A Study on Stress Intensity Factor of Outer Surface Crack along the
Direction of the Axis of Double Column
under Torsion Load for Dissimilar Material
S. Sasaki1 and T. Ezumi2
1
Department of Engineering, Shibaura Institute of Technology, 3-7-5 Toyosu , Koto-ku ,
Tokyo,108-8548 Japan
2
Department of Engineering, Shibaura Institute of Technology, 3-7-5 Toyosu, Koto-ku,
Tokyo,108-8548 Japan
ABSTRACT
This study is the purpose to obtain the stress intensity factor of the double column that is made of the dissimilar material by the
experiment. The authors used the model made of the epoxy resin like the test specimen, this is changed the rigidity that these
specimen are made to change the curning agency for the epoxy resin. In the case of changing the compounding ratio of the
hardner that is for the epoxy resin ,the rigidity of the epoxy resin changes. This is applied and experimented by the caustics
method and the stress frozen mehod of the photo elasticity method.
Introduction
Many structures is changing, the design by which the recyclability is valued at the same time as make to low-cost and lighten
advancing is requested strongly, and a lot of reproduction materials and the accumulating materials show the tendency to act.
The homogeneous of an internal defect and the material are often contained, avoided, and in these materials also which is a
new important problem without the carefully thinking . [1] The double columnar produced with epoxy resin is modeled when a
three-dimensional surface crack exists on the interfacial vicinity of a double column which consists of the dissimilar material,
and the experiment analysis by the stress frozen method of the photo elasticity method is done. [2]-[7]
This research assumed the case where a double column which axially had the crack acts a static load, evaluated the threedimensional surface crack when stress intensity factor KⅠ, KⅡand KⅢ existed at the same time single according to the stress
intensity factor, and elucidated the transformation style of crack up-to-date.
It was compared with the rigidity and the joint boundary radius etc. of the material examine the influence which causes for the
transformation style in the crack point as a result.
A double columnar was modeled by the epoxy resin when a three-dimensional crack existed in interfacial of the surface of a
double column which consisted of the dissimilar kind material, the stress frozen method of the photo elasticity method was
applied to the caustics method , and the experiment analysis was done for the obtaining the K value.
Calculation method of stress intensity factor
The method of calculating stress intensity factor KⅠ and KⅡ,n the photol elasticity method, and KⅢ is which uses the common
method in general. [8]
The relation of K /K is compared with the isochromatic fringec of angle θm and the stress intensity factor is shown in
expression (1) in generally in the crack point etc.
Stress intensity factor KⅠand KⅡare calculated from expression (2) with KⅡ/KⅠ=A.
Nm is the number of isochromatic fringe,αis photoelactic sensitivity, t is the thickness of the specimen, rm is the distance from
the furthest from the overhang point. KⅢ is obtained from expression (3).
Here, λ is λ = (Z 0+Z i) /Z i by the magnification of the
caustic image when the emanation light is used.
Z 0 is a distance of the test specimen and the screen, and Z i is the distances from the focus position of the settling point of
light to the test specimen.
In the actual experiment, it is Z i=1000 mm, and Z 0=2000 mm.
G of expression (3) is a horizontal elasticity coefficient, and KⅢ is obtained at once by measuring of maximum size Y max of
the caustic image. KⅠ, and KⅡ are calculated everything from an isochromatic fringe and KⅢ from the caustic image.
