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 statistical properties of self-affine surfaces in elastic contacts: A numerical investigation

In the last years, an increasing number of papers has been published in the field of contact mechanics between rough fractal surfaces. The increase in research is motivated by the wide variety of natural and industrial processes that involve formation of rough surfaces and interfaces, characterized by self-similarity or self-affine properties on multiple scales. In this paper, the contact between a linear elastic half-space and a rough self-affine fractal rigid surface is studied by employing a numerical method recently developed by the authors (Putignano et al., 2012). The paper aims at investigating the influence of surface parameters as fractal dimensions, mean square slope and mean square roughness on the relation between the contact area, the load and the average separation. The results show that, for relatively small loads, the real contact area–load relationship coefficient of proportionality $\kappa$ takes the universal value $\kappa =2$ independent of the statistical properties and fractal dimension D_f of the rough surface. This universal constant is just in between the two values predicted respectively by Bush et al. (1975) and Persson (2001). We also find that the average separation vs. load relation is affected by the fractal dimension D_f of the rough surface, as higher D_f lead to an increase of the average separation. Finally, in this work, we also study the behavior of the power spectral densities of the elastically deformed surface and of the distribution of local separations. We find that the trend of this quantities is in agreement with recent theoretical predictions.     

Evaluation of fractal terrain model for vehicle dynamic simulations

Fractals are a popular method for modeling terrains that include various scales. This paper investigates the effectiveness of using fractals for generating artificial terrains which can be used for vehicle simulations. The 3-D Weierstrass–Mandelbrot function was used to generate surfaces based on experimentally measured terrains. There is an exponential relationship between the root means squared elevation of the surfaces and the fractal scaling parameter. This relationship was used to determine the required fractal parameters to generate a surface with a desired roughness. A light detection and ranging (LiDAR) sensor coupled with a global positioning system (GPS) and inertial navigation system (INS) was used to measure two off road surfaces. The experimental terrain was then compared to the simulated terrain. Based on the comparison, the fractal model can capture the general roughness of the experimentally measured terrains as determined by the dynamic response of a suspension model. However, the fractal model fails to capture some of the nuances and non-periodic events observed in experimental terrains.     

关于虚拟激励法的一个注记

从具有随机频率与随机相位的随机谐和函数出发,证明了当随机频率与相位均为均匀分布而随机幅值与功率谱密度平方根成正比时,该随机过程的功率谱即精确地等于目标功率谱.进而表明:只需遍历频率区间,即可由单一谐和函数激励下的响应幅值给出响应的功率谱密度,从而揭示了虚拟激励法的物理意义.研究还表明:为了给出结构响应的功率谱密度,实际...     

颗粒物质的分形特征与其物理力学性质的关系探讨

从颗粒流的一般本构方程出发,利用室内大型相对密度仪和直接剪切仪,针对粒径分形分布对孔隙率和内摩擦角的影响进行了实验研究。结果表明,在相同实体密度、弹性恢复系数和几何平均中值粒径条件下,随粒径分维数的增大,孔隙率增大,而内摩擦角、颗粒浓度、剪切应力系数与正应力系数均减小。进而分析了粒径分形特性对颗粒流动行为的影响,得出粒径分维数小的颗粒系统剪切应力大,系统的流动性能差,鲁棒性强;反之,分维数大的颗粒系统剪切应力小,系统的流动性能好,敏感性强。对系统组构的分形特征影响系统颗粒物质崩塌流动行为的机制进行了初步探索,为颗粒流系统的研究提供了新思路。     

Direct specturm analysis of the state space for random vibration of MDOF linear damped system

