Evaluation of Mixed-Mode Thermal Stress Intensity Factor
At Various Temperatures
M. Suetsugu*1, K. Sekino*2, T. Nishinohara*3 and K. Shimizu*4
*1: Associate Professor, Department of Mechanical Engineering, Suzuka National College of
Technology, Shiroko, Suzuka, Mie 510-0294, Japan, e-mail: [email protected]
*2: Graduate School of Engineering, Kanto Gakuin University, 1-50-1, Mutsuura-higashi, Kanazawa-ku,
Yokohama 236-8501, Japan
*3: Graduate School of Engineering, Kanto Gakuin University, 1-50-1, Mutsuura-higashi, Kanazawa-ku,
Yokohama 236-8501, Japan (Present; Marui Industrial Co., Ltd.)
*4: Professor, College of Engineering, Kanto Gakuin University, 1-50-1, Mutsuura-higashi,
Kanazawa-ku, Yokohama 236-8501, Japan
ABSTRACT
Mixed-mode thermal stress-intensity factors are studied by using the method of caustics at various temperatures. First,
theoretical caustic patterns are shown in various cases of the type of optical system, the ratio of KII and KI and moreover the
optical property of materials. Next, the value of stress-intensity factor and the aspect of crack propagation are experimentally
and numerically investigated at various temperatures for the glass plates with an inclined artificial notch or a natural crack. It is
shown that there is much difference in the values of KI and KII and the crack propagation direction which are obtained at high
temperatures and at low temperatures. Moreover, crack opening displacement is evaluated by using the optical interferometric
technique at elevated temperatures. Consequently, it is revealed that the distribution of COD along the crack surface is not
simple because of the thermal expansion.
Introduction
Recently, it is very important to analyze the stress-strain condition of materials at various temperatures in the operation of
machines and nuclear reactors under severe loading condition. Especially, fracture behavior in the vicinity of a crack tip at
high and low temperatures should be studied in detail because cracked materials are often fractured by thermal stress. In the
past, attempts have been made for analyzing theoretically [1, 2] the thermal stress-intensity factor (SIF). On the other hand,
there are various papers on the experimental approach to the problems using the photoelastic method [3, 4] and caustic
method [5-7]. However, only a few studies have been conducted focused on the effect of crack width and on the crack
propagation behavior.
In this study, the effect of crack width on the mixed-mode thermal SIF and crack propagation phenomenon are studied by
using the method of caustics [8] and FEM analysis. The signs of plus-minus of KI and KII are easily determined from the shape
of caustic pattern, and the caustic technique is very useful compared with the photoelastic method from this standing point.
Firstly, theoretical caustic patterns which are obtained in various cases of the ratio of KII and KI, the type of optical system and
the property of materials are shown. Next, the value of thermal SIF and the aspect of crack propagation are investigated at
various temperatures for the glass plates with an inclined artificial notch or a natural crack. Moreover, crack opening
displacement (COD) which has an influence on thermal SIF is evaluated by the optical interferometric technique at elevated
temperatures.
Basic Principle of Caustics
Shape of the caustic pattern depends on the signs of plus-minus of KI and KII, the ratio of KII and KI, the optical properties of
materials and the optical system used in the experiment. The fundamental theory [9, 10] of caustics for the optically
anisotropic materials under mixed-mode loading condition in various cases can be given as follows.
For a Parallel or Divergent Light Source
Figure 1. Principle of Caustics
Figure 2. Optical systems forming caustic pattern
→
A collimated or divergent light beam is projected onto the cracked specimen as shown in Figure 1. The light vector W on the
→
screen is given by the following Equation using the deviation w of the light beam
→
W = λr + w,
(1)
where λ is the magnification factor of light as shown in Figure 2, and it is given by
λ = ( z i + z 0 ) / zi
.
(2)
The mapping equation of light on the screen is given by
1
⎧
⎫
3
3
⎛ξ ⎞ −
2
2
⎨ K I cos φ − K II sin φ ± ⎜ ⎟T 2 K I (3 cos φ − 3 cos 3φ ) + K I K II (− 4 sin φ + 12 sin 3φ ) + K II (7 cos φ + 9 cos 3φ ) ⎬ (3)
2
2
8
2π
⎝
⎠
⎩
⎭
1
⎫
z 0 c0 d − 32 ⎧
3
3
⎛ ξ ⎞ −2 2
2
r ⎨ K I sin φ + K II cos φ ± ⎜ ⎟T K I (3 sin φ − 3 sin 3φ ) − K I K II (4 cos φ + 12 cos 3φ ) + K II (13 sin φ + 9 sin 3φ ) ⎬
= λr sin φ −
2
2
2π
⎝8⎠
⎩
⎭
W x'1, 2 = λr cos φ −
W y'1, 2
z 0 c0 d
r
−
[
3
2
]
[
with
]
(
)
T = K I2 sin 2 φ + 2 K I K II sin 2φ + K II2 3 cos 2 φ + 1 .
