Buckling, post-buckling and delamination propagation in debonded composite laminates Part 2: Numerical applications

Available online at www.sciencedirect.com

Composite Structures 88 (2009) 131–146 www.elsevier.com/locate/compstruct

Buckling, post-buckling and delamination propagation in debonded composite laminates Part 2: Numerical applications S. Wang a,*, Y. Zhang b a

Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough LE11 3TU, UK b Doosan Babcock Energy Limited Technology and Engineering, Porterfield Road, Renfrew PA4 8DJ, UK Available online 19 February 2008

Abstract A layerwise B-spline finite strip method is developed with consideration of delamination kinematics to study the buckling, postbuckling and delamination propagation in debonded composite laminates under compression in Part 1 [Zhang Y, Wang S. Buckling, post-buckling and delamination propagation in debonded composite laminates Part 1: Theoretical development. Compos Struct 2009;88:121–30]. Extensive numerical applications are conducted in this Part 2 to validate the theoretical development. Comparisons are made between the present predictions and the existing numerical and experimental results. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Buckling; Post-buckling; Delamination; Interface springs; Energy release rates; Spline finite strip

1. Introduction The mechanical and numerical model developed in Part 1 of the paper [1] has been implemented in a computer software for buckling, post-buckling and delamination propagation in debonded composite laminates under in-plane loadings. The geometric nonlinear problem is solved using Newton–Raphson technique. To verify the capability of the model and study the concerned mechanical behaviour of debonded composite laminates, extensive numerical investigations are conducted and are presented in details in this Part 2 of the paper.

authors for the calculation of energy release rates of double cantilever beams (DCB) under pure modes I, II and mixed modes I and II fractures. Here we study the validity of the interface spring models in the prediction of buckling loads. Consider a unidirectional laminated plate containing a through-width delamination as shown in Fig. 1. The plate has dimensions of thickness h, length A, and width B. The delamination has length a and is located at the span centre with a distance d from the surface. The plate is clamped at both ends and is subjected to uniform end-shortening strain along the clamped ends. The material properties and the plate geometry are taken as

2. Numerical investigations

E1 ¼ 40:4G23 ; E2 ¼ 2:82G23 ; G12 ¼ G13 ¼ 1:62G23 ; m12 ¼ 0:29

2.1. Buckling analysis of a unidirectional laminated composite plate

and A=h ¼ 40;

The validity of the interface spring models has been thoroughly investigated in a related study [2] by the *

Corresponding author. Tel.: +44 1509 227252; fax: +44 15097275. E-mail address: [email protected] (S. Wang).

0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.02.012

d=h ¼ 0:3

and B=h ¼ 4

Along the width direction half of the plate is modelled using one quadratic strip. The number of strip sections used is forty. Three numerical layers are used. The interface where delamination occurs, is treated using two different

132

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

B d

h a A

Fig. 1. A delaminated laminate clamped at both ends and subjected to uniformly end shortening.

non-delaminated part as perfect bonded interface and does not use imaginary springs. The purpose of this test is to examine the validity of the interface spring model. Table 1 shows the normalised buckling load with respect to the perfectly bonded laminate for various delamination lengths. It is shown that the two approaches give very close results which compare well with comparative solutions [3]. Results in Table 1 is also graphically shown in Fig. 2 where the FEM results from [4] are also presented. 2.2. Post-buckling analysis of an isotropic square plate

Table 1 Normalised buckling loads for a delaminated plate with clamped ends (d/ h = 0.3) a/A

0 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000

Present FSM

1-D model [3]

Point spring model

No spring model

1.0000 0.9994 0.9992 0.9980 0.9923 0.9679 0.8654 0.5441 0.3585 0.2531 0.1881 0.1453 0.1156 0.0948

1.0000 1.0000 0.9999 0.9987 0.9933 0.9686 0.8586 0.5365 0.3541 0.2504 0.1863 0.1441 0.1147 0.0948

1.0000 1.0000 0.9998 0.9985 0.9924 0.9662 0.8582 0.5314 0.3469 0.2435 0.1804 0.1390 0.1105 0.0900

approaches. One calls for use of an interface spring model in which point springs are placed at the non-delaminated part with a stiffness value of 100E1. Another treats the

The comparative results are those of the analytical solution obtained by Yamaki [5] on the basis of CPT. For this problem, the transverse shearing effect is not expected to be significant. From the figure it can be seen a close comparison

Present layerwise FSM: without use of interface springs: using point interface springs: x

