epoxy laminates and tubes

Composites: Part B 61 (2014) 282–290

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Damage evolution under cyclic multiaxial stress state: A comparative analysis between glass/epoxy laminates and tubes M. Quaresimin a,⇑, P.A. Carraro a, L.P. Mikkelsen b, N. Lucato a, L. Vivian a, P. Brøndsted b, B.F. Sørensen b, J. Varna c , R. Talreja d a

Department of Management and Engineering, University of Padova, Stradella S. Nicola 3, Vicenza, Italy Department of Wind Energy, Technical University of Denmark, Risø Campus, DK-4000 Roskilde, Denmark Composite Centre Sweden, Luleå University of Technology, SE-971 87 Luleå, Sweden d Department of Aerospace Engineering, Texas A&M University, College Station, TX, USA b c

a r t i c l e

i n f o

Article history: Received 12 December 2013 Received in revised form 26 January 2014 Accepted 27 January 2014 Available online 7 February 2014 Keywords: A. Laminates B. Fatigue C. Damage mechanics Multiaxial fatigue

a b s t r a c t In this work an experimental investigation on damage initiation and evolution in laminates under cyclic loading is presented. The stacking sequence [0/h2/0/h2]s has been adopted in order to investigate the influence of the local multiaxial stress state in the off-axis plies and the possible effect of different thickness between the thin (2-plies) and the thick (4-plies) layers. Results are presented in terms of S–N curves for the initiation of the first cracks, crack density evolution, stiffness degradation and Paris-like curves for the crack propagation phase. The values of the off-axis angle h has been chosen in order to obtain local multiaxial stress states in the off-axis plies similar to those in previous studies for biaxially loaded tubes. Results concerning damage initiation and growth for these two specimen configurations are shown to be consistent for similar local multiaxial stress states. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The multiaxial fatigue behaviour of composite materials represents a topic of extreme importance and interest for many industrial fields. In fact, composite laminates are increasingly used for structural components subjected to cyclic loads as wind turbine blades, airplane wings and car frames just to name a few. In addition, the stress state is in general multiaxial, it being due to external loads in several directions (‘‘external’’ multiaxiality) or to the material anisotropy (‘‘internal’’ multiaxiality [1]). A recent review [1] points out the lack of reliable and general design tools for multidirectional laminates subjected to multiaxial fatigue loading. In addition, information on damage mechanisms and their evolution under cyclic loading, particularly in correlation with the local multiaxial conditions, are missing as well. However, the problem is not trivial at all, since the fatigue life of multidirectional laminates is characterised by several sources of damage, the progressive evolution of which causes the degradation of global stiffness and residual strength of the laminate. Some macroscopic evidences of this behaviour can be found in Refs. [2–6] among the others. Typically, the first damage mechanism occurring ⇑ Corresponding author. Tel.: +39 0444 998723. E-mail address: [email protected] (M. Quaresimin). http://dx.doi.org/10.1016/j.compositesb.2014.01.056 1359-8368/Ó 2014 Elsevier Ltd. All rights reserved.

since the earlier stages of fatigue life is the initiation and propagation of off-axis cracks, until a saturation condition of multiple off-axis cracks with near-regular spacing is reached. Another, and usually subsequent, type of damage is represented by the onset and propagation of delaminations. This behaviour has been experimentally documented by several works in the literature (see Refs. [7–12] for instance). In particular, it is of extreme importance to have models that enables the prediction of the onset and propagation of multiple off-axis cracks since these phenomena can be responsible of a pronounced stiffness reduction much before the final failure of the laminate. In addition, off-axis cracks act as stress concentrators for the 0° oriented plies, promoting fibre breaks in the vicinity of their tips. The initiation and propagation of off-axis cracks are matrix or fibre–matrix interface controlled phenomena, the prediction of which has to be based on models and criteria suitable to account for multiaxial stress states. In the authors’ opinion, reliable multiaxial criteria must be defined on the basis of the damage mechanisms at the microscopic scale, leading to the initiation and propagation of off-axis cracks. The lack of knowledge on such mechanisms and their dependence on the multiaxial condition is also highlighted in Refs. [1,13,14]. Accordingly, the need to improve the knowledge on this subject by means of dedicated and extensive experimental characterisations is clearly evident.

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Several procedures and specimens configurations for multiaxial testing of composites are reviewed in Refs. [15–17]. ‘‘Internal’’ multiaxial loading conditions can be achieved by means of off-axis flat specimens, while combined external loads (leading to an ‘‘external’’ multiaxiality) can be applied by means of cruciform or tubular specimens. With the aim to improve the understanding of the matrixdominated fatigue behaviour of UD laminates, Quaresimin and Carraro carried out an extensive experimental activity on glass/ epoxy tubes subjected to an ‘‘external’’ multiaxial condition, obtained by combined tension and torsion cyclic loads [13,14]. Quaresimin et al. [1] suggested to use the bi-axiality ratios to describe the local multiaxial stress state acting on a lamina

