NASA Technical Reports Server (NTRS) 20040085784: Analysis of the Effects of Residual Strains and Defects on Skin/Stiffener Debonding using Decohesion Elements

9 Pages·1.1 MB·English

by  NASA Technical Reports Server (NTRS)

ANALYSIS OF THE EFFECTS OF RESIDUAL STRAINS AND DEFECTS ON
SKIN/STIFFENER DEBONDING USING DECOHESION ELEMENTS

Carlos G. Dávila
Aerospace Engineer, Analytical and Computational Methods Branch, Senior Member, AIAA.
NASA Langley Research Center, Hampton, VA 23681
Pedro P. Camanho
Assistant Professor, University of Porto
Porto, Portugal

Abstract interaction of the interlaminar stresses in conjunction
with a characteristic distance.1,2 The characteristic
Delamination is one of the predominant forms of
distance is an averaging length that is a function of
failure in laminated composites especially when there
geometry and material properties, so its determination
is no reinforcement in the thickness direction. To
always requires extensive testing.
develop composite structures that are more damage
tolerant, it is necessary to understand how Most analyses of delamination growth apply a
delamination develops and how it can affect the fracture mechanics approach and evaluate energy
residual performance. A number of factors such as release rates, G, for self-similar delamination growth.
residual thermal strains, matrix curing shrinkage, and The G values are usually evaluated using the virtual
manufacturing defects affect how damage will grow in crack closure technique (VCCT) proposed by Rybicki
a composite structure. It is important to develop and Kanninen3. The approach is computationally
analysis methods that are computationally efficient that effective since the energy release rates can be obtained
can account for all such factors. The objective of the from only one analysis.
current work is to apply a newly developed decohesion In the present paper, an approach is proposed that is
element to investigate the debond strength of well suited to progressive failure analyses where
skin/stiffener composite specimens. The process of delaminations are present. The approach consists of
initiation of delaminations and the propagation of placing interfacial decohesion elements between
delamination fronts is investigated. The numerical composite layers. The proposed constitutive equations
predictions are compared with published experimental for the interface are phenomenological mechanical
results. relations between the tractions and interfacial
separations. With increasing interfacial separation, the
1. INTRODUCTION
tractions across the interface reach a maximum,
decrease, and vanish when complete decohesion
Interlaminar cracks (delaminations), as a result of
occurs. The work of normal and tangential separation
impact or a manufacturing defect, can cause a
can be related to the critical values of energy release
significant reduction in the compressive load-carrying
rates4.
capacity of a composite structure. The stress gradients
that occur near geometric discontinuities such as ply In order to predict the initiation and growth of
drop-offs, stiffener terminations and flanges, promote delamination, an 8-node decohesion element is
delamination initiation, trigger intraply damage developed and implemented in the ABAQUS finite
mechanisms, and cause a significant loss of structural element code5. The decohesion element is used to
integrity. The study of delamination mechanics may be model the interface between sublaminates or between
divided into the study of delamination initiation and two bonded components. The material response built
the analysis of delamination propagation. into the element represents damage using a cohesive
Delamination initiation analyses are usually based on zone ahead of the crack tip to predict delamination
stresses and use criteria such as the quadratic growth.
1
2. ELEMENT FORMULATION ∫ δfτ(δ)dδ=G [3]
C
0
2.1 Kinematics
where G is the critical energy release rate for a
C
A zero-thickness decohesion element with 8-nodes is particular mode, and δ f is the corresponding relative
proposed to simulate the resin-rich layer connecting displacement at failure (δ f=δ , δ , δ, δ , or δ in
pp pro lin Ne reg
two laminae of a composite laminate. The constitutive Figure 1).
equation of zero-thickness decohesion elements is
established in terms of relative displacements and σ
tractions across the interface. The vector defining the Perfectly plastic (pp)
Linear softening (lin)
relative displacement in global coordinates is: σ
c Progressive softening (pro)
Regressive softening (reg)
∆i =ui+ −ui− = Nkuk+i −Nkuk−i = Nkuki [1] Gc Needleman (Ne)
where N are standard Lagrangian shape functions, and
k 0
+ − pp pro lin Ne reg
u and u are the components of the displacements of
