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Study on the compressive strength of laminated composite with through-the-width delamination
COMPOSITE STRUCTURES Composite Structures 41 (1998) 229-241
Study on the compressive strength of laminated composite through-the-width delamination
with
Y.J Lee*, C.H. Lee, W.S. Fu Department of Naval Architecture and Ocean Engineering. National Taiwan lJniversi& Taipei 106, Taiwan
Abstract A nonlinear finite element code, DELAM3D, with a three-dimensional layered solid element based on the updated Lagragian formulation, is developed to simulate the compressive response of a laminated composite plate with multiple delaminations. An analytical model is established to characterize the mechanical behaviors such as postbuckling, contact of the delaminating interface, delamination growth and fiber-matrix failure. Double cantilever beam (DCB) and end notched flexure (ENF) tests, with T300/976 graphite/epoxy, are performed to verify the energy release rate of the material. Experiments with various crack numbers, sizes, locations and layer orientations have been conducted and compared with the numerical solution. Good agreement is expected. 0 1998 Elsevier Science Ltd. AI1 rights reserved.
1. Introduction The mechanical performance of laminated composites is outstanding in the fiber direction, but bonding between different layers depends only on the matrix. The matrix, compared with fiber direction, limits the strength of laminated composites. Laminated compo-
sites containing delaminations resulting from tools dropped or manufacture defects, can cause significant reductions in stiffness and strength under compression. The compressive behavior and strength of composites are extremely important in designing and maintaining structure. A number of investigations have been performed to study the delamination induced by the compression failure of laminated composites containing delaminations. For model simplification and calculation efficiency, most of the early studies [l] were performed under the thin-film assumption. This is appropriate for a true thin-film structure, but the responses of the interface between the delamination have been ignored. Moreover, it can not simulate the multiple dclaminations and the delamination growth. A finite element method with a two-dimensional quadrilateral element has been produced to study the compressive response of laminated composites that contain multiple delaminations [2]. Owing to the limitations of the four-point quadrilateral plane stress element used in that study, *Author to whom correspondence 0263-X223/98/$ - see front matter PII: SO263-8223(98)00015-4
should be addressed.
0 1998 Elsevier
Science
this method cannot be extended to simulate the laminated composite with a circular or elliptical delamination between layers. The objective of this study is to develop a three-dimensional finite element code, using an eight-node hexahedral element with an incompatible mode, to simulate the compression behavior of laminated composite with multiple delamination, and it can be extended to study the composite structure with a circular or elliptical embedded delamination, which can simulate more closely the damage induced from low-speed impact.
2. Experiment A series of experiments were performed to verify structure modeling and the computer code. T300/976 graphite/epoxy material is used for this test. Thin teflon films (0.051 mm) were embedded in the specimen to simulate the initial delamination. Each specimen contains one or two delaminations. All the specimens wcrc cured under a standard cycle and a C-SCAN system was used to inspect the specimens before testing. In order to obtain the critical energy release rate, the DCB/ENF methods were used to measure Glc and Gllc. A series of tests utilizing a total of 36 types (each type contains three specimens with the same configuration and ply orientation) of test specimen separated into six groups were designed for parameter study. The
Ltd. All rights reserved
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I: J. Lee et al. IComposite Structures 41 (1998) 229-241
the surface, u, is the displacement vector, s is the surface area, and v is the volume. The system is assumed to be conservative, so the equilibrium state is obtained by minimizing II. Invoking the stationary of II, i.e. 6II = 0, we obtain
applied d&pi.
HI(u)=
S, Si$e;jdv-[Y Ti.Gu;ds=O
(2)
3.2. Contact
76.2 mm 20 layers
4 25.4 mm
Fig. 1. Test specimen configuration
specimens were 203.2 mm long (50.8 mm for test region and 76.2 mm for the fixture to simulate fixed end boundary along each side), 25.4 mm wide with 20 layers (Fig. 1). It is noted that double slashes ‘//’ will be used to identify the location of the delamination. Groups l-3 are designed to study the influence of the compressive behavior on the delamination location along the thickness direction, the size and ply orientation. Groups 4-6 are used to observe the influence of the compressive response on the structure containing double delaminations, each group consisting of four types: two with different delamination length and the other two containing 0” and 45” ply orientation. The configuration and ply orientation of specimens used in this study are summarized in Table 1. T300/976 graphite/epoxy material properties were listed in Table 2. In this table the material properties are obtained from Ref. [2], but the GIJGllc values are obtained from the test result in this study. Two axial strain gages bonded on each side in the middle were used to monitor the compressive behavior during the test (Fig. 1). A MTS810 hydraulic control system with 0.12 mm/min displacement control rate was used for the test.
3. Analysis
In order to prevent the delamination surface from overlapping during post-buckling behavior, especially for the composite laminate containing multiple delaminations, a contact condition must be specified in the analysis. This condition assumes that the distance (g, a gap function) between two contact surfaces must be greater than or equal to zero. The gap function is defined as Ilg(u)II= IlXs-&II
(3)
where XJX,,, are the contact points on the slave/master surface. Since the contact behavior of two delamination surfaces in this study are not very complicated, and considering the computation efficiency, the pointto-point contact process was introduced. To accurately describe the contact surfaces, the local coordinate system (Fig. 2) is needed, so the gap function g can be rewritten as g(u) = (X, - &l).n
(5)
Normal contact is considered and the friction between contact surfaces is ignored. Therefore, the penalty method is adopted in this study and the total potential energy given in eqn (1) can be modified as
I=I=II+
+gTgds s
01
dl=l= 6l-I +
&gTGgds
(7)
3.1. Governing equation
The total potential energy can be described as II(u)=
[ W(e,Jdv-
[ T,.u,ds
(I)
where W ii’the strain ekrgy which can be expressed by the 2nd Piloa-Kirchhoff stress S, and Green’s strain tensor ei, relation as W(e,,) = (1/2)Si,, . eii, T is traction in
where E is the penalty parameter. It can be observed when E+CO, the gap function g must be zero to satisfy the constrained condition. Unfortunately, an unsatisfactory conditions will result when E approaches infinity. In order to solve eqn (7), E is chosen as a large constant to avoid producing an unsatisfactory condition. Because E is not infinity, an overlapping between
Y J. Lee et al./Composite
contact surfaces may occur. A square of the maximum element stiffness is used as an E value in this study, and this overlapping is accepted in engineering practice.
Considering the fact that the T300/976 composite material is brittle, the delamination growth is predicted based on linear elastic fracture mechanics. The strain energy release rate G~/G,,/G,,~ and represented mode
Table 1 Specimen configuration and ply orientation No
PIY
orientation ______-.
~._
Delamination length (mm)
231
Structures 41 (1998) 229-241
Table 2 Material properties for T300/976 graphite/epoxy _____-_ Ply longitudinal modulus, El, (GPa) Ply transverse modulus, Ex (GPa) Out-of-plane modulus, Es3 (GPa) Poisson’s ratio, qz . II Poisson s ratlo, v2) 3 Poisson s ratlo, I’,~ Inplane shear modulus, Glz (GPa) Out-of-plane shear modulus, Gz, (GPa) Out-of-plane shear modulus, Go (GPa) Ply 1ongitudinaI tensile strength, X, (MPa) Ply longitudinal compressive strength, X, (MPa) Ply transverse tensile strength, Yt (MPa) Ply transverse compressive strength, Y, (MPa) Inplane shear strength, Siz (MPa) Out-of-plane shear strength, Sr? (MPa) Out-of-plane shear strength, S,, (MPa) Critical energy release rate for mode I, Glc (J/m’) Critical energy release rate for mode II, Gllc (J/m’y
139.3 9.72 9.72 0.29 0.40 0.29 5.59 3.45 5.59 1517 1593 45 253 107 107 107 193 455
“Measure from this study.
