Fractal structure of cast iron and its strength characteristics

The manufacturing process of many products is highly complex due to its multifactorial [1] and multicriteria nature [2]. This complexity creates challenges in predicting product quality metrics, given the wide range of technological parameters that can influence these outcomes [3]. At the same time, direct testing of finished products is not always feasible , as it can compromise their integrity [4]. Consequently, various modeling methods are actively used in materials science to assess and predict the qualitative characteristics of cast iron rolls [5].

The most common models focus on analyzing the influence of the chemical composition of rolls on their properties [6], [7]. Additionally, many models investigate the impact of structural characteristics on the operational performance of rolls [8–10]. Most such models emphasize the relationship between mechanical properties and Euclidean parameters of structural elements (e.g., area, length, diameter) [11]. However, predictions based solely on Euclidean characteristics often fail to meet specific analytical needs [12], [13].

The study of the relationship between structure and material properties remains a cornerstone of materials science [14], as the choice of an appropriate method for identifying complex structural geometries is critical [15]. The limitations of formal methods [16] using Euclidean geometry in analyzing material structures can be addressed by applying approaches such as fractal analysis [17], multifractal analysis [18], wavelet transformation theory [19], and compromise property domain analysis [20], [21].

Given the inherent heterogeneity of roll cast iron structures, multifractal analysis is a suitable approach for their identification. Multifractal methods [22], [23], which analyze heterogeneous objects, utilize the spectrum of Rényi’s statistical dimensions [24], and have proven useful in predicting the mechanical properties of rolling mill rolls [25]. These methods have broad applications across various fields, providing valuable insights into complex systems.

For instance, multifractal detrended cross-correlation analysis (MF-DCCA) has been used to study price-volume correlations in the Chinese metal futures market, revealing long-range dependencies and non-Gaussian distributions [26]. Similarly, a forecasting model combining multifractal detrended fluctuation analysis (MF-DFA) and wavelet coherence demonstrated superior accuracy in predicting the behavior of gold, silver, and platinum prices [27]. In geoscience, multifractal methods were applied to humus geochemistry data, identifying transitions between background and anomalous metal concentrations near a smelter site [28].

The Pareto-lognormal distribution from multifractal modeling also revealed deviations in the sizefrequency distributions of Canadian ore deposits, influenced by historical mining practices [29]. Furthermore, multifractal models have characterized the vertical distributions of ore-forming elements in the Wandongshan deposit, linking mineralization patterns to geological features [30]. In finance, multifractal analysis has provided insights into the volatility of non-ferrous metal futures, with Aluminum futures exhibiting the highest complexity and Copper the lowest, suggesting different investment opportunities [31].

Several studies have also demonstrated the utility of multifractal analysis in various material science applications. One study [32] investigated the surface morphology and nanotexture of titanium thin films deposited by DC magnetron sputtering at different sputtering powers. The results showed that roughness and interface width increased with sputtering power, and multifractal analysis revealed improved surface homogeneity and reduced surface porosity at higher sputtering power, leading to greater topographic uniformity. In another study [33], MF-DFA was used to analyze the corrosion potential fluctuations (Ecorr) of copper patina over a 10-day period in Caribbean seawater. The results confirmed the multifractal behavior of Ecorr, with power-law relationships observed across different size ranges, and the corrosion process was linked to the release of copper ions and transformations in the copper patina.

In the field of welding, a study of the gas metal arc welding (GMAW) process [34] applied multifractal theory to investigate the chaotic dynamic behavior of welding current signals. The research found that changes in shielding gas flow and welding current influenced stability and droplet transfer behavior, with the multifractal spectrum closely related to these transitions. Another study explored the fractal properties of polarizability in graphite deposits [35], introducing a fractal method to predict graphite deposit properties. The study showed that the concentration-area method was superior for anomaly extraction, while the spectrum-area method was more effective for covering larger regions. The study also confirmed that the spectrum-area method's background mode could identify potential graphitebearing strata and predict high-grade targets for future exploration. Furthermore, a study combining Fourier Ptychographic Microscopy (FPM) with multifractal analysis [36] assessed the dose-dependent effects of copper on microalgae, providing high-resolution, multi-scale data. This demonstrated the suitability of fractal analysis for monitoring the impact of pollutants on aquatic life.

