Prediction of fatigue life to crack initiation in unidirectional plies containing voids

Composites Part A 127 (2019) 105638

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Prediction of fatigue life to crack initiation in unidirectional plies containing voids L. Maragoni, P.A. Carraro, M. Quaresimin

T

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Department of Management and Engineering, University of Padova, Stradella S.Nicola 3, 36100 Vicenza, Italy

A R T I C LE I N FO

A B S T R A C T

Keywords: B: Defects B: Fatigue B: Porosity/voids C: Crack C: Modelling

In the present work, a model is proposed to predict the life to crack initiation in porous unidirectional composite plies subjected to tensile fatigue loadings. The average values of the Local Maximum Principal Stress (LMPS) and Local Hydrostatic Stress (LHS) in a control volume of the matrix are proposed to be the driving forces to the initiation of fatigue cracks in off-axis plies. The experimental fatigue curves for the life to crack initiation of laminates characterised by different void contents are found to fall in the same scatter band once plotted in terms of the proposed driving forces, thus proving the capability of the proposed model to predict the life to crack initiation in porous laminates based on the behaviour of the void-free material. Finally, a parametric analysis is carried out to study the influence of the global void content and average void size on the reduction of the life to crack initiation and the applicability of the proposed model to different void distributions and shapes is discussed.

1. Introduction In the development of load bearing composite structures, a fundamental role in both manufacturing costs and design is played by manufacturing-induced defects. Such defects involve, among others, ply misalignment, fibre waviness, and incomplete cure, but the most common and difficult to avoid is the presence of porosity, due to the materials and processes involved. Voids may form due to different velocities of the resin flow within and between fibre tows when liquid resin is used, while in pre-preg processing they may originate from air entrapped during the stacking operation, from absorbed moisture and from the release of volatile components during the cure. Through dedicated experimental campaigns, voids were widely shown to decrease the static tensile [1–6], compressive [7–9], flexural [2,10,11] and inter-laminar [2,6,10,12–15] performances of composite laminates. Less efforts were instead devoted to study the effect of porosity on the fatigue behaviour, and in-depth analyses were carried out only in recent years [16–25]. Voids have been observed to shorten the fatigue life to crack initiation, to promote a faster damage evolution in terms of crack density and delaminations, and to lead to a premature final failure. The reader is referred to [6,24,25] for more extensive reviews on this topic. Given the detrimental influence of voids, to achieve the best

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mechanical properties it is desirable to keep their content to a minimum. The operations required to this end, however, may lead to an increase of the manufacturing costs, by including operations such as to degas liquid resin before infusion, to carefully control the resin flow, to apply intermediate vacuum during pre-pregs lay-up, and to use high pressures in autoclave. Unfortunately, even if all the precautions are taken, voids may form in specific locations due to the geometry of the part. Therefore, the knowledge of the relations between porosity and mechanical behaviour is of primary importance for the design of composite parts, for two main reasons: first, they enable a more reliable design, in which the material properties can be globally or locally changed based on the expected or allowable void content. Second, in combination with the relations between process parameters and porosity content, they can lead to a cost-effective production, that maximizes the performance/cost ratio. The easiest way to draw void-performance relations is through extensive experimental campaigns. Apart from being expensive and timeconsuming, their validity outside the investigated lay-up and manufacturing process needs to be verified. To overcome these issues, models to predict the mechanical behaviour in the presence of voids have to be developed. Also in this field, however, most of the efforts were directed towards the prediction of elastic properties and static strength or damage evolution ([4,6,13,14,26–30]).

Corresponding author. E-mail address: [email protected] (L. Maragoni).

https://doi.org/10.1016/j.compositesa.2019.105638 Received 21 January 2019; Received in revised form 26 July 2019; Accepted 17 September 2019 Available online 18 September 2019 1359-835X/ © 2019 Elsevier Ltd. All rights reserved.

