锡基巴氏合金磨损表面的分形与磨损率

利用结构函数分析法研究了滑动摩擦学系统中金属磨损表面轮廓线的分形特性。结果表明：磨损表面轮廓线在小于Ｓｍ的尺度上具有分形结构。采用结构函数法可以方便地确定粗糙表面轮廓线的分形参数，即垂直于滑动方向上磨损表面轮廓线的分维Ｄ可作为磨损表面分维的特征值，它与金属磨损率的变化有着密切的关系，最佳分维Ｄｏｐｔ值对应于材料的最低磨损率。     

Fractal Prediction Model of Spontaneous Imbibition Rate

By utilizing fractal dimension as one of the parameters to characterize rocks, a mathematical model was derived to predict the production rate by spontaneous imbibition. This fractal production model predicts a power law relationship between spontaneous imbibition rate and time. Fractal dimension can be estimated from the fractal production model using the experimental data of spontaneous imbibition in porous media. The experimental data of recovery in gas-water-rock and oil–water–rock systems were used to test the fractal production model. The rocks (Berea sandstone, chalk, and The Geysers graywacke) in which the spontaneous water imbibition experiments were conducted had different permeabilities ranging from 0.5 to over 1000 md. The results demonstrate that the fractal production model can match the experimental data satisfactorily in the cases studied. The fractal dimension data inferred from the model match were approximately equal to the values of fractal dimension measured using a different technique (mercury-intrusion capillary pressure) in Berea sandstone.     

粗糙表面的分形特征与分形表达研究

得用触针轮廓仪和数据采集系统对磨削和车削表面的粗糙轮廓曲线进行了测量，并就粗糙表面的分形特征作了分析与讨论，同时还提出了粗糙表面的特征粗糙度概念及其定义，并将其用表面粗糙度水平的表述。     

PEN薄带在沙尘环境中磨粒磨损分形特征研究

选取聚萘二甲酸乙二酯(PEN)薄带在ATM自动取款机摩擦学系统中进行了沙尘磨粒磨损试验,利用表面形貌仪以一定时间间隔测量薄带磨损形貌,采用结构函数法考察磨损形貌的分形特征,并利用扫描电子显微镜分析其磨损机理.结果表明:经过沙尘磨粒磨损的PEN薄带表面形貌表现出二重分形特征,沙尘粒子的犁沟和切削磨损及PEN材料的微观断裂是引起大尺度和小尺度区间分形特征的主要机理.     

Fractal dimensions of terrain profiles

Analysing terrain profiles of fields, roads, and other terrains, it was determined that terrain profiles are random and non-periodical. Mandelbrot has defined non-scaling, self-similar figures as fractals, and many investigators have tried to characterize natural forms and structures using fractal geometry. The work here investigates whether terrain profiles can be defined as fractals. Fractal dimensions of profiles were calculated. These were compared with a locus of Brownian motion further to investigate characteristics of terrain profiles. Fractals are defined to be self-similar and irregular. Measuring and analysing terrain profiles, it was established that the statistical characteristics of any part of a terrain profile are similar and that the statistical characteristics of profiles of any kind of terrain are similar irrespective of roughness. This means that terrain profiles are self-similar, and irregular. From these results, it was determined that terrain profiles are fractals. The fractal dimensions were calculated with a coarse-graining method and by Power Spectral Densities (PSD), and fractal dimensions by Scaling were between 1.1 and 1.8 and by PSD between 1.3 and 1.5. Using the locus of Brownian motion, fractal dimensions were 1.5 or slightly larger than those of the terrain profiles. Fractal dimensions for the locus of smoothed Brownian motion were nearly equal to terrain profiles. Therefore terrain profiles could be artificially generated from the locus of smoothed Brownian motion. It appears that terrain roughness is formed by random and non-periodical force.     

Correlation dimension of paddy soil strength in China

Embedding phase space R^m is reconstructed from the spatial series g(x) of cone indices measured in two paddy fields near Nanjing, China. The correlation dimension D₂^m for each field is derived from the correlation integral C^m(r) and the neighbours distance r in log–log scale. Results show D₂^m increases as m, and tends to 5.0, which expresses the estimate of correlation dimension for each soil strength profile measured.     

On star product fractal surfaces and their dimensions

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Self-affine cracks in a brittle porous material

Fractal mechanics and probabilistic methods are applied to an ideal porous material (gypsum rock). The behavior of the material fracture is brittle in essence, concluding that the lines of fracture propagation have self-affine invariance, due to the high rugosity of the crack developed within a wide but limited length interval of scale in the cracks. The validity of the mechanic equations of self-affine cracks are verified by a comparison between the predictions of self-affine fracture mechanics and the results of standardized mechanical tests, obtaining the fractal fracture toughness of gypsum rock.     

