EVALUATION OF KI-V RELATION IN CERAMICS BY USING DT TECHNIQUE
(CRACK LENGTH MEASUREMENT BY USING COMPLIANCE METHOD)
Hitoo TOKUNAGA*a, Kiyohiko IKEDA*b, Koichi KAIZU*b, and Hiroyuki KINOSHITA*b
*a
Ube National College of Technology, Ube-shi, 755-8555, Japan, [email protected]
*b
Faculty of Engineering, University of Miyazaki, Miyazaki, 889-2192, JAPAN
ABSTRACT
In order to evaluate crack growth characteristics of brittle materials as ceramics, the Double-Torsion (DT) technique was used
in combination with a compliance method. In the crack propagation test, compliance of the DT specimen was obtained from
the relation between the deflection at the loading point measured by the laser displacement meter and applied load. In addition
to that, the crack length was measured by using a grid pattern film method, which is developed by authors. A calibration curve
of the relative crack length, a/L, as a function of the specimen compliance was derived, and KI-V relation can be estimated by
using the calibration curve. The validity of this proposed method was evaluated by the crack propagation test on soda lime
glass. As a result, it was found that the crack length in slow crack growth in ceramics could be detected with high accuracy by
using the proposed method.
Introduction
For crack propagation test of ceramics, specimens with the long crack, such as Double-Torsion (DT)[1], Double Cantilever
Beam (DCB)[2]and Compact Tension (CT)[3] specimens are generally used. In particular, the DT technique has attractive
possibility for evaluation of slow crack growth behavior in brittle materials, because (1) the stress intensity factor KI is
independent of the crack length, (2) the specimen and loading geometries are simple and (3) slow crack growth data can be
obtained using the load relaxation method. However, it has the problem to consider that the load relaxation method is the only
and reliable method to evaluate crack growth behavior, because of the difficulty in crack length measurement. Also, variations
in slow crack growth data obtained by the load relaxation method appear in the literatures [4],[5]. On the other hand, as a
typical indirect method to monitor crack growth behavior continuously, compliance method is used for several fracture tests [6].
On applying the compliance method to crack length measurement, it is necessary to obtain the relationship between the
specimen compliance and crack length. However, in-situ observation of an extending crack tip is difficult on the DT specimen,
because of the crack tip being not exposed in the crack propagating. We have developed the crack length measurement using
grid pattern Au film [7], to monitor the slow crack growth behavior. In the grid pattern film method, crack growth can be
detected through the stair like voltage response by cutting each grid with crack growth.
The purpose of the present study is to propose a simple and practical method on evaluation of crack growth behavior in
ceramics, which combines the compliance method and grid pattern film method in the DT technique. In this paper, at first, the
measurement theory of proposed method is described. Next, the possibility of application compliance method to crack length
measurement is investigated on crack propagation test by the DT technique with soda-lime glass specimen. Finally, on the
basis of obtained crack growth data, crack growth characteristics of soda-lime glass is evaluated and measurement precision
and the validity of proposed method are shown.
Technique for crack length measurement
Crack length measurement by using grid pattern Au film[7]
The crack length measurement method using grid pattern Au film (grid pattern film method) is based on changes in the
electrical resistance of the material as shown in a commercial crack gauge. In this method, several numbers of thin Au film
grids are prepared on the crack growth path of specimen surface as shown in Fig. 1. The film is made by an ion sputtering
apparatus using a gold (Au) sputtering target. The thickness of the film is from seven to ten nanometers depending on the
sputtering time. Both ends of the film are connected to an electric circuit. Based on the law of Ohm, the electric resistance of
the film cut by the crack can be expressed approximately as the following equation.
R=
(1)
C
k
åW
i
+d
1
Crack propagation
direction
Terminal
Voltage
measurement system
Grid pattern film
Crack length
Precrack
Output voltage
where R is electric resistance of the film between the terminal plates, C is a constant depending on the material, length and
thickness of the film, k is the number of the grid which is not being cut by the crack, Wi is the width of grid i, δis the width of
the uncut part of partially cut grid. When the crack grows, each grid is cut by the crack and the electric resistance of the grid
pattern film increases. Particularly, in the grid pattern film method, the stair like change of electric resistance is obtained
though the output voltage response, as shown in Fig.1. Each rapid increase of the output voltage corresponds to the complete
cutting of each grid by the crack. Since the film thickness is extremely thin, this measurement system follows the slight grid
cutting in the crack growth sensitivity. So, if the grid width and location of film are known, which can be measured before or
after the experiment, the crack length can be monitored from the output voltage response.