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latter case [24,25,33]. This indicates that the same driving forces to fatigue crack initiation proposed in Ref. [34] can be used also in the presence of voids, once the perturbation of the local stresses caused by the porosity is properly taken into account. Therefore, a brief review of the damage-based criterion developed in [34] is reported hereafter, being it the base for the new model proposed here to predict fatigue crack initiation in the presence of cigar-like voids. After careful observations of damage mechanisms and fracture surfaces, the authors proposed the Local Maximum Principal Stress (LMPS) in the matrix to be the driving force to fatigue crack formation when a UD ply is subjected to sufficiently large shear stress to transverse stress ratios. Instead, when the stress state of a UD ply is close to pure transverse tension, they proposed that the crack formation is driven by the Local Hydrostatic Stress (LHS) in the matrix, extending the findings of Asp and co-authors for static loadings [37]. The transition from a LHSdominated to a LMPS-dominated loading condition in a ply was based on a characteristic value of the shear to transverse stress ratio (biaxiality ratio) λ12 = σ6/σ2, referred to as λ12*, that was found to be around 0.5. It was thus possible to collapse fatigue curves for crack initiation obtained for different off-axis angles (or different multiaxial stress states) under a given positive load ratio by plotting the curves in terms of LHS or LMPS. To predict fatigue crack initiation under any external stress state it is then sufficient to obtain experimentally only two fatigue curves, applying one LHS-dominated and one LMPS-dominated loading conditions (characterised by, respectively, λ12 < λ12* and λ12 > λ12*). The reader is referred to [34] for a full description of the model. The LMPS and the LHS values for a given material are meant as the peak values of the maximum principal stress and hydrostatic stress, respectively, in the matrix, calculated through FE analyses of a fibrematrix unit cell. However, the validity of such an approach was proved also for irregular fibre distributions [33]. In principle, the model proposed in [34] could be used also to predict the life to crack initiation for porous specimens under different multiaxial conditions, but two S-N curves (one for λ12 < λ12* and one for λ12 > λ12*) should be obtained experimentally for any void content, implying a large experimental effort. To extend the model so that the influence of homogeneously distributed cigar-like voids could be included, their influence on the local stress distributions in the matrix must be assessed, making use of Representative Volume Elements (RVEs) of the material instead of simple unit cells. In the presence of homogeneously distributed elongated cigar-like voids, commonly found in UD plies [19,24,25,27,32], the material RVE can be reasonably simplified to a 2-D square model including a single void, to which periodic boundary conditions are applied to account for the presence of other neighbouring voids, respecting the global void content. In this way, a uniform distribution of voids having different sizes is represented by a regular (square) pattern of voids having a uniform cross section, characterised by the average void diameter within a given ply. A tool to build micro-scale RVEs developed in a previous work [35] was then suitably modified to include a void of circular section at its centre (Fig. 1). Such RVEs can be analysed through FE analyses. Periodic boundary conditions are applied to the RVE as reported in Ref. [35] for the individual stress components acting on the ply of interest within a laminate and combined through the superposition principle. The remote stress components in a ply can be easily calculated, for instance with the Classical Lamination Theory (CLT). Indeed, CLT estimates with good accuracy the stress distribution among plies as long as no damage is present in the laminate, a condition that is met when applying the proposed model, that it is meant to predict the initiation of the first fatigue cracks. Once a FE analysis is carried out on a RVE, the local stress fields in the matrix in the absence and presence of voids are obtained. The proposed criterion to predict tensile fatigue crack initiation is based on the identification of an effective stress parameter acting in a control volume of the matrix. In particular, the average value of the LHS and

Only a few attempts have been made to predict the fatigue behaviour of composites in the presence of porosity, despite their large detrimental influence shown experimentally. By normalizing the maximum flexural cyclic stress to flexural static strength, de Almeida and Neto [10] found that the S-N curves for final failure of woven fabrics with different void content collapse into a single scatter band. Lambert and co-authors [19] observed a slightly linear correlation between the life to final failure of multidirectional laminates under fully reversed fatigue loading (R = −1) and the largest void in a critical location. Instead, no clear trend could be detected between fatigue life and global void content, indicating the importance of accounting for voids shape and distribution. Also Seon et al. [21] observed that the interlaminar tensile fatigue failure of curved beam specimens subjected to four-point bending load was related to the presence of an individual void in a critical location rather than to the global void content. To capture such an influence, they carried out FE analyses including the presence of the actual voids, revealed through μ-CT scans. A linear correlation was found between the calculated interlaminar tensile stress (ILTS) at the void tip and the fatigue life of the samples, whereas no clear trend was observed between the fatigue life and the ILTS calculated assuming void-free specimens. Before the final failure occurs in multidirectional composite laminates under cyclic loadings, other damage events take place, such as initiation and propagation of multiple off-axis cracks and delamination. If estimating the final failure is essential for a strength-driven design, predicting crack initiation and evolution is needed for stiffness-driven applications and in conservative approaches that require no damage at all to initiate in the part [31]. Therefore, the influence of voids on the whole damage progression should be suitably modelled for an advanced design of structural composite components. To the best of the authors’ knowledge, the scientific literature is currently lacking models to predict the life to fatigue crack initiation considering also the presence of porosity. To move a step forward and fill this gap, in the present work a model is developed to predict the life to crack initiation in composite UD plies subjected to tensile fatigue loadings containing homogeneously distributed cigar-like voids with a cross section comparable to that of the fibres, a common scenario in composite UD plies produced by several manufacturing processes [19,24,25,27,32]. Based on experimental observations of the micro-scale damage mechanisms, the fatigue crack formation is predicted through effective stress parameters calculated by means of Finite Element (FE) analyses on Representative Volume Elements (RVEs) of the material including porosity. As deeply discussed in the next Section, the proposed model is based on other recent works by the authors. As the same damage mechanisms at the micro scale were experimentally observed both in void-free [33] and porous specimens [24], the same stress parameters proposed in Ref. [34] are here linked to the life to crack initiation in the presence of porosity. To properly account for the effect of voids on those stress parameters, they are calculated in material RVEs capable of including voids, based on the procedure proposed in Ref. [35]. The model is validated against two sets of data obtained in a previous experimental campaign [25] and its application to different void morphologies and void distributions is eventually discussed. 2. Model development Predictive models should be based on the damage mechanisms occurring in the material to be reliable and applicable in the largest variety of cases [36]. The damage mechanisms leading to fatigue crack formation under tension-tension loadings, meant as the initiation of a visible crack involving the whole ply thickness and propagating along the fibre direction, have been observed to be similar in the absence and the presence of voids, even if the crack density evolution is faster in the 2

