A total fatigue life model for mode I delaminated composite laminates

International Journal of Fatigue 28 (2006) 33–42 www.elsevier.com/locate/ijfatigue

A total fatigue life model for mode I delaminated composite laminates Kunigal Shivakumara,*, Huanchun Chena, Felix Abalib, Dy Leb, Curtis Davisb a

Center for Composite Materials Research, NC A and T State University, 1601 E. Market Street, Greensboro, NC 27411, USA b William J. Hughes Technical Center, FAA, Atlantic City, NJ 08405, USA Received 16 June 2004; received in revised form 8 February 2005; accepted 2 April 2005 Available online 23 May 2005

Abstract Fracture mechanics based total fatigue life model for delaminated composite structures is presented. The model includes the delamination growth in three domains, namely subcritical, linear, and final fracture. The resistance increases due to matrix cracking and fiber bridging in the case of unidirectional composites; and tow splitting, separation, bridging and breaking in the case of woven/braided fiber composites were included through normalization of the applied load (GImax) by the instantaneous resistance value (GIR). The proposed method was applied to mode I loaded woven roving glass/vinyl ester delaminated composite. The ASTM standard mode I fracture test was conducted to determine the resistance (GIR) as a function of delamination extension. The fatigue onset life test was conducted to determine the threshold energy release rate, GIth. Constant amplitude cyclic opening displacement fatigue test was conducted to establish the delamination growth rate (da/dN) as a function of maximum cyclic energy release rate (GImax). The total life delamination growth rate was found to be da=dN Z AðGImax =GIR Þm ð1K ðGIth =GImax ÞÞD1 =ð1K ðGImax =GIR ÞÞD2 where the material constants A, m, D1, and D2 were 0.1, 5.4, 8 and 2, respectively. This equation was verified for a block loading and found to accurately predict the delamination length. q 2005 Elsevier Ltd. All rights reserved. Keywords: Delamination; Composite laminate; Mode I; Total life

1. Introduction Susceptibility to delamination is a major weakness of composite laminates. Knowledge of material’s resistance to interlaminar fracture and fatigue is essential to establish design allowable and damage tolerance guidelines for structures. Fracture mechanics based delamination growth models are required to predict fatigue life and establish suitable inspection intervals so that a delamination can be found and repaired long before it becomes critical or exceeds the residual strength of the component. Fatigue delamination growth laws that cover the threshold, the stable growth and the unstable fracture domains are needed for total life estimation. Such growth laws were proposed in the past for metallic materials [1,2] and are now becoming accepted in damage tolerant designs. Similar methodology is needed for composite laminates. Hypothetically, we can assume that the delamination growth rate has three domains, * Corresponding author. Tel.: C1 336 334 7411; fax: C1 336 256 0873. E-mail address: [email protected] (K. Shivakumar).

0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2005.04.006

namely subcritical (slow), linear, and unstable growth rate domains (see Fig. 1). The growth rate depends on microscope details of fiber architecture and resin properties in domain 1; on crack driving force (energy release rate G or DG) in domain 2; and on interlaminar fracture characteristics of the laminate in the unstable domain 3. In composite laminates substantial research [3–15] has been done on delamination growth laws in domain 2. The data has been expressed by power law equation in GImax or DGImax by curve fits. Research efforts also focused on studying the effect of stress ratio [3,4], matrix toughness [5,6], and pure and mixed mode stress states [4,6,7]. But none of these studies tried to model the all three domains of delamination growth rates. Furthermore, these studies ignored the effect of increased fracture resistance as the delamination grows. Increased fracture resistance with delamination growth is commonly observed in fracture tests of composite laminates because of fiber bridging and matrix cracking. Ignoring the resistance increase with delamination growth can severely underestimate the life of a component. Poursartip [14] was the first to recognize the importance of resistance curve and proposed GR normalized da/dN equation for edge delaminated composite specimens.

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K. Shivakumar et al. / International Journal of Fatigue 28 (2006) 33–42

Nomenclature a delamination length da/dN delamination growth rate GIC opening mode I interlaminar fracture toughness GImax maximum cyclic mode I energy release rate GIR opening mode I interlaminar fracture toughness resistance GIth threshold value of mode I energy release rate

His results showed that the GR normalization increased the exponent in the da/dN equation. Recently, Paris and O’Brien [15] proposed an approach for total da/dN equation that also included the resistance effect and hypothesized the total life concept but no data was presented. This paper explains the total life model including the resistance, presents a step by step approach of testing and data reduction results for woven-roving glass fiber/vinyl ester composite laminate subjected to mode I loading. Then the total life equation, and its verification for a cyclic block loading.