Group 1 (single delamination between layer 4/s) 10101 IW&,l 10102 [Od/%,] 10201 W/( * 30)dO,] 10202 Wi( + 30)dO,l 10301 [Wi//(f 45)dO,l 10302 Pdi~k 45)dO,l 10401 IO4N+ 6O)dO,l 10402 [04//Cf 6O)dO‘ll
19.05 38.10 25.40 38.10 25.40 38.10 25.40 38.10
Group 2 (single deiamination between layer S/9) 20101 [W/O,,1 20102 [0~/0,21 20201 [OJ( f 30)2//( + 30)4/0~1 20202 [Odl(k 30)2//( + 30)4/041 20301 I%/( f45)?ii( + 30)4/04] 20302 [04/i(_+45)2//( i 30)&&] 20401 [OJ( k Wzii( zt 30)4/04] 20402 [Od( + ~OM(t 30)4/04]
19.05 38.10 25.40 38.10 25.40 38.10 25.40 38.10
Group 3 (single delamination between layer 10/l 1) 30101 IO~d/O,~,] 30102 [O,~/O,~] 30201 [04/(* 30)3/1(rt 3O)JO,] 30202 [04/C_t 3Wi( + 30)~0/] 30301 [04/Ck 45)J/( i 30)3/010,] 30302 [04/(i 45)?f/( rt 3O)JO,] 30401 [04/fF WJ/( zk3O)JO,l 30402 [041(i ~)~/( zk3O)yiO+]
19.05 38.10 25.40 38.10 25.40 38.10 25.40 38.10
Group 4 (double delamination between layer 41.5and E/16) 40101 [04//0,d/041 12.7i25.40 40102 w/0ui/0,1 19.05/38.1 40301 12.7i25.40 [O$/(+ 45)d/O41 40302 lW/( F45)d/O~l 19.05138.1
I/II/III fracture mode are used as an index of delamination growth and can be expressed as [3]
lui(a+Aa)-u,(a+Aa)]cr,,da
[~+(a + Au) - ~,,‘(a+ Aa)]a,qnda
where ha is the area of the crack surface, r,s,n are the axes of the local coordinate system, a,,,, a,,, a, are the stresses at the crack front, u,f,u~,u,’ and Un 3U5 IAlr are the displacement of the upper and lower crack surfaces. Therefore, the strain energy release rate can be calculated at the delamination front. Considering the mixed fracture mode, the criterion for determining the delamination growth is selected as [4,5]
Group 5 (double delamination between Iayer 4/5 and S/9) 50101 Fbhw/0,21 12.7119.05 50102 w/w/0,21 25.40/3&l 50201 IWi( * 45)?//( + 45)4/O,] 12.1119.05 50202 25.40/3&t lW/( +45)2//t k45)4/041 Group 6(double delamination between layer 4/S and S/9) 60101 [04//04//0,*] 19.05112.7 60102 KbmMh21 38.1125.40 60201 19.05i12.7 wi//( + 45)?//( &45)4/O,] 60202 3&l/25.40 wit k 45)2//C+45)4/o,]
Fig. 2. Local coordinate for contact surface.
232
Y J. Lee et al.lComposite Structures 41 (1998) 229-241
In this study, it is assumed that Gtc, G,,,-, GIilt do not change during delamination growth and are independent of the layer-up orientation and sequence. Most research [2] has found that selecting c(= p = y = 1 would result in the best fit between experiment and numerical analysis. The delamination will start to grow when Ed,, > 1 and the stress distribution in the structure will be re-distributed. 3.4. Failure criteria and ~ro~~~ de~adati5~ In order to predict the final collapse behavior of the structure, the local fiber-matrix failure will be considered. The well-known Hashine [6] failure criterion is used to determine different types of failure. Four types of failure are considered in this study: fiber failure in tension/compression and matrix failure in tension/compression. These types of failures will lead to property degradation and the reduction of the global stiffness that can induce structure collapse. Three-dimensional Hashine criteria will be checked in each ply under the following modes: (a) Fiber failure in tension (a,, 20)
(10) where ~~~~ 212, r13 are the stresses in fiber direction, in-plane and interlaminar shear stress. X, is the ply tensile strength in the fiber direction and Sj2, Si3 are the ply in-plane, out-of-plane shear strength. (b) Fiber failure in compression (cJ,, 60)
(11) where XC is the ply compressive strength in fiber direction. If one of eqns (10) and (11) is satisfied, the property of this ply will be modified by setting Eli, GIZt v12,vz3,v13to zero. (c) Matrix failure in tension ((rZ2+ f13320)
where Y, is the ply transverse compressive strength. If one of eqns (12) and (13) is satisfied, the property of this ply will be modified by setting EZZ, v12, vu, v13to zero.
4. Finite element formulation 4.1. Layered solid element In order to accurately simulate the bending behavior and increase computation efficiency, a layered solid element with an incompatible mode was developed in this study. The displacement field of this eight-node isoparametric hexahedral element is assumed to be in the form [7] U = i UjNi(r,s,t) + $ UiPj(r,s,t) ,=I ,=I
where Uj is the nodal displacement, aj is the internal degree of freedom (in~mpatible mode), N, P are the shape function related to the degree of freedom U, and ai which can be expressed with the local coordinate system (r,s,t) as Ni(r,s,t) = t (1 r_+rrJ(l + SS{)(1 + tti)
(15)
P,(r,s,t) = (1 - 2) P2(r,s,t) = ( 1 - 2) P&r&) = (1 - 2)
(161
The element matrices are obtained by numerical integration. Gaussian quadrature is used in the plane of the layer and Simpson’s rule is used in the stacking (thickness) direction (Fig. 3). The Simpson’s integration used for this element following the rule will N8
N7
(12) Nl
where ~2~. cx3, ~23, cri2, ~~223 are the stress components in the on-axes ply coordinate system. k; is the ply transverse tensile strength and SZ3is the out-of-plane shear strength. (d) Matrix failure in compression (cr22+ 03360)
(14)
N3 N2
Fig. 3. Numerical integration for hexahedral element.
IT .I. Lee et aLlComposite
233
Structures 41 (1998) 229-241
a
achieve better agreement than following traditional Gaussian quadrature integration, where one element contains only one layer, from a series of numerical tests: 1. number of layers of element <8:3 integration points for each layer; 2. number of layers of element > 8:l integration point for each layer. Since this incompatible element has contained internal degrees of freedom and is independent of the other elements, the static condensation process is used to remove these degrees of freedom from element stiffness before assembling the global stiffness matrix.
N
E
M
Aa
Fig. 4. Finite element
model near crack front.
4.2. Equation of equilibrium The variational equation (eqn (7)) is balanced at the current configuration (t +At) in which the stress and strain are unknown. The updated Lagragian formulation [8] with Newton Raphson iteration is used to solve this nonlinear equation. The 2nd Piloa-Kirchhoff stress tensor and Green Lagragian strain tensor are employed to transform the Sij/e, (in t +At configuration) to the previous state SJe, (in t configuration). So, the integrals in the linearized equilibrium equation can be expressed as ,C,,,,,.e,.~.Ge,,.‘dV+
s ‘V
J I
-
I
(F”-FN),
of node FMN is the force difference d”,dN is the displacement of node M&.
M,N
From the concept of a similar approach, the stress and displacement distribution across the delamination front do not change after crack growth, i.e. FMNz FDE, the modified crack closure technique is performed and the energy release rates are modified as:
GIZ
T+ArT~.buf.~u~.r+A’ds
s
'~i,~Gre,/,dV+ Eg
a&? aurn
FF(dr
-df)
(17)
GII = lim 1
~0-0 2Aa
or in matrix form
+ [Kclu= Pea-Fine - Fan = R
lim -!-
Aa+O2Aa
FfE(df - d,M) Gn = lim 1 AU-O 2Aa
AIS
‘V
[Ktl
where
s ‘V
rc$$,dIs
+
=
‘~ij.G,)lij.‘dV
(19)
(18)
where [K,] is the tangent stiffness matrix, [KC] is the matrix from contact condition, {P,,,) is the external force vector, {Fin,} is the internal element force vector, {F,,,} is the contact force vector, {R) is the residual force vector.
Fr(dr
- d;M)
(20)
The above equations are very easy to calculate by the finite element method and only one analysis step is needed to obtain the nodal forces and displacements. Once the energy release rates are calculated, the delamination growth criterion is applied to each delamination front. If Edel > 1, the delamination growth starts.
4.3. Delamination growth 5. Numerical
The physical meaning of the Irwin’s crack closure integral [3] is that the total work required to close a very small crack (whose energy is nearly enough to extend the crack in the surface by the same size). The crack closure energy can be expressed by the products of forces and displacements as the following form from Fig. 4
verification
5.1. Double cantilever beam for mode I fracture
The configuration of the double cantilever beam test for a mode I fracture is shown in Fig. 5. The specimen used in the analysis is [O,,] laminate with T300/5208 material.
Y J. Lee et al.lComposite Structures 41 (1998) 229-241
234
Fig. 5. Configuration of DCB.
Gi, = 103 J/mm’, Gllc= 876 J/mm2 E,= 136 GPa, E,.=E:=
10 GPa, -1.50
G,Y,= G, = 5.7 GPaG, = 5.7 GPa,
-l.m
0.00 -0.50 MIDDLE OISPC. (mm)
0.50
1.m
Fig. 7. Comparison of present and ABAQUS analysis resutt.
v,,,= v,,,= 0.3 I ) v,*,= 0.40 The analysis agrees very well with the comparison results reported from Ref. [2] (Fig. 6). 5.2. Postbuckiing behavior of the embedded circular delamination with ABAQUS@fNTlJ results
A square lanlinated plate (100 x 100 mm with [(Oi- 45/90/45)&m stacking sequence) loaded with uniform compression with a 25 mm diameter delamination located between ply 4/5 is selected to demonstrate the postbuckling behavior simulation [9]. The characteristics of the ply are as follows: EL= 136GPa, ET= lOGPa, GLT= 5.7 GPa, v = 0.3 1 A commercial finite element program, ABAQUS@ with 20-node hexahedral element, is selected to compare with the present solution. Good agreement is shown in Fig. 7.
