Menger Sponge Research Articles

Effect of pore structure characteristics on the properties of red mud-based alkali-activated cementitious materials

This study used a mercury intrusion porosimeter (MIP) to establish a Menger sponge model and a thermodynamic fractal model to investigate the influence mechanism of changes in the pore structure of red mud-based alkali-activated cementitious materials under different Na2O contents on their mechanical properties and volume stability. The results indicate that the thermodynamic fractal model can better reflect the pore size distribution within the entire pore size measurement range, with a correlation coefficient R 2 maintained between 0.98347 and 0.99854. Moreover, the thermodynamic fractal model can well reflect the relationship between the pore structure characteristics of the matrix and its macroscopic properties, with the increase of Na2O content, the N-(C)-A-S-H gel content continuously increased, and the pore fractal dimension increased from 2.42609 to 2.72440, indicating a denser pore structure, which promoted the improvement of compressive and flexural strength of the matrix at 28 days while deteriorating volume stability.

Printing resolution effect on mechanical properties of porous boehmite direct ink 3D printed structures

Direct ink writing (DIW) is one of the facile and versatile manufacturing techniques for additive manufacturing of ceramics. 3D printed porous boehmite structures fabricated via DIW can offer significant benefits in terms of their high specific surface area and robust mechanical properties, rendering them suitable for applications like catalysis, adsorption, etc. The characteristics of the boehmite structures produced through 3D printing are significantly influenced by several parameters associated with the DIW process, including but not limited to solids concentration in the ink, resolution, and drying time. In this work, for the first time, we try to optimize the DIW parameters for 3D printing of porous boehmite structures to achieve better shape stability, resolution, and mechanical properties. A shear-thinning boehmite ink was prepared by peptization of boehmite powder in nitric acid and the rheological properties were optimized by varying concentrations of boehmite and nitric acid. Boehmite ink with a solid loading of 37 % and a pH of 4 was found to have the optimal rheological properties and printability. A Menger sponge porous structure was printed by DIW using the optimized ink with different sample sizes and resolutions. The resolution and size were optimized for better shape accuracy of printing and structural integrity. Crack-free boehmite structures were obtained by optimizing the drying time. Boehmite structures printed with higher resolution resulted in enhanced compressive strength and energy absorption, i.e., compressive strength and energy absorption of 0.5 mm resolution boehmite structure is enhanced by 208 % and 144 %, respectively, compared to the 1 mm resolution structure.

Torricelli’s Law in Fractal Space–Time Continuum

A new formulation of Torricelli’s law in a fractal space–time continuum is developed to compute the water discharge in fractal reservoirs. Fractal Torricelli’s law is obtained by applying fractal continuum calculus concepts using local fractional differential operators. The model obtained can be used to describe the behavior of real flows, considering the losses in non-conventional reservoirs, taking into account two additional fractal parameters α and β in the spatial and temporal fractal continuum derivatives, respectively. This model is applied to the flows in reservoirs with structures of three-dimensional deterministic fractals, such as inverse Menger sponge, Sierpinski cube, and Cantor dust. The results of the level water discharge H(t) are presented as a curve series, showing the impact and influence of fluid flow in naturally fractured reservoirs that posses self-similar properties.

The Menger curve and spherical CR uniformization of a closed hyperbolic 3-orbifold

The Menger curve and spherical CR uniformization of a closed hyperbolic 3-orbifold

Pore heterogeneity analysis and control mechanisms in Cambrian shale of the Shuijingtuo Formation, Yichang area, China

This study focuses on understanding the fractal characteristics and controlling factors of micropore structures within organic-rich shale of the Cambrian Shuijingtuo Formation in the Yichang area of Hubei Province. Mineralogy, petrology, and organogeochemical characteristics were confirmed through comprehensive testing methods, including whole-rock X-ray diffraction and organic geochemical analyses. Additional experiments included low-temperature carbon dioxide adsorption, low-temperature nitrogen adsorption, and high-pressure mercury injection. Fractal dimensions of micropores, mesopores, and macropores were calculated using the V-S, FHH, and MENGER sponge models, respectively. Results indicate that the Cambrian Shuijingtuo Formation represents a typical deposit from an alkaline water body, resulting in high-calcareous shale. Fractal dimensions were as follows: micropores (D1) ranged from 2.1138 to 2.3475 (average 2.2342), mesopores (D2) ranged from 2.5327 to 2.7162 (average 2.6171), and macropores (D3) ranged from 2.7361 to 2.9316 (average 2.82905). Correlations were observed between total organic carbon (TOC) content and Ro with D1 and D2 (positive) and D3 (negative). Shale pore volume and specific surface area exhibited positive correlations with D1 and D2 but negative correlations with D3. High bio-deposited silica positively influenced micropore and mesopore development, while clay mineral compaction and dehydration transformations favored macropore development. Carbonate minerals primarily contributed to regular macropores, with complex correlations involving fractal dimensions D1, D2, and D3. The research results provide theoretical support for analyzing pore fractal characteristics of shallow old Marine shale reservoirs and the prediction and development plan of high-quality reservoirs of the Shuijingtuo Formation in the Yichang area.

