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Krueger2006-Fracture-Composites.pdf
FRACTURE MECHANICS FOR COMPOSITES
STATE OF THE ART AND CHALLENGES
Ronald Krueger
National Institute of Aerospace, Hampton, Virginia, USA
For laminated composite materials, interlaminar fracture mechanics has proven useful for
characterizing the onset and growth of delaminations. To fully understand this failure mechanism, the total
strain energy release rate, GT, the mode I component due to interlaminar tension, GI, the mode II
component due to interlaminar sliding shear, GII, and the mode III component, GIII, due to interlaminar
scissoring shear, need to be calculated. In order to accurately predict delamination onset or growth for twodimensional problems, these calculated G components are compared to interlaminar fracture toughness
properties experimentally measured over a range from pure mode I loading to pure mode II loading.
It is state of the art to determine a quasi static mixed-mode fracture criterion by plotting the
interlaminar fracture toughness, Gc , versus the mixed-mode ratio, GII/GT, obtained from data generated
using pure Mode I Double Cantilever Beam (DCB) (GII/GT=0), pure Mode II End Notched Flexure (ENF)
(GII/GT=1), and Mixed Mode Bending (MMB) tests of varying ratios. A curve fit of these data is performed
to determine a mathematical relationship between Gc and GII/GT. Failure is expected when, for a given
mixed mode ratio GII /GT, the calculated total energy release rate, GT, exceeds the interlaminar fracture
toughness, Gc. Although several different types of test specimens have also been suggested for the
measurement of the mode III interlaminar fracture toughness property, an interaction criterion
incorporating the scissoring shear, however, has not yet been established and remains a challange.
The methodology described above has been extended to predict fatigue delamination onset life but to
date a standard only exists for the Mode I DCB test. Interlaminar fracture mechanics has also been used to
characterize the extension or growth of delaminations when subjected to fatigue loading. In analogy with
metals, delamination growth rate can therefore be expressed as a power law function. However, the
exponent is typically high for composite materials compared to metals and standards for the measurement
of fatigue delamination growth have not yet been established and remain a challange.
Today a variety of methods are used to compute the strain energy release rate based on results
obtained from finite element analysis. The virtual crack closure technique (VCCT) is widely used for
computing strain energy release rates based on results from continuum (2D) and solid (3D) finite element
analyses to provide results on the mode separation required when using the mixed-mode fracture criterion.
Although the original publication on VCCT dates back more than a quarter century state of the art
techniques typically require geometric non-linear finite element analyses with additional post processing
routines that are currently not an integral part in most commercial codes. ABAQUS recently announced the
release of a new add-on for ABAQUS 6.5 called VCCT for ABAQUS which is a first step in making
computational fracture mechanics for composite available and attractive to a larger user community. Other
large commercial finite element codes such as MSC NASTRAN or ANSYS, which are the frequently used
in industry, however, do not offer any choice for calculating mixed mode energy release rates today. The
implementation of methods to compute mixed mode energy release rates into these codes remains a
challenge.
To date interlaminar fracture mechanics has proven useful for characterizing the onset of
delaminations in composites and has been used with limited success primarily to investigate onset in
fracture toughness specimens and laboratory size coupon type specimens. Future acceptance of the
methodology by industry and certification authorities however, requires the validation and verification of
the methodology and successful demonstration on structural level. The objective of this presentation is to
demonstrate the state-of-the-art in the areas of delamination characterization, interlaminar fracture
mechanics analysis tools and demonstrate the application on structural level for which a panel was selected
which is reinforced with stringers. Full implementation of Interlaminar Fracture Mechanics (ILFM) in
design however remains a challenge and requires a continuing development effort of codes to calculate
energy release rates and advancements in delamination onset and growth criteria under mixed mode
conditions.