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Fig. 1. Examples of RVEs in the (a) absence and (b) presence of voids.

(2) A second consideration is made to identify the actual volume of material within a single strip that governs the crack formation process. Indeed, a fatigue crack under tension-tension loadings was experimentally observed to form both in the absence and presence of voids from the initiation and coalescence of local damage events at the micro-scale, such as matrix micro-cracks [33,24] and fibrematrix debonds [38]. Those events occur in narrow inter-fibre regions characterised by high stress concentrations. Thus, it is reasonable to assume that the driving force to a fatigue crack formation process under tension is represented by an average stress within a control volume (Vc) representative of those regions. This control volume can be identified once the Cumulative Distribution Functions (CDFs) of the LHS and LMPS in a single strip are known from a FE analysis (Fig. 3), by establishing a percentage of the matrix volume subjected to the highest stresses (right tails of the CDF curves). For instance, a Vc = 10% corresponds to the 10% of the matrix characterized by the most critical stress values. Fig. 4 shows examples of control volumes corresponding to different percentages of matrix volume. It is possible to notice that Vc values larger than 10% involve regions that are far away from inter-fibre locations, that are assumed not to contribute to the formation of a fatigue crack. Therefore, an upper limit to the control volume can be set to 10%. The specific value to be assigned to the control volume and its influence on the prediction capabilities of this approach will be discussed in the next section, showing a high robustness of the model in relation to this parameter.

LMPS in such a control volume are proposed as driving forces to fatigue crack initiation. To identify the control volume, two considerations are made: (1) First, under tension-tension fatigue loadings, cracks were experimentally observed not to significantly deviate from a straight path along the thickness of a UD ply in the absence of voids. Hence, the material volume actually involved in a matrix crack formation can be approximated to that of a strip along the material 3-direction (Fig. 2). Indeed, two locally damaged spots (as for instance two debonded fibres or matrix-micro-cracks) will not contribute to the formation of the same off-axis crack if they are sufficiently far apart from each other along the material 2-direction (Fig. 2). In the presence of porosity, a crack tends to connect voids along the ply thickness [25]. By representing a uniform void distribution with a regular (square) void pattern, the material volume involved in the crack formation process can still be assumed as a strip along the material 3-direction. The strip of material must have a finite width. A strip width of two times the fibre diameter appears to be in accordance with experimental observations (Fig. 2a). A deeper analysis on the influence of the strip width on the model predictions is made in Section 3, showing a negligible influence of this parameter when varied within a reasonable range. Therefore, after a FE analysis is carried out on the whole RVE, the stress analysis focuses on strips of material having a width of two fibre diameters (Fig. 2b).

In light of those considerations, the stress parameters driving the fatigue crack formation process under tension-tension loadings are identified as the average values of the LHS and LMPS in the above defined control volume Vc in a single strip of material, referred to as LHSav and LMPSav, both in the absence and presence of voids. In the former case those parameters are related to the irregular fibre distribution within a ply RVE only, while in the latter to both the irregular fibre distribution in the RVE and the presence of the void itself, that acts like a local stress concentrator. To account for the statistical variability of the fibre distribution along the 2 and 3 directions of a ply, several RVEs can be analysed. The effective stress parameters linked to the initiation of a fatigue crack are then defined as the mean values of LHSav and LMPSav among all the strips of several RVEs in the absence of voids, and among only the strips that involve voids in the presence of porosity. In the latter case, indeed, the strips containing voids represent the most favourable sites for crack initiation. Those effective stress parameters are referred to as LHS* and LMPS*. Which of the two to use in a ply is determined by the ply biaxiality ratio λ12 = σ6/σ2, where the