The Kinetics of Semi-Coke CO2 Gasification Based on Pore Fractal Growth

The gasification kinetics of semi-coke are an important research topic in the gasification process of semi-coke. The evolution of the pore structure is one of the most important factors affecting the gasification rate of semi-coke. In this paper, the pore fractal growth model was established based on the principle of pore fractal growth and the Sierpinski sponge structure. Three kinds of semi-coke raw materials were used to prepare porous carbon with different degrees of gasification. Combined with the TG curves of raw materials, the gasification kinetics based on the fractal model were verified. The curves of the gasification reaction rate and the specific surface area as a function of carbon conversion were consistent with the random pore model and experimental data, which verified the feasibility of the model. The pore fractal dynamic model could predict the change in the pore structure with carbon conversion during semi-coke gasification, so as to reveal the kinetic law of carbon gasification.

Doubling properties of self-similar measures and Bernoulli measures on self-affine Sierpinski sponges

Doubling properties of self-similar measures and Bernoulli measures on self-affine Sierpinski sponges

Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary

Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary

Understanding the impact of synthesis parameters on the pore structure properties of fly ash-based geopolymers

This paper systematically investigated the effects of alkali activator modulus (AM), alkali dosage (AD), and water-fly ash ratio (W/FA) on the pore volume, pore size distribution, pore volume fractal dimension (Dv), and pore surface fractal dimension (Ds) of fly ash-based geopolymer (FABGs) by orthogonal test and mercury intrusion porosimetry test. Dv and Ds of FABG can be calculated by the Menger sponge model and Zhang and Li's models, respectively. The relationship between multiple synthesis parameters and pore characteristics of FABGs was detailedly studied through range analysis and correlation detection analysis, which provides a basic understanding of the influence of multiple parameters on the pore characteristics of FABG, as well as the relationship between pore characteristics and mechanical properties. The results show that AM has the greatest effect on the volume proportion of gel pores and transition pores, and AD and W/FA have the greatest effect on the volume proportion of capillary pores and large pores, respectively. W/FA has the greatest effect on the porosity, and the porosity and the most probable aperture increase with the increment of W/FA and decrease with a rise in AD. It is worth noting that AM, AD, and W/FA have little effect on the Dv of larger pores, but have a greater on the Dv of smaller pores, with the influence order of AD-W/FA-AM. Ds of the large pores and capillary pores range between 2.70–2.90, and the Ds of the transition pores and gel pores are mostly greater than 3.00, which suggests that there may be more ink-bottle pores in transition pores and gel pores. In addition, there was a strong linear relationship between the porosity and compressive strength of FABGs, and the compressive strength of FABGs showed a decreasing trend with the increase of porosity. Compared with Dv, there is a stronger correlation between Ds and compressive strength, indicating that the roughness of the pore surface and the pore size distribution have a greater influence on the compressive strength of FABGs.

Investigation into compressive property, chloride ion permeability, and pore fractal characteristics of the cement mortar incorporated with zeolite powder