COMPUTATIONAL FRACTURE MECHANICS FOR COMPOSITES
STATE OF THE ART AND CHALLENGES1
Ronald Krueger
National Institute of Aerospace2, Hampton, Virginia, USA
ABSTRACT
Interlaminar fracture mechanics has proven useful for characterizing the onset of
delaminations in composites and has been used with limited success primarily to
investigate onset in fracture toughness specimens and laboratory size coupon type
specimens. Future acceptance of the methodology by industry and certification
authorities however, requires the successful demonstration of the methodology on the
structural level. In this paper, the state-of-the-art in fracture toughness characterization,
and interlaminar fracture mechanics analysis tools are described. To demonstrate the
application on the structural level, a panel was selected which is reinforced with
stringers. Full implementation of interlaminar fracture mechanics in design however
remains a challenge and requires a continuing development effort of codes to calculate
energy release rates and advancements in delamination onset and growth criteria under
mixed mode conditions.
1. BACKGROUND
Many composite components in aerospace structures are made of flat or curved panels
with co-cured or adhesively bonded frames and stiffeners. Over the last decade a consistent stepwise approach has been developed which uses experiments to detect the failure mechanism,
computational stress analysis to determine the location of first matrix cracking and
computational fracture mechanics to investigate the potential for delamination growth. Testing of
thin skin stiffened panels designed for aircraft fuselage applications has shown that bond failure
at the tip of the frame flange is an important and very likely failure mode. Debonding also occurs
when a thin-gage composite fuselage panel is allowed to buckle in service. A methodology based
on fracture mechanics [1] has proven useful for characterizing the onset and growth of
delaminations in composites and has been used with limited success to investigate delamination
onset and debonding in simple laboratory coupon type specimens [2, 3]. Future acceptance of a
fracture mechanics methodology by industry and certification authorities however, requires the
successful demonstration of the methodology on structural level.
The objective of this paper is to demonstrate the state-of-the-art in the areas of
delamination characterization, interlaminar fracture mechanics analysis tools and demonstrate
the application on the structural level for which a panel was selected which is reinforced with
stringers. The advances required in all three areas in order to reach the level of maturity desired
for implementation of this methodology for design and certification of composite components
are highlighted.
1
Presented at the NAFEMS Nordic Seminar: Prediction and Modelling of Failure Using FEA,
Copenhagen/Roskilde, Denmark, June 2006.
2
National Institute of Aerospace, 100 Exploration Way, Hampton, VA 23666-6147, USA
Email: [email protected]
2. METHODOLOGY
2.1. Interlaminar Fracture Mechanics
Interlaminar fracture mechanics has proven useful for characterizing the onset and
growth of delaminations [1, 4-6]. When using fracture mechanics, the total strain energy release
rate, GT, the mode I component due to interlaminar tension, GI, the mode II component due to
interlaminar sliding shear, GII, and the mode III component, GIII, due to interlaminar scissoring
shear, as shown in Figure 1, are calculated along the delamination. The calculated GI, GII, and
GIII components are then compared to interlaminar fracture toughness values in order to predict
delamination onset or growth. Today, the interlaminar fracture toughness values are determined
experimentally over a range of mode mixities from pure mode I loading to pure mode II loading
[7-10].
A quasi static mixed-mode fracture criterion is determined by plotting the interlaminar
fracture toughness, Gc, versus the mixed-mode ratio, GII/GT. The fracture toughness data is
generated experimentally using pure Mode I (GII/GT=0) Double Cantilever Beam (DCB), pure
Mode II (GII/GT=1) four point End Notched Flexure (4ENF), and Mixed Mode Bending (MMB)
tests of varying ratios as shown in Figure 2 for a carbon/epoxy material. A failure criterion – as
shown in Figure 2 - was suggested by Benzeggah and Kenane [11] using a simple mathematical
relationship between Gc and GII/GT
$ GII '*
Gc = GIc + (GIIc " GIc ) # & ) .