Time
Fig.1 Schematic diagram of the Au grid sputtered film.
Evaluation of crack length by using compliance method
In fracture testing, the potential of the compliance technique to monitor subcritical crack growth has been recognized. The
technique is relatively simple since it is based on measurement of two most basic quantities in the material testing, namely,
applied load and specimen deflection. In addition, this technique lends itself to the acquisition of automatic and continuous
data.
The compliance l of the specimen with the crack is defined as follows [8].
æaö D
l º lç ÷ =
èLø P
(2)
where, a is the crack length, L is the specimen length, P is the applied load and D is the deflection at the loading point. By the
determination of the function of l about the each fracture testing specimen such as CT and SEN, the crack length can be
estimated continuously.
In general, the crack length expression as a function of the normalizing compliance is given by [6],[8]
a L = b0 + b1× × U + b2 × U 2 + b3 × U 3 + b4 × U 4 + b5 × U 5
(3)
æ BED ö
U = fç
÷,
è P ø
(4)
where b0 -b5 are regression coefficients, E is Young’s modulus, and D is loading point displacement. The transfer function U is
employed to facilitate fit for a side range of compliance values corresponding to 0.2<a/L<0.975. Fit using several suitable
functions were attempted, and recently, the optimum fit was obtained from the following function.
ö
æ BED
U =ç
+ 1÷
÷
ç
P
ø
è
-1
(5)
By using Eq.(3), crack growth behavior during the crack propagation test can be monitored continuously through the
compliance measurement. If Eq.(3) is unknown for the specimen, it is necessary to calibrate Eq.(3) through the experimental
or analytical data. In the case of experimental calibration, some relationships between the crack length and compliance for the
specimen are necessary. By using these data, regression coefficients in Eq.(3) are determined from the least squares method.
Several crack length measurement methods by using compliance method have been established for common fracture
mechanics specimens, such as Compact Tension (CT), and Center Crack Tension (CCT) specimens. And the compliance is
commonly measured by using clip the gauge method or strain gauge [8].
Experimental Procedure
Material and specimen
The experiment to examine the validity of the proposed crack measurement method was conducted by using a commercial
soda-lime glass. This material has some merits for the examination because of the following reasons: (1) abundant of
subcritical crack growth data, (2) crystallinity of the specimen and easiness of crack tip confirmation by optical method and (3)
cheapness. The DT specimen geometry is a thin plate with the length L=100mm, width W=40mm and thickness t=3mm, as
shown in Fig.2. In addition, the pop-in pre-crack is introduced by four-point loading with the cross head speed (0.05mm/min).
W
L
d
dn
P
Wm
P/2
P/2
Fig.2 Schematic showing specimen and loading geometry of the Double-torsion specimen.
Crack propagation test by DT technique
The DT technique has been widely used to evaluate the relationship between the stress intensity factor KI and crack growth
rate da/dt of brittle materials, because of the several reasons as mentioned above.
The stress intensity factor KI of the DT specimen is given by [1]
é
ù
3
K I = PW m ê
ú
3
ëêWd d n (1 -n )x ûú
12
(6)
where P is the applied load, n is Poisson’s ratio and the other terms are shown in Fig.2. Also, according to Williams and Evans
[9], it was confirmed that there was a following liner relationship between the compliance l and crack length a of the DT
specimen,
l = D P = Ba + l0
(7)
where D is the deflection, B is the slope of the curve, and l0 is a constant.
To make use of these characteristics in the DT technique two approaches of constant load method or load incremental method
have been used for obtaining the information about K-V diagrams. In this method, a new method is proposed by combining the
crack length measurement method by grid pattern Au film with compliance calibration. Namely, the load is applied to DT
specimen with the constant loading rate of dP/dt=0.01N/s. The crack length increases linearly with the increase of the load,
and is monitored by both compliance and grid pattern film methods.