6. Comparison and parameter study In this section the data obtained from the experiment will be compared with the numerical solution output from the computer program ‘DELAM3D’. The strains shown in the load-strain curve were calculated
0.05’1mm away from the strain gage bounded surface to inflect the strain gage thickness effect. 6.1. Phenomenon description Typical experimental data from the specimen is shown in Fig. 8. Two different load processes are observed (path l-2-3-4 and 1-2-3’-4’) and each process can be divied into four paths. Path 1 corresponds to linear behavior in which the compressive strain is obtained from two strain gages. When load increases, postbuckling behavior can be found in thin sublaminate, but another strain remained in compression. As the load continues to increase, a sudden change of slope for the load-strain curve is observed which indicates the initialization of delamination growth. Depending on the delamination location in thickness, two types of load change would suddenly be observed (path 3 and 3’). In general, path 3 represents the stable growth of delamination and can always be found in the specimen containing thin sublaminate delamination (test group 1). In contrast, path 3’ can be found from the group 213 test result that contains a deeper delamination and the load drops very quickly, which is likely to lead to unstable delamination growth. Because of the unstable delamination growth, the delamination front reached the clamped boundary. So, in the final region path 4’, the load change was very slow and the strain increases were very rapid. Finally, collapse
x 0.20
cmcl open Dhphcnnsnt
0.24
d (mm)
Fig. 6. Comparison DCB result.
STRAIN
Fig. 8. A typical load-strain
response.
I: J. Lee
et al./Compsite
Structures
41 (1998)
235
229-241
..,
occurred. For the stable delamination growth case, the delamination grows step-by-step as the load increases. At times postbuckling behavior can be observed again. When the delamination front reaches the clamped boundary collapse occurs. In some of the experiment results, we do not see initial buckle and specimen collapse occuring suddenly after the initial delamination growth.
. :
6.2. Results comparison r(c4m
Although the analysis result does not coincide completely with the test data from the series of specimens in this study, the critical bifurcation (the load of initial local buckle, initial delamination growth and maximum capacity of structure) are agreeable with numerous cases. The following discussion will be based on the group defined in Table 1. GROUP 1: the majority of the analysis results shown in Figs 9-16 agree very well with the test data. For Figs 11 and 12, the analytical load of the initial buckle is quite different compared with the test result. It seems to be influenced by the bonding force between teflon and delamination surface which can make the initial buckle occurs very late, but the ~stbuckling
Fig. 9. Comparison between analysis and test results of a [OJOlh] composite with 19.05 mm delamination.
*3ocnm
_..
._.. ,....... ..,... ..~.... ....l......
_
:i
:
:
moo
404s
Fig. 11. Comparison between analysis and test [Od/(+3O)JO,] composite with 25.4 mm delamination.
results
of a
behavior then occures after the initial buckle. In Fig. 15, the initial buckle of the test occurs very early compared with the analysis and it will have a great influence on the postbuckling behavior, especially on the rn~irn~ capacity of load for the structure. GROUP 2: the postbuckling behavior in this group (Figs 17-24) is quite different from group 1 with the following properties: (1) the load of the local buckle is very close to the maximum capacity of the structure; (2) has an unstable initial delamination growth to reach the boundary; (3) after initial delamination
-mm
MCRO STRAIN
-mo
.,
,.
-ax?4
h&Os&E
4ooll
Fig. 12. Comparison between analysis and test [On( f 30)d04] composite with 38.1 mm delamination.
a000
results
of a
-.__
j i
0 -Moo
D
MCRO?hAIN
4ml
6000
Fig. 10. Comparison between analysis and test results of a (O&O,,] composite with 38.1 mm delamination.
4nm
-xJ!lo
Ii&R0
STi%
4004
Fig. 13. Comparison between analysis and test [OJI( _t45)dO4]composite with 25.4 mm delamination.
oam
results
of a
236
I: J. Lee et ai. IComposite Structures 41 (1998) 229-241
-1CW6
E f -6m
0 -36m
6
666U
6665
-woo
~*=~~~~~,~
Fig. 14. Comparison between analysis and test [O,N(k45)d04] composite with 38.1 mm delamination.
__ ...._.i” :. ..I ;
1
i
_ ._ _
I
:
1
-zooo
0
2000
4004
woo
8ooo
too00
MICROSTRAIN results
of a
growth, the load reduced slowly before final collapse. In Fig. 24, the load of the initial delamination growth is very close to the test data, but the strain value has many discrepancies. GROUP 3: in this group (Figs 2%32), the max. load of the laminate nearly equals the load of the initial delamination growth. After that, only the laminate containing a longer delamination length has postbuckling behavior. In another case, the initial buckle and final collapse occur simultaneously. In general, the
-Km0
1 -4004
Fig. 17. Comparison between analysis and test results of a [Oplia,,] composite with 19.05 mm delamination.
maximum load capacity of the laminate compared with the test data agree very well. GROUPS 4-6: according to the second delamination, the compressive behavior is much difference compared to the laminate containing only one delamination specimen. Although the interaction of the two delamination surfaces is very complicated and difficult to predict very well along the load-strain curve, many cases shown in Figs 33 and 34 of groups 4-6 agree with the special bifurcation (initial buckle, initial delamination growth and structure maximum capacity).
..: j
0 -4ma
4wo
-mm
M&O
.6wo
Fig. 15. Comparison between analysis and test [O.J( + 60)F/04]composite with 25.4 mm delamination.
*wo
-2000
0
6rJoJ
S&ii
2060
466cl
6WO
results
of a
-4ow
-MMI
2ooo
4606
m
RWCRODSTRAIR
Fig. 18. Comparison between analysis and test results of a [OpjlOlz] composite with 38.1 mm delamination.
6066
MICRO STRAIN
Fig. 16. Comparison between analysis and test [OJ/( ;t 6O)h/O4] composite with 38.1 mm delamination.
results
of a
Fig. 19. Comparison between analysis and test results [OJ/( +30)~//( f 30)J04] composite with 25.4 mm delamination.
of a
I! 1 Lee
/ ...~_.I.
-18wo
_
et al. IComposite
-...
Siructures
41 (1998)
237
229-241
:
ii __ .._. ;.
_,mo
.._.
i(
‘i
: .;.
I” P
-sow
i
1
-41 ..I Q
-2Ysl
j
0
;.
4ow
-2000
0 MICRO
20M STRAIN
40133
BOCIO
Fig. 20. Comparison between analysis and test results of a [OJ( F 30)&( +-30)JOa] composite with 38.1 mm delamination
6.3. Parameter study 6.3.1. The influence of the compressive behavior on the delamination length
There are two different delamination lengths of the same stacking sequence and delamination location between layers for all the groups l-3 defined in Table 1. The laminate containing a shorter delamination length has a higher initial buckling ioad, but unstable delamination growth appears. The maximum load
-m
-4fml
-2ccQ
0
woo
8&o
8ooo
MICRO~TRAIN
Fig. 21. Comparison between analysis and test results [Od/( &45)2//( +4S).+/0/0,] composite with 25.4 mm delamination.
-4ouo
-2vw
0
4000
6OW
of
8CQO
MlcRO%RAIN 22. Comparison between analysis and test results fO+‘(145)J/( zk4.%/041composite with 38.1 mm deIamination.
Fig.
4ooo
8Qw
MICROTRAIN
SW0
Fig. 23. Comparison between analysis and test results [O.J/(rt 60)~/( i: 60&/O&]composite with 25.4 mm delamination,
of a
capacity of the laminate depends on the delamination location in thickness direction. The maximum load for group 1 in which the delamination is located at l/S thickness does not obviously change when the delamination lengths are different, but the m~mum load for group 213 (Figs 17-32) with a shorter delamination length is greater than the laminate with a longer delamination by about 60-100%.
0
-2ow -4000
0
8&J
uno
MlcRo~mIrJ
Fig. 24. Comparison between analysis and test results [a,/‘(+ 60)~/( i_ 60)404]composite with 38.1 mm delamination,
-6oon
-4000
a00
6ooo
8000
of a
l!xJCa
IicRO~Td~
of
Fig. 25. Comparison between analysis and test results of a IO,&O,,J composite with 19.05 mm delam~ation.
