A METHOD TO MEASURE CRACK OPENING DISPLACEMENT OF
FAST PROPAGATING CRACK IN ARALDITE B
Shinichi Suzuki and Kazuya Iwanaga
Department of Mechanical Engineering
Toyohashi University of Technoloty
Hibarigaoka 1-1, Tempaku-cho, Toyohashi, Aichi, 441-8580, Japan
ABSTRACT
The method of crack opening displacement (COD) has been used to obtain the energy release rate of fast propagating cracks
just before and just after crack bifurcation. In the method, COD is measured on the microscopic photographs of the cracks.
But, in the case of cracks in Araldite B, the corner made by a crack surface and a specimen surface is chipped out from the
specimen, then, it is often difficult to measure CODs from the photographs of the cracks. The present study shows a method
to measure the COD in the region where COD is difficult to be measured directly on the photograph. The accuracy of the
method of the present paper is good enough that the measured CODs can give the energy release rate of the crack at
bifurcation with the accuracy of about 7%.
Introduction
When a brittle material breaks under external force, a fast propagating crack often appears whose speed is more than 200m/s.
The fast propagating crack bifurcates into two cracks if the crack speed is high enough [1-3]. Bifurcation is a characteristic
feature of fast propagating cracks, accordingly, many researchers have studied it theoretically and experimentally. But the
mechanism of bifurcation is not fully understood yet.
One of the most important problems to figure out the mechanism of rapid crack bifurcation is the continuity of energy release
rate at bifurcation [2,5]. The area of crack surface newly made by the unit length extension of the crack after bifurcation is
twice as large as that before bifurcation. Hence the energy release rate immediately after bifurcation becomes discontinuously
twice as large as that before bifurcation if the crack speed after bifurcation is the same as that before bifurcation. On the other
hand, if the energy release rate is continuous at bifurcation, the crack speed must decrease discontinuously at the bifurcation.
But, it was impossible to measure the energy release rate just after bifurcation, because strong interaction occurs between the
near-tip stress fields around the two branch crack tips since the distance between the two crack tips is very short.
Suzuki et al [2,3] recently proposed a method to measure the energy release rate of fast propagating cracks after bifurcation.
If the bifurcation angle θ is small and the lengths of the branch cracks are short (Fig. 1), the crack can be thought to be a
single crack, and the COD of the mother crack is proportional to the square root of the distance from the crack tip. Such
conditions are approximately satisfied just after crack bifurcation. Consequently, the energy release rate of the crack just after
bifurcation can be obtained by measuring the CODs of the mother crack with the formula of COD for a single crack. They
measured the energy release rate of fast propagating cracks in PMMA, and obtained the result that both of energy release rate
Mother crack
y
Branch cracks
COD
θ
x
Figure 1 Bifurcated crack.
Specimen
Specimen
Crack
Crack
Crack
Defect
Crack
COD
Incident light
Crack
Reflected light
COD
Incident light
Crack
Reflected light
(a)
(b)
Figure 2 Illumination of crack and specimen in high-speed holographic microscopy.
(a) PMMA and Homalite 100, (b) Araldite B.
COD δ1
Defect
D
δ2
Crack
Figure 3 Crack opening displacement (COD) and the width of defects.
and crack speed are continuous at bifurcation. In the measurement, they used the pulsed holographic microscopy to take
microscopic photographs of bifurcating cracks, and obtained energy release rate by measuring CODs of the cracks on the
photographs. Pulsed holographic microscopy is the only method to measure the energy release rate just after bifurcation.
Figure 2(a) shows the method to photograph fast propagating cracks by high-speed holographic microscopy. The figure
shows the cross-section of a specimen, and the crack propagates normal to the paper plane. When the crack is propagating,
the incident light emitted form a pulsed laser falls on the specimen surface perpendicularly. The reflected light beam from the
specimen surface is recorded and reconstructed as the object beam of holography. Thus the specimen surface is bright on
the photograph. On the other hand, the crack opening is dark on the photograph, because the crack opening doesn’t reflect
the incident light. As the result, COD is easily measured as the width of cracks dark on the photographs. The method has
been successfully applied to measure the CODs of the cracks in PMMA, Homalite 100 and AISI4340 steel [2,3,5].