k1 ¼

r2 r6 r6 ; k2 ¼ ; k12 ¼ r1 r1 r2

ð1Þ

where r1 and r2 are the longitudinal and transverse normal stress components and r6 is the in-plane shear stress in the material coordinates system. Quaresimin and Carraro [13] designed a specimen configuration that enables experimental studies of the effect of r2 and r6 on the fatigue life for transverse crack initiation. To this aim, tubular samples were produced, with the fibres oriented at 90° with respect to the axis of the tube. Combining tension and torsion cyclic loads by means of a biaxial testing machine, different values of k12 were obtained (0, 1 and 2). The presence of an increasing shear stress component was proved to have a detrimental effect on the S–N curves relating the maximum cyclic transverse stress r2,max to the number of cycles spent for the initiation of a transverse crack [13]. Since these specimens were made of unidirectional plies, this event led to the final failure of the tubes, preventing the authors to analyse the cyclic crack propagation phase. In Ref. [14] tubular specimens with lay-up [0F/90U,3/0F], where the subscripts F and U mean fabric and unidirectional, respectively, were produced and tested, focusing the attention on the initiation and propagation of cracks in the 90° UD layers. Four values of k12 were obtained (0, 0.5, 1 and 2) by means of combined tension–torsion loadings. A detrimental effect of the shear stress component was found both for the crack initiation process, analysed in terms of S–N curves for the onset of the first transverse crack, and the propagation phase, described by means of Paris-like curves relating the Crack Growth Rate (CGR) to the mode I Energy Release Rate (ERR), GI. In addition, SEM investigations on the fracture surfaces revealed the dependence of the damage modes at the microscopic scale on the shear stress contribution [14]. Indeed, rather smooth fracture surfaces were observed for specimens tested with a low shear stress contribution (k12 = 0 and 0.5), while a significant presence of matrix shear cusps between the fibres was found on tubes subjected to larger amount of shear stress (k12 = 1 and 2). The experimental results and the evidences presented in [13,14] are useful for the definition and validation of a damage based multiaxial criterion for crack initiation which is being presented in an incoming paper [18]. It is important to remind that the activity carried out in [13,14] is related to an ‘‘external’’ multiaxial condition, obtained by means of a biaxial testing machine. However, it would be definitely far easier and cheaper to test the multiaxial fatigue behaviour of composites by means of off-axis loaded flat coupons, taking advantage of the ‘‘internal’’ multiaxiality due to material anisotropy. UD offaxis specimens could be used for this purpose, and several works in the literature report S–N curves for different off-axis angles [19–21]. The main limit of this specimen configuration is that it does not allow to analyse the crack propagation phase, which can be studied, instead, by adding 0° layers constraining the offaxis plies. The propagation of matrix cracks in laminates has been studied in Refs. [22–25], but only in [23] an off-axis angle different

from 90° has been considered, which leads to a mixed I + II mode propagation. Although there are no specific reasons to believe that the material response should be different weather an ‘‘external’’ or an ‘‘internal’’ multiaxiality is applied, it is important to verify their equivalence with a dedicated experimental campaign. Accordingly, the aims of the present work are: (1) to design a specimen configuration for multidirectional flat laminates with off-axis layers oriented at an angle suitable to obtain ‘‘controlled’’ multiaxial conditions (k12 values), similar to those adopted in Ref. [14]; (2) to analyse the possible effect of the layer thickness on fatigue crack initiation and propagation, by including, in the same laminate, a ‘‘thick’’ and a ‘‘thin’’ layer with the same orientation; (3) to characterise the fatigue behaviour of the laminates in terms of first crack initiation, crack density evolution, stiffness degradation and crack propagation; (4) to compare the S–N curves for crack initiation and the curves for crack propagation with those obtained for tubular samples [14] under similar multiaxial conditions. To these aims, the same glass/epoxy pre-preg material of Ref. [14] has been used for [0/h2/0/h2]s flat specimens with off-axis angles h of 50° and 60°, in order to achieve biaxiality ratios k12 = 0.57 and 1.15, as will be better clarified later on. The results of the fatigue tests are compared to those for tubes [14] for k12 = 0.5 and 1. In addition, in the present work an efficient method for the calculation of the mode I and mode II components of the ERR, GI and GII, in off-axis layers is presented.

2. Materials, specimen preparation and test equipment E-glass/epoxy pre-preg ‘‘UE400-REM’’ by SAATI S.p.A. (Italy) was used in a [0/h2/0/h2]s specimen layup studying two different configurations using h = 50°or h = 60°, respectively. In the following, the two-plies and the central four-plies layers will be referred to as the thin and the thick layers, respectively. The laminates were manufactured using hand lay-up of the pre-preg followed by autoclave curing at 140 °C and pressure of 6 bars, resulting in lam inates with a total thickness of about th¼50 layup ¼ 4:5 mm and h¼60 tlayup ¼ 4:3 mm, respectively. The cured ply elastic properties for the UE400-REM were reported in Refs. [13,14]. Their values are listed in Table 1. The tensile apparent modulus in the x-direction, Ex, was measured for the produced laminates. A value of 17,660 ± 500 MPa was found for the [0/602/0/602]s specimens, which is in close agreement with the prediction based on the classical lamination theory (17,332 MPa), starting from the ply properties listed in Table 1. The Ex value for the [0/502/0/502]s laminates was found to be of 16,720 ± 570 MPa, lower than the expected value of 17,823 MPa estimated with the classical lamination theory. The reason for this disagreement was identified in a different fibre volume fraction for the two panels, though they were manufactured nominally with the same materials and process. By means of a thermo-gravimetric test, volume fractions of 0.45 and 0.38

