through pinholes in polymer electrolyte fuel cell membranes, Electrochim. Acta 80 (2012) 240–247. [20] S. Kreitmeier, G.A. Schuler, A. Wokaun, F.N. Büchi, Investigation of membrane degradation in polymer electrolyte fuel cells using local gas permeation analysis, J. Power Sources 212 (2012) 139–147. [21] A. Kusoglu, A.M. Karlsson, M.H. Santare, S. Cleghorn, W.B. Johnson, Mechanical response of fuel cell membranes subjected to a hygro-thermal cycle, J. Power
4. Conclusions In this study, the fatigue crack propagation behaviors of flat and slant cracks subjected to both uniaxial and in-plane biaxial loading conditions were investigated. The fatigue crack growth rates under different conditions were compared, and the crack paths were recorded and analyzed. It was found that the presence of transverse stress paralleling to crack acted to reduce fatigue crack growth rate. For flat cracks, the effects of stress ratio, phase difference, constant transverse stress and biaxial stress ratio on biaxial fatigue crack growth rate have been explored. Stress ratio and phase difference increased biaxial fatigue crack growth rate, whereas the fatigue crack growth rate was rarely influenced by the constant transverse stress and biaxial stress ratio. Such phenomena were ascribed to the fact that the biaxial fatigue crack growth rate was less sensitive after the transverse stress increased to a certain extent. For slant crack, the crack paths were compared and analyzed where crack branching was found under biaxial loading with 180° phase difference and crack curving was observed under uniaxial loading. The crack paths observed here were similar to the Y-shaped 64
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[33] ASTM E647–11, Standard Test Method for Measurement of Fatigue Crack Growth Rates, (2011) (West Conshohocken, PA). [34] R.C. McClung, G.G. Chell, D.A. Russell, G.E. Orient, A practical methodology for elastic-plastic fatigue crack growth, in: R.S. Piascik, J.C. Newman, N.E. Dowling (Eds.), Fatigue and Fracture Mechanics: 27th Volume, ASTM International, 1997, pp. 317–337. [35] X. Wang, Fully plastic J-integral solutions for surface cracked plates under biaxial loading, Eng. Fract. Mech. 73 (2006) 1581–1595. [36] P. Dai, M. Feng, Z. Li, Effects of the transverse stress on biaxial fatigue crack growth predicted by plasticity-corrected stress intensity factor, Int. J. Fatig. 61 (2014) 101–106. [37] K. Gotoh, T. Niwa, Y. Anai, Numerical simulation of fatigue crack propagation under biaxial tensile loadings with phase differences, Mar. Struct. 42 (2015) 53–70. [38] F. Erdogan, G.C. Sih, On the crack extension in plates under plane loading and transverse shear, J. Basic Engi. 85 (1963) 519–525. [39] S. Mall, V.Y. Perel, Crack growth behavior under biaxial fatigue with phase difference, Int. J. Fatig. 74 (2015) 166–172. [40] H. Kitagawa, R. Yuuki, K. Tohgo, M. Tanabe, ΔK-dependency of fatigue growth of single and mixed mode cracks under biaxial stress, in: K.J. Miller, M.W. Brown (Eds.), Multiaxial Fatigue, American Society for Testing and Materials, Philadelphia, 1985, pp. 164–183. [41] A. Sadeghi Alavijeh, M.A. Goulet, R.M.H. Khorasany, J. Ghataurah, C. Lim, M. Lauritzen, E. Kjeang, G.G. Wang, R.K.N.D. Rajapakse, Decay in mechanical properties of catalyst coated membranes subjected to combined chemical and mechanical membrane degradation, Fuel Cell. 15 (2015) 204–213. [42] C. Lim, L. Ghassemzadeh, F. Van Hove, M. Lauritzen, J. Kolodziej, G.G. Wang, S. Holdcroft, E. Kjeang, Membrane degradation during combined chemical and mechanical accelerated stress testing of polymer electrolyte fuel cells, J. Power Sources 257 (2014) 102–110. [43] S. Shi, A.Z. Weber, A. Kusoglu, Structure/property relationship of Nafion XL composite membranes, J. Membr. Sci. 516 (2016) 123–134.