3

50 ȝm

2

F

F

a)

b)

Fig. 2. Examples of (a) experimental fatigue crack in a 90° ply, and (b) strips where the stress analysis focuses in a RVE (between vertical lines). 3

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0.4 0.2

a

0 -50

0

50 LHS (MPa)

100

b

0

150

-50

0

50 LMPS (MPa)

100

150

Fig. 3. Cumulative distribution function (CDF) in a strip of (a) LHS in the 90° ply of a [0/902]S laminate under σx = 50 MPa, and (b) LMPS in the −45° ply of a [0/ 452/0/−452]S laminate under σx = 100 MPa (glass/epoxy, Vf = 0.55 [25]).

effort. To summarize, the initiation of a fatigue crack in a UD ply, meant as the formation of a visible crack involving the whole ply thickness, under tension-tension loadings and at a given load ratio, is proposed to be driven by the value of the LHS*, or LMPS*, depending on the stresses acting on the ply (λ12 < λ12* or λ12 > λ12*) [34], calculated by means of RVEs in a most critical percentage of the matrix volume (Vc) along vertical strips of material. The loads applied to the RVE are those acting on the UD ply for a given remote load, which can be easily calculated with the Classical Lamination Theory (CLT) and which also indicate whether the crack initiation is LHS- or LMPS-dominated. A schematic of the data processing flow is reported in Fig. 6. The input data required to calculate the entity of the proposed driving forces are the elastic properties of fibre and matrix, the fibre volume fraction, the global void content, and the average diameter of the void cross section, in addition to the laminate stacking sequence and the remote loads to which it is subjected. Once the driving force to fatigue crack initiation has been identified and calculated for both void-free and porous materials, its relationship with the number of cycles spent to initiate a crack has to be obtained. It is here reminded that the damage mechanisms at the micro-scale before the formation of a visible fatigue crack were found to be the same in the

ply stresses can be calculated with the Classical Lamination Theory (LHS* for λ12 < λ12*, LMPS* for λ12 > λ12*) [34]. Once the driving force has been identified, it must be checked that the size of the Volume Element is large enough for it to be representative with respect to that parameter. The size of a RVE is quantified by the ratio δ = S/Rf, where S is the RVE side length and Rf the average fibre radius. In the presence of voids, the RVE size is a direct function of the average void size and the global void content. Indeed, once the void area fraction Av and the average void diameter Dv are identified, the resulting RVE size is calculated as follows: 2

D π v 4

δ=

S = Rf

Av Df 2

=

Dv Df

π Av

(1)

For the void-free case, it was found that analysing a large number of small RVEs or a small number of large RVEs lead to the same LHS* and LMPS* for any value of the control volume Vc. For example, convergence to the same LHS* and LMPS* is reached if 10 RVEs with δ = 30, 5 RVEs with δ = 40, or 3 RVEs with δ = 80 are analysed (Fig. 5). A RVE size of δ = 40 was used in all the following analyses to have a good balance between number of analyses and computational

2%

Off-axis angle: 90° 5% 10% 20%

Within inter-fibre regions

2%

50%

Off-axis angle: -45° 5% 10% 20%

Involve areas far from Within inter-fibre regions inter-fibre regions

50%

Involve areas far from inter-fibre regions b)

a)

Fig. 4. Matrix regions (in black) in a strip characterised by (a) the highest LHS in the 90° ply of a [0/902]S laminate, and (b) the highest LMPS in the −45° ply of a [0/ 452/0/−452]S laminate (glass/epoxy, Vf = 0.55 [25]). 4

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120 80 60 40 20 a

0

0

90° ply in [0/902]S laminate

5

10

į = 30 (10 RVEs) Serie1 Serie2 į = 40 (5 RVEs) Serie3 į = 80 (3 RVEs

250 LMPS* (MPa)

LHS* (MPa)

300

į = 30 (10 RVEs) Serie1 Serie2 į = 40 (5 RVEs) Serie3 į = 80 (3 RVEs)

100

200 150 100 50 b

0

15

Vc (%)

0

-45° ply in [0/452/0/-452]S laminate

5

10

15

Vc (%)

Fig. 5. (a) LHS*, and (b) LMPS* as a function of the control volume Vc for different RVE size (δ). LHS* is calculated in the 90° ply of a [0/902]S laminate subjected to σx = 50 MPa, and LMPS* is calculated in the −45° ply of a [0/452/0/−452]S laminate subjected to σx = 100 MPa (glass/epoxy, Vf = 0.55 [25]).