238
Y J. Lee et al. /Composite Structures 41 (1998) 229-241
-6w
-5000
vi. 0
-icm
-xxx)
0
4wcl
~,C~~~~~
m
.j.
0 -8000-mm I,
8a%l
Fig. 26. Comparison between analysis and test results of a [O,,J/O,,] composite with 38.1 mm delamination.
6.3.2. The influence of the compressive behavior on the second de~~i~atio~ introduced
In this study, there are three groups defined in Table 1 which have two delaminations. From Figs 33-44, we can find that the second delamination has a great influence on the postbuckling behavior compared to the laminate with only one delamination. In Figs 10 and 34, the specimens have the same m~imum delamination length and only the load of the initial local buckle is similar, while the other bifurcations are quite different. The reduction ratio of the maximum load
~... ,_ ..i
i
:
:
t
,
I,
-4aw
f
i
. ../..
I
..I..
~
,
,
I,,
.r..
/‘...;
i
(
‘7’
I
4000
-Moo ~,~~*O~~~~
6006
6OW
Fig. 29. Comparison between analysis and test results [OJ( +45)?j-/( +45)r/OJ composite with 25.4 mm delamination.
capacity is not predictable, the load in Fig. 34 is about 80% of Fig. 10 and Fig. 36 is about 50% of Fig. 14. 63.3. The influence of the compressive behavior on the delamination locatd in the thickness direction In groups 1-3, the delaminations are located at l/6, 216, 316 of the thickness. From the comparison of Figs 9-32 we can find that the influence depends on the delamination length. For the cases with a smaller delamination length (Figs 9, 17 and 25) the load of the initial buckle increases when the delamination is _,f,@Jo
,.......
,.
. . . .. . . . . . .. . .. . .. ..i.
.j..
..-.
_...~
c...:
.
. .. .
..
I
--
:
-4WO
Fig. 27. Comparison between analysis and test results [04/(k 3O)J/( + 3O)JO,] composite with 25.4 mm delamination.
4*w
-2000
0
MICR~~RAIN 4000
6WO
of a
Fig. 28. Comparison between analysis and test results [04/(2 3O)J/( f 3O)JO,J composite with 38.1 mm delamination.
of a
-2wo
HI&to STG
4wo
:
6wo
Fig. 30. Comparison between analysis and test results [OJ( +45)J/( k45)JOJ composite with 38.1 mm delamination.
#aI
8000
of a
-4wo
-mJo
woo
4wo
of a
woo
MH;ROCJTRAIN
Fig. 31. Comparison between analysis and test results [OJ/( _t 60)3//( f 60)d04] composite with 25.4 mm delamination.
of a
239
I: J. Lee et aLlComposite Structures 41 (1998) 229-241
-2-X0
0
MEW SRGN
mm
ww
of a
located deeper and the maximum capacity has the same phenomena. This is because when the delamination is located deeper, the initial buckle, delamination growth and final collapse occurs simultaneously. When the delamination becomes longer(Figs 12, 18 and 26) the load of the initial buckle increases when delamination is located deeper, but the maximum load capacity in Fig. 10 (l/6 thickness) is greater than the others (2/6, 3/6 thickness). Fig. 16 with delamination located at l/6
0
Fig. 35. Comparison between analysis and test results [O.J/(+45)d/O,] composite with 12.7, 25.4 mm delamination.
Ii%0SG
ooo0
6.3.4. The influence of the compressive behavior on the stacking orientation From the results of groups l-3 (Figs 9-32) it is easy to observe that the load of the initial buckle and
0
wou
Fig. 33. Comparison between analysis and test results of a [OJ/O,d/O,] composite with 12.7, 19.05 mm delamination.
-2ooo
0
MG&O s&
woo
o&J
Fig. 34. Comparison between analysis and test results of a [O~/O,~/O//o,] composite with 19.05, 38.1 mm delamination.
of a
thickness has the maximum thickness for another side (5/6 thickness) and it has a higher buckling load.
-2MO 0
mO0
M?ROSTE
Fig. 32. Comparison between analysis and test results [04/(k60)~////(k60)304] composite with 38.1 mm delamination.
-2030
wm
0
-2oOa
2WO MICRO
4Ool STRAIN
Fig. 36. Comparison between analysis and test [OJ/( +45),J/O,] composite with 38.1 mm delamination.
-MM0
-4wo
-2ooo
8000
6000
2mI
results
4004
of a
6ooo
MlCROOSTRAlN
Fig. 37. Comparison behveen analysis and test results of a [OJ/OJO,,] composite with 12.7, 25.4 mm delamination.
Y J. Lee et al. /Composite Structures 41 (1998) 229-241
240
0
-2cm
6wO MzKl
EC00
-4m
0 MICROYrnIN
-2000
4000
go00
8000
STizd
Fig. 38. Comparison between analysis and test results of a [OJ/OJ/O,,] composite with 19.05, 38.1 mm delamination.
Fig. 41. Comparison between analysis and test results of a [Od/OJ/O,,] composite with 25.4, 12.7 mm delamination.
7. Conclusions delamination growth is decreased when ply orientation increased. This is because the stiffness of the whole structure decreases when the ply angle increases, and the structure easily deforms. In Figs 10, 12, 14 and 16 with ply angles of o”, 30”, 45” and 60”, the initial buckling load is 4000, 2600, 2300, and 2000 N and it is difficult to find a formulation which will fit. The other group experiences the same phenomena.
-4om
0
-2mo
4Ma
BOOD
In this study, an analytical model with the computer code was developed to predict the compressive behavior of the laminated composite plate with delamination. Based on the results of the test and analysis discussed previously, the following points can be made: 1. It will have unstable delamination growth when the delamination length is shorter or deeper. 2. For the laminated plate with shallow delamination, the load of the initial buckling, delamination growth
sow
MICROSTRAIN Fig. 39. Comparison between [OJ/( k45)J/( *45)410,] composite
analysis and test results of with 12.7, 25.4 mm delamination.
a
-2000
0
tG%o
smstt
Boo0
8aHl
Fig. 42. Comparison between analysis and test results of a [OJ/O,//O,,] composite with 38.1, 19.05 mm delamination.
-6mo
E
-arm
Bmo
0
Bwo
MEktl SlTEN Fig. 40. Comparison [OJ/( *45)d/( +45h04]
between analysis and test results of composite with 19.05, 38.1 mm delamination.
a
Fig. 43. Comparison between [O,//( +45)4/( + 45)4j04] composite
analysis and test results of with 25.4, 12.7 mm delamination.
a
Y J. Lee et al. /Composite
Structures 41 (1998) 229-241
241
Acknowledgements
This study was supported by the National Science Council, Republic of China, under the grant no. NSC 85-2611-E-002-021. References
PI Chai H, Babcock CD, Knauss WG. One dimensional modeling of 4000
-2000
0
2000
MICRO
4OW
6003
8000
STRAIN
Fig. 44. Comparison between analysis and test results of a [04//(*45)J/( +45),0,] composite with 38.1, 19.05 mm delamination.
and final collapse were predicted very well compared with the test result. For some cases, the maximum load capacity of the structure are greater than the load of the initial delamination of about 50% (Fig. 12) or greater than the initial buckling load about 500% (Fig. 16). It seems to be conservative if you do not use the maximum load for structure design. So, it is very important to find the bifurcation (initial buckle, initial delamination and collapse) from the compressive behavior of the structure analysis. 3. The compressive behavior becomes more complicated when the second delamination was introduced and it cannot be predicted from the laminated composite containing only one delamination.
failure in laminated plates by delamination buckling. Inter1981;17:(11): national Journal of Solids and Structures 1069-1083. PI Kutlu Z, Chang FK. Modeling compression failure of laminated composites containing multiple through-the-width delaminations. Journal of Composite Materials 1992;26:(3):350-387. 131 Rybickl EF, Kanninen MF. A finite element calculation of stress intensity factors by modified crack closure integral. Engineering Fracture Mechanics 1997;9:921-938. I41 Kanninen MF, Popela CH. Advanced Fracture Mechanics. Oxford Engineering Science Series. New York: Oxford University Press. [51 Hahu H.T. A mixed-mode fracture criterion for composite material. Composite Technology Review 1983;5:26-29. 161 Hashine Z. Failure criteria for unidirectional fiber composites. ASME Journal of Applied Mechanics 1980;47:329-334. [71 Bathe KJ. Finite Element Procedures in Engineering Analysis. New Jersey: Prentice-Hall, 1982. PI Talyor RL, Beresford J, Wilson EL. A non-conforming element for stress analysis. IJNME 1976;10:1211-1219. I91 Cochelin B, Potier-Ferry M. A numerical model for buckling and growth of delamination in composite laminates. Composite Methods in Applied Mechanics and Engineering 1991;89: 361-380.
