Fracture and Damage Behaviors of Concrete in the Fractal Space

The fracture toughness, the driving force and the fracture energy for an infinite plate with a fractal crack are investigated in the fractal space in this work. The perimeter-area relation is adopted to derive the transforma-tion rule between damage variables in the fractal space and Euclidean space. A plasticity yield criterion is introduced and a damage variable tensor is decomposed into tensile and compressive components to describe the distinct behaviors in tension and compression. A plastic damage constitutive model for concrete in the Euclidean space is developed and generalized to fractal case according to the transformation rule of damage variables. Numerical calculations of the present model with and without fractal are conducted and compared with experimental data to verify the efficiency of this model and show the necessity of considering the fractal effect in the constitutive model of concrete. The structural response and mesh sensitivity of a notched unre-inforced concrete beam under 3-point bending test are theoretical studied and show good agreement with the experimental data.

Stability analysis of a time-delayed Van der Pol–Helmholtz–Duffing oscillator in fractal space with a non-perturbative approach

The time-delayed fractal Van der Pol–Helmholtz–Duffing(VPHD)oscillator is the subject of this paper,which explores its mechanisms and highlights its stability analysis.While time-delayed technologies are currently garnering significant attention,the focus of this research remains crucially relevant.A non-perturbative approach is employed to refine and set the stage for the system under scrutiny.The innovative methodologies introduced yield an equivalent linear differential equation,mirroring the inherent nonlinearities of the system.Notably,the incorporation of quadratic nonlinearity into the frequency formula represents a cutting-edge advancement.The analytical solution's validity is corroborated using a numerical approach.Stability conditions are ascertained through the residual Galerkin method.Intriguingly,it is observed that the delay parameter,in the context of the fractal system,reverses its stabilizing influence,impacting both the amplitude of delayed velocity and the position.The analytical solution's precision is underscored by its close alignment with numerical results.Furthermore,the study reveals that fractal characteristics emulate damping behaviors.Given its applicability across diverse nonlinear dynamical systems,this non-perturbative approach emerges as a promising avenue for future research.

ON SPACES OF FRACTAL FUNCTIONS

In this paper we Ointroduce linear-spaces consisting of continuous functions whose graphs are the attactars of a special class of iterated function systems. We show that such spaces are finite dimensional and give the bases of these spaces in an implicit way. Given such a space, we discuss how to obtain a set of knots for whah the Lagrange interpolation problem by the space is uniquely solvable.

Fractal analysis of polyferric chloride-humic acid(PFC-HA) flocs in different topological spaces

The fractal dimensions in different topological spaces of polyferric chloride-humic acid （PFC-HA） flocs, formed in flocculating different kinds of humic acids （HA） water at different initial pH （9.0, 7.0, 5.0） and PFC dosages, were calculated by effective densitymaximum diameter, image analysis, and N2 absorption-desorption methods, respectively. The mass fractal dimensions （De） of PFC-HA floes were calculated by bi-logarithm relation of effective density with maximum diameter and Logan empirical equation. The Df value was more than 2.0 at initial pH of 7,0, which was 11% and 13% higher than those at pH 9.0 and 5.0, respecively, indicating the most compact flocs formed in flocculated HA water at initial pH of 7.0. The image analysis for those flocs indicates that after flocculating the HA water at initial pH greater than 7.0 with PFC flocculant, the fractal dimensions of D2 （logA vs. logdL） and D3 （logVsphere vs. logdL） of PFC-HA floes decreased with the increase of PFC dosages, and PFC-HA floes showed a gradually looser structure. At the optimum dosage of PFC, the D2 （logA vs. logdL） values of the flocs show 14%-43% difference with their corresponding Dr, and they even had different tendency with the change of initial pH values. However, the D2 values of the floes formed at three different initial pH in HA solution had a same tendency with the corresponding Df. Based on fractal Frenkel-Halsey-HiU （FHH） adsorption and desorption equations, the pore surface fractal dimensions （Ds） for dried powders of PFC-HA flocs formed in HA water with initial pH 9.0 and 7.0 were all close to 2.9421, and the Ds values of flocs formed at initial pH 5.0 were less than 2.3746. It indicated that the pore surface fractal dimensions of PFC-HA floes dried powder mainly show the irregularity from the mesopore-size distribution and marcopore-size distribution.

