Key Takeaways & Executive Findings
- •• Fractal analysis provides a quantitative mathematical framework for describing complex, irregular metal surface and interface geometries that conventional Euclidean methods cannot capture. • The review systematically covers six categories of metal material surfaces/interfaces (precipitates, grain boundaries, deposited films, fractures, machined surfaces, wear surfaces) and links fractal dimension to material properties. • The establishment of quantitative structure–property relationships through fractal dimensions enables predictive analysis of metal material processing and performance across nano-to-micro scales. • Future work should focus on revealing the deep influence mechanism between fractal dimensions and properties, and expanding research scope, imaging techniques, and fractal calculation methods.
Abstract
As a mathematical analysis method, fractal analysis can be used to quantitatively describe irregular shapes with self-similar or self-affine properties. Fractal analysis has been used to characterize the shapes of metal materials at various scales and dimensions. Conventional methods make it difficult to quantitatively describe the relationship between the regular characteristics and properties of metal material surfaces and interfaces. However, fractal analysis can be used to quantitatively describe the shape characteristics of metal materials and to establish the quantitative relationships between the shape characteristics and various properties of metal materials. From the perspective of two-dimensional planes and three-dimensional curved surfaces, this paper reviews the current research status of the fractal analysis of metal precipitate interfaces, metal grain boundary interfaces, metal-deposited film surfaces, metal fracture surfaces, metal machined surfaces, and metal wear surfaces. The relationship between the fractal dimensions and properties of metal material surfaces and interfaces is summarized. Starting from three perspectives of fractal analysis, namely, research scope, image acquisition methods, and calculation methods, this paper identifies the direction of research on fractal analysis of metal material surfaces and interfaces that need to be developed. It is believed that revealing the deep influence mechanism between the fractal dimensions and properties of metal material surfaces and interfaces will be the key research direction of the fractal analysis of metal materials in the future.
1. Introduction
In research on metal materials, various surface and interface characteristics with certain regularity and relatively complex structures are encountered, such as grain boundaries [1–3], phase boundaries [4], and fracture surfaces [5–6]. These surface and interface characteristics significantly affect the processing and performance of metal materials; however, they are difficult to describe using conventional Euclidean geometry. Typically, the spatial distribution regularities of shapes can only be described qualitatively using words, or simple size or density differences can be used to characterize shape distribution characteristics. Therefore, researchers are seeking new ways to describe the shapes of these metal material surfaces and interfaces.
The term “fractal” was proposed and used by French mathematician Benoit B. Mandelbrot in 1973. Fractal patterns are usually described as a rough or fragmented geometric shape that can be split into multiple parts, each of which is at least a reduced-size copy of the whole [7]. The term “fractal” is mainly used to describe irregular shapes with self-similar (or self-affine) properties to establish a quantitative expression link for the potential connection between irregular shapes and the actual performance of the target object. Currently, fractal analysis is widely used in the fields of medicine, geophysics, signal processing, mineralogy, and materials science.
Because fractal analysis can describe the regular characteristics of the aforementioned related metal material surfaces and interfaces, it has also been promoted and applied in the field of metal materials. Linking fractal analysis to the properties of metal materials makes it an important mathematical tool for analyzing metal materials. Fractal analysis was used to characterize the regularity of the spatial distribution of the shape of metal materials and to calculate the fractal dimension. The shape characteristics of metal materials can be quantitatively described using the fractal dimension as a parameter to describe the spatial characteristics of metal materials on planes and curved surfaces. The proposed method establishes the quantitative relationship between shape characteristics and various properties of metal materials to realize systematic and in-depth research on metal materials.
At present, the primary research objects are two-dimensional and three-dimensional surfaces and interfaces at various scales (i.e., nanometer to micrometer) in fractal analysis research on metal materials, including grain boundary spatial distribution shapes, fracture surfaces, and surfaces. The surface and interface patterns of different metal materials can be determined using different imaging methods. Atomic force microscopy (AFM) with atomic-level resolution and optical three-dimensional imaging instruments with micron-level resolution are all used to collect images of the three-dimensional surfaces of metal materials [8–11].
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Qinjin Dai, Xuefeng Liu, Xin Ma, Shaojie Tian, Qinghe Cui (2025). Research status and prospects of the fractal analysis of metal material surfaces and interfaces. Journal of Mineral Metallurgy and Materials Science. https://doi.org/10.1007/s12613-024-2961-1
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Frequently Asked Questions
What is fractal analysis and why is it used for metal surfaces?
Fractal analysis is a mathematical method that quantitatively describes irregular, self-similar or self-affine shapes. In metal materials, it characterizes the complex geometries of surfaces and interfaces—such as grain boundaries, fracture surfaces, and precipitate interfaces—that are difficult to describe with conventional Euclidean geometry, thereby establishing quantitative relationships between shape features and material properties.
Which metal material surfaces and interfaces are reviewed in this paper?
The paper reviews fractal analysis applied to metal precipitate interfaces, metal grain boundary interfaces, metal-deposited film surfaces, metal fracture surfaces, metal machined surfaces, and metal wear surfaces.
How is fractal dimension related to metal material properties?
The fractal dimension serves as a quantitative parameter to describe the spatial characteristics of metal material surfaces and interfaces. By linking the fractal dimension to properties such as mechanical strength, wear resistance, and fracture behavior, researchers can establish predictive models connecting microstructure geometry to macroscopic performance.
What imaging methods are commonly used for fractal analysis of metal surfaces?
The paper mentions atomic force microscopy (AFM) for atomic-level resolution and optical three-dimensional imaging instruments for micron-level resolution. These techniques capture 3D surface images that are then processed using fractal analysis methods.
What are the future research directions for fractal analysis of metal materials?
Future directions include expanding the research scope to more material systems and interface types, improving image acquisition and resolution, refining fractal calculation methods, and most importantly, revealing the deep influence mechanism between fractal dimensions and material properties to enable broad engineering applications.
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