In the case of fast propagating cracks in Araldite B, however, the corner made by a crack surface and a specimen surface is
chipped off, and the defects appear as shown in Fig.2(b). The defects don’t reflect the incident light perpendicularly to the
specimen surface, and then, the defects are dark on the photographs. The brightness of crack surface depends on the shape
of the crack surface and the defect there, thus, crack surface is not always photographed brightly and clearly. As the result, it
is often difficult to measure CODs from microscopic photographs of the fast propagating cracks in Araldite B.
The present paper shows a method to approximately measure CODs of cracks in Araldite B even in the region where CODs
are difficult to be measured directly from the microscopic photographs.
Theory of Measurement
The mean value and the standard deviation of the width of defect
Figure 3 shows that D is the sum of the crack opening displacement, COD, and the widths, δ1 and δ2, of the defects, that is,
(1)
D = COD + 2δ
δ +δ
δ≡ 1 2 ,
2
(2)
where δ is the mean value of δ1 and δ2, and is named “defect width” at the position of measurement. The mean value and the
standard deviation of the defect width δ is obtained through the following procedure.
(1) Measure both COD and D at the positions where the crack surface is photographed brightly and clearly enough to
measure COD.
(2) Obtain δ with Eq.(1).
(3) Measuring COD and D at various positions, obtain the mean value δ and the standard deviation Δδ of the defect width δ
through the following equations,
1
( Di − CODi )
2
1 N
δ = ∑ δi
N i =1
1 N
2
Δδ =
∑ δi − δ
N i =1
δi =
(
(3)
(4)
)
(5)
where i is the number labeled to data, Di and CODi are the ith measured values of D and COD, and N is the number of data.
COD measurement in the region of unclear crack surface
In the region where crack surfaces are not clear enough to measure CODs, the present paper measures D instead of COD.
Then the approximate value COD* of COD is obtained by subtracting twice the mean value δ of the defect width from the
measured D.
COD* = D − 2δ
(6)
Using Eq.(1), (6) and Eq.(5), one can derive the following equation,
(
)
2
4(δ − δ ) = 2Δδ .
COD * −COD = 2 δ − δ
(COD * −COD )2
=
(7)
Eq.(7) says that the root mean square of (COD * −COD ) is twice as large as the standard deviation Δδ . Consequently, the
approximate value, COD*, of COD obtained from Eq.(6) is close to the true COD, if the scattering of the defect width δ is much
smaller than the value of COD. If COD* is approximately equal to COD, then, one can obtain the dynamic stress intensity
factor and energy release rate from the COD*.as follows
Energy release rate
The energy release rate G(v) of a fast propagating crack is obtained by measuring COD through the following equations [2-4],
G (v ) =
1
μ
K I (v ) =
AI (v) =
L (v ) =
AI (v ) K I (v ) 2
(8)
π
μ COD 1
8 1 − η1
r L (v )
α1 (1 − α 2 2 )
4α1α 2 − (1 + α 2 2 ) 2
2α1 (α12 − α 2 2 )
4α1α 2 − (1 + α 2 2 ) 2
α1 = 1 − (v c1 )2
α 2 = 1 − (v c2 )2
η1 =
η
1+η
where KI(v) is the dynamic stress intensity factor, AI(v) and L(v) are the functions describing the dynamic effect, r is the
distance from the crack tip, v is the crack speed, μ is the modulus of rigidity, η is the Poisson’s ratio, and c1 and c2 are the
longitudinal and transverse wave speed respectively. When COD* is approximately equal to COD, then one can obtain the
energy release rate G(v) by substituting COD* into Eq.(8) instead of COD.
Experimental Method
The present study uses Araldite B plate specimens of 50mm long, 325mm wide and 3mm thick. Uniform tensile stress is
applied to the specimen. A crack propagates at a speed more than 400m/s, and bifurcates into two cracks in the observation
area around the center of the specimen.
When propagating in the observation area, the crack is successively recorded as three holograms with the optical system of
high-speed holographic microscopy [2,3,6]. The real images of the crack are reconstructed from the holograms, and are
magnified and photographed with a conventional microscope.