Table 1 Properties of UE400-REM cured ply, taken from Refs. [13,14], adopted for [0/602/0/ 602]s specimens. E1 (MPa)

E2 (MPa)

m12

G12 (MPa)

Avg. ply thickness (mm)

34,860 ± 2365

9419 ± 692

0.326 ± 0.015

3193

0.35

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were measured for the [0/602/0/602]s and [0/502/0/502]s specimens, respectively. Therefore, for the [0/502/0/502]s laminate the micromechanical formulation proposed in Ref. [26] was adopted for the estimation of the ply elastic properties, based on the measured fibre volume fraction of 0.38 and typical glass and epoxy elastic properties (Efibre = 72000 MPa, mfibre = 0.2, Ematrix = 4200 MPa, mmatrix = 0.34). The results are show in Table 2. Using these values as input for the classical lamination theory, the predicted Ex for the [0/502/0/502]s specimens is equal to 16,300 MPa, in good agreement with the experimental measurements. It is also worth mentioning that the average ply thickness is equal to 0.35 and 0.375 for the [0/602/0/602]s and [0/502/0/ 502]s specimens, respectively. Eventually, micrographic analyses revealed that in the [0/502/0/ 502]s samples the thickness of the thick 50° layer is 1.88 (instead of 2) times that of the thin one. This has to be accounted for in the following sections of the paper, mainly for the calculation of the ERR for the crack propagation analysis in Section 5. The cured laminates were cut into uniaxial tensile test specimens with the geometry L  W = 300  24 mm2 (including 75 mm tabs at each end) and polished on both edges. Uniaxial fatigue tests were performed using a servo-hydraulic test machine with hydraulic grips, see Fig. 1. The deformation was measured using two back to back mounted 25 mm/±2.5 mm extensometers and a load cell of 100 kN. Cyclic loads were applied under load control with a frequency adapted to the load level in the range of 3–5 Hz following a sine variation of the load with a R-value of R = Pmin/Pmax = 0.1. Based on the initial specimen stiffness of the individual test samples, the maximum load level, Pmax, was set aiming for different initial maximum strain values in the range of 0.4–1.1%. As the stiffness of the specimens decreased throughout the fatigue test due to damage evolution (see later in Section 3) the strain value increased during the test while the prescribed load amplitude was kept constant. The glass fibre reinforced epoxy is semitransparent and by using a uniformly distributed light source it was possible to observe and follow crack initiation and growth throughout the fatigue tests. To avoid the extensometers to influence the tunnel crack measurements, the extensometers were mounted at the

bottom end of the test samples as shown in Fig. 1. The tunnel crack measurements were based on a 10.2 MP digital reflex camera used in an automatic images acquisition system (image acquisition was triggered by the computer program controlling the testing machine). Thereby, it was possible to extract the crack density and crack growth of the individual tunnel cracks as a function of number of fatigue cycles. The biaxiality ratios k12 and k1 (Eq. (1)) can be used to quantify the degree of multiaxiality of the stress state in the off-axis layers, r1, r2 and r6 being defined according to the reference system in Fig. 2b. Since the matrix-dominated fatigue behaviour is of interest for the present work, particular attention is focused on the biaxiality ratio k12. The dependence of the stress components and k12 on the orientation h of the off-axis layers is shown in Fig. 2a and b. The curves are based on a full 3-dimentional finite element calculations, but could also be based on a 2-D laminate calculation as well. Thermal stresses were included in the analysis. The commercial code Abaqus 6.11-1Ò was adopted for FE modelling. Eight nodes C3D8RH elements were used, with 4 divisions for each ply. Fig. 2 was obtained with the elastic properties listed in Table 2, whereas a transversely isotropic behaviour was assumed for the out-ofplane properties (E3 = E2, G13 = G12, m13 = m12, m23 = 0.4, G23 = 2900 MPa). As shown in Fig. 2a, the transverse stress r2 is positive (tensile) and has an increasing value for increasing angles above 30° while it has a small negative (compressive) value for angles below 30° with a vanishing stress for h = 0°. The shear stress r6 vanishes for off-axis angle of 0° and 90°, while it has a maximum around 45–50°. In the present study, the two angles h = 50° and h = 60° are chosen such that the biaxiality ratio k12 only slightly depends on the precise value of the angle thus being insensitive to small possible errors in the ply orientation during the hand lay-up process. The biaxiality ratios for the two lay-ups were found to   be kh¼50 ¼ 1:15 and kh¼60 ¼ 0:57, which are also useful for com12 12 parisons with the initiation and propagation results on tubular samples [14] tested at k12 = 0.5 and 1 under tension/torsion loading. The r1 component in the laminates is such that biaxiality   ratios kh¼50 ¼ 0:67 and kh¼60 ¼ 2:14 are achieved. In both cases the 1 1 longitudinal stress r1 is of the same order of magnitude of the transverse stress and therefore a matrix-dominated behaviour, mainly driven by the transverse and shear stresses, is expected.

