The relation between the strain energy release in fatigue and quasi-static crack growth

The relation between the strain energy release in fatigue and quasi-static crack growth

Engineering Fracture Mechanics 145 (2015) 86–97 Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsev...

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Engineering Fracture Mechanics 145 (2015) 86–97

Contents lists available at ScienceDirect

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

The relation between the strain energy release in fatigue and quasi-static crack growth Lucas Amaral ⇑, Liaojun Yao, René Alderliesten, Rinze Benedictus Structural Integrity & Composites Group, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands

a r t i c l e

i n f o

Article history: Received 25 March 2015 Received in revised form 8 July 2015 Accepted 10 July 2015 Available online 18 July 2015 Keywords: Strain Energy Release Rate Delamination Fatigue Quasi-static

a b s t r a c t This work proposes to use an average Strain Energy Release Rate (SERR) to characterise similarly fatigue and quasi-static delamination growth. Mode I quasi-static and fatigue tests were performed. The quasi-static crack extension was considered as a low-cycle fatigue process, discretized to different levels and correlated to the fatigue data. Fracture surfaces were analysed and damage mechanisms were related to average SERRs for each case. The strain energy released in crack extension showed to be dependent on the decohesion mechanisms, and it is demonstrated how the values of the SERR for fatigue and quasi-static loading can be linked through physical principles. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The number of applications of Carbon Fibre Reinforced Polymers (CFRP) in aerospace structures is increasing significantly because of the necessity of lighter and, at the same time, more damage tolerant and durable structures. CFRP are attractive for aerospace applications because of their high specific strength and stiffness [1]. However, their application in primary structures is limited by the poor interlaminar strength [2], which causes delamination to be the most frequently observed damage mode in CFRP structures [3]. Therefore, several studies have been conducted to assess delamination growth in composite structures [1–11]. The appropriate similitude parameter that should be used for the assessment of fatigue delamination is still under discussion [12]. Some authors use the maximum Strain Energy Release Rate (SERR) Gmax to characterise delamination under p p p fatigue loading [13], while others prefer the SERR range DG = Gmax–Gmin [14] or even D G = ( Gmax– Gmin)2 as a parameter that describes the similitude [10]. Amongst these propositions, some authors propose obtaining an actual SERR from measured data only, and not from a theoretical model. The procedure consists in measuring, during a fatigue test, the crack length a, the displacement d, the force P and the number of cycles N. With these data it is possible to obtain a graph plotting da/dN versus dU/dN. In this presentation of the data, the SERR dU/dA is obtained from the inverse of the slope of the curve, defined by Eq. (1), where b is the width of the specimen. It is notable that this procedure is based on an energy balance, and it accounts for the stress ratio in its definition, often collapsing fatigue curves for different stress ratios [15–18].

G ¼

1 dU=dN dU ¼ b da=dN dA

⇑ Corresponding author. E-mail address: [email protected] (L. Amaral). http://dx.doi.org/10.1016/j.engfracmech.2015.07.018 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.

ð1Þ

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Nomenclature A a b G Gmax Gmin G⁄ N P U

delamination area (m2) delamination length (m) width (m) Strain Energy Release Rate (J/m2) Strain Energy Release Rate at maximum fatigue load (J/m2) Strain Energy Release Rate at minimum fatigue load (J/m2) average Strain Energy Release Rate over the cycle (J/m2) number of cycles force (N) strain energy (J)

Greek symbols DG Strain Energy Release Rate range (J/m2) d displacement (m) Subscripts crit critical max maximum min minimum on onset

