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Transverse crack growth behavior considering free-edge effect in quasi-isotropic CFRP laminates under high-cycle fatigue loading
Composites Science and Technology 69 (2009) 1388–1393
Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Transverse crack growth behavior considering free-edge effect in quasi-isotropic CFRP laminates under high-cycle fatigue loading Atsushi Hosoi a,*, Yoshihiko Arao a, Hiroyuki Kawada b a b
Graduate School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
a r t i c l e
i n f o
Article history: Received 5 December 2007 Received in revised form 22 April 2008 Accepted 7 September 2008 Available online 13 September 2008 Keywords: B. Fatigue C. Transverse cracking B. Durability C. Stress concentration
a b s t r a c t The high-cycle fatigue characteristics focused on the behavior of the transverse crack growth up to 108 cycles were investigated using quasi-isotropic carbon fiber reinforced plastic (CFRP) laminates whose stacking sequence was [45/0/45/90]s. To assess the fatigue behavior in the high-cycle region, fatigue tests were conducted at a frequency of 100 Hz in addition to 5 Hz. In this study, to evaluate quantitative characteristics of the transverse crack growth in the high-cycle region, the energy release rate considering the free-edge effect was calculated. Transverse crack growth behavior was evaluated based on a modified Paris law approach. The results revealed that transverse crack growth was delayed under the test conditions of the applied stress level of rmax/rb = 0.2. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction CFRPs are expected to replace metal materials in various fields in the future, especially, as the structures of airplanes, automobiles, trains and so on, since they have excellent mechanical properties, such as lightness, high strength and good moldability. In particular, when CFRP laminates are used as components in automobiles and high-speed trains, they are subjected to a cyclic loading of over 108 cycles. Moreover, it is noted that high-cycle fatigue fractures are the main causes of structural damage. Thus, it is important to investigate the high-cycle fatigue characteristics of CFRP laminates to establish the long-term reliability. Many studies about fatigue damage mechanisms have been conducted. Typical damage characteristics in CFRP laminates are intralaminar cracks followed by delamination and finally fiber raptures due to being subjected to cyclic loading. The intralaminar crack initiates first and causes the delamination due to the stress concentration at the matrix crack tip. Therefore it is important to focus on the behavior of the transverse crack growth in high-cycle regions to establish the long-term reliability of CFRP laminates. Many studies have been conducted on the transverse crack growth behavior [1–11]. Nairn [5] calculated the energy release rate associated with initiation of the new microcrack between existing the microcracks considering the residual thermal stress with a variational approach proposed by Hashin [6]. Moreover, Liu and Nairn [7] assessed the growth of transverse crack density in cross-ply * Corresponding author. Tel.: +81 3 5286 3261; fax: +81 3 5273 2667. E-mail address: [email protected] (A. Hosoi). 0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.09.003
laminates during fatigue loading by applying the analysis proposed by Nairn [5]. They showed that the master Paris law plot gave a characterization of microcrack formation during fatigue loading. Yokozeki et al. [8] researched transverse cracks propagating in the width direction of specimens with several CFRP cross-ply laminates and quasi-isotropic laminates during fatigue loading. They showed that the derived energy release rate was independent of the crack length, and the transverse crack growth in the width direction could be evaluated with the modified master Paris law plot. However, these studies researched the fatigue characteristics up to 106 cycles. Hosoi et al. [9] evaluated transverse crack propagation with quasi-isotropic [45/0/45/90]s CFRP laminates by conducting high-cycle fatigue test up to 108 cycles. They indicated that the lower threshold of transverse crack propagation exists by using the modified Paris law approach. It is known that the transverse crack growth behavior is different depending on the stacking sequence of laminates. In [0m/90n]s CFRP laminates, the transverse cracks propagate to the specimen width simultaneously with the initiation of the transverse cracks. On the other hand, in [45/0/45/90]s CFRP laminates, the transverse cracks begin to propagate to the specimen width once the small transverse cracks increase at the specimen edges because the axial normal stress applied in the 90° ply increases at the specimen edges by the free-edge effect. Yokozeki [10] evaluated such transverse crack growth behavior with virtual crack closure method (VCCM). In the present study, considering free-edge effect, the energy release rate that was associated with the formation of small transverse cracks was derived with the interaction between transverse crack formation and propagation. Moreover, under the
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The material system used in this study was T700S/2500. The stacking sequence of laminates was [45/0/45/90]s. All specimens were 210 mm long, 30 mm wide, and 1.0 mm thick with 55 mm GFRP end-tabs leaving a 100 mm gauge section. Young’s modulus and tensile strength of the laminates were 45.3 GPa and 826 MPa, respectively. The mechanical properties of the specimen are shown in Table 1. Tensile fatigue tests were conducted at room temperature with a sine waveform under load control conditions using a hydraulic-driven testing machine. All tests were run at a stress ratio of R = 0.1 and the selected maximum stress levels were 20–60% of the static tensile strength, rb. When the fatigue test was conducted at a frequency of 5 Hz, the applied stress level was set within rmax/ rb = 0.2–0.6. At a frequency of 100 Hz, the applied stress level was set within rmax/rb = 0.2–0.3 by considering the effect of the specimen temperature rising under fatigue tests. Then, it was shown that the influence of the frequency on the behavior of transverse crack growth was small when fatigue tests were conducted under the conditions that the temperature rising of the specimens was small [9]. To observe damage behavior, both an optical microscope and soft X-ray photography were used. The edges of the specimens were polished and it was checked that initial defects did not exist in the specimens before the fatigue tests. 3. Experimental results The quasi-isotropic CFRP laminates show a highly complex fracture mode as follows [9]. In the primary stage of the fatigue test, transverse cracks were initiated in the 90° ply. Then, the number of transverse cracks reached saturation state at the specimen edges. Almost simultaneous with the saturation of transverse cracks, delamination in the interlaminar area of the 45°/90° plies or the 90°/90° plies, and the 45° ply from the transverse crack tip of the 90° ply occurred at the specimen edges. The transverse cracks of 90° ply and the delamination that initiated at the specimen edges gradually propagated to the width direction of the specimen with the increase of the number of cycles. Moreover, matrix cracks of the 45° were observed at the edges. Finally, the specimen radically fractured by the fibers breaking in the 0° ply. In this study, a key focus was the transverse crack growth in the 90° plies. The behavior of transverse crack growth is shown Figs. 1 and 2. Fig. 1 shows the transverse crack density initiated at the specimen edge as a function of the number of cycles normalized by fatigue life predicted with a S–N curve. Fig. 2 shows the average transverse crack length normalized by specimen width as a function of the number of cycles normalized by fatigue life with the S–N curve. From the two graphs, it was found that the transverse crack propagated in the width direction of the specimen once it
Table 1 Mechanical properties of T700S/2500 unidirectional composite Longitudinal Young’s modulus Transverse Young’s modulus In-plane shear modulus In-plane Poisson’s ratio Out-of-plane Poisson’s ratio Longitudinal thermal expansion coefficient Transverse thermal expansion coefficient Manufacturing temperature a
Assumed value.