% GT (
(1)
In this expression, GIc and GIIc are the experimentally-determined fracture toughness data for
! mode I and II as shown in Figure 2. The factorTM" was determined by a curve fit using the
Levenberg-Marquardt algorithm in KaleidaGraph . Failure is expected when, for a given mixed
mode ratio GII/GT, the calculated total energy release rate, GT, exceeds the interlaminar fracture
toughness, Gc.
!
In order to predict delamination onset or growth for three-dimensional problems,
however, the entire failure surface Gc=Gc (GI, GII, GIII) as shown in Figure 3 is required.
Although several specimens have been suggested for the measurement of the mode III
interlaminar fracture toughness property [12, 13], an interaction criterion incorporating the
scissoring shear has not yet been established and remains a challenge. Currently, the edgecracked torsion test (ECT) is being considered for standardization [14-16].
The methodology has been extended to predict fatigue delamination onset life [17-19].
To date, a standard only exists for the Mode I DCB test [20] although mixed-mode onset data
have been generated [21]. Interlaminar fracture mechanics has also been used to characterize the
extension or growth of delaminations when subjected to fatigue loading [22]. In analogy with
metals, delamination growth rate can therefore be expressed as a power law function. However,
the exponent is typically high for composite materials compared to metals [23, 24]. To date,
standards for the measurement of fatigue delamination growth have not yet been established but
development is currently being discussed in standard developing organizations3 and few results
have been published [25]. The discussion of load history effects and spectrum loading on
delamination growth are in its infancy [26]. In view of the uncertainties related to the high
exponents of the delamination grow rate, it has been suggested to design to levels below a
threshold stain energy release rate to ensure no delamination growth.
2.2. Analysis Tools
Several methods have emerged in the past for computing strain energy-release-rates for
delamination growth in a wide variety of composite structures. The methods are primarily based
on analytical, closed form solutions or finite element analysis.
One method – described in detail in reference [27] - is based on a sublaminate analysis
which treats portions of a laminate as higher-order plates. The plates may be stacked such that
the displacements and tractions are identical at the shared interfaces. By assuming a constant
cross-section in one dimension, the resulting systems of governing differential equations can be
solved in closed form. The plates may be coupled end-to-end, allowing complex structures to be
modeled in a manner similar to finite element analysis. The method has been implemented into a
commercial software (SUBLAM) that allows the computation of mixed-mode strain energy
release rates. Through superposition, general loading conditions can be applied, and coupling
with general purpose finite element codes can be accomplished.
In the past decade, Davidson et. al. developed two-dimensional and three-dimensional
crack tip elements [28-30]. The crack tip elements provide analytical solutions for strain energy
release rates and mode mix using plate theory-based near-tip forces and moments. These latter
quantities may often be obtained from global, undamaged finite element models of the structure
of interest. In comparison to finite element modeling, the crack tip element analyses provide a
significant computational advantage for predicting energy release rates.
Several methods are documented in the literature to compute the strain energy release
rate based on results obtained from finite element analysis. The finite crack extension method
[31, 32] requires two complete analyses. In the model, the crack gets extended for a finite length
prior to the second analysis. The method provides one global total energy release rate as global
forces on a structural level are multiplied with global deformations to calculate the energy
available to advance the crack. The virtual crack extension method [33-42] requires only one
complete analysis of the structure to obtain the deformations. The total energy release rate or Jintegral is computed locally at the crack front, and the calculation only involves an additional
computation of the stiffness matrix of the elements affected by the virtual crack extension. The
method yields the total energy release rate as a function of the direction in which the crack was
extended virtually, yielding information on the most likely growth direction. Modifications of the
method have been suggested in the literature to allow the mode separation for two-dimensional
3
The Composite Materials Handbook MIL-17; http://www.mil17.org/
ASTM International, Committee D30 on Composite Materials; http://www.astm.org/
European Structural Integrity Society (ESIS), TC4: Polymers and Polymer Composites; http://www.esisweb.org/
analysis [43, 44]. An equivalent domain integral method which can be applied to both linear and
nonlinear problems and additionally allows for mode separation was proposed in [45, 46]. A
comprehensive overview of different methods used to compute energy release rates is given in
[47]. New methods to compute the strain energy release rate based on results obtained from finite
element analysis have also been published recently [48-50].
