#
Spatial Series and Fractal Analysis Associated with Fracture Behaviour of UO_{2} Ceramic Material

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## Abstract

**:**

_{2}ceramic materials have been analysed. In this paper, we introduce some algorithms and develop a computer application based on the time-series method. Utilizing the embedding technique of phase space, the attractor is reconstructed. The fractal dimension, lacunarity, and autocorrelation dimension average value have been calculated.

## 1. Introduction

^{238}U (99.28% abundance),

^{235}U (0.71% abundance), and

^{234}U (0.0054% abundance). Classified in the periodic table as an actinide, uranium is generally a solid body at room temperature [1]. Uranium is a naturally radioactive element, from the physics viewpoint. It powers nuclear reactors in the form of nuclear fuel and helps to make atomic bombs (still improperly called), but more precisely, named nuclear bombs, because fission is a nuclear process.

_{2}), recognized as nuclear fuel, is stuffed with supplementary ions of oxygen in the meshes of the network, it can form nonstoichiometric compounds (e.g., UO

_{2+x},), of which the composition may change with the function of exterior environmental conditions, among which we enumerate temperature itself and the partial pressure of oxygen. The fracture comportment of a sintered ceramic UO

_{2}substance has been studied in light of microstructural (micro porosity, grain size, etc.) parameters, with everything being in the function of the most adequate composition delivered and the final architecture. Utilizing SEM images as an investigation method, the fracture properties have been evaluated and compared for different microstructural conditions present in the same sample of solid ceramic materials and in a sintered UO

_{2}pellet specimen. As a general conclusion, we can consider that the fracture strength in the low-density area was superior in contrast to the that of the high-density area. Among other things, this was assigned to fissure-type deflection and bifurcation at the grain boundary, expected as owed to the porosity presence. This paper realizes an investigation of the uranium dioxide SEM pictures by utilizing the time series evaluation procedures and fractal analysis, a natural prolongation of a usual research executed before but on ductile materials [5,6,7,8].

_{2}) as nuclear fuel and ceramic materials in general. The second section focuses on providing theoretical support regarding the fractal dimension, lacunarity, and time (spatial) series. The third section introduces the results obtained and elaborates on them in a discussion. Finally, the paper concludes in the fourth and last section devoted to the conclusions.

## 2. Theoretical Background in Brief

#### 2.1. Fractal Dimension and Lacunarity

#### 2.2. Time (Spatial) Series

- (1)
- $K\subset X$ is a nonempty set;
- (2)
- K is closed;
- (3)
- K is invariant, i.e., $T\left(x\right)\subset X$, for all $x\in K$.

^{1}, x

^{2}, …, x

^{n}), called random variables, a set of functions depending on certain spatial coordinates (x

^{1}, x

^{2}, …, x

^{n}).

^{1}, x

^{2}, …, x

^{n}) argument fluctuates only on an ordinary Cartesian grid/lattice. Utilizing the appropriations of the linear (Hilbert) space connected to the series of data, the notions of novelty and a complete nondeterministic series are highlighted [15].

#### 2.3. SEM Picture Exploration

_{2}SEM pictures, we used computer programming initially created for metallic or alloy materials but subsequently excellently adapted to ceramic materials, a software application that generates a time series associated with the image, then reconstructs the associated attractor, and finally computes its autocorrelation dimension [21]. The procedure for investigating a SEM picture (micrograph) is debuted by loading an image bitmap version in the computer software application used. The first step in our consideration is to generate the weighted fractal dimensions map (WFDM) through which the potential modified structures themselves are revealed (conformable to a precedent article [15]). The second step to follow is to produce a real spatial series for a picture-selected zone, as follows: the initial picture is cut into slices that are approximatively 12–16 pixels deep; by placing all these fragments/pieces together, we procure an entire tape/strip. The spatial series s(t) is acquired by calculating the mean value of the grey level for every pixel column within the tape. The investigation of these nonlinear data suites starts with the attractor reconstruction by embedding the spatial series in an upper dimensional phase space. We establish a reasonable time delay $\tau >0$ from the beginning and then, continuing, for a determined embedding dimension d, we take into account the collection/set

## 3. Results and Discussion

_{2}ceramic material [23]. We emphasise/mention that the sorting of the micrographs with the referenced areas was executed as stated by the WFDM method [15]. Conforming to the mentioned procedure, three sets of characteristic images are studied as much as possible [25,26].