Study on the compressive strength of laminated composite through-the-width delamination
with
Y.J Lee*, C.H. Lee, W.S. Fu Department of Naval Architecture and Ocean Engineering. National Taiwan lJniversi& Taipei 106, Taiwan
Abstract A nonlinear finite element code, DELAM3D, with a three-dimensional layered solid element based on the updated Lagragian formulation, is developed to simulate the compressive response of a laminated composite plate with multiple delaminations. An analytical model is established to characterize the mechanical behaviors such as postbuckling, contact of the delaminating interface, delamination growth and fiber-matrix failure. Double cantilever beam (DCB) and end notched flexure (ENF) tests, with T300/976 graphite/epoxy, are performed to verify the energy release rate of the material. Experiments with various crack numbers, sizes, locations and layer orientations have been conducted and compared with the numerical solution. Good agreement is expected. 0 1998 Elsevier Science Ltd. AI1 rights reserved.
1. Introduction The mechanical performance of laminated composites is outstanding in the fiber direction, but bonding between different layers depends only on the matrix. The matrix, compared with fiber direction, limits the strength of laminated composites. Laminated compo-
sites containing delaminations resulting from tools dropped or manufacture defects, can cause significant reductions in stiffness and strength under compression. The compressive behavior and strength of composites are extremely important in designing and maintaining structure. A number of investigations have been performed to study the delamination induced by the compression failure of laminated composites containing delaminations. For model simplification and calculation efficiency, most of the early studies [l] were performed under the thin-film assumption. This is appropriate for a true thin-film structure, but the responses of the interface between the delamination have been ignored. Moreover, it can not simulate the multiple dclaminations and the delamination growth. A finite element method with a two-dimensional quadrilateral element has been produced to study the compressive response of laminated composites that contain multiple delaminations [2]. Owing to the limitations of the four-point quadrilateral plane stress element used in that study, *Author to whom correspondence 0263-X223/98/$ - see front matter PII: SO263-8223(98)00015-4
should be addressed.
0 1998 Elsevier
Science
this method cannot be extended to simulate the laminated composite with a circular or elliptical delamination between layers. The objective of this study is to develop a three-dimensional finite element code, using an eight-node hexahedral element with an incompatible mode, to simulate the compression behavior of laminated composite with multiple delamination, and it can be extended to study the composite structure with a circular or elliptical embedded delamination, which can simulate more closely the damage induced from low-speed impact.
2. Experiment A series of experiments were performed to verify structure modeling and the computer code. T300/976 graphite/epoxy material is used for this test. Thin teflon films (0.051 mm) were embedded in the specimen to simulate the initial delamination. Each specimen contains one or two delaminations. All the specimens wcrc cured under a standard cycle and a C-SCAN system was used to inspect the specimens before testing. In order to obtain the critical energy release rate, the DCB/ENF methods were used to measure Glc and Gllc. A series of tests utilizing a total of 36 types (each type contains three specimens with the same configuration and ply orientation) of test specimen separated into six groups were designed for parameter study. The
Ltd. All rights reserved
230
I: J. Lee et al. IComposite Structures 41 (1998) 229-241
the surface, u, is the displacement vector, s is the surface area, and v is the volume. The system is assumed to be conservative, so the equilibrium state is obtained by minimizing II. Invoking the stationary of II, i.e. 6II = 0, we obtain
applied d&pi.
HI(u)=
S, Si$e;jdv-[Y Ti.Gu;ds=O
(2)
3.2. Contact
76.2 mm 20 layers
4 25.4 mm
Fig. 1. Test specimen configuration
specimens were 203.2 mm long (50.8 mm for test region and 76.2 mm for the fixture to simulate fixed end boundary along each side), 25.4 mm wide with 20 layers (Fig. 1). It is noted that double slashes ‘//’ will be used to identify the location of the delamination. Groups l-3 are designed to study the influence of the compressive behavior on the delamination location along the thickness direction, the size and ply orientation. Groups 4-6 are used to observe the influence of the compressive response on the structure containing double delaminations, each group consisting of four types: two with different delamination length and the other two containing 0” and 45” ply orientation. The configuration and ply orientation of specimens used in this study are summarized in Table 1. T300/976 graphite/epoxy material properties were listed in Table 2. In this table the material properties are obtained from Ref. [2], but the GIJGllc values are obtained from the test result in this study. Two axial strain gages bonded on each side in the middle were used to monitor the compressive behavior during the test (Fig. 1). A MTS810 hydraulic control system with 0.12 mm/min displacement control rate was used for the test.
3. Analysis
In order to prevent the delamination surface from overlapping during post-buckling behavior, especially for the composite laminate containing multiple delaminations, a contact condition must be specified in the analysis. This condition assumes that the distance (g, a gap function) between two contact surfaces must be greater than or equal to zero. The gap function is defined as Ilg(u)II= IlXs-&II
(3)
where XJX,,, are the contact points on the slave/master surface. Since the contact behavior of two delamination surfaces in this study are not very complicated, and considering the computation efficiency, the pointto-point contact process was introduced. To accurately describe the contact surfaces, the local coordinate system (Fig. 2) is needed, so the gap function g can be rewritten as g(u) = (X, - &l).n
(5)
Normal contact is considered and the friction between contact surfaces is ignored. Therefore, the penalty method is adopted in this study and the total potential energy given in eqn (1) can be modified as
I=I=II+
+gTgds s
01
dl=l= 6l-I +
&gTGgds
(7)
3.1. Governing equation
The total potential energy can be described as II(u)=
[ W(e,Jdv-
[ T,.u,ds
(I)
where W ii’the strain ekrgy which can be expressed by the 2nd Piloa-Kirchhoff stress S, and Green’s strain tensor ei, relation as W(e,,) = (1/2)Si,, . eii, T is traction in
where E is the penalty parameter. It can be observed when E+CO, the gap function g must be zero to satisfy the constrained condition. Unfortunately, an unsatisfactory conditions will result when E approaches infinity. In order to solve eqn (7), E is chosen as a large constant to avoid producing an unsatisfactory condition. Because E is not infinity, an overlapping between
Y J. Lee et al./Composite
contact surfaces may occur. A square of the maximum element stiffness is used as an E value in this study, and this overlapping is accepted in engineering practice.
Considering the fact that the T300/976 composite material is brittle, the delamination growth is predicted based on linear elastic fracture mechanics. The strain energy release rate G~/G,,/G,,~ and represented mode
Table 1 Specimen configuration and ply orientation No
PIY
orientation ______-.
~._
Delamination length (mm)
231
Structures 41 (1998) 229-241
Table 2 Material properties for T300/976 graphite/epoxy _____-_ Ply longitudinal modulus, El, (GPa) Ply transverse modulus, Ex (GPa) Out-of-plane modulus, Es3 (GPa) Poisson’s ratio, qz . II Poisson s ratlo, v2) 3 Poisson s ratlo, I’,~ Inplane shear modulus, Glz (GPa) Out-of-plane shear modulus, Gz, (GPa) Out-of-plane shear modulus, Go (GPa) Ply 1ongitudinaI tensile strength, X, (MPa) Ply longitudinal compressive strength, X, (MPa) Ply transverse tensile strength, Yt (MPa) Ply transverse compressive strength, Y, (MPa) Inplane shear strength, Siz (MPa) Out-of-plane shear strength, Sr? (MPa) Out-of-plane shear strength, S,, (MPa) Critical energy release rate for mode I, Glc (J/m’) Critical energy release rate for mode II, Gllc (J/m’y
139.3 9.72 9.72 0.29 0.40 0.29 5.59 3.45 5.59 1517 1593 45 253 107 107 107 193 455
“Measure from this study.