1

Normalised Buckling Load

Here, we consider an isotropic square plate with side-tothickness ratio of A/h = 120. The plate is simply supported in such a way that around all the sides the out-of-plane deflection is constrained while the in-plane displacements are free. The Poisson ratio is taken to be m = 1/3. This example was investigated with analytical method by Yamaki [5]. In the present LW FSM analysis, along the crosswise x2 direction half the plate is modelled using four quadratic strips with six strip sections. Through the thickness one and three numerical layers are used, respectively. For the case of one numerical layer, shear correction factor k2 = 0.8333 is used. Fig. 3 shows the behaviours of compressive force- end shortening and compressive load-central deflection. The load factor is defined as Z Z A X b=2 A T 11 dx1 dx2 F ¼ 2 3 p Eh b=2 0

Beam analysis[3]: FEM [4]:

0.8

+ o d

h

0.6

a A

0.4

A/h=40 d/h=0.3

0.2

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalised Crack Length a/A Fig. 2. Buckling load of a delaminated laminate with clamped ends.

0.9

1

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

133

1.5

Present LW FSM: N=1 N=3

Load factor F

1

0.5

0

0

0.5

1

1.5

2

2.5

3

End shortening strain

x 10 -3

1.6

Yamaki [5] Present LW FSM: N=1 N=3

1.4

Load factor F

1.2

1

0.8

0.6

0.4

0.2

0

0.5

1

1.5

2

2.5

3

3.5

4

Central deflection w/h Fig. 3. Response of a simply supported isotropic square plate under end shortening (a) compressive load-end shortening behaviour and (b) compressive load-central deflection behaviour.

exists between the results obtained by the present LW FSM using one and three numerical layers. Use of three numerical layers tends to give slight softer results than use of one numerical layer. The present results are close to but slight softer than those of Yamaki. 2.3. Compressive behaviour of delaminated composite laminates under end shortening In this subsection the concern is with the behaviour of delaminated laminated plate under progressive uniform

end-shortening strain. Numerical analysis is attempted to obtain knowledge about: (1) buckling loads and post-buckling response of delaminated plates, (2) delamination propagation during post-buckling, (3) interaction of delaminations during post-bukling. For the purposes, numerical analysis is first carried out to determine the initial buckling load and mode shape. Then postbuckling analysis is followed to investigate the

134

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

compression responses of the plates. For each loading increment, the nonlinear equations of equilibrium are solved and then the energy release rates are calculated to determine the delamination propagation. If the delamination criterion defined by Eq. (29) in Part 1 of the paper is satisfied the delamination is extended by one spline section length along the strip longitudinal direction. With the updated delamination length the nonlinear equations of equilibrium are solved again. This procedure is repeated until no further delamination propagation is detected. It should be noted that only the delamination failure mode is considered in the numerical applications presented here, in spite of the fact that matrix and fibre failure modes may occur during the compressive processing and they definitely affect the compressive behaviour to some content. The problems considered here are those of T300/976 laminated composite plates under progressive uniform end-shortening. The plate has length A equal to 2 in. (50.8 mm) and is clamped at two ends. As shown in Fig. 4, the delamination configuration of the laminated plates considered include two types, i.e., one through-width delamination with length a1 and two through-width delaminations with length a1 and a2. The delaminations are located at the centre of the plate span. The lay-ups from bottom to top of the laminates are [04/012//04], [04//012// 04] and [04/(±45)6//04]. Here the double slashes ‘‘//” implies delaminated interface. All plies have equal thickness. Numerical studies are carried out on four cases of geometrical configurations that are detailed in Table 2. For all of those examples experimental data for compression load-

x2 x3

Top

x1 h a2 B

a1 A

Fig. 4. Configuration of delaminated composite laminate under uniformly end-shortening at two clamped ends.

Table 2 Geometrical configuration of panels considered Case

Lay-ups

a1 (in., mm)

a2 (in., mm)

Thickness h (in., mm)

1 2 3 4

[04/012//04] [04/012//04] [04//012//04] [04/(±45)6//04]

1.50, 0.75, 1.50, 1.00,

0, 0 0, 0 0.75, 19.05 0, 0

0.102, 0.100, 0.101, 0.100,

38.1 19.05 38.1 25.4

2.59 2.54 2.57 2.54

Table 3 Material properties for T300/976 graphite/epoxy lamina Ply longitudinal modulus Ply transverse modulus Out-of-plane modulus Inplane shear modulus Out-of-plane shear modulus Poisson’s ratio

