Mechanical properties of construction steel under rapid assessment

Meeting technological demands necessitates the development of advanced materials with tailored performance characteristics. Evaluating these characteristics relies on a thorough analysis of composition, structure, and properties [1]. A critical challenge in materials science is establishing precise correlations between structural features and the physical and mechanical properties of materials [2]. Addressing this challenge requires integrating technological innovations [3] with advanced mathematical modeling techniques [4]. At different structural scales, distinct elements contribute uniquely to the material’s overall properties, emphasizing the need for comprehensive analytical approaches.

Traditional Euclidean descriptors, such as length, area, and volume, often fail to capture the inherent complexity of material structures and their influence on properties [5], [6]. This limitation arises from the irregularities and heterogeneity present in real micro- and macrostructures. Fractal geometry has emerged as a transformative tool, offering a more nuanced characterization of these complexities. By introducing fractional dimensions, fractal analysis accurately represents the intricate geometry of material structures [7], [8].

Recent research has confirmed the ubiquity of fractal characteristics in the structures of various materials, including metals [9], [10], concrete [11]–[13], composites [14]–[16], and minerals [17], [18]. Despite these advancements, there remains a significant gap in the development of unified, systematic algorithms for applying fractal analysis to explore specific relationships between structural features and material properties. This gap highlights the need for a comprehensive framework that integrates multifractal analysis with advanced methodologies to improve characterization and predictive modeling.

Multifractal analysis has gained substantial attention in recent years [19]. It enables the assessment of structural heterogeneity in materials with varying degrees of geometric complexity, opening new avenues in materials science. This methodological framework not only facilitates the development of materials with tailored performance characteristics but also enhances the ability to predict their behavior under real-world conditions [20].

This study aims to advance the understanding of material heterogeneity through a detailed examination of cast iron and concrete. The research focuses on developing and applying algorithms that significantly enhance the accuracy of performance predictions by investigating the relationships between structural parameters, geometric features, and mechanical properties.

The need for specific algorithms arises from the complexity of structural heterogeneity, which involves multiple influencing parameters [21], [22], [23]. Each structural element has its own fractal dimension, and their collective interaction determines the material's spectrum of properties. Current approaches to fractal and multifractal analyses often fail to adequately address the interplay of multiple structural parameters and their cumulative impact on material properties. Moreover, the influence of structural heterogeneity [24], as described by a spectrum of multifractal dimensions, on specific performance characteristics remains underexplored, particularly for materials with intricate internal geometries, such as cast iron and concrete.

The objective of this study is to address this gap by developing and validating algorithms that integrate multifractal analysis to:

• 1.    Accurately characterize the heterogeneity of material structures.

• 2.    Identify and prioritize the critical structural parameters affecting material performance.

• 3.    Establish robust correlations between fractal descriptors of structure and physical-mechanical properties.

• 4.    Provide predictive insights into the behavior of materials under real-world conditions.

Material complexity necessitates specialized analytical approaches that account for all significant parameters. Fractal formalism offers a robust framework for describing geometric features and their impact on physical and mechanical characteristics. However, its effective application requires expert analysis to identify the most critical structural parameters and their contributions to the formation of material properties.

• 2    Materials and Methods

In the context of fractal modeling of material structures and properties, there is no universal methodological approach. Instead, individual components of developed algorithms exist that, when combined, allow for the creation of comprehensive models:

• 1.    Calculation of the fractal dimension D of the object under study using the well-established Hausdorff method [25]–[27].

• 2.    Utilization of a fractal-type model based on the determination of the self-similarity of the object (scale invariance) [28,29].

• 3.    Determination of the representation scale of the fractal object (structural scale), enabling the consideration of multi-scale aspects of the material’s structure.

• 4.    Verification of the model’s conformity to conditions related to sensitivity, defined by the formula [30]: K = Y - Y + i|/1 * , " X i ₊₁| , where Y and Y 2 are quality indicators at two arbitrarily selected points of the object, and D ₁ and D ₂ are the fractal dimensions of the structural elements at these points.

• 5.    Selection of an objective function (quality criterion), variables (fractal dimensions of structural elements), and reference points in the state space of the studied model.

• 6.    Formalization of results, including the selection of an appropriate model that describes the relationship between the fractal structure of the material and its properties [31].

• 7.    Assessment of the degree of heterogeneity of the fractal object using Renyi's formula to determine the object’s membership in a multifractal [32].

• 8.    Interpretation of the obtained results, allowing for the linking of simulation data with real material properties and the development of recommendations for optimization.

The outlined stages emphasize the complexity and multifaceted nature of fractal modeling. This approach integrates both theoretical and applied aspects, making it a universal tool for analyzing complex material structures. However, each model must be adapted to specific tasks and conditions, which requires in-depth expert analysis and careful validation.

Fractal modeling is becoming increasingly in demand due to its ability to account for structural heterogeneity and self-similarity, which enhances the prediction of materials' operational characteristics such as strength, wear resistance, and durability.

In some cases, structural heterogeneity is effectively captured using multifractal analysis [33]. This method allows for the consideration of the complexity of structural elements, relying on the spectrum of Renyi’s statistical dimensions. The multifractal approach provides a more precise description of heterogeneities that significantly influence the physical-mechanical properties of materials.

For instance, the study [34] established a relationship between the spectrum of Renyi’s statistical dimensions and the mechanical properties of cast iron rolls. The following formula was used to calculate the spectrum:

N ln ∑ p_i

D ( q ) =-- lim i ---,                                          ⁽¹⁾

q - 1 δ→∞ lnδ where δ represents the linear dimensions of a square cell covering the cast iron structure, and pi is the probability of finding a point (pixel in the digital system) of a structural element in the i-th cell with dimensions δ.

