Influence Of Fractal Dimension Research Articles

Study on the influence of fractal dimension and size effect of coarse aggregate on the frost resistance of hydraulic concrete

In cold regions, hydraulic concrete structures are exposed to harsh environmental conditions that greatly impact their durability and safety. The role of aggregates in influencing concrete performance is paramount. This study aims to investigate how the fractal distribution and size of aggregates affect the frost resistance of hydraulic concrete, with air void characteristics, mass loss rate, Elastic Modulus of dynamic, and strength serving as primary references. Two experiment series, fractal distribution influence and aggregate size influence of concrete frost resistance, were conducted to analyze the frost resistance of hydraulic concrete with different aggregates. Based on the fractal theory, four grading fractal distributions were utilized to fractalize aggregate gradation, aiming to investigate the influence of aggregate fractal dimension on the frost resistance of hydraulic concrete. Simultaneously, concrete specimens of fully graded aggregate and wet-screened aggregate were also constructed to simultaneous freeze-thaw cycle tests, the impact of aggregate size on concrete's frost resistance performance was also studied by analyzing the air void structure in the frost-resistant concrete specimens. Results indicate that the specimens exhibit varying frost resistance performance under different aggregate gradation fractals, ranking from strongest to weakest as follows: D=2.5, D=2.3, D=2.1, D=2.7, which is attributed to the differences of air void spacing factors. Increasing aggregate size leads to greater internal structural heterogeneity in concrete, resulting in an increase in defects and weak points. This causes microcracks to develop more rapidly in the transition zone of fully graded concrete during freeze-thaw cycles, leading to more severe damage. The findings of this study can serve as a fundamental scientific basis for the designing and constructing of hydraulic concrete structures in cold areas.

Simulation and analysis of micro-textured rough curved surface based on fractal characterization method

PurposeMicro-texture is processed on the surface to reduce the friction of the contact surface, and its application is more and more extensive. The purpose of this paper is to create a texture function model to study the influence of surface parameters on the accuracy of the simulated surface so that it can more accurately reflect the characteristics of the real micro-textured surface.Design/methodology/approachThe microstructure function model of rough surfaces is established based on fractal geometry and polar coordinate theory. The offset angle θ is introduced into the fractal geometry function to make the surface asperity normal perpendicular to the tangent of the surface. The 2D and 3D contour surfaces of the surface groove texture are analyzed by MATLAB simulation. The effects of fractal parameters (D and G) and texture parameter h on the curvature of the surface micro-texture model were studied.FindingsThis paper more accurately characterizes the textured 3D curved surface, especially the surface curvature. The scale coefficient G significantly affects curvature, and the influence of fractal dimension D and texture parameters on curvature can be ignored.Originality/valueThe micro-texture model of the rough surface was successfully established, and the range of fractal parameters was determined. It provides a new method for the study of surface micro-texture tribology.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-09-2023-0298/

Fractal model of thermal contact conductance of the involute arc cylindrical gear considering friction coefficient

PurposeThe purpose of this study is to propose a fractal model of thermal contact conductance (TCC) of the involute arc cylindrical gear considering friction coefficient.Design/methodology/approachThe influences of fractal dimension, fractal roughness and surface modification coefficient on the TCC of the rough surface were studied.FindingsThe results indicate that increasing the fractal dimension or reducing the fractal roughness enhances the TCC of the rough surface, and raising the surface correction coefficient contributes to this improvement.Originality/valueIn this work, the novelty of the work is that the authors first established a novel fractal model of TCC of the involute arc cylindrical gear considering friction coefficient. The achievements of this study provide some theoretical basis for the investigation of the TCC of the involute arc cylindrical gear.