(4)
In Equation (3), ξ is the coefficient of anisotropy effect of the material, d is the specimen thickness and c0 is the optical
constant of caustics. By using Equations (5) and (6), Equation (3) is written as Equation (7).
μ=
K II
KI
(5),
N=
z 0 c0 d
2π
(6)
3
− ⎧
3
3
⎫
Wx'1, 2 = λr cos φ − K I Nr 2 ⎨− cos φ + μ sin φ ± ξ ⋅ Bx (μ , φ )⎬
2
2
⎩
⎭
W y'1, 2
(7)
3
2
3
3
⎧
⎫
= λr sin φ − K I Nr ⎨− sin φ − μ cos φ ± ξ ⋅ B y (μ , φ )⎬
2
2
⎩
⎭
−
By zeroing of the Jacobian determinant, D=0, in Equation (7), the initial curve r0 on the specimen forming the caustic pattern
on the reference plane is obtained by
2
r01, 2
⎫5
⎧ 3NK I
H1, 2 (μ ,φ , ξ )⎬ ,
=⎨
⎭
⎩ 2λ
and the caustic pattern which is formed on the screen for r0 is defined by
(8)
⎧
2 ⎡
3
3
⎤⎫
(
)
Wx'1, 2 = λ (r0 )1, 2 ⎨cos φ −
φ
μ
φ
ξ
B
μ
φ
cos
sin
,
−
+
±
x
⎥⎦ ⎬ = λ (r0 )1, 2 (E + F )
3H1, 2 ⎢⎣
2
2
⎩
⎭
.
⎧
2 ⎡
3
3
⎤⎫
Wy'1, 2 = λ (r0 )1, 2 ⎨sin φ −
− sin φ − μ cos φ ± ξBy (μ , φ )⎥ ⎬ = λ (r0 )1, 2 (I + J )
⎢
3H1, 2 ⎣
2
2
⎦⎭
⎩
(9)
In Equation (9), E, F and I, J correspond to the first and second terms in each Equation, respectively, and this convenient
expression is used in the followings. In the Equations mentioned above, Bx,y(μ,φ) and H1,2(μ,φ,ξ) are the functions of μ, φ and
ξ. By using the Equation (9), the caustic patterns for various conditions are shown below.
B
KI >0, KII >0
KI <0, KII <0
KI >0, KII <0
KI <0, KII >0
Wx'1, 2 = λ ( r0 )1, 2 (E + F )
W x'1, 2 = λ ( r0 )1, 2 (E − F )
W x'1, 2 = λ ( r0 )1, 2 (E + F )
W x'1, 2 = λ ( r0 )1, 2 (E − F )
W y'1, 2 = λ ( r0 )1, 2 (I + J )
W y'1, 2 = λ ( r0 )1, 2 (I − J )
W y'1, 2 = λ ( r0 )1, 2 (I + J )
W y'1, 2 = λ ( r0 )1, 2 (I − J )
(10)
(11)
(12)
Figure 3 illustrates an example of the caustic patterns calculated for N=1, KI =1 and ξ =0.3.
For parallel or divergent light source
For convergent light source
Figure 3. Example of caustic patterns (N =1, KI =1, ξ =0.3, λ =1)
(13)
For a Convergent Light Source
When a convergent light source as shown in Figure 4 is used, the magnification factor of
light λ is
λ = (zi − z 0 ) / zi ,
(14)
and caustic patterns in this case are given by Equations (15), (16), (17) and (18).
Figure 4. Optical system for a
convergent light
KI >0, KII >0
KI <0, KII <0
KI >0, KII <0
KI <0, KII >0
W x'1, 2 = − λ ( r0 )1, 2 (E − F )
W x'1, 2 = − λ ( r0 )1, 2 (E + F )
W x'1, 2 = − λ ( r0 )1, 2 (E − F )
W x'1, 2 = − λ ( r0 )1, 2 (E + F )
W y'1, 2 = − λ ( r0 )1, 2 (I − J )
W y'1, 2 = − λ ( r0 )1, 2 (I + J )
W y'1, 2 = − λ ( r0 )1, 2 (I − J )
W y'1, 2 = − λ ( r0 )1, 2 (I + J )
(15)
(16)
(17)
(18)
Calculated caustic patterns are shown in Figure 3. As shown in Figure 3, it is seen that the caustic patterns obtained by a
convergent light source are just reverse with those obtained by a parallel or divergent light source in the signs of KI and KII.
