Explanation of scaling phenomenon based on fractal fragmentation

A significant ‘size effect' is observed in tensile strength of solid particles, such as ice, rock, ceramics and concrete: the tensile strength is not independent of the fragment size, but decreases with increasing size. The Weibull statistical theory was universally used to calculate the size effect observed in solid particles. Recent developments in fractal theory suggest that fractals may provide a more realistic representation of solid particles. In this paper, the scaling phenomenon of ice mechanics is explained using the fractal model for ice particle fragmentation. The Weibull statistics is modified using the fractal crushing of ice, and is compared with the conventional one. Goodness-of-fit statistics show that the modified Weibull statistics fits the experimental data of ice much better than the conventional one. The modified Weibull statistics has only one parameter, the fractal dimension of the fragment size distribution, which has a general value of 2.50 for the ice fragmentations.     

A fractal study of sound propagation characteristics in roughened porous materials

The influence of rough elements of porous materials on sound propagation characteristics in roughened porous materials is investigated. Rough surface topography on the roughened porous materials is simulated by the fractal geometry theory, in which relative roughness is defined clearly as a function of the fractal dimension, porosity, average diameter and diameter ratio of the hexagonal elements on pore wall of the porous materials. Based on the sound propagation model of the smooth porous materials, a sound propagation model of the roughened porous materials is built. The effective density and bulk modulus, acoustic impedance and propagation constant, flow resistivity and sound absorption coefficient of the roughened porous materials are derived and discussed as a function of the relative roughness. It is demonstrated that the sound absorption coefficient of the roughened porous materials is improved as the relative roughness increases. The model predictions for the sound absorption coefficient of the roughened porous materials are well agreed with that of the existing test results.     

Implication of fractal geometry for fluid flow properties of sedimentary rocks

It is demonstrated that a certain amount of order can be extracted from an apparently random distribution of pores in sedimentary rocks by exploiting the scaling characteristics of the geometry of the porespace with the help of fractal statistics. A simple fractal model of a sedimentary rock is built, and is tested against both the Archie law for conductivity and the Carman-Kozeny equation for permeability. We demonstrate how multifractal scaling of pore-volume can be used as a tool for rock characterization by computing its experimentalf() spectrum, which can be modelled by a simple two-scale Cantor set.     

爆破地震信号的分形盒维数值分析

在岩石场地进行了单段深孔爆破试验,获得了具有该场地特征的爆破地震波传播规律。从理论上推导了爆破地震波振动强度衰减系数K和与分形盒维数D以及盒维数计算中-lg(k) lgNk的双对数拟合直线方程参数b的关系。采用了适用于爆破地震波曲线双尺度特征的矩形盒模型计算了爆破地震波的盒维数D。数据分析表明:在同一场地条件下,爆破地震波的盒维数比较稳定,且爆破地震波振动强度的场地衰减指数与D为两倍的关系;药量和距离对盒维数拟合直线方程参数b的影响明显,且其关系与场地衰减指数对爆破地震波峰值强度的作用相近。通过数据分析,获得了参数b与爆破地震波振动峰值A的关系式:b=0.689lgA+3.0669,其相关系数为0.93。     

A fractal analysis of subcooled flow boiling heat transfer

A fractal model for the subcooled flow boiling heat transfer is proposed in this paper. The analytical expressions for the subcooled flow boiling heat transfer are derived based on the fractal distribution of nucleation sites on boiling surfaces. The proposed fractal model for the subcooled flow boiling heat transfer is found to be a function of wall superheat, liquid subcooling, bulk velocity of fluid (or Reynolds number), fractal dimension, the minimum and maximum active cavity size, the contact angle and physical properties of fluid. No additional/new empirical constant is introduced, and the proposed model contains less empirical constants than the conventional models. The proposed model takes into account all the possible mechanisms for subcooled flow boiling heat transfer. The model predictions are compared with the existing experimental data, and fair agreement between the model predictions and experimental data is found for different bulk flow rates.     