(GPa) (GPa) (GPa)
(106 K1) (106 K1) (K)
131.2 9.25 3.50 0.37 0.49a 0.3 36.5 403
Transverse crack density mm
2. Experiment
2.0
1.5 f= 5 Hz σ max /σ b=0.6 f= 5 Hz σ max /σ b=0.5 f= 5 Hz σ max /σ b=0.4 f= 5 Hz σ max /σ b=0.3 f= 5 Hz σ max /σ b=0.2 f=100 Hz σ max /σ b=0.3 f=100 Hz σ max /σ b=0.2
1.0
0.5
0 10 -7
10 -6
10 - 5
10 -4
10 -3
10 -2
10 -1
10 0
Number of cycles normalized by fatigue life predicted with S-N curve N/N f Fig. 1. Transverse crack density observed at specimen edge as a function of number of cycles normalized by fatigue life predicted with S–N curve.
Average transverse crack length normalized by specimen width
high-cycle fatigue test up to 108 cycles, the growth behavior of small transverse crack was evaluated with the modified Paris law approach.
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A. Hosoi et al. / Composites Science and Technology 69 (2009) 1388–1393
1.0 0.8 0.6 f= 5 Hz σ max/σ b =0.6 f= 5 Hz σ max/σ b =0.5 f= 5 Hz σ max/σ b =0.4 f= 5 Hz σ max/σ b =0.3 f= 5 Hz σ max/σ b =0.2 f=100 Hz σ max/σ b =0.3 f=100 Hz σ max/σ b =0.2
0.4 0.2 0 10 -7
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
10 0
Number of cycles normalized by fatigue life predicted with S-N curve N/N f Fig. 2. Average transverse crack length propagating in width direction of specimen as a function of number of cycles normalized by fatigue life predicted with S–N curve.
saturated at the specimen edge. As shown in Fig. 3, the small transverse crack growth behavior before propagating to the width direction of the specimen was evaluated quantitatively in this study. 4. Derivation of the energy release rate considering the freeedge effect In this study, a unit cell model for the transverse crack in [(S)/ 90]s laminates was considered. (S) is any orthotripic sub-laminate. The energy release rate associated with the formation of new small transverse crack between the existing small transverse cracks was calculated. A schematic illustration for calculation is shown in Fig. 4. In conventional analysis, the energy release rate that was associated with the new microcracking was calculated as the crack passed through the width direction of a specimen at the same time as its initiation [5]. However, the behavior of the transverse crack growth is different depending on the configuration of laminates or the material properties. It is known that the free-edge effect causes the initiation of the small transverse crack [10]. Hence, the energy release rate for the small transverse crack initiated at the specimen edge between the existing small cracks was derived considering the free-edge effect. By the energy balance, the energy release rate due to the formation of a new microcrack between the existing microcracks is derived following Nairn [11,12]. The energy release rate for propagation of a microcrack of current area, A, is
Gm ðAÞ ¼
oU ext oU oA oA
ð1Þ
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A. Hosoi et al. / Composites Science and Technology 69 (2009) 1388–1393
Small transverse cracks
6mm
2mm
Loading direction Fig. 3. Observation of the small transverse cracks with soft X-ray photography. (rmax/rb = 0.4, n = 1500 cycles and f = 5 Hz).
Small transverse crack
2W
[S]
[90]
[S]
[S]
Loading direction
[S]
[90]
s 2s
x
s Δa t2
t1
z
B
y Fig. 4. Schematic illustration of representative element of the laminate. (A) Two microcracks in the 90° ply. (B) The formation of a microcrack between the existing microcracks.
where, Uext is external work and L is total strain energy. Here, external work is expressed as
U ext ¼ 2r0 BWu
ð2Þ
where, r0 is axial stress, and B, W and u are specimen thickness, half of specimen width and the total axial displacement between the two existing microcracks, respectively. The total axial displacement can be written as
u ¼ 2s
r0 EA
þ aA T
ð3Þ
where, EA and aA are the effective axial modulus and thermal expansion coefficient of the laminate. Moreover, T is the temperature difference between the test temperature and residual stressfree temperature. On the other hand, total strain energy can be express as a sum of mechanical strain energy and thermal strain energy
1 r20 U ¼ 4sBW þ U res 2 EA
ð4Þ
where Ures is the residual strain energy per unit volume. Substituting Eqs. (2) and (4) into Eq. (1), the energy release rate is expressed as
G ¼ 4sBW
1 2 o 1 o o r0 r T a ð Þ U þ 0 res 2 oA EA oA A oA
ð5Þ
Following Nairn [11,12], the effective thermal expansion coefficient, aA and the residual strain energy per unit volume, Ures, can be derived. Substituting them into Eq. (5), the energy release rate is expressed in terms of effective modulus,
DaTE0 G ¼ 2sBW r0 C 1 Eð90Þ xx
!2
o 1 oA EA
ð6Þ
where, ðSÞ Da ¼ að90Þ xx axx ; E0 ¼
ðSÞ Eð90Þ 1 1 ð1 þ kÞE0 xx þ kExx ; C 1 ¼ ð90Þ þ ðSÞ ¼ ð90Þ ðSÞ 1þk Exx kExx kExx Exx
ð7Þ ð90Þ xx ,
a
ðSÞ xx ,
Eð90Þ xx
EðSÞ xx
a and are the thermal expansion coefficients and axial modulus in 90° ply and sub-laminate. k = ts/t90, and E0 is the rule-of-mixtures axial modulus. By simple laminated plate calculation, the applied stress partitioned into the 90° ply is expressed as
r0ð90Þ ¼ xx
Eð90Þ xx r0 E0
ð8Þ
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r0ð90Þ xx
Eð90Þ xx
ðSÞ kEð90Þ xx Exx DaT ð1 þ kÞE0
o 1 oA EA
ð9Þ
The term of the thermal residual stress is equal to the residual thermal stress applied in 90° ply of the undamaged laminate. Thus, Eq. (9) can be written as
G ¼ 2sBW
E0
!2
Eð90Þ xx
1 Ið90Þ 2 o rxx oA EA
ð10Þ
Ið90Þ