2.2.1 Virtual Crack Closure Technique
For delaminations in laminated composite materials where the failure criterion is highly
dependent on the mixed-mode ratio (as shown in Figure 2), the virtual crack closure technique
(VCCT) has been widely used for computing energy release rates [51, 52]. Results based on
continuum (2-D) and solid (3-D) finite element analyses provide the mode separation required
when using the mixed-mode fracture criterion.
The mode I, and mode II components of the strain energy release rate, GI, GII are
computed as shown in Figure 4 for a 2-D four-node element as an example of VCCT. The terms
X'i , Z'i are the forces at the crack tip at nodal point i and u'l, w'l and u'l*, w'l* are the
displacements at the corresponding nodal points l and l* behind the crack tip. For geometric
nonlinear analysis where large deformations may occur, both forces and displacements obtained
in the global coordinate system need to be transformed into a local coordinate system (x', z')
which originates at the crack tip as shown in Figure 4. The local crack tip system defines the
tangential (x', or mode II) and normal (z', or mode I) coordinate directions at the crack tip in the
deformed configuration. The extension to 3-D is straightforward and the total energy release rate
GT is calculated from the individual mode components as GT =GI +GII +GIII, where GIII =0 for
the two-dimensional case shown in Figure 4.
Although the original publication on VCCT dates back a quarter century [51], the virtual
crack closure technique has been used mainly by scientists in universities, research institutions
and government laboratories and is usually implemented in their own specialized codes or used
in post-processing routines in conjunction with general purpose finite element codes. Until
recently, FRANC2D, developed and owned by the Cornell Fracture Group (CFG) at Cornell
University, has been the only publicly available, highly specialized finite element code that uses
the virtual crack closure technique [53, 54]. To date, the virtual crack closure technique has not
been implemented into the other large commercial general-purpose finite element codes such as
the MSC line of products, ANSYS® or PERMAS. Only ABAQUS® started to offer VCCT for
ABAQUS® in 2005 as a separate addition to their ABAQUS® Standard software [55]. The
implementation allows for the calculation of the mixed-mode energy release rate at the crack tip
in 2D models and along the delamination front in shell and 3D solid models. Crack and
delamination propagation analysis are possible based on different failure criteria including the
one shown in equation (1).
Lately, the HyperSizer® bonded joint analysis capability, which is based on analytical
(non-FEA) formulations has been extended to include the virtual crack closure technique for
predicting crack growth [56]. New methods to compute mixed-mode energy release rates
suitable for the application with the p-version of the finite element method have also recently
been developed [57]. Some modified and newly developed formulations also allow applications
independent of the finite element analysis and are suitable for boundary element analysis [54,
58].
2.2.2 A Global/Local Shell 3D Modeling Technique
Built-up structures are traditionally modeled and analyzed using plate or shell finite
elements to keep the modeling and computational effort affordable. Computed mixed mode
strain energy release rate components, however, depend on many variables such as element order
and shear deformation assumptions, kinematic constraints in the neighborhood of the
delamination front, and continuity of material properties and section stiffness in the vicinity of
the debond when delaminations or debonds are modeled with plate or shell finite elements [5961]. These problems may be avoided by using three-dimensional models. Since many layers of
brick elements through the thickness are often necessary to model the individual plies, however,
the size of finite element models required for accurate analyses may become prohibitively large.