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Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
Biaxial fatigue crack propagation behavior of perfluorosulfonic-acid membranes
T
Qiang Lin, Shouwen Shi∗, Lei Wang, Xu Chen, Gang Chen∗∗ School of Chemical Engineering and Technology, Tianjin University, Tianjin, 300072, China
H I G H L I G H T S
G RA P H I C A L AB S T R A C T
biaxial effect on fatigue crack • In-plane propagation behavior is investigated. introduction of transverse stress • The retards fatigue crack growth. and curved cracks are ob• Branched served under biaxial loading condition.
crack growth paths are dis• Fatigue cussed from a mechanical perspective. after ex-situ fatigue test is si• Crack milar in geometry to that after in-situ test.
A R T I C L E I N F O
A B S T R A C T
Keywords: Nafion Biaxial Fatigue crack propagation PEM fuel cells Crack path
Perfluorosulfonic-acid membranes have long been used as the typical electrolyte for polymer-electrolyte fuel cells, which not only transport proton and water but also serve as barriers to prevent reactants mixing. However, too often the structural integrity of perfluorosulfonic-acid membranes is impaired by membrane thinning or cracks/pinholes formation induced by mechanical and chemical degradations. Despite the increasing number of studies that report crack formation, such as crack size and shape, the underlying mechanism and driving forces have not been well explored. In this paper, the fatigue crack propagation behaviors of Nafion membranes subjected to biaxial loading conditions have been investigated. In particular, the fatigue crack growth rates of flat cracks in responses to different loading conditions are compared, and the impact of transverse stress on fatigue crack growth rate is clarified. In addition, the crack paths for slant cracks under both uniaxial and biaxial loading conditions are discussed, which are similar in geometry to those found after accelerated stress testing of fuel cells. The directions of initial crack propagation are calculated theoretically and compared with experimental observations, which are in good agreement. The findings reported here lays the foundation for understanding of mechanical failure of membranes.
1. Introduction Perfluorosulfonic-acid (PFSA) membranes are the most widely used ion-conductive membranes in polymer electrolyte fuel cells (PEFCs) for their superior mechanical and electrochemical properties [1,2]. In PEFCs, electrodes are separated by the membrane that is used as a solid-
∗
state electrolyte for proton transport and an electrical insulator for electron insulation. In addition, the membrane also functions as a separator to prevent reactants mixing and a structural framework to support catalysts [3]. Therefore, the structural integrity of the membrane is of vital importance for maintaining high performance and durability of PEFCs. During fuel cell practical operation, the structural
Corresponding author. Corresponding author. E-mail addresses: [email protected] (S. Shi), [email protected] (G. Chen).
∗∗
https://doi.org/10.1016/j.jpowsour.2018.02.002 Received 14 November 2017; Received in revised form 31 January 2018; Accepted 2 February 2018 0378-7753/ © 2018 Elsevier B.V. All rights reserved.
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Fig. 1. Different biaxial loading paths: (a) represents uniaxial loading; (b)–(d) represent biaxial loading at different stress ratios where stresses of x and y directions are always the same; (e) represents 180° out-of-phase biaxial loading; (f)–(g) represent biaxial loading at biaxial stress ratios of 1.5 and 0.5, respectively; (h) represents constant stress of x-direction during biaxial loading; (i)–(j) represent in-phase and 180° out-of-phase biaxial loading of slant cracks.