i i
the top and bottom surfaces, respectively. Figure 1. Softening constitutive models.
For a general element shape and alignment, the
normal and tangential relative displacements δ must 2.3 Mixed-Mode Delamination Criterion
s
be determined in local coordinates. The transformation
In structural applications of composites,
between the global and the local coordinate fame is
delamination growth is likely to occur under mixed-
accomplished using the rotation tensor θ defined in
si
mode loading. Therefore, a general formulation for
Ref. 6:
decohesion elements must deal with mixed-mode
delamination growth problems.
δ =θ∆ =θ N u =B u
s si i si k ki sik ki [2]
Under pure mode I, II or III loading, the onset of
damage at the interface can be determined simply by
2.2 Constitutive Equation
comparing the tractions with their respective
allowable. Under mixed-mode loading, however,
The need for an appropriate constitutive equation in
damage onset may occur before any of the stress
the formulation of the decohesion element is fun-
components involved reach their respective allowable.
damental for an accurate simulation of the interlaminar
The mixed-mode criterion proposed assumes that
cracking process. A constitutive equation is used to
relate the traction, τ, to the relative displacement, δ, at damage initiation can be predicted using the quadratic
failure criterion:
the interface. Some softening models that have been
proposed are shown in Figure 1. One characteristic of
all softening models is that the cohesive zone can still  τ 2 τ 2 τ 2
 3  + 2 + 1 =1 [4]
transfer load after the onset of damage (δ 0 in Figure 1).  T   S   S 
For pure mode I, II or III loading, after the interfacial
normal or shear tractions attain their respective where τ is the normal traction, and τ and τ are the
3 2 1
interlaminar tensile or shear strengths, the stiffnesses transverse tractions. T and S are the nominal normal
are gradually reduced to zero. The area under the tensile and shear strengths, respectively. The operator
traction-relative displacement curves is the respective <x> is defined as x if x>0, and 0 otherwise.
(mode I, II or III) fracture energy. Using the definition The delamination mechanisms in mode II and mode
of the J integral7, it can be shown that for small III are assumed to be the same. Therefore, mode III
cohesive zones4: can be combined with mode II by using a total tan-
gential displacement δ defined as the norm of the
shear
2
two orthogonal tangential relative displacements δ The exponent α in the power law is usually selected
1
and δ. to be either 1 or 2, in which case the criterion is a two-
2
parameter interaction law with parameters G and
IC
δ = δ2+δ2 [5] G . A recently proposed criterion, the B-K criterion8,
shear 1 2 IIC
is established in terms of the single-mode fracture
The total mixed-mode relative displacement δ is toughnesses G and G and a parameter η:
m IC IIC
defined as:
η
G 
G =G +(G −G ) II  [12]
δm = δ3 2+δs2hear [6] T Ic IIc Ic GT 
where δ 3 is the relative opening (mode I) Reeder9 applied the B-K criterion to mixed-mode
displacement. Using the same penalty stiffness K in test results of AS4/3501-6 and obtained a best fit using
P
modes I and II, the tractions before the onset of η=1.45, which is shown in Figure 2.
damage are: The linear (α=1) and quadratic (α=2) criteria, which
are also shown in Figure 2, do not correlate well with
τ =K δ, i=1,2,3 [7]
i P i the experimental results for this material. Therefore, in
order to accurately account for the variation of fracture
The single-mode failure initiation displacements are
toughness as a function of mode ratio, the B-K
then:
criterion is implemented here.
S
δ0 =δ0 =δ0 =
2 1 shear K
P [8] 0.7
T
δ0 = m
3 m 0.6
KP G, N/ C Test (Reeder)
If the relative opening displacement δ 3 is not zero, Rate, 0.5
the mode mixity can be expressed by: ase 0.4
e
Rel η=1.45
β=δsδh3ear [9] al Energy 00..23 α=2 α=1
c
The mixed-mode damage initiation displacement is Criti
0.1
obtained by substituting Eqs. 5-9 into 4, which gives:
0
0 0.2 0.4 0.6 0.8 1
1+β2
δm0 =δ10 δ30 (δ0)2 +(βδ0)2 [10] G I I/GT
1 3
Figure 2. G versus G /G mode ratio for AS4/3501-6.
C II T
The criteria used to predict delamination
propagation under mixed-mode loading conditions are
For the bilinear traction-relative displacement
generally established in terms of the energy release
softening law assumed here, the critical energy release
rates and fracture toughness. The most widely used
rates in mode I and mode II are:
criteria to predict the interaction of the energy release
rates in mixed-mode is the power law given by the
Tδf Sδf
expression: G = 3 ; G = 2 [13]
Ic 2 IIc 2
 G α  G α where δf and δf are the ultimate opening and
 I  + II  =1 [11] 3 2
   