As a non-homogeneous porous material, the microstructure of cementitious composites critically affects the mechanical and durability properties. This paper aims to analyze the effect mechanism of cement replacement by zeolite powder on the microscopic pore structure characteristics, macroscopic mechanical behavior, and durability of mortar to reduce cement consumption and improve mortar performance. The compressive strength and chloride ion permeability of mortar incorporated with zeolite powder were experimentally determined, and the effects of zeolite powder content on the composition of cement hydration products and pore structure were further analyzed by X-ray diffraction (XRD), scanning electron microscope (SEM), energy dispersive spectrometer (EDS), and mercury intrusion porosimetry (MIP). The pore structure characteristics were quantitatively analyzed by fractal theory. The results shows that incorporating zeolite powder in cement mortar fills the pores, promotes secondary hydrate formation, reduces the Ca/Si ratio, refines the pore size, and reduces the porosity of large pores. Thus, the internal pore structure of the matrix is significantly improved, reducing the pore fractal dimension and improving the compressive strength and chloride ion penetration resistance. 15% cement replacement zeolite powder has been found optimal in reducing the porosity and chloride ion permeability coefficient while enhancing the compressive strength. The pore structure fractal dimensions calculated by the Menger sponge fractal model can quantitatively describe the pore structure distribution characteristics, which can help ascertain the relationship between zeolite powder content, pore structure quantitative characteristics, and macroscopic mechanical behaviors.

Improved deep artificial neural network-powered prediction of extreme mechanical performances of fractal architectures with high hierarchical rank

This work explores the mechanical performances, including anisotropy, wave propagation, and buckling resistance capability of the Menger sponge and Jerusalem fractal structures, with the high rank, using an improved deep artificial neural network (ANN) model inserted by a loop with multiple conditions. Where, the effective mechanical attributes, including Young’s modulus, shear modulus, and Poisson’s ratio of the Menger sponge structure with rank 4 are attained using high-performance computer cluster (HPCC) that are then used to train and also validate the ANN model. The accuracy of the deep ANN model trained by data of fractal structures with and without rank 4 is verified by the numerical model and experiment with the good agreement found or the maximum percent difference being 15.6% where the numerical and experimental data don’t belong to its training samples. Remarkably, the deep ANN model demonstrates an extremely low cost, but very fast compared with the numerical and experimental works. Namely, the improved deep ANN model implemented by a regular computer can accurately predict the mechanical attributes of Menger sponge fractal structures with full ranks in 3 min. While it takes a week for numerical work using HPCC or experiment method to produce the same mechanical properties of the Menger sponge with only rank 4, displaying at least thousands of times slower than the ANN model. The ANN results indicate that the onset of anisotropy of considered fractal structures is from rank 1, and the reversal of anisotropy will take place upon an increase in the rank while such reversal is not seen for the case of lattice structure. The anisotropy reversal could be one of main reasons why the fractal structure tends to have a high rank in nature. In addition, the buckling stress and phase wave velocity are observed to decrease with rising rank, with a nonlinear pattern.

Bose-Einstein Condensation of q-Deformed Bosons Harmonically Trapped on Sierpiński Carpet and Menger Sponge

Bose-Einstein Condensation of q-Deformed Bosons Harmonically Trapped on Sierpiński Carpet and Menger Sponge

A combinatorial Fredholm module on self-similar sets built on $n$-cubes

We construct a Fredholm module on self-similar sets such as the Cantor dust, the Sierpiński carpet and the Menger sponge. Our construction is a higher dimensional analogue of Connes' combinatorial construction of the Fredholm module on the Cantor set. We also calculate the Dixmier trace of two operators induced by the Fredholm module.

Effective thermo-electro-mechanical properties of Menger sponge-like fractal structures: a finite element study

Menger sponges are hierarchical structures with tunable mechanical and electrical properties. In this work, different orders (0th, 1st, 2nd and 3rd) of hierarchical structures were studied for their effective properties by square, circular and hexagonal-shaped cavities. The elastic modulus, Poisson’s ratio, thermal and electrical conductivities were investigated as a functions of the density. The variation of normalized parameters with normalized density for square, cylindrical, and hexagonal-shaped cavities was used to obtain the empirical relations. The normalized specific modulus and Poisson’s ratio were validated using available analytical models for all cavities. The normalized Poisson’s ratio, thermal conductivity and electrical conductivity decreased with a reduction in the effective density. The effect of a different cavity (square, cylindrical and hexagonal) on the Menger sponge’s mechanical and electrical behaviour shows variation after the effective density falls below 0.8. Menger sponge with a square cavity shows the maximum decrement in thermal and electrical conductivity among other cavities with increasing order of structure. Menger sponge with hexagonal cavity consists of least reduced normalized thermal and electrical conductivity with decreasing effective density. An increment in the order of fractals leads to a near-zero value for Poisson’s ratio. These structures can be used for medical, aerospace, and industrial applications according to the properties required in different applications.