These findings highlight the significant potential of multifractal theory in predicting quality criteria for cast iron rolls. The primary advantage of multifractal analysis lies in its ability to analyze the dimensional spectrum of structural elements, accounting for the complex geometry and heterogeneity of material structures. By offering insights into the relationships between microstructure and operational characteristics, such as strength, wear resistance, and thermal stability, multifractal analysis presents an advancement over traditional methods based on Euclidean characteristics. Unlike conventional approaches, the multifractal method more accurately evaluates local structural features that have a critical impact on mechanical properties.

However, a comprehensive exploration of how multifractal techniques can be applied to predict the hardness of cast iron rolls—specifically those made from SPHN and SSHNF grades—has not yet been fully addressed. There is a need to better understand how multifractal parameters, such as fractal dimensions, correlate with traditional structural characteristics and how these relationships can be used to develop more accurate predictive models.

Although multifractal analysis has been applied to a variety of materials and processes, its connection to the hardness of cast iron rolls remains underexplored. This gap presents a unique opportunity for further research, where multifractal methods can be employed to examine the correlation between microstructure and hardness. By addressing this gap, a more robust framework for assessing the quality of cast iron rolls can be developed. This review aims to fill this void by synthesizing existing research and presenting a novel approach to understanding the influence of multifractal dimensions on material properties, with particular emphasis on SPHN and SSHNF grades of cast iron. The findings of this research could advance the field of materials science and provide valuable insights for the metallurgical industry, particularly in enhancing the operational performance and durability of cast iron rolls used in rolling mills.

In conclusion, the adoption of multifractal analysis in materials science offers a promising foundation for developing innovative technologies that improve the production of cast iron rolls with enhanced operational parameters. This method serves not only as a powerful research tool but also as a significant step toward improving product quality in the metallurgical industry.

• 2    Materials and Methods

This study examines the influence of structural elements on the hardness of SPHN and SSHNF grades of cast iron rolls used in section rolling mills. Structural evaluation was conducted using both traditional quantitative metallography methods in compliance with current standards and multifractal analysis. SPHN-grade rolls are designed for use in roughing stands of small and medium-section mills, as well as tube rolling mills. SSHNF-grade rolls are applied in roughing and edging stands of section mills.

SPHN-grade cast iron rolls are classified as section rolls (S) and have a structure that includes lamellar graphite (P). The working layer surface is alloyed with chromium (C) and nickel (N). SSHNF-grade cast iron rolls are also section rolls (S), but their structure contains spheroidal graphite (S), and the surface of the working layer is further alloyed with chromium (C), nickel (N), and vanadium (F). This dual approach enables a deeper understanding of how structural features influence the operational properties of rolls and provides a foundation for further optimization of their composition and manufacturing technologies.

Table 1 presents the chemical composition of the cast irons after two melts without heat treatment.

Table 1. Chemical composition of the working ball of cast iron rollers

Cast Iron Rolls	C	Si	Mn	P	S	Cr	Ni
SPHN-43	3.52	1.00	0.59	0.075	0.016	0.72	1.31
SSHNF-47	3.05	1.35	0.49	0.056	0.014	0.60	1.17

For this study, 20 samples were selected from the working layer of the barrels of SPHN-43 and SSHNF-47 rolls in accordance with normative requirements ( Figure 1 ). This sampling method ensures representativeness and analytical accuracy since the material areas studied form the basis of the rolls' functional properties. Such preparation contributes to more accurate interpretations of the material's composition and structure concerning their influence on operational characteristics.