mechanical behaviors under biaxial loading conditions before small cracks or defects were formed, i.e. the crack initiation period. However, such a period could not represent the whole lifetime of the proton exchange membrane in that more time was needed for cracks to propagate till failure. The fuel cells were found to run for additional 50% of the whole lifetime after the diagnostic of pinholes [27]. Such a phenomenon was ascribed to the presence of water that covered the surface of pinhole or cracks, preventing further crossover of reactants [19,27,28]. Till now, only the fatigue crack propagation behaviors of PFSA membrane under uniaxial loading conditions have been experimentally investigated [29]. In most cases, the fatigue crack propagation behaviors were studied through numerical simulations [30,31]. To have a better understanding of the fatigue crack propagation behaviors of PFSA membrane during fuel cell practical operation conditions, it is instructive to study the fatigue crack propagation behaviors under biaxial loading conditions. Hence, the objective of this paper is to investigate the impact of biaxial stress state on fatigue crack propagation behaviors, with the aim of finding the underlying mechanism of crack growth under practical operation conditions of fuel cells. For this reason, the fatigue crack growth under different biaxial loading conditions will be compared, and the fatigue crack growth paths will be discussed. The findings in this paper are expected to provide insight into the understanding of failure mechanism of fuel cell membranes, which will potentially help optimize and develop ion-conductive membranes of higher mechanical durability.
integrity of membrane is frequently impaired by combined chemical and mechanical degradation which manifest themselves as membrane thinning, creep, delamination, cracks, pinhole etc. [4]. Among them, cracks formation and development in the membrane are considered as the lifetime-limiting failure mode for fuel fell applications. Failure analysis is a common method to explore the possible failure mechanism of fuel cell during fuel cell durability research, where different techniques have been applied before, during and after fuel cell applications. Prior to fuel cell operation, cracks were found on the surfaces of membrane electrode assemblies which were thought to arise from manufacturing process and identified as the potential source for pinhole formation [5]. Defects in fuel cell catalyst layers could be detected during manufacturing process with the assist of infrared thermography as a quality control tool [6]. The same technique could also be applied on membrane defects detection during fuel cell operation [7–14]. During post-mortem analysis of failure membrane after fuel cell operation, scanning electron microscopy (SEM) might be the most widely used technique. Cracks of different lengths were found, ranging from several microns to hundreds of microns and even larger than 1 cm [15–19]. Additionally, cracks of different geometries and features were also found where both throughout and partially throughout cracks were observed in through-plane directions [8,16,20], and both branched and curved cracks were found in in-plane directions [8,17]. Regardless of the increasing number of studies concerning membrane failure modes and features, the underlying driving force and mechanism for pre-exist defect/crack to propagate is still unclear. During fuel cell operation, the catalyst coated membrane is sandwiched between two gas diffusion layers where the planar constraint keeps the membrane in a nearly in-plane strain state. Upon water sorption or desorption, the constraint prevents membrane from swelling or shrinking, thus generating in-plane biaxial compressive or tensile stresses [21]. Hence, compared to uniaxial loading conditions, the biaxial loading conditions represent more closely to the actual stress state of membrane during fuel cell practical operation. The equibiaxial stress state had been produced by pressurized blister method where the strength, fatigue, creep behaviors of proton exchange membranes [22–24] had been investigated. To bridge the gap between stress-strain responses and fatigue behaviors, the in-plane biaxial cyclic mechanical behavior was also studied using cruciform specimens [25], and an elastic-viscoplastic model was established to describe the biaxial behavior [26]. Such investigations were mainly concerned with the
2. Experiment 2.1. Materials and preparation In this study, Nafion® 212 membranes in protonated (H+) form from Dupont were used with the nominal thickness of 50 μm. Specimens were cut into cruciform shape using a sophisticated molding cutting die with the central gauge area dimension of 30 mm × 30 mm. Detailed description of cruciform specimen could be found in our previous work [25]. During biaxial fatigue crack propagation tests, center-cracked tension (CCT) specimens were used. The specimens were made by cutting notches at the middle of cruciform specimens using fresh razor blade to serve as stress concentration sites. Two kinds of notches were made: one is the notch that was either parallel or perpendicular to 59
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J = Je + Jp
loading directions, while the other one was the slant notch that was 45° degree relative to each loading direction, as shown in Fig. 1 (a) and (i), respectively. Here, the stress paralleling to crack length was termed as σx and that being perpendicular to crack length was termed as σy, as shown in Fig. 1 (a). Biaxial experiments were conducted with biaxial cyclic testing system (IPBF-300, CARE Measurement & Control Co., Ltd.) [25,32]. The specimens were firstly cycled at low stress level to make pre-cracks from notches. The stress level was selected to make sure that the plastic zone ahead of the crack tip was not too large. In this study, the maximum stress at this stage was selected to be 4 MPa, and the stress ratio which represents the ratio between minimum and maximum stress in one loading direction (R=σmin/σmax) was chosen to be 0.1. During this process, the biaxial stress ratio λ, which represents the ratio between σx and σy at the same time (λ=σx/σy), was chosen to be 1. Crack length was calculated as the sum of notch length and pre-crack length, and was measured by a travelling microscope with a resolution of 2 μm. Afterward, the specimens were subjected to cyclic biaxial stress loading, and the number of cycles as well as corresponding crack length were recorded.