The relationship between them is also understood easily by comparing Equations (10)-(13) with (15)-(18).
Figure 3 also shows that we can obtain the double caustic curves as shown in Figure 5 by choosing the suitable optical system,
and the values of KI and KII are determined from the following Equations.
KI =
K II = μ ⋅ K I
(20),
1.671 1
⋅
z 0 d c0 λ1.5
K eff =
⎛D
⋅ ⎜⎜ in
⎝ δ in
K I2 + K II2
⎞
⎟⎟
⎠
2 .5
⋅
1
(19)
1+ μ 2
(21),
r0 =
Din
δ in ⋅ λ
(22)
The ratio μ can be determined by measuring the ratio of ((Dout)max－(Dout)min)/(Dout)max, and the
value of δin in Equation (19) is sequentially obtained through the ratio μ. Keff in the Equation (21)
is the effective stress-intensity factor and r0 in Equation (22) is the size of initial curve. In this
study, the value of K is corrected through the relationship between r0/d and KIexp/KIth when
Figure 5. Caustic pattern
the size of r0 is small to evaluate the exact K-value [8].
under mixed-mode
condition
Test Specimen
The rectangular specimens of glass plates shown in Figure 6 are prepared. As shown in Figure 6, a natural crack or an
artificial notch is produced in the specimen with various inclined angles θ=30°, 45°, 70°and 90°. The notch tip has a
semicircular shape with radius ρ=0.15mm. The length “a” is 12mm for a natural crack and 6mm for an artificial notch. By
10
2
carrying out the compressive test of SEN specimen, the values of c0 and ξ are found to be －0.073×10－ (m /N) and 0.3,
respectively.
Figure 6. Shape of specimen
Figure 7. Experimental setup
Experimental Apparatus and Test Procedure
The experimental setup for the study of thermal SIF is shown in Figure 7. In addition to the parallel laser light source shown in
Figure 7, divergent or convergent light source is properly employed. The bottom surface of the specimen is placed on the
heater or cooling device settled to the testing temperature, and the variation of caustic pattern formed by the thermal stress is
recorded. The cooling device using liquid nitrogen is made in our laboratory.
Experimental Results and Discussion
Thermal SIF of Notch
At High Temperatures
Example of caustic patterns for glass plate with notch at T=673K is
shown in Figure 8. These patterns are obtained by using a convergent
light source, and therefore, the sigh of KI is negative. As shown in
Figure 8(b), caustic patterns formed by an inclined notch are
asymmetric because of the mixed-mode loading.
Moreover, by
comparing the caustic patterns at t =5s and t =180s in Figure 8(b), it is
seen that the direction of deviation of these two caustic patterns is
opposite. This means that the sign of KII varies from positive to
negative with time t. Figures 9 and 10 show the variations of KI and KII
with time t, respectively. The values of KI and KII are increased with
time until t =30s, and thereafter these values are decreased. It is also
revealed in Figure 10 that the value of KII is increased with time as the
positive value at first, and the sign of KII is reversed after that. Figure
11 is the relationship between (KI)max and (KII)max and notch angle θ at
T=673K. The Figure shows that the value of KI becomes maximum at
θ=90° and the value of KII becomes maximum at θ≒45°.
0
2
T=673K
0
II
-4
70°
(K )
(MPa・m1/2)
-2
30°
45°
1
)
70°
90°
II
(K )
I
-4
T=673K
-6
T=673K
-6
-2
0
30
60
90 120 150 180
t (s)
Figure 9. KI versus t for notch
max
(K
-1
max
-2
I,II max
30°
45°
K (MPa・m1/2)
I
K (MPa・m1/2)
0
Figure 8. Caustic patterns for notch at T=673K
(Convergent light)
0
30
60
90 120 150 180
t (s)
Figure 10. KII versus t for notch
30
40
50
60
70
80
90
θ (°)
Figure 11. (KI)max and (KII)max versus θ
At Low Temperatures
Caustic patterns for glass plate with notch at T=233K are shown in Figure 12.