Group 1 (single delamination between layer 4/s) 10101 IW&,l 10102 [Od/%,] 10201 W/( * 30)dO,] 10202 Wi( + 30)dO,l 10301 [Wi//(f 45)dO,l 10302 Pdi~k 45)dO,l 10401 IO4N+ 6O)dO,l 10402 [04//Cf 6O)dO‘ll
19.05 38.10 25.40 38.10 25.40 38.10 25.40 38.10
Group 2 (single deiamination between layer S/9) 20101 [W/O,,1 20102 [0~/0,21 20201 [OJ( f 30)2//( + 30)4/0~1 20202 [Odl(k 30)2//( + 30)4/041 20301 I%/( f45)?ii( + 30)4/04] 20302 [04/i(_+45)2//( i 30)&&] 20401 [OJ( k Wzii( zt 30)4/04] 20402 [Od( + ~OM(t 30)4/04]
19.05 38.10 25.40 38.10 25.40 38.10 25.40 38.10
Group 3 (single delamination between layer 10/l 1) 30101 IO~d/O,~,] 30102 [O,~/O,~] 30201 [04/(* 30)3/1(rt 3O)JO,] 30202 [04/C_t 3Wi( + 30)~0/] 30301 [04/Ck 45)J/( i 30)3/010,] 30302 [04/(i 45)?f/( rt 3O)JO,] 30401 [04/fF WJ/( zk3O)JO,l 30402 [041(i ~)~/( zk3O)yiO+]
19.05 38.10 25.40 38.10 25.40 38.10 25.40 38.10
Group 4 (double delamination between layer 41.5and E/16) 40101 [04//0,d/041 12.7i25.40 40102 w/0ui/0,1 19.05/38.1 40301 12.7i25.40 [O$/(+ 45)d/O41 40302 lW/( F45)d/O~l 19.05138.1
I/II/III fracture mode are used as an index of delamination growth and can be expressed as [3]
lui(a+Aa)-u,(a+Aa)]cr,,da
[~+(a + Au) - ~,,‘(a+ Aa)]a,qnda
where ha is the area of the crack surface, r,s,n are the axes of the local coordinate system, a,,,, a,,, a, are the stresses at the crack front, u,f,u~,u,’ and Un 3U5 IAlr are the displacement of the upper and lower crack surfaces. Therefore, the strain energy release rate can be calculated at the delamination front. Considering the mixed fracture mode, the criterion for determining the delamination growth is selected as [4,5]
Group 5 (double delamination between Iayer 4/5 and S/9) 50101 Fbhw/0,21 12.7119.05 50102 w/w/0,21 25.40/3&l 50201 IWi( * 45)?//( + 45)4/O,] 12.1119.05 50202 25.40/3&t lW/( +45)2//t k45)4/041 Group 6(double delamination between layer 4/S and S/9) 60101 [04//04//0,*] 19.05112.7 60102 KbmMh21 38.1125.40 60201 19.05i12.7 wi//( + 45)?//( &45)4/O,] 60202 3&l/25.40 wit k 45)2//C+45)4/o,]
Fig. 2. Local coordinate for contact surface.
232
Y J. Lee et al.lComposite Structures 41 (1998) 229-241
In this study, it is assumed that Gtc, G,,,-, GIilt do not change during delamination growth and are independent of the layer-up orientation and sequence. Most research [2] has found that selecting c(= p = y = 1 would result in the best fit between experiment and numerical analysis. The delamination will start to grow when Ed,, > 1 and the stress distribution in the structure will be re-distributed. 3.4. Failure criteria and ~ro~~~ de~adati5~ In order to predict the final collapse behavior of the structure, the local fiber-matrix failure will be considered. The well-known Hashine [6] failure criterion is used to determine different types of failure. Four types of failure are considered in this study: fiber failure in tension/compression and matrix failure in tension/compression. These types of failures will lead to property degradation and the reduction of the global stiffness that can induce structure collapse. Three-dimensional Hashine criteria will be checked in each ply under the following modes: (a) Fiber failure in tension (a,, 20)
(10) where ~~~~ 212, r13 are the stresses in fiber direction, in-plane and interlaminar shear stress. X, is the ply tensile strength in the fiber direction and Sj2, Si3 are the ply in-plane, out-of-plane shear strength. (b) Fiber failure in compression (cJ,, 60)
(11) where XC is the ply compressive strength in fiber direction. If one of eqns (10) and (11) is satisfied, the property of this ply will be modified by setting Eli, GIZt v12,vz3,v13to zero. (c) Matrix failure in tension ((rZ2+ f13320)
where Y, is the ply transverse compressive strength. If one of eqns (12) and (13) is satisfied, the property of this ply will be modified by setting EZZ, v12, vu, v13to zero.
4. Finite element formulation 4.1. Layered solid element In order to accurately simulate the bending behavior and increase computation efficiency, a layered solid element with an incompatible mode was developed in this study. The displacement field of this eight-node isoparametric hexahedral element is assumed to be in the form [7] U = i UjNi(r,s,t) + $ UiPj(r,s,t) ,=I ,=I
where Uj is the nodal displacement, aj is the internal degree of freedom (in~mpatible mode), N, P are the shape function related to the degree of freedom U, and ai which can be expressed with the local coordinate system (r,s,t) as Ni(r,s,t) = t (1 r_+rrJ(l + SS{)(1 + tti)
(15)
P,(r,s,t) = (1 - 2) P2(r,s,t) = ( 1 - 2) P&r&) = (1 - 2)
(161
The element matrices are obtained by numerical integration. Gaussian quadrature is used in the plane of the layer and Simpson’s rule is used in the stacking (thickness) direction (Fig. 3). The Simpson’s integration used for this element following the rule will N8
N7
(12) Nl
where ~2~. cx3, ~23, cri2, ~~223 are the stress components in the on-axes ply coordinate system. k; is the ply transverse tensile strength and SZ3is the out-of-plane shear strength. (d) Matrix failure in compression (cr22+ 03360)
(14)
N3 N2
Fig. 3. Numerical integration for hexahedral element.
IT .I. Lee et aLlComposite
233
Structures 41 (1998) 229-241
a
achieve better agreement than following traditional Gaussian quadrature integration, where one element contains only one layer, from a series of numerical tests: 1. number of layers of element <8:3 integration points for each layer; 2. number of layers of element > 8:l integration point for each layer. Since this incompatible element has contained internal degrees of freedom and is independent of the other elements, the static condensation process is used to remove these degrees of freedom from element stiffness before assembling the global stiffness matrix.
N
E
M
Aa
Fig. 4. Finite element
model near crack front.
4.2. Equation of equilibrium The variational equation (eqn (7)) is balanced at the current configuration (t +At) in which the stress and strain are unknown. The updated Lagragian formulation [8] with Newton Raphson iteration is used to solve this nonlinear equation. The 2nd Piloa-Kirchhoff stress tensor and Green Lagragian strain tensor are employed to transform the Sij/e, (in t +At configuration) to the previous state SJe, (in t configuration). So, the integrals in the linearized equilibrium equation can be expressed as ,C,,,,,.e,.~.Ge,,.‘dV+
s ‘V
J I
-
I
(F”-FN),
of node FMN is the force difference d”,dN is the displacement of node M&.
M,N
From the concept of a similar approach, the stress and displacement distribution across the delamination front do not change after crack growth, i.e. FMNz FDE, the modified crack closure technique is performed and the energy release rates are modified as:
GIZ
T+ArT~.buf.~u~.r+A’ds
s
'~i,~Gre,/,dV+ Eg
a&? aurn
FF(dr
-df)
(17)
GII = lim 1
~0-0 2Aa
or in matrix form
+ [Kclu= Pea-Fine - Fan = R
lim -!-
Aa+O2Aa
FfE(df - d,M) Gn = lim 1 AU-O 2Aa
AIS
‘V
[Ktl
where
s ‘V
rc$$,dIs
+
=
‘~ij.G,)lij.‘dV
(19)
(18)
where [K,] is the tangent stiffness matrix, [KC] is the matrix from contact condition, {P,,,) is the external force vector, {Fin,} is the internal element force vector, {F,,,} is the contact force vector, {R) is the residual force vector.
Fr(dr
- d;M)
(20)
The above equations are very easy to calculate by the finite element method and only one analysis step is needed to obtain the nodal forces and displacements. Once the energy release rates are calculated, the delamination growth criterion is applied to each delamination front. If Edel > 1, the delamination growth starts.
4.3. Delamination growth 5. Numerical
The physical meaning of the Irwin’s crack closure integral [3] is that the total work required to close a very small crack (whose energy is nearly enough to extend the crack in the surface by the same size). The crack closure energy can be expressed by the products of forces and displacements as the following form from Fig. 4
verification
5.1. Double cantilever beam for mode I fracture
The configuration of the double cantilever beam test for a mode I fracture is shown in Fig. 5. The specimen used in the analysis is [O,,] laminate with T300/5208 material.
Y J. Lee et al.lComposite Structures 41 (1998) 229-241
234
Fig. 5. Configuration of DCB.
Gi, = 103 J/mm’, Gllc= 876 J/mm2 E,= 136 GPa, E,.=E:=
10 GPa, -1.50
G,Y,= G, = 5.7 GPaG, = 5.7 GPa,
-l.m
0.00 -0.50 MIDDLE OISPC. (mm)
0.50
1.m
Fig. 7. Comparison of present and ABAQUS analysis resutt.
v,,,= v,,,= 0.3 I ) v,*,= 0.40 The analysis agrees very well with the comparison results reported from Ref. [2] (Fig. 6). 5.2. Postbuckiing behavior of the embedded circular delamination with ABAQUS@fNTlJ results
A square lanlinated plate (100 x 100 mm with [(Oi- 45/90/45)&m stacking sequence) loaded with uniform compression with a 25 mm diameter delamination located between ply 4/5 is selected to demonstrate the postbuckling behavior simulation [9]. The characteristics of the ply are as follows: EL= 136GPa, ET= lOGPa, GLT= 5.7 GPa, v = 0.3 1 A commercial finite element program, ABAQUS@ with 20-node hexahedral element, is selected to compare with the present solution. Good agreement is shown in Fig. 7.