Structural heterogeneity is a key factor determining the properties of materials, such as strength, hardness, plasticity, and wear resistance. In real materials, structural elements such as grains, inclusions, pores, and microcracks are distributed heterogeneously. This heterogeneity influences the material's behavior under load and its operational characteristics.

The use of multifractal analysis allows for the assessment of the level of heterogeneity and the description of the material's structure at various scale levels. The Renyi spectrum helps determine the degree of influence of individual structural elements on the overall material properties:

• -    Plasticity and strength . Structural heterogeneity, such as an increase in pore quantity or size, can reduce the material's plasticity and ultimate strength.

• -    Wear resistance . Materials with high structural heterogeneity may exhibit improved or degraded wear resistance depending on the type of inclusion distribution and their interaction with the matrix.

The application of multifractal analysis in materials science enables:

• 1.    Development of new materials with enhanced properties through structural optimization. For example, the creation of cast iron with a specific distribution of graphite inclusions to improve wear resistance.

• 2.    Evaluation of material quality. The analysis of the Renyi spectrum helps identify defects and predict their impact on the material's operational properties.

• 3.    Development of material behavior models. Incorporating heterogeneity parameters into computational models enhances their predictive accuracy.

Thus, accounting for structural heterogeneity through the multifractal approach significantly expands the possibilities in materials science. This method not only deepens the understanding of the nature of complex materials but also enables targeted management of their properties to address contemporary engineering challenges.

• 3    Results and Discussion

The pearlitic cast irons alloyed with chromium and nickel of the SSHN grade were studied (Fig. 1).

Fig. 1 - Microstructure of SSHN cast iron: a - spheroidal graphite; b - cast iron matrix (coarse and medium dispersion pearlite + cementite)

The calculated multifractal characteristics, D ₁₀₀ and D _{- 100} , which reflect the heterogeneity of cast iron with a pearlitic matrix, are presented.

• -    The dimension D ₁₀₀ reflects the distribution characteristics of the material in the least populated areas of the structure. These zones are often associated with a potential decrease in mechanical strength or the appearance of regions sensitive to external impacts.

• -    The dimension D _{- 100} determines the properties of the densest areas of the structure, where the material exhibits high strength and density. Such areas are key to ensuring high hardness, wear resistance, and other operational characteristics.

Relatively high correlation coefficients were recorded between the dimensions of spheroidal graphite (Fig. 2a) and chromium carbides (Fig. 2b) for SSHN rolls and the hardness (HSD) of the material, calculated by the Shore method, which measures the hardness based on the rebound height of the hammer using a portable durometer.

These results underscore the importance of multifractal analysis in studying the relationship between structural characteristics and mechanical properties of materials. The graphite and carbide dimensions, calculated using the multifractal approach, reflect the complexity and heterogeneity of the distribution of these structural elements, which significantly influence hardness.

a

Fig. 2 - Relationship between the heterogeneity indices of the cast iron structure elements and hardness values

The reduction in the heterogeneity indices of the multifractal dimensions D ₁₀₀ and D _{- 100} indicates an increase in the structural order of the cast iron, which positively influences its strength.

In the study [35], a multifractal analysis of the macrostructure of plasticized cement mortar was conducted. By analyzing the statistical dimensions of the Renyi spectrum, it was found that the macrostructure of the mortar contains heterogeneous zones dominated by sand or cement stone. Regression analysis of the statistical dimensions led to the development of models for predicting flexural strength. Among the models considered, a relatively high correlation was identified: R² = 0.75 and R² = 0.81 for the fractal dimensions D ₀ of sand areas and pores, respectively. A relationship was also established between strength and the dimension of the light regions of the structure containing sand ( D ₁₀₀ , R² = 0.77) and the dark areas dominated by cement stone ( D _{- 100} , R² = 0.66).

The applicability of the multifractal model for predicting the hardness of cast iron with spheroidal graphite and a pearlitic matrix is confirmed by the obtained modeling results. The study revealed a high sensitivity of hardness values to the heterogeneity parameters of the material's structure, which justifies and demonstrates the effectiveness of using this model.

• 4    Conclusions

• 1.    The application of the multifractal model has demonstrated its high effectiveness for assessing the hardness of cast iron with spheroidal graphite and a pearlitic matrix and has highlighted its promising potential in materials science. This approach allows for a deeper understanding of the relationship between structural heterogeneity and mechanical properties, which is crucial for evaluating material performance.

• 2.    The relatively high values of the coefficient of determination (R² = 0.89...0.94) confirm the reliability of the proposed models. These models show a strong correlation between the multifractal dimensions of the material’s structure and its hardness, underscoring the precision of the multifractal analysis in capturing the complexity of cast iron microstructures.

• 3.    The multifractal model has proven to be highly applicable in materials science, particularly in the development of predictive models for mechanical properties based on the structural features of materials. This makes it an important tool for advancing the understanding of material behavior at different scales and improving the performance of cast iron and similar materials.

• 4.    The proposed models are suitable for industrial applications, especially in the rapid quality control of cast iron. The ability to assess the material’s heterogeneity and predict its mechanical properties through the multifractal dimensions offers a practical and efficient approach for real-time quality

• 5.    The findings from this study suggest that the multifractal model can be extended to other materials and processes, further enhancing its potential as a tool for evaluating and optimizing materials. By integrating multifractal analysis with other material characterization techniques, more comprehensive models can be developed to predict a broader range of material properties, thus contributing to the improvement of industrial material design and performance.

monitoring in manufacturing processes, ensuring the consistency and reliability of the produced material.

• 5    Fundings

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