Delamination characteristics of double-cantilever beam and end-notched flexure composite specimens

Composites

Science and Technology 56 (1996) 451-459 0 1996 Elsevier Science Limited

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in Northern

Ireland.

SO266.3538(96)00001-Z

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DELAMINATION CHARACTERISTICS OF DOUBLECANTILEVER BEAM AND END-NOTCHED FLEXURE COMPOSITE SPECIMENS

C. T. Sun & S. Zheng School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana 47907-1282,

USA

(Received 14 March 1995; revised version received 1.5November 1995; accepted 14 December 1995) analytical solution to simplify the data reduction procedure. Since simple beam theory cannot adequately simulate the deformation near the crack tip, a great deal of effort’-‘” has been devoted to a search for a better model (such as advanced beam models) for calculating the strain energy release rate at a DCB crack front. As for the ENF specimen test, owing to the sensitivity of compliance with respect to crack length,’ beam theories with appropriate correction factors are used as data reduction too1s.l”’ However, it has been pointed out that strain energy release rate distribution along the straight crack fronts of a DCB specimen is not uniform,14-17 thus casting doubt on the suitability of beam theories as data reduction tools. It has been noted that delamination in a composite laminate usually occurs at the interface of different ply orientations. DCB and ENF experiments are also used to determine the critical strain energy release rate at the interface of different ply orientations.5*” Experimental results from these tests often revealed that critical strain energy release rates thus obtained depend on the lay-up sequence of the test specimen.‘3”36 Such a lay-up sequence dependent interlaminar fracture toughness should not be regarded as a true material property before the influence of other factors is determined. One of the variables that require more careful study is the effect of stacking sequence on the strain energy release rate distribution at the crack fronts of DCB and ENF specimens. In fact, this subject is more complicated than it appears. In a general laminated DCB or ENF specimen, the strain energy release rate at the crack front varies (and is skewed) across the specimen width. Consequently, the crack front is also curved and skewed. Thus, before beam theories are used to reduce experimental data, we must evaluate the validity of the beam-based solutions and provide proper interpretations of these solutions. The behavior of strain energy release rate at the crack front of DCB and ENF specimens has been investigated by a number of researchers. Among them, Crews et ~1.‘~ used a three-dimensional analysis

Abstract The distributions of strain energy release rate, G, at the

crack fronts of double-cantilever beam (DCB) and end-notched flexure (ENF) specimens have been analysed by means of the plate finite element. A boundary layer phenomenon in the distribution of G at the crack front is found. The applicability of beam theories as data reduction tools for DCB testing is examined. The effect of a curved crack front on the use of beam theories in the calculation of strain energy release rate for a DCB specimen is also discussed. Except for unidirectional and cross-ply laminates, the distribution of strain energy release rate at the crack front is found to be skewed, and a parameter is introduced to measure this skewness. Recommendations on DCB specimen design are made for the fracture toughness test to minimize the non-uniform and skewness effect. Finally, it is found that the boundary layer effect in the ENF specimen is not as severe as that in the DCB specimen. 0 1996 Elsevier Science Limited Keywords:

double-cantilever beam specimen, endnotched flexure specimen, strain energy release rate, delamination 1 INTRODUCTION Fracture mechanics has found extensive applications in damage analysis of composite laminates, especially in delamination analysis. One of the most important parameters in the application of fracture mechanics in composite structures is the strain energy release rate. In order to determine the critical strain energy release rate, fracture experiments must be performed. The double-cantilever beam (DCB) and the end-notched Ilexure (ENF) specimens are the most popular specimen configurations in the experimental determination of mode I and mode II interlaminar fracture toughnesses.1-7 A common procedure for data reduction of DCB tests is to use a compliance calibration technique known as Berry’s method.3%8As an alternative, it would be desirable to have an 451

