Fractal and Fractional in Construction Materials

1. Introduction

Construction materials, including concrete, cement mortar, asphalt, ferric metal, fiber-reinforced materials, bamboo, polymer, etc., have been widely used in civil engineering. In recent years, fractal theory has been widely adopted in many research fields, such as civil engineering and materials science, to probe the origin of material properties (such as rheology, permeability, diffusivity, and thermal transportation). Fractal geometry is a new branch of nonlinear science, proposed and fundamentally established in the 1970s, focusing on the irregularities, as well as haphazard phenomena and self-similarity in nature.

Concrete structures exposed to harsh conditions, particularly those containing sulfate and chloride ions, are susceptible to the infiltration of harmful agents, leading to performance degradation and a significantly reduced service life. To resolve this, silanes have been employed as protective agents for porous cementitious materials in building applications. Acoustic emission (AE) monitoring offers real-time tracking of damage and crack propagation in loaded materials. Researchers have successfully used changes in AE signals as precursors to structural instability for safety assessments [1]. These signals, generated by the expansion and propagation of internal cracks during loading, exhibit nonlinear variation. Fractal theory effectively describes the spatiotemporal distribution of AE sequences and characterizes fracture evolution during material damage, as demonstrated by Yang et al. [2] in their multi-fractal analysis of AE parameter evolution in concrete. However, while current AE research largely focuses on conventional concrete failure, there is a notable lack of studies on uniaxial compression AE characteristics and fracture precursors of silane-impregnated concrete subjected to sulfate attack, highlighting a critical research gap.

Ultra-high-performance concrete (UHPC) has gained significant attention in modern civil engineering. While traditional UHPC relies on quartz sand and river sand as aggregates, manufactured sand (MS) is increasingly used due to its consistent supply and environmental advantages. However, MS’s properties vary significantly based on its processing techniques, impacting UHPC’s matrix pore structure and overall performance. Although the effects of MS on UHPC’s packing and pore space are recognized, research is limited on pore space distribution, concentration, influencing mechanisms, and the quantitative relationship between pore structure and micromechanical properties for different MS types. Furthermore, traditional testing methods inadequately characterize UHPC’s pore structure, highlighting the need for advanced techniques. In recent years, fractal analysis and multifractal theory have emerged as powerful mathematical tools for understanding complex natural phenomena. By quantifying self-similarity and heterogeneous distributions [3], these approaches offer novel methodologies for deciphering the microstructural topology of cementitious systems within building materials science [4].

Surface cracks in concrete structures are critical indicators of internal damage, and their complexity (distribution, branching, tortuosity) is linked to structural performance [5]. Fractal theory, particularly the fractal dimension, offers a powerful tool for quantifying this complexity, enabling novel approaches to structural health monitoring and damage assessment [6], as well as unique insights into crack evolution [7]. The multifractal analysis of crack features shows promise for the rapid evaluation of concrete’s mechanical performance, advancing both fractal theory and crack analysis. Recently, integrating fractal theory with digital image processing and machine learning has further expanded crack analysis [8]. The box-counting method is widely used for fractal dimension calculation due to its simplicity and compatibility with image data. Despite the advancements in applying the box-counting method to concrete cracks and establishing links between fractal dimension and structural damage, the systematic selection of scale parameters, addressing counting origin sensitivity, and optimizing box-counting dimension computation remain critical gaps.

The macroscopic properties of cement-based composites are closely related to their complex microscopic structure, which exhibits uncertainty, irregularity, and nonlinear characteristics that are challenging to describe geometrically. As a result, fractal theory is extensively applied to quantify the microscopic structure of cement-based composites. In the application of fractal theory, techniques like small-angle X-ray scattering (SAXS) or neutron scattering (SANS) [9] are used to measure particle structure at the nanoscale, while nitrogen adsorption–desorption (NAD) [10] and mercury intrusion porosimetry (MIP) [11] are used to determine the micro-pore structure. Scanning electron microscopy (SEM) [12] and computer simulations are utilized to measure surface and internal cracks. Based on these microscopic structural data, numerous scholars have used fractal theory to study the macroscopic performance of cement-based composites, but their conclusions are sometimes different or even contradictory. Existing reviews discussing the use of fractal dimension for characterizing microstructures and evaluating macroscopic performance have several shortcomings: a chaotic classification of fractal dimensions, a lack of a systematic process for establishing fractal dimensions, no comparative analysis of different models with the same fractal dimension, an incomplete evaluation of the correlation between different microstructures and macroscopic performance, and a lack of discussion on the application of fractal dimension in evaluating high-temperature resistance.

This Special Issue, “Fractal and Fractional in Construction Materials”, focuses on the investigation and application of fractals and fractionals in construction materials worldwide. This Special Issue gathers four papers on fractal and fractional theories in the research field of construction materials. An overview of these papers is given as follows.