2. Methodology The methodology proposed is general and is applicable to both pure and mixed mode loading. However, the description presented here is for mode-I loaded case for simplicity and clarity. The delamination driving force is

Domain 1

Subcritical growth region

2

G dominate growth region

3

Unstable region

PImax R dImin dImax D

mode I load at dImax ratio of minimum to maximum cyclic displacement minimum value of cyclic displacement maximum value of cyclic displacement delamination length correction factor for modified beam theory

expressed by energy release rate GI, the material resistance by GIR, and the delamination growth rate by da/dN. 2.1. Assumptions The following assumptions are made in developing the da/dN equation. 1. Interlaminar fracture resistance (GIR) increases with delamination length (a) because of matrix cracking and fiber bridging in the case of unidirectional composites; tow cracking, multiple delaminations, tow bridging and tow breaking in the case of woven/braided fiber composites. The resistance curve GIR (a) can be expressed as a function of initiation fracture toughness GIC and the delamination extension (a–a0) as shown in Fig. 2. The initial delamination length is a0. 2. The da/dN is proportional to the driving force GImax and inversely proportional to the delamination resistance GIR at the current delamination location. 3. The da/dN is bounded by two extreme values of GImax; the threshold energy release rate GIth at or below which da/dN is nearly zero (no growth) and the GIR at which da/dN is very large or unstable (infinite). Note that GIR is a function of delamination growth. 4. The value of GIth can be determined by fatigue delamination onset test as per ASTM D6115.

da dN GIR = f (GIC, a)

Resistance GIR

GImax/GIR

1.0

Threshold (GIth/GIC)

Fig. 1. Hypothetical plot of da/dN versus GImax/GIR.

a0 Delamination length, a

Fig. 2. Typical delamination growth resistance curve.

K. Shivakumar et al. / International Journal of Fatigue 28 (2006) 33–42

2.2. Approach As in metallic material (for example [16]), da/dN data for composite laminates also falls into three domains, namely, subcritical or slow crack growth domain, GImax controlled domain (or Paris’domain), and the unstable or fracture domain (see a typical Fig. 1). From the assumptions 1 and 2, the da/dN in domain 2 is written in a power law form as:
 da G ðaÞ m Z A Imax (1) dN GIR ðaÞ where A and m are material constants to be determined from the curve fit to the test data. In the subcritical domain, the da/dN (assumption 3) varies between zero (when GImax!GIth) and a value that matches domain 2. The form of the da/dN equation can be written as:   da GImax m ZA ½1 K ðGIth =GImax ÞD1  (2) dN GIR The exponent D1 is determined from curve fit to the test data. In the unstable domain, da/dN varies between N (when GImaxZGIR) and the transition value that matches Eq. (1) in domain 2. The form of the da/dN equation can be written as:   da GImax m 1 ZA (3) dN GIR ½1 K ðGImax =GIR ÞD2  The exponent D2 is a material parameter determined from the curve fit to the test data. Finally, the combined da/dN equation that covers all three domains is   D1   m 1 K GGIth Imax da GImax  ZA (4)  D2  dN GIR GImax 1 K GIR The values of A, m, D1 and D2 are the material parameters to be determined by curve fit to the fatigue test data. 2.3. Test procedure for establishing the parameters a. Conduct mode-I fracture test using the DCB specimen and establish resistance curve GIR(a) as per ASTM D5528 [17]. Fit a GIR versus a equation. Many times a power law equation fits the data very well. b. Conduct ASTM D-6115 [18] fatigue onset life test and establish the GIth by a curve fit and a chosen onset life criteria. c. Conduct a fatigue test on a virgin DCB specimen with GImax value about 0.3 GIC till the compliance change is about 2% for unidirectional and 5% for textile preform composites. This ensures a natural delamination to

d.

e. f.

g. h.

i. j.