452

C. T. Sun, S. Zheng

to calculate the strain energy release rate in a DCB specimen. By attributing the variation of strain energy release rate, G, to anticlastic curvature, they employed Poisson’s ratio, -v>,~,to determine qualitatively the anticlastic effect. Davidson,” Davidson and Shapery,16 and Nilssoni7 analysed the growth of the DCB crack front and found it to be curved and thumb-nail shaped. Davidson and Shapery16 analysed the strain energy release rate distribution for a laminated DCB specimen. By examining the transition from plane strain to plane stress, they proposed a parameter D, = D&/D11D22 to describe the variation of strain energy release rate at the crack front, where D12, D1, and Dz2 are bending stiffness components in the laminate D matrix. When dealing with laminated composite specimens, all of the above analyses have assumed symmetry in the width direction with respect to the X axis. Consequently, their results showed that the distribution of G along the crack front was symmetric with respect to the X axis (see Fig. 1) even for angle-ply laminated composite specimens.i4yt5 In the present paper, a double-plate model” has been used to perform the delamination crack analysis. Both isotropic and composite DCB specimens were investigated. The boundary layer in the distribution of G was identified. The applicability of beam theories as data reduction tools has also been addressed and the effect of a curved crack front on the use of beam theories for calculating strain energy release rate at a DCB crack front has been examined. A parameter is proposed to account for the stacking sequence effect on the distribution of the strain energy release rate. A design recommendation is made to minimize the variation and skewness of the strain energy release rate distribution for a composite specimen. Finally, the ENF specimen is analysed and conclusions drawn. 2 PROBLEM DESCRIPTION The dimensions of the DCB specimen are shown in Fig. 1. At the split ends of the DCB specimen, uniform displacements in the 2 direction are applied

1*P

t-l a

L

Fig. 1. DCB specimen

configuration.

Fig. 2. ENF specimen

configuration.

symmetrically for specimens with symmetric arms. For ._ specimens with unsymmetric arms, symmetric forces are applied while the displacement components in the 2 direction for nodes at the end of each DCB arm are kept uniform. This kind of loading is typical of the usual DCB test loading condition. Unless otherwise specified, the end load used in the analyses of both DCB and ENF specimens (see Fig. 2) is 32 N/m. For the numerical analysis, the double-plate model proposed by Zheng and Sun’* was employed to calculate the strain energy release rate. In the double-plate model, a delaminated composite plate is modeled by two separate Mindlin plates. In the intact region, the two plates are ‘tied’ together to ensure displacement continuity; in the delaminated region, the two plates are not constrained except for the contact conditions. A crack closure method was used to calculate the strain energy release rate at the crack front. By comparing with the three-dimensional finite this element solutions for a DCB specimen, double-plate model was shown to be computationally efficient and accurate.” Two different finite element meshes were used throughout this study: one with 12 elements along the width and the other with 24 elements. The element size at the crack tip is 1% of the crack length. Linear except for laminated analyses were performed specimens with asymmetric lay-up sequences, for which geometric non-linearity was considered. For asymmetric laminates, the geometrically non-linear effect is magnified by the extension-bending coupling, and large deflection theory must be employed.” For isotropic and cross-ply composite specimens, only half of the specimen was modeled in the numerical simulation on account of the symmetry. For specimens containing angle plies, the whole specimen was modeled. The eight-node isoparametric element (element type S8R) in the commercial finite element code ABAQUS2’ was used to perform delamination crack analysis. The non-penetration condition within the crack region was imposed through the use of gap elements (element type GAPUNI). Table 1 lists the properties of the materials considered in this paper.