KⅡ
2
= (cot 2 θ m ±
3
KⅠ
cot
2θ m +
2
3
)
4
N m (2πrm )1 / 2
KⅠ =
⋅⋅⋅⋅


αt[(sin θ m + 2 A cos θ m ) 2 + A 2 sin 2 θ m ]1 / 2 

AN m (2πrm )1 / 2

KⅡ＝
αt[(sin θ m + 2 A cos θ m ) 2 + A 2 sin 2 θ m ]1 / 2 
K
Ⅲ
= 0 . 5986
GY
3 / 2
max
(1 )
⋅ ⋅ ⋅ ⋅ (2)
⋅⋅⋅⋅
Z oλ1/2
(1)
(2)
(3)
(3)
Table 1 Property of epoxy resin by ultrasonic inspection
Mixture ratio
100:25
100:30
100:35
100:40
Modulus of longitudinal elasticity E (GPa)
3.93
4.06
4.13
4.25
Modulus of transverse elasticity G (GPa)
1.45
1.48
1.51
1.55
Table 2 Epoxy resin mixture ratio and relation of bond interface radius
Araldaite: Hardner
Inner Phase
Outer Phase
G
in
/G
out
R
in
[mm]
R
out
[mm]
in
/R
Type A
100：40
100：30
1.05
12.5
25
0.5
Type B
100：35
100：30
1.02
12.5
25
0.5
Type C
100：30
100：30
1.00
12.5
25
0.5
Type D
100：25
100：30
0.98
12.5
25
0.5
Fixed
Free
200
100
Outer phase (GⅠ)
50
Inner phase (GⅡ)
Din
Dout
A
Rin
Rout
A
Crack
out
Length 100mm
Width 0.15mm
Fig.1 Shape of specimen with crack
No.1 No.2 ･･････････････････ No.11
Load 2.12.N・m
Fixed
1255
5
Temperature (℃)
Load
All slice thickness 5mm
Slow down
(-3℃/h)
65
23
χ
0
L
2
35
15
7
Fig.3 Stress freezing cycle
Fig.2 Slice specimen
Test specimen
Balancer
Free
Fixed
Load
Fig.4 Static torsion tester
Slice No.1
Slice No.2
Fig. 5 Isochromatic fringe pattern and caustics pattern (Type A)
Slice No.3
39
Time (h)
Slice No.1
Slice No.3
Slice No.2
Fig. 6 Isochromatic fringe pattern and caustics pattern (Type B)
Slice No.1
Slice No.2
Slice No.1
Slice No.2
Fig. 7 Isochromatic fringe pattern (Type C)
Fig. 8 Isochromatic fringe pattern (Type D)
Ymax
Y
r0
X
Fig.9 Theoritical caustic pattern for mode Ⅲ
r0
0
Crack
Fig.10 Sketch of caustic pattern for mode Ⅲ
Test specimen and experiment method
Mixture ratio of the hardner for the epxy resin are 100:25,100:30,100:35,100:40.
Modulus of longitudinal elasticity of these specimen are E (GPa) 3.93, 4.06, 4.13 and 4.25. The modulus of transverse
elasticity G (GPa) 1.45, 1.48, 1.51 and 1.55.
In this experiment, the Young's modulus of epoxy resin was changed as a dissimilar material was used.
It was used that the Young's modulus changed by changing the mixture ratio of the quantity of the curing agent with epoxy
resin. To see the influence given to the interface, each mixture ratio is four kinds.
Table 1 is the mechanical property of the epoxy resin by ultrasonic inspection.
The HARDNER is the curing agent of the epoxy resin.
Moreover, each mixing ratio, the outside diameter of a double column, and the inside diameter size are shown in Table 2.
A crack axially was inserted on the surface of the test specimen.
The test machine was processed by using epoxy resin to see the influence given to K value of an interfacial position like the
column of outside diameter D = 50 mm, length L = 20 mm, joint boundary radius R in =25 mm, 30mm, 35mm, and 40mm, and
the crack was inserted width W =0.15 mm, depth D =5 mm, and length L =100 mm in the direction of the axis with the miller
cutter of 0.15mm in the thickness of the blade.
The test specimen shape is shown and how to take the slice is shown in Figure 2 and Figure 1.
The specimens was sliced in a vertical direction axially.
After both sides of the slice splinter had been polished with the grinding paper, these were smoothly finished up with the
grinding liquid to obtain a plain image.
An optical distance of incidence light changes by the transformation on the side and the change in the refractive index, and, as
a result, the image on the screen is formed.