to crack initiation can be found for a given load ratio by calibration on two experimental fatigue curves for crack initiation of the void-free material, obtained respectively under a LHS-dominated (λ12 < λ12*) and a LMPS-dominated (λ12 > λ12*) loading condition. Plies at 90° and 45° can be tested for this purpose, as discussed in Ref. [31]. As shown in the next section, once this relation is found, it is possible to

absence and in the presence of voids [24,33], with voids only acting as local stress concentrators, promoting a faster damage evolution at the micro-scale and leading to an earlier off-crack initiation. Thus, the driving force to fatigue crack formation (in our case LHS* and LMPS*) is expected to be the same in void-free and porous laminates. As suggested in Ref. [34], the relation between LHS* and LMPS* and the life

Fig. 6. Schematic of the data processing for the calculation of the proposed driving force to fatigue crack initiation. 5

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4 3.5 3 2.5 2 1.5 1 0.5 0

20

LHS= 1.19 MPa for all Ȝ12 values

Ȝ12 = 0 Serie1 Ȝ12 = 0.25 Serie2 Ȝ12 = 0.5 Serie3

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Ȝ12 = 2 Serie1 Ȝ12 = 3 Serie2 Ȝ12 = 5 Serie3

10 5

b

0 0

15

Vc (%)

5

10

15

Vc (%)

Fig. 7. LHS* and LMPS* as a function of Vc for different biaxiality ratios that lead to equal values of LHS and LMPS in the previous model [34].

different void contents were obtained in laminates manufactured adopting different process parameters. For the [0/902]S lay-up, voids were found to be homogeneously distributed along the specimens. In the −45° ply of the [0/452/0/ −452]S stacking sequence, voids were instead rather concentrated at regular intervals along the length of the specimens, and the values reported in Table 1 refer to the void-containing regions. Within such regions, voids could be considered as homogeneously distributed, so both for the 90° and the −45° plies the assumption of uniform void distribution may be applicable. Even if in reality the void size follows a statistical distribution, it is reasonable to assume that a crack on its path along the ply thickness involves smaller and larger voids, and its formation is therefore affected by an average void size, such as that considered in the material RVE. In Ref. [25], the initiation of the first six cracks was considered for each specimen as data for life to crack initiation, since they were sufficiently far apart from each other to be considered as independent events. The detrimental influence of voids is clearly proved by the shift of the crack initiation curves from right to left as the void content is increased (see Figs. 8a and 9a). It was also proved by micrographic images that the first cracks in the porous specimens were originated within strips of material including voids, confirming them as the cause of the shorter life to crack initiation [25]. The 90° and the −45° plies were respectively LHS-dominated LMPSdominated [34]. 5 RVEs with δ = 40 were analysed for the void-free configuration, whereas for the porous specimens the analysis was carried out on 5, 30, and 9 RVEs respectively for Av = 0.34, 6.7 and 2.1. In all cases, the number of analysed RVEs was sufficient to have LHS* and LMPS* convergence. For the porous materials, the RVE size was calculated from the void area fraction Av and the average void diameter Dv through Eq. (1). An element size of 0.5 μm was used in all cases and a perfect bond was assumed between fibre and matrix. Periodic boundary conditions were applied to the RVEs as described in Ref. [35]. As shown in Figs. 8 and 9, the S-N curves for fatigue crack initiation obtained for different void content fall into the same narrow scatter band when plotted in terms of LHS* and LMPS*. This is confirmed by the low values of Tσ (ratio between the curves for 90% and 10% probabilities of survival), that results equal to 1.209 and 1.206 for the LHS*- and LMPS*-dominated loading conditions, respectively. Although a few void contents were available for the same material to validate the proposed model, it is worth pointing out that the microstructures tested in [25] refer to a void-free material, to a small void content commonly accepted for high quality standards (< 1%), and to a large porosity amount which may lead to a quality rejection. Thus, the model was validated over a large overall range of conditions, and its predictive capability is expected to be unchanged for other porosities, at least within the large tested range. In Figs. 8, and 9, a control volume of Vc = 5% was taken. As discussed earlier, an upper limit of Vc = 10% was considered, since the fatigue crack formation is believed to be driven by the local stress concentration in the inter-fibre regions along the thickness of a ply, and