自贡市古镇开敞空间分形秩序特征研究

开敞空间是古镇空间的重要组成部分,文章基于古镇边界形态量化分类结果,分别选取团状古镇、指状古镇、带状古镇为研究对象,运用计盒维数法,分析其开敞空间分维值和分形秩序。研究结果表明:古镇开敞空间分形秩序特征具备层级连续,面积递减;层级数量指数式迭代增长;古镇空间自相似性高,形状边界与自然环境紧密相关等特点。古镇总体分维值较高,是良好的分形体,则它们的分形秩序特征具备一定参考性,基于此,提出古镇开敞空间分形秩序优化建议。

基于分形理论的城市配网地理空间-网架布局一致性评估方法

我国国土空间规划要求电力部门展开空间布局规划。首先,将分形理论应用于描述配网网架的拓扑图,并给出相应的分形维数定义与计算方法。其次,在提出只考虑城市配网拓扑特性评估方法的基础上,综合考虑其负荷特性对分形维数指标进行加权,实现从空间利用程度和中心集聚程度两个角度来评估城市配网地理空间与网架布局之间的一致性。最后,以某地区实际网架为例,将所定义的量化指标应用于评估该地区配网网架布局与地理空间之间的一致性,并对几个地区实际网架进行综合比较分析,验证了所提方法的合理性及可行性。

城市居住区三维绿色空间影响空气温湿度的机制研究

以郑州市居住区为例,采用手持式路昌LM-8000四合一环境测量仪,对郑州市区中不同方位、不同高度、不同布局形式分类的6个居住区微气候环境进行观测,对气温、相对湿度等气象数据进行分析,并探讨绿色空间布局形式影响空气温湿度的机制。结果表明:居住区白天相对湿度的变化幅度高于温度;测量日内,高层小区和多层小区的温湿度达到最大值的时间不同;在静风条件下,植物蒸散和阴影影响的范围为10~50 m,相对湿度的影响范围为10~50 m;居住区相对湿度最小值、均值分别与测点10 m范围内绿色空间的分形维数有关;相对湿度最大值与50 m范围内绿色空间的形状指数有关;白天时段,湿度的变化与50 m范围内的形状指数和连接度指数有关。

统计学与动力学方法结合的渭河上游干旱研究

为更加准确地识别和预测干旱事件,以林家村水文站1960—2020年长系列日径流数据为基础,将统计学和动力学方法相结合,分别计算水文干旱概率P、径流分维数D_(2)、径流李雅普诺夫指数LE等参数,并建立D_(2)-P,LE-P函数关系,从混沌动力学角度求解干旱概率;由径流Z指数序列重构相空间,并在相空间中对未来干旱事件进行预测。研究结果表明:统计参数P与混沌参数D_(2)、LE具有较好的线性关系,回归方程的决定系数R^(2)值分别为0.93和0.83;在预测区间内,混沌预报模型预测2021年的径流Z指数为1.69,其对应的水文干旱状态为“无旱”,该结论与实际水文情势相吻合,预测效果良好。研究方案可为干旱理论提供一种新的借鉴。

Fractal Method for Statistical Analysis Geological Data

This paper establishes the phase space in the light of spacial series data , discusses the fractal structure of geological data in terms of correlated functions and studies the chaos of these data . In addition , it introduces the R/S analysis for time series analysis into spacial series to calculate the structural fractal dimensions of ranges and standard deviation for spacial series data -and to establish the fractal dimension matrix and the procedures in plotting the fractal dimension anomaly diagram with vector distances of fractal dimension . At last , it has examples of its application .