where, rxx is a sum of mechanical stress and residual thermal stress in the undamaged laminate. Thus, it is found that the energy release rate associated with transverse crack growth depends on the stress applied in 90° ply and the change of effective axial modulus associated with transverse crack growth. In this study, the small transverse cracks caused in the specimen edges are considered. Yokozeki [10] showed that the transverse cracks in quasi-isotropic [45/0/45/90]s laminates once halted without passing through the specimen width simultaneously with the transverse crack initiation by calculating the energy release rate associated with transverse crack propagation with VCCM. The reason why the small transverse cracks are once caused at the specimen edges is the increase of the axial stress applied in 90° ply by free-edge effect. Considering the increase of axial applied stress in 90° ply in the vicinity of specimen edges, the energy release rate associated with the small transverse crack formation is expressed as
G ¼ 2sBW
E0
!2
Eð90Þ xx
r0Ið90Þ xx
2 o 1 oA EA
ð11Þ
0Ið90Þ
where, rxx is a sum of the mechanical stress, the residual thermal stress and the incremental stress due to the free-edge effect of 90° ply. As shown in Fig. 4, new small transverse crack area is expressed as
dA ¼ 2t90 2Da
ð12Þ
where, 2t90, and Da are the thickness of 90° ply and the length of small transverse crack. Moreover, it was assumed that the specimen width was sufficiently great comparing with the length of a crack. Substituting Eq. (12) into Eq. (11), the energy release rate is expressed as
BW E0 G¼ 4qt 90 Da Eð90Þ xx
!2
ð1 þ kÞW E0 ¼ 2qDa Eð90Þ xx
2
r0Ið90Þ D xx
!2
1 EA
0Ið90Þ 2
rxx
1 1 EA ðq=2Þ EA ðqÞ
ð13Þ
where, q = 1/2s. Moreover, the effective laminate modulus in the axial direction, EA ðqÞ, is expressed with the rule-of-mixture as
DaEeff TC ðqÞ þ ðW DaÞE0 EA ðqÞ ¼ W
ð14Þ
where, Eeff TC ðqÞ is the laminate modulus in the x direction when the aspect ratio of transverse cracks that pass through the width direction of the specimen is q. Eeff TC ðqÞ can be calculated by using the method proposed by Joffe and Varna [13]. The energy release rate associated with the new small transverse crack considering the residual thermal stress and the fee-edge effect can be calculated 0Ið90Þ by using Eq. (13). However, rxx is unknown in Eq. (13). Thus, in this study the energy release rate associated with the new microcrack formation considering the free-edge effect was calculated
with the theories based on the interaction between the initiation and the propagation of transverse cracks. The detail is described in the next section. 5. Interaction between the initiation and the propagation of the transverse cracks In this study, it was observed that a transverse crack initiated at the specimen edges, and then propagated to the width of the specimen after the number of transverse cracks reached a saturation state at the specimen edges as shown in Figs. 1 and 2. Fig. 5 shows that the typical relationship between the density of transverse crack initiated at the specimen edge, and the normalized length of transverse cracks propagated to the width of the specimen, as a function of cycles under the test condition at the applied stress level of rmax/rb = 0.4. In this study, the energy release rate associated with the formation of new transverse cracks considering the free-edge effect was calculated with the interaction between the initiation and the propagation of the transverse crack. The applied stress in 90° ply considering the free-edge effect was defined based on the theories as follows. The transverse cracks that caused at the edges of the specimen started to propagate to the specimen width at the transverse crack density of 1.5 mm1 at the applied stress level of rmax/rb = 0.4 as shown in Fig. 5. That is to say, the energy release rate range (ERRR) associated with the formation of new transverse crack must be equal to the ERRR associated with the propagation of the transverse cracks when the density of the transverse crack reaches 1.5 mm1. The energy release rate associated with the transverse crack propagation was calculated with the approach proposed by Hosoi et al. [9]. The energy release rate is expressed as
t1 þ t2 Gt ¼ 2q1 t1
t EðuÞ r0 2 xx Daxx T t1 þ t2
!2
E0 Ec
! ð15Þ
ðExx Þ2
where t1, t2 and q1 are thickness of 90° ply and undamaged plies, and ð1Þ ðuÞ the crack density of 90° plies, respectively. Then, Daxx ¼ axx axx is the difference between the axial direction thermal expansion coefficients of the 90° ply and undamaged plies, and T = TS T0 is the difference between the test temperature TS and the residual stress-free temperature T0. Moreover, EðuÞ xx ; Exx , Ec and E0 are the rule-of-mixtures axial modulus of the uncracked plies, the effective laminate modulus in the axial direction, the axial laminate modulus that transverse cracks pass through the specimen width and the undamaged laminate modulus in the axial direction, respectively. Thus, using the condition that the ERRR by new crack formation is equal to the ERRR by the crack propagation at the transverse crack density of 1.5 mm1, the applied stress of 90° plies consider-
2.0
Crack density Crack length
saturation
1.0 0.8
1.5
0.6 1.0 0.4 0.5
0.2 propagation
0 10 0
10 1
10 2
10 3
10 4
10 5
0 10 6
Average transverse crack length normalized by specimen width
E0
!2
-1
G ¼ 2sBW
!2
Transverse crack density mm
Substituting Eq. (8) into Eq. (6),
Number of cycles N Fig. 5. Transverse crack density and normalized transverse crack length as a function of cycles under the test condition of rmax/rb = 0.4.