GIc  GIIc  tangential displacements, respectively. The mode I and
3
mode II energies released at failure are computed with:
from:
∂B 
K =∫ D B  spy u +B dA [20]
G =∫δm3fτ dδ kizv A sr rzv ∂uki yp sik
I 3 3
0 [14]
δ2f
G =∫ m τ dδ The irreversibility of the damage process is taken
II 2 2
0
into account by using the maximum value of the rela-
where δ3f and δ2f are the mode I and mode II rela- tive displacement, δmax, rather than the current value.
m m m
tive displacements at failure under mixed-mode The integration is performed numerically using a
loading. Using the definition of the mixed-mode rela- Newton-Cotes integration, which has been shown to
tive displacement δ m in Eq. 6 and the mode mixity perform better than Gaussian integration in problems
ratio β given by Eq. 9, one can solve for the ultimate involving strain softening.4 The element proposed is
relative displacement δf . For the B-K criterion in Eq. implemented in ABAQUS5 as a user-defined element.
m
12, the ultimate relative displacement is then:
3. ELEMENT VALIDATION
 η
2  β2 
δf = G +(G −G )   [15] Before simulating structural components, the
m Kpδm0  IC IIC IC 1+β2  proposed approach is validated by comparing the
predictions with experimental data obtained for
Having defined the relative displacement corre-
double-cantilever beam (DCB−mode I loading), end-
sponding to damage onset, δ0, and total decohesion,
m notched flexure (ENF−mode II loading) and mixed-
δf , the constitutive equation can be defined as: mode bending (MMB−mixed-mode I and II) tests in
m
unidirectional AS4/PEEK carbon-fiber reinforced
τs =Dsrδr [16] composite. The geometry of the specimens and the
material properties are presented in Ref. 6. The
The constitutive operator D is defined in Ref 6 as:
sr experimental tests were performed at different
 G /(G+G ) ratios, ranging from pure mode I loading
II I II
 δ K , δmax ≤δ0 to pure mode II loading.
 sr p m m
Dsr =δsrKp(1−d)Kp +d Kpδs3 −−δδ3 , δm0 <δmmax <δmf The initial size of the delamination is simulated by
  3  placing open decohesion elements along the length
 −δ
δ δ 3 K , δmax ≥δf corresponding to the initial delamination of each
 s3 3r −δ3 p m m specimen. These elements are capable of dealing with
[17]
the contact conditions occurring for mode II or mixed-
where δ is the Kronecker delta, δmax is the mode I and II loading, therefore avoiding
sr m
interpenetration of the delamination faces. The
maximum value (over time) of the relative
numerical predictions and the experimental data for the
displacement, and
test cases simulated are shown in Fig. 3.
( )
δf δmax −δ0 [ ] It can be concluded that a good agreement between
d = m (m m), d∈ 0,1 [18]
δmax δf −δ0 the numerical predictions and the experimental results
m m m
is obtained. The largest difference (8.3%) corresponds
The element stiffness matrix is obtained using the to the case of an MMB test specimen with G /(G+G )
II I II
Principle of Virtual Work: =20%.
K u = f [19]
kizv zv ki
4
Load, N Strain gages
203 mm
ENF Numerical
Experimental
25.4 mm
MMB (G/G=80%) 42 mm
II T
50 mm
Tension Test
MMB (G/G=50%) Extensometer
II T
P
DCB
MMB (G/G=20%)
II T
127 mm
Bending Test
Displacement, mm
Figure 3. Experimental and predicted load- P
displacement relations.
Figure 4. Skin-stiffener specimen configuration (not
to scale).
4. ANALYSIS OF SKIN-STIFFENER Table 1. Material Properties of IM6/3501-6
DEBONDING Unidirectional Graphite/Epoxy10
E E =E ν =ν ν G =G G
Most composite components in aerospace structures 11 22 33 12 13 23 12 13 23
(GPa) (GPa) (GPa) (GPa)
are made of panels with co-cured or adhesively bonded
144.7 9.65 0.3 0.45 5.2 3.4
frames and stiffeners. Testing of stiffened panels has
shown that bond failure at the tip of the stiffener flange
Table 2. Properties of the interface10
is a common failure mode. Comparatively simple
specimens consisting of a stringer flange bonded onto G (N/mm) G (N/mm) T (MPa) S (MPa) η
Ic IIc
a skin have been developed by Krueger et al. to study 0.075 0.547 61 68 1.45
skin/stiffener debonding10. The configuration of the