35

initiate yet the resistance change from GIC is very small and GIRZGIC can be assumed. Conduct a constant amplitude displacement controlled fatigue test starting with a GImax value that is slightly lower than GIC. A good starting value is about GImaxZ 0.8GIC. Continue the test till the delamination growth rate becomes very small (for example, da/dN!10K7 in/cycle). Repeat the test at least for three specimens and plot the data as da/dN versus (GImax/GIR) on a log–log graph. Divide the plot into three domains by visual inspection (see Fig. 1). The middle linear region and the two ends, namely subcritical and unstable regions. Perform least square log–log fit to the domain 2 data and determine the constants A and m. Determine D1 of Eq. (2), using already determined A and m by fitting the equation to domains 1 and 2 data. A trial and error approach works well. Finally determine D2 of Eq. (3) by fitting the equation to domains 2 and 3 data as in step ‘h’. Using the constants determined above plot Eq. (4) and compare it with the test data. If the fit is not satisfactory repeat steps ‘f’ through ‘j’.

3. Material system and specimen configuration The material system chosen was woven roving E-glass fiber supplied by Fiber Glass Industries (FGI) with FGI’s super 317 sizing for ease of handling, fast wet out, and compatibility with vinyl ester resins. The fabric designation was FGI 1854 with 18 Oz/square yard areal weight and unbalanced construction. About 59% of fibers in warp (0-deg) direction and the remaining in fill (90-deg) direction. The vinyl ester resin used was Dow Chemicals Derakane 510A-40 brominated for fire resistance property. The resin has 350 cps viscosity at room temperature, which is ideally suited for vacuum assisted resin transfer molding (VARTM). The 510A-40 matrix has specific gravity of 1.23, tensile modulus and strength of about 3.4 GPa and 73 MPa, respectively, flexural modulus and strength of about 3.6 GPa and 125 MPa, and heat distortion temperature of 225 8F. A 10 ply 0-deg laminate with a 0.5 mil Kaptan insert at the mid-plane was fabricated by the VARTM process explained in reference [19]. The Kapton insert was to create a delamination. One-half mil thickness is best suited for the study. The panel was about 813 mm long, 318 mm wide with a delamination length of 64 mm. Edges of the panel were trimmed about 25 mm all around to make sure that the edges were parallel and perpendicular to the weft (0-deg) direction of the fibers. The specimens were machined using the diamond tipped wheel and finished with surface grinder. The specimen length (L), width (b), thickness (h) and delamination length (a0) were about 25.4 cm, 38, 5 and 38 mm,

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Table 1 Specimen number and test Test

Specimen number

a0 (mm)

b (mm)

h (mm)

L (mm)

Fracture

W-1 W-2 W-3 W-4 W-5 WF-6 WF-7 WF-8 WF-12

36.96 36.42 37.47 37.54 37.80 37.19 36.60 37.29 36.58

38.03 38.07 38.03 38.07 38.05 37.97 37.85 37.87 37.90

4.95 4.98 4.95 4.95 5.04 4.98 4.85 5.00 5.08

247.90 251.33 255.14 252.78 241.81 240.94 256.24 253.87 240.03

Fatigue

respectively. A list of specimen number, their dimensions and the tests conducted is given in Table 1. After the specimen was machined, the loading hinges were mounted on the outer faces of the specimen at the debond end. The hinges were located 25 mm from the debond end to achieve the delamination length (a0) of about 38 mm that is the same as the specimen width (b). The aluminum hinges were mounted using 3M DP-460 two-part epoxy adhesive. The adhesive was cured at 60 8C for 2 h. The specimen configuration and loading are shown in Fig. 3.

4. Fracture test The fracture test was carried out in a MTS test machine using a 890 N (200-lb) load cell. The piano hinge tabs of the specimen were mounted in the hydraulic grips of the load frame. The tests were conducted by displacement control loading with a cross-head rate of 0.5 mm/min. Load and cross-head displacement were recorded throughout the test. A magnifying camera was mounted on a traversing stand (both vertical and horizontal) with vernier scale. The camera was connected to a TV monitor to locate and track the delamination tip. At the start of each test the delamination-tip location was noted. The cross-head displacement was started and the delamination-tip propagation was monitored through the camera. The specimen was loaded at a constant cross-head

