A J INTEGRAL APPROACH FOR MEASURING COHESIVE LAWS USING A
MODIFIED DCB SANDWICH SPECIMEN
C. Lundsgaard-Larsen and C. Berggreen
Department of Mechanical Engineering
Technical University of Denmark
Dk-2800 Kgs. Lyngby
Denmark
B.F. Sørensen
Risø National Laboratory
Technical University of Denmark
DK-4000 Roskilde
Denmark
ABSTRACT
Extraction of cohesive laws are conducted for interfaces in sandwich structures. Separation between face and core are driven
by pure bending moments applied to a double cantilever beam (DCB) specimen. By varying the ratio between moments
applied to the two beams the test is conducted for different mode mixities. The sandwich specimen consists of glass fiber
faces and Divinycell H200 foam core. Arbitrary stiffening of the sandwich specimen with steel bars adhered to the faces
reduces rotations and ensures that the method is useable for a wide range of materials. The J integral is employed and the
opening of the pre-crack tip is recorded by a commercial digital photogrammetry measurement system. Cohesive laws are
extracted by differentiating J with respect to the normal and tangential opening of the pre-crack tip. Some results are presented
and discussed.
Introduction
Defects in sandwich structures are often inevitable and can originate both from manufacture and use. An important defect type
is separation of face and core (debonding). In a sandwich structure debondings can arise in production when an area between
face and core has not been primed properly resulting in a lack of adhesion. In use, damages such as impact loading due to
collision with objects might result in formation of a debond crack. With debonds present the structure might fail under loads
significantly lower than those for an intact sandwich structure [1,2]. A debond crack can propagate self similarly or kink away
from the interface into either the face or core. Kinking is governed by the mode mixity of the loading and the properties of the
face, core and adhesive. The criticality of an existing crack can be highly dependent on the crack propagation path, since the
fracture toughness of the face, core and interface are very different. As the crack propagates in the interface or laminate the
fibers can form a bridging zone behind the crack tip. This can increase the fracture toughness significantly [3,4] since the
bridging fibers provide closing tractions between the debonded faces. Cohesive zone models may be an effective way of
treating mechanisms such as fiber bridging, plasticity and friction if the size of the process zone is not small compared to any
relevant specimen dimension [5]. The cohesive stresses between layers can represent several combined fracture
mechanisms. By relating the J integral to the normal and tangential opening of the initial crack tip, a cohesive law can be
determined [6,7].
Analyzing the problem
This study concerns crack propagation in sandwich interfaces. This often entails large scale fiber bridging, which increases the
size of the process zone beyond a point where linear elastic fracture mechanics (LEFM) is applicable. First consider a crack at
the interface of a sandwich specimen, see Figure 1. The problem is assumed two-dimensional, and plane conditions prevail.
y
x
δn*
*
δ
=
Gti p
Glocal
δ0
δt*
L
Figure 1: Process zone of a crack in a sandwich specimen subjected to mixed mode loading
As the crack propagates the tip of the pre-crack opens. The opening displacement of the pre-crack tip δ is separated in a
*
normal and tangential opening displacement component, δ n and
δ t* . The crack opening is Mode I dominated if δ n* is large
*
*
*
compared to δ t and mode II dominated if δ t is large compared to δ n . As the crack opening reaches a certain level δ o fibers
*
break and the process zone is fully developed (steady state). The length of the process zone at steady state is L. The process
zone consists of the crack tip where the material cracks and separates and a zone behind the crack tip where large scale fiber
bridging develops, which provide closing tractions between the crack faces. The J integral is evaluated in closed form along a
path following the outer edges of the specimen. The J integral is given as function of geometry, elastic properties and applied
moments, which is described in detail in [8].
Since the pre-crack is made with teflon film there are initially no fibers connecting the crack faces so only the crack tip
contributes to the fracture toughness:
J R = J tip
where
JR
is the J value as the crack propagates, and
(1)
J tip is the J integral evaluated around the crack tip. If the crack
propagates self similar in the interface and does not kink into the laminate or foam,
the test (for constant
J tip
can be assumed constant throughout
δ n* / δ t* ratio). With fibers bridging between the separating crack faces the fracture resistance (equal to
J R ) increases, since fiber bridging in the process zone contributes to the crack growth resistance. This can be written as
J R = J pz + J tip
where
(2)
J pz is the contribution to J from bridging fibers in the process zone. As the crack opening reaches δ o
the fracture
toughness reaches steady state, see Fig. 1 . The cohesive law in terms of normal and shear stress as function of crack
opening is found from differentiating
JR
σ n (δ , δ ) =
*
n
*
t
with respect to
J R (δ n* , δ t* )
δ n*
δ n* and δ t*
∧
respectively [6,7].