Ply longitudinal tensile strength Ply longitudinal compressive strength Ply transverse tensile strength Ply transverse compressive strength Ply shear strength Critical energy release rate for mode I Critical energy release rate for mode II

E1 E2 E3 G12 G13 G23 m12 m13 m23 Xt Xc Yt Yc S GIC GIIC

20.2 msi, 139.3 GPa 1.41 msi, 9.72 GPa 1.41 msi, 9.72 GPa 0.81 msi, 5.58 GPa 0.81 msi, 5.58 GPa 0.5 msi, 3.45 GPa 0.29 0.29 0.4 220 ksi, 1517 MPa 231 ksi, 1593 MPa 6.46 ksi, 44.54 MPa 36.7 ksi, 253.0 MPa 15.5 ksi, 106.9 MPa 0.50 lbf/in., 87.6 N/m 1.8 lbf/in., 315.2 N/m

Note: 1 msi = 1000 ksi = 106 lbf/in.2 = 6.895 GPa, 1 lbf = 4.448 N.

1 in. = 25.4 mm,

surface strain histories are available in Ref. [6], which will be used for comparison. The width of the specimen is not given in Ref. [6]. A width of 0.2 in. (5.08 mm) is used in the present analysis. The material properties used are the same as those in Ref. [6], which are given in Table 3. In the present studies, half of the plate in the width direction is modelled using one quadratic strip with 64 spline sections and three numerical layers. The numerical results for the strains at the surface are calculated at 0.002 in. (0.05 mm) away from the outer surface to allow for the strain gauge thickness [6]. 2.3.1. Unidirectional laminated plate with a single delamination Figs. 5–7 show the compression responses of a [04/012// 04] composite plate containing a single delamination with length of 1.5 in. (38.1 mm). Figs. 5 and 6 give the axial compression load-end shortening strain behaviour and load-deflection histories. The comparison between the predicted and experimental strain histories at the centre of the top surface and the bottom surface are presented in Fig. 7. The normalised energy release rates against the respective critical values are shown in Fig. 8 with respect to end shortening strain. These figures show that an initial buckling occurs at an end-shortening strain above 5  104. The detailed numerical calculations give its value, ec = 5.729  104, corresponding to a compression load of 1180 lbf/in. (207 N/ mm). This initial buckling is a stable local buckling of the thinner upper sub-laminate as clearly shown in Fig. 6. The axial stiffness of the whole laminate has a considerable change due to this local buckling as shown in Fig. 5 because of the reduction of axial stiffness of the buckled upper sub-laminate. In the post-local-buckling stage, significant transverse deflection is mainly in the thinner upper sub-laminate as observed in Fig. 6, and the lower sub-laminate has a very small deflection. Detailed numerical results

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

135

-9000 -8000

Compression load F (lbf/in)

-7000 -6000 -5000 -4000 a1

-3000

2"

-2000

[04 /012 //04 ] a1=1.5" h=0.102"

-1000 0 0

-1

-2

-3

-4

-5

-6

End shortening strain

-7 -3

x 10

Fig. 5. Longitudinal compression load vs end shortening strain behaviour of a [04/012//04] laminate containing an initial 1.5 in. (38.1 mm) delamination (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm).

-9000 -8000

Compression load F (lbf/in)

-7000 -6000

Top

-5000

Bottom

-4000

[04 /012 //04 ] a1=1.5" h=0.102"

-3000 -2000

a1 2"

-1000 0 -0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Central deflection (in) Fig. 6. Load-deflection histories of a [04/012//04] laminate containing an initial 1.5 in. (38.1 mm) delamination subjected to end shortening strain (1000 lbf/ in. = 175.1 N/mm, 1 in. = 25.4 mm).

also show that both of the upper and lower sub-laminates move upward initially and as the end shortening strain increases to a slightly larger value than the buckling end shortening strain ec, the deflection of the lower sub-lami-

nate changes from upward to downward (snap-through), but it is still very small compared with the upper sub-laminate deflection. This phenomenon is also observed in the case of [04/012//04] composite plate containing a 0.75 in.