6. Comparison and parameter study In this section the data obtained from the experiment will be compared with the numerical solution output from the computer program ‘DELAM3D’. The strains shown in the load-strain curve were calculated
0.05’1mm away from the strain gage bounded surface to inflect the strain gage thickness effect. 6.1. Phenomenon description Typical experimental data from the specimen is shown in Fig. 8. Two different load processes are observed (path l-2-3-4 and 1-2-3’-4’) and each process can be divied into four paths. Path 1 corresponds to linear behavior in which the compressive strain is obtained from two strain gages. When load increases, postbuckling behavior can be found in thin sublaminate, but another strain remained in compression. As the load continues to increase, a sudden change of slope for the load-strain curve is observed which indicates the initialization of delamination growth. Depending on the delamination location in thickness, two types of load change would suddenly be observed (path 3 and 3’). In general, path 3 represents the stable growth of delamination and can always be found in the specimen containing thin sublaminate delamination (test group 1). In contrast, path 3’ can be found from the group 213 test result that contains a deeper delamination and the load drops very quickly, which is likely to lead to unstable delamination growth. Because of the unstable delamination growth, the delamination front reached the clamped boundary. So, in the final region path 4’, the load change was very slow and the strain increases were very rapid. Finally, collapse
x 0.20
cmcl open Dhphcnnsnt
0.24
d (mm)
Fig. 6. Comparison DCB result.
STRAIN
Fig. 8. A typical load-strain
response.
I: J. Lee
et al./Compsite
Structures
41 (1998)
235
229-241
..,
occurred. For the stable delamination growth case, the delamination grows step-by-step as the load increases. At times postbuckling behavior can be observed again. When the delamination front reaches the clamped boundary collapse occurs. In some of the experiment results, we do not see initial buckle and specimen collapse occuring suddenly after the initial delamination growth.
. :
6.2. Results comparison r(c4m
Although the analysis result does not coincide completely with the test data from the series of specimens in this study, the critical bifurcation (the load of initial local buckle, initial delamination growth and maximum capacity of structure) are agreeable with numerous cases. The following discussion will be based on the group defined in Table 1. GROUP 1: the majority of the analysis results shown in Figs 9-16 agree very well with the test data. For Figs 11 and 12, the analytical load of the initial buckle is quite different compared with the test result. It seems to be influenced by the bonding force between teflon and delamination surface which can make the initial buckle occurs very late, but the ~stbuckling
Fig. 9. Comparison between analysis and test results of a [OJOlh] composite with 19.05 mm delamination.
*3ocnm
_..
._.. ,....... ..,... ..~.... ....l......
_
:i
:
:
moo
404s
Fig. 11. Comparison between analysis and test [Od/(+3O)JO,] composite with 25.4 mm delamination.
results
of a
behavior then occures after the initial buckle. In Fig. 15, the initial buckle of the test occurs very early compared with the analysis and it will have a great influence on the postbuckling behavior, especially on the rn~irn~ capacity of load for the structure. GROUP 2: the postbuckling behavior in this group (Figs 17-24) is quite different from group 1 with the following properties: (1) the load of the local buckle is very close to the maximum capacity of the structure; (2) has an unstable initial delamination growth to reach the boundary; (3) after initial delamination
-mm
MCRO STRAIN
-mo
.,
,.
-ax?4
h&Os&E
4ooll
Fig. 12. Comparison between analysis and test [On( f 30)d04] composite with 38.1 mm delamination.
a000
results
of a
-.__
j i
0 -Moo
D
MCRO?hAIN
4ml
6000
Fig. 10. Comparison between analysis and test results of a (O&O,,] composite with 38.1 mm delamination.
4nm
-xJ!lo
Ii&R0
STi%
4004
Fig. 13. Comparison between analysis and test [OJI( _t45)dO4]composite with 25.4 mm delamination.
oam
results
of a
236
I: J. Lee et ai. IComposite Structures 41 (1998) 229-241
-1CW6
E f -6m
0 -36m
6
666U
6665
-woo
~*=~~~~~,~
Fig. 14. Comparison between analysis and test [O,N(k45)d04] composite with 38.1 mm delamination.
__ ...._.i” :. ..I ;
1
i
_ ._ _
I
:
1
-zooo
0
2000
4004
woo
8ooo
too00
MICROSTRAIN results
of a
growth, the load reduced slowly before final collapse. In Fig. 24, the load of the initial delamination growth is very close to the test data, but the strain value has many discrepancies. GROUP 3: in this group (Figs 2%32), the max. load of the laminate nearly equals the load of the initial delamination growth. After that, only the laminate containing a longer delamination length has postbuckling behavior. In another case, the initial buckle and final collapse occur simultaneously. In general, the
-Km0
1 -4004
Fig. 17. Comparison between analysis and test results of a [Oplia,,] composite with 19.05 mm delamination.
maximum load capacity of the laminate compared with the test data agree very well. GROUPS 4-6: according to the second delamination, the compressive behavior is much difference compared to the laminate containing only one delamination specimen. Although the interaction of the two delamination surfaces is very complicated and difficult to predict very well along the load-strain curve, many cases shown in Figs 33 and 34 of groups 4-6 agree with the special bifurcation (initial buckle, initial delamination growth and structure maximum capacity).
..: j
0 -4ma
4wo
-mm
M&O
.6wo
Fig. 15. Comparison between analysis and test [O.J( + 60)F/04]composite with 25.4 mm delamination.
*wo
-2000
0
6rJoJ
S&ii
2060
466cl
6WO
results
of a
-4ow
-MMI
2ooo
4606
m
RWCRODSTRAIR
Fig. 18. Comparison between analysis and test results of a [OpjlOlz] composite with 38.1 mm delamination.
6066
MICRO STRAIN
Fig. 16. Comparison between analysis and test [OJ/( ;t 6O)h/O4] composite with 38.1 mm delamination.
results
of a
Fig. 19. Comparison between analysis and test results [OJ/( +30)~//( f 30)J04] composite with 25.4 mm delamination.
of a
I! 1 Lee
/ ...~_.I.
-18wo
_
et al. IComposite
-...
Siructures
41 (1998)
237
229-241
:
ii __ .._. ;.
_,mo
.._.
i(
‘i
: .;.
I” P
-sow
i
1
-41 ..I Q
-2Ysl
j
0
;.
4ow
-2000
0 MICRO
20M STRAIN
40133
BOCIO
Fig. 20. Comparison between analysis and test results of a [OJ( F 30)&( +-30)JOa] composite with 38.1 mm delamination
6.3. Parameter study 6.3.1. The influence of the compressive behavior on the delamination length
There are two different delamination lengths of the same stacking sequence and delamination location between layers for all the groups l-3 defined in Table 1. The laminate containing a shorter delamination length has a higher initial buckling ioad, but unstable delamination growth appears. The maximum load
-m
-4fml
-2ccQ
0
woo
8&o
8ooo
MICRO~TRAIN
Fig. 21. Comparison between analysis and test results [Od/( &45)2//( +4S).+/0/0,] composite with 25.4 mm delamination.
-4ouo
-2vw
0
4000
6OW
of
8CQO
MlcRO%RAIN 22. Comparison between analysis and test results fO+‘(145)J/( zk4.%/041composite with 38.1 mm deIamination.
Fig.
4ooo
8Qw
MICROTRAIN
SW0
Fig. 23. Comparison between analysis and test results [O.J/(rt 60)~/( i: 60&/O&]composite with 25.4 mm delamination,
of a
capacity of the laminate depends on the delamination location in thickness direction. The maximum load for group 1 in which the delamination is located at l/S thickness does not obviously change when the delamination lengths are different, but the m~mum load for group 213 (Figs 17-32) with a shorter delamination length is greater than the laminate with a longer delamination by about 60-100%.
0
-2ow -4000
0
8&J
uno
MlcRo~mIrJ
Fig. 24. Comparison between analysis and test results [a,/‘(+ 60)~/( i_ 60)404]composite with 38.1 mm delamination,
-6oon
-4000
a00
6ooo
8000
of a
l!xJCa
IicRO~Td~
of
Fig. 25. Comparison between analysis and test results of a IO,&O,,J composite with 19.05 mm delam~ation.