(a)

Table 2 Estimated properties, adopted for [0/502/0/502]s specimens. E1 (MPa)

E2 (MPa)

m12

G12 (MPa)

Avg. ply thickness (mm)

30,620

8620

0.2985

3250

0.375

(b)

Fig. 1. Fatigue test set-up and acquisition system.

Fig. 2. (a) Normalised stress components and (b) biaxiality ratio, k12 , as a function of the angle h calculated from a full 3-D finite element model.

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3. Crack initiation, multiplication, growth and stiffness degradation The first damage mode occurring during fatigue life was the onset of off axis cracks mainly at the edges. As expected, the first crack initiation was followed by multiple onsets in the whole specimen length. As far as these initiation sites were far enough from each other, the cycles spent for each crack onset could be considered as independent fatigue results, accounting for the statistical distribution of fatigue strength along the specimen length. Fig. 3a and b show the S–N data (i.e., the number of load cycles to the initiation of the first cracks) for each load level, for h = 50° and 60°, respectively. The curves are presented in terms of the maximum cyclic transverse stress r2 in the off-axis layers. It can be observed that the initiation data for the thick and the thin layers are reasonably well described by the same scatter band. With increasing number of cycles the number of cracks progressively increased, and the initiated cracks propagated along the longitudinal direction of the layer. Often in the literature the crack multiplication phenomenon is described in terms of evolution of the crack density, meant as the number of cracks divided by the length of the observation zone, measured perpendicularly to the propagation direction. Referring to the geometry shown in Fig. 4, the above mentioned crack density q can be calculated as follows.

q¼

Total N of cracks jACj

ð2Þ

For a better estimation of the state of damage the crack length should be incorporated in the definition of the crack density, as already highlighted in [14,27–29]. Accordingly the weighted crack density has been defined as a weighted number of cracks per unit length in the direction orthogonal to the crack path, see Fig. 4. Defining the crack density, the cracks are weighted with their length. This is done by classifying the cracks falling into 8 groups of crack lengths (12.5%, 25%, 37.5%, . . . , 100%) where 100% is a crack running throughout the full width of the test sample. The weighted crack density, qw, is then calculated as

(a)

Fig. 4. A representative crack density image showing both cracks in the thin and thick off-axis layers.

qw ¼

25% 100% 0:125N12:5% crack þ 0:25N crack þ    þ 1:00N crack jACj

ð3Þ

Using Eq. (3) together with the output from the automatic images acquisition system it was then possible to plot the evolution of the crack density in the thin and thick layer for the two cases, see Fig. 5. As one can expect, the crack density increases as the applied initial strain increases. At high number of cycles, the entity of which depends on the load level, a saturation condition is reached. The ratio between the saturation values of the crack density in the thin and the thick layers is found to be approximately inversely proportional to the ratio between the layers’ thickness. The consequence of the increasing the crack density in the offaxis layers is a stiffness degradation of the test samples. Fig. 6 shows the measured stiffness degradation as a function of the number of cycles for the test specimens. It can be seen that there is a strong dependence of the stiffness reduction on the load level and that the stiffness reduction occurs rather early in the fatigue life. The lower bound for the stiffness degradation can be estimated with the ply discount method and it corresponds to the case where the 0° – layers carry all the load. For h = 50° and h = 60° this corresponds to a stiffness reduction of Ex/Ex0 = 0.63 and Ex/Ex0 = 0.65, respectively. The stiffness is seen to saturate on a higher level indicating that the off-axis layers still carry some load even in the saturation condition. 4. Crack propagation

(b)

Fig. 3. S–N curves for the initiation of the off-axis cracks for (a) [0/502/0/502]s and (b) [0/602/0/602]s specimens.

In addition to the crack density, the output from the automatic images acquisition system can be used to determine the crack propagation rate. During the post-processing of the test results, isolated cracks initiated in the first stages of each test were selected for the analysis of the propagation phase. Based on the finite element simulation described in the next section, a crack can be considered as isolated and non-interacting with the neighbouring cracks as long as the ratio between the orthogonal distances to these neighbouring cracks, L, is larger than 4 times the thickness h of the layer. In addition, as shown later the energy release rate for tunnelling cracks is independent on the crack length if the tip is far enough from the edges of the specimen. Indeed, Fig. 7 shows a typical measurement giving a rather linear relation between the crack length a and the number of cycles of propagation Np, the Crack Growth Rate (CGR) being equal to the curve’s slope. The measured CGRs of the thin and thick layers are plotted in Fig. 8 as a function of the initial strain level. Each point corresponds to a single crack propagating with a constant rate. It is seen that, for a fixed applied strain level, the crack growth rate in the thin plies is significantly lower than in the thick ones.