predict, for the same load ratio, the life to crack initiation in the presence of cigar-like voids for any combination of global void content and average void size. 3. Model validation The first step to validate the proposed model is to check its capability to predict the life to crack initiation for a void-free material subjected to different values of the biaxiality ratio λ12 = σ6/σ2 as the model proposed in Ref. [34]. To verify this, it is sufficient to prove that different combinations of σ6 and σ2 that lead to equal LMPS in the original model using a fibre-matrix unit cell, lead also to equal LMPS* (although the values of LMPS and LMPS* are expected to be different). This analysis has also to be carried out for different control volumes Vc. Fig. 7 shows that the curves of LHS* and LMPS* as a function of Vc are overlapped for different biaxiality ratios that lead to an equal value of LHS and LMPS, respectively, as calculated in Ref. [34]. In particular, the equivalence for LHS* is trivial as its value depends only on σ2 and not on σ6. Hence, the equivalence of the present model to the one of Ref. [34] to predict life to crack initiation for different biaxiality ratios when no voids are present is proved. Moreover, this equivalency appears to hold for any value of the control volume even beyond the considered limit Vc ≤ 10%, indicating a low sensitivity of the proposed model to this parameter, as discussed later on. As stated, the present generalization of our model should be capable of accounting for homogeneously distributed cigar-like voids in the prediction of the life to crack initiation. To validate this capability, a comparison with fatigue crack initiation data obtained for the same material in the absence and presence of voids is needed. In Ref. [25], the authors produced [0/902]S and [0/452/0/−452]S glass/epoxy laminates by vacuum resin infusion by using different process parameters, obtaining both void-free and porous specimens. A fibre volume fraction of 0.55 was measured for all the laminates. Tensile fatigue tests (load ratio R = 0.05) were carried out on the specimens, and damage evolution was monitored throughout the tests, observing the influence of voids on the life to crack initiation, crack growth rate, and crack density evolution in the 90° ply and in the −45° ply. The void content was measured by micrographic images, and quantified by the void area fraction Av and the average void diameter Dv. As an alternative to optical microscopy, μ-CT could be used to obtain such data. Table 1 shows the void content features of the laminates. For the [0/902]S lay-up, two Table 1 Void area fraction (Av) and average void diameter (Dv) of specimens manufactured in Ref. [25]. For the cross-ply configuration, two different void contents were obtained in laminates manufactured adopting different process parameters. Off axis angle

90°

90°

−45°

Av (%) Dv (μm)

0.34 43

6.7 45

2.1 49

6

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90

predictions of the model in the range Vc ≤ 10% has to be analysed. Figs. 10 and 11 show the fatigue curves for fatigue crack initiation of the same experimental data in terms of LHS* and LMPS* calculated for Vc = 1%, 2%, 5%, and 10%. It is possible to observe that the slope of the fatigue curve and the value of Tσ are not sensibly affected by the choice of Vc. This indicates that the model is robust with respect to the parameter Vc, as the predictions only marginally depend on its exact value, at least when Vc is limited to the inter-fibre regions. The only visible effect of Vc is a shift of the fatigue curve. This is expected, as Vc is the most critical volume of the matrix. Therefore, the larger Vc, the smaller the average stress calculated within it, resulting in a lower intercept of the S-N curve (Figs. 10 and 11). Clearly, depending on the chosen value of the control volume Vc, the S-N curve for a given material will change. Hence, to use the proposed model to estimate the life to crack initiation in a porous component it is only required to be consistent once the choice of Vc is made. Another robustness check must be carried out on the value of the strip width over which LHS* and LMPS* are calculated. As shown in Fig. 12, the influence of the strip width on the LMPS* appears to be negligible both in the void-free and the porous configurations, highlighting a very low dependency of the proposed model also on this parameter. Similar trends are obtained also for LHS*. In the presence of voids, the maximum strip width was chosen to be as large as the average void diameter, not to include material regions that are shielded by the void itself.

a

ıx (MPa)

80 70 60

void-free Av=0.34% Av=6.7%

50 40 1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

Life to crack initiation (cycles)

LHS* (MPa)

100

10 1E+2

Vc = 5% b

Void-free Av=0.34% Av=6.7% 1E+3

LHS* = 109ÂN-0.0936 Tı=1.209

1E+4

1E+5

1E+6

1E+7

4. Discussion

Life to crack initiation (cycles) Fig. 8. Fatigue curves for life to crack initiation in the 90° ply of [0/902]S glass/ epoxy laminates, (a) in terms of global stress σx [25], and (b) in terms of the proposed driving force to fatigue crack initiation.