This paper proceeds from the general case of the unsymmetric linearized multi-degrees of freedom (MDOF) systems. By adopting the general complex modal theory of the state space, the response analysis for a system subjected to random excitation of the same source is carried out using as a kind of direct spectrum analysis method in frequency domain. With the input of power spectral density function given, the explicit expression of the power spectral density function matrix of the output response can be obtained. By taking Fourier inverse transform, the integrated expressions of the correlation function matrix and of the spectrum moment matrix are obtained. Comparing with the time domain method, this method enjoys the merit of visualization and avoids the procedure of transformation from the obtained response correlation function to be solved for the output spectrum utilizing Fourier transform. This paper has extended the application range of the traditional frequency domain analysis method. The mean square values and variety of statistical values can be obtained conveniently. This method and the time domain method are different in approach but equally satisfactory in their results. The project is supported by National Natural Science Foundation of China and National Education Commission Science Foundation of China     

A slip-line plasticity analysis of abrasive wear of a smooth and soft surface sliding against a rough (fractal) and hard surface

Abrasive wear of a soft and smooth surface sliding against a rough (fractal) and hard surface was analyzed by the slip-line theory of plasticity. The analysis is based on a slip-line model of a rigid spherical asperity (wear particle) plowing through a soft surface and removing material by microcutting. Integration of this single-contact model into a three-dimensional contact mechanics analysis of an abrasive surface exhibiting multi-scale roughness described by fractal geometry yielded relationships of the abrasive wear rate and wear coefficient in terms of the interfacial shear strength (adhesion), topography (fractal) parameters of the hard/rough surface, elastic–plastic material properties of the soft/smooth surface, and total normal load. Analytical results from the single-contact analysis provide insight into the deformation of a perfectly plastic material caused by the abrasive action of a rigid asperity/wear particle under different normal load and interfacial friction (adhesion) conditions. The dependence of the abrasive wear rate and wear coefficient on normal load (global interference), roughness of the abrasive surface, elastic–plastic material properties of the abraded surface, and interfacial shear strength (lubrication effect) is interpreted in the context of numerical results obtained for representative ceramic/ceramic, ceramic/metallic, and metal/metal sliding systems.     

摩擦信号分形维数与载荷和速度的关系研究

在立式万能摩擦磨损试验机上对GCr15/45#钢摩擦副进行不同接触载荷和滑动速度下的摩擦磨损试验,对初始滑动距离1 000 m内的摩擦系数信号进行数据采集,运用结构函数测度方法对摩擦系数信号进行分形表征,计算不同载荷和速度下的分形维数,研究摩擦信号的分形维数随载荷和速度的变化规律.结果表明,摩擦系数信号具有显著的分形特征,其分形维数随载荷和速度而变化.在相同载荷下,分形维数随滑动速度增加而增大;在相同速度下,分形维数随着接触载荷增加而增大.磨损率与摩擦系数信号的分形维数正相关.     

A slip-line plasticity analysis of sliding friction of rough surfaces exhibiting self-affine (fractal) behavior

A plasticity analysis of sliding friction of rough (fractal) surfaces sliding against smooth surfaces was developed based on a slip-line model of a rigid spherical asperity (wear particle) plowing and cutting through a soft semi-infinite medium. Solutions of the fraction of fully plastic asperity microcontacts responsible for the evolution of friction and energy dissipation were obtained in terms of the total normal load (global interference), interfacial adhesion characteristics, topography (fractal) parameters of the hard surface, and elastic–plastic material properties of the soft surface. This was accomplished by incorporating the slip-line model of a single microcontact into a friction analysis of sliding surfaces demonstrating multi-scale roughness. Numerical results provide insight into the effects of global interference (normal load), fractal parameters (surface roughness) of the hard surface, interfacial shear strength (adhesion), and material properties of the soft surface on plastic deformation at the microcontact level, global coefficient of friction, and frictional energy dissipated during sliding.     

