# Analysis of the fractal dimension of grain boundaries of AA7050 aluminum alloys and its relationship to fracture toughness

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## Abstract

Quantitative analysis of the grain boundaries in partially recrystallized microstructures of heat-treated 7050 aluminum alloys has been performed. Fractal dimensions of the extracted grain boundaries were calculated by box-counting method. Five different types of tear-tested materials rolled in different processes each at two orientations of 0° and 90° were studied. Efforts were made to connect the fractal dimensions of grain boundaries in the crack propagation direction to the fracture toughness (unit propagation energy, UPE, in tear test). The results show that there is a linear correlation between UPE and the fractal dimensions of the grain boundaries along the crack propagation direction for both 0° and 90° samples. The dependence corresponds well with the observation of transition from intergranular fracture to transgranular fracture with the increase of UPE. Quantitative analysis has also been performed on the micrographs to estimate the degree of recrystallization and the grain size in crack growth direction. No correlation between the fraction of recrystallized grains and the UPE could be detected.

## Keywords

Fracture Toughness Fractal Dimension Intergranular Fracture Intergranular Crack Crack Growth Behavior## Introduction

The concept of fractal geometry introduced by Mandelbrot [1] has been applied extensively to describe complex and irregular patterns in natural science. In the science of metallic materials fractal analysis has been conducted to quantify the shape of various microstructural features, for instance dislocation patterns [2, 3, 4], powder particles [2], dendritic structures [2], slip lines [2], precipitates [2, 5, 6, 7], void structures [2, 7], fracture surfaces [8, 9], and grain boundaries [10, 11, 12, 13].

In polycrystalline metallic materials the structure of grain boundaries plays an important role in defining the mechanical properties [14]. Since grain boundaries are the preferred sites for crack nucleation and growth, intergranular fracture is the main origin of brittleness of engineering materials [15]. The prevention of intergranular fracture for materials at high temperature applications can be obtained by strengthening the grain boundaries [16] by modification of the grain boundary morphology. It has been reported that cold work, hot work, or heat treatments can generate serrated grain boundaries in alloys, which improve the high temperature strength of the materials, especially the creep rupture and high temperature fatigue properties, compared to the alloys with straight grain boundaries [10, 17, 18, 19, 20]. The effect of grain boundary serration on creep properties of metallic materials has been investigated for several systems; austenitic AISI 316 stainless steel [11, 21] and Ni-based and Co-based superalloys [18, 22, 23, 24, 25]. In austenitic heat resisting steels formation of rugged grain boundaries by grain boundary reaction precipitates has been reported to improve the high temperature low cycle fatigue strength of [20]. The improved high temperature resistance of these materials is mainly attributed to reduction of grain boundary sliding [26], decrease in stress concentration at grain boundary triple points [17], crack path deflection which hinders the crack growth process [27, 28], and the lengthening of the fracture path [17, 28]. However, no research has been found that surveyed the effect of grain boundary serration on mechanical properties of metallic materials at room temperature.

In this article the structure of serrated grain boundaries of heat-treated 7050 aluminum alloys rolled according to different hot rolling strategies are examined quantitatively. The term “grain boundary” refers to the grain boundary profile between recrystallized and unrecrystallized regions in two dimensional section of the microstructures [29]. Since these alloys are widely used for manufacturing the structural components of aircraft where fracture toughness and damage tolerance properties are important [30, 31], investigating the correlation between the degree of serration of the grain boundaries and fracture toughness of the materials is of scientific and technological significance.

*D*≤ 2), this item is shortly called the fractal dimension of the grain boundary which has been estimated in the scale length range from 1 μm to 1 mm. Various methods are available for measuring the fractal dimension of objects. In the current work the box-counting method [1, 32, 33, 34, 35] has been applied for fractal dimension estimation of grain boundaries, since it is easy and automatically computable for digital images of the microstructures. However, this technique needs proper implementation of image processing to extract grain boundaries. In this method meshes of increasing size cover the digital image whose fractal dimension is to be determined and number of boxes containing at least one pixel of the grain boundary are counted. The procedure starts for a box size of one pixel and is repeated for increasing box sizes, until the largest box fits the whole image. Figure 1 shows schematically the box-counting procedure to obtain the fractal dimension of a grain boundary. The number of boxes containing at least one pixel of the grain boundary,

*N*(

*r*), relates to the size of the boxes,

*r*, following a power-law

*N*

_{0}is a constant. Fractal dimension (

*D*) is determined from the regression slope of the

*ln*(

*N*(

*r*)) versus

*ln*(

*r*).

The purpose of the present study is making a quantitative fractal dimension estimation on the grain boundaries of partially recrystallized 7050 aluminum alloys. The effect of grain boundary serration on fracture toughness of the materials is investigated.

## Experimental procedure

### Materials

In total five different types of AA7050 plates with identical chemical composition were received from our material supplier in the form of tear-tested samples. All plates had received 50 % thickness reduction during the multipass assymetric hot rolling processes and further T74 solution treatment and artificial aging. The heat treatment yielded specimens that are partially recrystallized.

### Image processing of microstructure image

- 1.
Thresholding: This process transforms a gray-scale image with 256 gray levels into a binary one including pixel values of 0 and 1, black and white. The proper threshold level is defined by the user to get the desired grain boundary as a well-defined continuous profile.

- 2.
Impurities and particles removal: To this aim the region filling operation is used. This operation identifies the pixels constituting holes within connected components in the binary image and fills them.