rate and the load and displacement values were recorded continuously. The load-displacement data was recorded when the visual onset of delamination movement was observed on the edge of the specimen. The initial loading was stopped after an increment of delamination growth of about 5 mm. The specimen was unloaded at a constant cross-head rate of up to 25 mm/min. After unloading, the position of the delamination tip was marked on both edges of the specimen. If the delamination length on the two edges differs by more than 2 mm, the test is invalid. The specimen was reloaded at the same constant cross-head speed of 0.5 mm/min without stopping or unloading until the final delamination length increment is reached. The load and displacement values were recorded at every 1 mm increment in the first 5 mm delamination growth increment. Subsequently, load and displacement data were recorded at every 5 mm delamination growth, until the delamination had propagated for 45 mm and again at every 1 mm increment for the last 5 mm of delamination propagation. The load versus displacement curves for all five FGI1854/Dow 510A-40 specimens are shown in Fig. 4. The initial slopes of the curves are almost identical. The load-displacement response is like typical of brittle matrix composite laminates. The camera mounted to capture the edge view of the specimen recorded interesting failure modes such as tow cracking/splitting and separation, fiber bridging and fiber breakage. These pictures are shown in Fig. 5. Fig. 6 shows the Scanning Electron Microscopy Fracture Modes: • Tow cracking/Splitting & Separation • Fiber bridging • Multiple cracks •Fiber breakage

180

P

hh 2h

load, N

120

60

b

0

P

ao

Fig. 3. Specimen configuration.

0

10

20

30

40

Load point displacement, mm Fig. 4. Load-displacement response for glass/vinyl ester composite.

K. Shivakumar et al. / International Journal of Fatigue 28 (2006) 33–42

37

Fig. 5. Edge failure modes of woven-roving glass/vinyl ester laminate.

images of fracture surfaces that reveal the tow splitting, fiber breakage, and fiber–matrix interfacial fracture. The energy release rate GI was calculated from the modified beam theory equation [17] GI Z 3Pd=2bðaC jDjÞ where PZload, dZload point displacement, bZspecimen width, and aZdelamination length. The parameter D is a delamination length correction parameter for not perfectly built-in condition of the DCB. The D was determined from the specimen compliance and delamination length data [17]. Energy release rate GIR was plotted against delamination propagation da (Za–a0) as shown in Fig. 7. A power law equation was fitted to the data and the resulting resistance equation is: GIR Z 0:35 C 1:23ða K a0 Þ0:31

(5)

Increased resistance was caused by tow splitting, separation, bridging and breakage, and multiple delaminations. The initial fracture toughness GIC was determined by

the load at the point where delamination was visually observed to grow on the edge of the specimen. The GIC of this material is 0.35 kJ/m2.

5. Fatigue onset life 5.1. Test Fatigue delamination growth onset life tests were conducted as per ASTM D-6115 [18] to determine the threshold energy release rate value GIth. All fatigue tests were conducted using specimens made from same panel (batch) that was used for fracture tests. An expression relating the load-point displacement dImax and the GImax at the onset of fracture (or at the visible delamination propagation) was determined. For a chosen value of maximum cyclic energy release rate ratio fZGImax/(GIC),

Fig. 6. Fracture surface of glass/vinyl ester laminate (SEM).

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K. Shivakumar et al. / International Journal of Fatigue 28 (2006) 33–42 1.0

slowly reduced to zero and the stroke controlled fatigue test was started. The load and displacement were recorded at every cycle using a Vishey 5000 data acquisition system. The test was continued for predetermined number of cycles (see Table 2). The same test procedure was repeated for other values of dImax. All the data recorded were reduced and listed in Table 2.

GIR = 0.35+1.23(a-a0)0.31

0.8

GIR kJ/m2

0.6 0.4 0.2 0.0

5.2. Test results 0

10

20

30

40

50

60

Delamination growth (a-a0), mm Fig. 7. Energy release rate versus delamination growth for glass/vinyl ester laminate.

the dImax was calculated from the equation: sffiffiffiffiffiffiffiffiffiffiffiffi  GImax a0 C D 2 dImax Z dIC GIC aIC C D

(6)

where a0 is the initial delamination length for this fatigue onset test, aIC is the initial delamination length in the fracture test, D is the delamination length correction parameter determined from the fracture test, and dIC is the load-point displacement when the delamination starts to grow in the fracture test. The cyclic loading ratio (R) selected was 0.1, from that dImin was determined (dIminZRdImax). Values of dImax for (GImax/GIC) values of 0.4, 0.3, 0.25, 0.2 and 0.17 were calculated and the approximated values used for testing are listed in Table 2. The table also lists the width and delamination length of the specimens tested. The loading hinges on the specimen were mounted in the loading machine grips and specimen was aligned and centered. Before gripping, the end of the specimen opposite to grips was supported to assist in the mounting process. The test machine was set to the stroke loading at a frequency of 1 Hz between dImax and 0.1 dImax. At first, the specimen was displaced to dImax and the load PImax was recorded. From these values and the specimen geometry, the initial compliance at NZ1 and the actual GImax were calculated and tabulated (Table 2). The machine displacement was