136

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

-9000 -8000

Compression load F (lbf/in)

-7000 a1

-6000

2"

-5000

[04 /012 //04] a1=1.5" h=0.102"

-4000 -3000 -2000

× -1000 0 -1

0

1

2

Present Experiment [6]

3

4

5

6 -3

Strain

x 10

-9000 -8000

Compression load F (lbf/in)

-7000 -6000 -5000 -4000 -3000 -2000

[04 /012 //04 ] a1=1.5" h=0.102"

-1000 0 -5

-4

-3

Present: Experiment [6]:

-2

-1

0

Strain

1

2

3

× Δ

4

5 -3

x 10

Fig. 7. Comparison between the predicted and the experimental strain histories of a delaminated [04/012//04] laminate under compression (a1 = 1.5 in. (38.1 mm)) (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm) (a) top surface and (b) bottom surface.

(19.05 mm) delamination and can be explained by theoretical analysis as in Ref. [7]. As shown in Fig. 7a, the present analysis underestimates the initial local buckling load in comparison with the experimental data. The possible reason is due to the sticking of the specimen’s sub-laminates through Teflon film during manufacturing, thus increasing the buckling load [6]. As shown in Fig. 8, the normalised energy release rates are zero before the initial local buckling

as expected. Their afterwards variations with end shortening strain can be understood with the following explanation. After the initial local buckling, both resultant compressive axial forces and bending moments appear on the cross sections of the two sub-laminates at the delamination front. The difference in the increase rates of the two axial forces produces mode II fracture while the bending moments produce both mode I and II fractures. It is

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

137

1.6 1.4

a1 2"

Energy release rate

1.2

[04 /012 //04] a1=1.5" h=0.102"

1 0.8

GI /GIC+ GII/GIIC GII/GIIC

0.6 0.4

GI/GIC 0.2 0

0

-0.5

-1

-1.5

-2

End shortening strain

-2.5

-3

-3.5

x 10

-3

Fig. 8. Energy release rate at the front of the delamination for a [04/012//04] laminate containing a 1.5 in. (38.1 mm) delamination under end shortening strain.

observed from Fig. 5 that the axial stiffness of post local buckling is fairly constant before delamination propagation starts. Moreover, Fig. 6 shows that significant deflection only occurs at the upper sub-laminate. Therefore, it can be concluded that the resultant axial force in the lower sub-laminate increases linearly with the increase in the end shortening strain and provide primarily the axial stiffness of the whole laminate, whilst the axial force in the upper sub-laminate increases non-linearly at first and then decreases after certain level of shortening strain due delamination growth. The two bending moments are small and increase nonlinearly with a slow rate. Consequently, the normalised mode II energy release rate increases much more rapidly than the normalised mode I energy release rate as shown in Fig. 8 and dominates the delamination growth. The delamination is predicted to start propagation at an end shortening strain near to 2.5  103 (compression load 4460 lbf/in. (781 N/mm)) and to extend to the ends of the plate at end shortening strain near to 3.4  103 (compression load 5795 lbf/in. (1015 N/mm)) in a stable manner. It is noted that in Fig. 8 seven growth steps are taken corresponding to one spline section growth (i.e. 2 in./64=0.03125 in. (0.79 mm)) in each of the first six step and two spline section unstable growth in the last step. When more growth steps are taken the zig-zag line will approach to a horizontal line. It is noted that the mode I energy release rate remains fairly constant in each growth step and approaches to zero at the clamped end and the mode II energy release rate increases steadily in each growth step reflecting the steady increase of the resultant axial force in the lower sub-laminate. It is also observed that the axial stiffness has a slight reduction due to delam-

ination growth as shown in Fig. 5. Also, as shown in Figs. 6 and 7, corresponding to each step of delamination growth, the total compressive load has a slight drop, the central deflection of the upper sub-laminate has a step increase while its top surface tensile strain has a step drop, and the central deflection of the lower sub-laminate has a step drop while its bottom surface compressive strain has a step increase. Eventually a neutral global buckling of sub-laminate occurs at end shortening strain near to 4.80  103 (compression load 8044 lbf/in. (1409 N/mm)). The predicted global buckling load overestimates the collapse load as seen in Fig. 7. A possible cause can be either the validity of the clamped end conditions at such high compression load in the experiments or the material failure at the clamped ends due to the approaching delamination. In Fig. 7a it is shown that the present prediction for the top surface strain (the upper sub-laminate) is reasonablly close to the experimental results until the delamination extends to the clamped ends. The stable delamination growth is also predicted well. For a [04/012//04] composite plate containing a single delamination with a length of 0.75 in. (19.05 mm), the compression responses are shown in Figs. 9–12. Among them, Figs. 9 and 10 give the axial compression load-end shortening strain behaviour and load-deflection histories, respectively. The comparison between the predicted and experimental surface strain histories at the plate centre are shown in Fig. 11. The variation of normalised energy release rates with respect to end shortening strain is given in Fig. 12. Buckling analysis indicates that a stable local buckling occurs in the upper sub-laminate at an end shortening strain of ec = 2.024  103 (corresponding to a