453

DCB and ENF composite specimens

Table 2. Average strain energy release rate G,, ( X lo-’ J/m*) for aluminum DCB with various widths

Table 1. Material properties

Material

Material constants

E = 71 GPa, Y = 0.3 E = 34 GPa, Y = 0.3 E,= 134 GPa, E, = 13 GPa,

Aluminum Resin Graphite/epoxy

b (mm)

12.5

25

50

100

200

300

600

G,”

9.61

9.47

9.23

9.05

8.95

8.92

8.91

G,, = 6.4 GPa, vi2 = 0.34

3 BOUNDARY

LAYER

3 shows the strain energy release rate distributions across the width in aluminum DCBs with various beam widths, b. All of these specimens have the same thickness h = 1.65 mm. In these plots, the strain energy release rate is normalized with respect to the average value, G,,, listed in Table 2. A distinct feature of these distribution curves is the identical boundary layer near the edge of the beam. Within this boundary layer, the strain energy release rate exhibits significant variation. Away from the boundary layer, the value of G approaches a constant value of 8.91 X 10-2J/m2. For this example, we estimate the thickness of the boundary layer to be about 20h. For DCBs of smaller widths, the two boundary layers along the two beam edges may merge, and a constant G may not exist as shown in Fig. 4. This boundary layer is caused by the anticlastic curvature associated with bending-bending coupling in the X and Y directions.14 It exists in both isotropic and composite specimens. Figure 5 shows the boundary layer effect for a laminated DCB specimen with the lay-up sequence [03/906/03] for both arms. The average strain energy release rates are 6.126 x 10e2, 6.121 X 10mm2 and 6.120 X lo-* J/m* for b = 100, 200 and 300mm, respectively. Apparently, the boundary layer thickness in this case is smaller than Figure

that for the aluminum specimen. To obtain a uniform strain energy release rate along the crack front, the boundary layer effect must be eliminated. For isotropic materials, this can be achieved by imposing the boundary conditions for cylindrical bending, i.e. suppressing the rotation at the edge of the DCB specimen (+1 = 0 at y = *b/2). Alternatively, we can choose a material with a vanishing Poisson ratio. However, for composite DCB specimens, the boundary layer effect cannot be eliminated by the above methods. For illustration, Fig. 6 shows the G distribution along the crack front for a composite DCB specimen with zero Poisson ratio (lay-up sequence [8.3672/58.367,], for each arm). The distribution of G is not uniform (denoted by ‘normal’ in Fig. 6). Even with the edge rotation suppressed (denoted by ‘no edge rot’ in Fig. 6), the G distribution still varies across the beam width. 4 APPLICABILITY

OF BEAM THEORIES

4.1 Straight crack front As analytical tools, beam theories12,i3 are usually employed to calculate the strain energy release rate for DCB specimens. However, from the discussion above it is noted that the strain energy release rate at the crack front is not a constant. This prompts us to investigate what the results of beam models represent. 1.4

1.2

0.8 4

$ Q 0.6

0.6

c3 0.4

1

+b=30Omm +b=20OOmlU +b=10omm

_--1

0

20

2

40

60

80

Y/h Fig.

3. Strain energy release rate in aluminum large width.

4

6

8

10

12

14

5

100

DCB with

Fig. 4. Strain energy

Y/b release rate in aluminum small width.

DCB with

454

C. T. Sun, S. Zheng

1.2

Table 3. Comparison of G ( X lo-’ J/m2) DCB with b = 25 mm Method

Plane stress”

for aluminum

Plane strain”

Other (Gay)

0.8 is CO.6 a

Simple beam Sun and

e,

G

eb

9.40 9.80

0.7% 3.5%

8.56 8.92

9.6% 5.8%

9.66

2.0%

8.79

6.8%

Pandey ” Olsson’” Present

0.4 0.2 0

I 20

0 Fig. 5.

G

40

Y/h

-+

b=300mm

+

b=200mm

+

b=l OOmm

60

80

100

Strain energy release rate for [0,/90,/O,] laminated DCB.