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(a)

(a)

[0/502/0/-502]s

1.00

0.6% 0.65%

Ex/Ex0

0.90 0.7%

0.80

0.9%

0.8%

1.0%

1.1%

0.70

ply discount

0.60 0.0E+00

5.0E+05

1.0E+06

1.5E+06

Number of cycles N

(b)

(b)

[0/602/0/-602]s

1.00

0.4%

0.5%

Ex/Ex0

0.90

0.6% 0.9%

0.7%

0.80

0.8% 1.0%

1.1%

0.70

ply discount

0.60 0.0E+00

5.0E+05

1.0E+06

1.5E+06

Number of cycles N

(c)

Fig. 6. Stiffness degradation during fatigue tests for (a) [0/502/0/502]s and (b) [0/ 602/0/602]s specimens.

crack length, a [mm]

30

a = 6.57E-04·Np + 0.477

25

(c)

20

(b)

15 10 5

(a)

0 0

(d)

20000

40000

60000

propagation cycles Np

load direction a a (c)

a (a)

(b)

Fig. 7. Example of crack propagation curve in the thick layer for the [0/502/0/502]s lay-up, and e = 0.7%.

Fig. 5. Weighted crack density for [0/502/0/502]s specimen in the thin (a) and thick (b) layer and [0/602/0/602]s specimen for the thin (c) and thick (d) layer.

5. Calculation of the energy release rate With the aim to analyse in detail the crack propagation phase, the Energy Release Rate (ERR) was calculated for the cracks propagating in the ±h layers of the lay-ups under investigation. Finite Element (FE) analyses were carried out with the code Abaqus 6.11-1Ò. First, an entire specimen with one crack only was modelled, in order to compute the ERR associated to an isolated, non-interacting crack. The Abaqus’ J-integral calculation feature was adopted for the ERR calculation. 8-nodes C3D8RH elements were used with an element size of 0.0125 mm at the crack front. The J-integral was calculated through circular paths at 61 locations along the thickness of the cracked layer which was divided in 30 elements

along the thickness direction. Convergence was found to occur at the 5th circular contour. The value of the J-integral associated to a certain crack was averaged over the 61 calculation points through the thickness direction z, according to Eq. (4).

J¼

1 h

Z

h

JðzÞdz

ð4Þ

0

where h is the layer thickness and z is the out of plane coordinate axis. In the simulations, a straight crack front was used even though this does not represent the correct crack front shape. In reality, the shape should be determined such that J(z) = Gc, Gc being the critical ERR which is, in general, function of the mode mixity. For a fixed mode mixity, this corresponds to a constant value of J(z). Nevertheless, for a general case the mode mixity may vary in the thickness direction resulting in a critical energy release rate which may vary too. For the pure mode I case, Nakamura and Kamath [30] showed

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1.E+0

CGR [mm/cycle]

(a)

[0/502/0/-502]s

thin thick

1.E-1 1.E-2 1.E-3 1.E-4 1.E-5 1.E-6 0.4

1.2

0.8

1.6

initial strain level, %

(b) CGR [mm/cycle]

1.E+0

[0/602/0/-602]s

thin thick

1.E-1 1.E-2 1.E-3 1.E-4 1.E-5 1.E-6 0.3

0.6

0.9

1.2 1.5

initial strain level, % Fig. 8. Steady-state crack growth rate versus the initial strain level tests for (a) [0/502/0/502]s and (b) [0/602/0/602]s specimens.

that the average J-integral value (calculated with the stress intensity factor) is rather independent on the actual shape of the steady state crack front indicating that a straight crack front can be used. A similar observation was made here comparing the average value of J for the straight front with the corresponding value obtained for a non-straight crack front. Different crack length values were considered to evaluate the possible influence on the ERR. The results of this analysis for the [0/502/0/502]s specimens are presented in Fig. 9 where the crack length is normalised to the layer thickness. The ERR was computed for both the thick and thin layers. It can be noticed that the ERR has an increasing trend for short cracks but after a length approximately twice the layer thickness it reaches a steady-state value. Steady-state cracking is well-known for tunnelling cracks in constrained layers [31]. As a consequence, the crack growth process occurs in a Steady State (SS) condition for the larger part of the propagation phase. This explains also why the crack length versus fatigue life curves discussed in paragraph 4 are characterised by a linear trend (see Fig. 7). If the ERR remains constant during the propagation phase there is no reason for a crack to increase its growth rate, other than variations of the local toughness due to the presence of defects or to the local microstructure. The J-integral approach has the drawback that it is suitable to compute the total ERR (Gtot = GI + GII) and not to decouple it in the mode I and mode II components GI and GII. It was shown [14] that the resistance to crack propagation depends indeed on the

Total SERR [kJ/m2]