The polishing of metallographic specimens for optical microscopy was performed using a polishing wheel with diamond paste containing abrasive particles of 10 and 5 μm. For etching, a 4% nitric acid solution in alcohol was used. Microstructure analysis was conducted using a Neophot 2 optical microscope (Germany) equipped with an Olympus C-50 digital camera for capturing images. Electronic images of the microstructure were saved in BMP format with 256 grayscale gradations and a resolution of 2500 × 1800 pixels, ensuring high image detail.

Figure 2 shows the microstructure of SPHN-43 cast iron rolls under various magnifications. The analysis revealed that the material consists of a pearlitic matrix ( Figures 2a, 2b, 2c ) alloyed with chromium and nickel, along with a moderate content of carbides (Fe ₃ C) ( Figure 2b ). Graphite inclusions in the cast iron structure have a lamellar shape ( Figure 2c ), characteristic of this alloy type.

a,x900                                               b,x500

c,x900                                               d,x500

Fig. 2 - SPHN structure (2a, 2b), SSHNF structure (2c, 2d)

This microstructural analysis is an essential step in understanding the relationship between the chemical composition, microstructural features, and operational properties of cast iron rolls, providing a basis for subsequent application of multifractal analysis methods.

To perform multifractal analysis on microstructure images, a methodology was developed based on using the statistical sum Spq , which accounts for the complex structure of the fractal subsets of the i=1

object. These subsets are characterized by variable dimensions, where the scaling exponent q varies over a wide range, from -да to +да. This approach offers a comprehensive representation of the microstructure's properties through the analysis of the Renyi spectrum D(q) (1), reflecting a set of dimensions with distinct physical meanings.

1        ln ∑ ^N p i ^q

D ( q ) =      ⋅ lim ^{i = 1}

q - 1 δ→∞ lnδ where:

• •    δ is the size of the square grid cell superimposed on the digital image of the microstructure,

• •    p_i is the probability of a point (or pixel, in the case of a digital image) of the object falling into the i - th cell with a linear size δ ,

• •    q is a parameter that adjusts the sensitivity of the analysis to point density distribution in different regions of the object.

As δ → 0 , the structure's detail is maximized, allowing for the consideration of fine point (pixel) distribution features. Thus, the calculation of the multifractal spectrum D ( q ) becomes the basis for quantitatively evaluating the complex characteristics of cast iron microstructure.

Physical Significance of Renyi Spectrum Dimensions:

• 1.    Fractal dimension D ₀ characterizes the compactness of point distribution in space. At q = 0 , all structural elements are considered, regardless of their density, which makes this dimension essential for describing the overall space occupancy of the investigated object.

• 2.    Information dimension D ₁ represents the entropy of the system and is calculated at q = 1. This parameter indicates the rate at which the amount of information required to determine the position of a point increases as the cell size decreases ( δ → 0). It reflects the density distribution of points and the structural heterogeneity.

• 3.    Correlation dimension D ₂ describes the probability that two points of the object are located within the same grid cell. This dimension is associated with the internal correlation of structural elements and helps identify patterns in the distribution of denser regions.

• 4.    Dimension D _∞ represents the sparsest areas of the object, corresponding to the lightest structural elements. This parameter describes regions with minimal density, such as voids or areas with low material concentration.

• 5.    Dimension D _-∞ captures the properties of the densest regions of the object, corresponding to the darkest structural elements. It characterizes the maximum concentration of points, such as carbide inclusions or zones with high graphite saturation.

3 Results and Discussion

To establish the relationship between the hardness properties of the studied cast iron rolls and the characteristics of their structure, a quantitative metallographic analysis of the microstructure and a multifractal analysis of structural elements were performed using formula (1).

The analysis involved the following stages:

• 1.    Quantitative metallographic analysis was conducted to measure key microstructural characteristics, including phase fractions, dimensions of graphite inclusions, and carbide particles. The analysis adhered to standard methodologies, ensuring the objectivity of the obtained data.