In general, fatigue and quasi-static delamination growth are evaluated with different methods. For quasi-static delamination growth, the SERR is calculated just before the crack propagates [19]. This value is generally referred to as the onset value Gon. Meanwhile, fatigue delamination is usually assessed through the relation of a SERR based parameter (Gmax, DG or p D G) with da/dN, or via delamination resistance curves [4–11,20]. Although several studies have performed both quasi-static and fatigue tests [2,21–25], a clear relation between what is done for both loading conditions does not seem to be available. Moreover, although the energy balance introduced by Griffith [26] proposed a release of strain energy per unit area of crack independently of the load, the SERR parameter that is used nowadays to assess crack extension seems to be regarded as dependent on the load type. 2. Problem statement The question that arises is where do these quasi-static and fatigue SERR definitions meet? These different approaches to assess fatigue and quasi-static delamination complicate the establishment of a correlation between the energy released by crack growth in both loading conditions. Moreover, some authors [7] normalise the SERR used to characterise fatigue delamination data with a critical SERR calculated from quasi-static tests. This seems to imply that there is a straightforward correlation between the crack growth resistance in fatigue and in quasi-static loading. However, the exact nature of such correlation has not been established yet. Thus, the questions that need to be answered are: what are the differences in the energy dissipation in delamination growth in quasi-static and fatigue loading, and to what mechanisms should such differences be attributed? For that reason it is proposed here to analyse the quasi-static data with the same procedure as proposed in [16], described by Eq. (1). Assessing both quasi-static and fatigue delamination data with the same procedure may shed light on how these parameters of similitude may correlate. Thus, the objective of this study is to correlate quasi-static and fatigue loading using identical energy balance principles. To this end, the difference in the energy released in both fatigue and quasi-static loading conditions is characterised and related to fracture surfaces observed with microscopy. 3. Hypotheses 3.1. Analysing quasi-static data as low-cycle fatigue data A schematic load–displacement curve is shown in Fig. 1(a) for a typical mode I quasi-static test performed on a CFRP double cantilever beam (DCB) specimen in displacement controlled conditions, according to ASTM D5528-01 standard [19]. In this illustration, Point 1 represents the conditions just before the test starts. When the applied force P is increased and reaches a critical value, Point 2, crack growth occurs, which under displacement control condition causes a decrease in

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Fig. 1. (a) Illustration of a quasi-static force–displacement curve and (b) determination of dU in a quasi-static test increment.

the applied force. The system is thus taken to Point 3, because the machine keeps imposing a displacement upon the test specimen and the crack propagation is not perfectly continuous. Subsequently, there is additional work applied to the specimen with the test machine, which increases the applied force P once more. In the illustration this is represented by moving from Point 3 to Point 4. This incremental decrease in force by crack extension and increase in force by application of additional work to the specimen is repeated continuously in a gradual negative slope of the curve. In this regard it is important to note that, for different materials, the load–displacement curve may be different. Particularly, the alternation seen in these experiments between increase and decrease in load, during quasi-static crack extension, may be attributed to the slip–stick phenomenon observed in toughened composites. Nevertheless, the energy dissipation during quasi-static crack extension can still be compared to the unit crack growth observed in the test specimen in a consistent manner. Therefore, the quasi-static test seems to allow representation of the data as low-cycle fatigue behaviour. For example, each drop and increase in the load can be considered a cycle N, and an strain energy U can be associated with each N, as illustrated in Fig. 1(b). 3.2. da/dN versus dU/dN: a physical SERR Such as introduced in Section 1, measuring the crack length a, the displacement d, the force P and the number of cycles N allows to graph a plot between the crack growth rate and the change in strain energy with the number of cycles: da/dN versus dU/dN. Fatigue data has been shown to align linearly in this type of presentation of data, under different stress ratios [15,16]. Suppose a straight line is fit to this data, such as illustrated in Fig. 2. The SERR of a crack extension is obtained by Eq. (1). Therefore, the crack extension can be characterised by a single physical parameter, the average SERR over the cycle, G⁄. One should note that G⁄ is a parameter from a different nature than Gmax and DG. The maximum SERR and the SERR range describe a theoretical energy release that can be calculated even if there is no crack extension. In this case, what

Fig. 2. da/dN versus dU/dN: illustration of a straight line adjusted to the data.