Calculating fractal parameters from low-resolution terrain profiles

Driver comfort on rough terrain is an important factor in the off-road performance of wheeled and tracked ground vehicles. The roughness of a terrain has typically been quantified by the U.S. Army as the root-mean-square elevation deviation (RMS) of the terrain profile. Although RMS is an important input into many mobility calculations, it is not scale invariant, making it difficult to estimate RMS from low resolution terrain profiles. Fractal parameters are another measure of roughness that are scale invariant, making them a convenient proxy for RMS. While previous work found an empirical relationship between fractal dimension and RMS, this work will show that, by including the cutoff length, an analytic relationship between fractal properties and RMS can be employed. The relationship has no free parameters and agrees very well with experimental data - thus providing a powerful predictive tool for future analyses and a reliable way to calculate surface roughness from low-resolution terrain data in a way that is scale invariant. In addition, we show that this method applies to both man-made ride courses and natural terrain profiles.     

粗糙表面测量轮廓的分形插值模拟

引入粗糙表面的分形插值理论，提出用分形插值函数模拟粗糙表面轮廓的方法。并以车削，磨削及磨损表面为研究对象，对粗糙表面进行了分形插值模拟，系统地研究了影响其模拟模拟效果的因素和规律。发现可以采用分形插值理论来模拟或表征具有分形特征的粗糙表面。     

Analytical solutions for anomalous transport of volatile pollutants in nonaqueous-phase liquid contaminated soil

In this paper, we study a fractal model for the transport of a volatile component from a nonaqueous-phase liquids (NAPL) trapped in homogeneous soil. By introducing a kind of new integral transform in fractal space, analytical solutions of fractal model are given. Numerical results are presented graphically for various values of fractal dimension.     

分形力学的数学基础

在分形空间考虑的力学称为分形力学。由于分形理论的特点,分形力学的描述需要引入新的空间概念、新的数学手段和方法。本文从介绍空间、测度和维数入手,引入了分形空间,Ｂｅｓｏｖ空间和分形插值方法,进而介绍了在分形边界和分形体中力学量和力学定律的定义。      

Anomalous diffusion in fractal porous medium

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Fractal characteristics of rocks fractured under tension

Fractal geometry can be useful for explaining the fracture behavior and rock properties. The fractal properties of rock fracture surface developed under tension were examined. Seven different rock samples were selected for the tests. An automated surface scanning device was used to map the fractured surfaces. Variogram analysis (VA) (for 1D self-affine sets) and power spectral density (PSD) measurement (for 2D self-affine sets) were applied to calculate the fractal dimension. On a comparative basis, there exists a trend between the fractal dimension and loading rate. The profiles in the loading direction yield higher fractal dimensions indicating the anisotropic feature of fractal. The fractal dimensions obtained by PSD and VA display a relationship for grain size and porosity. Higher porosity samples give different fractal dimensions for upper and lower fractures surfaces.     

The influence of the soil stratification on free field traffic-induced vibrations

Summary  This paper deals with the influence of the soil stratification on the free field vibrations generated by the passage of a vehicle on an uneven road. A two-stage solution procedure is applied for the numerical prediction of the free field traffic-induced vibrations. First, a 2D vehicle model is used for the calculation of the axle loads from the longitudinal road profile. Next, the free field response is calculated with the dynamic Betti-Rayleigh reciprocity theorem, using a transfer function between the road and the receiver. The dynamic road-soil interaction problem is solved with a substructure method. The road is modelled as a beam of infinite length, while the boundary element method, based on the Green's functions for a horizontally layered linear elastic halfspace is used for the soil. The influence of the soil stratification is demonstrated by a numerical example where the free field vibrations during the passage of a vehicle on a traffic plateau are calculated. Three different cases are considered for the layering of the soil: a homogeneous halfspace, a layer built in at its base and a layer on a halfspace. Special emphasis goes to the dynamic interaction between the road and the soil. It is demonstrated that the stratification of the soil has a considerable influence on both the peak particle velocity and the frequency content of the free field vibrations. Received 28 November 2000; accepted 24 April 2001     

Fractal analysis of Mesoamerican pyramids

A myth of ancient cultural roots was integrated into Mesoamerican cult, and the reference to architecture denoted a depth religious symbolism. The pyramids form a functional part of this cosmovision that is centered on sacralization. The space architecture works was an expression of the ideological necessities into their conception of harmony. The symbolism of the temple structures seems to reflect the mathematical order of the Universe. We contemplate two models of fractal analysis. The first one includes 16 pyramids. We studied a data set that was treated as a fractal profile to estimate the Df through variography (Dv). The estimated Fractal Dimension Dv = 1.383 +/- 0.211. In the second one we studied a data set to estimate the Dv of 19 pyramids and the estimated Fractal Dimension Dv = 1.229 +/- 0.165.     

