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A new fractal model for anisotropic surfaces

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1 Introduction
Rough surface analysis and characterization play key roles in several areas of engineering and science including the design of superconductors, machine design, materials science, scattering of electromagnetic waves, surface contact and wear mechanics and tribology. The investigation of topography of engineering surfaces has a rather long and distinguished history; more recent developments in this field have been profoundly influenced by the statistically-basedformulations of LonguetHiggins[1] and Nayak[2], among others. But statisticalmethods have several shortcomings (such as nonstationarity of distributions,multiscales and instrument dependence of measurements), and this has lately given rise to an alternative approach to the analysis of engineering surfaces using fractal geometry [3,4]. This fractal approach, fueled by a number of notable successes over the last dozen years such as those reported by Mandelbrot[5], Gagnepain & Roques-Carmes[6], Thomas[7], Thomas & Thomas[8], Roques-Carmes et al.[9], Ling[10] and Majumdar & Tien[ll], has now become quite popular for investigations which have led to some useful advances in the theory and applications of rough surfaces as, for example, in Zhou et al.[12, 13] and Lopez et al.[14]. The Weierstrass-Mandelbrot function studied in Berry & Lewis[15], with its inherent fractal nature, has served as a starting point for many informative and useful investigations of rough surfaces via their profiles. These surface profiles represent the variation of surface topography along one-dimensional intersections of the surface with planes perpendic-
A
NEW
FRACTAL
MODEL
Байду номын сангаасFOR
ANISOTROPIC
SURFACES
D. BLACKMOREt AND G. ZHOU~t
~Department of Mathematics, New Jersey Institute of Technology, Newark, New Jersey, USA $Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, USA
ABSTRACT: A new fractal-based functional model for anisotropic rough surfaces is used to devise and test two methods for the approximate computation of the fractal dimension of surfaces, and as an instrument for simulating the topography of engineering surfaces. A certain type of statistical self-affinity is proved for the model, and this property serves as the basis for one of the methods of approximating fractal dimension. The other technique for calculating fractal dimension is derived from a Htflder type condition satisfied by the model. Algorithms for implementing both of these new schemes for computing approximate values of fractal dimension are developed and compared with standard procedures. Both the functional model and its corresponding modified Gaussian height distribution are used for simulating fractal surfaces and several examples are adduced that strongly resemble some common anisotropic engineering surfaces. ~ 1998 Elsevier Scicncc Ltd
ular to the planar domains over which the surface is defined. In [16], we postulated a generalized Weierstrass-Mandelbrot type of surface profile model that possesses most of the known theoretical and physical features found in engineering surfaces, and using this model we were able to prove that the probability density function for the surface height distribution along a profile must be of the form of a Gaussian distribution multiplied by a convergent power series expanded about the mean height. If a surface is isotropic, a surface profile through a point suffices to completely characterize the local surface topography, but this is certainly not the case for anisotropic surfaces. Yet, the surface height distribution of anisotropic surfaces has also been observed to have the same type of slightly biased Gaussian form exhibited by their profiles. We have just developed [17] a fractal-based functional model that captures most of the features of anisotropic rough surfaces and from this proved that the height distribution is of the same type as that obtained for surface profiles [16]. In this paper we shall show how certain properties of our new model for anisotropic surfaces such as a form of self-attlnity and a HOlder type condition can be used for the approximate calculation of the fractal dimension, and we shall demonstrate how the model and its height distribution can be applied to the simulation of engineering and natural surfaces. We begin in Section 2 with a brief description of the functional model and its corresponding surface height distribution. Next, in Section 3, we give a proof of the exact sense in which the model is self-affine. In Section 4 we describe how the self-afflnity and H~lder properties of the surface function can be employed to compute the

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材料断口中的分形(英文)

材料断口中的分形(英文)

Fractals in Material BreaksWhen materials break, they often leave behind jagged and irregular surfaces. These surfaces, known as fracture surfaces, contain a wealth of information about the behavior of the material leading up to the break. One particular aspect that researchers have been studying is the presence of fractals in these surfaces.Fractals are mathematical patterns that repeat at different scales. They are often found in nature, such as in the branching of trees or the coastline of a beach. However, they can also be observed in man-made materials, especially when those materials are subjected to stress and strain.Researchers have found that the presence of fractals in material fracture surfaces can give valuable insights into the material's mechanical properties. By analyzing the fractal dimension of the surface, they can determine the level of complexity and irregularity in the material's structure.Fractals can also help predict the behavior of materials under different conditions. By studying the fractal patterns of a material before and after exposure to stress, researchers can better understand how the material responds to different forces and how it might fail under certain conditions.In addition to mechanical properties, fractals in material breaks can also be used for forensic analysis. By studying the fractal patterns left behind on a surface, investigators can determine the type and amount of force that was exerted on the material, helping to identify the cause of failure.Overall, the study of fractals in material fracture surfaces is an exciting area of research with many potential applications. By understanding the patterns and behaviors of materials on a fractal level, we can better design and utilize these materials in industry, while also gaining a deeper understanding of the world around us.。