(1)
where Je and Jp are the elastic and plastic portion of J J-integral, respectively. For power-law materials, the J expression could be expanded in the general form [34]. n+1
J=
K2 ⎧ F 2 ⎛ n − 1 ⎞ ⎡ (σ∞/ σ0)2 ⎤ ⎫ σ + ασ0 ε0 ah1 ⎛ ∞ ⎞ 1+ 2 ⎥⎬ ⎢ + + E′ ⎨ C n 1 1 ( σ / σ ) ⎝ ⎠ 2 0 ∞ ⎝ σ0 ⎠ ⎣ ⎦⎭ ⎩ ⎜
⎟
(2)
where C2=2 and E’ = E for plane stress conditions. F represents the geometry factor in elastic K expression. The strain-hardening exponent n and the coefficient α, ε0, and σ0 are determined from uniaxial stressstrain curve with Ramberg-Osgood (power-law) constitutive relationship. The fully plastic factor, h1, is obtained from the Electric Power Research Institute (EPRI) handbooks. For in-plane biaxial tension loading conditions, the effects of multiaxiality are accounted for by changing h1 expression, where the biaxial solution h1′ is given by Refs. [34,35].
h1′ = h1 ( 1 − λ + λ2 )
n−1
(3)
For cyclic loading conditions, the estimate of ΔJ expression is similar to that of monotonic J expression but with single value of σ and ε replaced by their ranges, Δσ and Δε. Combining Eq. (3), then Eq. (2) is transformed into
2.2. Biaxial fatigue crack propagation tests The biaxial fatigue crack propagation tests consist of several loading paths as illustrated in Fig. 1. Fatigue crack propagation under uniaxial loading was firstly performed to set as the baseline for subsequent comparison, as shown in Fig. 1 (a). To investigate the effect of stress ratio, biaxial fatigue crack propagation tests were carried out at three different stress ratios, e.g. 0.1, 0.3 and 0.5, at the biaxial stress ratio of 1, as shown in Fig. 1 (b)-(d). Considering the fact that stresses in two directions does not always function at the same time, the effect of phase difference was studied by performing 180° out-of-phase biaxial fatigue crack propagation experiments where 180° phase difference exists between stresses of two directions (Fig. 1 (e)). In the above experiments, the maximum stress experienced at two loading directions are the same. However, there are cases where the maximum stresses at two loading directions are not the same. Hence, biaxial fatigue crack propagation experiments were carried out at different biaxial stress ratios, e.g. 0.5 and 1.5, as shown in Fig. 1 (f)-(g). As an extreme condition where the stresses paralleling to crack length might keep constant during fuel cell operation, the impact of constant stress on biaxial fatigue crack propagation was thus investigated by keeping σx as 6 MPa during experiments (Fig. 1 (h)). The flat cracks aforementioned are ideal assumptions where stress in both directions are either parallel or perpendicular to crack length. More common conditions are slant cracks where an angle (not 0°) exists between crack length and loading condition, as shown in Fig. 1 (i)-(j). In such cases, the biaxial fatigue crack propagation behaviors are different with flat cracks as the slant cracks are subjected to a mixed loading mode. Therefore, the biaxial fatigue crack propagation behaviors of slant cracks subjected to biaxial loadings were studied at different loading phase differences (0° and 180°). For comparison purpose, the fatigue crack propagation behavior under uniaxial loading was also investigated which is not shown in Fig. 1. For slant cracks subjected to biaxial loading, crack lengths were recorded as the crack projection lengths on two loading directions. Triangular waveform was used during all tests with the time period of 2.8 s for each cycle. All experiments were conducted at ambient room condition (25 °C/60% RH). The crack length and fatigue crack propagation rate were estimated using the 7-point incremental polynomial method as recommended by ASTM E647-11 [33].