Vehicular traffic effects on development and yield of corn (maize)

A 52 plot experiment was performed during the growing season of 1977 in a Ste. Rosalie clay soil, using a randomized complete-block design with 13 treatments of machinery traffic within each of four blocks. Three vehicle contact pressures, four numbers of tractor passes, and a control of zero traffic were used to relate the growth and yield variables to wheel traffic and the resulting soil compaction. The number of days required for (a) plant emergence, (b) tasselling, and (c) silking were minimum in moderately compacted plots. The plant growth rate monitored at 41, 54, 68, and 105 days from the seeding time was different from plot to plot. Growth models at different times of the season were derived in terms of the wheel traffic variables, and plant and ear moisture content variation with traffic treatment was investigated. Yield and ear yield increased with increases in machine contact pressure and passes reaching a peak around 500 kPa and dropped off for further increases. The reduction in yield was over 35% in some cases, suggesting that careful traffic planning is essential to obtain better production in agricultural fields. Prediction models were obtained for all the plant growth characteristics in terms of traffic variables. A relation for yield in terms of soil bulk density was established for a dry season.     

Fractal geometry and fracture of rock

The irregular and rough profile of fracture surfaces of rock can be regarded as having self-similarity statistically. The authors apply a new geometry—fractal geometry to describe this irregularity. Fractal models of the transgranular fracture and the combined intergranular and transgranular fracture are established. The fractal character of fracture surface of rock is examined by electron scanning and optical fractographic analyses. Finally, the relation between fractal dimension and macromechanics quantities of rock fracture is obtained.     

Simulating shear behavior of a sandy soil under different soil conditions

Understanding of soil shear behavior is very important in the field of agricultural machinery and soil dynamics. In this study, a discrete element model was developed using a simulation tool, Particle Flow Code in Three Dimensions (PFC^3D). The model simulates direct shear tests of soil and predicts soil shear behavior, in terms of shear forces and displacements. To determine and calibrate model parameters (stiffness of particles, strength and stiffness of bond between particles), laboratory direct shear tests were conducted to examine effects of soil moisture content and bulk density on shear behaviors of a sandy soil. Three soil moisture levels (0.02%, 13.0%, and 21.5%) and four bulk density levels (0.99, 1.28, 1.36, and 1.50 Mg/m³) were used in the tests. The test results showed that in general drier and denser soil conditions produced higher shear forces. Based on the test results, the bond strengths of the model particles were determined from soil cohesion and internal friction angle. The model particle stiffness was calibrated based on the yield forces from the tests. The calibrated particle stiffness varied from 1.0 × 10³ to 8.2 × 10³ N/m, depending on soil moisture and density levels. The bond stiffness calibrated was 1.0 × 10⁷ Pa/m for all soil conditions.     

Experimental criteria for the determination of fractal parameters of premixed turbulent flames

The influence of spatial resolution, digitization noise, the number of records used for averaging, and the method of analysis on the determination of the fractal parameters of a high Damköhler number, methane/air, premixed, turbulent stagnation-point flame are investigated in this paper. The flow exit velocity was 5 m/s and the turbulent Reynolds number was 70 based on a integral scale of 3 mm and a turbulent intensity of 7%. The light source was a copper vapor laser which delivered 20 nsecs, 5 mJ pulses at 4 kHz and the tomographic cross-sections of the flame were recorded by a high speed movie camera. The spatial resolution of the images is 155 × 121 m/pixel with a field of view of 50 × 65 mm. The stepping caliper technique for obtaining the fractal parameters is found to give the clearest indication of the cutoffs and the effects of noise. It is necessary to ensemble average the results from more than 25 statistically independent images to reduce sufficiently the scatter in the fractal parameters. The effects of reduced spatial resolution on fractal plots are estimated by artificial degradation of the resolution of the digitized flame boundaries. The effect of pixel resolution, an apparent increase in flame length below the inner scale rolloff, appears in the fractal plots when the measurent scale is less than approximately twice the pixel resolution. Although a clearer determination of fractal parameters is obtained by local averaging of the flame boundaries which removes digitization noise, at low spatial resolution this technique can reduce the fractal dimension. The degree of fractal isotropy of the flame surface can have a significant effect on the estimation of the flame surface area and hence burning rate from two-dimensional images. To estimate this isotropy a determination of the outer cutoff is required and three-dimensional measurements are probably also necessary.