138

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

-8000

Compression load F (lbf/in)

-7000 -6000 -5000 -4000 a1

-3000

2"

-2000

[04 /012//04] a1=0.75" h=0.100"

-1000 0 0

-1

-2

-3

-4

-5

-6

End shortening strain

-7

x 10

-3

Fig. 9. Longitudinal compression load vs end shortening strain behaviour of a [04/012//04] laminate with an initial 0.75 in. (19.05 mm) delamination (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm).

-8000

Compression load F (lbf/in)

-7000 -6000 -5000 -4000

× o •

a1

-3000

2"

Top Bottom Unstable delamination growth

-2000 -1000 0 -0.06

[04 /012 //04] a1=0.75" h=0.100" -0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Central deflection (in.) Fig. 10. Load-deflection histories of a [04/012//04] laminate containing an initial 0.75 in. (19.05 mm) delamination subjected to end shortening strain (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm).

compression load 4089 lbf/in. (716 N/mm)). As expected it is much later than that of the previous example where the delamination has a doubled length. The critical end shortening strain is about three and half times the previous one. As loading continues, cracking energy accumulates at the

delamination fronts. It is observed from Fig. 9 that the axial stiffness has no significant change after the initial local buckling compared with the stiffness change in Fig. 5, which implies no significant reduction in the axial stiffness of the upper sub-laminate. Consequently, the two resultant

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

139

-8000

Compression load F (lbf/in)

-7000 -6000

Present: × • Unstable delamination growth Experiment [6]: Δ

-5000 -4000 -3000

[04 /012 //04] a1=0.75" h=0.100"

a1

-2000

2"

-1000 0 -3

-2

-1

0

1

2

3

4

5

Strain

6

x 10

-3

-8000

Compression load F (lbf/in)

-7000 -6000 -5000

Present: × • Unstable delamination growth Experiment [6]: Δ

-4000 -3000 -2000 -1000 0 -6

[04 /012 //04] a1=0.75" h=0.100"

-4

-2

0

2

Strain

4

6

8

x 10

-3

Fig. 11. Comparison between the predicted and the experimental strain histories of a delaminated [04/012//04] laminate under compression (a1 = 0.75 in. (19.05 mm)) (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm) (a) top surface and (b) bottom surface.

axial forces on the cross sections of the two sub-laminates at the delamination front have a close increase rates and thus have no much contribution to mode II fracture while the two bending moments on the cross sections are the major contributors to both mode I and II fractures. Bearing in mind the fact that mode I fracture has a much smaller critical value of energy release rate than the mode II has, it is expected that the normalised mode I energy release rate is much larger than the normalised mode II as shown in Fig. 12. Both normalised energy release rates increase with

the increase in the end shortening strain. As the end shortening strain increases near to 2.7  103 (compression load 5342 lbf/in. (935 N/mm)), the delamination starts to propagate from the initial length of 0.75 in. (19.05 mm) to about 1.625 in. (41.275 mm) in an unstable manner corresponding to fourteen spline sections growth (i.e. 14  2 in./ 64 = (1.625  0.75)/2) at each delamination front. These fourteen spline sections growth is clearly shown in Fig. 10 on the right branch indicating the variation of the central deflection of the upper sub-laminate with respect

140

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

2 1.8 a1

1.6 2"

Energy release rate

1.4

[04 /012 //04] a1=0.75" h=0.100"

1.2 1

GI/GIC+ GII /GIIC

0.8

GII/GIIC GI /GIC

0.6 0.4 0.2

GI /GIC

GII/GIIC

0 -1.6

-1.8

-2

-2.2

-2.4

-2.6

-2.8

-3

-3.2

-3.4

-3.6

x 10 -3

End shortening strain

Fig. 12. Energy release rate at the front of the delamination for a [04/012//04] laminate containing a 0.75 in. (19.05 mm) delamination under end shortening strain.