#### Final Discussions

_{2}, using the fractal dimension of the image and its lacunarity. This information, obtained through the fractal analysis, is closely related to highlighting the type of fracture (brittle in our case) and the microcracks produced in the material. As can be seen, there is a direct connection with the microdeformations present on the image in the area without significant tearing of the material and a directly proportional increase in the lacunarity in the area with the rupture produced.

## 4. Conclusions

_{2}material, using the fractal analysis technique and time (spatial) series, have been investigated.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The time series generated by the selected area in Figure 1.

**Figure 7.**Primary processing of the selected image 1: (

**a**) original image (the portion in the yellow border); (

**b**) the grayscale version; and (

**c**) the grayscale version without luminance.

**Figure 8.**Secondary processing of the selected image 1: (

**a**) binarized version; (

**b**) application of the mask. A threshold of 25 was used for binarization.

**Figure 12.**The selection of the modified area (WFDM) for Figure 10.

**Figure 13.**The gravity poles of the modified area for Figure 10.

**Figure 14.**The time series generated by the selected modified area for Figure 10.

**Figure 16.**Primary processing of the selected image 2: (

**a**) original image (the portion in the yellow border); (

**b**) the grayscale version; and (

**c**) the grayscale version without luminance.

**Figure 17.**Secondary processing of the selected image 2: (

**a**) binarized version; (

**b**) application of the mask. A threshold of 25 was used for binarization.

**Figure 21.**The selection of the modified area (WFDM) for Figure 19.

**Figure 22.**The gravity poles of the modified area for Figure 19.

**Figure 23.**The time series generated by the selected modified area for Figure 19.

**Figure 25.**Primary processing of the selected image 3: (

**a**) original image (the portion in the yellow border); (

**b**) the grayscale version; and (

**c**) the grayscale version without luminance.

**Figure 26.**Secondary processing of the selected image 3: (

**a**) binarized version; (

**b**) application of the mask. A threshold of 25 was used for binarization.

Name | Fractal Dimension | Standard Deviation | Lacunarity |
---|---|---|---|

Image 1 | 1.8220 | ±0.3440 | 0.0357 |

Name | Fractal Dimension | Standard Deviation | Lacunarity |
---|---|---|---|

Image 2 | 1.7751 | ±0.3363 | 0.0359 |

Name | Fractal Dimension | Standard Deviation | Lacunarity |
---|---|---|---|

Image 3 | 1.8103 | ±0.3508 | 0.0375 |

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**MDPI and ACS Style**

Paun, M.-A.; Paun, V.-A.; Paun, V.-P.
Spatial Series and Fractal Analysis Associated with Fracture Behaviour of UO_{2} Ceramic Material. *Fractal Fract.* **2022**, *6*, 595.
https://doi.org/10.3390/fractalfract6100595

**AMA Style**

Paun M-A, Paun V-A, Paun V-P.
Spatial Series and Fractal Analysis Associated with Fracture Behaviour of UO_{2} Ceramic Material. *Fractal and Fractional*. 2022; 6(10):595.
https://doi.org/10.3390/fractalfract6100595

**Chicago/Turabian Style**

Paun, Maria-Alexandra, Vladimir-Alexandru Paun, and Viorel-Puiu Paun.
2022. "Spatial Series and Fractal Analysis Associated with Fracture Behaviour of UO_{2} Ceramic Material" *Fractal and Fractional* 6, no. 10: 595.
https://doi.org/10.3390/fractalfract6100595