Development of an improved three-dimensional rough discrete fracture network model: Method and application

Structure plane is one of the important factors affecting the stability and failure mode of rock mass engineering. Rock mass structure characterization is the basic work of rock mechanics research and the important content of numerical simulation. A new 3-dimensional rough discrete fracture network (RDFN3D) model and its modeling method based on the Weierstrass-Mandelbrot (W-M) function were presented in this paper. The RDFN3D model, which improves and unifies the modelling methods for the complex structural planes, has been realized. The influence of fractal dimension, amplitude, and surface precision on the modeling parameters of RDFN3D was discussed. The reasonable W-M parameters suitable for the roughness coefficient of JRC were proposed, and the relationship between the mathematical model and the joint characterization was established. The RDFN3D together with the smooth 3-dimensional discrete fracture network (DFN3D) models were successfully exported to the drawing exchange format, which will provide a wide application in numerous numerical simulation codes including both the continuous and discontinuous methods. The numerical models were discussed using the COMSOL Multiphysics code and the 3-dimensional particle flow code, respectively. The reliability of the RDFN3D model was preliminarily discussed and analyzed. The roughness and spatial connectivity of the fracture networks have a dominant effect on the fluid flow patterns. The research results can provide a new geological model and analysis model for numerical simulation and engineering analysis of jointed rock mass.

Pore Permeability Model Based on Fractal Geometry Theory and Effective Stress

AbstractA reasonable coal seam permeability model should be established to accurately estimate the extraction effectiveness of coalbed methane (CBM). Existing permeability models typically ignore the influence of pore structure parameters on the permeability, leading to an overestimation of the measured permeability, and consequently, the CBM production cannot be effectively predicted. This paper presents a novel permeability model based on discrete pore structures at the micro–nano scale. The model considers the interaction between the pore fractal geometry parameters, coal deformation, and CBM transport inside these pores. The contributions of key pore geometry parameters, including the maximum pore diameter, minimum pore diameter, porosity, and fractal dimensions, to the initial permeability were investigated. A numerical analysis showed that the influence of fractal dimension on the permeability is finally reflected in the influence of pore structure parameters. The initial permeability is exponential to the minimum pore diameter and proportional to the maximum pore diameter and porosity. In addition, the macroscopic permeability of the coal is positively correlated with the maximum pore diameter, minimum pore diameter, and porosity, with the minimum pore diameter having the most significant influence on the permeability evolution process. This research provides a theoretical foundation for revealing the gas flow mechanism within coal seams and enhancing the extraction effectiveness of CBM.

The Influence of the Fractal Dimension on the Mechanical Behaviors of the Soil–Rock Mixture: A Case Study from Southwest China

As the typical multi-phase geotechnical material, the particle size distribution of the natural soil–rock mixture (S–RM) has a significant impact on the structural and mechanical properties. The coarse grain content used in the laboratory and simulation tests falls short of accurately describing the particle size distribution feature of the entire material. The main subject of this article is the influence of the fractal dimension on mechanical behaviors based on the fractal theory. The double fractal characteristics were principally discussed along with the typical particle size distribution characteristics of the S–RM in the Three Gorges Reservoir and southwest China. The influence of the various fractal dimensions on the mechanical behaviors of S–RM was then investigated using three groups of large–scale triaxial tests, and the responses of the linear and nonlinear strength indexes were analyzed. The results show that the stress–strain curves of S–RM in the hyperbolic shape are visible under various confining pressure, and the nonlinear strength characteristics can be observed. The coarse grain content exhibits a negative correlation to the average fraction dimension. The difference between the coarse and fine grain fraction dimensions becomes considerably more obvious as the coarse grain content increases, which also increases the error when using the average fractal dimension. The voids between the coarse grains cannot be filled with the fine grains as the grain coarseness grows, resulting in a loose structure and a contact frictional effect, which lowers cohesion and raises the peak friction angle.

SIMULATION OF CHLORIDE ION TRANSPORT AND EROSION IN FRACTAL POROUS MEDIA UNDER MULTI-FIELD COUPLING

Concrete is an important part of submarine building structure, which is eroded by hydrostatic pressure and chloride ion. In this study, concrete is regarded as a dually-porous structure composed of pores and fractures. Considering hydrostatic pressure and fractal characteristics of concrete structure, a fractal model is established for chloride ion diffusion under the dual drive of concentration and pressure. Considering the effective stress caused by water pressure in pores of concrete, a fluid–solid model for chloride ion migration is proposed. The fluid–solid coupling fractal model for chloride ion migration was verified by comparing with the experimental data. The numerical simulation results show that: (1) the influence of fractal dimension on the chloride ion diffusion coefficient and chloride ion concentration is nonlinear; (2) compared with low hydrostatic pressure, high hydrostatic pressure has more significant effects on chloride ion concentration and diffusion coefficient; (3) when the effective stress and water pressure in pores are not considered, the diffusion coefficient of chloride ion is always a constant.