COD
Mother crack
COD
Crack tip
COD
(a)
Crack tip
B
C
(b)
Crack 3
Crack 2
Crack 1
1 mm
(c)
Figure 4 A rapidly bifurcating crack in Araldite B plate specimen. Crack opening displacement, COD, is measured on
the photographs.
Results
Figure 4 shows the microscopic photographs of a rapidly bifurcating crack in Araldite B specimen taken by high-speed
holographic microscopy. Figure 4(a), (b) and (c) are the first, the second and the third frame, respectively. The frame interval
is about 10μs. From the photographs, CODs are measured along the cracks.
Figure 5 represents an example of cracks whose crack surfaces are clearly photographed. The magnified photograph of the
crack opening is shown in Fig.5(b). In the photograph, the pointed by an arrow is the crack surface, not the outer boudarys of
dark region. As described in Fig.1(b), the defects appear at the corner of the specimen surface and crack surfaces. The
defects are the dark region outside the crack surfaces in Fig.5(b). One must carefully distinguish the true crack opening
displacement, COD, and the width D that includes the defect widths of δ1 and δ2.
In order to obtain the mean value δ and the standard deviation Δδ of the defect width δ, the crack opening displacement,
COD, and D are measured in the region where crack surfaces are clearly photographed as shown in Fig.5. The measured
COD and D give the defect width δ through Eq.(1). The number of measurement point is 103 on six cracks. One of the
measurement points is indicated in Fig.5(b).
One of
measurement
points
Crack
surface
n
2 mm
Width of
defect δ2
0.2 mm
(a)
δ [μm]
(b)
Figure 5 Magnified photograph of crack in Araldite B
Figure 6 Measurement result of width of defect
Figure 6 shows the histogram of the defect width δ. The horizontal axis is the defect width δ, and the vertical axis is the
frequency. The mean value δ of the defect width is 44μm, and the standard deviation Δδ is 9μm, respectively.
δ = 44μm ,
Δδ = 9μm .
(9)
Figure 7(a), (b) and (c) are the measurement results of COD and COD* of the cracks shown in Fig.4(a), (b) and (c),
respectively. The horizontal axis is the distance r from a crack tip, and the vertical axis indicate COD or COD*. The open
circles denote the CODs measured on the microscopic photographs in Fig.4 directly, and the dark circles denote the COD*s
obtained through Eq.(6) and (9) by measuring D on the photographs in Fig.4. Both of COD and COD* are measured at the
same positions along the cracks.
The graphs in Fig.7 clearly shows that the COD*s are in good agreement with CODs. This is caused by the fact that the
standard deviation of the defect width, Δδ = 9μm , is much smaller than the values of COD that are from 70 to 500μm. This
result and Fig.7 say that the scattering of COD* due to the scattering of δ is less than 10% in the region of r greater than 2mm
where COD is greater than 180μm.
The lines in Fig.7 are determined from the COD data by the least square method, with assumed that the lines are proportional
to r . Figure 7 designates that COD*s are proportional to r as well as CODs. This is due to the fact that the scattering of
COD*s caused by the scattering of the defect widths is much smaller than the values of CODs.
COD or COD* [mm]
COD or COD* [mm]
COD or COD* [mm]
(a)
(b)
(c)
Figure 7 COD and COD* versus r
Figure 8 Energy release rate G as a function of crack tip position.
One can obtain the energy release rate G(v) from the measured COD*s through Eq.(8), because the COD*s are proportional to
r . Figure 8 shows the energy release rate G(v) obtained from COD*s as a function of crack tip position whose origin is at
the bifurcation point of the cracks. The dark symbols are G(v) obtained from COD*s, and the open symbols are G(v) obtained
from CODs. They are in good agreement and the error is about ± 7%. The graph indicates that the energy release rate of
rapidly bifurcating crack in Araldite B is continuous at bifurcation.
Conclusions
(1) It is possible to obtain the energy release rate of rapidly bifurcating cracks in Araldite B by measuring the approximate
values COD* of COD.
(2) The measurement accuracy of energy release rates obtained from COD* is about ± 7%.
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