Box Dimension Research Articles

Investigation of three-dimensional model reconstruction and fractal characteristics of crack propagation in jointed sandstone

The presence of random joints, cracks, and other defects significantly affects the meso-damage mechanism and macro-mechanical behavior of the rock. This study employed micro-CT scanning, digital image processing (DIP), and the rock failure process analysis system (RFPA3D) to reconstruct a genuine mesostructure, creating a three-dimensional (3D) numerical model of jointed sandstone. Under uniaxial stress, this model facilitated the meso-damage evolution process of prefabricated cracks in sandstone with varying dip angles. Additionally, the influence of jointed sandstone heterogeneity and prefabricated cracks with various dip angles on its failure mode and meso-damage mechanical properties were investigated. Utilizing the MATLAB platform, a 3D box dimension algorithm was developed to analyze the fractal characteristics of the mesodamage evolution in the sample. This algorithm enabled the quantitative characterization of the meso-damage evolution of sandstone. This study categorized three types of sandstone final failure modes: composite shear failure, shear failure along the joint surface, and tensile failure. Additionally, linear variations in the elastic modulus and compressive strength of the jointed sandstone were observed with increasing prefabricated fracture inclination, highlighting significant anisotropy. The presence of joints was found to induce and control the failure mode of sandstone. The meso-damage evolution process of sandstone was described in terms of the fractal dimension, indicating that more severe damage corresponded to a larger fractal dimension. This approach offers a novel statistical method for studying the progression of rock damage.

Multifractal characteristics of soil particle size distribution of abandoned homestead reclamation under different forest management modes

In this study, fast-growing poplar reclaimed from abandoned homestead in Xixian New District, Xi’an City, Shaanxi Province, was used as the research object to explore the multi-fractal characteristics of soil particle size distribution under different management modes of abandoned land (control), irrigation, fertilizer irrigation and mixed fertilizer irrigation. The results showed that the mean values of soil clay, silt and sand in abandoned land were 14.58%, 81.21% and 4.22% respectively, 14.08%, 79.92% and 5.99% under irrigation, 15.17%, 81.19% and 3.64% under fertilizer irrigation, and 16.75%, 80.20% and 3.05% in mixed fertilizer treatment. From 40 cm, with increasing soil depth, soil clay particles increase under irrigation, fertilizer irrigation, and mixed fertilizer irrigation modes. The single fractal dimension of soil particle size distribution (D) in each treatment ranges from 2.721 to 2.808. At 60–100 cm, D shows fertilizer irrigation > mixed fertilizer irrigation > irrigation > abandoned land, indicating that fertilization and irrigation can increase the fine-grained matter of deep soil particles and reduce soil roughness. Compared with abandoned land, under irrigation, fertilizer irrigation and mixed fertilizer modes the capacity dimension (D0), entropy dimension (D1), correlation dimension(D2), shape characteristics of the multifractal spectrum (Δf) and overall inhomogeneity of the soil particle size distribution (D0–D10) indicate an uneven distribution of soil particle size; fractal structure characteristics of soil (D−10–D0) indicate a simplified soil structure, and degree of dispersion of soil particle size distribution (D1/D0) indicates that soil particle size is distributed in dense areas. Pearson correlation analysis showed that D was significantly correlated with clay, sand, D0–D10, soil organic matter (SOM) and soil available phosphorus (SAP) (P < 0.05). Stepwise regression analysis showed that clay was the main controlling factor of D and D0–D10 changes. The research results can provide some potential indicators for the quality evaluation of abandoned homestead reclamation.

General fractal dimensions of graphs of products and sums of continuous functions and their decompositions

This study takes a broad approach to the fractal geometry problem and proposes an intrinsic definition of the general box dimensions and the general Hausdorff and packing dimensions by taking into account sums of the type ∑ih−1(sg(ri)) for some prescribed functions h, g and for all positive real s. Our primary aim is to conduct a more comprehensive exploration of the fractal dimensions exhibited by graphs resulting from the products and sums of two continuous functions. This article is entirely devoted to the investigation of this specific problem domain. In the course of our research, we furnish explicit formulas for computing both the general upper box dimension and the general lower box dimension associated with the graphs formed by the products and sums of two continuous functions. Additionally, we shed light on the decomposition of continuous functions into the sum of two continuous functions, employing the framework of the general lower and upper box dimensions. Furthermore, we establish an upper bound for the general upper box dimension applicable to each element within a finite ring of polynomials derived from continuous functions defined over the real number field R. Our inquiry also extends to the examination of general Hausdorff and packing dimensions for the graphs generated through both the summation and multiplication of two continuous functions.