A possible way to correlate the results of beam models and the present plate model is to compare the values of the strain energy release rate from beam models with the average strain energy release rate from the plate model. Note that two different values of strain energy release rate can be obtained by using beam theories, corresponding to the cases of plane strain and plane stress, respectively. The expression of strain energy release rate for a simple beam model is expressed as:

Table 3 presents a comparison of the results of strain energy release rate from the simple beam model and from some advanced beam models12,13 with the

9.47

average strain energy release rate from the present analysis for an aluminum DCB with b = 25 mm. From the results in Table 3, one may conclude that the simple beam model with the plane stress assumption provides a result that is closest to the present analysis; however, this is not always the case. In fact, the result from the simple beam model just happens to be close to the present result for the case b = 25 mm. If we calculate the average strain energy release rate for different beam widths, we would find that G actually varies with beam width. Figure 7 shows the variation of average strain energy release rate with beam width. Several observations are noted in Fig. 7. First, the average strain energy release rate for aluminum DCB decreases as the beam width, b, increases. Second, the limits are estimated as lim Gay = G& = 9.72 X lop2 J/m2 h+O

(2)

lim G,, = GrV = 8.91 X 1O-2 J/m2 h-m

(3)

Since the case b -+O corresponds to plane stress and b + ~0 corresponds to plane strain, we can now

0.9

0.6

--c no edge rot

4

0.5

-15

-10

Fig. 6. Strain energy

-5

&n)

5

10

0

15

y release

rate distribution

DCB of [8.367,/58-367,], lay-up.

of a composite

Fig. 7.

100

200 b/h

300

400

Average energy release rate versus beam width for aluminum DCB.

455

DCB and ENF composite specimens

compare the results from the beam theories with those of the present analysis (Table 4). It is evident from Table 4 that the advanced beam models of Sun and Pandey” and Olsson13 provide far closer results as compared with the present results than the simple beam model. The present results lie between the results obtained from the Sun-Pandey model and those obtained from the Olsson model. It is interesting to note that Gz, corresponds to the constant value of G in DCB specimens with zero Poisson ratio and GY&corresponds to the constant value of G in DCB specimens with edge rotation suppressed. For practical purposes, when using a beam model to calculate the strain energy release rate for DCB specimens, the advanced beam models (such as those of Sun and Pandey and Olsson) are not necessarily better than the simple beam model as compared with the plate model. As can be seen in Tables 3 and 4, for the commonly used testing specimen with b = 25 mm, the simple beam model with plane stress assumption may yield better agreement with the plate model. However, in order to reduce the boundary layer effect, specimens with large beam widths are recommended if beam theories are to be used as data reduction tools. In this situation, a plane strain assumption is selected and the advanced beam models should be used. 4.2 Curved crack front Since strain energy release rate varies across beam width, the actual DCB crack front after crack growth could be curved rather than straight.15,‘7,21 By calculating the total strain energy release rate (TG) which is obtained by integrating G along the crack front, Nilsson17 concluded that TG could be 24% higher at the instant when the curved crack front is fully developed and is propagating than at the instant when a straight crack front starts to grow initially at the center of the DCB. In this section, we shall examine the effect of curved crack front on the use of beam theories in calculating strain energy release rate. Consider an aluminum DCB specimen, as shown in Fig. 1, with L = 100 mm, b = 25 mm, h = 1.65 mm. In order to find the actual curved crack front, an effort was made to search for the crack front where strain Table 4. G ( X 10-zJ/mZ) for DCB in plane stress and plane strain Method

Simple beam Sun and Pandey12 Olsson’” Present (G,,)