For detailed modeling and analysis of the delaminations, the shell/3D modeling technique
will reduce the modeling time since existing plate or shell models may be modified to shell/3D
models. This is a considerable advantage compared to the creation of an entirely new threedimensional finite element model. The technique will also reduce computational time because
only a relatively small section of interest needs to modeled with solid elements keeping the
number of unknowns small. The technique combines the accuracy of the full three-dimensional
solution with the computational efficiency of a plate or shell finite element model and has been
demonstrated for various applications such as fracture toughness characterization specimens [62]
and on the coupon level for the skin/stringer separation specimen [63, 52]
3. STRINGER STIFFENED PANEL
For the demonstration of a fracture mechanics methodology on the structural level, a
stringer stiffened panel as shown in Figure 5, was selected and analyzed. The square panel is
made of carbon/epoxy tape and reinforced with three stringers made of carbon/epoxy plain
weave fabric. The stiffened panel is bolted to a steel picture frame and subjected to pure in-plane
shear loading. During manufacturing, an artificial defect had been placed at the termination of
the center stiffener. Sufficient shear loading causes the panel to buckle as shown in Figure 5. The
resulting out-of-plane deformation causes skin/stringer separation at the location of the initial
defect as shown in the detail of the deformed mesh in Figure 5. A total of eight delamination
lengths between 81.9 and 355.6 mm were modeled. The initial length corresponds to the length
of the insert used to create an initial defect at the termination of the center stringer. Additional
lengths were chosen to study the change in energy release rate distribution across the width of
the stringer with increasing delamination length. The mixed-mode strain energy release rates
were calculated using the virtual crack closure technique across the width of the stringer foot. A
failure index was calculated by correlating the results with the mixed-mode failure criterion of
the graphite/epoxy material.
3.1 Finite Element Model
The global model representing the steel loadframe, the graphite tape/epoxy panel and the
graphite fabric/epoxy stringers was created using standard S4 shell elements available in the
finite element software ABAQUS®. To accurately model earlier tests, which were performed
under constant displacement control, uniform displacements u,v were applied at one corner node
to introduce shear as shown in Figure 5. The inplane displacements u,v were suppressed at the
diagonally opposite corner, and the out of plane displacements w were suppressed along all four
edges across the entire width of the steel load frame.
At the stringer termination, shell elements representing the foot of the stringer and the
panel were removed from the original shell model. The shell elements used to model the flange
web and hat were not removed. A local 3D FE-model containing a straight delamination front
was generated using C3D8I solid brick elements and inserted into the global shell model as
shown in the detail of Figure 5. The local 3D model consists of an intact section and a
delaminated section with a fine mesh around the delamination front. The plane of delamination is
located at the bondline between stringer foot and the panel. The delamination was modeled as a
discrete discontinuity using two unconnected nodes with identical coordinates, one on each side
of the delamination. A refined mesh as shown in Figure 5 was used along the stringer boundary
in order to capture edge effects. Using the finite sliding option available in ABAQUS®, contact
was modeled between the delaminated surfaces to avoid interpenetration during analysis. The
local 3D model was connected with the shell model using the shell to solid coupling option in
ABAQUS® which allows the connection between non-conforming shell and solid models. For
the entire investigation of the panel buckling, the ABAQUS® geometric nonlinear analysis
procedure was used. Additional modeling details are described in reference [64].
3.2 Calculation of Mixed-Mode Strain Energy Release Rates and Failure Indices
The virtual crack closure technique (VCCT) – discussed earlier - was used to calculate
the mode contributions GI, GII and GIII, the total energy release rate GT=GI+GII+GIII, as well as
the mixed mode ratios GS/GT along the delamination front across the width of the stringer for all
delamination lengths modeled. Here, GS denotes the sum of the in-plane shearing components
GII+GIII. For two-dimensional analyses, where GIII=0, this definition is equal to the commonly
used definition of the mixed mode ratio, GII /GT. For three-dimensional analysis, which also
yields results for the scissoring mode GIII, the modified definition of GS is introduced since a
mixed-mode failure criterion, which accounts for all three modes is currently not available.