Compliance calibration
To obtain the crack length from the change in compliance, the relationship between the relative crack length and compliance
as shown in Eq.(3), that is calibration curve, is needed. Regression coefficients in Eq.(3) are somewhat dependent on the test
conditions, such as the precrack length and the applied load. Therefore, in-situ calibration during the crack propagation test is
conducted by combining the compliance measurement with grid pattern film method for each DT specimen. Figure 3 shows
the schematic diagram for the compliance and crack length measurements in the DT technique. The displacement of the
loading point, D, can be measured by the laser displacement meter, and the applied load, P, can be measured by using the
load cell. The crack length, a, can be measured by using the grid pattern film method. Some measurements of the relative
crack length, a/L, and specimen compliance, D/P, were independently expressed as a function of time t as shown in Fig.4(a)
and (b). The transfer function U obtained from these specimen compliance are plotted as a function of relative crack length
a/L as shown in Fig.4(c). On the based of the least square fit in the full set of U and a/L data, regression coefficients in Eq.(3)
can be determined and the calibration curve can be obtained. By using the calibration curve crack length can be monitored
continuously over a wide range of the crack growth.
Pre-crack
Crack growth path
Voltage
Measurement
System
a
Pre-crack
P
Grid Pattern Metal Film
Tension side
of DT specimen
1 2
i
n
Grid Number
L (length of specimen)
Output Voltage
Beam path
CCD laser displacement meter
t1 t2
ti
tn
Loading Time t
Fig.3 Schematic diagram of the crack length measurement
a L = b0 + b1U + b2U 2
+b3U 3 + b4U 4
(c)
ai/L
a1 /L
Relative
crack length a/L
(b)
an /L
a2/L
Compliance D/P
Relative
crack length a/L
(a)
U1 U2
Ui
Un
Transfer function U
t1
t2
ti
Loading time t
tn
Fig.4 Compliance calibration.
Results and Discussion
Relationships between compliance and crack length
The crack propagation tests are conducted under constant load and constant loading rate conditions. Figures 4(a) and (b)
show the change of applied load P, displacement D of the loading point, and relative crack length a/L as a function of loading
time t in the constant loading test and constant loading rate test, respectively. Figures 5(a) and (b) show the relationship
between compliance and crack length in these two kinds of loading tests, respectively. In the constant load test (P = 39N), it is
found that the deflection and relative crack length increase linearly with time in the a/L range of 0.4 to 0.7. Therefore, it is
confirmed that the stress intensity factor KI is constant in that range, because the crack growth rate da/dt is constant.
According to Trantina[10], it is shown that the stress intensity factor KI of the DT specimen is independent on the crack length
a in the range of 0.55W<a<L-0.65W, and the constant KI range obtained from the test is in good agreement with that range.
On the other hand, in the constant loading rate test (dP/dt = 0.01N/s), it is found that the relationship between the relative
Applied load P N
38
(1)
36
34
32
Applied load P N
40
(1)
40
30
30
60
60
(2)
50
40
30
20
50
Deflection D mm
Deflection D mm
(2)
40
30
20
10
Relative crack length a/L
1
(3)
0.8
0.6
0.4
0
500
Loading time t s
1000
1
Rrelative crack length a/L
(3)
10
0.8
0.6
0.4
0
(a)Constant load test
(P=39N)
500
1000
Loading time t s
1500
(b) Constant loading rate test
(dP/dt=0.01N/s)
Fig.4 Relationships between (1)applied load , (2) deflection of loading point, (3) relative crack length and loading time.
-6
-6
[´10 ]
[´10 ]
D/P=Ba+ l
l0=-6.12923635e-07
B=2.19160994e-05
1
Compliance D/P m/N
Compliance D/P m/N
D/P=Ba+ l0
0.5
0.04
0.05
0.06
0.07
Crack length a m
(a)Constant load test
0.08
1
l=-6.50043304e-07
B=2.45843223e-05
0.5
0.04
0.05
0.06
0.07
Crack length a m
(b) Constant loading rate test
Fig.5 Relationships between compliance and crack length.
