AN APPLICATION OF IMAGE-PROCESSING TO STRESS
MEASUREMENT BY COPPER PLATING FOIL
(ON THE EFFECT OF FREQUENCY, STRESS RATIO AND STRESS
WAVEFORM)
M. Sugiura1，R. Ozeki2 and M. Seika3
1.Professor, Department of Mechanical Engineering, Daido Institute of Technology,
Nagoya,Japan, [email protected]
2.Graduate Student, Department of Mechanical Engineering, Daido Institute of
Technology, Nagoya, Japan
3.Emeritus Professor, Nagoya University, Nagoya, Japan
ABSTRACT
Strain gages made of copper plating foil are devised for measuring the elastic surface stress of machine parts in operation.
The elastic stress is measured by observing slip bands in the bonded foil. Calibration studied by cyclic tension test with
various frequencies, various stress ratios and different stress waveforms is performed on round steel bars with plating foil. It
is verified that the relation between the threshold stress on the first appearance of slip bands and the number of cycles is
not affected by the stress ratios and stress waveforms but rather by the frequencies. Using a computer image-processing
system, the density of slip bands in a microscopic image of bonded foil is analyzed automatically and quantitatively. The
relation between the stress amplitude and the density of slip bands for a constant number of cycles is examined under
various frequencies, various stress ratios and different stress waveforms. In order to examine the accuracy of this method,
the peak stresses in grooved shafts under cyclic tension test are obtained using the present method. The results are
compared with those derived from the design formulas by Roark and Young. It is confirmed that this method provides
accurate calibration value for measuring cyclic stress.
Introduction
Strain gages made of copper plating foil were devised for measuring the elastic surface stress of machine parts in operation
[1]. When copper plating foil is attached on the surface of a specimen subjected to repeated load, grown grain or slip bands
occur in the foil resulting from the repeated strain. These phenomena are based on the repeated maximum shearing stress
acting on the specimen, and the elastic surface stress of the specimen can be measured by observing grown grain or slip
bands. The slip bands are caused by fatigue of the grown grains and can usually be observed by an optical microscope at a
magnification of 100 times. The density of slip bands in a given microscopic field depends on the amplitude of cyclic stress
and the number of cycles. Hence, for a specified number of cycles, the stress in a specimen can be determined from the
density of slip bands in the bonded foil. The density of slip bands in a photomicrograph has been evaluated by the ratio of
area of slip band to the whole area. In this paper, calibration studied by cyclic tension with various frequencies, various
stress ratios and different stress waveforms is performed on round steel bars with copper plating foil. The relation between
the threshold stress for the first appearance of slip bands and the number of cycles is examined under various frequencies,
various stress ratios and different stress waveforms. After reviewing the techniques proposed in previous papers [2-4], a
new method is developed for measuring the density of slip bands in the bonded foil. This method directly processes a
microscopic image of slip bands using an image-processing system with a personal computer, and the density of slip bands
is analyzed automatically and quantitatively. In order to examine the accuracy of this method, the peak stresses in grooved
shafts under cyclic tension test are obtained using the present method. The results are compared with those derived from
the design formulas by Roark and Young [5].
Experimental Procedure
Testing Machine and Test Specimen
A hydraulic servo-type cyclic tension fatigue testing machine (98 kN) was used for cyclic tension tests with a constant load
amplitude. Using drawn of carbon steel rods annealed after heating at 900℃ for one hour, the tapered calibration
specimens shown in Figure 1 and the grooved specimens shown in Figure 2 and Table1 were made.
φ25
d
φ38
φ14
φ10.5
S
16
39
Copper Plating Foil
210
16
ρ
D=18
h
.5
R1
5
C1
φ25
.