For each nodal point along the delamination front, the critical energy release rate Gc was
calculated from the mixed mode failure criterion for graphite/epoxy (Figure 2)
# GS & 4.46
Gc = 207.7 + 1126.8 " % ( .
$ GT '
(2)
for the computed mixed-mode ratio GS/GT at each point. Subsequently, the failure index GT/Gc
! was determined with the assumption that delamination propagation occurs for
GT
" 1.
Gc
(3)
For all delamination lengths modeled, the computed failure indices were calculated for
! every fifth load increment plus the final increment and plotted versus the dimensionless
coordinate s across the width of the stringer b (see detail in Figure 5)
s( y ) =
y " y0
;
b
0.0 # s # 1.0 .
(4)
At the left edge of the stringer, the nodal point coordinates are equal to y=y0 which yields s=0.0,
! and the right edge nodal point coordinates are equal to y=yb which results in s=1.0 as depicted in
Figure 5.
The calculated failure index for delamination length a=81.9 mm is shown in Figure 6 for
selected analysis increments only. The failure index peaked at the edges (s=0.0 and s=1.0) with
an additional peak around the center (s~0.5) underneath the stringer. Early in the analysis (small
increment numbers), which corresponds to small applied displacements (u=v), the failure index
GT/Gc is well below unity across the entire width. This result indicates that the delamination is
not going to grow. With increasing load, the failure index approaches unity first near one edge
where failure is expected to initiate. Generally, for the next load increment, the index is well
above unity across the entire width. For the longer delaminations (a=101.6 mm and
a=355.6 mm) which are associated with a different global buckling pattern, the distribution
across the width changes, the failure index peaks in the center underneath the stringer web and is
reduced toward the edges as discussed in detail in reference [64].
A different way to visualize the results is to plot the critical displacement, i.e. the applied
displacement (u or v) for which GT/Gc=1.0, at the center of the specimen (s=0.5) versus the
delamination length as shown in Figure 7. The critical displacements for delamination lengths of
a=81.9 mm, 88.9 mm, and 94.9 mm, were almost identical. For the longer delaminations
modeled (a=101.6 mm up to 355.6 mm), the critical displacements are significantly lower, which
suggests that rapid delamination progress is to be expected once the delamination starts to
propagate for a critical applied displacement. Analysis and result details are described in
reference [64].
CONCLUDING REMARKS
For laminated composite materials, fracture mechanics has proven useful for
characterizing the onset of delaminations in composites. To fully understand this failure
mechanism, the mixed-mode strain energy release rates need to be calculated and compared to
interlaminar fracture toughness properties experimentally measured over a range from pure mode
I loading to pure mode II loading.
It is state of the art to determine mode I fracture toughness using Double Cantilever
Beam (DCB) and mode II fracture toughness using End Notched Flexure (ENF) tests. The
Mixed Mode Bending (MMB) tests is used to determine the fracture toughness of varying
mixed-mode ratios. Although several different types of test specimens have also been suggested
for the measurement of the mode III interlaminar fracture toughness property, an interaction
criterion incorporating the scissoring shear, has not yet been established and remains a challange.
The methodology described above has been extended to predict fatigue delamination
onset life, but to date a standard only exists for the Mode I DCB test. Interlaminar fracture
mechanics has also been used to characterize the extension or growth of delaminations when
subjected to fatigue loading. Standards for the measurement of fatigue delamination growth have
not yet been established and remain a challenge.
Today, a variety of methods are used to compute the strain energy release rate based on
results obtained from finite element analysis. The virtual crack closure technique (VCCT) is
widely used for computing mixed-mode strain energy release rates based on results from finite
element analyses. Currently, VCCT has only been implemented into the commercial finite
element software ABAQUS®whereas other large commercial finite element codes do not offer
any choice for calculating mixed mode energy release rates today. The implementation of
methods to compute mixed mode energy release rates into these codes remains a challenge.