2. Overview of This Special Issue

To systematically examine the uniaxial compression acoustic emission (AE) characteristics and fracture precursors of silane-impregnated concrete in sulfate erosion environments, a multi-method analysis incorporating an AE b-value, energy concentration patterns, and multifractal theory was employed in Zhang, W. et al.’s article (contribution 1). Sulfate-exposed concrete was used to analyze mechanical loading responses and damage precursors in silane-treated and untreated specimens, and the multifractal characteristics of AE sequences were evaluated. The results establish a framework for assessing silane protection and identifying damage precursors in concrete structures, offering practical implications for tunnel lining preservation and methodological references for maintaining concrete in aggressive environments like chloride-laden and marine conditions. The study provides an AE-based approach for detecting destabilization precursors in silane-impregnated concrete under sulfate erosion, addressing the accelerated degradation faced by concrete infrastructure in sulfate-rich environments.

To further elucidate the influence of manufactured sand (MS) on the pore structure of ultra-high-performance concrete (UHPC) matrices, multifractal theory was applied to analyze the pore characteristics of MS-based UHPC in Wang, X. et al.’s article (contribution 2). In their study, five commercially available MS types with distinct properties were selected, and UHPC specimens with varying binder-to-sand ratios were prepared. Flowability, expansion time, compressive strength, flexural strength, and pore distribution were evaluated. The pore structures of different MS types under varying water-to-binder ratios were systematically characterized using multifractal parameters. SM-type sands showed more uniform pore distributions, better structures, and homogeneity due to finer particles, moderate stone powder, and cleanliness. High/low stone powder and low cleanliness led to pore aggregation. This study links MS variations to UHPC pore structure fractal characteristics and relates microscopic pore structure to macroscopic strength using multifractal analysis. Pore distribution aligned with multifractal analysis, demonstrating stone powder’s filler effect. A linear relationship was found between the multifractal parameters and mechanical properties.

To address the critical limitations in applying the box-counting method to concrete cracks, Wang, J. et al.’s article (contribution 3) investigated the effects of scale parameters and counting origins on the box-counting fractal dimension and its engineering application in concrete beam crack analysis. The aim was to determine optimal parameter combinations and counting strategies, providing practical suggestions for applying fractal theory to structural crack analysis. The study systematically investigated the effects of four minimum measurement scales and three cutoff scales through analyses of classical fractal images and crack image samples. Subsequently, the FPS method was used to select counting starting points and compare the effectiveness of different counting optimization strategies. Finally, the research findings were validated through static load tests on pre-stressed concrete beams. This work advances standardization in fractal analysis, enhances reliability in concrete crack assessment, and provides critical support for the practical application of fractal theory in structural health monitoring and damage evaluation.

Zhang, P. et al.’s review (contribution 4) examines the application of fractal analysis to cement-based composite microstructures and its correlation with macroscopic performance. Their study categorizes commonly used testing methods into particle distribution (e.g., laser granulometry), pore structure (e.g., mercury intrusion porosimetry), and fracture analysis (e.g., computed tomography). The review systematically outlines a detailed process for applying these testing methods, processing the results, building models, and calculating fractal dimensions. It compares the applicability of different fractal dimension calculation models, discusses the range of fractal dimensions they establish, and analyzes their advantages and disadvantages. Furthermore, this research studies the relationship between the fractal dimension of cement-based composites’ microstructures and their macroscopic properties, such as compressive strength, corrosion resistance, impermeability, and high-temperature resistance. The underlying principles influencing the positive or negative correlation between fractal dimension and macroscopic performance are discussed and elucidated. This comprehensive review provides researchers with practical methods and models for quantitatively analyzing the microscopic structural parameters of cement-based composites and offers a pathway for the non-destructive assessment of their macroscopic performance.

3. Conclusions

It is hoped that the collected four papers will inspire new ideas and provide a foundational guide for researchers studying fractal and fractional methods in construction materials. The purpose of this Special Issue is to promote the deeper and broader investigation and application of fractal theory in the field of construction materials research.

Despite significant progress in applying fractal theory to construction materials research, future efforts should address the following limitations:

(1)

Nanoscale Resolution: improve the resolution and accuracy of testing methods and fractal models to address structural issues at the nanoscale.

(2)

Unified Standards: establish unified relationships or standards between fractal dimensions and macroscopic properties to enable the comparison of results from different fractal models.

(3)

In-Depth Property Analysis: conduct more comprehensive fractal studies of mechanical properties and durability to determine accurate quantitative relationships between fractal dimensions and micro/macrostructural performance.

(4)

Advanced Techniques: employ multifractal spectrum analysis to quantify complexity differences and develop machine learning-based dynamic scale parameter optimization algorithms to improve the accuracy and scope of fractal dimensions.