The ASTM D-6115 recommends two criteria for determining the onset life, namely 1 and 5% increase of compliance compared to the compliance at NZ1. Because the composite considered in this study is a woven roving fiber system the 1% criterion is too conservative. Therefore 2, 5, 7 and 10% increases in compliance were chosen to establish the onset life. Table 3 summarizes the onset lives for the specimen tested at various GImax/GIC ratios. The table includes the data based on 1, 2, 5, 7 and 10% criteria. GImax versus log N was examined for 2–10% onset criteria, 5 and 7% compliance change criteria were chosen to establish GIth. Fig. 8 shows the plot of GImax versus log N data for 5 and 7% criteria. O’Brien [7] suggested that GImax versus log N between 100%N%106 can be represented as a linear equation and the GIth can be calculated at NZ106 from that equation. For the present material, the linear equation did not fit the data well. An alternative power law fit, though complex, represented the data well and the equation is GImaxZ0.34(log NC1.00)K0.97 (see Fig. 8). Surprisingly, the equation agreed very well for both 5 and 7% compliance change criteria. The threshold energy release rate (GIth) calculated from the equation for NZ106 cycles was 0.15 GIC. This GIth was used in the development of constants for delamination growth rate equation (Eq. (4)).

6. Fatigue delamination growth rate test 6.1. Test The fatigue delamination growth test set-up and testing were similar to those of the onset life test. The test was

Table 2 Fatigue onset test data for FGI 1854/510A-40 glass/vinyl ester composite Specimen #

WF6A WF7A WF8A WF9A WF10A WF11A WF12A WF13A WF14A

B (mm)

37.97 37.85 37.87 37.92 37.95 37.87 37.90 37.97 37.95

a0 (mm)

37.19 36.60 37.29 36.65 36.65 36.07 36.58 36.75 36.91

Intended

Applied

Measured

Actual

GImax/(GIC)

dImax (mm)

Nmax

da (mm)

PImax (N)

GImax (kJ/m2)

GImax/(GIC)

0.40 0.40 0.30 0.30 0.25 0.25 0.20 0.20 0.17

2.19 2.19 1.93 1.93 1.77 1.77 1.58 1.58 1.52

5200 3000 3000 3000 5000 5000 10,000 10,000 500,000

1.52 2.92 0.00 0.76 0.00 0.00 0.00 0.00 –

68.18 70.87 58.80 66.78 57.19 54.73 52.35 50.51 38.73

0.140 0.144 0.111 0.125 0.100 0.097 0.082 0.079 0.059

0.407 0.416 0.322 0.364 0.290 0.282 0.238 0.228 0.170

K. Shivakumar et al. / International Journal of Fatigue 28 (2006) 33–42

39

Table 3 Fatigue onset lives for FGI 1854/510A-40 glass/vinyl ester composite Specimen #

GImax/(GIC)calc

N1%

N2%

N5%

N7%

N10%

WF6A WF7A WF8A WF9A WF10A WF11A WF12A WF13A WF14A

0.407 0.416 0.322 0.364 0.290 0.282 0.238 0.228 0.170

19 10 2 4 6 6 16 13 –

149 36 14 24 79 46 180 368 –

507 193 185 446 1389 2653 1930 6713 30,000

863 234 641 832 2456 4560 4697 – –

1840 435 2850 1817 4830 7941 8585 – –

conducted under constant amplitude cyclic displacement loading at the frequency range of 1–4 Hz with R ratio of 0.1. All fatigue delamination growth test specimens were precracked by conducting fatigue tests at load equivalent to GImaxZ0.3GIC till the specimens had a compliance change of about 5%. New delamination lengths were measured and recorded. Delamination growth rate data was generated by applying the dImax corresponding to (GImax/GIC) of 0.5, 0.6, 0.7, 0.8, 1.0, 1.2, 1.4, 1.5, 1.6, 1.8, 2.0 and 2.2. Loads and delamination lengths were measured for specified number of load cycles. The load cycle interval DN was selected based on simple doubling scheme starting with DNZ1. Once DN reached 10,000, it was incremented only when no measurable delamination propagation was observed in that interval. At each DN the test was stopped. Results of N, DN, a, Da, PImax and dImax were recorded for every DN selected. From these results, da/dN, GImax, and GIR were calculated. Imax dImax The GImax was calculated using GImax Z 3P and DZ 2bðaCjDjÞ 3.28 obtained from the fracture test.