A-Zee量子场论笔记完全版

A-Zee量子场论笔记完全版

Lecture Notes of Quantum Field Theory
Prof. Anthony Zee. (recorded by Gao Ping) June 27, 2012
Lecture 1
1.1 Introduction
At first, I will introduce a cube of physics shown below, where the horizontal axis is about the speed of motion in a physical system, leftside for slow and rightside for fast, and where the vertical axis is about the scale of a physcical system, upside for small and downside for big. Considering this property, we can fill the cube with dominant theories in physics, namely left-downside for classical physics, right-downside for special relativity, left-upside for quantum mechanics and finally right-upside for quantum field theory(QFT), a marrigae of special relativity and quantum mechanics. However, with all above, we can just fill up a planar table but not a cube, which must have the other dimension behind. And thus it is gravity theory, such as general relavity. It is challenging to build a quantum theoretical gravity today. Roughly, we can say Einstein has two daughters, special relativity and general relativity, among which one has married with quantum but the other has not. small big Quantum Mechanics Classical Physics slow Quantum Field Theory Special Relativity fast
高三英语材料科学单选题60题

高三英语材料科学单选题60题

高三英语材料科学单选题60题1. In the field of materials science, the "atom" is considered as the basic ______.A. unitB. elementC. particleD. molecule答案：A。

本题考查名词词义辨析。

选项A“unit”有“单位；单元”的意思，“atom”（ 原子）被视为材料科学中的基本“单位”，符合语境。

选项B“element”指化学元素；选项C“particle”侧重于粒子；选项D“molecule”指分子。

在材料科学中，原子通常被描述为基本单位，其他选项不符合。

2. The study of materials science often involves the analysis of various ______.A. substancesB. compoundsC. mixturesD. materials答案：D。

此题考查名词的含义。

选项A“substances”指物质；选项B“compounds”指化合物；选项C“mixtures”指混合物；选项D“materials”指材料。

材料科学的研究对象主要是各种“材料”，其他选项虽然也与物质相关，但不如“materials”直接和准确。

3. When testing the properties of a new material, the ______ of heat transfer is an important factor.A. processB. methodC. wayD. means答案：A。

本题考查名词的用法。

选项A“process”强调过程；选项B“method”侧重方法；选项C“way”比较常用，指方式、道路；选项D“means”意为手段、工具。

在测试新材料性能时，热传递的“过程”是重要因素，A 选项最符合题意。

Is the Fractal Model Appropriate for Terrain

Is the Fractal Model Appropriate for Terrain


Figure 1: Fractal surface without power modification.
Figure 2: Non fractal (oscillatory) surface.
Figure 3: Power-modified fractal landsape.
dependent features. The nondifferentiability of the fractal models results in terrains that in the author’s opinion appear too rough at small scales (e.g., Figure 1 and [11]). Most of the terrain figures accompanying this article were produced using two-dimensional variations of a “Markaussian” autocorrelation R(τ ) = exp(−|τ | α ) with 1 < α < 2, which produces noises falling between the extremes of the Gaussian (too smooth) and Markovian (usually too rough) textures shown in Figure 6. The desirability of scale-dependent features for visual modeling is also obvious: most visual phenomena provide at least an approximate sense of scale (atmospheric phenomena are perhaps the exception again). As an illustration, one would be surprised to find a terrain such as Figure 3 at one’s feet, and the scale-dependent depressions in Figure 4 certainly do not detract from the realism of that terrain. The lack of scale-dependent features can be a practical problem in visual terrain simulation, since without such detail the observer cannot judge the distance to the “ground”. We conclude that, although a single descriptor such as the fractal dimension or spectrum exponent may be adequate for some classification purposes, the validity of the fractal model is not well established for many natural phenomena, and it is evident that the visual characteristics of many phenomena cannot be adequately differentiated using only this model. For example, it was found in [1] that an airport runway had a fractal dimension identical to that of some topographical data, the interpretation being that this was due to the attenuated low-frequency variation of the runway. Spectral modeling with Gaussian processes permits the description of a variety of phenomena, including fractal noises as a special case. Important perceptual characteristics of the noise, such as scale, period of oscillation, and directional tendencies are directly reflected in the noise autocorrelation function. Other characteristics such as dominant scales of detail and the small-scale or high-frequency behavior of the noise are easier to specify in the frequency domain, using the intuitive interpretation of the spectrum as the amount of detail at each scale. These comments originally appeared in section 6.1 of the academic paper [2].