FRACTAL与物理学

对分形（ｆｒａｃｔａｌ）思想的起源、发展和分形、分维（ｆｒａｃｔａｌ ｄｉｍｅｎｓｉｏｎ）的含义作了简略的介绍。鉴于物理学中对空间认识的改变而引起的物理学的发展，试探性地将分形、分维思想用来对物质世界的一些物质结构和物质运动现象作定性解释。

克拉玛依油田七西区三叠系下克拉玛依组储层微观非均质性定量评价

克拉玛依油田七西区三叠系下克拉玛依组(简称克下组)油藏开发面临单井产量衰减快、含水高,水驱开发效果差的难题,剩余油分布规律不明确,需要加强目的储层微观非均质性评价。利用岩心观察、薄片分析等研究手段,对储层岩石学及储集特征进行分析;以压汞实验分析为基础,利用“主要流动空间”和“分形维数”分别从“量”和“形”的角度关注目的储层孔隙结构非均质性程度,定量表征储层微观非均质性。结果表明:克下组储层岩性以砂砾岩为主,储集空间类型多样,以剩余粒间孔为主,储层以低孔特低渗为主。“主要流动空间”反映渗流能力贡献率为95%的主要流动空间仅占整体储集空间的不到33%;受微裂缝影响,“分形维数”分布于2.6~2.7的样品孔隙结构相对较好,整体反映出储层孔隙结构非均质性很强,利用上述两个参数对储层微观非均质性进行合理评价,为后续高效开发提供地质依据。

Quantum Fractals and the Casimir-Dark Energy Duality—The Road to a Clean Quantum Energy Nano Reactor

Based on Witten’s T-duality and mirror symmetry we show, following earlier work, the fundamental complimentarity of the Casimir energy and dark energy. Such a conclusion opens new vistas in cold fusion technology in the wider sense of the word which we tackle via fractal nano technologies leading to some design proposals for a nano Casimir-dark energy reactor.

Pinched Material Einstein Space-Time Produces Accelerated Cosmic Expansion

An instructive analogy between the deformation of a pinched elastic cylindrical shell and the anti-gravity behind accelerated cosmic expansion is established. Subsequently the entire model is interpreted in terms of a hyperbolic fractal Rindler space-time leading to the same robust results regarding real energy and dark energy being 4.5% and 95.5% respectively in full agreement with all recent cosmological measurements.

闽南传统村落核心保护区边界空间形态量化研究——以泉州晋江梧林村为例

乡村振兴的热潮推动了传统村落的保护规划工作,随即产生了村落分区保护的概念,即针对不同区域实行不同的保护规划实施手段,但由于缺乏对核心保护区边界空间的考虑,导致该范围内出现建筑形式混乱、协调性差、空间断层、空间的性质和需求不符及街巷路网可达性不足等现象。本文选取中国传统村落福建省泉州市晋江市梧林村为研究对象,运用分形理论和空间句法对梧林村保护规划实施前后核心保护区边界空间形态进行量化分析,从中总结出适合闽南传统村落核心保护区边界空间的更新策略,以期为传统村落的保护规划编制与实施提供思路和参考。

切缝药包爆破定向断裂机理及围岩损伤特性分析

依据弹性波动理论,结合冲击波在岩石或类岩石介质中的传播规律,建立了切缝药包爆破时切缝方向与非切缝方向的孔壁峰值应力、粉碎区和裂隙区范围之间的比例关系.利用AUTODYN软件建立了切缝药包爆破模型,在其切缝方向与非切缝方向各等间距设置5个测点,对测点处的应力峰值、爆破振动速度峰值以及峰值对应的时间进行了分析.而后基于淮南矿区顾北煤矿的巷道爆破试验,进行了普通药包、切缝药包以及不同周边孔间距的爆破试验,研究了切缝药包以及周边孔间距的大小对围岩损伤度的影响.结果表明,使用切缝药包爆破时,在其切缝方向会产生爆轰产物射流和应力集中,同时会减弱非切缝方向的应力峰值和爆破振动速度,延缓非切缝方向的能量传播速度,减弱非切缝方向的能量传播大小,从而达到定向断裂的目的;通过现场试验可以发现,使用切缝药包比使用普通药包爆破后,围岩的损伤度下降了30%以上;当使用切缝药包爆破时,随着周边孔间距的增大,爆后围岩损伤度呈下降趋势.