140

Modelling the relations between void content and long-term properties of composite materials can be of great help in both the advanced design and the definition of a cost-effective manufacturing of a composite part. In addition to being in good agreement with experimental data, the present model has the advantage of requiring a small experimental effort to be applied. It is indeed sufficient to have only two fatigue curves for crack initiation for the void-free material (one under LHS*-dominated and one under LMPS*-dominated loading conditions), as for the original model [34], to be able to estimate fatigue crack initiation in the presence of cigar-like voids for any combination of global void content and average void diameter. By comparing the predicted life to crack initiation in the presence of voids with the behaviour of the void-free material, design plots such as those reported in Fig. 13 can be drawn. In these plots, the life reduction for the initiation of the first cracks (Ni/Ni,void free) is reported as a function of the global void content and the void size, quantified respectively by the void area fraction Av and by the ratio between the average void diameter (Dv) and the average fibre diameter (Df). It is interesting to note that the largest part of the drop in performance is predicted to be given by the “introduction” of the first voids in the ply even if in small amounts (Av = 0.5%), and that the increase in the detrimental influence gets smaller with increasing global void content. For a given void content, a larger void size implies a smaller number of voids, as they are farther away from each other. From Fig. 13 it is also possible to notice that few large voids (greater Dv/Df) tend to be slightly more detrimental than many small voids for the same Av. This is in line with Lambert and co-author’s findings [19], who observed a linear relation between the largest void in a critical location and the final failure of glass/epoxy laminates. Finally, as expected, the larger influence of voids on the 90° ply compared to the −45° ply, observed experimentally, appears to hold for all the analysed void contents. Small irregularities in the trends of Fig. 13 can be attributed to the randomness of the generated RVEs. As a last note, it is highlighted that care should be taken when analysing RVEs with high Av and small Dv, as they lead to very small RVE size containing only a few fibres and may be not fully representative of the material microstructure. Plots as those reported in Fig. 13 can be used as an easy-to-read reference in the design of parts where voids are expected to occur and

a

ıx (MPa)

130 120 110 100 90 1E+2

void-free Local Av=2.1% 1E+3

1E+4

1E+5

Life to crack initiation (cycles) 400

LMPS* (MPa)

LMPS* = 303ÂN-0.135 Tı=1.206

40 1E+2

Vc = 5% b

Void-free Local Av = 2.1% 1E+3

1E+4

1E+5

Life to crack initiation (cycles) Fig. 9. Fatigue curves for life to crack initiation in the −45° ply of [0/452/0/ −452]S glass/epoxy laminates, (a) in terms of global stress σx [25], and (b) in terms of the proposed driving force to fatigue crack initiation.

a Vc of 10% already involves matrix regions that are far from those locations. Nonetheless, the influence of this parameter on the 7

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200

200

Vc = 1% a

Void-free Av=0.34% Av=6.7%

20 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 Life to crack initiation (cycles)

Void-free Av=0.34% Av=6.7%

20 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 Life to crack initiation (cycles) 100

Vc = 5% c

Void-free Av=0.34% Av=6.7%

LHS* = 141ÂN-0.0935 Tı=1.193

Vc = 10% d LHS* = 89.2ÂN-0.0936 Tı=1.224

LHS* (MPa)

LHS* (MPa)

100

Vc = 2% b

LHS* (MPa)

LHS* = Tı=1.186

LHS* (MPa)

169ÂN-0.0927

LHS* = 109ÂN-0.0936 Tı=1.209

10 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 Life to crack initiation (cycles)

Void-free Av=0.34% Av=6.7%

10 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 Life to crack initiation (cycles)

Fig. 10. Robustness of the model against the choice of Vc: S-N curves of 90° ply in [0/902]S glass/epoxy laminates.

40 1E+2

Void-free Local Av = 2.1%

1E+3 1E+4 1E+5 Life to crack initiation (cycles)

Void-free Local Av = 2.1%

30 1E+2

1E+3 1E+4 1E+5 Life to crack initiation (cycles)

Vc = 10% d

LMPS* = 251ÂN-0.135 Tı=1.207

LMPS* (MPa)

LMPS* (MPa)

LMPS* = Tı=1.206

Void-free Local Av = 2.1%

300

Vc = 5% c

303ÂN-0.135

Vc = 2% b

LMPS* = 377ÂN-0.131 Tı=1.192

40 1E+2

1E+3 1E+4 1E+5 Life to crack initiation (cycles)

400

40 1E+2

400

Vc = 1% a

LMPS* = 458ÂN-0.131 Tı=1.192

LMPS* (MPa)

LMPS* (MPa)

400

Void-free Local Av = 2.1% 1E+3 1E+4 1E+5 Life to crack initiation (cycles)

Fig. 11. Robustness of the model against the choice of Vc: S-N curves of −45° ply in [0/452/0/−452]S glass/epoxy laminates.