• 2.    Multifractal analysis employed a statistical sum to determine the dimensional spectrum of structural elements. This approach accounted for the heterogeneity in the distribution of structural components, which is typical for complex cast irons.

The results of the quantitative analysis of structural elements in SPHN and SSHNF rolls are presented in Table 2. These results illustrate the primary characteristics of the pearlitic matrix, lamellar and spheroidal graphite, and carbide inclusions, as well as the degree of their structural organization.

Table 2. The results of the quantitative analysis of the elements of the roll structure

Sample No.	Pearlitic Matrix				Carbides (Cementite, Fe 3 C)				Graphite Inclusions
	<	Q s Ф Ф	c 0 Ф .2 E Q a 2 ш Ф 0.	c 0 Ф W Q Q Ф Q- c Ф	co <	Ф 5 СП 0 0 0 СП о 9 и г О Qi	О N S Е с а о -с го с о о с О < m 0	2 c ° £ Ф O) 0 c С Ф 2 -1 ф or	(0 2 <	Ф 2 О <0 ф 2 ё< 2 £ Ф O'	E £ c Ф -J 2 (0 О	CD 3 Ф Q. О — ф ф O' -1
SPHN
1	70	P70	1.5	PD1.4	27	C25	12960	Cp13000	3	PG4	96	0 ■0 0 CL
2	65		1.3	PD1.4	32		14230		3		75
3	61		0.9	PD1	37		15290		2.5		72
4	67		1.0	PD1	31		13550		2		74
5	69		1.1	PD1	28		11680		3		88
SSHNF
1	62	P70	0.7	PD0.5	34	C25	11500	Cp13000	4	SG6	35	LO 0 w
2	73		1.1	PD1	24		12000		3		45
3	71		0.9	PD1	22		9500	Cp6000	5		54
4	75		1.1	PD1	20		12670	Cp13000	4		59
5	68		1.2	PD1	23		14570		5		51

The relationship between hardness and traditional structural characteristics of the cast iron was analyzed by comparing the hardness indices with corresponding structural metrics. It was found that the pairwise correlation coefficients for hardness prediction using traditional structural characteristics, such as length, diameter, and area, were R ² = 0.79 and R ² = 0.94 for models (2) and (3), respectively.

HSD = 3.2268 x S a - 127.27; R 2 = 0.94,

where S_a is the area of carbide inclusions (cementite).

HSD = - 9.2642 x d + 539.52; R 2 = 0.79, where d is the diameter of spheroidal graphite.

In other cases, the pairwise correlation coefficients remain relatively low, ranging between 0.4 and 0.7. This indicates the limitations and incompleteness of formal axiomatic approaches when describing structural elements with complex geometric configurations, particularly when traditional analysis methods are employed. These results underscore the necessity of adopting more precise and comprehensive approaches for evaluating material characteristics.

To partially address the identified limitations of the existing mathematical models, multifractal formalism was utilized. This method enables a deeper investigation of the spatial heterogeneity and multilevel nature of the studied material. Multifractal analysis allows consideration of both macrostructural and microstructural features, facilitating a more detailed characterization of correlation properties and improving the accuracy of predictions for the performance characteristics of cast iron rolls.

Mathematical models of the fractal type are presented below (Fig.3).

Fig. 3 - The relationship between hardness indices and the dimensions of carbides (a and b) and the dimensions of graphite (c and d) in SPHN -43 rolls

• -    Equation a describes the relationship:

HSD = 9.5238 x D1 + 35.81; R2 = 0.89, where D1 represents the informational dimension of carbides.

• -    Equation b expresses the correlation:

HSD =-25.000 x D2 + 94.500; R 2 = 0.95, where D2 is the correlation dimension of carbides.

• -    Equation c establishes the dependence:

HSD = 12.172 x D0 + 27.808; R 2 = 0.82, where D0 denotes the fractal dimension of graphite.

• -    Equation d highlights the relationship:

HSD = 13.832 x D-100 +15.711; R 2 = 0.87, where D-100 represents the dimension of graphite at q = -100.