R1
d=11.5
Figure 1. Tapered calibration specimen for tension (Dimensions in mm)
39
210
Figure 2. Grooved specimen
(Dimensions in mm)
Table 1. Dimensions of grooved specimen
Specimen
(a)
(b)
(c)
(d)
D
(mm)
18
18
18
18
d
(mm)
11.5
11.5
11.5
11.5
ρ
(mm)
4
6
8
10
h
(mm)
3.25
3.25
3.25
3.25
D/d
ρ/d
h/ρ
1.57
1.57
1.57
1.57
0.35
0.52
0.70
0.87
0.81
0.54
0.41
0.33
Copper Plating Foil and Its Adhesion
A stainless-steel plate polished by buffing was coated with a copper plating under the conditions shown in Table 2, and a
sheet of plating foil about 10 µm thick was made by stripping the deposited layer from the stainless-steel plate. The
copper plating foil was cut into a number of small rectangular pieces (about 3 mm ×40 mm) and squares (about 1 mm×1
mm, 2 mm×2mm) to make foil gages. Four pieces of rectangular foil were used for the calibration specimen and bonded to
four symmetrical sides of the tapered part (Fig. 1), while four pieces of square foil were used for the grooved specimen and
bonded to four symmetrical positions on the periphery along the root of the groove. The finally deposited surface of the
plating foil was selected as the bond surface of the foil gage. An adhesive based on a cyanoacrylate resin (CC-33A) was
used to attach the foil gage, and the adhesive layer was about 2 µm in thickness.
Table 2. Composition of plating solution and plating conditions
Copper sulfate CuSO4・5H2O
250 g
Sulfuric acid H2SO4
80 g
Distilled water H2O
1L
Bath temperature
23 ℃
Deposited time
20 min
Current density
300 A/m2
Bath voltage
0.5 V
Measurement of Threshold Stress
The threshold stress σp for the first appearance of slip bands in the foil gage was measured using the tapered calibration
specimen with rectangular foil under cyclic tension with various frequencies, various stress ratios and different stress
waveform. Slip bands were observed with an optical microscope at a magnification of 100 times. When slip bands in the
bonded foil began to appear at the distance S (Fig.1) under a constant load amplitude W and a specified number of cycles
N, the threshold stress σp in cyclic tension was calculated by the following formulas:
σp =
4W
πd 2
(1)
where d 〔mm〕was the diameter at the position. The distance S〔mm〕was determined as the average of the measured
values in the four pieces of bonded foil. Referring to Figure 3, the stress ratio R was represented by the following formulas:
R=
σ min
σ max
(2)
Figure 3. Schematic diagram of cyclic tension
The relation between the threshold stress σp and the number of cycles N for various frequencies ( f =10 ～ 40 Hz), various
stress ratios (R = 0 ～ 0.3) and different stress waveforms (sine and triangle wave) were obtained.
Image-processing System
Figure 4 shows the configuration of the image-processing system. A microscopic image of slip bands was transmitted from
a CCD camera attached to the optical microscope (100 times) as images signal. This picture was A/D converted and stored
in an image memory. One pixel in the image memory consisted of 256-graduated brightness. The data stored in the image
memory were read and processed by personal computer, and the results were forwarded to the image memory. The images
from the TV camera and the image memory were displayed by a TV monitor. The software used for the image-processing
was「Win Roof」by Mitani Co., Ltd (Japan). The density of slip bands in a microscopic image was obtained by the ratio of the
region of slip bands (f pixels), to the whole region of the image (F pixels), which is denoted by the slip-band density r (%)
as :
⎛ f ⎞
r = ⎜ ⎟ × 100(% )
⎝F⎠
(3)
Image input
Processing
range setting
Change to
black and white
Extraction
Elimination of
minute particle
Measurement
Figure 4. Configuration of image processing system
For the foil bonded to the tapered calibration specimen, the whole region of the original image, F, was equal to 212×280
pixels (about 0.51 mm×0.67 mm); whereas the region of slip bands, f, in the original image was determined after eliminating
minute regions under 50 square pixels, which might be made from minute flaws and the like in the foil occupying by the
whole region of the original image.