n+1
ΔJ =
ΔK 2 ⎧ F 2 ⎛ n − 1 ⎞ ⎡ (σ∞/ σ0)2 ⎤ ⎫ Δσ + 4ασ0 ε0 ah1′ ⎛ ∞ ⎞ 1+ 2 ⎥⎬ ⎢ + + E′ ⎨ C n 1 1 ( σ / σ ) 2σ0 ⎠ ⎝ ⎠ 2 0 ∞ ⎝ ⎣ ⎦ ⎭ ⎩ ⎜
⎟
(4) 3. Results and discussion 3.1. Biaxial fatigue crack growth rate The effect of stress ratio on biaxial fatigue crack grow rate is investigated as shown in Fig. 2. At the same ΔJ, the biaxial fatigue crack growth rate is the smallest when R value equals 0.1 and increases with
2.3. Determination of ΔJ-integral The J-integral is used as a parameter to describe elastic-plastic fatigue crack growth, which could be estimated by general engineering approach as:
Fig. 2. Biaxial fatigue crack growth rate at different stress ratios. The fatigue crack growth rate under uniaxial loading condition (R = 0.3) is shown here for comparison.
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the increase of R, similar with the stress ratio dependence during uniaxial loading conditions [36]. It is interesting to note that after increasing R value to 0.3, further increasing R value to 0.5 does not cause any increase in biaxial fatigue crack growth rate, implying that the biaxial fatigue crack growth rate is not sensitive to stress ratio at this stage. A power law relationship exists between fatigue crack growth rate da/dN and ΔJ, as follows:
da/ dN = C (ΔJ )m
(5)
where C and m are material constants. The parameter C obtained for membranes tested at R = 0.1, 0.3 and 0.5 are 3.45, 4.05 and 5.26, respectively. To illustrate the impact of stress of the transverse direction (σx) on fatigue crack growth rate, the fatigue crack growth rate under uniaxial loading condition at the stress ratio of 0.3 is also shown for comparison. In general, the fatigue crack growth rate under uniaxial loading conditions is larger than that under biaxial loading conditions, indicating that the presence of stress paralleling to crack length serves to hinder crack growth. Such a finding implies that the fatigue crack growth rate reported before under uniaxial loading conditions might be non-conservative, especially when considering the actual stress state during practical operation of fuel cells. Due to the inhomogeneous water distribution and water transportation kinetics, different locations in the membrane might experience different humidity cyclings, and it is much likely that a phase difference of resultant stress between different locations exists, particularly between different loading directions. The influence of phase difference between two loading directions is also studied as shown in Fig. 3. The phase differences of 0° (in-phase) and 180° (out-of-phase) are investigated and compared. It is found that fatigue crack growth rate of 180° phase difference is larger than that of 0° difference, and this discrepancy decreases with the increase of ΔJ. Such an observation is consistent with the observation for metals where the fatigue crack growth rate is found to be the largest for 180° phase difference and decreases with the decrease of phase difference [37]. It is important to note that for biaxial loading with 180° phase difference, the biaxial stress ratio is not maintained constant through one cycle even though the maximum stress at two directions are the same, which might
Fig. 4. Comparison of biaxial fatigue crack growth rates at constant and cycling transverse stress. Inset shows the enlarged view where the line in the inset figure is to guide for eyes.