To date, interlaminar fracture mechanics has been used with limited success primarily to
investigate onset in fracture toughness specimens and laboratory size coupon type specimens.
Future acceptance of the methodology by industry and certification authorities requires the
validation and verification of the methodology and successful demonstration on the structural
level. The skin/stringer separation of a graphite/epoxy composite panel reinforced with three
stringers and subjected to pure shear loading was analyzed to demonstrate the state-of-the-art
application on the structural level. Full implementation of Interlaminar Fracture Mechanics in
design, however remains a challenge and requires advancements in delamination onset and
growth criteria under mixed mode conditions and continuing development effort of codes to
calculate energy release rates.
ACKNOWLEDGEMENTS
The skin/stringer separation analysis was supported by The Boeing Company and the
Aviation Applied Technology Directorate under Technology Investment Agreement
No. DAAH10-02-2-0001 as part of the Survivable, Affordable, Repairable, Airframe Program
(SARAP).
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P
Q
P
Interlaminar tension
Mode I
τ
Interlaminar sliding shear
Mode II
Interlaminar scissoring shear
Mode III
Figure 1: Fracture Modes.
1600
DCB, Mode I
MMB, Mode I and II
G
4ENF, Mode II
IIc
1200
GC,
J/m2
experimental data
mean values
Gc = GIc + (GIIc-GIc)(GII/GT)n
800
400
G
GIc
207.7
GIIc
1334.5
η
4.46
Ic
G = G + (GIIc-GIc)(GII/GT)η
c
0
0
0.2
Ic
0.4
0.6
Mixed Mode Ratio GII/GT
0.8
1
Figure 2: Mixed-mode fracture criterion for a toughened carbon/epoxy.
GIIIc
GI/ GT =0
mode II/III interaction only
GIIc
GII/ GT =0
mode I/III interaction only
GT
GIII/GT
1.0
Failure surface
Gc =Gc(GIc, GIIc, GIIIc)
GII/GT
1.0
GIc
GIII / GT =0
mode I/II interaction only
as shown in Figure 2.
0
Figure 3: Mixed mode failure criterion for modes I, II and III.
∆a
local crack tip system
z',w',Z'
∆a
undeformed state
(outline)
crack closed
x',u',X'
a
Z'i
i
w'
k
u'
z,w,Z
X'i
w' *
global system
*
u'
x,u,X
X,Y,Z: forces
u,v,w: displacements
*
GI = – Z'i ( w'l – w'l * ) / ( 2Δa )
deformed state
GII = – X'i ( u'l – u'l * ) / ( 2Δa )
Figure 4: Virtual Crack Closure Technique (VCCT).
panel made of carbon/epoxy
tape
steel load frame
stringer made of
carbon/epoxy fabric
u=v=0
u=v=6.35 mm
w=0
detail of local 3D insert
w=0
z,w
w=0
y,v
u,v,w: displacements
x,u
y=yb
s=1.0
y=y0
s=0.0
intact section
s
b
delamination front
Figure 5: Finite Element model of stiffened panel (1016 mm x 1016 mm) and load frame.
5
increment / appl. displ.
20/0.052
25/0.061
30/0.095
35/0.157
4
3
GT/G
c
2
1
0
1
0.8
0.6
0.4
coordinate s, -
0.2
0
Figure 6. Computed failure index across the width of the stringer
for selected increments for delamination length a=81.9 mm.
3.0
2.5
a=81.9 mm
a=88.9 mm
a=94.6 mm
G /G =1.0
T
c
2.0
Applied
displacement
u, mm
a=355.6 mm
a=279.4 mm
1.5
a=127.0 mm
a=203.2 mm
1.0
a=101.6 mm
0.5
0.0
0.0
50.0
100.0
150.0
200.0
250.0
Delamination length a, mm
300.0
Figure 7. Applied external displacement at delamination onset
for different delamination lengths modeled.
350.0
400.0
Fly UP