- Research Article
- Open Access

# Cyclic Biaxial Stress Measurement Method Using the Grain Growth Direction in Electrodeposited Copper Foil

- Yuichi Ono
^{1}Email author, - Cheng Li
^{1}and - Daisuke Hino
^{1}

**2010**:928216

https://doi.org/10.1155/2010/928216

© The Author(s) 2010

**Received:**28 December 2009**Accepted:**11 April 2010**Published:**17 May 2010

## Abstract

A method that uses grain growth direction in electrodeposited copper foil to measure cyclic biaxial stress is examined in this paper. The grain growth direction is measured by image processing software after a cyclic loading test for various biaxial stress ratios is carried out. Since the grain growth occurs in two directions and its directions correspond closely with the direction of maximum shearing stress when the biaxial stress ratio is negative, the principal stress can be measured using Mohr's stress circle. On the other hand, when the biaxial stress ratio is positive, above-mentioned feature does not occur. Therefore, the first principal stress can be measured based on the grain growth density. The number of grains necessary to measure the biaxial stress is estimated by a statistical approach.

## Keywords

- Principal Stress
- Growth Direction
- Circular Hole
- Copper Foil
- Stainless Steel Plate

## 1. Introduction

The copper electroplating method is used to measure cyclic stress that causes metal fatigue [1–3]. If copper foil adhered to a machine element is subjected to repeated loads, grain growth occurs in the copper foil. Since the grain growth density is controlled by the maximum shearing stress and the number of cycles, the maximum shearing stress can be measured based on the grain growth density in the prescribed number of cycles [4]. This method has the advantage of detecting stress in microscopic regions like the stress concentration region. Moreover, this method can be easily applied to rotating machines and machine elements in sealed casings, since it does not need an output line like an electrical resistance strain gauge.

Since the principal stresses that are important for evaluating metal fatigue cannot be detected by this method, a new method using copper foil with circular holes has been developed [4, 5]. However, this new method is somewhat complex, because the grain growth length at hole edges as well as the grain growth density in the copper foil must be measured. This also means that two kinds of copper foils (foil with and without circular holes) are necessary for the principal stress measurement.

From the above viewpoint, we examined the principal stress measurement using only one piece of foil without circular holes. To do this, we focused on the grain growth direction, since the growth direction of an individual grain is expected to correspond closely with the direction of maximum shearing stress. The principal stress measurement becomes possible using this feature as described in the next chapter. First, we proposed the principal stress measurement method based on the grain growth direction. Second, we investigated the relative frequency distribution of the grain growth direction for various biaxial stress conditions. Finally, the number of grains necessary to measure the principal stress was estimated by regarding the relative frequency distribution as the normal distribution.

## 2. Biaxial Stress Measurement Method

*C*in addition to in the case of . Namely, determining the sign of biaxial stress ratio is required to measure the first principal stress, since the basic equation to obtain the first principal stress is different. Figure 2 illustrates grain growth in copper. The grain growth caused by cyclic stress in copper is considered to be a kind of thermal recrystallization [6]. The dislocation movement caused by mechanical rather than thermal energy results in the grain growth. Since the shearing stress is responsible for the dislocation movement, the grain growth is considered to be controlled by the shearing stress. occurs in the direction that divides the and directions in two when as shown in Figure 1(a). Therefore, the plane in which the shearing stress becomes the maximum is an -plane, as shown in Figure 2(a). On the other hand, occurs in the direction that divides the and directions in two when as shown in Figure 1(b). So, the plane in which the shearing stress becomes the maximum is an -plane, as shown in Figure 2(b). In addition, there are two directions where the shearing stress reaches the maximum, and the interval between the two is 9 , as understood from Mohr's stress circle shown in Figure 1. As a result, the grain growth observed by electrochemical polishing occurred in the two directions and the interval between the two is 9 when . On the other hand, the grains do not show the above-mentioned features on the -plane when , since grains grow in the thickness direction. Therefore, the sign of biaxial stress ratio can be determined by using these characteristics.