Lipschitz classification of Bedford-McMullen carpets with locally uniform horizontal fibers

It is well known that the Hausdorff dimension and box dimension of a Bedford-McMullen carpet coincide if and only if the carpet has uniform horizontal fibers. Yang et al. [17] gave a complete Lipschitz classification of such carpets which are totally disconnected and possessing vacant rows. In this paper, we show that if we replace the uniform horizontal fibers condition by a locally uniform horizontal fibers condition, the conclusions of Yang et al. still hold.

On the Minkowski dimension of some invariant sets of dynamical systems

Abstract. It is well known that a fractal set is not a submanifold of the ambient space. However, fractals arise as invariant subsets even in infinitely smooth conditions and the Minkowski dimension serves in this case as a characteristic of complexity of this scale. For example, when the equilibrium state during the Andronov-Hopf bifurcation losses its stability, the closure of the non-singular trajectory is a parametrically defined curve of the fractal type. In this work the fractal dimension of such curves is calculated. In addition, special two-parameter family of functions is studied such that Minkowski dimension of their graphs varies from 1 to 2. The obtained result allows us to implement a regular dynamic system with an isolated hyperbolic point such that the closure of two-dimensional stable manifold of this point may have Minkowski dimension greater than 2. To calculate the graph dimension, the segment of the argument defining the graph is split into two parts. The dimension of the first part of the graph can be estimated from above by direct calculation of the corresponding curve’s length. The upper estimation of the other part’s dimension is provided by means of the area of rectangle containing this curve. The lower estimation of the Minkowski dimension is based on calculating the cardinality of ε -distinguishable set of graph points.

Study on the effect of wood admixture on the physical and mechanical properties of corn cob ecological recycled concrete

This paper investigates the physical and mechanical properties of ecological concrete using corn cob pellets, shavings, and wood chips. Wood chips and shavings were partially used as substitutes for corn cob particles to enhance the physical and mechanical properties of the eco-concrete. The impact of substitution on the macroscopic physical and mechanical properties is examined based on various pore structures. The pore size distribution was analyzed using the non-destructive testing technique of nuclear magnetic resonance (NMR). The pore complexity is analyzed through fractal dimension analysis. The pore distribution was assessed using magnetic resonance imaging (MRI). The roughness of pore distribution was analyzed by combining gray theory and counting the box dimension. The addition of biomass particles results in a 22-fold increase in the content of harmful pores, which can be quantitatively and visually observed using NMR and MRI techniques. This is the reason of compressive strength deterioration. Additionally, the optimal substitution of wood chips is determined. With a maximum biomass admixture of 50%, the uniaxial compressive strength increased by 18.9% (to 2.565 MPa, which meets the strength requirements of self-bearing wall insulation blocks). This study aims to establish a foundation for utilizing ecological concrete in agricultural facilities and rural wall materials.

Self-similarity study based on the particle sizes of coal-series diatomite

Coal-series diatomite (CSD) is widely distributed in China and has poor functional and structural properties and exhibits limited utilization of high value-added materials, resulting in a serious waste of resources and tremendous pressure on the environment. Moreover, due to differences in the mineralogical characteristics of CSD, different particle size scales (PSSs) have different functional structures and exhibit different self-similarities. In this study, we took CSD as the research object and PSS as the entry point and carried out a self-similarity study based on gas adsorption and an image processing method to illustrate the microstructures and self-similarities of different PSSs. The results showed that the pore structure of the CSD was dominated by mesopores and macropores and basically lacked micropores. The fractal dimensions were calculated with the Frenkel-Haisey-Hill (FHH) model and Menger model, and the DF1 values for − 0.025 mm and − 2 mm were 2.51 and 2.48, respectively, and the DM1 values were 3.75 and 3.79, respectively, indicating that the mesopore structure of the fine PSS was complex, whereas macropores were present in the coarse PSS. MATLAB was programmed to obtain grayscale thresholds, binarized images, grayscale histograms, three-dimensional (3D) reconstruction images and box dimensions, which enabled us to observe the microstructures and self-similarities of the CSD. Self-similarity studies based on particle sizes are very important for functional application of CSD.Please note that article title mismatch between MS and JS we have followed MS, kindly check and cofirm.Yes, I have checked and confirmed.Kindly check and confirm corresponding author mail id are correctly identified.Yes, I have checked and confirmed.