Therefore, the experiment order gives the annealing processing by the first experimenting on optical elasticity, and using the
same test specimen afterwards.
Three dimensional stress frozen method becomes of the state of the rubber elasticity by using the characteristic of epoxy
resin in general at a usual temperature of about 120℃ or more.
The strain caused in the rubber elasticity region is fixed as it is the time when the temperature is made to descend up to the
room temperature more than this temperature like the load, and the transformation is keeping on the original state even if the
load is detached.
This is the reason that this character was used.
That is, to apply the caustics method to the slice splinter of three dimensional model to which the stress is frozen; The freezing
stress was opened the annealing doing when the stress froze the slice splinter without the surface transformation finished up
and almost same cycle.
The specimen recovers in the state before the frozen slice, and the surface transformation is caused.
The sliced specimen takes a picture of the image with a reflection type caustic image formation device after the transformation
corresponding to the torque at the load is caused in the slice splinter.
Figure 3 shows the stress frozen cycle.
The static torsion examination tester was made for the experimental device.
It is a load examination machine which fixes the left side so that the test specimen should not receive a stress axially, makes
free axially right, rotates only on the other hand, and adds the load by a force couple.
The torques put on the examination machine were united with all 2.12N・m.
It was assumed the screw stop of six places with the examination machine and the model.
Figure 4 shows the rough sketch of the static twist examination machine.
0.6
0.5
Stress intensity factor
K Ⅲ(MPa.mm1/2 )
Stress intensity factor
K Ⅰ, K Ⅱ（MPa.mm1/2 ）
0.6
KⅠ
KⅡ
0.4
0.3
0.2
0.1
0.5
0.4
0.3
0.2
0.1
0
0
0
0.2
0.4
0.6
0.8
1
Dimensionless distance (x/L)
Fig.11 Relations between KⅢdimensionless distance
(Type B)
0
0.2
0.4
0.6
0.8
Dimensionless distance (x/L)
Fig.12 Relations between KⅠ,KⅡ and
dimensionless distance (Type B)
1
0.6
Stress intensity factor
K Ⅰ,K Ⅱ(MPa.mm1/2 )
Stress intensity factor
K Ⅲ(MPa.mm1/2 )
0.6
0.5
0.4
0.3
0.2
0.1
0.5
KⅠ
KⅡ
0.4
0.3
0.2
0.1
0
0
0
0.2
0.4
0.6
0.8
0
1
Ｄｉｍｅｎｓｉｏｎless ｄｉｓｔａｎｃｅ (x/L)
0.4
0.6
0.8
1
Dimensionless distance (x/L)
Fig.13 Relations between KⅢ and dimensionless distance
(Type B)
Fig.14 Relations between KⅠ,KⅡ and
dimensionless distance (Type C )
0.6
0.6
0.5
Stess intensity factor
K Ⅰ,K Ⅱ(MPa.mm1/2 )
Stress intensity factor
K Ⅲ (Mpa.mm1/2 )
0.2
0.4
0.3
0.2
0.1
0.5
KⅠ
KⅡ
0.4
0.3
0.2
0.1
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Dimensionless distance (x/L)
Dimensionless distance (x/L)
Fig.15 Relations between KⅢ and dimensionless distance
( Type C)
Fig.16 Relations between KⅠ,KⅡ and
dimensionless distance ( Type D)
Stress intensity factor
K Ⅲ(MPa.mm1/2 )
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Dimensionless distance (x/L)
Fig.17 Relations between KⅢ and dimensionless distance
( Type D)
Experiment result
Figure 5 and Figure 6 are the sliced model after the stress of a double column freezes of the dissimilar material, and are the
potoelasticity methods of the center part of the crack and are an example of the image of the caustics method.
In slice 1, it is a color line stripes photograph, and slice 2 is an example of the expansion image in the crack part as it depends
on the photo elasticity method.
Slice 3 is a image of caustic image KⅢ in the crack part in slice 1 which gives the annealing processing again after an
isochromatic fringe image is taken, opens the freezing stress, and takes a picture by the reflection type caustics method.