Formation (with free edge effect) Formation (without free edge effect) Propagation
-1
Energy release rate range ΔG J/m
120
Crack density growth rate mm /cycle
A. Hosoi et al. / Composites Science and Technology 69 (2009) 1388–1393 2
1392
100 Intersection
Free-edge effect
80
60
40 0
0.5
1
1.5
Transverse crack density mm
2
100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10
-1
with free-edge effect (5 Hz) with free-edge effect (100 Hz) without free-edge effect (5 Hz) without free-edge effect (100 Hz)
50
100
500
1000
Energy release rate range J/m2
Fig. 6. Energy release rate range as a function of transverse crack density under the test condition of rmax/rb = 0.4.
Fig. 7. The transverse crack growth rate as a function of the energy release rate range by comparing the results of the consideration of the free-edge effect with non-consideration of the free-edge effect.
0Ið90Þ
ing free-edge effect, rxx , was calculated with Eqs. (13) and (15). In the case of the applied stress level of rmax/rb = 0.3, the value of the saturation state was about 1.0 mm1 as shown in Fig. 1. Hence the ERRR associated with the formation of new transverse crack at the condition was calculated to be equal to the ERRR associated with the transverse crack propagation at the crack density of 1.0 mm1. Figure 6 shows two results in comparison to the transverse crack propagation. That is to say, the figure shows the comparison of the ERRRs associated by the formation of new transverse cracks whose length is Da as a function of crack density with the ERRR associated by the propagation from transverse cracks whose length is Da as a function of crack density. In this study, the energy release rate was calculated as Da = 0.42 mm from the average of experimental results. The solid line shows the ERRR for the formation of new small cracks considering the free-edge effect, the dash-dotted line shows the ERRR for the formation of new small cracks without considering the free-edge effect and the dashed line shows the ERRR associated with the propagation of cracks. In Fig. 6, comparing the ERRR associated with the crack formation considering the free-edge effect (the solid line) with the ERRR associated with the crack propagation (the broken line), it is explained that the transverse crack of the length of Da is caused at the specimen edge until the crack density of 1.5 mm1 because the value of the ERRR for the crack formation is greater than that for the crack propagation, and then the transverse cracks propagate, after the density of the transverse crack reaches 1.5 mm1. On the other hand, comparing the energy release rate range associated by the crack formation without considering the free-edge effect (the dash-dotted line) with the ERRR associated with the crack propagation (the dashed line), the values of the ERRRs are almost equal until the crack density of 1.0 mm1. The results explain that cracks propagate to the width of a specimen at the same time as the crack initiates in the laminates which the free-edge effect is small such as cross-ply laminates which have thick 90° plies in the middle of laminates or GFRP laminates. 6. Evaluation with modified Paris law approach The transverse crack growth behavior was evaluated by the modified Paris law approach as
dDTC ¼ AðDGÞn dN DG ¼ Gmax Gmin
ð16Þ ð17Þ
where DTC is the transverse crack density, and A and n are constants. Figure 7 shows the transverse crack growth rate as a function of the ERRR by comparing the results of the consideration of the free-edge
effect with non-consideration of the free-edge effect under the test conditions of the frequency of 5 Hz and 100 Hz. From the result shown in Fig. 7, it was observed that the transverse crack growth rate was considerably delayed under the applied stress level of rmax/rb = 0.2. The results imply that a threshold exists for the transverse crack initiation. Then, it was found that the energy release rate associated with the formation of new crack was estimated to be smaller when the free-edge effect was not considered. 7. Conclusion The growth behavior of small transverse cracks was evaluated under high-cycle fatigue loadings up to 108 cycles. The following results are obtained in this study. The energy release rate associated with the small transverse crack formation could be derived with the relationship that the energy release rate associated with the formation of new transverse crack is equal to that associated with the propagation of transverse cracks when the transverse cracks begin propagating to the width direction of the specimen. Moreover, it was implied that the lower threshold of transverse crack initiation exists by evaluating the transverse crack growth behavior with the modified Paris law approach. Acknowledgements The authors wish to thank Dr. Ogasawara of Japan Aerospace Exploration Agency and Dr. Yokozeki of University of Tokyo for help to use the soft X-ray photography. This work was supported in a part of the 21st century Center of Excellence program. References [1] Masters JE, Reifsnider KL. An investigation of cumulative damage development in quasi-isotropic graphite/epoxy laminates. ASTM STP 1982;775:40–62. [2] Nairn JA. Matrix microcracking composites. In: Talreja R, Månson JAE, editors. Polymer matrix composites; 2000. p. 403–32. [3] Tong J. Characteristics of fatigue crack growth in GFRP laminates. Int J Fatigue 2002;24:291–7. [4] Ogihara S, Takeda N, Kobayashi S, Kobayashi A. Damage mechanics characterization of transverse cracking behavior in quasi-isotropic CFRP laminates with interlaminar-toughened layers. Int J Fatigue 2002;24:93–8. [5] Nairn JA. The strain energy release rate of composite microcracking: a variational approach. J Compos Mater 1989;23:1106–19 [and errata: J Compos Mater 1990;24:223–4]. [6] Hashin Z. Analysis of cracked laminates: a variational approach. Mech Mater 1985;4:121–36. [7] Liu S, Nairn JA. Fracture mechanics analysis of composite microcracking: experimental results in fatigue. In: Proceedings of the fifth technical conference on composite materials. East Lansing, Michigan: American Society for Composites; 1990. p. 287–95.
A. Hosoi et al. / Composites Science and Technology 69 (2009) 1388–1393 [8] Yokozeki T, Aoki T, Ishikawa T. Fatigue growth of matrix cracks in the transverse direction of CFRP laminates. Compos Sci Technol 2002;62: 1223–9. [9] Hosoi A, Arao Y, Karasawa H, Kawada H. High-cycle fatigue characteristics of quasi-isotropic CFRP laminates. Adv Compos Mater 2007;16:151–66. [10] Yokozeki T. Multiple ply matrix cracking behavior in composite laminates. PhD Thesis. University of Tokyo, Department of Aeronautics and Astronautics; 2004 [in Japanese].
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[11] Nairn JA. Some new variational mechanics results on composite microcracking. In: Proceedings of 10th international conference on composite materials (ICCM-10). Whistler BC, Canada; 1995. p. 423–30. [12] Nairn JA. Fracture mechanics of composite with residual thermal stresses. J Appl Mech 1997;64:804–10. [13] Joffe R, Varna J. Analytical modeling of stiffness reduction in symmetric and balanced laminates due to cracks in 90° layers. Compos Sci Technol 1999;59:1641–52.
Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Transverse crack growth behavior considering free-edge effect in quasi-isotropic CFRP laminates under high-cycle fatigue loading Atsushi Hosoi a,*, Yoshihiko Arao a, Hiroyuki Kawada b a b
Graduate School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
a r t i c l e
i n f o
Article history: Received 5 December 2007 Received in revised form 22 April 2008 Accepted 7 September 2008 Available online 13 September 2008 Keywords: B. Fatigue C. Transverse cracking B. Durability C. Stress concentration
a b s t r a c t The high-cycle fatigue characteristics focused on the behavior of the transverse crack growth up to 108 cycles were investigated using quasi-isotropic carbon fiber reinforced plastic (CFRP) laminates whose stacking sequence was [45/0/45/90]s. To assess the fatigue behavior in the high-cycle region, fatigue tests were conducted at a frequency of 100 Hz in addition to 5 Hz. In this study, to evaluate quantitative characteristics of the transverse crack growth in the high-cycle region, the energy release rate considering the free-edge effect was calculated. Transverse crack growth behavior was evaluated based on a modified Paris law approach. The results revealed that transverse crack growth was delayed under the test conditions of the applied stress level of rmax/rb = 0.2. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction CFRPs are expected to replace metal materials in various fields in the future, especially, as the structures of airplanes, automobiles, trains and so on, since they have excellent mechanical properties, such as lightness, high strength and good moldability. In particular, when CFRP laminates are used as components in automobiles and high-speed trains, they are subjected to a cyclic loading of over 108 cycles. Moreover, it is noted that high-cycle fatigue fractures are the main causes of structural damage. Thus, it is important to investigate the high-cycle fatigue characteristics of CFRP laminates to establish the long-term reliability. Many studies about fatigue damage mechanisms have been conducted. Typical damage characteristics in CFRP laminates are intralaminar cracks followed by delamination and finally fiber raptures due to being subjected to cyclic loading. The intralaminar crack initiates first and causes the delamination due to the stress concentration at the matrix crack tip. Therefore it is important to focus on the behavior of the transverse crack growth in high-cycle regions to establish the long-term reliability of CFRP laminates. Many studies have been conducted on the transverse crack growth behavior [1–11]. Nairn [5] calculated the energy release rate associated with initiation of the new microcrack between existing the microcracks considering the residual thermal stress with a variational approach proposed by Hashin [6]. Moreover, Liu and Nairn [7] assessed the growth of transverse crack density in cross-ply * Corresponding author. Tel.: +81 3 5286 3261; fax: +81 3 5273 2667. E-mail address: [email protected] (A. Hosoi). 0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.09.003
laminates during fatigue loading by applying the analysis proposed by Nairn [5]. They showed that the master Paris law plot gave a characterization of microcrack formation during fatigue loading. Yokozeki et al. [8] researched transverse cracks propagating in the width direction of specimens with several CFRP cross-ply laminates and quasi-isotropic laminates during fatigue loading. They showed that the derived energy release rate was independent of the crack length, and the transverse crack growth in the width direction could be evaluated with the modified master Paris law plot. However, these studies researched the fatigue characteristics up to 106 cycles. Hosoi et al. [9] evaluated transverse crack propagation with quasi-isotropic [45/0/45/90]s CFRP laminates by conducting high-cycle fatigue test up to 108 cycles. They indicated that the lower threshold of transverse crack propagation exists by using the modified Paris law approach. It is known that the transverse crack growth behavior is different depending on the stacking sequence of laminates. In [0m/90n]s CFRP laminates, the transverse cracks propagate to the specimen width simultaneously with the initiation of the transverse cracks. On the other hand, in [45/0/45/90]s CFRP laminates, the transverse cracks begin to propagate to the specimen width once the small transverse cracks increase at the specimen edges because the axial normal stress applied in the 90° ply increases at the specimen edges by the free-edge effect. Yokozeki [10] evaluated such transverse crack growth behavior with virtual crack closure method (VCCM). In the present study, considering free-edge effect, the energy release rate that was associated with the formation of small transverse cracks was derived with the interaction between transverse crack formation and propagation. Moreover, under the
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The material system used in this study was T700S/2500. The stacking sequence of laminates was [45/0/45/90]s. All specimens were 210 mm long, 30 mm wide, and 1.0 mm thick with 55 mm GFRP end-tabs leaving a 100 mm gauge section. Young’s modulus and tensile strength of the laminates were 45.3 GPa and 826 MPa, respectively. The mechanical properties of the specimen are shown in Table 1. Tensile fatigue tests were conducted at room temperature with a sine waveform under load control conditions using a hydraulic-driven testing machine. All tests were run at a stress ratio of R = 0.1 and the selected maximum stress levels were 20–60% of the static tensile strength, rb. When the fatigue test was conducted at a frequency of 5 Hz, the applied stress level was set within rmax/ rb = 0.2–0.6. At a frequency of 100 Hz, the applied stress level was set within rmax/rb = 0.2–0.3 by considering the effect of the specimen temperature rising under fatigue tests. Then, it was shown that the influence of the frequency on the behavior of transverse crack growth was small when fatigue tests were conducted under the conditions that the temperature rising of the specimens was small [9]. To observe damage behavior, both an optical microscope and soft X-ray photography were used. The edges of the specimens were polished and it was checked that initial defects did not exist in the specimens before the fatigue tests. 3. Experimental results The quasi-isotropic CFRP laminates show a highly complex fracture mode as follows [9]. In the primary stage of the fatigue test, transverse cracks were initiated in the 90° ply. Then, the number of transverse cracks reached saturation state at the specimen edges. Almost simultaneous with the saturation of transverse cracks, delamination in the interlaminar area of the 45°/90° plies or the 90°/90° plies, and the 45° ply from the transverse crack tip of the 90° ply occurred at the specimen edges. The transverse cracks of 90° ply and the delamination that initiated at the specimen edges gradually propagated to the width direction of the specimen with the increase of the number of cycles. Moreover, matrix cracks of the 45° were observed at the edges. Finally, the specimen radically fractured by the fibers breaking in the 0° ply. In this study, a key focus was the transverse crack growth in the 90° plies. The behavior of transverse crack growth is shown Figs. 1 and 2. Fig. 1 shows the transverse crack density initiated at the specimen edge as a function of the number of cycles normalized by fatigue life predicted with a S–N curve. Fig. 2 shows the average transverse crack length normalized by specimen width as a function of the number of cycles normalized by fatigue life with the S–N curve. From the two graphs, it was found that the transverse crack propagated in the width direction of the specimen once it
Table 1 Mechanical properties of T700S/2500 unidirectional composite Longitudinal Young’s modulus Transverse Young’s modulus In-plane shear modulus In-plane Poisson’s ratio Out-of-plane Poisson’s ratio Longitudinal thermal expansion coefficient Transverse thermal expansion coefficient Manufacturing temperature a
Assumed value.