Plane stress

Plane strain

G,

e,

GE

eh

9.40 9.80 9.66 9.72

3.3% 0.8% 0.6% 0.0%

8.56 8.92 8.79 8.91

3.9% 0.1% 1.3% 0.0%

k

4

ae

Fig. 8. Curved crack front in aluminum DCB. energy release rate equals its critical value, G,, corresponding to the critical loading. Note that G, was chosen to be 1600 J/m* for the present simulation. Figure 8 shows a schematic representation of a curved crack front, which yields the critical loading P = 3940 N/m and the tip displacement 6 = 7.1 mm. At the edge of the DCB specimen the crack length is ae = 50.9 mm, and at the center the crack length is a’ = 52.7 mm. In fracture toughness tests on DCB specimens, the crack length is usually measured at the edge of the specimen (ae). If we adopt the straight crack front assumption as commonly practiced, we can calculate the average strain energy release rate by using the double-plate model for critical load P = 3940 N/m and crack length ae = 50.9 as GiV = 1479 J/m*. This value is 6% lower than G, at the curved crack front. For comparison, the simple beam model predicts GP = 1513 J/m*, which is 5% lower than this G,. However, if we use the critical tip displacement S = 7-l mm to calculate the strain energy release rate at the straight crack front, we find Gi, = 1751 J/m2, which is 9% higher than the critical strain energy release rate at the curved crack front. Again, for comparison, it is noted that the simple beam model predicts G” = 1803 J/m*, which is 13% higher than G,. Thus, if a straight crack front is assumed, an equivalent crack length, aeq, should be used. The above discrepancies between the experimentally determined critical strain energy release rate, GfV, (determined by using the critical load) and G!& (determined by using the critical tip displacement) is due to the fact that the measured crack length, ae, underestimates the equivalent crack length, seq. This discrepancy between GrV and G& can best be explained by using the simple beam theory. From simple beam theory, the strain energy release rate at the DCB crack front can be expressed in terms of load, P, and tip displacement, 6, (with crack length, a) as:

G”S

=_

3S2Eh3 4a4

(5)

C. T. Sun, S. Zheng

456 respectively. have:

By using the equivalent crack length, we G = 12P2(aeq)* c

Eb2h3

Since, ae I aeq, from following bounds:

=-

-+-

3cS2Eh3

(6)

4(a”q)4

eqns (4)-(6)

we have the

Gap I G, 5 Gas

(7)

Thus, by using critical load to calculate the strain energy release rate, the result would be lower than the corresponding critical strain energy release rate, G,, at the curved crack front, and using the critical tip displacement would lead to a higher value. It is worth noting that the current ASTM standard for DCB test data reduction procedure does not use an analytical solution based on beam theories. Instead, it uses a compliance calibration technique known as Berry’s method?” Specifically, the formula for the experimentally determined critical strain energy release rate is: G=

S2 dC _h!!!=__=__ b da

2bC2 da

P*dC 2b da

(8)

where U is the strain energy and C is the compliance calibrated by an empirical formula. Since the compliance, C, is calibrated experimentally, it includes the effect of the curved crack front. However, in eqn (8), the quantity b should be replaced with the length of the curved crack front. Without such a correction, eqn (8) would overestimate the strain energy release rate. 5 SKEWED

L2, s=O.32

G DISTRIBUTION

For cross-ply and unidirectional laminated composite DCB specimens, the distribution of strain energy release rate at the crack front is similar to that in an isotropic specimen, i.e. it exhibits the boundary layer effect and its distribution is symmetric with respect to the X axis. Thus, the conclusions drawn above for isotropic DCB specimens are still applicable. For other types of laminates, even if they are symmetric the strain energy release rate and balanced, distribution across the width may be quite different. The main reason is, in general, that bending-twisting coupling is present in the laminates. For unidirectional and cross-ply laminates, owing to the absence of bending-twisting coupling, the analysis can be performed by using half of the specimen. However, caution must be exercised when such a procedure is employed in the analysis of other types of laminates. For illustration, consider the following symmetric and balanced lay-up sequences: Ll, [ f 45,],; and L2, two specimens are almost [ f @i2/ f 451,. These identical except for lay-up sequence. However, they have quite different bending-twisting coupling. The

-15 Fig.

-10

9. Variation

-5

0 y hm)

5

in G along crack front composite DCB (large s).