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do these calculations represent? Meanwhile, G⁄ is a parameter that describes the energy released by crack growth and is calculated from crack extensions. The basic assumption in the da/dN versus dU/dN plot is that when data are aligned along the same slope, this means they have the same release of strain energy per crack increment, dU/da. Thus, the same amount of energy dissipation corresponds to the same amount of crack growth or crack surface. Regarding this point, it is worth noting that the physical SERR works as a basis theory and not as a prediction model. p Prediction models, based on Gmax, DG or D G give an estimation of the crack growth rate for a given load cycle, but they do not explain crack growth, as they do not have a physics based theory behind them. Meanwhile, a basis theory explains the crack propagation based on physical principles, but does not necessarily allow the prediction of the crack growth rate. Therefore, once this basis theory is consolidated, efforts shall be directed to developing prediction models based on this theory. Furthermore, since the composite material used in this study presents neither relevant plasticity nor other significant energy dissipation mechanisms, any curve adjusted to the data in the da/dN versus dU/dN plot must go through the origin, because it is deemed impossible to dissipate energy without extending a crack. 3.3. The SERR and the fracture surfaces The energy released per unit area in crack growth is related to the amount of damage created. Decohesion implies breaking bonds, which requires energy. If more decohesion happened, more energy was released per unit area. Thus, the fracture surface generated in a crack extension and the strain energy released during this crack extension are related. Therefore, as data aligned on the same slope in the da/dN versus dU/dN plot present the same amount of energy dissipation per area, they are expected to present similar fracture surfaces as well. Any clear differences in the fracture surfaces, such as rougher or smoother features, would indicate a change in the damage state. A change in the damage state indicates that more, or less energy, was consumed during crack surface extension. This would be equivalent to data located in a different position in the da/dN versus dU/dN plot. Fracture surfaces obtained from quasi-static and fatigue mode I crack extensions usually present as characteristic features matrix cleavage, fibre imprints, fibre bridging, broken fibres and shear cusps [27]. Each one of these features is considered to contribute with the release of energy during crack extension. Thus, the fracture surfaces that present more of these features are expected to be related with higher values of dU/dA. In case of the comparison between two fracture surfaces with the same damage features, for example presenting only fibre imprints, the roughness of the fracture surface is expected to relate to the energy release. As an example consider Fig. 3, which illustrates a straight line that fits linearly a set of data. Data located to the right of this straight line would represent crack extensions that consumed more energy per crack area, and are thus expected to present rougher fracture surfaces or more damage features. Data located to the left of this straight line would represent crack extensions that consumed less energy per crack area, being expected to present smoother fracture surfaces or less damage features. 4. Data integration Five mode I quasi-static tests were performed in CFRP DCB specimens. All test specimens were unidirectional and manufactured from the same material batch (M30SC-150-DT 120-34 F) with 32 layers. A 13 lm Polytetrafluoroethylene (PTFE) film was used as initial crack. The tests were reported in [15,28].

Fig. 3. Illustration of linear fit through the data. Data to the right of the fit consumed more energy per crack area.