混凝土剪切断裂能与分形维数关系的研究

混凝土断裂试件断口蕴涵着丰富的力学信息，而传统的欧几里得几何难以对复杂的断口形貌进行描述，本文引入分形几何理论对断口形貌进行研究，结合宏观的断裂参量进行数值分析，从而找到了混凝土断裂微观现象与宏观力学量联系的桥梁，本文分析了混凝土四点剪切加载试件的断裂性能，计算了断裂韧度、断裂能等相关参数，并通过扫描电子显微镜对断口形貌观测，运用分形理论计算了此类断口的分形维数，分析了Ⅱ型断裂断裂能和分形维数之间的关系。     

Tensile fracture behavior of heterogeneous materials based on fractal geometry

Many of heterogeneous structural materials, like concrete, have different behavior under tensile stresses in comparison to their behavior under compressive stresses. The aim of this paper is to interpret behavior of such materials subjected to tensile stresses, by using newly introduced concept of fractal geometry. In the first part of this paper, tensile behavior of granular composites has been studied by using fractal geometry. It is shown that the fractality of the cross section in this kind of composites can be used to interpret the size effect on tensile strength. In fact, this work is a modification with innovations on the previous studies on fractal based size effect.This hypothesis that the fracture surfaces of quasi-brittle materials are fractals has been verified by several investigations. Accordingly, in the other part of this paper, softening process in heterogeneous materials is studied. Resulting from presented approach, a new softening curve for quasi-brittle materials is proposed. This new softening curve is denominated “Quasi-fractal softening curve” and is consisted of two parts, a linear portion in beginning part and an exponential portion in rest of the curve. This makes it very compatible to the pre-existing softening curves.     

Scale dependent variability of soil electrical conductivity by indirect measures of soil properties

Knowledge of scale dependent variation of soil properties is important where upscaling and generalization from plot scale studies to field and larger scale is desired. We used conventional statistics, geostatistics, and fractal analysis to characterize and compare the apparent soil electrical conductivity (ECa) of six contiguous agricultural fields each ranging between 9.5 and 14.0 ha in size. Factor analysis revealed that ECa was strongly related to ammonium extractable K, organic matter (OM), pH and Bray-2 Phosphorus, but not to ammonium extractable Ca and sum of bases. All six fields were spatially structured and well described by exponential semivariograms. Fractal dimensions estimated from the linear portion of the semivariogram using a linear plateau model were statistically different (p = 0.05) among some of the fields, and the differences may have been caused by management differences. Fractal analysis identified at least two scales of variation for the fields. The first scale of variation, common to all six fields, was for distances less than 9 m. The second scale of variation was for distances ranging between 9 and 46 m (field NC), 9 and 79 m (fields SC and SW); and 9 and 126 m (field SE). Two of the fields (fields NW and NE) did not have a plateau on the log–log plot of the semivariograms, indicating a scaling behavior at larger distances. The study showed that although the semivariogram forms are similar among the six fields, the rate of change of the semivariograms (as indicated by the fractal dimension) differs for some of the fields at distances greater than 9 m.     

基于分形理论的煤层气非达西流动分析

在考虑了煤层的分形特征和启动压力梯度影响的基础上,建立了无限大煤层中气体低速非达西流动的数学模型,并求得了量纲为一的井底压力的Laplace空间解析解,并根据数值求解结果绘制了典型的井底压力动态曲线。研究结果表明:在定产量生产的情况下,分形维数和量纲为一的启动压力梯度对早期井底压力动态无显著影响;在生产的中后期,由于受二者的影响,压力导数曲线上的径向流水平直线段消失;量纲为一的井储系数的影响主要表现在续流阶段,而吸附系数则主要影响煤基质向裂隙扩散的过渡阶段。     

Effect of localized faults on chaotic vibration of rolling element bearings

This paper investigates the effect of localized faults on the chaotic vibration of rolling element bearings. The presence of chaotic behavior is demonstrated using experimental vibration data. A nonlinear mathematical model is developed that captures bearing dynamics. The numerical simulations of the model agree with the experimental evidence and provide insight into the bearings chaotic response in a wide range of rotational speeds. The bearing chaotic behavior is quantified using the Lyapunov exponent and correlation dimension. It is further shown that these measures can be exploited in detecting bearing failure.     