0.8

mode mixity (defined as MM = GII/Gtot) and this makes it important to know the mode decomposition, besides the total value of the ERR. It is also worth noting that, as the mode mixity is based on the ERR components, this is a better parameter with respect to the k12 ratio to quantify the local multiaxial conditions when analysing the propagation phase. To overcome the J-integral limitation, we propose here to calculate the steady-state energy release rate by energy accounting. The proposed approach is a generalisation of an approach developed for mode I tunnelling cracking by Suo [32]. As described above, when the crack is long in comparison with the layer thickness, cracking occurs under steady-state. The cracking is 3-dimensional but the shape of the crack front and the stress field around the crack front remain invariant as the crack propagates across the specimen width (the shape will be such that at every point along the crack front the energy release rate is equal to the fracture energy of the associated mode mixity). Therefore, the energy change for a unit crack advancement is equivalent to removing a slice of material of unit width far ahead of the crack tip and inserting a slice of material of unit width far behind the crack tip. For fixed grips, the change in the potential energy between the two states can be determined as the work required to close the crack and re-establish the stress state far ahead of the crack. Consequently, the average energy release rate (averaged across the layer thickness) can be calculated as

Gtot;ss ¼ GI;ss þ GII;ss ¼

1 2h

Z 0

h

r2 dn ðzÞdz þ

1 2h

Z

h

r6 dt ðzÞdz

ð5Þ

0

where z is the through-the-thickness coordinate and varies from 0 to the ply thickness h. In Eq. (5), the stress components r2 and r6 are those acting in the considered layer far ahead of the crack tip while dn and dt are the normal and tangential crack faces opening displacements far behind the crack tip, (see Fig. 10). This approach enables the calculation of the energy release rate without consideration of the near-crack tip stress field. The first integral in Eq. (5) can be considered the mode I energy release rate, GI, and the second integral can be taken as the mode II energy release rate, GII. For the application of Eq. (5), FE models were realised with a single crack in the +h or h layers. A crack length equal to 50% of the maximum crack length was modelled. The cracked layers were meshed with 30 elements in their thickness, the confining layers with 10 elements and eventually 4 divisions were adopted for the remaining layer. The stress components (r2 and r6), which are constant in the thickness direction in the corresponding offaxis layer, can be easily computed by means of the classical lamination theory or extracted from the FE model far ahead of the crack

= 1.1% thick

0.6

Jss

thin

0.4 0.2 0.0

0

2

4

6

8

10

12

14

16

a/h Fig. 9. Total ERR against normalised crack length, lay-up: [0/502/0/502]s, e = 1.1%.

Fig. 10. Schematic of the crack opening and sliding displacements (crack out of scale) for the calculation of the steady state ERR components.

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tip. The crack opening and sliding profile are extracted from the nodal displacements at the crack surface away from the crack front and far enough from the edges of the plate. An advantage of using the energy approach (Eq. (5)) instead of the J-integral approach is that the values needed for Eq. (5) are more straight forward to extract from the finite element model and that these values do not depend at all on the actual shape of the crack front. Consequently, Eq. (5) will give the correct result even though the shape of the crack front in the FE model is different from the crack front shape that would be encountered in the real situation. In addition the total ERR can be decoupled into its mode I and II contributions by independent calculation of the two integrals in Eq. (5). The results for the steady state values of GI, GII and Gtot (GI,ss, GII,ss, Gtot,ss) are reported in Table 3 for the thick and thin layers of both the specimens configurations. It is also shown that the total steady state ERR computed with Eq. (5) is consistent with the steady state value of the J-integral, shown in Fig. 9, their difference being lower than 3%. It is clear from Eq. (5) that the ERR components can be seen as the product between the stress components in the uniform far field and the average crack face opening and sliding displacements. The latter depend on the thickness of the cracked layer and also on the stiffness of the constraining plies, as highlighted by Varna et al. [33] and Singh and Talreja [34]. In the case of the present lay-up the constraining effect provided by the surrounding plies is reasonably expected to be very similar for the thin and thick layers. In such a case, dimensional analysis reveals that the ERR must scale according to

Gtot;ss /

r 2x h

ð6Þ

E0

 x is the remote applied stress (averaged across the laminate where r thickness) or a stress component of the layer. It is therefore convenient to normalise the energy release rate as follows, according to Hutchinson and Suo [35]:

f ¼

Gtot;ss  E0 r22  h

ð7Þ

where E0 ¼ E2 =ð1  m221 Þ. The normalised parameter f does not depend on the stress level, identified by the transverse stress r2, and on the layer thickness h. However, in general f does depend on the thickness, stiffness and orientation of the constraining layers and – possibly only weakly – on the thickness and orientation of the other plies in the laminate. As can be observed from Table 3, the value of f is very similar for the thin and thick plies for each lay-up (a 1% difference can be considered negligible), thus confirming that the constraining effect for the thin and thick layers can be considered to be the same. Eventually, as the two laminates have different lay-up and the parameter f is related to the total ERR, its value is clearly not the same for the two laminate configurations. As shown in Section 3, multiple cracking occurred during fatigue life. This produces a shielding effect, the consequence of which is a lower value of the steady state ERR as the crack density increases. This phenomenon has already been shown in [31,32]. In the present work the normalised driving force f was computed