10 of

15 angle-ply

composite material properties are listed in Table 1; the ply-thickness is 0.127 mm. For these laminates, the distributions of G are shown in Fig. 9. It is clear that G is not symmetrically distributed with respect to the center line. In fact, the G distribution is highly skewed. This skewness can be qualitatively characterized by the parameter:

where D16 and Dll are bending stiffness components in the D matrix of the laminate. It was found that the larger the s value, the more severe the skewness. Figure 10 shows additional examples of the G distribution at 45/45 and O/O interfaces in laminates L3 to L6 as given in Table 5. The skewed behavior of the G

Fig.

10. Variation

in G along crack front composite DCB (small s).

of

angle-ply

457

DCB and ENF composite specimens Table 5. Lay-up sequences for the upper arms of DCB specimen and value of s

Notation

Lay-up sequence

s

I* 4533,

O-5019 0.3153 0 0.10 0.167 0.250

Ll L2 L3 L4 L.5 L6

[ f 45*/ f 451s

Pl1*

distribution is caused by bending-twisting coupling. The validity of the parameter s as an indicator of the skewness of the G distribution is verified by the examples in Figs 9 and 10 for s values ranging from 0 to 0.5. In fact, careful selection of laminate lay-up sequences can ensure that s = 0 for the specimen, thus eliminating the skewness of the G distribution. For example, as long as the lay-up sequence for both arms of the DCB specimen is anti-symmetric (e.g. [ - O/O] or [OJ -@JO,/ -@,I, etc.), then we always have s = 0 (because D16 = 0) for the specimen. Figure 11 shows the G distribution for a DCB consisting of a [ - 301301 upper arm and a [4.5/ -451 lower arm. Apparently G achieves a symmetric distribution. 6 DESIGN

RECOMMENDATION

To use the DCB specimen to measure fracture toughness, ideally the strain energy release rate should be uniform along the crack front in order to utilize the beam model to reduce the experimental data. However, as discussed above, in a composite laminate

DCB specimen, the G distribution is not uniform and may be highly skewed. As a result, the crack front of the DCB specimen may be curved and skewed, which makes the interpretation of test data ambiguous. Since the crack length at the two edges of the DCB specimen may be different, and neither of these two crack lengths is an appropriate representation of the equivalent crack length, the resulting experimental data may not produce accurate fracture toughness values. Meanwhile, since the crack front curvature and skewness depend on the lay-up sequence of the specimen, these may contribute to the dependency of the experimental fracture toughness on the lay-up a good DCB specimen sequence. Consequently, should both minimize the variation and skewness of its G distribution. In order to minimize the variation of G, Davidson’s suggested minimizing the parameter D, = D:2/D11 Dz2. From lamination theory, we have: D, = i k$, Q~(z: - ~2~))

where n is the number of laminae and the definition of @ can be found in the Appendix. Consider the following:

Note that & = u, - u, cos 4@/,2 u, - u,. Thus, D12 is a minimum at cos 401, = 1, i.e. Ok = 0” or 90”. Further numerical examination reveals that D, achieves its minimum when Ok = 0. Thus, the 0” unidirectional laminate DCB specimen should have the smallest G variation across the crack front. To minimize the skewness of the G distribution, we should minimize the parameter

-15

-10

-5

Y(k) 5

lo l5

Fig. 11. G distribution at crack front for [ T 30/ f 451 (with

s =O).

(9)

016

s = _ . Obviously, I DII I the best lay-ups are those yielding D16 = 0. These include O”, 90”, cross-ply and antisymmetric laminates. Our task is to find the laminates that minimize the skew behavior of the strain energy release rate along the DCB crack front and at the same time keep its variations across the width as small as possible. On the basis of this design guide, the following lay-up sequence is recommended for testing 0(1)/O(2) interfacial fracture toughness: [ - O~‘~/O,/O”~//O~*~/ OJ - @‘*‘I (s = 0), wh ere /I indicates the location of the crack and n is a large integer that makes the behavior of this specimen approach that in [0,/O//O/O,]. Figure 12 shows the behavior of the G distribution for the lay-up [ - 30/0,,/30//45/02,/ - 451, which is designed for testing the 30/45 interfacial toughness. For the purpose of comparison, the G distribution in the [O20/O//O/O,,] specimen is also shown. It is evident that the design objective is achieved.