- 3.
Labeling: This operation analyses an input binary image so as to identify all the connected objects in the image. In the current work, in order to preserve every structural detail of the boundaries the 8-connection concept is applied, meaning that a given foreground pixel is considered as part of the same object if it has at least one neighboring foreground pixel to the north, south, east, west, north-east, north-west, south-east or south-west of itself. A group of pixels which are all connected to each other in this way is differentiated from others by giving it a unique label.

- 4.
Canny edge detection on extracted grain: This method produces a one pixel wide edge, while preserving the structural properties of the desired grain.

Since the main crack grows parallel to the S and L directions in 90° and 0° specimens, respectively (see Fig. 2), the fractal dimension of grain boundaries between recrystallized and unrecrystallized regions is characterized along S (90°) and L (0°) directions. For this purpose rectangular masks having aspect ratio of 3 (300 × 100 pixels) were located on the grain boundary, while the long axis of the box was always oriented in the crack propagation direction. Then the part of recrystallized grain boundary fitted in the mask was cut for fractal analysis. In this work 10 partial grain boundaries for each sample were chosen, cropped, and processed following the procedure described above. Box-counting measurements were then performed on the binary images of one pixel wide grain boundaries in the following manner.

### Box counting

To implement the box-counting method to extracted grain boundaries a MATLAB program was developed based on the theoretical considerations discussed earlier. The original size of the images which are cropped from the micrographs according to the mask size is 300 × 100 pixels for the horizontal masks, and 100 pixels × 300 pixels for the vertical masks. Grids of increasing size in the order of \(r = 1, 2, 4 \ldots 2^n\) were used to cover the image.* N* is defined as the smallest integer such that max (size(*I*)) ≤ 2^{ n }, where max(size(*I*)) is the maximum dimension of the input image. In order to fulfill max(size(*I*)) ≤ 2^{ n }, images are padded with zeros to size 2^{9} over each dimension. Meshes of increasing size starting from one pixel cover the digital images and number of boxes containing at least one pixel of the grain boundary are counted. The procedure is repeated for increasing box sizes until the largest box fits the whole image and returns 1 for the box number, in this case the largest box size is 512 pixels. The fractal dimension corresponding to the slope of the ln(*N*(*r*)) versus ln(*r*) plot is calculated by least-squares linear fitting. The program has been validated using test images of Koch curves with known fractal dimension and was found to be accurate. The indicated procedure was followed to estimate fractal dimension of 10 randomly selected grain boundaries for each studied sample. Results show that the correlation coefficients of all fitted lines show a value of more than 0.99. The overall fractal dimension for each sample is averaged over 10 test results.

## Results

Results of box counting and UPE data

Samples | Average FD | FD standard deviation | Average UPE (KJ/m2) | UPE standard deviation |
---|---|---|---|---|

1—90° | 1.032 | 0.012 | 76 | 11.26 |

2— 90° | 1.053 | 0.025 | 130 | 22.51 |

3—90° | 1.094 | 0.027 | 151 | 12.01 |

4—90° | 1.023 | 0.012 | 36 | 7.09 |

5—90° | 1.049 | 0.023 | 119 | 17.03 |

1—0° | 1.019 | 0.010 | 23 | 7.23 |

2— 0° | 1.023 | 0.016 | 60 | 7 |

3—0° | 1.036 | 0.021 | 118 | 13.74 |

4—0° | 1.018 | 0.015 | 21 | 5.03 |

5—0° | 1.027 | 0.012 | 65 | 26.15 |

The UPE data for every sample were averaged over three test results meeting all criteria for valid Kahn tear-tested load-displacement curves and the average values are shown in Table 1. Standard deviations are also reported. Samples 3—90° and 4—0° show the highest and the lowest UPE, respectively. For each rolling process 90° samples have higher UPE values than 0° samples. In the next section, the obtained results will be interpreted to justify the observed a correlation between UPE and fractal dimension of the grain boundaries.

## Discussion

### Effect of fractal dimension of grain boundaries on fracture toughness of the material

### Effect of other microstructural features on fracture toughness of the material

In order to investigate possible relations between the fracture toughness and other attributes of the partially recrystallized microstructure of aluminum alloys, the degree of recrystallization and the recrystallized grain size in crack growth direction on crack plane has been measured for microstructures studied at a magnification of 20× . It has been claimed that these two parameters affect fracture toughness of 7XXX series aluminum alloys [31, 44].

## Conclusions

Quantitative characterization of high angle grain boundaries carried out on Kahn tear-tested specimens has shown that there is a strong correlation between UPE data of Kahn tear test and the fractal dimension of the grain boundaries aligned in crack propagation direction calculated by box-counting tool. Fractographic observations show that with the increase of UPE the transition takes place from intergranular fracture to transgranular fracture. Grain boundaries with low fractal dimension, thus low UPE, possess intergranular dominated fracture. Further increasing the fractal dimension shows sharp increasing of UPE with a slight increase in fractal dimensions due to higher fraction of transgranular fracture. Toward highly irregular grain boundaries the fracture mode appears to be transgranular dominated, thus not dramatically affected by fractal dimension of the grain boundaries any more. There is no clear correlation between the fraction of recrystallized grains and the UPE. The effect of the grain size in the fracture direction seems to be a secondary effect.

## Notes

### Acknowledgements

The authors acknowledge a valuable discussion with Dr. R.C. Alderliesten on the interpretation of data. One of the authors, X. Wu, acknowledges the financial support by the foundation Materials Innovation Institute (M2i) during the execution of the work.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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**Open Access**This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.