the compliance calibration method with data smoothing can be used to predict delamination length as a function of N. Edge views of the fatigued specimen are shown in Fig. 10. Tow splitting and separation, fiber bridging and fiber breakage, which are the same failure modes as in fracture tests were observed. Fig. 11 is the scanning electron microscopy images of fatigue delamination propagation surfaces of the specimen. Detailed features of tow splitting, fiber breakage and fiber–matrix interfacial fracture are seen in these images. Features of the failure mode in fatigued specimen (Figs. 10 and 11) are similar to those of fractured specimen (see Figs. 5 and 6). These features justify that the material resistance increases as the delamination front propagates under both fracture and fatigue loadings. The resistance equation for fatigue test is similar to that for fracture test. Therefore, the use of Eq. (5) for GIR and the normalization of GImax by GIR fatigue data reduction is a valid approach.

6.2. Test results

7. Fatigue delamination growth rate equation

Fig. 9 shows the plot of da/dN versus GImax/GIR data for four different specimens. The GIR was calculated from Eq. (5) at the current delamination lengths. One of the reasons for the large data scatter may be starting and stopping the test at every DN to measure the delamination length and compliance. Other methods, such as

The test data and the equation are bounded by the limits GImaxZGIth and GImax/GIRZ1. The GIth from the onset life test was 0.15 GIC and the upper limit GImax/GIRZ1 is from the condition that when GImaxZGIR the delamination growth rate is infinite or unstable. The test data between the above limits in Fig. 9 was divided into three parts by 100

0.6 0.5

Static

10

5%

100

7%

–1

10

Equation

0.4

GIC =0.35 kJ/m2

GImax kJ/m2 0.3

1–

GIth GImax

1–

GImax GIR

8

2

–2 da/dN 10 mm/cycle 10–3

10–4

0.2

0.97

GImax = 0.34 (logN+1.00)

10–5 10–6

0.1 0.0

da G = 0.1 Imax GIR dN

5.4

10–7 0.1

0

1

2

3 logN

4

Fig. 8. Variation of onset life with GImax.

5

6

GIth / GIR

0.2

0.3

0.4

0.5 0.6 0.70.8 0.9 1

GImax /GIR Fig. 9. Comparison of total delamination growth rate equation with the test data.

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K. Shivakumar et al. / International Journal of Fatigue 28 (2006) 33–42

Fig. 10. Edge failure modes of fatigued specimens.

Fig. 11. Fatigue delamination propagation surfaces.

visual inspection, middle linear region (domain 2) and two ends, namely subcritical (domain 1) and unstable regions (domain 3). A least square log–log equation fit was performed for the middle region of 0.2%GImax/GIR%0.7. The constants A and m were determined to be 0.1 and 5.4, respectively. The exponent D1 in Eq. (2) was determined by trial and error approach to best fit the data in domains 1 and 2. The D1 was found to be 8. A similar approach was used to determine D2 in domains 2 and 3 and D2 was found to be 2. The final equation of da/dN that covers all three domains is:   8   5:4 1 K GGIth Imax da GImax  Z 0:1 (7)  2  dN GIR GImax 1 K GIR Fig. 9 compares the Eq. (7) with the test data. The sensitivity of the parameters was determined by varying A from 0.10 to 0.13, m from 5.3 to 5.5, D1 from 4 to 8 and D2 from 2 to 4. Fig. 12 compares the sensitivity of A and m with the test data and Fig. 13 compares that for D1 and D2. For the selected range of parameters the Eq. (7) fits the test data very well and the material parameters are insensitive to small changes in their values.