Relationship between fractal feature and compressive strength of fly ash-cement composite cementitious materials

Microscopic pore structure and compressive strength of fly ash-cement composite cementitious materials at 20 °C and 60 °C measured by mercury intrusion porosimetry and compressive strength test. Microscopic pore structure was analyzed from porosity, pore volume, total pore surface area, average pore diameter, median pore diameter, most probable diameter, threshold diameter and fractal feature. The fractal dimensions of each specimen were calculated by Menger Sponge. The regression analysis between fractal dimension and pore structure parameters were fitted by linear function, quadratic polynomial function, power function and exponential function, respectively. Eventually, the relationship between pores structure and compressive strength was characterize by the mathematical model between the optimal comprehensive parameter of pores structure and compressive strength. And the T-test and F-test were used to reveal the significance of the regression parameter and function. Research results showed that the fractal curves of all specimens consist of four segments. The pores have obvious fractal features in Region I and Region IV, while presented no-fractal feature in Region II and Region III. In Region I and Region IV, fractal dimension followed the exponential function with average pore diameter, median pore diameter, most probable diameter and threshold diameter, consistent with quadratic polynomial function with porosity, pore area and pore volume, and accords with the positive power exponential function with compressive strength. Thus, the morphology of hierarchical aggregates expressed by fractal dimension can reflect the relationship compressive strength and pore structure in fly ash-cement composite cementitious materials more directly.

Multi-scale pore fractal characteristics of differently ranked coal and its impact on gas adsorption

Well-developed pores and cracks in coal reservoirs are the main venues for gas storage and migration. To investigate the multi-scale pore fractal characteristics, six coal samples of different rankings were studied using high-pressure mercury injection (HPMI), low-pressure nitrogen adsorption (LPGA-N2), and scanning electron microscopy (SEM) test methods. Based on the Frankel, Halsey and Hill (FHH) fractal theory, the Menger sponge model, Pores and Cracks Analysis System (PCAS), pore volume complexity (Dv), coal surface irregularity (Ds) and pore distribution heterogeneity (Dp) were studied and evaluated, respectively. The effect of three fractal dimensions on the gas adsorption ability was also analyzed with high-pressure isothermal gas adsorption experiments. Results show that pore structures within these coal samples have obvious fractal characteristics. A noticeable segmentation effect appears in the Dv1 and Dv2 fitting process, with the boundary size ranging from 36.00 to 182.95 nm, which helps differentiate diffusion pores and seepage fractures. The D values show an asymmetric U-shaped trend as the coal metamorphism increases, demonstrating that coalification greatly affects the pore fractal dimensions. The three fractal dimensions can characterize the difference in coal microstructure and reflect their influence on gas adsorption ability. Langmuir volume (VL) has an evident and positive correlation with Ds values, whereas Langmuir pressure (PL) is mainly affected by the combined action of Dv and Dp. This study will provide valuable knowledge for the appraisal of coal seam gas reservoirs of differently ranked coals.

Experimental investigation of the pore fractal characteristics and damage degradation mechanism of sandstone after cyclic Freeze‒thaw treatments

To reveal the microscopic pore characteristics and strength weakening mechanism of rock under the influence of cyclic freeze-thaw treatment, uniaxial compression tests and MIP (mercury intrusion porosimetry) were conducted on red sandstone specimens with different numbers of freeze-thaw cycles (0, 20, 40, 60, and 80). The experimental results showed that cyclic F-T (freeze-thaw) treatment can lead to an increase in porosity and the expansion of the pore structure. Specifically, the porosity increased by nearly 30.22% from 8.467% without freeze-thaw to 11.026% after 80 freeze-thaw cycles. The distribution ratios of mesopores (2–10 μm) and micropores (less than 0.1 μm) changed from 0.7% to 19.9% without freeze‒thaw to 12.6% and 11.6% after 80 freeze‒thaw cycles, which showed an expansion trend from small to large pores. For the PFD (pore fractal dimension) of sandstone, three fractal models (capillary pressure model, Menger sponge model and thermodynamic model) were used to calculate the pore fractal dimension of different freeze-thaw cycles, which gradually increased with the increase in freeze‒thaw cycles. Specifically, the PFD increased by 3.02, 3.01 and 2.65% after 20, 40, 60, and 80 freeze-thaw cycles, respectively. Additionally, the PFD calculated by the capillary pressure model and Menger sponge model had piecewise characteristics, which can better characterize pore fractal characteristics with pore diameters less than 2 μm. In summary, based on thermodynamic theory and pore water frost heaving deformation theory, a freeze-thaw damage degradation model based on the porosity of a fractured rock mass was established, and the theoretical value obtained was in good agreement with the experimental results.