Experimental Results and Discussion
Threshold Stress σp
Threshold Stress σp (MPa)
Figure 5 shows the relation between the threshold stress σp for the first appearance of slip bands and the number of cycles
N obtained by cyclic tension tests with various frequencies (f =10～40 Hz), various stress ratios (R = 0～0.3) and different
stress waveforms (sine and triangle wave). From this figure, it is found within the limits of this experiment that the threshold
stress σp is not affected by the stress ratios and stress waveforms but rather by the frequencies. When the frequency
decreases, the slip band reading occurs, and threshold stress σp decreases and can be measured under low stress.
The σp- N curve shown in Figure 5 can be applied to the copper plating foil gage as a calibration value based on the
threshold stress for the first appearance of slip bands.
As an example, Figure 8 shows the change of appearance of slip bands due to the difference of stress amplitude σ which
was taken from the calibration specimen tested for N = 4 ×106 , R = 0.1, f = 40 Hz and sine wave. The density of slip bands
is sparser with a decrease in σ. The slip bands in Figure 8 ④ correspond to those at the threshold stress σp.
40Hz
350
●＝ 40Hz (R=0 ～ 0.2)
○＝ 30Hz (R=0 ～ 0.3)
■＝ 20Hz (R=0 ～ 0.2)
□＝ 10Hz (R=0 ～ 0.2)
30Hz
300
10Hz
20Hz
2
4
6
Number of cycles N
Figure 5.
sine,triangle wave
sine,triangle wave
sine,triangle wave
sine,triangle wave
8
10
[×106]
Threshold stress σp versus number of cycles N for various frequencies, various stress ratios and different
stress waveforms
Measurement of Slip Band Density
The rectangular foil bonded to the tapered calibration specimens, which were tested for N=4 ×106, R=0.1, different stress
waveform (sine and triangle wave) and various frequencies (f =10～40 Hz), were used to measure the density of slip bands.
Four positions in the bonded foil were selected at about equal intervals along the axis of the calibration specimen, and the
microscopic images of slip bands at each position were analyzed by the image-processing system to obtain the slip band
6
density r (%). As an example, Figure 8 shows a photomicrograph of slip bands and their binary images at N= 4 ×10 , R =
0.1, f=40 Hz and sine wave, where theσvalue at r =1.58 % is the threshold stress σp . On the other hand, using the method I
stated above, figure 6 shows the σ- r curves for the change in each frequency by a constant number of cycles (Ｎ＝4
6
×10 ) and stress ratios (R=0.1). Even if the slip band density is the same as in this figure, the stress amplitude σ decreases
as the frequency is lower. It is thought that a slip band reading becomes slippery if the frequencies become low. In addition,
Figure 7 shows the relation between stress amplitudeσand slip band density r for a constant number of cycles (N = 3 ×106 )
and frequencies (f=30 Hz), and investigated about the influence of the stress ratio and the stress wave pattern in detail.
Within the limits of this experiment the σ-r relations for various stress ratios and stress waveforms can be considered to
approximately agree, and the σ-r curve is not taken to be affected by the stress ratio R and stress waveforms. Therefore,
the σ-r curve shown in Figure 6 and 7 can be applied to the copper plating foil gage as a calibration value.
Stress amplitude σ (MPa)
450
N=4×10 6
R=0.1
400
◇10Hz
◆10Hz
△20Hz
▲20Hz
□30Hz
■30Hz
○40Hz
●40Hz
350
300
0
10
20
30
sine wave
triangle wave
sine wave
triangle wave
sine wave
triangle wave
sine wave
triangle wave
40
slip band density r(%)
Stress amplitude
σ (MPa)
Figure 6. Slip band density versus stress amplitude σ for various frequencies f and different stress waveforms
450
400
N=3 × 10 6
350
● R=0, sine wave
▲ R=0.2, sine wave
■ R=0.3, sine wave
○ R=0, triangle wave
△ R=0.2, triangle wave
□ R=0.3, triangle wave
300
250
0
10
20
Slip band density
r
30
(%)
Figure 7. Slip band density r versus stress amplitude σ for various stress ratios R and different stress waveforms
Measurement of Peak Stresses in Grooved Shafts
The peak stresses in grooved shafts under tension were measured with specimens shown in Figure 2 and Table 1.