(GPa) (GPa) (GPa)
(106 K1) (106 K1) (K)
131.2 9.25 3.50 0.37 0.49a 0.3 36.5 403
Transverse crack density mm
2. Experiment
2.0
1.5 f= 5 Hz σ max /σ b=0.6 f= 5 Hz σ max /σ b=0.5 f= 5 Hz σ max /σ b=0.4 f= 5 Hz σ max /σ b=0.3 f= 5 Hz σ max /σ b=0.2 f=100 Hz σ max /σ b=0.3 f=100 Hz σ max /σ b=0.2
1.0
0.5
0 10 -7
10 -6
10 - 5
10 -4
10 -3
10 -2
10 -1
10 0
Number of cycles normalized by fatigue life predicted with S-N curve N/N f Fig. 1. Transverse crack density observed at specimen edge as a function of number of cycles normalized by fatigue life predicted with S–N curve.
Average transverse crack length normalized by specimen width
high-cycle fatigue test up to 108 cycles, the growth behavior of small transverse crack was evaluated with the modified Paris law approach.
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A. Hosoi et al. / Composites Science and Technology 69 (2009) 1388–1393
1.0 0.8 0.6 f= 5 Hz σ max/σ b =0.6 f= 5 Hz σ max/σ b =0.5 f= 5 Hz σ max/σ b =0.4 f= 5 Hz σ max/σ b =0.3 f= 5 Hz σ max/σ b =0.2 f=100 Hz σ max/σ b =0.3 f=100 Hz σ max/σ b =0.2
0.4 0.2 0 10 -7
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
10 0
Number of cycles normalized by fatigue life predicted with S-N curve N/N f Fig. 2. Average transverse crack length propagating in width direction of specimen as a function of number of cycles normalized by fatigue life predicted with S–N curve.
saturated at the specimen edge. As shown in Fig. 3, the small transverse crack growth behavior before propagating to the width direction of the specimen was evaluated quantitatively in this study. 4. Derivation of the energy release rate considering the freeedge effect In this study, a unit cell model for the transverse crack in [(S)/ 90]s laminates was considered. (S) is any orthotripic sub-laminate. The energy release rate associated with the formation of new small transverse crack between the existing small transverse cracks was calculated. A schematic illustration for calculation is shown in Fig. 4. In conventional analysis, the energy release rate that was associated with the new microcracking was calculated as the crack passed through the width direction of a specimen at the same time as its initiation [5]. However, the behavior of the transverse crack growth is different depending on the configuration of laminates or the material properties. It is known that the free-edge effect causes the initiation of the small transverse crack [10]. Hence, the energy release rate for the small transverse crack initiated at the specimen edge between the existing small cracks was derived considering the free-edge effect. By the energy balance, the energy release rate due to the formation of a new microcrack between the existing microcracks is derived following Nairn [11,12]. The energy release rate for propagation of a microcrack of current area, A, is
Gm ðAÞ ¼
oU ext oU oA oA
ð1Þ
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Small transverse cracks
6mm
2mm
Loading direction Fig. 3. Observation of the small transverse cracks with soft X-ray photography. (rmax/rb = 0.4, n = 1500 cycles and f = 5 Hz).
Small transverse crack
2W
[S]
[90]
[S]
[S]
Loading direction
[S]
[90]
s 2s
x
s Δa t2
t1
z
B
y Fig. 4. Schematic illustration of representative element of the laminate. (A) Two microcracks in the 90° ply. (B) The formation of a microcrack between the existing microcracks.
where, Uext is external work and L is total strain energy. Here, external work is expressed as
U ext ¼ 2r0 BWu
ð2Þ
where, r0 is axial stress, and B, W and u are specimen thickness, half of specimen width and the total axial displacement between the two existing microcracks, respectively. The total axial displacement can be written as
u ¼ 2s
r0 EA
þ aA T
ð3Þ
where, EA and aA are the effective axial modulus and thermal expansion coefficient of the laminate. Moreover, T is the temperature difference between the test temperature and residual stressfree temperature. On the other hand, total strain energy can be express as a sum of mechanical strain energy and thermal strain energy
1 r20 U ¼ 4sBW þ U res 2 EA
ð4Þ
where Ures is the residual strain energy per unit volume. Substituting Eqs. (2) and (4) into Eq. (1), the energy release rate is expressed as
G ¼ 4sBW
1 2 o 1 o o r0 r T a ð Þ U þ 0 res 2 oA EA oA A oA
ð5Þ
Following Nairn [11,12], the effective thermal expansion coefficient, aA and the residual strain energy per unit volume, Ures, can be derived. Substituting them into Eq. (5), the energy release rate is expressed in terms of effective modulus,
DaTE0 G ¼ 2sBW r0 C 1 Eð90Þ xx
!2
o 1 oA EA
ð6Þ
where, ðSÞ Da ¼ að90Þ xx axx ; E0 ¼
ðSÞ Eð90Þ 1 1 ð1 þ kÞE0 xx þ kExx ; C 1 ¼ ð90Þ þ ðSÞ ¼ ð90Þ ðSÞ 1þk Exx kExx kExx Exx
ð7Þ ð90Þ xx ,
a
ðSÞ xx ,
Eð90Þ xx
EðSÞ xx
a and are the thermal expansion coefficients and axial modulus in 90° ply and sub-laminate. k = ts/t90, and E0 is the rule-of-mixtures axial modulus. By simple laminated plate calculation, the applied stress partitioned into the 90° ply is expressed as
r0ð90Þ ¼ xx
Eð90Þ xx r0 E0
ð8Þ
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r0ð90Þ xx
Eð90Þ xx
ðSÞ kEð90Þ xx Exx DaT ð1 þ kÞE0
o 1 oA EA
ð9Þ
The term of the thermal residual stress is equal to the residual thermal stress applied in 90° ply of the undamaged laminate. Thus, Eq. (9) can be written as
G ¼ 2sBW
E0
!2
Eð90Þ xx
1 Ið90Þ 2 o rxx oA EA
ð10Þ
Ið90Þ