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Each of the five sets of quasi-static data was discretized in order to calculate dU/dN. This discretization was performed as explained in Fig. 4(a), in which each shaded area is a dU in one cycle. As the quasi-static data is being treated as a low-cycle fatigue, each shaded area in Fig. 4(a) can then be considered a dU/dN. As an outcome of this discretization, a cloud of points was obtained when plotting the results in terms of da/dN versus dU/dN. This is shown by the blue markers in Fig. 5. The average of these points, in da/dN and dU/dN, resulted in one point in the graph plotting da/dN versus dU/dN, shown by the red marker in Fig. 5. However, this result is not enough to enable full understanding of the trend of dU/da with respect to an increasing da/dN, because a single data point contains no information on the slope. In which fashion would this point move if da/dN was increased or decreased? Thus, new ways of analysing the same data with different discretization methods were necessary. Therefore, in order to analyse the trend of the data properly, each of the five sets of data was discretized in four different ways. The data was discretized to different levels, considering different number of cycles for the same crack growth, such as illustrated in Fig. 4(a) to (d). In Fig. 4(a), for example, 3 cycles were considered, and thus 3 values of dU/dN and da/dN were calculated. Meanwhile, the exact same crack was considered to be grown in 6 cycles in Fig. 4(b), resulting in 6 values of dU/dN and da/dN. In each of these integrations what differs is the step da in crack growth that is used to calculate dU/dN. In this illustration each shaded area limited by dashed lines represents a dU/dN. In Fig. 4(b) and (c), for example, the steps da are smaller than in Fig. 4(a). Meanwhile, in Fig. 4(d), the step in da is bigger than in Fig. 4(a). It is important to note that, although each set of data is discretized with four different procedures, each set still represents the same crack extension. In other words, the fracture surface and the energy spent in creating this fracture surface quasi-statically is the same, independently of the way dU/dN was calculated. This is observed by the fact that all integration procedures, for a given test specimen, yield the same value for the sum of the individuals dU/dN. This can be easily observed in Fig. 4: the sum of the shaded areas always results the same total area. This shows that the procedure is consistent, because the total energy spent in extending the crack is the same for a given dataset. Therefore, the average values of dU/dN and da/dN were calculated with these discretization procedures and plotted together. Data obtained from five specimens were discretized in four different increments, yielding 20 points in the da/dN versus du/dN plot. These points were plotted together, and a linear regression was used to produce the best linear fit by the minimisation of the sum of the square of the error. The result, presented in Fig. 6, shows a good correlation (i.e. coefficient of determination R2 = 0.9894). This graph shows that a quasi-static test can be analysed as low-cycle fatigue in a consistent manner, and the SERR can be easily calculated from the slope of the linear fit, which is 1/(dU/da). Therefore, the actual SERR in the mode I quasi-static fracture is given by Eq. (2).

Gquasi-static ¼ 611:6 J=m2

ð2Þ

5. Linking quasi-static and fatigue SERR: energy characterisation Considering the quasi-static test as a low-cycle fatigue has the advantage of enabling the gathering of data from both fatigue and quasi-static tests in the same format, i.e. the da/dN versus dU/dN plot. In this plot the SERR can be easily calculated from the slope of the curve, as explained in the previous section for the quasi-static SERR. It is important to note that with this procedure a real physical SERR is obtained directly from measured data and not from a theoretical model [16]. Fatigue tests were performed at three stress ratios (0.1, 0.5 and 0.7) in DCB specimens made from the same material and with the same dimensions as the ones used in the quasi-static tests. These were reported in [15]. The fatigue data is also fit by a linear function in a da/dN versus dU/dN plot. Fig. 7(a) shows the plot of both quasi-static and fatigue linear fits. The common procedure at this moment would be to fit both fatigue and quasi-static datasets with the same linear function. However, as explained in Section 3.3, the functions that fit the data must start at the origin of axes, because it is deemed impossible to dissipate energy without extending a crack. In Fig. 7 it is notable that the quasi-static data is shifted to the right of the fatigue data. This indicates more energy is dissipated per area in the quasi-static crack growth. In other words, less energy is required for the same amount of crack growth in fatigue than under quasi-static loading. A reason for this lower energy dissipation during fatigue crack growth may be attributed to the frequent change in the local delamination fronts under repeated cycles, so that the crack growth tracks the least resistant path. Thus, it can be argued that growing a crack in fatigue is more efficient than growing it quasi-statically, as it is shown by Eq. (4). In addition, this difference in the energy released during crack extension implies that different mechanisms of decohesion contributed to the energy dissipation, as seen in Fig. 8. Scanning Electron Microscopy (SEM) was performed after the tests, and the results are reported in this study. The fracture surface corresponding to the fatigue loading presents less damage features than the quasi-static one, resulting in less energy consumption per crack extension, i.e. smaller dU/dA. This is shown in Fig. 8(b). The main features visible on the fracture surface of the fatigue specimen are fibre imprints and cusps. Although cusps are typical features of mode II crack extension, they are commonly observed in mode I fracture surfaces and they occur due to the local shear induced by fibres being pulled from the surfaces during crack opening [27]. Meanwhile, the fracture surface that is the result of quasi-static crack extension, seen in Fig. 8(a), presents broken fibres, matrix cleavage, fibre imprints and cusps. The fibre-bridging process plays an important role in the energy consumption in crack extension and, consequently, in the fracture surface appearance. For the tests performed, fibres would bridge throughout several different

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Fig. 4. Integration of the quasi-static data as low-cycle fatigue data – each shaded area delimited by dashed lines is considered a dU/dN.