These patterns are obtained by using a collimated light source, and therefore, the
sigh of KI is positive. Figures 13 and 14 show the variations of KI and KII with time
t, respectively. Figure 15 is the relationship between (KI)max and (KII)max and notch
angle θ at T=233K. This Figure shows that the value of KI becomes maximum at
θ=90° and the value of KII becomes maximum at θ≒45°, and this tendency is
similar to that one at high temperatures.
Figure 12. Caustic patterns for notch at T=233K
(Collimated light)
1.5
0
1.5
-0.4
)
II
30°
45°
70°
90°
30°
45°
0
70°
-0.6
0
10
20
30
t (s)
40
50
Figure 13. KI versus t for notch
(K )
I max
1
I,II max
0.5
-0.2
(K )
II max
0.5
(K
K (MPa・m1/2)
I
K (MPa・m1/2)
1
(MPa・m1/2)
T=233K
T=233K
T=233K
0
0
10
20
30
t (s)
40
50
30
40
50
60
70
80
90
θ (°)
Figure 15. (KI)max and (KII)max versus θ
Figure 14. KII versus t for notch
Example of caustic patterns at T=123K is shown in Figure 16. In both cases of (a) and (b) in the Figure, the third frame shows
the critical state of the stationary notch, and the fourth frame shows the running crack. The variation of Kef f -value with time is
shown in Figure 17. In the Figure, the values of Keff at the initiation of crack extension are denoted by the arrows, and the
fracture toughness value at 123K is 1.33MPa・m1/2 on average.
Keff (MPa・m1/2)
2
1.5
1
0.5
45°
90°
0
0
Figure 16. Caustic patterns for notch at T=123K
(Collimated light)
10
20
30
t (s)
40
50
Figure 17. Keff versus t for notch at T=123K
Thermal SIF of Natural Crack
At High Temperatures
Figure 18 shows the variations of KI and KII with time t obtained from the caustic patterns for glass plates with a natural crack
at 373K. Figure 19 shows the relationship between (KI)max and (－KII)max and notch angle θ.
0.2
I
-K
0.1
II
0
0.3
(K )
I max
0.2
)
30°
45°
70°
90°
30°
45°
70°
I,II max
I
(K
K
0.3
(MPa・m1/2)
0.4
II
K , K (MPa・m1/2)
0.4
(- K )
0.1
II max
0
0
30
60
90 120 150 180
t (s)
Figure 18. Variations of KI and KII with time t
for natural crack at T=373K
30
40
50
60
70
80
90
θ (°)
Figure 19. (KI)max and (－KII)max versus θ at T=373K
Keff (MPa・m1/2)
1
0.8
0.6
0.4
30°
45°
70°
90°
0.2
0
0
Figure 20. Caustic patterns for natural crack at T=473K
(Divergent light)
10
20
30
t (s)
40
50
60
Figure 21. Keff versus t for natural crack at T=473K
Figure 20 shows an example of caustic patterns at the initiation of crack extension at T=473K. In both cases of (a) and (b) in
the Figure, the third frame shows the critical state of the stationary crack, and the fourth frame shows the running crack. We
can see in the Figure 20(b) that the state of inclined crack tip is under mixed-mode loading condition before crack extension,
however, it becomes under pure mode I loading condition after crack extension. Figure 21 shows the variation of Kef f-value
with time. In the Figure, the values of Keff at the initiation of crack extension are denoted by the arrows, and the fracture
1/2
toughness value at T=473K is 0.69MPa・m on average. Since the caustic patterns shown in Figure 20 are obtained by
using a divergent light source in this case, the sigh of KI is positive.
At Low Temperatures
Caustic patterns for glass plate with natural crack at T=123K are shown
in Figure 22. In both cases of (a) and (b) in the Figure, the second
frame shows the critical state of the stationary crack, and the third frame
shows the running crack.
These patterns are obtained by using a
collimated light source, and therefore, the sigh of KI is positive. The
1/2
fracture toughness value of Keff at T=123K is 0.9 MPa・m on average.
Behavior of Crack Extension
At High Temperatures
Because of existence of compressive stress near the crack tip, crack
extension from a notch is not occurred under high temperature condition.
Figure 23 shows the state of crack extension from natural crack at
T=473K, and Figure 24 gives the relationship between α and inclined
angle θ. α is the direction of the crack at the beginning of crack
extension and α’ is the direction of σθmax which is theoretically calculated
for the value of μ. It is revealed in Figure 24 that the direction of crack
propagation for glass plate is in accordance with the theory of σθmax [11].