-5000 -4500

Compression load F (lbf/in)

-4000 -3500 -3000

a1 -2500

h -2000

a2

-1500

2"

[04 //012 //04] a1=1.5" a2=0.75" h=0.101"

-1000 -500 0 0

-0.5

-1

-1.5

-2

-2.5

End shortening strain

-3

-3.5

-4

x 10

-3

Fig. 13. Longitudinal compression load vs end shortening strain behaviour of a [04//012//04] laminate containing two delaminations (initial length a1 = 1.5 in. (38.1 mm), a2 = 0.75 in. (19.05 mm)) (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm).

to the growth. A load drop is obvious, which is also a distinct feature of Fig. 9. The unstable growth is, of course also shown in Fig. 12 where the spike is its trade mark in which both energy release rates increase first and then drop. As unstable growth continues the mode I crack driving energy gradually loses its strength while the mode II crack takes upper hand. Finally, a stable growth is

observed. When end shortening strain increases to 3.6  103 (compression load 5996 lbf/in.(1050 N/mm)) the delamination reaches the clamped ends. Fig. 11 shows the results of surface strains from present prediction and the experimental data [6]. The unstable growth is also observed from experiments. Starting from the stable delamination propagation stage, the compression

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

responses are similar to those of the previous example. The final global buckling occurs at end shortening strain near to 4.63  103, corresponding to a compression load 7620 lbf/ in. (1334 N/mm). As expected, the final global buckling load is close to the previous example of [04/012//04] plate containing a 1.5 in. (38.1 mm) delamination. In Fig. 11a it is seen that the numerical prediction for the top surface strain (the thinner sub-laminate) compares well with the experimental results until the delamination extending to the clamped ends. Numerical results on both of the unstable and the stable delamination growth are also in agreement with the experimental results. Again, the predicted global buckling load overestimates the collapse load as seen in Fig. 11. 2.3.2. Unidirectional laminated plate with two delaminations In order to demonstrate the effect of the interaction between delaminations on the compression response of delaminated composite plates, numerical analysis is next performed to investigate the compression behaviour of a [04// 012//04] composite plate containing two delaminations with lengths of 1.5 in. (38.1 mm) and 0.75 in. (19.05 mm). The results are presented in Figs. 13–16. The predicted compression load-end shortening strain behaviour is shown in Fig. 13. The load-deflection histories are recorded in Fig. 14. Fig. 15 gives the predicted strain histories at the plate centre compared with experimental results. Fig. 16 presents the normalised energy release rates with respect to end shortening strain. The calculated critical buckling end-shortening strain is ec = 5.621  104 (corresponding to a load of 1179 lbf/in.

141

(206 N/mm)) due to a stable local buckling of the upper sub-laminate. It is very close to that of the [04/012//04] plate containing a 1.5 in. (38.1 mm) delamination. In the initial post-buckling states, significant transverse deflection is mainly in the upper sub-laminate that moves upward. Slight downward deflections exist in the middle and the lower sub-laminates, and the middle and the lower sublaminates keep in separation. As the end shortening strain increases, the downward deflection in the lower sub-laminate becomes significant as shown in Fig. 14, although it is still small compared with the deflection of the upper sub-laminate. Meanwhile, the deflections in both of the upper and middle sub-laminates keep increasing, but the deflection in middle sub-laminate is still faint. The longer delamination (1.5 in (38.1 mm)) starts propagation when end shortening strain is increased near to 2.6  103 (compression load 4517 lbf/in. (791 N/mm)), slight later than the [04/012//04] plate containing a single 1.5 in. (38.1 mm) delamination. With continual increase in end shortening strain to 2.8  103 (compression load 4756 lbf/in. (833 N/mm)), the longer delamination stably grows to a length about 1.63 in. (41.4 mm) (2 spline sections growth at each delamination front) and then the shorter delamination (0.75 in. (19.05 mm)) begins growth. Immediately, the longer delamination extends to the clamped ends (6 spline sections growth ) in a mode II delaminated fracture and the shorter one extends nearly to the clamped ends in an unstable manner (20 spline sections growth). Correspondingly, a load drop is seen in Fig. 13 and a sudden increase in deflections is observed in Fig. 14. Global buckling occurs immediately following the unstable delamination growth.

-5000 -4500

Compression load F (lbf/in)

-4000 -3500 -3000 a1

-2500

h

-2000

a2 2"

-1500 -1000 -500 0 -0.08

× + o •

[04 //012 //04] a1=1.5" a2=0.75" h=0.101" -0.06

-0.04

-0.02

0

0.02

Top sublaminate Mid sublaminate Bottom Unstable delamination growth 0.04

0.06

0.08

Central deflection (in.) Fig. 14. Load-deflection histories of a [04//012//04] laminate containing two delaminations (initial length a1 = 1.5 in. (38.1 mm), a2 = 0.75 in. (19.05 mm)) subjected to end shortening strain (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm).