238
Y J. Lee et al. /Composite Structures 41 (1998) 229-241
-6w
-5000
vi. 0
-icm
-xxx)
0
4wcl
~,C~~~~~
m
.j.
0 -8000-mm I,
8a%l
Fig. 26. Comparison between analysis and test results of a [O,,J/O,,] composite with 38.1 mm delamination.
6.3.2. The influence of the compressive behavior on the second de~~i~atio~ introduced
In this study, there are three groups defined in Table 1 which have two delaminations. From Figs 33-44, we can find that the second delamination has a great influence on the postbuckling behavior compared to the laminate with only one delamination. In Figs 10 and 34, the specimens have the same m~imum delamination length and only the load of the initial local buckle is similar, while the other bifurcations are quite different. The reduction ratio of the maximum load
~... ,_ ..i
i
:
:
t
,
I,
-4aw
f
i
. ../..
I
..I..
~
,
,
I,,
.r..
/‘...;
i
(
‘7’
I
4000
-Moo ~,~~*O~~~~
6006
6OW
Fig. 29. Comparison between analysis and test results [OJ( +45)?j-/( +45)r/OJ composite with 25.4 mm delamination.
capacity is not predictable, the load in Fig. 34 is about 80% of Fig. 10 and Fig. 36 is about 50% of Fig. 14. 63.3. The influence of the compressive behavior on the delamination locatd in the thickness direction In groups 1-3, the delaminations are located at l/6, 216, 316 of the thickness. From the comparison of Figs 9-32 we can find that the influence depends on the delamination length. For the cases with a smaller delamination length (Figs 9, 17 and 25) the load of the initial buckle increases when the delamination is _,f,@Jo
,.......
,.
. . . .. . . . . . .. . .. . .. ..i.
.j..
..-.
_...~
c...:
.
. .. .
..
I
--
:
-4WO
Fig. 27. Comparison between analysis and test results [04/(k 3O)J/( + 3O)JO,] composite with 25.4 mm delamination.
4*w
-2000
0
MICR~~RAIN 4000
6WO
of a
Fig. 28. Comparison between analysis and test results [04/(2 3O)J/( f 3O)JO,J composite with 38.1 mm delamination.
of a
-2wo
HI&to STG
4wo
:
6wo
Fig. 30. Comparison between analysis and test results [OJ( +45)J/( k45)JOJ composite with 38.1 mm delamination.
#aI
8000
of a
-4wo
-mJo
woo
4wo
of a
woo
MH;ROCJTRAIN
Fig. 31. Comparison between analysis and test results [OJ/( _t 60)3//( f 60)d04] composite with 25.4 mm delamination.
of a
239
I: J. Lee et aLlComposite Structures 41 (1998) 229-241
-2-X0
0
MEW SRGN
mm
ww
of a
located deeper and the maximum capacity has the same phenomena. This is because when the delamination is located deeper, the initial buckle, delamination growth and final collapse occurs simultaneously. When the delamination becomes longer(Figs 12, 18 and 26) the load of the initial buckle increases when delamination is located deeper, but the maximum load capacity in Fig. 10 (l/6 thickness) is greater than the others (2/6, 3/6 thickness). Fig. 16 with delamination located at l/6
0
Fig. 35. Comparison between analysis and test results [O.J/(+45)d/O,] composite with 12.7, 25.4 mm delamination.
Ii%0SG
ooo0
6.3.4. The influence of the compressive behavior on the stacking orientation From the results of groups l-3 (Figs 9-32) it is easy to observe that the load of the initial buckle and
0
wou
Fig. 33. Comparison between analysis and test results of a [OJ/O,d/O,] composite with 12.7, 19.05 mm delamination.
-2ooo
0
MG&O s&
woo
o&J
Fig. 34. Comparison between analysis and test results of a [O~/O,~/O//o,] composite with 19.05, 38.1 mm delamination.
of a
thickness has the maximum thickness for another side (5/6 thickness) and it has a higher buckling load.
-2MO 0
mO0
M?ROSTE
Fig. 32. Comparison between analysis and test results [04/(k60)~////(k60)304] composite with 38.1 mm delamination.
-2030
wm
0
-2oOa
2WO MICRO
4Ool STRAIN
Fig. 36. Comparison between analysis and test [OJ/( +45),J/O,] composite with 38.1 mm delamination.
-MM0
-4wo
-2ooo
8000
6000
2mI
results
4004
of a
6ooo
MlCROOSTRAlN
Fig. 37. Comparison behveen analysis and test results of a [OJ/OJO,,] composite with 12.7, 25.4 mm delamination.
Y J. Lee et al. /Composite Structures 41 (1998) 229-241
240
0
-2cm
6wO MzKl
EC00
-4m
0 MICROYrnIN
-2000
4000
go00
8000
STizd
Fig. 38. Comparison between analysis and test results of a [OJ/OJ/O,,] composite with 19.05, 38.1 mm delamination.
Fig. 41. Comparison between analysis and test results of a [Od/OJ/O,,] composite with 25.4, 12.7 mm delamination.
7. Conclusions delamination growth is decreased when ply orientation increased. This is because the stiffness of the whole structure decreases when the ply angle increases, and the structure easily deforms. In Figs 10, 12, 14 and 16 with ply angles of o”, 30”, 45” and 60”, the initial buckling load is 4000, 2600, 2300, and 2000 N and it is difficult to find a formulation which will fit. The other group experiences the same phenomena.
-4om
0
-2mo
4Ma
BOOD
In this study, an analytical model with the computer code was developed to predict the compressive behavior of the laminated composite plate with delamination. Based on the results of the test and analysis discussed previously, the following points can be made: 1. It will have unstable delamination growth when the delamination length is shorter or deeper. 2. For the laminated plate with shallow delamination, the load of the initial buckling, delamination growth
sow
MICROSTRAIN Fig. 39. Comparison between [OJ/( k45)J/( *45)410,] composite
analysis and test results of with 12.7, 25.4 mm delamination.
a
-2000
0
tG%o
smstt
Boo0
8aHl
Fig. 42. Comparison between analysis and test results of a [OJ/O,//O,,] composite with 38.1, 19.05 mm delamination.
-6mo
E
-arm
Bmo
0
Bwo
MEktl SlTEN Fig. 40. Comparison [OJ/( *45)d/( +45h04]
between analysis and test results of composite with 19.05, 38.1 mm delamination.
a
Fig. 43. Comparison between [O,//( +45)4/( + 45)4j04] composite
analysis and test results of with 25.4, 12.7 mm delamination.
a
Y J. Lee et al. /Composite
Structures 41 (1998) 229-241
241
Acknowledgements
This study was supported by the National Science Council, Republic of China, under the grant no. NSC 85-2611-E-002-021. References
PI Chai H, Babcock CD, Knauss WG. One dimensional modeling of 4000
-2000
0
2000
MICRO
4OW
6003
8000
STRAIN
Fig. 44. Comparison between analysis and test results of a [04//(*45)J/( +45),0,] composite with 38.1, 19.05 mm delamination.
and final collapse were predicted very well compared with the test result. For some cases, the maximum load capacity of the structure are greater than the load of the initial delamination of about 50% (Fig. 12) or greater than the initial buckling load about 500% (Fig. 16). It seems to be conservative if you do not use the maximum load for structure design. So, it is very important to find the bifurcation (initial buckle, initial delamination and collapse) from the compressive behavior of the structure analysis. 3. The compressive behavior becomes more complicated when the second delamination was introduced and it cannot be predicted from the laminated composite containing only one delamination.
failure in laminated plates by delamination buckling. Inter1981;17:(11): national Journal of Solids and Structures 1069-1083. PI Kutlu Z, Chang FK. Modeling compression failure of laminated composites containing multiple through-the-width delaminations. Journal of Composite Materials 1992;26:(3):350-387. 131 Rybickl EF, Kanninen MF. A finite element calculation of stress intensity factors by modified crack closure integral. Engineering Fracture Mechanics 1997;9:921-938. I41 Kanninen MF, Popela CH. Advanced Fracture Mechanics. Oxford Engineering Science Series. New York: Oxford University Press. [51 Hahu H.T. A mixed-mode fracture criterion for composite material. Composite Technology Review 1983;5:26-29. 161 Hashine Z. Failure criteria for unidirectional fiber composites. ASME Journal of Applied Mechanics 1980;47:329-334. [71 Bathe KJ. Finite Element Procedures in Engineering Analysis. New Jersey: Prentice-Hall, 1982. PI Talyor RL, Beresford J, Wilson EL. A non-conforming element for stress analysis. IJNME 1976;10:1211-1219. I91 Cochelin B, Potier-Ferry M. A numerical model for buckling and growth of delamination in composite laminates. Composite Methods in Applied Mechanics and Engineering 1991;89: 361-380.