The correlation coefficients R 2 = 0.82....0.95 (Equations 4-7) for predicting the hardness of SPHN-43 cast iron rolls indicate a higher degree of accuracy compared to traditional quantitative metallographic methods. This highlights the feasibility of employing multifractal theory to assess the quality of cast iron rolls used in section rolling mills.

A decrease in hardness values is observed with increasing informational (Figure 3a) and correlation (Figure 3b) dimensions of carbides. As established in studies [6], [7], these structural parameters significantly affect the distribution of graphite within the material's volume. Conversely, an increase in the dimension of lamellar graphite within the analyzed area of the polished section (Figures 3c and 3d) leads to a rise in hardness values. This is attributed to the fact that cast iron containing graphite with an equilibrium geometric shape exhibits better mechanical properties compared to cast iron with a complex lamellar form.

In lamellar graphite, up to 50% of the strength of the metallic matrix is utilized, while its plastic properties remain largely unrealized [37]. This is partially explained by the fact that the boundaries of lamellar graphite typically act as stress concentrators. An increase in the content of lamellar graphite weakens the cast iron, which negatively impacts its mechanical properties. Therefore, the mechanical properties of cast iron are influenced by the content, size, and distribution of graphite, as well as the geometric configuration of its inclusions.

To describe the geometry of the structure, a computational metric is specified, and the accuracy of dimension calculations depends on the choice of this metric. Determining the optimal metric is a key step in developing methodologies for predicting the properties of cast iron. The application of multifractal analysis in this context provides the means for precise quantitative characterization of the material's structural features and enhances methods for predicting its performance characteristics.

The correlation coefficients for the hardness prediction models of rolls with a spheroidal graphite structure (SSHNF-47, Figure 4) ranged from 0.90 to 0.99 (Equations 8–10). These values confirm the high adequacy of using fractal geometry to model the structure and properties of metals.

Fig. 4 - Dependence of hardness indicators on the dimensions of carbides (a, b) and the dimensions of graphite (c) in SSHNF-47 rolls

• a)    HSD = 18.912 x D0 + 17.132; R2 = 0.99,

where D ₂ is the correlation dimension of carbides.

• b)    HSD = 14.241x D100 + 34.229; R 2 = 0.93,

where D ₁₀₀ is the dimension of carbides at the exponent 100.

• c)    HSD = 29.452xD0 - 3.9247;R2 = 0.90,

where D ₀ is the fractal dimension of graphite.

The application of fractal geometry enables the consideration of complex spatial and scale characteristics of structural elements, significantly enhancing the accuracy of mechanical property predictions. The spherical shape of graphite ensures a more uniform distribution of internal stresses within the metallic matrix, reducing the likelihood of localized deformations and fractures.

Furthermore, the high correlation coefficients highlight the robustness of fractal analysis methods against variations in material structure, making them indispensable for a comprehensive modeling approach. This advancement creates new opportunities for predicting the performance characteristics of metals and designing materials with improved mechanical properties.

• 4    Conclusions

Thus, the integration of fractal analysis methods into metallographic studies opens opportunities to improve the accuracy of modeling and optimize production processes related to the creation of high-quality cast iron products. The following conclusions can be drawn:

• 1.    The metric used for analysis determines the sensitivity of models to the features of structural elements, such as size, shape, and distribution of inclusions.

• 2.    Correctly defining the metric ensures optimal alignment between calculated and experimental data, which is critical for predicting the material's operational properties.

• 3.    The application of a multifractal spectrum of dimensions provides a deeper understanding and modeling of complex systems, such as the structure of cast iron. This approach allows for the description of multi-scale material properties and the consideration of structural heterogeneities.

• 4.    Therefore, selecting the appropriate metric becomes a crucial step in developing predictive models aimed at enhancing the quality and reliability of cast iron products.

• 5    Fundings

  This research was partially by Ministry of Science and Higher Education of Russian Federation (funding No FSFM-2024-0025).