where, rxx is a sum of mechanical stress and residual thermal stress in the undamaged laminate. Thus, it is found that the energy release rate associated with transverse crack growth depends on the stress applied in 90° ply and the change of effective axial modulus associated with transverse crack growth. In this study, the small transverse cracks caused in the specimen edges are considered. Yokozeki [10] showed that the transverse cracks in quasi-isotropic [45/0/45/90]s laminates once halted without passing through the specimen width simultaneously with the transverse crack initiation by calculating the energy release rate associated with transverse crack propagation with VCCM. The reason why the small transverse cracks are once caused at the specimen edges is the increase of the axial stress applied in 90° ply by free-edge effect. Considering the increase of axial applied stress in 90° ply in the vicinity of specimen edges, the energy release rate associated with the small transverse crack formation is expressed as
G ¼ 2sBW
E0
!2
Eð90Þ xx
r0Ið90Þ xx
2 o 1 oA EA
ð11Þ
0Ið90Þ
where, rxx is a sum of the mechanical stress, the residual thermal stress and the incremental stress due to the free-edge effect of 90° ply. As shown in Fig. 4, new small transverse crack area is expressed as
dA ¼ 2t90 2Da
ð12Þ
where, 2t90, and Da are the thickness of 90° ply and the length of small transverse crack. Moreover, it was assumed that the specimen width was sufficiently great comparing with the length of a crack. Substituting Eq. (12) into Eq. (11), the energy release rate is expressed as
BW E0 G¼ 4qt 90 Da Eð90Þ xx
!2
ð1 þ kÞW E0 ¼ 2qDa Eð90Þ xx
2
r0Ið90Þ D xx
!2
1 EA
0Ið90Þ 2
rxx
1 1 EA ðq=2Þ EA ðqÞ
ð13Þ
where, q = 1/2s. Moreover, the effective laminate modulus in the axial direction, EA ðqÞ, is expressed with the rule-of-mixture as
DaEeff TC ðqÞ þ ðW DaÞE0 EA ðqÞ ¼ W
ð14Þ
where, Eeff TC ðqÞ is the laminate modulus in the x direction when the aspect ratio of transverse cracks that pass through the width direction of the specimen is q. Eeff TC ðqÞ can be calculated by using the method proposed by Joffe and Varna [13]. The energy release rate associated with the new small transverse crack considering the residual thermal stress and the fee-edge effect can be calculated 0Ið90Þ by using Eq. (13). However, rxx is unknown in Eq. (13). Thus, in this study the energy release rate associated with the new microcrack formation considering the free-edge effect was calculated
with the theories based on the interaction between the initiation and the propagation of transverse cracks. The detail is described in the next section. 5. Interaction between the initiation and the propagation of the transverse cracks In this study, it was observed that a transverse crack initiated at the specimen edges, and then propagated to the width of the specimen after the number of transverse cracks reached a saturation state at the specimen edges as shown in Figs. 1 and 2. Fig. 5 shows that the typical relationship between the density of transverse crack initiated at the specimen edge, and the normalized length of transverse cracks propagated to the width of the specimen, as a function of cycles under the test condition at the applied stress level of rmax/rb = 0.4. In this study, the energy release rate associated with the formation of new transverse cracks considering the free-edge effect was calculated with the interaction between the initiation and the propagation of the transverse crack. The applied stress in 90° ply considering the free-edge effect was defined based on the theories as follows. The transverse cracks that caused at the edges of the specimen started to propagate to the specimen width at the transverse crack density of 1.5 mm1 at the applied stress level of rmax/rb = 0.4 as shown in Fig. 5. That is to say, the energy release rate range (ERRR) associated with the formation of new transverse crack must be equal to the ERRR associated with the propagation of the transverse cracks when the density of the transverse crack reaches 1.5 mm1. The energy release rate associated with the transverse crack propagation was calculated with the approach proposed by Hosoi et al. [9]. The energy release rate is expressed as
t1 þ t2 Gt ¼ 2q1 t1
t EðuÞ r0 2 xx Daxx T t1 þ t2
!2
E0 Ec
! ð15Þ
ðExx Þ2
where t1, t2 and q1 are thickness of 90° ply and undamaged plies, and ð1Þ ðuÞ the crack density of 90° plies, respectively. Then, Daxx ¼ axx axx is the difference between the axial direction thermal expansion coefficients of the 90° ply and undamaged plies, and T = TS T0 is the difference between the test temperature TS and the residual stress-free temperature T0. Moreover, EðuÞ xx ; Exx , Ec and E0 are the rule-of-mixtures axial modulus of the uncracked plies, the effective laminate modulus in the axial direction, the axial laminate modulus that transverse cracks pass through the specimen width and the undamaged laminate modulus in the axial direction, respectively. Thus, using the condition that the ERRR by new crack formation is equal to the ERRR by the crack propagation at the transverse crack density of 1.5 mm1, the applied stress of 90° plies consider-
2.0
Crack density Crack length
saturation
1.0 0.8
1.5
0.6 1.0 0.4 0.5
0.2 propagation
0 10 0
10 1
10 2
10 3
10 4
10 5
0 10 6
Average transverse crack length normalized by specimen width
E0
!2
-1
G ¼ 2sBW
!2
Transverse crack density mm
Substituting Eq. (8) into Eq. (6),
Number of cycles N Fig. 5. Transverse crack density and normalized transverse crack length as a function of cycles under the test condition of rmax/rb = 0.4.