σ t (δ , δ ) =
*
n
*
t
J R (δ n* , δ t* )
δ t*
An example of deriving the cohesive law for a test specimen is given in the result section.
(3)
Test method
The test method is based on a double cantilever beam (DCB) specimen loaded by moments, see Figure 2. A detailed
description of the test rig can be found in [9]. The test is conducted under displacement control in a tensile test machine, and
the moment ratio is kept constant throughout one test. Stable crack growth is accommodated, since the J integral is
independent of crack length for a DCB specimen loaded by pure moments [6,7].
P
l1
l2
M1
M2
y
x
Figure 2: Schematic of a DCB sandwich specimen loaded by moments.
The moments are obtained from forces acting on horizontal bars attached to the top of the specimen. The load is applied by a
single wire, hereby ensuring that the four horizontal forces are equal in size. The moments acting on the left and right beam
are determined from the force in the wire and the distance between rollers. It is seen that
M 1 = pl1
M 2 = pl 2
M 2 are moments acting on the right and left beam respectively positive in counter clockwise direction.
The ratio between M 1 and M 2 are changed by adjusting the relative distance between rollers. Furthermore the direction of
where
M1
(4)
and
the moments can be reversed by changing the mounting direction of the wire. If moments with opposite signs are applied e.g.
M 1 / M 2 = −1 crack opening in the normal direction is dominating (Mode I). If moments with the same sign are applied the
crack opening in tangential direction is more dominating (Mode II). It is possible to vary the loading and hereby the
normal/tangential crack opening ratio
δ n* / δ t*
to almost any preferred value. More details on the test method and analysis of
the specimen are found in [8].
Measuring crack opening
The opening of the initial crack tip in the normal and tangential direction is measured. For previous experiments concerning
crack propagation in single laminate specimen this is successfully conducted by inserting pins into the material measuring
displacements with extensometers [9]. For sandwich specimens, the core is soft and it is tedious to fasten pins close to the
crack tip. Instead displacements are measured with a commercial digital photogrammetry system ARAMIS, where the
displacement of several points are found through digital photogrammetry. A speckled pattern is applied to the surface of
specimen, which allows the system software to track 3D displacements of any points on the specimen surface. The set-up of
the specimen and cameras is seen in Figure 3.
steel wire
steel wire
roller
camera
specimen
Figure 3: The deformation of the sandwich specimen is tracked by two digital cameras
The accuracy of the digital photogrammetry equipment is tested before conducting the experiment. Five pictures are taken of
the undeformed specimen while moving the cameras approximately 10 mm for each new picture. Two points are selected on
the specimen, and the system identifies these points in each picture. The distance between the two identified points are
compared for each picture and it is found that the deviation is below 2 μm. This accuracy is however dependant on the
speckled pattern and the size of the measurement area. Further details regarding the measurement procedure can be found in
[8].
Results
Results in terms of measured fracture toughness and derived bridging laws are described in the following. The J value and
opening of the initial crack tip (end of the teflon film) are recorded during the test, and J as function of
δ * for one specimen is
2
plotted on Figure 4. Initially J increases to approx 670 J/m with only a small opening displacement at the crack tip. Then the
crack propagates and it can be deducted, that the energy dissipated near the crack tip is 670 J/m2 since no fiber bridging is
present in the pre-crack. The crack propagates in jumps of 2-3 cm, which is observed as sudden drops in J value, see dashed
line in Figure 4. A script is used to extract the top points of the curves, which is the J value as the crack propagates, and the
extracted data is used to derive bridging laws. The fracture resistance (equal to J) increases as the crack propagates since
large scale fiber bridging is developing behind the crack tip. As the fracture toughness reaches a plateau, steady state is
obtained, and the process zone simply translates along the interface as the crack propagates.
1200
1000
J [J/m2]
800
600
400
200
original data
extracted data
0
0
2
4
6
*
δ [mm]
8
10
12
Figure 4: Measured and extracted J data as function of crack opening
δ*
By varying the moment ratio
varied. As
M 1 / M 2 the ratio between normal and tangential opening of the pre-crack tip δ n* / δ t* is also
M 1 / M 2 increases δ n* / δ t* decreases, and the loading becomes more mode II dominating. The fracture
toughness J as function of the pre-crack tip opening
δ * is illustrated in Figure 5 for different moment ratios going from -0.5 to
1.34, where -0.5 is mode I dominated and 1.34 is more mode II dominated. Each load case is represented by 2-3 specimens
loaded identically.