500

a

LMPS* (MPa)

LMPS* (MPa)

400

Strip width/fibre diameter: 2 4 8

300 200 100 0

void-free 0

5

10

15

400 350 300 250 200 150 100 50 0

b

Strip width/fibre diameter: 1 2 3

Av = 2.1% 0

5

Vc

10 Vc

8

15

Fig. 12. Influence of the strip width on LMPS* as a function of the control volume Vc in (a) void-free and (b) porous configurations. LMPS* is calculated in the −45° ply of a [0/452/0/−452]S laminate subjected to σx = 100 MPa (glass/epoxy, Vf = 0.55 [25]). Similar trends are obtained for the LHS* calculated in the 90° ply of a [0/902]S laminate made with the same material [25].

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90° ply in [0/902]S

100

Av=4% Av=2% Av=1% Av=0.5%

80 60

Ni/Ni,void free (%)

Ni/Ni,void free (%)

100

40 20 a

0 0

1

2

3 Dv/Df

4

5

-45° ply in [0/452/0/-452]S

80 60

Av=4% Av=2% Av=1% Av=0.5%

40 20 b

0 6

0

1

2

3 Dv/Df

4

5

6

Fig. 13. Predicted reduction of life to crack initiation as a function of global void content and average void size for (a) 90° ply in [0/902]S laminates and (b) −45° ply in [0/452/0/−452]S laminates (glass/epoxy, Vf = 0.55 [25]).

conditions. As shown by a parametric study, it appears that even a small amount of voids is sufficient to sensibly reduce the life to crack initiation, in accordance with experimental data. Also, for a given global void content, few large voids tend to be more detrimental than many small voids. Finally, even if the proposed model was developed for homogenously distributed voids, the same criterion is potentially applicable to other void distributions and more complicated void shapes, once they have been suitably modelled in a RVE. The proposed model requires a relatively small experimental effort for its calibration, namely one fatigue curve for first crack initiation under LHS*-dominated loading condition and one under LMPS*-dominated loading condition. The model can be useful in the definition of design tools capable of accounting for the local microstructure, which may lead to an advanced design of composite components and to the definition of a cost-effective composite manufacturing.

be tolerated. They can also be useful in the development of manufacturing processes that maximize the performance/cost ratio. To this end, however, more information is needed linking the process parameters and their specific costs to the porosity in the material. It is also worth mentioning that the behaviour of the void-free material is not the only possible reference. Indeed, the S-N curve in the presence or even absence of voids can be estimated through the proposed model also from that of a porous laminate, provided that its global void content and average void size for that laminate are known. To have the most general validity, a predictive model concerning the influence of porosity should be capable of accounting for void shape, size and distribution. The proposed model was developed to predict fatigue crack initiation in UD plies containing homogeneously distributed cigar-like voids with a cross-section that can be roughly approximated as circular and a few times larger than that of the fibres (typical voids obtained in laminates made of UD plies). However, it may be applicable also in other cases: if cigar-like voids are concentrated in certain regions, the material properties can be changed just locally, as seen for the −45° ply in the previous section. If the distribution of cigarlike voids could not be approximated to uniform, larger RVEs could be built accommodating multiple voids that are distributed following a desired spatial distribution. The strips of those larger RVEs could then be analysed, implying on the other hand a higher computational effort. In the same way, if few large voids of irregular shape are present in the material, such as those found in [6], they can be included in RVEs with their actual shape. To reduce the computational cost, it could be possible to separately model fibres and matrix only in the vicinity of the voids, where LMPS* and LHS* will be calculated, and homogenizing the other regions of material. Therefore, even if the current approach was conceived and validated here for a specific void configuration, it is potentially applicable to other microstructural conditions.

Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgements The authors wish to acknowledge the support and the computational resources made available by the High Performance Computing Lab at the Department of Management and Engineering (DTG), cofunded by the University of Padova in the framework of the program “Scientific research instrumentation 2015”. The financial support to the activities, received from the project “CARR_SID_03 - Advanced methods for health monitoring in primary composite structures” by the University of Padova is also acknowledged.

5. Conclusions References In the present work, a model was developed to predict the life to crack initiation in UD plies containing homogeneously distributed cigarlike voids under tensile loading at a given load ratio. The model is an extension of the criterion developed by the authors in Ref. [34]. It proposes, as driving force to fatigue crack formation, the average values of the Local Hydrostatic Stress (LHS*) and the Local Maximum Principal Stress (LMPS*) in a control volume Vc of the matrix along vertical strips of materials, calculated by means of FE analyses of periodic Representative Volume Elements (RVEs). For two different stacking sequences, namely [0/902]S and [0/452/ 0/−452]S, the experimental fatigue curves for the life to crack initiation of void-free and porous laminates were found to fall into the same scatter band when plotted in terms of the proposed driving forces. This indicates that with the proposed criterion the fatigue crack initiation in the presence of voids can be predicted based on the behaviour of the void-free material, under both LHS*- and LMPS*-dominated loading

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