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In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the For each Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.

Many cases have indicated that the conductivity and permeability of porous media may decrease to zero at a nonzero percolation porosity instead of zero porosity. However, there is still a lack of a theoretical basis for the percolation mechanisms of the conductivity and permeability. In this paper, the analytical percolation expressions of both conductivity and permeability are derived based on fractal theory by introducing the critical porosity. The percolation models of the conductivity and permeability were found to be closely related to the critical porosity and microstructural parameters. The simulation results demonstrated that the existence of the critical could lead to the non-Archie phenomenon. Meanwhile, the increasing critical porosity could significantly decrease the permeability and the conductivity at low porosity. Besides, the complex microstructure could result in more stagnant pores and a higher critical porosity. This study proves the importance of the critical porosity in accurately evaluating the conductivity and permeability, and reveals the percolation mechanisms of the conductivity and permeability in complex reservoirs. By comparing the predicted conductivity and permeability with the available experimental data, the validity of the proposed percolation models is verified.

Underground coal mining leads to serious surface deformation, which negatively affects the physical properties of soils. Soil particle size distribution (PSD) is one of the most basic soil physical characteristic that influences other important properties, such as soil hydraulics and thermodynamics. Understanding the spatial variability of the soil PSD in subsided land can provide targeted guidance for land reclamation. In this study, we conducted a quantitative study on the spatial variability of the soil PSD in the Pingshuo mining area on the Loess plateau, Shanxi Province in China, and explored the effects of subsidence and reclamation on the soil PSD. A plot experiment, including one unmined plot, one subsided plot, and one reclaimed plot, was performed in Anjialing No.3 underground coal mine in the, Pingshuo mining area. Four multi-fractal parameters of the soil PSD—D(0), D(1), Δα(q), and Δf(α)—were analyzed at the three sample sites. The joint multi-fractal method was carried out to analyze the spatial correlation of the soil PSD to further reveal the impacts of coal mining subsidence and land reclamation on the soil PSD. The multi-fractal method can reflect the local non-uniformity and heterogeneity of the soil PSD, while the joint multi-fractal approach can illustrate the correlation of the soil PSD between different soil depths. The range and spatial variability of the soil PSD increased due to coal mining subsidence and the impact of subsidence on the spatial disturbance of the surface soil PSD was greater than that of the deeper layers. The spatial correlation of clay in subsided land (21.00–34.34%) was larger than those of unmined land (12.36–16.37%) and reclaimed land (15.08–19.50%), and the degree of correlation was lower in 0–20 cm (21.68%) and 20–40 cm than in 40–60 and 60–80 cm soil layers (34.34%). Whereas, for silt and sand, the correlation was smaller. Land reclamation decreased the spatial variability of the soil PSD, which was near that of the unmined land after reclamation.

Cement-based materials, including cement and concrete, are the most widely used construction materials in the world. In recent years, the investigation and application of fractal theory in cement-based materials have attracted a large amount of attention worldwide. The microstructures of cement-based materials, such as the pore structures, the mesostructures, such as air voids, and the morphological features of powders, as well as the fracture surfaces and cracks, commonly present extremely complex and irregular characteristics that are difficult to describe in terms of geometry but that can be studied by fractal theory. This paper summarizes the latest progress in the investigation and application of fractal theory in cement-based materials. Firstly, this paper summarizes the principles and classification of the seven fractal dimensions commonly used in cement-based materials. These fractal dimensions have different physical meanings since they are obtained from various testing techniques and fractal models. Then, the testing techniques and fractal models for testing and calculating these fractal dimensions are introduced and analyzed individually, such as the mercury intrusion porosimeter (MIP), nitrogen adsorption/desorption (NAD), and Zhang’s model, Neimark’s model, etc. Finally, the applications of these fractal dimensions in investigating the macroproperties of cement-based materials are summarized and discussed. These properties mainly include the mechanical properties, volumetric stability, durability (e.g., permeability, frost and corrosion resistance), fracture mechanics, as well as the evaluation of the pozzolanic reactivity of the mineral materials and the dispersion state of the powders.

The work is intended to summarize the recent progress in the work of fractal theory in packaging material to provide important insights into applied research on fractal in packaging materials. The fractal analysis methods employed for inorganic materials such as metal alloys and ceramics, polymers, and their composites are reviewed from the aspects of fractal feature extraction and fractal dimension calculation methods. Through the fractal dimension of packaging materials and the fractal in their preparation process, the relationship between the fractal characteristic parameters and the properties of packaging materials is discussed. The fractal analysis method can qualitatively and quantitatively characterize the fractal characteristics, microstructure, and properties of a large number of various types of packaging materials. The method of using fractal theory to probe the preparation and properties of packaging materials is universal; the relationship between the properties of packaging materials and fractal dimension will be a critical trend of fractal theory in the research on properties of packaging materials.

Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed. nema

Knowledge of the behavior of highly compacted expansive clays, as an engineered barrier, in disposal of high-level nuclear waste (HLW) systems to prevent the pollution due to migration of radionuclide is extremely essential. The prominent properties of globally and widely used bentonites have been extensively studied during past two decades. In China, GaoMiaoZi (GMZ) bentonite is the first choice as a buffer or backfill material for deep geological repositories. This review article presents the recent progresses of knowledge on water retention properties, hydromechanical behavior, and fractal characteristics of GMZ bentonite-based materials, by reviewing 217 internationally published research articles. Firstly, the current literature regarding hydrogeochemical and mechanical characteristics of GMZ bentonite influenced by various saline solutions are critically summarized and reviewed. Then, the role of osmotic suction π alongside the application of surface fractal dimension D s is presented from the standpoint of fractal theory. Finally, the strength characteristics of GMZ bentonites using fractal approach have been discussed. Furthermore, this study sheds light on gaps, opportunities, and further research for understanding and analyzing the long-term hydromechanical characteristics of the designed backfill material, from the standpoint of surface fractality of bentonites, and implications of sustainable buffer materials in the field of geoenvironmental engineering.

Red-bed argillaceous siltstone is a common soft rock in the drawdown area of water diversion project extending from the Yangtze to Huaihe Rivers, with the characteristics of water softening and disintegration, which directly threatens the stability and safety of the diversion project. To better understand the effect of cyclic drying–wetting on the disintegration characteristics and mechanism, disintegration experiments were conducted on the red-bed argillaceous siltstone from the Tongcheng area of the water diversion project extending from the Yangtze to Huaihe Rivers. Experimental results indicated that, with an increasing number of drying–wetting cycles, the red-bed argillaceous siltstone was gradually crushed, large particles gradually transformed into small particles. A microstructural analysis showed that a continuous drying–wetting process resulted in the sample surface becoming disordered and complicated, and new micro-fractures and pores were generated. Notable changes in the concentrations of ions in the soaking solutions indicated continuous dissolution of the minerals, and a large amount of mineral loss under the action of cyclic drying–wetting. Furthermore, the evolution of disintegration parameter further indicated that the disintegration of red-bed argillaceous siltstone was gradually intensified by the increasing number of drying–wetting cycles. The fractal dimension D and the incremental surface energy gradually increased with an increase in the number of drying–wetting cycles. Thus, the proposed energy dissipation model effectively describes the disintegration characteristics of red-bed argillaceous siltstone under the cyclic drying–wetting, and thus, it can be used to guide engineering practices.

Superconductivity is analysed based on the product-like fractal measure approach with fractal dimension α introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media. Our study shows the emergence of a massless state at the boundary of the superconductor and the simultaneous occurrence of isothermal and adiabatic processes in the superconductor depending on the position of the electrons. Several physical quantities were found to be position-dependent comparable with those arising in heavy doping and p–n junction. At the boundary of the superconductor, a shrinkage of the magnetic field was observed, leading to a scenario equivalent to the Meissner–Ochsenfeld effect. An enhancement of the London penetration depth is revealed and such an improvement was observed in pnictides at the onset of commensurate spin-density-wave order inside the superconducting phase at zero temperature. The Bardeen–Cooper–Schrieffer theory was also analysed and the appearance of zero-energy states is detected. Nucleation of superconductivity in a bulk was also studied. The system acts as a quantum damped harmonic oscillator and our analysis showed that type-I superconductivity occurs when κ < 2 / ( 1 + α ) , whereas type II occurs for κ > 2 / ( 1 + α ) , where κ is the Ginzburg–Landau parameter. The transition at the passage from the ‘genuine’ to the ‘intermediate’ type-I estimates 0.767767 < α ≤ 1 .

The meshing surfaces of a gear pair are rough from a microscopic perspective and the surface topography will affect the dynamic response. To study the influence of real surface topography on the gear system dynamic performance, this paper establishes a 3‐degree of freedom transverse‐torsional dynamic model with regard to the morphology of the interface. By fractal theory, the expression of backlash between gears is modified based on the height of asperities. The time‐varying stiffness is calculated according to the fractal method rather than assuming a constant, which is more realistic. The dimensionless dynamic differential equations are established and solved with surface topography affected backlash function and time‐varying stiffness. The dynamic response of the gear system with respect to fractal dimension and fractal roughness is analyzed.