Formation (with free edge effect) Formation (without free edge effect) Propagation
-1
Energy release rate range ΔG J/m
120
Crack density growth rate mm /cycle
A. Hosoi et al. / Composites Science and Technology 69 (2009) 1388–1393 2
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100 Intersection
Free-edge effect
80
60
40 0
0.5
1
1.5
Transverse crack density mm
2
100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10
-1
with free-edge effect (5 Hz) with free-edge effect (100 Hz) without free-edge effect (5 Hz) without free-edge effect (100 Hz)
50
100
500
1000
Energy release rate range J/m2
Fig. 6. Energy release rate range as a function of transverse crack density under the test condition of rmax/rb = 0.4.
Fig. 7. The transverse crack growth rate as a function of the energy release rate range by comparing the results of the consideration of the free-edge effect with non-consideration of the free-edge effect.
0Ið90Þ
ing free-edge effect, rxx , was calculated with Eqs. (13) and (15). In the case of the applied stress level of rmax/rb = 0.3, the value of the saturation state was about 1.0 mm1 as shown in Fig. 1. Hence the ERRR associated with the formation of new transverse crack at the condition was calculated to be equal to the ERRR associated with the transverse crack propagation at the crack density of 1.0 mm1. Figure 6 shows two results in comparison to the transverse crack propagation. That is to say, the figure shows the comparison of the ERRRs associated by the formation of new transverse cracks whose length is Da as a function of crack density with the ERRR associated by the propagation from transverse cracks whose length is Da as a function of crack density. In this study, the energy release rate was calculated as Da = 0.42 mm from the average of experimental results. The solid line shows the ERRR for the formation of new small cracks considering the free-edge effect, the dash-dotted line shows the ERRR for the formation of new small cracks without considering the free-edge effect and the dashed line shows the ERRR associated with the propagation of cracks. In Fig. 6, comparing the ERRR associated with the crack formation considering the free-edge effect (the solid line) with the ERRR associated with the crack propagation (the broken line), it is explained that the transverse crack of the length of Da is caused at the specimen edge until the crack density of 1.5 mm1 because the value of the ERRR for the crack formation is greater than that for the crack propagation, and then the transverse cracks propagate, after the density of the transverse crack reaches 1.5 mm1. On the other hand, comparing the energy release rate range associated by the crack formation without considering the free-edge effect (the dash-dotted line) with the ERRR associated with the crack propagation (the dashed line), the values of the ERRRs are almost equal until the crack density of 1.0 mm1. The results explain that cracks propagate to the width of a specimen at the same time as the crack initiates in the laminates which the free-edge effect is small such as cross-ply laminates which have thick 90° plies in the middle of laminates or GFRP laminates. 6. Evaluation with modified Paris law approach The transverse crack growth behavior was evaluated by the modified Paris law approach as
dDTC ¼ AðDGÞn dN DG ¼ Gmax Gmin
ð16Þ ð17Þ
where DTC is the transverse crack density, and A and n are constants. Figure 7 shows the transverse crack growth rate as a function of the ERRR by comparing the results of the consideration of the free-edge
effect with non-consideration of the free-edge effect under the test conditions of the frequency of 5 Hz and 100 Hz. From the result shown in Fig. 7, it was observed that the transverse crack growth rate was considerably delayed under the applied stress level of rmax/rb = 0.2. The results imply that a threshold exists for the transverse crack initiation. Then, it was found that the energy release rate associated with the formation of new crack was estimated to be smaller when the free-edge effect was not considered. 7. Conclusion The growth behavior of small transverse cracks was evaluated under high-cycle fatigue loadings up to 108 cycles. The following results are obtained in this study. The energy release rate associated with the small transverse crack formation could be derived with the relationship that the energy release rate associated with the formation of new transverse crack is equal to that associated with the propagation of transverse cracks when the transverse cracks begin propagating to the width direction of the specimen. Moreover, it was implied that the lower threshold of transverse crack initiation exists by evaluating the transverse crack growth behavior with the modified Paris law approach. Acknowledgements The authors wish to thank Dr. Ogasawara of Japan Aerospace Exploration Agency and Dr. Yokozeki of University of Tokyo for help to use the soft X-ray photography. This work was supported in a part of the 21st century Center of Excellence program. References [1] Masters JE, Reifsnider KL. An investigation of cumulative damage development in quasi-isotropic graphite/epoxy laminates. ASTM STP 1982;775:40–62. [2] Nairn JA. Matrix microcracking composites. In: Talreja R, Månson JAE, editors. Polymer matrix composites; 2000. p. 403–32. [3] Tong J. Characteristics of fatigue crack growth in GFRP laminates. Int J Fatigue 2002;24:291–7. [4] Ogihara S, Takeda N, Kobayashi S, Kobayashi A. Damage mechanics characterization of transverse cracking behavior in quasi-isotropic CFRP laminates with interlaminar-toughened layers. Int J Fatigue 2002;24:93–8. [5] Nairn JA. The strain energy release rate of composite microcracking: a variational approach. J Compos Mater 1989;23:1106–19 [and errata: J Compos Mater 1990;24:223–4]. [6] Hashin Z. Analysis of cracked laminates: a variational approach. Mech Mater 1985;4:121–36. [7] Liu S, Nairn JA. Fracture mechanics analysis of composite microcracking: experimental results in fatigue. In: Proceedings of the fifth technical conference on composite materials. East Lansing, Michigan: American Society for Composites; 1990. p. 287–95.
A. Hosoi et al. / Composites Science and Technology 69 (2009) 1388–1393 [8] Yokozeki T, Aoki T, Ishikawa T. Fatigue growth of matrix cracks in the transverse direction of CFRP laminates. Compos Sci Technol 2002;62: 1223–9. [9] Hosoi A, Arao Y, Karasawa H, Kawada H. High-cycle fatigue characteristics of quasi-isotropic CFRP laminates. Adv Compos Mater 2007;16:151–66. [10] Yokozeki T. Multiple ply matrix cracking behavior in composite laminates. PhD Thesis. University of Tokyo, Department of Aeronautics and Astronautics; 2004 [in Japanese].
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[11] Nairn JA. Some new variational mechanics results on composite microcracking. In: Proceedings of 10th international conference on composite materials (ICCM-10). Whistler BC, Canada; 1995. p. 423–30. [12] Nairn JA. Fracture mechanics of composite with residual thermal stresses. J Appl Mech 1997;64:804–10. [13] Joffe R, Varna J. Analytical modeling of stiffness reduction in symmetric and balanced laminates due to cracks in 90° layers. Compos Sci Technol 1999;59:1641–52.