explain the different biaxial fatigue growth rates. The dependence of fatigue crack growth rate on phase difference might change with the applied stress level, where the fatigue crack growth rate was found to increase with phase difference at low applied stress level while the opposite was true at higher applied stress level [36]. Nevertheless, the fatigue crack growth rate of biaxial 180° phase difference is still lower than that of uniaxial loading. To investigate how biaxial fatigue crack growth rate changes at constant transverse stress, the fatigue crack growth rates at constant and cycling transverse stress are compared, as shown in Fig. 4. The difference of fatigue crack growth rate between these two conditions is not significant, and the inset figure shows the enlarged view. Even though the difference is minor, the fatigue crack growth rate at the constant transverse stress is still larger than that at cycling transverse stress. The constant transverse stress could be seen as an extreme of the cycling stress where the amplitude is zero and the mean stress is the maximum stress. Hence, the slight increase in fatigue crack growth rate is consistent with our observation in Fig. 2 where the biaxial fatigue crack growth rate does not increase with stress ratio after the stress ratio reaches 0.3. For constant transverse stress, the stress ratio is not constant but rather larger than 0.3. As the fatigue crack growth rate is not sensitive to stress ratio at this stage, the fatigue crack growth rate at constant maximum transverse stress shows no big difference with that at cycling transverse stress. Aside from the phase difference induced by inhomogeneous water distribution, stress difference also exists as a result of water content difference. Therefore, the effect of biaxial stress ratio on biaxial fatigue crack propagation rate is investigated, as shown in Fig. 5. Fatigue crack growth rates at biaxial stress ratio of 0.5, 1 and 1.5 are compared. To serve as a baseline, the fatigue crack growth rate at uniaxial loading which corresponding to biaxial stress ratio of 0 is also compared. Increasing biaxial stress ratio from 0 to 0.5 significantly reduces the fatigue crack growth rate, which is in agreement with the finding in Fig. 2 that the presence of transverse stress acts to retard fatigue crack growth. Further increasing biaxial stress ratio from 0.5 to 1 and 1.5
Fig. 3. Comparison of biaxial fatigue crack growth rate at different phase differences.
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Fig. 6. Comparison of biaxial fatigue crack growth rates of slant crack at different phases.
KI = KII =
σ πa {(1 + λ ) + (1 2 σ πa {(1 − λ )sin 2 2
− λ )cos 2 β } β}
(6)
For flat cracks where β is 0, KII reduces to 0, the cracks are only subjected to mode I growth and the resultant crack paths are straight, which is not discussed here. However, for slant cracks, KI and KII are no longer 0 except for λ = 1, and the cracks are subjected to a mixed mode growth. At this time, the fatigue crack propagation paths are no longer straight any more. The fatigue crack propagation paths for slant cracks under biaxial and uniaxial loading conditions are shown in Fig. 7. As the precrack is generated by subjecting notch to in-plane biaxial loading condition where λ = 1, the precrack path is thus straight. As soon as the biaxial loading with 180° phase difference is applied, the crack starts to deviate from straight path. More importantly, the original single crack splits into two cracks when the phase difference biaxial stress is applied and grows without further branching. The direction of initial crack propagation path with respect to precrack is measured to be 40°. It should be noted that this angle only represents the angle between the initial stage of crack and precrack, and the region of the initial stage is relatively small in comparison with the overall crack length due to the continuous changing of crack path directions during fatigue crack growth. However, the region is still visible with closer inspection. The theoretical value of initial crack propagation θ in mixed fracture mode (I and II) has been derived by Erdogan and Sih [38]:
Fig. 5. Comparison of biaxial fatigue crack growth rates at different biaxial stress ratios.