Influence of nonlinearity to box-counting dimensions of spiral orbits for two-dimensional differential systems

This study focuses on spiral orbits approaching the origin of two-dimensional nonautonomous nonlinear systems. The main result explains that the fractal dimensions (box-counting dimensions) of spiral orbits of the systems on the phase plane can be derived from the power of the nonlinear term. The examples given show that when the coefficients are power functions, the balance between their power and the power of the nonlinear term determines the fractal dimension. In addition, some numerical simulations are also included.

Damage evaluation of rock blasting based on multi-fractal study

This study shows the failure mechanism of rock under blasting loads by CT scanning and utilizing fine sandstone as a test specimen. The results indicated that the 3D reconstruction technique can be employed to construct macroscopic fissures in the rock. The complexity of macroscopic fissures in the rock after blasting is quantitatively evaluated using the 3D box-counting dimension. In addition, a multi-fractal method is proposed for comprehensively evaluating macroscopic fissures and microscopic damage of rock after blasting. Compared to the numerical simulation results, the multi-fractal dimension represents the cracks and damage of rock after blasting more precisely than the simple fractal dimension. The blasting experiments of the jointed sandstone demonstrated that the 3D box dimension is suitable for evaluating the 3D macroscopic fractures of the rock after blasting and that it is a straightforward method of macroscopic evaluation. The multi-fractal dimension can evaluate the macroscopic fissures and microscopic damage of various CT scanning rock layers with greater precision.

Applications of the Information Dimension in Detecting Border Perturbations

This study utilizes the concept of information dimension based on Shannon and Tsallis' entropy to analyze the contours of flat objects. Our objective is to employ the information dimension for detecting perturbations in borders. We create examples of squares with slight border perturbations, possessing the same mass and sharing identical box-counting dimensions yet exhibiting distinct information dimensions. This construction was devised with the understanding that entropy is responsive to the image frequency within each box. Consequently, the information dimension provides a more precise index of fractal shapes when compared to the box-counting dimension.

Numerical investigations and design optimization of cryogenic plate-fin heat exchanger in Electric Arc Furnace using radiative heat transfer model

The most crucial element affecting cryogenic systems’ energy efficiency is heat exchangers. In this article, a model for cryogenic plate-fin heat exchanger (PFHE) under thermal-hydraulic performance with various blends is proposed. To reduce five objective functions total heat transfer rate, total weight, total mass flow rate, number of entropy generating units, and whole annual cost the study carried out the PFHE’s optimal design. This study innovatively focuses on optimizing PFHE design for cryogenic conditions using Opposition Learning with the Beetle Swarm algorithm. The proposed geometry addresses non-uniform temperature distribution, emphasizing thermal-hydraulic properties in various blends. Additionally, a second-order semi-discretization radiative heat transfer approach enhances boiler simulation accuracy. A multi-objective optimization of Opposition Learning with Beetle Swarm (MO-OBBS) optimization approach was suggested in the article for the design of PFHE. Despite the significant potential for renewable energy the low-temperature waste heat treatment process is rarely examined from a sustainability perspective. Consequently, the study proposed a second-order semi-discretization radiative heat transfer method to simulate a furnace using a water heat recovery process. The cooling panel’s numerical analysis was conducted using computed temperature fields. The performance of the various algorithms is compared using the Freidman rank-sum test to determine the statistical significance of the findings obtained. In logarithmic coordinates, the proposed method’s faster convergence is compared to GQB-PSO, SLTLBO, Jaya, CKKO, and OLE-SCA algorithms. The proposed method exhibits a maximum average increase of 115.39% for PFHE design. When the numerical simulation results are compared to measurements of output water temperature, the accuracy is highly improved.

Numerical treatment of entropy generation and Bejan number into an electroosmotically-driven flow of Sutterby nanofluid in an asymmetric microchannel

This article aims to investigate the electrokinetic peristaltic flow of the Sutterby nanofluid through an asymmetric microchannel with a porous medium and the Joule heating parameter. The magnetic field is applied in the transverse direction and the electric field on the flow direction. The flow model consists of the continuity, momentum, heat, and electric potential equations, with appropriate boundary conditions. The Debye-Hückel linearization approximation is considered. Under the long wavelength and small Reynolds number approximation, the lengthy equations were reduced. The resultant coupled equations are numerically solved by the renowned NDSolve coding using Mathematica software. Entropy and Bejan number are also incorporated in this study. The velocity, temperature, and the phenomenon of entrapment are analyzed in depth with the aid of graphical representations. Streamlines of various waveforms are also discussed. The analysis discovered that the velocity of the fluid enhanced for porosity parameter and reverse effects is observed on the quantity of the magnetic parameter.