The fractal research and predicating on the times series of sunspot relative number

In this paper, with the theory of nonlinear dynamic systems, It is analyzed that the dynamic behavior and the predictabilityfor the monthly mean variationsof the sunspot relative number recorded from January 1891 to December 1996. Inthe progress, the fractal dimension (D=3. 3±0.2) for the variation process wascomputed. This helped us to determine the embedded dimension [2×D+1]=7.By computing the Lyapunov index (λ₁=0.863), it was indicated that the variationprocess is a chaotic system. The Kolmogorov entropy (K=0.0260) was also computed, which provides, theoretically, the predicable time scale. And at the end, according to the result of the analysis above, an experimental predication is made,whose data was a part cut from the sample data.     

Effects of land use patterns on soil aggregate stability in Sichuan Basin, China

Soil aggregate stability as a key indicator of soil structure, is a product of interactions between soil environment, management practices, and land use patterns. The objective of this study was to analyze the impact of various land use patterns on soil aggregate stability in Sichuan Basin of southwestern China. The dry- and water-stable aggregate size distributions were determined by manual dry sieving procedure and Yoder＇s wet sieving procedure, respectively, while microaggregates and its mechanical and chemical stabilities by Kachisky＇s method, oscillator method, and citrate-dithionate （C-D） reagent method, separately. The results indicated that fractal dimension and surface fractal dimension were useful indicators to reflect soil aggregate distribution. Land use patterns have an obvious influence on soil aggregate stability. In the study area, water stability, mechanical stability, and chemical stability followed the sequence, Barren land 〉 forestland 〉 orchard 〉 cropland, and the original stability and collapse velocity were sensitive to soil properties and soil structure. The difference of aggregate stability under different land use patterns is mainly due to the intensity of human disturbance and cultivation. Improper land use patterns will lead to breakdown of unstable aggregates, producing finer and more-easily transportable particles and microaggregates. In the future, inappropriate cultivation and land use patterns should be changed to protect soil structure, to improve soil aggregate stability and soil fertility in Sichuan Basin.     

Effects of land use patterns on soil aggregate stability in Sichuan Basin, China

Soil aggregate stability as a key indicator of soil structure, is a product of interactions between soil environment, management practices, and land use patterns. The objective of this study was to analyze the impact of various land use patterns on soil aggregate stability in Sichuan Basin of southwestern China. The dry- and water-stable aggregate size distributions were determined by manual dry sieving procedure and Yoder's wet sieving procedure, respectively, while microaggregates and its mechanical and chemical stabilities by Kachisky's method, oscillator method, and citrate-dithionate (C-D) reagent method, separately. The results indicated that fractal dimension and surface fractal dimension were useful indicators to reflect soil aggregate distribution. Land use patterns have an obvious influence on soil aggregate stability. In the study area, water stability, mechanical stability, and chemical stability followed the sequence, Barren landforestlandorchardcropland, and the original stability and collapse velocity were sensitive to soil properties and soil structure. The difference of aggregate stability under different land use patterns is mainly due to the intensity of human disturbance and cultivation. Improper land use patterns will lead to breakdown of unstable aggregates, producing finer and more-easily transportable particles and microaggregates. In the future, inappropriate cultivation and land use patterns should be changed to protect soil structure, to improve soil aggregate stability and soil fertility in Sichuan Basin.     

A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

In this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell.     

A molecular mechanics study on size-dependent elastic properties of single-walled boron nitride nanotubes

We present an analytical study for the elastic properties of single-walled boron nitride nanotubes via a molecular mechanics model. Closed-form expressions for Young's modulus, Poisson's ratio and surface shear modulus are derived as functions of the nanotube diameter. The results are helix angle sensitive and comparable to those from ab initio calculations. This work is a first effort to establish analytical model of molecular mechanics for composite nanotubes and reveals the dissimilarities between size-dependent elastic properties of carbon and boron nitride nanotubes.     