考虑摩擦因素的结合面切向接触刚度分形预估模型及其仿真分析

为建立更完善和精确的结合面接触刚度模型,本文根据分形理论和摩擦学原理,从微观角度建立了考虑摩擦因素的结合面切向接触刚度分形预估模型.通过数值仿真分析研究了接触载荷、分形维数、摩擦系数和接触面积等因素对结合面切向接触刚度的影响.分析结果表明:结合面切向接触刚度随法向载荷和分形维数的增加而增大,而随分形尺度参数的增大而减小;摩擦系数对结合面切向接触刚度的影响较大,不同实际接触面积下的切向刚度相差较大;当分形维数较小时,摩擦系数对结合面切向刚度的影响将降低.这些研究对于进一步开展结合面的动力学特性研究具有重要意义.     

FRACTAL DESCRIPTION OF ROCK FRACTURE BEHAVIOR

岩石可压裂性的定量评价是非常规油气资源开采中一个重要且复杂的问题,利用已有岩心实验数据,考虑岩石的岩性及压裂裂隙的分形特征,建立了岩石压裂裂隙的分维数、岩石抗压强度和裂隙面密度之间的定量关系,并根据所采岩心,利用自行开发的软件计算出其压裂后分形维数,验证了裂隙越复杂,分形维数越大. 提出采用压裂裂隙的分维数来表征岩石的可压裂性, 为定量分析和评价岩石可压裂性与储层开采价值提供了一个新的思路.     

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.     

Predicting mobility performance of a small,lightweight track system using the computer-aided method NTVPM

This paper describes the results of a study of applying the physics-based, computer-aided method – the Nepean Tracked Vehicle Performance Model (NTVPM), originally developed for evaluating the mobility of large, heavy tracked vehicles, to predicting the performance of a small, lightweight track system on sandy soil. The objective is to examine the applicability of NTVPM to predicting the cross-country performance of small, lightweight tracked vehicles on deformable terrain. The performance of the track system predicted by NTVPM is compared with experimental data obtained in a laboratory soil bin by the Robotic Mobility Group, Massachusetts Institute of Technology. It is shown that the correlation between the tractive performance predicted by NTVPM and that measured is reasonably close, as indicated by the values of the coefficient of correlation, coefficient of determination, root mean squared deviation, and coefficient of variation. The results of this study provide evidence for supporting the view that physics-based methods, such as NTVPM, that are developed on the understanding of the physical nature and detailed analysis of vehicle–terrain interaction, are applicable to large, heavy, as well as small, lightweight vehicles, provided that appropriate terrain data are used as input.     

Fractal dimension as a characteristic of deformation stages of austenite stainless steel under tensile load

A technique for defining the fractal dimension of the deformed specimen is considered by using surface relief images obtained from a scanning electron microscope. By subjecting the austenitic stainless steel specimen to a step-wise increase in load, the fractal dimension is observed. Surface mesostructure could be characterized by discrete fractal dimension. The spectrum of the fractal data agrees with the self-affinity function calculated from the universal material constant. A correlation is established between the step-wise increase in the fractal dimension and the specific energies dissipated in the deformed specimen.     

双旋转附属圆柱对主圆柱尾流控制的影响

基于Fluent软件平台,采用数值模拟方法对非稳态圆柱体结构尾流流动特性进行了研究。对在Re=50～200范围内,双旋转附属圆柱的转速对主圆柱体尾流流动特性的影响进行了分析。研究结果表明:随着附属圆柱旋转速率的增加,主圆柱体表面所受阻力系数平均值与均方根值、升力系数均方根值均会减小。同时,旋转速率的变化对柱体结构表面压力分布的影响显著,压力系数在附属圆柱的位置产生了跳跃性变化。另外,当附属圆柱转速达到临界值时,尾流涡街变窄,涡脱落现象消失,并且系统的能量效率到达最佳状态。     