specimens studied in Ref. 10 is shown in Figure 4. The
specimens are 203 mm-long, 25.4 mm-wide. Both skin A complete analysis of the delamination growth in
and flange were made from IM6/3501-6 the tension specimen requires a high degree of
graphite/epoxy prepreg tape with a nominal ply complexity. For instance, Krueger10 developed highly
thickness of 0.188 mm. The skin lay-up consisting of detailed two-dimensional models using up to four
14 plies was [0/45/90/-45/45/-45/0] and the flange elements per ply thickness. The approach taken here,
s
lay-up consisting of 10 plies was [45/90/-45/0/90]. on the other hand, is to determine if it is possible to
s
The properties of the unidirectional graphite/epoxy predict the debond load of the specimen using
and the properties of the interface reported in Ref. 10 decohesion elements in a much coarser three-
are shown in Tables 1 and 2, respectively.
dimensional model. To keep the modeling difficulties
The relative mode I and mode II displacements for
low and the approach applicable to larger problems,
damage onset are δ30=61×10-6mm and the model that was developed uses only two brick
δ0=68×10-6mm, respectively. The corresponding elements through the thickness of the skin, and another
2
ultimate relative displacements are two through the flange. The complete model consists
δf =2.46×10-3mm and δf =17.5×10-3 mm. The of 1,002 three-dimensional 8-node C3D8I elements
3 2
parameter η=1.45 for the B-K criterion is taken from and 15,212 degrees of freedom. To prevent
test data for AS4/3501-6. delamination from occurring at both ends of the flange
simultaneously, model symmetry was reduced by
5
Extensometer
modeling the tapered end of the flange with a refined
P P
mesh on one side and a coarser mesh on the other. This
z
Coarse end Refined end
model does not contain any pre-existing delaminations. x
Residual thermal effects are simulated by perform-
Corner not
ing a thermal analysis step before the mechanical loads yet debonded
are applied. The same coefficients of thermal
expansion (α=-2.4x10-8/ºC and α=3.7x10-5/ºC) are
11 22
applied to the skin and the flange, and the temperature
Before separation After separation
difference between the curing and room temperature is
∆T=157 ºC. The flange has more 90º plies than 0º
Figure 5. Deformed plots of skin/flange debond model.
plies and the skin is quasi-orthotropic, so it is expected
that residual thermal stresses are present at their It can be observed that good accuracy in the pre-
interface at room temperature. diction of debond loads is obtained in the 2 cases
investigated. Without thermal loads (∆T=0 ºC), the

4.1 Tensile loading debond load is approximately equal to the maximum
value obtained in the tests. The thermal loads (∆T=157
The specimen configuration corresponding to tensile
ºC) cause an 8.5% reduction in the predicted debond
loading is shown in Figure 5. The mechanism for
load, which is now well within the range of the
debonding under axial load is primarily by shear lag.
experimental results.The predicted stiffness of the
Deformed plots of the finite element model imme-
specimen is also in good agreement with the
diately before and after flange separation are shown in
experimental data until a relative displacement of
Figure 5. It can be observed that only the refined end
approximately 0.06 mm is reached. For displacements
of the flange separates. A detailed analysis of the
higher than 0.06 mm, the experimental data shows a
results indicated that the coarse end of the flange also
stiffening effect. This effect is due to the fact that the
softens, but that the separation of the flange at the re-
relative displacement is measured using an
fined end relieves the stresses at the coarse end. It is
extensometer placed between two points of the
interesting to note that static tests exhibit debond of
specimen. With increasing loads, the specimen’s
only one end while fatigue tests induce debonding at
curvature also increases due to the higher bending
both ends10.
moments, and the extensometer no longer measures the
Note that the debond growth is not symmetric across relative displacements in the global x direction shown
the width: the debond initiates on the left corner of the in Figure 5.
flange, as shown in Figure 5, due to the unsymmetry

introduced by the terminated plies at the flange tapered
ends. Table 3. Experimental and numerical results (tension).
The load-extensometer measurement predictions are
Exper. Analysis ∆T=0 ºCAnalysis ∆T=157 ºC
compared with the experimental data in Figure 6. The
Load Load Error Load Error
initiation of delamination is marked by the sharp
Damage
breaks in the extensometer readings. Krueger et al.10 initiation 20.9 kN NA NA NA NA
defined the damage initiation load as the load corre- load