Normal contact stiffness model for fractal surfaces considering scale dependence and friction behavior

The normal contact stiffness on rough surfaces has a significant impact on the interfacial dynamic characteristics of mechanical joints. Based on the fractal geometry theory, this work improves the contact stiffness modeling by considering the complete contact characteristics of multiple asperities and frictional factor. First, to include the property that the critical contact areas of asperities are scale-dependent, the model for the contact stiffness is formed on each scale and the relationship between total stiffness and load on rough surfaces is obtained by the summation of all length scale asperities. Second, friction factor is taken into account in the revised normal contact stiffness model, where a contact friction coefficient is introduced into the equation to incorporate the effects of friction force. Moreover, the influences of fractal dimension D and fractal roughness G on normal contact stiffness and friction coefficient are investigated. Through the comparison of numerical and experimental results, the revised model is shown to be more reasonable and presents higher consistency of the stiffness versus load curve than the original model. It is therefore concluded that a complete and more accurate model for normal contact stiffness is proposed in this work with precise modeling of scale-dependent contact characteristics and friction behavior taken into account, which is critical for exact estimation of rigidity of contact surfaces in industrial applications.

A Prediction Model of Electrical Constriction Resistance (ECR) between Dry-Friction Rough Surfaces Based on Three-Dimensional (3-D) Fractal Theory

In this paper, firstly, based on the fractal theory, Hertz contact theory and fractal network, the prediction model of the ECR for rough interface is established in consideration of the modified hypothesis, 3-D interface topography and elastic-plastic deformation. Secondly, using the engineering data of material, the influences of fractal dimension D, fractal roughness G , dimensionless loading force F* and friction coefficient μ on the ECR are analyzed by numerical simulation. Finally, compared with the classical models, the tendency of established model is same as that of other models. During the design stage of electrical equipment, the present model of the ECR can provide a theoretical reference for the accurate ECR prediction of rough interface.

Influence of Fractal Dimension on Network Anomalies Binary Classification Quality Using Machine Learning Methods

In this paper, it is proposed to improve the efficiency of binary classification of network traffic anomalous behavior by introducing an additional informative feature – fractal dimension. The overall effectiveness of the proposed method is estimated by evaluating the quality of binary classification using the algorithms Decision Tree Classifier, Random Forest and Ada Boost on the example of using the NSL-KDD database. It is shown that adding the fractal dimension in the binary classification of attacks, gives improvement of the precision metric in average by 6%, and for AUC-ROC about 10% for all considered classification algorithms. Furthermore, introduction of fractal dimension as an additional feature has allowed to significantly reduce the time of training and testing of binary classification. So, for the Random Forest algorithm, the decrease in processing time was more than 3 times, and for the Decision Tree Classifier more than 2 times.

Numerical simulation and laboratory measurements on the influence of fractal dimension on the acoustic beam modulation of a Polyadic Cantor Fractal lenses

The possibility of modulating the ultrasound beam produced by a transducer through lenses has become an important task. The fact that these lenses are flat, facilitates their design, construction and applications. If, in addition, the design itself incorporates geometries that affect differently the foci profile on the axial axis of the lens, improvements become more significant. In this work the design of a flat lens based on a Polyadic Cantor fractal geometries is presented. The influence of the so-called fractal dimension on the modulation of the ultrasonic beam to obtain the foci is analysed. In this paper, experimental results under controlled conditions are presented. The numerical solutions obtained have been validated.