6
Cyclic tension tests under N = 3×10 , R = 0 and sine wave were performed on specimens with four pieces of square foil
bonded to the root of the groove. Using microscope images of slip bands in the square foil and the image-processing
system, the slip band density r (%) was obtained by Equation(3), where the values of F and f were evaluated by the same
procedure as the rectangular foil bonded to the tapered calibration specimen. The value of r was determined as the average
of the measured values in the four pieces of bonded foil. Figure 9 shows a photomicrograph of slip bands at the root of the
groove of specimen(c) and its binary image at R = 0 and a load amplitude W = 29.38 kN. The slip band density r at the root
of the groove is 17.0 %, and so the peak stress σmax at the root of the groove is presumed to be 381 MPa from the σ- r curve
shown in Figure 7. When the nominal stress σ0 is represented by the tensile stress in a round bar with a diameter d
subjected to a tensile load W, which is written as:
_
σ0 =
4W
πd 2
(4)
The stress-concentration factor α is obtained by the formula:
α=
σ max
σ0
(5)
Substituting W = 29.38 kN and d = 11.5 mm for Equation(4), the value of σ0 is 283 MPa, and the value of α is equal to
1.35. The values of α for the other grooved specimens can also be obtained in the same way.
The measured values of α of the grooved shafts under various curvatures ρ are shown in Figure 10. The results are in
good agreement with those derived from the design formulas by Roark ＆ Young［5］.
①σ=449.8 MPa
③σ=376.0 MPa
r =30.31 %
r =13.11 %
②σ=408.6 MPa
r =25.95 %
④σp=349.6 MPa
r =1.58 %
6
Figure 8. Appearance of slip band density at each point (N=4×10 ，R=0.1, f=40 Hz, sine wave)
σmax = 381 MPa
r = 17.0 %
Stress concentration factorα
Figure 9. Original image and extracted image after image processing of grooved specimen(c)
1.7
● Measured value
○ R.J.Roark et al
D/ｄ=1.57
1.6
1.5
1.4
1.3
0.2
0.4
0.6
0.8
1
Radius of curvature versus smaller diameter
Figure 10. Stress concentration factor α of grooved specimens under tension
Conclusions
A new method for automatic measurement of the density of slip bands in copper plating foil was successfully developed
using an image-processing system with a personal computer. The slip band densities in plating foil bounded to round steel
bars under cyclic tension with various frequencies, various stress ratios and different stress waveforms were accurately
obtained, and it was verified that the relation between the threshold stress for the first appearance of slip bands and the
number of cycles is not affected by stress ratios and stress waveforms but rather by the frequencies. The peak stresses in
grooved shafts subjected to cyclic tension under different curvatures were measured using the proposed method, and
accurate results were obtained. It was found that an application of computer image-processing to the observation of slip
bands in copper plating foil is very useful for improving practical usage of copper plating foil strain gages.
References
1. H. Okubo and K. Hosono., Memoirs of the Faculty of Eng, Nagoya Univ. 14, 77-82 (1962).
2. A. Kato, “ Stress Measurements by Copper Electroplating Aided by a Personal Computer”, Experimental Mechanics, 27,
132-137 (1987).
3. M. Sugiura and M.Seika.,” An Application of Computer Image－processing and Film Replica Technique to the Copper
Electroplating Method of Stress Analysis”, Experimental Techniques, 18, 33-36 (1994).
4. M. Sugiura, A. Arakawa,Y. Aoyama and M. Seika., Proceeding of International Conference on Advanced Technology in
Experimental Mechanics, ATEM’03, Nagoya, Japan, CD-ROM (2003).
th
5. Roark, R.J ＆ W.C. Young ., “ Formulas for stress and strain”, 5 ed., New York, McGraw-Hill, 599 (1975).