sponding to the first load-drop before flange debond- Flange
22.7 kN 23.6 kN 4.0 % 21.6 kN -4.8 %
debond
ing. The numerical predictions and experimental re-
load
sults are shown in Table 3.
6
30 Table 4 Experimental and numerical results
(3-point bending).
Predicted debond load, 24 kN
25
Range of debond load for all tests Exper. Analysis ∆T=0 ºC Analysis ∆T=157 ºC
e, kN 20 Predicted debond load, 22 kN Load Load Error Load Error
orc Damage
n F 15 initiation 428 N 517 N 19.4 % 419 N -2.1 %
ctio ∆Τ=0 load
a
Re 10 ∆Τ=157 Flange
Experimental debond 463 N 568 N 22.7 % 514 N 11.0 %
5 load
0
0 0.05 0.1 0.15 0.2
Extensometer measurement (mm)
Figure 6. Predicted and experimental extensometer
Deformed d=6.5 mm.
measurements.
Load
d
4.2 Three-point bending
The three-point bending test specimen simulated is
Undeformed
shown in Figure 4. The mechanism for the initiation of
debonding under three-point bending is a combination Figure 7. Deformed plots of skin/flange debond
of peel and interlaminar shear. Deformed plots of the model (3-point bending).
finite element model simulating the three-point
bending load case is shown in Figure 7. Debonding
between the skin and the flange is considered to occur 600
when a through the width crack is predicted. The pre- 514kN((∆∠TT==115577 )ºC)
500
dicted and experimental load displacement relation is N
shown in Figure 8. A comparison between the pre- ce, N400
dloiacdtesd i sa nshdo ewxnp einri mTaebnltea l4 d. amage initiation and debond action ForctionForce,300 419kN((∠∆TT==115577) ºC)
Good accuracy in the prediction of both damage ReRea200
∆∠T=∀(cid:0)0 ºC
initiation and debond loads is obtained only when re-
100 ∠∆T∀=(cid:0)1(cid:0)5(cid:0)7 ºC
sidual thermal stresses are taken into account. Without
Experimental
thermal stresses (∆T=0º C), the debond load is 0
0 1 2 3 4
overpredicted by 22.7%. After debonding, the analysis DDiissppllaacceemmenetn(mt,m m)m
including residual thermal stresses predicts a stable
Figure 8. Predicted and experimental load-
crack growth until a load of 572 N is attained. In
displacement relation.
contrast, the test specimens exhibited an unstable crack
growth of approximately half of the span of the flange.
In order to capture the unstable crack growth a more
4.3 Effects of Bond Defects
refined mesh, able to predict the condition of increase
of energy release rate with increasing debond length at
The manufacturing processes of laminated com-
failure (neglecting R-curve effects), ∂G∂a>0, would posite materials can lead to the existence of defects in
be required. The stiffness of the skin-stiffener the bonds. The debonds can be caused by air entrapped
specimens is predicted accurately for the two between plies or by inclusions in the prepreg.
conditions simulated. Furthermore, accidents such as tool drops, or other
7
low-velocity impact loads, often lead to debonds that loads for specimens with and without pre-existing de-
are difficult to detect. Under the previous circum- fects are shown in Table 5.
stances, cracks will be present in the material before
the application of the design loads. For damage toler-
Table 5. Predicted debond loads (tension).
ance considerations, it is therefore important to assess
the structural behavior of composite materials in the No pre-crack Case A Case B Case C
presence of pre-existing debonds. Flange debond
21.6 20.7 17.3 19.1
load (kN)
Difference (%) - -4.2 -19.9 -11.6
0.67 mm
22
8.5 mm 20
18
Defect A
16
N
e, k 14
orc 12
F
on 10
eacti 8 No pre-defect
R Case A
6
Case B
4 Case C
Defect B
2
8.5 mm 0
0 0.025 0.05 0.075 0.1 0.125 0.15
Extensometer measurement (mm)
Figure 10. Predicted load-extensometer measurement
for undamaged and damaged specimens
under tension.
8.5 mm
Defect C From the observation of Figure 10 and Table 5, it is
clear that both the stiffness and the debond loads of the
specimens are reduced by the pre-existence of bond
defects. The effects of the pre-existing defects are
more pronounced when the defects are located at the
Figure 9. Location of defects.
specimen’s edges (cases B and C). The largest reduc-
The procedure previously proposed is used to tion of stiffness and debond load (19.9%) corresponds
simulate the response of tension-loaded skin-stiffener to case B, when the defect is placed at the position