100 k cycles each with all tests conducted under constant amplitude displacement loading with RZ0.1. The first segment is increasing GImax from 0.3 to 0.5 GIC loading, the second is constant GImaxZ0.5 GIC loading, the third segment is decreasing GImax from 0.5 to 0.3 GIC loading. Since GImax loading cannot be applied directly, a stepped (step of 10 k cycles) constant amplitude cyclic displacement loading with dImax calculated from Eq. (6) was used. In Fig. 14, the broken line represents the intended loading while the solid line with symbol represents the actual loading. The actual loading includes initial, average, and final G values for each 10 k step. The solid line represents the average loading. Many reasons contributed to differences between the intended and the actual loading. However, the average loading was used in the calculation of delamination lengths. 100 10 100 10–1 10

Eq. (7) was verified for a block fatigue loading shown in Fig. 14. The block loading chosen has three segments of

5.4 1 –

GIth GImax

1–

GImax GIR

G da = 0.1 Imax GIR dN

–2

da/dN mm/cycle 10–3

8

2

10–4 10–5 10–6

8. Verification of the equation for a block loading

A = 0.10 to 0.13 m = 5.3 to 5.5

10–7 0.1

GIth / GIR

0.2

0.3

0.4

0.5 0.6 0.7 0.80.9 1

GImax /GIR

Fig. 12. Sensitivity of A and m parameters on predicted delamination growth rates.

K. Shivakumar et al. / International Journal of Fatigue 28 (2006) 33–42 100 10

da GImax = 0.1 GIR dN

100

5 .4

10–1

1–

GIth GImax

1–

GImax GIR

from the experiment and Eq. (7) were 54.13 and 52.78 mm, respectively. The difference between these two is about 2.5%.

8

D2 = 2

2

–2 da/dN 10 mm/cycle 10–3

9. Conclusions D2 = 4

10–4 10–5 10–6 10–7 0.1

D1 = 8 D1 = 4

GIth / GIR

0.2

0.3

0.4

0.5 0.6 0.7 0.8 0.9 1

GImax /GIR

Fig. 13. Sensitivity of D1 and D2 parameters on predicted delamination growth rates.

0.6 Intended

0.5 0.4

Actual

GImax /GIC 0.3

Average 0.2 0.1 0.0

0

100

200

300

N, k cycles

Fig. 14. Constant displacement amplitude block loading.

The delamination lengths were predicted by integrating the growth rate equation: ð 300k da (8) a Z a0 C dN NZ0 The prediction was performed starting from initial crack length of a0Z37.16 mm and with average linear variation of GImax with N. The delamination lengths were calculated for every 1, 10, and 100 load cycle increments. The results were nearly same for all these cases analyzed. Fig. 15 compares the prediction with the test data. The final delamination lengths 60 da G = 0.1 I max GI R dN

50

41

5.4

1

GIth GImax

1

G Imax G IR

8

2

Proposed and developed a total fatigue life model for laminated composite structures subjected to mode-I fracture loading. The model includes the delamination growth in subcritical, linear, and final fracture domains. The resistance increases due to matrix cracking and fiber bridging in the case of unidirectional composites; and tow splitting, separation, bridging and breaking in the case of woven/ braided fiber composites were included through normalization of the equation by the instantaneous resistance value (GIR). The proposed method was applied to woven roving glass/vinyl ester delaminated composite panel. The ASTM standard’s mode I fracture test was conducted to determine the GIR as a function of delamination extension and fatigue onset life test was conducted to determine the threshold energy release rate, GIth. The resistance equation was GIRZ 0.35C1.23(aKa0)0.31 and GIthZ0.15 GIC, where GIC is 0.35 kJ/m2. Constant amplitude cyclic opening displacement fatigue test was conducted to establish the delamination growth rate (da/dN) equation as a function of maximum cyclic energy release rate (GImax). The total life delamination growth rate was found to be da=dN Z AðGImax =GIR Þm ! ð1K ðGIth =GImax ÞD1 Þ=ð1K ðGImax =GIR ÞD2 Þ where the material constants A, m, D1, and D2 were 0.1, 5.4, 8 and 2, respectively. This equation was verified for a block loading and found to accurately predict the delamination length.

Acknowledgements The authors acknowledge the support of FAA Technical Center through contract DT FA 03-01-C0034 and Offices of Naval Research through a grant N 00014-01-1-1033, Dr Yapa Rajapakse, program manager. The authors would also thank Dr O’Brien, US. Army Research Laboratory, Vehicle Technology Directorate, NASA Langley Research Center and Dr Sheprekevich, FAA Tech center for very helpful technical suggestions. Authors thank Fiber Glass Industries for supplying the glass fabrics.

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a, mm 30

References

20 10 0 0

100

200

300

N, k cycles

Fig. 15. Comparison of predicted delamination growth with measured values.

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