*σ*

_{ x }and a shearing stress as shown in Figure 3. Mohr's stress circle of this small element is shown in Figure 4. Since the first principal stress and the second principal stress have opposite signs, the biaxial stress ratio becomes negative. The angle between the axial direction and the direction of principal stress is expressed as

## 3. Experimental Procedures

### 3.1. Test Specimen and Testing Machine

A copper foil was obtained as follows. A stainless steel plate (200 mm 100 mm 1 mm) was electroplated with copper sulfate solution [1]. Since the stainless steel plate is polished by buffing before plating, the deposited layer can easily strip from the stainless steel plate. This deposited layer is called a copper foil. All subsequent experiments were carried out by cutting this single foil to small pieces. The copper foil was about 20 m thick and the initial grain size was about 1 m [4]. This grain size is considerably smaller than the grown grain size.

*C*measured by a strain gauge rosette was 1.0. Images of grown grains were captured using a personal computer from a digital camera installed on an optical microscope (200x magnification), and we used image processing software to measure the grain growth direction.

Mechanical properties of Ti-6Al-4V alloy.

Proof stress [MPa] | Tensile strength [MPa] | Elongation [%] |
---|---|---|

946 | 1033 | 15.2 |

Biaxial stress ratio obtained by a strain gauge rosette.

Machine type | Schenck | Servo-hydraulic | ||||
---|---|---|---|---|---|---|

| 30 | 45 | 60 | 75 | 90 | — |

| 0.16 | 0.33 | 0.52 | 0.72 | 1.0 | 1.0 |

### 3.2. Experimental Procedure

## 4. Results and Discussion

### 4.1. Statistical Distribution of the Grain Growth Direction

### 4.2. Biaxial Stress Measurement Using the Grain Growth Direction

Biaxial stress ratio in each experimental condition.

| 0.16 | 0.33 | 0.52 | 0.72 | 1.0 | 1.0 |
---|---|---|---|---|---|---|

[%] | 3.11 | 4.13 | 4.50 | 4.78 | 4.85 | 1.99 |

| 99.3 | 91.9 | 94.6 | 96.1 | 94.7 | 53.1 |

[rad] | 2.99 | 2.69 | 2.48 | 2.22 | 1.88 | — |

[%] | 2.98 | 2.82 | 2.96 | 3.09 | 3.03 | — |

### 4.3. Estimation of the Number of Grains Necessary to Measure the Biaxial Stress

Since this method has the advantage of enabling measurements of the stress in a microscopic region, it is preferable to reduce the number of measured grains as much as possible. The number of grains necessary to determine the sign of biaxial stress ratio is only about 50, since the feature shown in Figure 9 is the same within the range from 50 to 150. Therefore, we pay attention to the number of grains necessary to measure the biaxial stress ratio. Namely, necessary to keep prescribed accuracy in the stress measurement can be statistically estimated.

^{2}, even when a condition requires many samples ( , %). Therefore, this new method can detect the cyclic biaxial stress in a small area using only one piece of copper foil.

Number of necessary grains (Confidence value: 95%).

| 0.16 | 0.33 | 0.52 | 0.72 | 1.0 | |
---|---|---|---|---|---|---|

[%] | 3 | 330 | 182 | 71 | 48 | 44 |

5 | 117 | 66 | 26 | 18 | 16 | |

7 | 61 | 34 | 13 | 9 | 8 |

## 5. Conclusions

We examined a method that uses the growth direction of grains in copper foil to measure cyclic biaxial stress. The number of grains necessary to measure the biaxial stress was also estimated statistically.

- (1)
When the biaxial stress ratio is negative, peaks of the relative frequency distribution of the grain growth direction corresponded well with the direction of maximum shearing stress, and the interval from one peak to another peak was almost 9 .

- (2)
The above-mentioned features are not recognized when the biaxial stress ratio is positive. Therefore, the sign of biaxial stress ratio is determined by using these features.

- (3)
The principal stress was obtained with Mohr's stress circle and the peak of the sin curve obtained by approximating the relative frequency distribution when the biaxial stress ratio is negative.

- (4)
The grain growth direction within the range of 4 from one peak of the distribution followed the normal distribution. Therefore, the number of grains necessary for the principal stress measurement could be estimated to the demanded accuracy.

- (5)
The first principal stress obtained by this new method agreed well with the result obtained by a strain gauge rosette. The area necessary for the principal stress measurement was only 5 mm

^{2}. - (6)
Since this method can measure the principal stress with only one piece of foil, it is more efficient than conventional methods.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.