composite structures with bond defects. The three corresponding to the onset of debonding in the un-
cases shown in Figure 9 are investigated. The three damaged case.
cases correspond to 8.5 mm x 0.67 mm debonds
located at different positions along the skin-flange 5. CONCLUDING REMARKS
interface. The defects are simulated by placing open
The debond strengths of skin/stiffener specimens
decohesion elements at the corresponding positions.
loaded in tension and in three-point bending are in-
The analyses include the effects of residual thermal
vestigated using a cohesive-zone model. The good
stresses.
agreement between the experimental and predicted
The predicted load-displacement relations for an location for crack initiation indicate that the proposed
undamaged specimen and for the three cases under approach can capture the onset of damage in structures
investigation is shown in Figure 10. The predicted where the exact location of an initial crack may be
debond loads and the difference between the debond difficult to determine a priori.
8
For a skin-stiffener specimen under tension, good 4. Camanho, P.P., Dávila, C.G., and Ambur, D.R.,
"Numerical Simulation of Delamination Growth
agreement between the experimental results and the
in Composite Materials," NASA-TP-2001-
numerical predictions is obtained. The stiffness of the
211041, Hampton, VA, 2001.
specimen, the location of crack initiation, and debond
5. ABAQUS V5.8 Users Manual, Hibbitt, Karlsson
loads are in good agreement with published experi-
& Sorensen, Inc., 1998.
mental data. Neglecting residual thermal stresses leads
6. Camanho, P.P. and Dávila, C.G., "Mixed-Mode
to a 4.8% overprediction of the debond load. Decohesion Elements for the Simulation of
Delamination in Composite Materials," NASA-
For a skin/stiffener specimen under three-point
TM-2002-211737, Hampton VA, 2002.
bending, good agreement between the experimental
7. Rice, J.R., "A Path Independent Integral and the
and the numerical damage initiation and debond loads
Approximate Analysis of Strain Concentration by
is obtained only when residual thermal stresses are
Notches and Cracks," Journal of Applied
taken into account in the analysis. This result indicates Mechanics, 1968, pp. 379-386.
that the residual thermal stresses of the flange and the 8. Benzeggagh, M.L., and Kenane, M.,
skin can cause a substantial reduction in the debond "Measurement of Mixed-Mode Delamination
strength of the specimens when loaded in three-point Fracture Toughness of Unidirectional
Glass/Epoxy Composites with Mixed-Mode
bending. The simulation of the unstable crack growth
Bending Apparatus," Composites Science and
following debonding under three-point bending re-
Technology, Vol. 56, No. 4, 1996, pp. 439-449.
quires a more refined mesh.
9. Reeder, J.R., "An Evaluation of Mixed-Mode
The effects of pre-existing bond defects in a Delamination Failure Criteria," NASA-TM
specimen loaded in tension are investigated for a 104210, Hampton, VA, 1992.
8.46 mm x 0.7 mm defect located at three different po- 10. Krueger, R., Cvitkovich, M.K., O'Brien, T.K.,
sitions. In all cases a reduction of both stiffness and and Minguet, P.J., "Testing and Analysis of
Composite Skin/Stringer Debonding under
debond load are predicted. The importance of the lo-
Multi-Axial Loading," J. Composite Materials,
cation of the defect is reflected in the range of pre-
Vol. 34, No. 15, 2000, pp. 1263-1300.
dicted reduction of debond load, from 4.2% up to
19.9%. It is concluded that the approach proposed here
can be used in the failure prediction of composite skin-
stiffener structural components when debonding is the
leading failure mechanism.

REFERENCES
1. Camanho, P.P., and Matthews, F.L.,
"Delamination Onset Prediction in Mechanically
Fastened Joints in Composite Laminates,"
Journal of Composite Materials, Vol. 33, No.
10, 1999, pp. 906-927.
2. Dávila, C.G., and Johnson, E.R., "Analysis of
Delamination Initiation in Postbuckled Dropped-
Ply Laminates," AIAA Journal, Vol. 31, No. 4,
1993, pp. 721-727.
3. Rybicki, E.F., and Kanninen, M.F., "A Finite
Element Calculation of Stress Intensity Factors
by a Modified Crack Closure Integral,"
Engineering Fracture Mechanics, Vol. 9, 1977,
pp. 931-939.
9