Gas transport in self-affine rough microchannels of shale gas reservoir

The matrix of shale gas reservoirs is always characterized by rough surfaces and the roughness of the topography significantly influences the gas flow characteristics. However, most existing models are based on smooth surface. Moreover, the roughness effect leads to a complex velocity distribution and can influence the apparent permeability through its own transport mechanism. In this study, a generalized lattice Boltzmann method with nonequilibrium extrapolation boundary conditions was developed to simulate the rarefied shale gas flow through nanochannels with rough surface. Fractal geometry was introduced to model the rough surface, and the roughness was characterized by two parameters, namely, relative roughness height and fractal dimension. Triple-effect permeability estimation model was adopted to calculate the apparent permeability accounting for multiple roughness characteristics including surface tortuosity, hydraulic tortuosity, and local roughness. Effects of fractal dimension, relative roughness, and tortuosity on velocity distribution were investigated. The effect of pressure and roughness characteristic parameters on the apparent permeability at different Knudsen numbers was also studied. The results showed that the influence of fractal dimension and relative roughness on the permeability was affected by the rarefaction effect. The roughness effect on gas flow behavior became more evident when associated with the enhanced rarefaction effect. Moreover, relative roughness was found to play a significant role in determining the permeability in the transient flow regime.

Experimental study on mechanical property and pore structure of concrete for shotcrete use in a hot-dry environment of high geothermal tunnels

In order to examine the deterioration mechanism and improvement measures in 100 °C hot-dry environments of high geothermal tunnels, the mechanical experiments and pore structure determinations were performed to study the macroscopic mechanical properties and pore structure characteristics of concrete for shotcrete use. The results show that: (1) in the hot-dry environments, the compressive strength and splitting tensile strength of basic mix proportion condition (BP) both decrease sharply after 7 days. By adding steel fibers or basalt fibers into concrete mix, the two kinds of strength of concrete are obviously improved, and the effect of steel fiber is significantly better than that of basalt fiber. Moreover, the improving effect of waved steel fiber among three kinds of fibers on compressive strength is the best. Compared with that of BP, the compressive strength of waved steel fiber condition (SF-W) increases by 17.2%, 37.5%, and 119.4% at 1-day, 7-day, and 28-day age respectively. The improving effect of hooked end steel fiber among three kinds of fibers on splitting tensile strength is the best. Compared with that of BP, the splitting tensile strength of hooked end steel fiber condition (SF-HE) increases by 88.5%, 72.6%, and 110.6% at 1-day, 7-day, and 28-day age respectively. (2) The median pore diameter and average pore diameter of BP at each age are higher than those of mixed fiber conditions at corresponding age. With the increase of age, the total porosity and the porosity of harmful pore (pore diameter >100 nm) of BP condition in the hot-dry environments increase rapidly. The addition of fiber materials effectively reduces the total porosity and the porosity of harmful pore, and the optimization effect of steel fiber on pore structure is better than that of basalt fiber. Compared with BP, the total porosities of SF-HE and SF-W decrease by 54.6% and 60.1%, and the porosities of harmful pore of SF-HE and SF-W decrease by 73.7% and 77.5% respectively. (3) The compressive strength and splitting tensile strength of double mix of steel fibers and silica fume condition (SF-HE + Si) are obviously lower than those of SF-HE, and the total porosity and the porosity of harmful pore are obviously higher than that of SF-HE in the hot-dry environments. (4) Compared with Menger sponge model, the fractal dimension based on thermodynamic method can describe the pore size distribution in the whole test range more comprehensively, and it is more suitable for solving the fractal dimension of concrete in this kind of environments. (5) The multi-factor strength mathematical model considering the influence of fractal dimension and composite porosity is in good agreement with the experimental results, thus, it can accurately be used to describe the quantitative relationship between strength and pore structure parameters.

The influence of fractal dimension in the microcontact of three-dimensional elastic-plastic fractal surfaces

Microcontact affects the fabrication and assembly of MEMS/NEMS system, the contact between thin film microcavity and microcantilever beam, and the motion of microstructure. In this work, in the microcontact of three-dimensional elastic-plastic Weierstrass-Mandelbrot (W-M) fractal surfaces, influence of fractal dimension was studied based on a comprehensive contact model. With increasing fractal dimension, maximum microcontact force in the plastic deformation zone shows parabolic change; comparably, the intermediate force in elastic zone parabolically varies. For both the forces, the minimum values are obtained when the fractal dimension is 2.5. Besides, in the plastic deformation zone, the real contact areas increase with the fractal dimension. Experiments were completed to compare with the numerical analysis. The results show that the simulated contact force curve is in line with the experimental load curve when Young’s modulus E and hardness H are equal to the actual measured values. Nevertheless, it will greatly deviate from the experimental load curve when E and H differ from the measured values.