空孔与运动裂纹相互作用的动焦散线实验研究

为研究预制裂纹不同偏移距离时运动裂纹与空孔的相互作用规律,采用动态焦散线实验系统,将预制裂纹的偏移距离设定为唯一变量,对含空孔的有机玻璃(PMMA)试件进行冲击三点弯实验。研究表明,存在两个临界距离：（6 mm (2 R )、9 mm (3 R )）,在该偏移距离下,裂纹扩展轨迹、动态断裂特性发生显著改变：(1) 预制裂纹偏移距离不大于3 mm时,裂纹贯穿空孔,发生二次起裂,且二次起裂的速度与应力强度因子显著大于一次起裂,无偏移时裂纹轨迹的分形维数为最小值;(2) 偏移距离增大至6 mm时,裂纹不再贯穿空孔,空孔对裂纹先吸引后排斥,裂纹速度与应力强度因子先减小后增大,裂纹轨迹的分形维数达到最大值;(3) 偏移距离大于6 mm时,空孔对裂纹的吸引作用逐渐减小,大于9 mm后,空孔对裂纹的吸引不再显著,裂纹起裂后即向落锤加载方向扩展直至贯穿试件。

     

枪械射击残留物分布密度的分形研究

为了推测发射枪械类别、弹种、枪械射击距离,利用X光摄影记录六种枪械发射子弹射击残留物的金属微粒密度与射击距离之间的关系,关系曲线表明二者是非线性的且具有分形特征;应用分形理论和线性回归原理,对射击残留物中的金属微粒分布密度进行计算和分析。结果表明,同种子弹由不同种枪发射时,其分形维数相应不同但相差较少;而不同种子弹由不同种枪发射时,其分形维数不同且相差较大。因此分形维数可以用来描述枪械发射子弹时射击残留物中的金属微粒密度分布情况。     

A comparative study of ECNC and CNC suspensions: effect of salt on rheological properties

This paper addresses the effect of monovalent (Na⁺) and divalent (Ca²⁺) ions on the shear viscosity and viscoelastic properties of two different aqueous suspensions of nanoparticles, cellulose nanocrystals (CNCs), and electrosterically stabilized nanocrystals of cellulose (ECNCs). ECNC is similar as CNC, but with carboxylated cellulose chains protruding from both endcaps. The different suspensions were studied in the semi-dilute regime, which corresponded to concentrations ranging from 0.5 to 8 wt% for CNCs and from 0.6 to 9 wt% for ECNCs. As the charges on CNCs are presumably distributed all along the crystal domains and the CNC have a twist in their backbone structure, their suspensions shifted to a cholesteric state as the volume fraction increased while ECNC suspensions did not. This is because the charges on ECNCs are mainly distributed at the endcaps of the particles and the protruding chains expel each other. On the one hand, it was demonstrated that at moderate ionic strength (I = 20 mM), CNC suspensions formed gels even at really low concentrations due to agglomeration. Calcium chloride (CaCl₂) had a greater effect than sodium chloride (NaCl) on both shear and viscoelastic properties due to stronger network formation. On the other hand, ECNCs could withstand much higher ionic strengths than CNCs. NaCl had no effect other than making the ECNC particles precipitate above a concentration of 200 mM, while CaCl₂ made ECNC suspensions turn into gels due to a bridging effect with their carboxylic acid groups at a concentration of [Ca2+] ≈ [COOH]/2.     

Evolution of properties in hydration of cements—A numerical study

The present work develops a numerical method for analysis of the microstructure and property evolution in the hydration of the cement. A time-dependent micro-mechanical model is established to investigate the microstructure development and the effective property evolution of the cement paste, while the input parameters of the model are based on experimental data. It is assumed that the cement paste composite consists of the anhydrous cement particles, cement gel and pores. The cement particles have a periodically spatial array and are wrapped by the cement gel. The Young’s modulus and Poisson’s ratio of the cement paste are calculated by direct average method and two-scale expansion method. The comparisons between the numerical results and experimental data show that this model can simulate the evolution of the microstructure and properties during the hydration of the cements quite satisfactorily.