with the J-integral approach considering several values of the crack density. Regularly spaced cracks with the same length, high enough to provide a steady state growth condition, were modelled along the entire specimen and the steady state value of the J-integral was computed and then normalised according to Eq. (7). The results for the [0/502/0/502]s specimens are plotted in Fig. 11 against L/h, where L is the crack spacing equal to the inverse of the crack density. It is worth mentioning that the mode decomposition could be performed also for the parameter f, normalising the mode I and II ERRs by the transverse and shear stress, respectively. This calculation was not carried out in the present work since the J-integral technique was used for the computation of the total ERR in the presence of multiple cracks. Therefore, in the multiple cracking regime the mode I and II components were not calculated and are not known. However the analysis presented above and in Fig. 11 on the parameter f was simply aimed to understand the maximum value of crack density (minimum crack spacing) ensuring cracks propagating in a non-interactive regime. To this aim, it can be observed that when L/h is larger than four the f value is constant and equal to that computed for an isolated crack. This allowed us to consider, for the analysis of the steady state propagation phase, the first few cracks initiated in each specimen, as long as the crack density was low enough, and therefore each crack could be considered as independent and non-interacting with the neighbouring cracks. 6. Crack growth rates versus energy release rate After calculating the steady state value of the ERR, the crack propagation data reported in Fig. 8 can be presented in terms of Paris-like curves relating the crack growth rate to the total ERR, Gss. In Fig. 12a and b the results are reported for h = 50° and 60°, respectively. It can be seen that in both cases the series of data for the thick and thin plies are reasonably well collapsed in a single scatter band. This allows us to conclude that the steady state ERR can be regarded as the driving force for the propagation of tunnelling cracks, as it is suitable to capture correctly the influence of the layer thickness on the CGR. This implies that it is possible to transfer crack growth rate data from one ply thickness to another providing that the mode mixity is the same. This suggests that a rational approach to cyclic crack growth of off-axis plies is to obtain the crack growth rate data (for various mode mixities) experimentally for a single ply thickness only and then using these data to predict the crack growth rate of other ply thicknesses using the fracture mechanics scaling shown in Eq. (6). In Fig. 12a and b all data for the thick and thin plies have been analysed together and fitted with the typical power law CGR ¼ D  Gntot;ss represented by a solid line. The 90–10% scatter bands are also calculated. The coefficients of the Paris-like curves are listed in Table 4. 7. Comparison with external multiaxial fatigue results In this section the fatigue test results of the present activity are compared to those obtained by Quaresimin and Carraro on tubular specimens, made of the same pre-preg material [14]. Glasss/epoxy tubes with lay-up [0F/90U,3/0F], where the subscripts F and U mean

Table 3 ERR components, J-integral and f parameter for e = 1.1%. Lay-up

[0/502/0/502]s [0/602/0/602]s

Thick

Thin

GI,ss (kJ/m2)

GII,ss (kJ/m2)

Gtot,ss (kJ/m2)

Jss (kJ/m2)

f

GI,ss (kJ/m2)

GII,ss (kJ/m2)

Gtot,ss (kJ/m2)

Jss (kJ/m2)

f

0.207 0.527

0.382 0.246

0.589 0.773

0.607 0.797

1.94 1.03

0.112 0.263

0.206 0.123

0.318 0.386

0.326 0.399

1.96 1.03

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2.0

[0/502/0/-502]s

thick (L12 12 = 1.15, R = 0.1) thin (L12 = 1.15, R = 0.1) 12

1.0

[MPa]

1.5

50°

L 0

2

4

6

x

8

10 1E+02

10

L/h

(b)

1E+04

1E+05

1E+06

1E+07

100

[MPa]

[0/602/0/-602]s

1.E+0

[0/502/0/-502]s thin (MM =0.65) thick (MM = 0.65) 1.E -2 1.E -1

2,max

CGR [mm/cycle]

1E+03

Nf

Fig. 11. Normalised steady-state energy release rate f versus the normalised crack spacing L/h, lay-up: [0/502/0/502]s.

(a)

tubes (L12 = 1, R = 0.05) 12

2,max

x

h

0.5 0.0

100

(a)

thin (L12 = 0.57, R = 0.1) 12 thick (L12 12 = 0.57, R = 0.1)

1.E -3

tubes (L12 12 = 0.5, R = 0.05)

10 1.E+00

1.E -4

1.E+02

1.E+04

1.E+06

Nf

1.E -5 1.E -6 0.01

0.1

1

Fig. 13. Comparison between the SN data for the first cracks initiation for tubes [14] and present results for (a) [0/502/0/502]s and (b) [0/602/0/602]s specimens.

1

constraining effect provided by the surrounding layers which are slightly different for the two kinds of samples and the presence of a certain amount of longitudinal stress r1 in the flat coupons. It can also be noted that the slope of the curves is consistent, and this is a very important issue since it represents the rate of the damage evolution during fatigue life, which is found to depend on the local multiaxial stress state, no matter if it is originated by external multiaxial loads (tubes) or by the material anisotropy (flat specimens). The crack growth data for the flat specimens presented in Section 6 in terms of the total ERR are also compared, in Fig. 14, to

(b)

1.E+0

CGR [mm/cycle]

Gss = GI,ss+GII,ss [kJ/m2]

1.E-1 1.E-2

[0/602/0/-602]s thin (MM=0.32) thick (MM=0.32)