1200
1500
J [J/m2]
800
2
J [J/m ]
1000
600
400
500
200
M /M = − 0.5
1
0
0
1000
5
M /M = 0.5
2
10
1
0
0
15
5
2000
1500
1500
J [J/m2]
2000
2
J [J/m ]
δ* [mm]
1000
500
20
25
1000
500
M1/M2 = 1.0
0
0
10
15
δ* [mm]
2
5
10
M1/M2 = 1.34
15
0
0
δ* [mm]
Figure 5: Extracted J data as function of crack opening
1
2
3
δ* [mm]
4
5
δ * for multiple moment ratios
By comparing the different sub-diagrams on Figure 5 it is found, that generally the steady state fracture toughness value
increases as the loading becomes more mode II dominating ( M 1 / M 2 increases). A relatively large scatter is observed for
specimens with the same loading, indicating that the spatial material variation is large.
The cohesive laws are found from partial differentiation using equation (3), and the obtained cohesive laws regarding stress in
the normal direction are shown in Figure 6.
4
12
4
x 10
12
M1/M2 = − 0.5
8
2
6
4
2
8
6
4
2
0
0
0.5
δ*n
1
[mm]
0
0
1.5
5
1
2
δ*n [mm]
3
4
x 10
x 10
2
6
M1/M2 = 1.0
M1/M2 = 1.34
5
σ [N/m2]
1.5
4
3
n
1
n
σ [N/m2]
M1/M2 = 0.5
10
σn [N/m ]
2
σn [N/m ]
10
x 10
2
0.5
1
0
0
0.5
δ*
n
1
[mm]
1.5
2
0
0
1
2
δ* [mm]
3
n
Figure 6: Cohesive stress in the normal direction as function of crack opening δ n
*
The cohesive laws plotted in Figure 6 illustrate the stress as function of opening in the direction normal to the crack surfaces,
see Figure 1. The plot indicates that the process consists of two phases. Initially as the material separates, the stress is
relatively high, but drops rapidly as the crack propagates. Since fiber bridging is present, the stress does not vanish completely
but decreases gradually as the crack opens further.
For comparison results are found qualitatively similar to mode-I fracture of polymer composites in Li et al [10], where the
cohesive law is obtained by matching numerical results to experimental observations.
Discussion
A method for extracting mixed mode cohesive laws for sandwich structure using a modified DCB specimen is proposed and
analyzed. It is believed, that the method is of high practical usage, since the method is applicable for a wide range of material
combinations and thicknesses due to the arbitrary stiffening of the faces. Further the method utilizes the J integral, and no
finite element model is necessary. However, the method bears some uncertainties, for which some are described below.
The displacement is found by tracking points initially located near the tip of the pre-crack. A small error will be created since
these points (B1 and B2) are in practice not perfect coincident, but separated approx. 2 mm on the unloaded specimen. This is
necessary since points very close to the opened crack faces can not be followed by the optical system. Thus the deformation
of the material between the points (B1 and B2) will create an error dependant on the deformation level. The tip of the pre-crack
is located between points B1 and B2, thus stresses between the points are maximum at the time where the crack starts
propagating. The magnitude of this error is not considered in this paper, but some remarks are found in [8].
It is believed that the test method of applying pure moments to the DCB specimen ensures steady state crack propagation,
since J is independent of crack length [8]. However, if the fracture toughness is dependant on the crack velocity (e.g. velocity
weakening) or if the material contains large heterogenous toughness variations e.g. by spread through-thickness-
reinforcements, the crack might still propagate rapidly. This is to some degree observed for tests conducted in this study, since
the crack propagates in jumps of 2-3 cm.
Conclusion
A test method for extracting cohesive laws for sandwich interfaces are described. The sandwich specimen can be loaded in
(almost) any desired mixed mode condition. The opening of the pre-crack tip is measured by an optical measurement system,
and by employing the J integral in closed form the fracture toughness as function of crack opening is determined. The
cohesive law is found by differentiating J with respect to crack opening, and results in terms of cohesive laws show that the
two distinct processes are taking place. Near the crack tip the material separates at a relatively high stress, and behind the
crack tip in the process zone the stress level gradually decreases due to the bridging fibers between the cracked faces.
Acknowledgements
The supply of specimens from Kockums AB in Karlskrona, Sweden, is highly appreciated.
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