One of the example of an isochromatic fringe photograph of type C and D was similarly shown in Figure 7 and in Figure 8.
Figure 9 is a theoretical image of KⅢ, and Figure 10 is a sketch chart of the caustic image of KⅢ obtained because of the
experiment.
Each graph is the result of the sliced specimen that the entire crack length is cut in 5mm of thickness, was experimented and
the stress intensity factor were calculated from theresult of experiment. As the definition of the stress intensity factor, KⅠ is
mode Ⅰ(opening type), KⅡ is mode Ⅱ (shearing type) and KⅢ is mode Ⅲ ( tearing type).
Figure 11 is an obtained result from the Type A that is the relation of the dimensionless distance like the distance x/L in the
graph.
Ttype A test specimen and the relation of K were shown from the caustics methods. It has been understood that the
tendency of K Ⅲ became of small according to the center of the crack.
About K Ⅰand K Ⅱ, these were not possible to confirm the isochromatic fringe image because the image was very small.
Figure 12 and Figure 13 are the result of type B was shown.
K Ⅰand K Ⅱare small, but K Ⅱ is very small, so it is thought that the K Ⅱ has the changeless tendency.
K Ⅲ shows the tendency like type A, and the vicinity of the center is too changeless.
Figure 14 and Figure 15 are the one that the tendency to type C was shown.
K Ⅰand K Ⅱ are small, but K Ⅰ is slightly big. It has been understood that K Ⅰis especially small in case of comparing with the
K Ⅰ of Figure 12.
The tendency to same value K Ⅲ was almost the same tendency as type A and type B.
Figure 16 and Figure 17 are like type D was shown in similar.
It is understood that the tendency of KⅠand KⅡ is stronger than that of type B and type C in type D and the tendency is
remarkably more different.
The tendency to KⅢ was almost the same tendency as type A, type B, and type C.
On the other hand, the change in KⅠ by changing G in/ Gout showed the tendency which decreased when KⅠ was large, and
the ratio of the modulus of longitudinal elasticity was large when it was large, and the ratio of the modulus of longitudinal
elasticity was small.
It has been understood that the influence of KⅠ is large from this in the point of strength in a double column.
In type B, type C, and type D, the high degree of the isochromatic fringe number in the crack point can be confirmed, and it
has been understood that it operates and same stress influences to the crack by non-symmetry .In a double column, the
transformation outside respect was confirmed to the crack point since the annealing was given to the test specimen to which
the optical elasticity experiment was finished, and the obtained caustic image was corresponding to the theoretical image of
modeⅢtransformation.
The difference compared with the rigidity of the material which composes a double column understands and it has been
understood that the influence by the difference of an inside, outside rigidity is large of influence by causing for the
transformation style of the crack of the joint field side neighborhood in mode Ⅰand modeⅡ.
Conclusion
In this research, a double column that was made of the epoxy resin was produced, the surface axially crack was inserted on
the surface of the specimen, the static torsion load was put, and it was experimented.
The influence which caused it for the crack point was examined, therefore the following knowledge was obtain –ed by the
diameter ratio of inside and outside compared with the horizontal elasticity coefficient using the optical elasticity method
together with the caustics methods.
1.
The influence of the value of KⅠ by the change compared with the modulus of longitudinal elasticity is large in a double column
which has the surface axially crack, and the value of KⅡ has understood the influence by the change compared with the
modulus of longitudinal elasticity is small.
2.
KⅢ is large, and is small in both ends in distance x / L = 0.5 of making to dimensionless (axis center).
The tendency as which any type was the same was shown.
In a double column which has the surface axially crack, it has been understood the value of KⅠ is to be able to control by
putting the material with a different elasticity rate internally.
3.
The influence of the value of KⅠ and KⅡ by changing Rin /Rout compared with an inside and outside radius is large in a double
column which has the surface axially crack, and it has been understood that the influence on KⅢ is small.
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