Fig. 5. Quasi-static dataset 1: the blue markers show the points that resulted from the integration procedure 4, and the red marker shows the average. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 6. Linear fit through the average values of dU/dN and da/dN obtained by the integration of the 5 sets of data with 4 different procedures.

crack growth rates. However, it was observed that at low values of da/dN these bridging fibres do not break. Therefore, as they remain in their elastic behaviour, there is no energy release [15]. Meanwhile, at large values of da/dN, going towards the quasi-static failure, the bridging fibres break, releasing energy. Thus, the broken fibres and the matrix cleavages are the reason for the higher energy consumption in quasi-static crack extension. These decohesion mechanisms explain why the quasi-static crack extension requires a larger dU/dA. Moreover, as shown in Fig. 7(b), the crack growth resistance is not the same under fatigue and quasi-static loading conditions. However, normalising the fatigue SERR by a SERR obtained quasi-statically is still possible, if this normalisation is performed with parameters obtained by identical energy balance principles. Gmax and Gon, or Gcrit for the case of the maximum SERR at quasi-static failure, are different parameters that do not describe the same physical process. Meanwhile, the average SERR G⁄ is an identical parameter for both quasi-static and fatigue crack extensions, and allows such normalisation. Characterising the difference in the energy released in quasi-static and fatigue loading allows the calculation of the actual SERR for each case. From the linear fit of the fatigue data it is possible to calculate that:

Gfatigue ¼ 289:7 J=m2

ð3Þ

Gquasi-static ¼ 2:1 Gfatigue

ð4Þ

Fig. 7. Relation between fatigue and quasi-static conditions: (a) log scale and (b) linear scale.

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Fig. 8. Quasi-static and fatigue fracture surfaces: roughness pattern.

Fig. 9. Fatigue data at lower crack growth rate: less energy spent due to non-damaged fibre bridging.

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Eq. (4) shows that to extend a crack by a unit area dA in fatigue releases approximately half the strain energy to extend a crack by the same dA quasi-statically. This explains why there is crack growth in fatigue at values of SERR lower than the critical SERR calculated in quasi-static tests. Therefore, one should not just fit a single curve through both fatigue and quasi-static datasets, but characterise the offset from fatigue to quasi-static crack extension from an energy point of view, taking into account the damage mechanisms associated with the fractured material. The linear slope of the fatigue fit seems to correlate to lesser extent with the data at low values of da/dN. As it is observed in Fig. 9, the points are slightly shifted to the left of the theoretical straight line. This indicates that less energy was released per crack extension under fatigue loading at low values of da/dN. Thus, a smoother fracture surface is expected to be obtained from these data points. Indeed, this trend is confirmed by Fig. 10, which shows the fracture surfaces for fatigue specimens at low values of crack growth rate. When fracture surfaces obtained at a low crack growth rate are compared with fracture surfaces obtained at a high crack growth rate one observes a difference in the roughness pattern. The fracture surfaces at low crack growth rates are smoother, and present basically fibre imprints as the main features. Meanwhile, the fracture surfaces at higher crack growth rates, where the linear fit presents a better correlation with the data, are rougher and present more damage features, such as fibre-imprints and cusps. Therefore, the difference in the SERR due to the mechanism of cusps formation, at higher values of da/dN, causes fatigue data to correlate to lesser extent with the linear fit at low values of crack growth rate. Following this trend, the fatigue data was then divided into two groups, according to similarities encountered on their fracture surfaces. Data below a crack growth rate around 107 m/cycle presented similar smooth fracture surfaces, dominated by fibre imprints, as in Fig. 10(a) and (b). Data above this crack growth rate presented similar rough fracture surfaces, dominated by features as fibre imprints and cusps, as in Figs. 8(b) and 10(c) and (d). Data with a crack growth rate below 107 m/cycle can be fitted by a different linear function, as shown in Fig. 11. The data with a crack growth rate above 107 m/cycle can still be fitted by the same linear function as before. Therefore, from the linear fit in Fig. 11(b), the SERR for the fatigue data with crack growth rates below 107 m/cycle is given by Eq. (5).