does not cause significant increase in biaxial fatigue crack growth, indicating that the biaxial fatigue crack growth rate is not sensitive to biaxial stress ratio at this regime. It seems that the retardation effect of transverse stress on fatigue crack growth rate reaches a plateau after the biaxial stress ratio reaches 0.5. The insensitivity of fatigue crack growth rate on stress ratio (Fig. 2) accounts for the insensitivity of fatigue crack growth rate on mean stress, as the amplitude difference at different stress ratios are already accounted for by ΔJ. The insensitive of fatigue crack growth rate in this case also reflects the insensitivity of fatigue crack growth rate on mean stress. Hence, combining the results from Figs. 2–5, it is not hard to find out that even though discussing about different factors, the underlying controlling mechanism is the same. The investigations mentioned above are all dealing with ideal conditions where the crack is either parallel or normal to loading directions. However, in most cases during fuel cell practical operation, the orientation of crack relative to biaxial loading directions is random. It is most likely that an angle exists between crack and two loading directions. Thus, the biaxial fatigue growth rates of slant cracks might characterize more accurately the fatigue crack behavior during fuel cell practical operation conditions. The biaxial fatigue crack growth rates for slant cracks at different phases are compared in Fig. 6. As it is complicated and almost impossible to calculate ΔJ in this case, only the half crack length is compared as a function of the number of cycles. Different from the phase dependence of biaxial fatigue crack growth rate in flat cracks, almost no difference in fatigue crack growth rate is observed at different phase differences. Repeated experiments were also conducted (not shown here) to verify the reproducibility of experimental results. In addition, the fatigue crack growth rate under biaxial loading conditions is lower than that under uniaxial loading, similar with the observation for flat cracks.
⎛1 − θ = 2 arctan ⎜ ⎜ ⎜ ⎝
1+8 K 4 KII I
KII 2 KI
( )
⎞ ⎟ ⎟ ⎟ ⎠
(7)
The ratio between KII and KI is given by Ref. [39].
σy − σx KII = KI σy + σx
(8)
The theoretical initial crack propagation values during the loading history calculated using the above equations are shown in Fig. 7 (b). The maximum value in a cycle equals to +41.8° and the minimum value equals to −41.8°, which means that the crack could either propagate in the direction of +41.8° with respect to the direction of precrack, or in the direction of −41.8°. Such an explanation well illustrates the branching behavior of precrack and the symmetry of two branching cracks. With regard to direction of initial crack propagation, the measured value is close to the theoretical value. Following initial crack branching and propagation, the stress states of the two branching crack are different from the initial ones. It was reported in the literature that the angle of crack propagation during subsequent cycles had only one
3.2. Crack paths of slant cracks under biaxial loading For a sheet with an inclined crack subjected to in-plane biaxial stress fields, the crack tip elastic stress intensity factors KI and KII are given by
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Fig. 7. Crack propagation paths for slant cracks under biaxial loading and uniaxial loading conditions: (a) shows the crack path of slant crack under biaxial loading with 180° phase difference and (b) is the corresponding theoretical value of direction of initial crack propagation during a cycle; (c) shows the crack path for slant crack under uniaxial loading conditions and (d) is the corresponding theoretical value of direction of initial crack propagation during a cycle.
precracks. The slant crack subjected to biaxial loading with 180° phase difference exhibits branched crack, while that subjected to uniaxial loading exhibits bent crack. Singh et al. [17] examined the crack geometry of crack in membranes after accelerated stress testing of fuel cells through non-invasive and non-destructive 3D X-ray computed tomography. The obtained results is shown in Fig. 8 (c) where I-shaped and Y-shaped cracks were found after subjecting membranes to combined mechanical and chemical degradation. By comparison, it is interesting to note that the branched crack (Fig. 8 (a)) formed in this study is similar in geometry to the Y-shaped crack reported by Singh et al., while the bent crack (Fig. 8 (b)) is similar to the I-shaped crack. In this sense, our findings provides a mechanical foundation which serves as an explanation for crack formation and growth mechanism. This would be helpful for further understanding of crack formation in membranes in
maximum absolute value [39], explaining the no further branching behavior of the cracks. For slant crack subjected to uniaxial loading, the fatigue crack path is shown in Fig. 7 (c) where the direction of initial crack propagation is found to be 54°. Different from the branching behavior of precrack under biaxial loading, the precrack in this case shows no branching behavior. Using the above equation, the theoretical value of initial crack propagation equals to −53.1° and keeps constant during one cycle, in well agreement with the observation results, which also explains the no branching behavior of precrack. 3.3. Comparison between in-situ and ex-situ fatigue crack growth To have a full picture of fatigue crack propagation paths, Fig. 8 schematically illustrates the overall paths including initial notch and
Fig. 8. Schematic illustration of crack propagation path under (a) biaxial loading with 180° phase difference conditions and (b) uniaxial loading conditions, inspired by Kitagawa et al. [40]; (c) shows the two type of cracks (Y-shaped and I-shaped) found in membranes after accelerated stress testing of fuel cells, reprinted from Ref. [17], Copyright (2017), with permission from Elsevier.