142

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

-5000 -4500

Compression load F (lbf/in)

-4000

Present: × • Unstable delamination growth Experiment [6]: Δ

-3500 -3000 -2500 -2000

[04 //012 //04] a1=1.5" a2=0.75" h=0.101"

-1500

a1

-1000

h a2

-500

2" 0 -1

0

1

2

3

4

5

Strain

x 10

-3

-5000 -4500

Compression load F (lbf/in)

-4000 -3500 -3000 -2500 -2000

Present: × • Unstable delamination growth Experiment [6]: Δ

-1500 -1000 -500 0 -2

-1

0

1

2

Strain

3

4

5

6

x 10

-3

Fig. 15. Comparison between the predicted and the experimental strain histories of a [04//012//04] laminate under compression (a1 = 1.5 in. (38.1 mm), a2 = 0.75 in. (19.05 mm)) (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm) (a) top surface and (b) bottom surface.

Compared with the results of the [04/012//04] plates containing a single delamination, the post-buckling response of the upper sub-laminate resembles the response of the thinner sub-laminate of the [04/012//04] plate with a single 1.5 in. (38.1 mm) delamination until unstable delamination growth starting. However, the post-buckling response of the lower sub-laminate is quite different from the response of the thinner sub-laminate of the [04/012//04] plate with a single 0.75 in. (19.05 mm) delamination due to the interaction between the delaminations. As seen in Fig. 15 the

numerical results of the surface strain histories compare well with the experimental results. The prediction of the final failure load is in excellent agreement with the experimental data. 2.3.3. Angle-ply laminated plate with a single delamination For a [04/(±45)6//04] composite plate containing a single 1 in. (25.4 mm) delamination, the numerical results of the compression responses are given in Figs. 17–20. The compression load-end shortening strain behaviour is shown in

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

143

a1 h

1

a2 2"

Energy release rate

GI /GIC+ GII /GIIC

[04 //012 //04] a1=1.5" a2=0.75" h=0.101"

0.8

0.6

GII /GIIC 0.4

GI /GIC 0.2

0

0

-0.5

-1

-1.5

-2

-2.5

End shortening strain 2.5

-3

x 10

-3

a1 h a2

Energy release rate

2

1.5

2"

[04 //012 //04] a1=1.5" a2=0.75" h=0.101"

1

GI /GIC+ GII /GIIC

GI /GIC

0.5

GII /GIIC 0 -0.5

-1

-1.5

-2

End shortening strain

-2.5

-3

-3.5

x 10 -3

Fig. 16. Energy release rate at the front of the delamination for a [04//012//04] laminate containing two delaminations under end shortening strain (initial length a1 = 1.5 in. (38.1 mm), a2 = 0.75 in. (19.05 mm)).

Fig. 17. The load-deflection histories are in Fig. 18. The predicted surface strain histories at the plate centre are compared with the experimental data in Fig. 19. The normalised energy release rates with respect to end shortening strain are shown in Fig. 20. Again, a stable local buckling of the upper sub-laminate occurs at end shortening strain ec = 1.172  103 (corresponding to load 1142 lbf/in. (200 N/mm)). A significant reduction of axial stiffness is observed of the laminate from Fig. 17. With increasing the end shortening strain to

1.7  103 (compression load 1493 lbf/in. (261 N/mm)), the delamination starts to propagate and immediately extends from the initial length of 1 in. to about 1.44 in. (36.6 mm) (7 spline sections growth). Load-dropping due to the unstable delamination growth can be seen in Fig. 17. Then with the increase of end shortening strain, the delamination gradually grows to the clamped ends. From Fig. 20 it can be seen that the mode I fracture dominates the unstable delamination initiation and it plays a more important role in the early stage of stable delamina-

144

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

-3000

a1 h

Compression load F (lbf/in)

-2500

2"

[04 /(±45)6 //04] a1=1.00" h=0.100"

-2000

-1500

-1000

-500

0 0

-0.5

-1

-1.5

-2

-2.5

-3

-3.5

-4

-4.5

End shortening strain

-5

x 10 -3

Fig. 17. Longitudinal compression load vs end shortening strain behaviour of a [04/(±45)6//04] laminate containing an initial 1 in. (25.4 mm) delamination (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm).