Three-dimensional fractal model of normal contact damping of dry-friction rough surface

According to the influence of the normal contact damping of joint surfaces on the dynamic characteristics of high-precision machinery (machine tool, robot, etc.), in this article, a three-dimensional fractal model of normal contact damping of dry-friction rough joint surfaces based on Hertz theory and fractal theory is established. The three-dimensional surface topography is constructed, according to the modified double variable Weierstrass–Mandelbrot function. The fractal model of strain energy [Formula: see text], dissipated energy [Formula: see text], damping loss factor [Formula: see text], and normal contact damping [Formula: see text] are deduced in detail, and the influence of fractal parameters and dynamic friction coefficient [Formula: see text] on them are simulated. The simulation curves show that strain energy [Formula: see text], dissipated energy [Formula: see text], damping loss factor [Formula: see text], and normal damping [Formula: see text] increase with the increase in the fractal roughness G; the influence of fractal dimension D on [Formula: see text] is more changeable, first [Formula: see text] decreases with the increase in D and then increases. [Formula: see text], [Formula: see text], and [Formula: see text] increase with the increase in D and [Formula: see text], respectively; the effect of [Formula: see text] on [Formula: see text] is not obvious, so simple change in [Formula: see text] has no significant change in [Formula: see text]; a comparative analysis of the theoretical calculation of normal damping and experimental results show that their general trend is consistent, and they increase with the increase in total normal contact load P, the relative error is 5%–25%. The theoretical model can provide reference for the design of normal contact damping of the joint surfaces.

Fractal characteristics of coal samples utilizing image analysis and gas adsorption

Coal is a porous medium with fractal characteristics. In order to investigate the effect of fractal dimensions on methane adsorption capacity, fractal characteristics of 11 coal samples were analyzed, using scanning electron microscopy (SEM) and low-pressure nitrogen gas adsorption (LP-N2GA). Data from SEM image analysis and LP-N2GA experiments were applied to assess the heterogeneities of pore distribution (D1) and the irregularities of coal surface (D2) on the basis of box-counting method and Frenkel–Halsey–Hill (FHH) theory, respectively. The relationship between fractal dimensions and coalification was investigated. Based on the physical description of fractal surfaces and pore distributions, the influence of fractal dimensions (both D1 and D2) on CH4 adsorption characteristics was also discussed. The results indicate that these coal samples have different CH4 adsorption characteristics and fractal geometries, with D1 ranging between 1.5380 and 1.8267, and D2 varying from 2.2656 to 2.6541. The U-shaped curve relationship between D values (including D1 and D2) and volatile matter (Vdaf) is observed, demonstrating that coalification makes coal surfaces and pore networks comparatively smoother and more regular for lower rank coals (Vdaf>15%), but rougher and more complex for higher rank coals (Vdaf<15%). The Langmuir volume (VL) shows a positive linear correlation with the fractal dimension D2 values, but little correlation with D1 values. While, the Langmuir pressure (PL) is affected by both D1 and D2. Fractal dimensions comprehensively reflect the difference in physical properties of coal, which can be used to evaluate CH4 adsorption capacity. Fractal analysis is of great significance for better understanding of the surface irregularity and methane storage capacity of a coal reservoir.

Gas flow through rough microchannels in the transition flow regime.

A multiple-relaxation-time lattice Boltzmann model of Couette flow is developed to investigate the rarified gas flow through microchannels with roughness characterized by fractal geometry, especially to elucidate the coupled effects of roughness and rarefaction on microscale gas flow in the transition flow regime. The results indicate that the surface roughness effect on gas flow behavior becomes more significant in rarefied gas flow with the increase of Knudsen number. We find the gas flow behavior in the transition flow regime is more sensitive to roughness height than that in the slip flow regime. In particular, the influence of fractal dimension on rarefied gas flow behavior is less significant than roughness height.