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and I-shaped cracks reported by other researchers. The mechanism of crack branching was revealed through theoretical calculation, and the theoretical values of direction of initial crack propagation were in good agreement with experimental observations. In conclusion, this study demonstrated how crack paths evolved in response to different external loading conditions and the underlying mechanisms, which could potentially benefit the understanding of membrane failure mechanism and optimization of fuel cell.
that most reports in the literature only discussed the crack geometry [4,15,17,19,20], such as size and shape, while the underlying mechanism on how the crack propagated is left unsolved. If the membrane is loaded only under uniaxial loading conditions, it is impossible to form branched cracks, whether for flat or slant initial cracks. However, the above explanation might be one of the possible reasons. In a subsequent study, Singh et al. [8] analyzed cracks on membranes subjected to pure mechanical and pure chemical degradations so as to uncouple the factors that affect membrane failures. It was found that the Y-shaped cracks did not appear after pure mechanical degradation, but instead only present under the combination of chemical and mechanical degradation. After pure mechanical degradation, only the I-shaped cracks were found on membrane surfaces. They attributed such findings to the embrittlement induced by chemical stressors. Indeed, chemical degradation attacks the main chain and side chains, causing loss in macromolecular entanglements and resulting in ductile to brittle transition [41,42]. The embrittlement of membranes makes it easier for crack propagation due to the reduction in toughness. From our findings, the phenomena mentioned above could be explained from another perspective. As seen from Fig. 8, the fatigue crack growth rate under uniaxial loading conditions is much higher than that under biaxial loading conditions. Thus, under the same stress level, it is harder for cracks to develop under biaxial loadings. As a result, cracks under uniaxial loading condition grow into larger and discernable cracks, while those under biaxial loading condition might scarcely to grow. This condition could be alleviated with the embrittlement of membrane where it is easier for cracks to grow under both uniaxial and biaxial loading conditions. The presence of branching cracks or Y-shaped cracks in our study might be ascribed to the large crack size or high stress level used, which imposes stronger impetus for cracks to propagate. Even though the potential factors affecting crack paths have been well explored by Singh et al. [8,17], the underlying driving forces for cracks branching and growing are identified in this paper. The questions as to why crack branching is observed for cracks, how flat or slant cracks respond to external uniaxial or biaxial loading conditions, and what are the fatigue crack propagation rates under those conditions have been clearly answered and clarified. Thus the findings in our paper could be helpful for understanding crack growth mechanism and even membrane mechanical failure mechanism, which could aid the design of new membrane materials and optimization of fuel cells. For instance, anisotropic materials with preferential mechanical orientation, such as Nafion XL membrane [43], could be used and oriented at a certain direction during fuel cell assembling according to the possible crack path obtained in this study, thus increasing the resistance for cracks to develop and propagate at certain directions.
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4. Conclusions In this study, the fatigue crack propagation behaviors of flat and slant cracks subjected to both uniaxial and in-plane biaxial loading conditions were investigated. The fatigue crack growth rates under different conditions were compared, and the crack paths were recorded and analyzed. It was found that the presence of transverse stress paralleling to crack acted to reduce fatigue crack growth rate. For flat cracks, the effects of stress ratio, phase difference, constant transverse stress and biaxial stress ratio on biaxial fatigue crack growth rate have been explored. Stress ratio and phase difference increased biaxial fatigue crack growth rate, whereas the fatigue crack growth rate was rarely influenced by the constant transverse stress and biaxial stress ratio. Such phenomena were ascribed to the fact that the biaxial fatigue crack growth rate was less sensitive after the transverse stress increased to a certain extent. For slant crack, the crack paths were compared and analyzed where crack branching was found under biaxial loading with 180° phase difference and crack curving was observed under uniaxial loading. The crack paths observed here were similar to the Y-shaped 64
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