-3000

× o •

Compression load F (lbf/in)

-2500

Top Bottom Unstable delamination growth

-2000

-1500

[04 /(±45)6 //04] a1=1.00" h=0.100"

-1000

a1 h

-500

2" 0 -0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Central deflection (in.) Fig. 18. Load-deflection histories of a [04/(±45)6//04] laminate containing an initial 1 in. (25.4 mm)delamination subjected to end shortening strain (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm).

tion growth. The finally global buckling load obtained from numerical analysis is about 2605 lbf/in. (456 N/ mm), which is very close to the experimental data as seen in Fig. 19 where it is shown that the numerical prediction for the surface strain histories compare well with the experimental results.

3. Concluding remarks The validity of the new developed LW B-s FSM capability in Part 1 [1] has been tested by extensive numerical applications including buckling ,post-buckling and delamination propagation in debonded composite laminates.

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

145

-3000

Compression load F (lbf/in)

-2500

Present: × • Unstable delamination growth Experiment [6]:

-2000

-1500

-1000

a1

[04 /(±45)6 //04] a1=1.00" h=0.100"

-500

0 -2

-1

0

1

2

h 2" 3

4

5

6

x 10 -3

Strain -3000

Compression load F (lbf/in)

-2500

-2000

[04 /(±45)6 //04] a1=1.00" h=0.100"

-1500

-1000

-500

0 -3.5

Present: × • Unstable delamination growth Experiment [6]:

-3

-2.5

-2

-1.5

Strain

-1

-0.5

0

x 10

-3

Fig. 19. Comparison between the predicted and the experimental strain histories of a delaminated [04/(±45)6//04] laminate under compression (a1 = 1 in. (25.4 mm)) (1000 lbf/in. = 175.1 N/mm, 1 in. = 25.4 mm) (a) top surface and (b) bottom surface.

Comparisons have been made between present predictions and experimental results and agreements are generally good. It has been shown that the present computational model is effective and efficient. From the numerical results the following remarks can be made. (1) The initial buckling load of delaminated composite laminates are often much lower than the global buckling load. To determine the load capacity of the delaminated composite laminates, post-buckling analysis is necessary.

(2) The delamination propagation has significant effect on the post-buckling response and the strength of the laminates. (3) Unstable delamination propagation is often caused by mode I fracture while the mode II fracture often leads to stable delamination growth as far as all the applications considered here are concerned. (4) Due to the interaction between the delaminations, the postbuckling response of the composite laminate with multiple delaminations can be quite different from that with a single delamination.

146

S. Wang, Y. Zhang / Composite Structures 88 (2009) 131–146

1.6 a1

1.4

h 2"

Energy release rate

1.2

GI /GIC+ GII /GIIC

[04 /(±45)6 //04] a1=1.00" h=0.100"

1 0.8

GI /GIC

0.6 0.4

GII /GIIC 0.2 0

0

-0.5

-1

-1.5

-2

-2.5

-3

End shortening strain

-3.5

-4

-4.5

-5 -3

x 10

Fig. 20. Energy release rate at the front of the delamination for a [04//(±45)6//04] laminate containing a 1 in. (25.4 mm) delamination under progressive end shortening strain.

The computational model developed in this paper has not taken the contact at a delamination interface into account. To deal with the contact problem, a possible approach is to develop a penalty function method by inserting the developed interface spring models at the overlapped area iteratively. That is beyond the consideration of this paper. References [1] Zhang Y, Wang S. Buckling, post-buckling and delamination propagation in debonded composite laminates Part 1: Theoretical development. Compos Struct 2009;88:121–30.

[2] Wang S, Zhang Y. Prediction of delamination growth in DCB composite beamsusing imaginary springs, in preparation. [3] Simitses GJ, Sallam S, Yin WL. Effect of delamination of axially loade homogeneous laminated plates. AIAA J 1985;23:1437–44. [4] Kutlu Z, Chang KK. Composite panels containing multiple throughthe-width delaminations and subjected to compression Part I: analysis. Compos Struct 1995;31:273–96. [5] Yamaki N. Postbuckling behaviour of rectangular plates with small initialcurvature loaded in edge compression. J Appl Mech 1959;26: 407–14. [6] Kutlu Z, Chang KK. Composite panels containing multiple through-the-width delaminations and subjected to compression Part II: experiments and verification. Compos Struct 1995;31: 297–314. [7] Yin WL, Sallam SN, Simitses GJ. Ultimate axial load capacity of a delaminated beam-plate. AIAA J 1986;24:123–8.