0.08
crack length and loading time is not linear, and the crack growth rate increases with loading time. Therefore, it can be
considered that the relationship between stress intensity factor and crack growth rate over the wide range of KI can be
obtained from a single experiment. However, as shown in Fig4(b), it is difficult to evaluate the relationships between da/dt and
KI in detail by only grid pattern film method, because the number of crack growth data points, which are obtained by the grid
pattern film method, are small. Then, by using these obtained results, relationships between the compliance D
 /P and crack
length a are investigated, and these relations are shown in Figs.5(a) and (b), respectively. As shown in Fig.5, in both the
constant load and constant loading rate test, the linear relations between D/P and a are confirmed. The straight lines in Fig.5
are obtained by least square fit of Eq.(3) to the data. Because the fit is reasonable, it is considered that the compliance data
obtained by our method is valid.
1
2+b
3
4
0.8
Relative crack length a/L
1+b1*U+b2*U 3*U +b4*U
b1=2.33916164e+03
b2=-2.92639482e+06
b3=1.08375941e+09
b4=-1.31986044e+11
0.6
0.4
0.002
0.0025
Transfer function U
(Compliance)
Fig.6 Compliance calibration
m/s
0.8
0.6
0.4
0
0.0015
Crack Growth rate da/dt
Relative crack length a/W
Estimation of compliance curve and KI-V diagram
Compliance calibration is conducted in the manner described above. Figure 6 shows the relations between transfer function U
and relative crack length a/L. Also, the curve in Fig.6 is obtained by the least square fit of Eq.(3) to the data. From this fitting,
regression coefficients of Eq.(3) can be obtained and the expression of the relative crack length as a function of compliance is
determined. Then, by substituting D/P data obtained from the crack propagation test into the expression, the crack length can
be obtained. Figure 7 shows the relationships between the relative crack length and loading time. In Fig.7, The plots are the
measurements by the grid pattern film method and the solid line is the approximate curve by the compliance method,
respectively. As shown in Fig.7, it is confirmed that the crack growth data can be monitored continuously by using the
proposed method over a whole region of the DT specimen. Figure 8 shows the log-log plots of KI-da/dt relation obtained by
using crack growth data shown in Fig.7. The stress intensity factor is determined by substituting the applied load P in Eq.(4),
and crack growth rate is determined by the slope of a-t relation. The obtained KI and da/dt are in the range of 0.4 to 0.6
1/2
-7
-4
MPam , and 10 to 10 m/s, respectively. The crack growth parameter, n, determined from the slop of KI-da/dt relation is
about 13. These results are in good agreement with the results reported by Wiederhorn[2]. These facts show the validity of
proposed crack measurement method, it is valuable to evaluate slow crack growth characteristics over a wide range of the
crack growth rate in ceramics by using the simple measurement system and a single experiment.
10-3
10-4
n=13
10-6
10-7
400
600
Loading timet s
800
Fig.7 Relationship between a/L and t
10-2
10-5
200
0.3
0.4
0.5 0.6 0.7 0.8
Stress Intensity Factor KI MPam1/2
Fig.8 Relationship between da/dt and KI
Conclusions
In this study, to evaluate slow crack growth characteristics in ceramics, a new method, which is based on Double-torsion
technique, was proposed. In the proposed method, crack growth behavior is monitored by using the crack length measurement
method which combines the compliance method and grid pattern film method. To examine the accuracy and practical
effectiveness of the proposed method, the crack propagation test was performed with a commercial soda-lime glass specimen.
The following conclusions were obtained.
(1) It is confirmed that the compliance transition of the double-torsion specimen during the crack propagation test can be
measured by using laser displacement meter.
(2) It is found that the slow crack growth in ceramics can be measured with high accuracy by using the proposed method.
Therefore, crack growth characteristics can be evaluated over a wide range of the crack growth rate from experimental data of
one time.
References
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[2] Wiederhorn, S. M., J. Am. Ceram. Soc., vol.50, 407-414, 1967
[3] Ogawa, T., J. Soc. Mater. Sci. Jpn., vol.40, 1479-1484, 1991 [in Japanese]
[4] Osamu, S., J. Soc. Mater. Sci. Jpn., vol.37, 152-158, 1987 [in Japanese]
[5] Ikeda, K., et al., JSME Series A, vol.63, 493-498, 1997 [in Japanese]
[6] Takagi, S., Nakano, H., Journ. of JSPE, vol.68, 811-816, 2002 [in Japanese]
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[8] Saxena, A. and Hudak J.Jr., Int. Journ. of Fracture, vol.14, 453-468, 1978
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[10] Trantina, G.G., J. Am. Ceram. Soc., vol.60, 338-341, 1977