Fig. 10. Fracture surfaces at different crack growth rates. Roughness and number of damage features increase for higher crack growth rates da/dN, indicating larger energy release.

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Fig. 11. Relation between fatigue and quasi-static loading conditions: divided fatigue data – (a) log scale and (b) linear scale.

Fig. 12. Fracture surfaces: from the smoother to the rougher – an indication of the energy released.

Gda 6107 dN

m=cycle

¼ 171:2 J=m2

ð5Þ

In the analyses of the SEM pictures it becomes obvious that the energy released by a crack extension relates with the roughness pattern and the damage features that appear on the fracture surface. This is shown in Fig. 12. This picture brings together the fracture surfaces presented in this work, starting from the smoother in Fig. 12(a), at low crack growth rates, and increasing in the number of fractographic features until the quasi-static fracture surface, shown in Fig. 12(f). It is observed that mode I crack extensions start with cohesive fracture and smooth fracture surfaces, and other features as cusps, fibre

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breakage and matrix cleavage appear as the crack growth rate increases towards a quasi-static failure. The decohesion mechanisms of fibre breakage and matrix cleavage, encountered on the quasi-static fracture surface, are dominant energy dissipation mechanisms. The fatigue data may, indeed, be divided in other intermediate groups, according to small differences encountered in their fracture surfaces. Although these groups of data may be fitted with higher coefficients of determination, they will follow a pattern of increasing roughness with increasing crack growth rate, until a new microscopic feature can be observed on the fracture surface. 6. Conclusions Quasi-static data was consistently treated as low-cycle fatigue, which allowed a comparison between mode I crack extensions in fatigue and quasi-static loading. A real physical SERR can be consistently obtained from a da/dN versus dU/dN plot, for both loading conditions. The average SERR over a cycle, G⁄, obtained by energy balance principles, can be used to characterise fatigue and quasi-static crack extensions. The SERR range or the maximum SERR are parameters that do not maintain the similitude principle. Although these parameters can give a crude estimation of the crack growth rate that is expected from a certain load, for materials where the relation with the crack growth rate was established, the discussion about which of them should be used can be misleading. They do not describe uniquely the load cycle and are not based on the physics of the problem. Thus, once the physical SERR describes the crack growth as a basis theory, efforts shall now be directed to developing a prediction model based on this theory. Furthermore, the SERR depends on the damage state of the fracture surface. Therefore, the energy released during crack growth is a characteristic of the damage mechanisms observed on the fracture surface, and not of the loading condition. The values of the SERR for fatigue and quasi-static loading conditions can be linked. The lower limit is given by fatigue loading at low da/dN values, which present the lowest SERR and, consequently, the smoother fracture surfaces. As the crack growth rate increases, the damage mechanism starts to change, and more energy is released in fracture. The upper limit is given by the quasi-static fracture, which presents the largest SERR due to matrix cleavage and fibre breakage. Acknowledgement This work was financially supported by CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico – Brazil and by China Scholarship Council. References [1] Khan R. Mode I Fatigue delamination growth in composites [PhD Thesis]. Delft: Delft University of Technology; 2013. [2] Hojo M, Ando T, Tanaka M, Adachi T, Ochiai S, Endo Y. Modes I and II interlaminar fracture toughness and fatigue delamination of CF/epoxy laminates with self-same epoxy interleaf. Int J Fatigue 2006;28:1154–65. 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