Right-angled Coxeter groups with non-planar boundary

We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle-free defining graphs such that the associated right-angled Coxeter group has boundary a Menger curve.

Multi-scale pore structure characteristics and main controlling factors analysis of Longtan formation shale in Northwest Guizhou

Marine-continental transitional shale strata are widely distributed in China and have high gas potential. The Longtan shales are a typical marine-continental transitional coal-bearing shale system in Northwest Guizhou. Low-temperature N2 adsorption experiments and high-pressure mercury injection experiments were carried out on the unweathered shale samples from Well JS-1. The multi-scale fractal dimensions were calculated based on the MESP (Menger-sponge) model, MESA (mercury saturation) model, and FHH (Frenkel-Halsey-Hill) model respectively. The macropores (≥100 nm) are mainly inorganic with a fractal dimension D1 between 2.8628–3.2057, indicating the macropore structure is relatively complex. The proportion of the macropores in total pores is not high in comparison to mini-micropores (<50 nm), and the content of brittle minerals and pyrite mainly controls its structure. Among the mesopores (50–100 nm), inorganic pores are still the primary pore type, but the proportion of organic pores increases. The fractal dimension D2 is between 2.2125–2.3016. It has medium complexity, and the proportion in the total pores is slightly higher than that of macropores. The influence of organic matter abundance on mesopore structure is greater than that in macropore because of the increase in the proportion of organic pores, making the controlling mechanism of mesopore structure more complicated. Mini-micro pores are mainly organic pores. The fractal dimension D3 (fractal dimension of mini-micropores under low relative pressure, P/P0 ≤0.5) ranges between 2.6709–2.8648, and D3’ (fractal dimension of mini-micropores under high relative pressure, P/P0 >0.5) ranges between 2.6661–2.9256, indicating complex pore structures and rough surfaces. Mini-micropore accounts for the highest proportion of the total pores and its structure is mainly controlled by the abundance of organic matter. The pore structures of macropores and mesopores greatly influence the proportion of desorption gas in shale, while the structure of mini-micropores can control the maximum adsorbed gas volume of shale.

STUDY ON FRACTAL CHARACTERISTICS OF FISSURE SPACE STRUCTURE AND TIGHT SOLID STRUCTURE OF COAL

The microstructure of a coal is a kind of fissured fractal tight porous media. The formation and action mechanism between its space and solid structure has a strong fundamental significance for engineering seepage problem. Currently, the research and experimental theory of the quantitative characterization of the fissure space structure is relatively complete. However, the physical meaning of the fractal dimension used to characterize structural features remains unclear. At the same time, the tight feature is attributed to the original complexity, and fractal topography dominates the behavioral complexity. Therefore, it is of great significance to clarify the distribution of tight solid structure from the fractal perspective for clarifying the complexity of coal structure. In this paper, the proportion of space structure and solid structure in coal body is accurately measured by scanning electron microscope (SEM) experiment combined with nuclear magnetic resonance (NMR) experiment, and a theoretical model which can be used to measure the fractal dimension of solid structure is proposed based on Menger sponge theory. In the process, the physical meaning of fractal dimension of the fissure space structure measured by the image analysis method and fluid intrusion method is proposed. The research shows that for the fissure space structure, the fractal dimension of fissure space structure calculated based on the SEM image and the box dimension method is between 1 and 2, showing an obvious positive correlation with the proportion of the fissure space structure area. The fractal dimension based on NMR [Formula: see text] spectrum curve fitting mainly characterizes the complexity of the pore size distribution. The fractal dimension increases with the increasing complexity of pore size distribution. For the tight solid structure, the fractal dimension obtained by theoretical calculation is less than 3, the compound fractal scaling law. The two-dimensional fractal dimension obtained through experiments and related algorithms is greater than 2, which do not conform to the fractal scaling law in general, but it can represent the proportion of solid structure in a limited scale as the theoretical fractal dimension.