1.E-3 1.E-4 1.E-5 1.E-6 0.01

0.1

Gss = GI,ss+GII,ss [kJ/m2]

Table 4 Coefficients of the Paris-like curves (when Gtot,ss in kJ/m2 and CGR in mm/cycles). Lay-up

n

D50%

D90%

[0/502/0/502]s [0/602/0/602]s

4.41 3.79

0.671 0.816

1.646 4.124

CGR [mm/cycle]

fabric and unidirectional, respectively, were tested with a load ratio R = 0.05. The attention was focused on the initiation and propagation of cracks in the 90° UD plies, made of the UE-400/REM tape. In Fig. 13a and b the S–N data for the initiation of the first crack in the 90° layers of the tubes, taken from Ref. [14], are shown and compared to those for the first cracks initiation in the [0/h2/0/ h2]s laminates presented in Section 3. The tubes’ results for k12 = 1 are compared to those for the flat laminates with h = 50°, corresponding to a biaxiality ratio of 1.15, whereas the data for the tubes with k12 = 0.5 are compared to those of the flat specimens for h = 60°, with biaxiality ratio equal to 0.57. In both cases it can be seen that the results are reasonably consistent. The positions of the S–N data for the tubes are only slightly lower than those for the flat laminates, and this can be due to small differences in the cured ply properties (fibre volume fraction for instance), the

(a)

1.E+0 1.E -1 1.E -2

thin (+50 , MM =0.65) thick (-50 , MM = 0.65) tubes (MM = 0.56)

1.E -3 1.E -4 1.E -5 1.E -6 0.01

0.1

1

Gss = GI,ss+GII,ss [kJ/m2]

(b) CGR [mm/cycle]

Fig. 12. Steady-state crack growth rates as a function of the energy release rate for (a) [0/502/0/502]s and (b) [0/602/0/602]s specimens in terms of Gtot,ss and associate 10–90% scatter bands.

1.E+0 1.E -1 1.E -2

thin (+60 , MM=0.32) thick (-60 , MM=0.32) tubes (MM=0.25)

1.E -3 1.E -4 1.E -5 1.E -6 0.01

0.1

1

Gss = GI,ss+GII,ss [kJ/m2] Fig. 14. Comparison between the steady-state crack growth data for tubes [14] and present results for (a) [0/502/0/502]s and (b) [0/602/0/602]s specimens.

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those obtained for the tubes with a similar multiaxial condition, i.e. k12 = 1 and 0.5 (MM = 0.56 and 0.25) [14]. Also in this case it can be seen that the results for similar values of the MM are reasonably consistent, confirming that the ERR is the driving force for the crack propagation, for a given mode mixity. It can be concluded that the damage initiation and evolution under fatigue loading depends on the local multiaxial conditions only, expressed in terms of the ratios between the ply stresses in the material reference system, or in terms of the mixity ratio between the mode I and II components of the ERR. As a consequence, the characterisation of the multiaxial fatigue behaviour of composite laminae and laminates can be carried out with flat coupons, taking advantage of the local multiaxial stress state originated by the material anisotropy. This is far easier than adopting more complicated test configurations in which external loads are applied in multiple directions (tubes under tension/torsion for instance). 8. Conclusions The results of an extensive experimental campaign on multidirectional glass/epoxy laminates have been presented and discussed. The aim of the work was to compare the evolution of fatigue damage in these laminates with that measured on tubes tested under tension–torsion loading conditions. The lay-up of the laminates was designed in such a way to introduce a local multiaxial stress state comparable to that present in the tubes subjected to external multiaxial loading. A theoretical approach for the calculation of the average steady-state energy release rate of a tunnelling crack propagating under mixed mode was presented. The approach enables the assessment of the average mode mixity at the crack front. The comparison of S–N data to the initiation of fatigue cracks and Paris-like propagation curves indicates that the evolution of fatigue damage in multidirectional laminates tested under uni-axial cyclic loading and tubes tested under external multi-axial (tension–torsion) loading is basically the same, provided that the local multiaxial stress state described in terms of biaxiality ratios is similar. This result is very important in view of an extensive characterisation of the influence of multiaxial stress state on the fatigue response of composite laminates. Finally, it was found that the difference in the steady-state growth rates of tunnelling crack could be understood by the way the steady-state energy release rate theoretically scales with ply thickness. Acknowledgments The present paper is the result of the collaborative work of an international group of researchers (the ‘‘Multiaxial Fatigue team’’) aiming to improve the understanding of the multiaxial fatigue behaviour of composites with the final aim of defining suitable and damage-based criteria and design tools. Two authors (L.P. Mikkelsen and B.F. Sørensen) were partially supported by the Danish Centre for Composite Structures and Materials for Wind Turbines (DCCSM), Grant No. 09-067212 from the Danish Strategic Research Council. References [1] Quaresimin M, Susmel L, Talreja R. Fatigue behaviour and life assessment of composite laminates under multiaxial loadings. Int J Fatigue 2010;32:2–16. [2] Diao X, Ye L, Mai YW. A statistical model of residual strength and fatigue life of composite laminates. Compos Sci Technol 1995;54:329–36.

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