Figure 22. Caustic patterns for natural crack
at T=123K (Collimated light)
α and α' (°)
30
α
20
10
α'
0
30
Figure 23. Aspect of crack extension for various
inclined cracks at T=473K
40
50
60
θ (°)
70
80
90
Figure 24. Comparison of crack extension direction α
with direction α’ of σθmax
At Low Temperatures
Figure 25 shows the state of crack extension from notch and natural crack at T=123K. As shown in the Figure, the direction of
crack propagation is abruptly curved just after the slight crack extension. To consider this phenomenon, FEM analysis is
carried out for the glass plates with a natural crack. Figure 26 shows the distribution of σx along the crack line at T=373K and
T=123K. σx is the normal stress perpendicular to the crack line. It is recognized in the Figure that the compressive stress is
occurred in front of crack tip at T=123K, and the turning of crack propagation is caused by this compressive stress.
5
123K
373K
σx (MPa)
4
3
2
1
0
-1
0
Figure 25. Aspect of crack extension for various
inclined cracks at T=123K
10
20
30
40
Distance from crack tip (mm)
Figure 26. Distribution of σx along crack line
Evaluation of COD by Optical Interferometric Method
As mentioned above, thermal SIF is strongly affected by the condition of crack opening,
and therefore, it is important to evaluate COD exactly.
Figure 27 shows the principle of optical interferometric method [12] for evaluating COD
of a transparent material. The incident light impinges on the crack surfaces, A and B,
and interference fringes are generated by reflected rays. When the difference of optical
path length reaches to the wavelength λ0 of incident light, an interference fringe is
generated, and the relation between λ0 and crack opening displacement δ is
δ = (λ0 2 )m ,
(23)
where m is fringe number.
Figure 27. Principle of optical
interferometric method
The rectangular specimen similar to Figure 6 is prepared using plate of
heat-resistant glass. The angle of θ in Figure 6 is 90°, and a natural
crack is inserted at the center of the specimen by thermal stress. Figure
28 shows the experimental apparatus of optical interferometric method.
In the optical interferometric method, the specimen is irradiated with the
white light beam through a beam splitter, and the incident light reflects
on the crack surfaces. Interferometric fringe pattern is recorded by CCD
camera.
Example of interferometric fringes for crack opening measurements is
shown in Figure 29. Interferometric fringes in the Figure are obtained
from the cracked specimen heated to T=363K. Isopachic lines of COD
measured from Figure 29 are shown in Figure 30. As shown in the
Figure, the maximum value of COD is occurred at the middle portion of
crack length, and COD at the bottom of the specimen is approximately
zero. This phenomenon is caused by closing behavior of the crack
surfaces induced by the thermal expansion at the bottom surface of the
specimen.
Figure 28. Experimental apparatus for
interferometric technique
Distance from crack tip x(mm)
Crack tip
Figure 29. An example of interferometric fringe
pattern at T=363K (t =20mm)
Width of crack W(mm)
0 5 10 15 20
0
5
10
15
20
Crack tip
m=1(COD=0.32μm)
m=2(COD=0.63μm)
m=3(COD=0.94μm)
m=4(COD=1.27μm)
m=5(COD=1.58μm)
25
30
35
Figure 30. Isopachic lines of crack opening
displacement
Conclusions
Mixed-mode thermal SIF for the cracked glass plates is determined at high and at low temperatures by using the method of
caustics. The following results are obtained in this study.
(1) Theoretical caustic patterns are shown for the various cases of the signs of KI and KII, the ratio of KII and KI, the type of
optical system and moreover the optical property of materials.
(2) In case of heated plate, it is revealed that the sign of KI for a notch is negative, whereas that for a natural crack is positive
by comparing the experimental and the theoretical caustic patterns.
(3) The values of KI and KII for an inclined crack or notch become maximum at θ=90° and θ≒45°, respectively.
(4) In case of cooled plate, the sign of KI for both a notch and a crack is positive.
(5) Crack propagation is observed for a crack at high temperatures and for both a crack and a notch at low temperatures.
Measured values of K at the crack propagation under these conditions are almost the same as those of static fracture
toughness KIc.
(6) The direction of crack propagation is in accordance with the theory of σθmax at both high and low temperatures. The
direction of crack propagation, however, is abruptly curved just after the slight crack extension at low temperatures. This
phenomenon is caused by the existence of compressive stress.
(7) The distribution of COD of glass plate deformed by the thermal expansion is determined using the optical interferometric
method.
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