FRACTAL PORE STRUCTURE MODEL AND MULTILAYER FRACTAL ADSORPTION IN SHALE

The complex structure and surface property of porous media have significant impact on its accumulation and adsorption capacity. Based on the fractal theory, this paper presents a fractal pore structure model for shales. The effect of different pore structures on fractal dimension is discussed, and the influence of fractal dimension and pore size distribution on porosity is also analyzed. It is shown that the fractal dimension D decreases with the increase of structure parameter q/m for a certain pore diameter ratio, and porosity has positive relationship with fractal dimension. This paper also presents a multilayer fractal adsorption model which takes into account the roughness of adsorption surface by using fractal theory. With the introduction of pseudo-saturated vapor pressure in the supercritical temperature condition, the proposed adsorption model can be applied into a wider range of temperature. Based on the low-pressure nitrogen adsorption and methane isothermal adsorption experiments, the effect of fractal dimension on the adsorption behavior of shales is discussed. Fractal dimension has significant impact on the surface adsorption property and adsorption layer number n. The monolayer saturated adsorption volume Vm increases with the increase of D, while parameter C has the opposite variation trend. Finally, the optimal combination of fractal parameters for describing pore structure of shale samples is selected.

Fractal surface synthesis based on two dimensional discrete Fourier transform

The discrete Fourier transform(DFT) is used for fractional Brownian motion(FBM) surface synthesis in tribology(i.e., contact, sliding, and sealing, etc). However, the relationship between fractal parameters(fractal dimension and scale factor) and traditional parameters, the influence of fractal parameters on surface appearance, have not been deeply discussed yet. These lead to some kind of difficulty to ensure the synthesized surfaces with ideal fractal characteristic, required traditional parameters and geometric appearance. A quantitative relationship between fractal parameters and the root mean square deviation of surface (Sq) is derived based on the energy conservation property between the space and frequency domain of DFT. Under the stability assumption, the power spectrum of a FBM surface is composed of concentric circles strictly, a series of FBM surfaces with prescribed Sq could be synthesized with given fractal dimension, scale factor, and sampling numbers, but the ten-point height(Sz), the skewness(Ssk) and the kurtosis(Sku) are still in random, where the probability distributions of Sz and Ssk are approximately normal distribution. Furthermore, by iterative searching, a surface with desired Abbott-Firestone curve could be obtained among those surfaces. An intuitive explanation for the influence of fractal dimension and scale factor on surface appearance is obtained by discussing the effects on the ratio of energy between high and low frequency components. Based on the relationship between Sq and surface energy, a filtering method of surface with controllable Sq is proposed. The proposed research ensures the synthesized surfaces possess ideal FBM properties with prescribed Sq, offers a method for selecting desired Abbott-Firestone curve of synthesized fractal surfaces, and makes it possible to control the Sq of surfaces after filtering.

Elastoplastic Fractal Model for Tangential Contact Damping Energy Dissipation of Machine Joint Interfaces

Based on contact fractal theory,modified MB fractal model,and considering the mechanism of energy dissipation and the elastoplastic deformation regime(not only fully elastic or plastic regime or plastic and elastic regime) of plane joint interfaces,an elastoplastic fractal model,considering the influence of domain extension factor on micro-contact size distribution,is proposed.Numerical simulations show the influence of elastoplastic deformation regime on dimensionless tangential damping dissipation is not obvious when fractal dimension D value between 1.1 and 1.4 compared with only considering elastic and plastic deformation regime,while if is found that dimensionless tangential damping dissipation considering elastoplastic deformation regime less than that of only considering elastic and plastic deformation regime when fractal dimension D value between 1.4 and 1.9.There is a concave nonlinear relation between dimensionless tangential damping dissipation and dimensionless total normal load,and the influence of fractal dimension D on dimensionless tangential damping dissipation is different when D=1.1-1.4,D=1.4-1.6,D=1.6-1.8,D=1.8-1.9,respectively.Besides,dimensionless tangential damping dissipation increases with the increment of fractal roughness,plastic index,and the ratio of total normal load to total tangential load.