Study on Mechanical Model for Postpeak Shear Behavior of Rock Joints Based on Degradation Characteristics of the 3D Morphology

The 3D morphology of the joint surface significantly influences the shear behavior of the jointrock. Constant normal load (CNL) direct shear tests with different shear displacement were conducted to understand the shear stress changing with joint roughness and damage degree during shear. The rough joint specimens were prepared using 3D scanning and printing techniques, and shear tests with different normal stresses and shear displacements were performed. Four different parameters and the damaged area quantitatively described by the image binarization and box dimension were calculated and compared to study the roughness evolution of joint surfaces. The experimental results demonstrated that the roughness parameter and shear stress decrease and approach constant values with increasing shear displacement. A JRC degradation model was presented based on regression analyses to evaluate the JRC values of rock joints under various displacements to replace it in the JRC–JCS model. Additionally, a new postshear behavior modeling was proposed for rock joints based on surface degradation characteristics under various initial joint roughness coefficients (JRC0) and normal stress. The stress–displacement curves resulting from the proposed modified model work well in predicting the postpeak stress–displacement curve, which can prove the effectiveness of the postpeak shear behavior modeling.

Limit theorems for a class of processes generalizing the U-empirical process

In this paper, we develop theory and tools for studying U-processes, a natural higher-order generalization of the empirical processes. We introduce a class of random discrete U-measures that generalize the empirical U-measure. We establish a Glivenko-Cantelli and a Donsker theorem under conditions on entropy numbers prevalent in the theory of empirical processes. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The uniform limit theorems discussed in this paper are key tools for many further developments involving empirical process techniques. Our results are applied to prove the asymptotic normality of Liu's simplicial median. We conclude this paper by extending Anscombe's central limit theorem to encompass randomly stopped U-processes, building upon its application to randomly stopped sums of independent random variables.

Hausdorff dimension and upper box dimension of a class of homogeneous Moran sets

ABSTRACT In this paper, we construct a class of special homogeneous Moran sets which is called the { m k } -homogeneous Moran sets by the connected components, and obtain the Hausdorff dimension and the upper box dimension of the { m k } -homogeneous Moran sets under some conditions.

Characterization of the Fine-Scale Evolution of Damage in Shale under the Influence of Two-Way Stress Differences Based on CT Images and Fractal Theory—The Example of the Anba Dyke in the Wufeng–Longmaxi Formation

To better understand the influence of different levels of two-way stress differences on the development of damage in Anchang diametral laminar shale in the northern Qianbei area, a numerical model of laminar shale with a representative fine-scale structure was established by using RFPA3D-CT. A triaxial compression test was conducted; a three-dimensional mesoscale fracture box dimension algorithm based on digital images was generated by using MATLAB R2020b; and the fractal characteristics were quantitatively analyzed. The results showed that under the influence of the horizontal stress ratio and two-way stress, the greater the two-way stress is, the more notable the plastic characteristics of specimen damage are, and the higher the residual strength is. The specimens with lower two-way stress exhibited obvious brittle damage characteristics. The difficulty degree of complex fracture network formation increased with the increase in the horizontal tension ratio, and the degree of increase in the fracture network complexity gradually decreased. At a horizontal stress ratio of 1.25, the fractal dimension was the highest, which indicates that the cracks were the most pronounced. Fracture formation after specimen damage was the most common phenomenon. Under the condition of a lower horizontal stress ratio, a large number of fracture structures could be generated in shale specimens after damage, promoting the expansion of natural fractures.

Thermo-entropy analysis of water-FMWCNT nano-fluid flow in a backward-facing channel with obstacle