C. T. Sun, S. Zheng

458

distribution similar to that for the 0” laminate as indicated by specimen Lc in Fig. 13. Thus, the same design lay-up for DCB specimen is also recommended here. 8 CONCLUSIONS The strain energy release rate distributions for DCB and ENF specimens have been analysed. The following conclusions have been obtained: . 0.2 -

I -15

-10

#-, ” /

-+-

o/o

+

design

-5

Fig. 12.

5

10

15

Design recommendation.

7 ENF SPECIMEN The ENF specimen in Fig. 2 has also been analysed by the double-plate model. In order to avoid interpenetration within the delamination region, a smooth contact condition was assumed. That is, nodes at top and bottom plates within the delamination region were assumed to have the same displacement in the 2 direction. The variation of the G distribution at the crack front of the ENF specimen behaves more or less like that in the DCB specimen. Figure 13 shows the G distribution at the crack front for three ENF specimens: La, [016//016]; Lb, [ f 4&/ F 45,// f 453/ r 45,]; and Lc, [ - 30/0,,/30//45/0,,/ - 451. It is noted that the boundary layer effect in the ENF specimen is not as pronounced as that in the DCB specimen. However, the G distribution is still skewed for the angle-ply laminate (see Lb in Fig. 13). By adopting the same design procedure as for the DCB specimen, we can virtually eliminate the skewness and make the G

1 ,

+-La +-Lb -tLc

n

6 --

,

5 --

l

l

A boundary layer effect causes the strain energy release rate to vary along the straight crack front. The thickness of this boundary layer is within 10 times the thickness of the DCB specimen away from the edge for the aluminum specimen. While assuming that the crack front is straight, the beam theories (advanced models) are adequate for calculating the strain energy release rate for DCB specimen as long as the beam width is large enough to represent plane strain. However, for commonly used specimens, the beam theories may cause errors in the determination of critical strain energy release rate. Because of the curved crack front in the actual test specimen, the use of critical loading in conjunction with beam theory will underestimate critical strain energy release rate, G,, while the use of critical displacement will overestimate G,. For laminated specimens containing angle plies, the distribution of strain energy release rate is skewed along the crack front. The degree of skewness depends on the lay-up sequence. A parameter has been proposed for the qualitative measurement of the skewness of G. The lay-up sequence [ - 0~1~/0,/0~‘~//0~2~/0~/ - O@‘] is recommended for testing O(‘)/O@) interfacial fracture toughness. The boundary layer in the ENF specimen is smaller than that in the DCB specimen. The distribution of G at the crack front is qualitatively similar to that in the DCB specimen, which is varied and skewed, and the same design lay-up is recommended for ENF testing involving angle-ply interface.

ACKNOWLEDGEMENT

d 3

This work is supported by NASA Langley Research Center under grant no. NAG-l-1323 to Purdue University. Drs Jerry Housner and John Wang are technical monitors. REFERENCES

-15

-10

-5

Fig. 13. ENF G distribution

5

10

at crack front.

15

1. Russell, A. J. & Street, K. N., Factors affecting the interlaminar fracture energy of graphite/epoxy laminates. ICCM-IV, Tokyo, 1982, pp. 279-86.

459

DCB and ENF composite specimens J. R., Camin, R. A., 2. Wilkins, D. J., Eisenmann, Margulis, W. S. & Benson, R. A., Characterizing delamination growth in graphite/epoxy. ASTM STP 775,1982, pp. 168-83. of fracture surface energies 3. Berry, J. P., Determination by the cleavage technique. .I. Appl. Phys., 34 (1963) 62-8.