A fractal reconstruction method for LDV spectral analysis

An unequally spaced fractal interpolation method is developed to reconstruct the low data rate laser Doppler velocimetry (LDV) signal for better spectral results. The reconstructed result not only augments the data points to improve the statistics but also retains the turbulent fractal dimension of the flow. In other words, the interpolated data are flow turbulence-correlated data which are physically reasonable. This method is demonstrated on the LDV spectral measurements in the shear layer of a jet flow and the results are compared against the hot-wire spectral data. When the data rate of the LDV signal is low, the fractal interpolation method can suitably restore the major spectral peaks while the conventional sample-and-hold fails. Effects of the data rate, data record length, sampling rate and the fractal dimension on the accuracy of the spectral reconstruction are also discussed. A version of this paper was presented at the 4th International Symposium on Transport Phenomena in Heat and Mass Transfer, July 14–18, 1991, Sydney, Australia     

Compressive strength of fibre composites with random fibre waviness

The compressive strength of unidirectional long fibre composites is predicted for plastic microbuckling from a random two-dimensional distribution of fibre waviness. The effect of the physical size of waviness is addressed by using couple stress theory, with the fibre bending resistance scaling with the fibre diameter d. The predicted statistical distribution of compressive strength is found using a Monte Carlo method. An ensemble of fibre waviness profiles is generated from an assumed spectral density of waviness and the compressive strength for each such realisation is calculated directly by the finite element method. The average predicted strength agrees reasonably with practical values, confirming the hypothesis that microbuckles can be initiated by fibre misalignment. It is found that the probability distribution of strength is well matched by a Weibull fit, and the dependence of the Weibull parameters upon the spectral density of waviness is determined. For the practical range of fibre distributions considered, it is concluded that the strength depends mainly upon the root mean square amplitude of fibre misalignment, with the shape of the power spectral density function playing only a minor role. An engineering model for predicting the compressive strength is proposed, akin to weakest link theory for materials containing flaws. A specimen containing randomly distributed waviness is examined to locate regions of high-fibre misalignment. The strength of each of these weak regions is estimated from a look-up table derived from calculations with idealised circular or elliptical patches of waviness. The strength of the composite is given by the failure stress associated with the weakest such patch. For random distributions of waviness, the predictions using this engineering approach are in good agreement with the direct calculations of strength using the finite element method.     

稳定分层流密度界面处湍流混合与分形结构

采用室内混合箱研究稳定分层流(上层淡水、下层盐水) 无剪切密度界面处的湍流混合与分形结构. 湍流通过浸没在盐水层中的振动格栅产生, 密度界面结构通过在盐水层中添加荧光剂或染料可视化, 共进行12 组实验. 实验观测并记录了:(1) 淡、盐水密度界面距混合箱底部的平均高程(h);(2) 淡、盐水层的密度(ρ₀,ρ), (3) 淡、盐水密度界面. 其中, 淡、盐水密度界面通过照片、录像进行记录. 观测结果用于计算:(1) 盐水层密度;(2) 卷挟速度, (3) 整体理查孙数(Ri_o), (4) 二维、三维密度界面, (5) 二维、三维密度界面的分形维度. 结果分析发现:(1) 湍流卷挟率随Rio 增大而减小, 并且满足Ri_o的-3=2 或-7=4 幂律;表明随着湍流强度的减弱, 混合的速度也越来越缓慢;(2) 二维密度界面分形维度大于1, 三维密度界面分形维度大于2;表明二维、三维密度界面存在分形结构;(3) 分形维度随Ri_o的增大而减小;表明随着湍流强度的减弱, 密度界面也越来越趋于光滑.      

单丝探头测量数据分析方法

本文介绍了用单丝热线探头测量三维湍流流场时的数据分析方法。文中着重研究了联系相关值和均方根值的相关系数及瞬时速度的泰勒级数的收敛性问题，认为在高湍流度流场中Ｋｉｎｇ的相关系数假定值宜作适当的修正，而当泰勒级数不满足余项收敛条件，本文提出了一种简便的计算方法，并用这种方法处理了圆柱体近尾区的实验数据，其结果与Ｏｎｇ用图表法处理实验数据的结果相符