Thermo hydraulic analysis of nano-fluid flow phenomena through backward-facing channel has currently gained attention of the researchers, and is applied in a variety of engineering applications. Hydro-thermal properties of water/FMWCNT nano-fluid flow in a backward-facing channel embedded with an obstacle under uniform heat flux have been studied numerically with the variations in the values of Reynolds numbers ( Re ) , weight percentage of carbon nano-particle ( wt . % ) , and differently shaped obstacles. The governing equations (continuity, x-momentum, y-momentum, and energy) have been solved using the finite volume approach, and fluent software has been used to visualize the simulation results. Impact of various forms of obstacle (plane, elliptical, and triangular), the ratio of obstacle height to width (A/B), the ratio of obstacle pitch to width ( B 1 / B ) , wt . % , and the values of Re on the different rheological behaviour of nano-fluid flow have been investigated. For all the forms of obstacle configuration, the distributions of velocity streamlines, temperature contours, absolute pressure drop ( Δ p ) , average friction factor ( F ) , local Nusselt number ( Nu ( x ) ) , skin friction coefficients ( c f ( x ) ) , average Nusselt number ( N u avg ), and pumping power ( P ) have been demonstrated for several values of Re ( 1 , 50 and 150 ) , and wt . % ( 0 , 0.12 and 0.25 ) . For several values of Re and wt . % , generations of frictional entropy ( N frictional ) , thermal entropy ( N thermal ) , and Bejan number ( Be ) have been explored. It seems that the presence of obstacle, and the increase in the values of Re , wt . % causes the rise of the values of Δ p , P , and N u avg than the case of smooth channel. Moreover, at wt . % = 0.25 and Re = 150 , it has been found that the % increase in N u avg becomes 47.37 in the case of backward-facing channel with plane obstacle (A/B = 1) as compared to the case of smooth channel. Furthermore, at a constant value of wt . % it has been found that the generation of N thermal decreases but the generation of N frictional entropy increases with the rise in the values of Re .

Forensic Analysis of the Abraham Lincoln Assassination: An On-Site Study of the Presidential Box at Ford's Theatre.

On April 14, 1865, Abraham Lincoln was assassinated while watching a play from the Presidential Box at Ford's Theatre in Washington, DC. There is still controversy concerning the findings of Lincoln's autopsy. The physicians that attended the autopsy documented that the bullet entered the left occipital region of the brain, but opinions differ as to the path the bullet took through the brain. The official autopsy report documented that the bullet traveled through the left brain and did not cross the midline. Others who watched the autopsy claimed that the bullet entered on the left side of the president's brain, crossed the midline, and ended up just above the orbit on the right. In this manuscript, we reviewed all of the statements of the witnesses to the assassination in an effort to reconstruct the approach that John Wilkes Booth, the assassin, took through the Presidential Box as he approached the president. In addition, we conducted an on-site analysis of the shape and dimensions of the Presidential Box at Ford's Theatre to support the approach that Booth took. Based on this forensic analysis, we provide supportive evidence that the findings of the official autopsy report are accurate; that is, the bullet that entered the president's left brain stayed on the left and did not cross the midline.

Estimation of the Compressive Strength of Cardboard Boxes Including Packaging Overhanging on the Pallet

In this study, a numerical investigation was conducted on a verified packaging model, which sticks out beyond the pallet base area, which will evidently weaken its load-bearing capacity. This could lead to damage of the protected goods transported within this packaging. It might also result in the unnecessary overengineered design of the packaging, particularly when the potential for overhanging is anticipated beforehand, but its exact extent is not known. The article analyzed hundreds of cases, varying in terms of packaging dimensions (from 150 mm up to 600 mm), the extent of protrusion beyond the edge of the pallet (from 1% to 50% of box dimensions) and the use of various corrugated boards (B-, C-, EB- and BC-flute), in order to assess the decrease in the load-bearing capacity of the packaging compared to reference packaging, which was not overhanging on a pallet. For instance, it appeared that the decrease in the load-bearing capacity of the packaging when overhanging was insensitive to the corrugated cardboard material used. Additionally, the decrease in box strength was rapid while overhanging, even for a small value of overhanging.

Microbic flow analysis of nano fluid with chemical reaction in microchannel with flexural walls under the effects of thermophoretic diffusion

The current investigation examines the peristaltic flow, in curved conduit, having complaint boundaries for nanofluid. The effects of curvature are taken into account when developing the governing equations for the nano fluid model for curved channels. Nonlinear & coupled differential equations are then simplified by incorporating the long wavelength assumption along with smaller Reynolds number. The homotopy perturbation approach is used to analytically solve the reduced coupled differential equations. The entropy generation can be estimated through examining the contributions of heat and fluid viscosities. The results of velocity, temperature, concentration, entropy number, and stream functions have been plotted graphically in order to discuss the physical attributes of the essential quantities. Increase in fluid velocity within the curved conduit is noticed for higher values of thermophoresis parameter and Brownian motion parameter further entropy generation number is boosted by increasing values of Grashof number.