4. Robinson, P. & Song, D. Q., A modified DCB specimen for Mode I testing of multidirectional laminates. J. Comp. Mater., 26 (1992) 1554-77.

5. Polaha, J. J., Effect of interfacial ply orientation on the fracture toughness of a laminated graphite/epoxy composite. AIAA-94-1537-CP, 1994, pp. 1707-16. 6. Ishikawa, H., Koimai, T. & Natsumura, T., Interlaminar fracture toughness of Mode I and Mode II in CFRP laminates. Proc. 2nd Int. Symp. on Composite Materials and Structures, ed. C. T. Sun & T. T. Loo. Beijing, 1992, pp. 340-5. 7. Whitney, J. M., Browning, D. E. & Hoogsteden, W., A double cantilever beam test for characterizing Mode I delamination of composite materials. J. Reinf: Plast. Comp., 1 (1982) 297-313. 8. ASTM standard test method for Mode I interlaminar fracture toughness of unidirectional continuous fiber reinforced composite materials. ASTM Standard D5528-94A, ASTM, Philadelphia, PA, 1994. 9. Kageyama, K., Kobayashi, T. & Chou, T.-W., An analytical compliance method for Mode I interlaminar fracture toughness testing of composites. Composites, 18

width on deflection orthotropic double

Mater., 22 (1988) 641-56.

17. Nilsson, K. F., On growth of crack fronts in the DCB test. Comp. Engng, 3 (1993) 527-46. 18. Zheng, S. & Sun, C. T., A double plate finite element model for impact induced delamination problems. Comp. Sci. Technol., 53 (1995) 111-18. 19. Sun, C. T. & Chin, H., On large deflection effects in unsymmetric cross-ply composite laminates. J. Comp. Mater., 22 (1988) 1045-59. 20. Hibbitt, Karlsson and Sorenson Inc., ABAQUS, Version 5.3. 21. De Kalbermatten, T., Jaggi, R., Flueler, P., Kausch, H. H. & Davies, P., Microfocus radiography studies during Mode I interlaminar fracture test on composites. J. Mater. Sci. Lett., 11 (1992) 543-6. APPENDIX The definitions

of 05 in eqn (9) are given below.

Q:, = u, + u, cos 20,, + u, cos 40/( @,

= u, - u, cos 20/, + u, cos 4Ok

Q’;2 = u, - u, cos 40/, 1 Q:, = - sin 203, + U, sin 401, 2

(1987) 393-9.

10. Hashemi, S., Kinloch, A. J. & Williams, J. G., Corrections needed in double-cantilever beam tests for assessing the interlaminar failure of fiber composites. J.

U, = 8 (3Qu + 3822 + 2Q12 + 4QwJ

Mater. Sci. Lett., 8 (1989) 125-9.

11. Chatterjee, S. N., Analysis of test specimens for interlaminar Mode II fracture toughness: part I. Elastic laminates. J. Comp. Mater., 25 (1991) 470-93. 12. Sun, C. T. & Pandey, R. K., Improved method for calculating strain energy release rate based on beam theory. AIAA .I., 32 (1994) 184-9. 13. Olsson, R., A simplified improved beam analysis of the DCB specimen. Comp. Sci. Technol., 43 (1992) 329-38. 14. Crews, J. H., Shivakumar, K. N. & Raju, I. S., Strain energy release rate distribution for double cantilever beam specimen. AZAA J. (1991) 1686-91. B. D., An analytical 15. Davidson, investigation of delamination front curvature in double cantilever beam specimens. J. Comp. Mater., 24 (1990) 1124-37. 16. Davidson, B. D. & Schapery, R. A., Effect of finite

and energy release rate of an cantilever specimen. J. Comp.

U, =; (QII - Q,,)

U3

=

;


U~=~(Q,,+Q~Z+~Q,,-~Q~~) Us = d (QII + Q22 - 2Q12 + 4Qs) Ok = ply stiffness.

angle

for

kth

layer,

Q, = reduced

lamina