固结压力对泥岩填料孔隙分布及土水特征曲线的影响

为研究固结压力对泥岩填料孔隙分布以及土水特征曲线(SWCC)的影响,以非饱和泥岩填料为研究对象,采用压汞试验分析不同固结压力作用下的土体微观孔隙分布,探究孔隙随荷载的演变规律,并结合毛细原理提出基于压汞实验计算SWCC的方法。结果表明:土体中收缩孔隙的存在使得进-退汞曲线表现出显著的滞回性,且瓶颈效应使得压汞试验中土体小孔隙分布密度偏大,大孔隙分布密度偏小;选择分形曲线的转折点作为泥岩填料孔隙划分界限,可将其孔隙划分为5个部分,孔隙含量的变化以大中孔隙体积被压缩为主,小微和超微孔隙几乎不受固结压力的影响;干密度为1.5 g·cm^(-3)时不同固结压力作用下的土水特征曲线可采用对数曲线表示。

On the Need for Fractal Logic in High Energy Quantum Physics

Modern advances in pure mathematics and particularly in transfinite set theory have introduced into the fundamentals of theoretical physics many novel concepts and devices such as fractal quasi manifolds with non-integer (Hausdorff) dimension for its geometry as well as infinite dimensional wild topology and non classical fuzzy logic. In the present work transfinite fractal sets and fuzzy logic are combined to enable the introduction of a new theory termed fractal logic to the foundation of high energy particle physics. This leads naturally to a new look at quantum gravity. In particular we will show that to understand and develop quantum gravity we have to bring various fields together, particularly fractals and nonlinear dynamics as well as sphere packing, fuzzy set theory, number theory and quantum entanglement and irrationally q-deformed algebra.

Chaotic Fractals at the Root of Relativistic Quantum Physics and Cosmology

At its most basic level physics starts with space-time topology and geometry. On the other hand topology’s and geometry’s simplest and most basic elements are random Cantor sets. It follows then that nonlinear dynamics i.e. deterministic chaos and fractal geometry is the best mathematical theory to apply to the problems of high energy particle physics and cosmology. In the present work we give a short survey of some recent achievements of applying nonlinear dynamics to notoriously difficult subjects such as quantum entanglement as well as the origin and true nature of dark energy, negative absolute temperature and the fractal meaning of the constancy of the speed of light.

Fractal and Fractional Diffusion Equations of Price Changing of Commodity

In this paper, three types of modeling of diffusion equations for price changing of commodity are studied. In which, the partial derivatives of price of commodity respected to time on the left hand side are integer-derivative, fractal derivative, and fractional derivative respectively;while just a second order derivative respected to space is considered on the right hand side. The solutions of these diffusion equations are obtained by method of departing variables and initial boundary conditions, by translation of variables, and by translation of operators. The definitions of order of commodity x and the distance between commodity?xi and xj are defined as [1]. Examples of calculation of price of pork, beef and mutton mainly due to price raising of pork in 2007-07 to 2008-02 inChina are given with same market data as [1]. Conclusion is made.

区域人居单元的概念、机制与特征

中国传统人居环境具有城乡连续一体的特征,表现为“城镇—乡村—田野”的连续空间谱系。中国在经历了城乡二元发展之后,进入了“城乡中国”的区域一体化阶段。面向区域城市化的外部趋势,基于“城乡中国”的本土背景,提出“区域人居单元”概念,从文化地理学视角分析区域人居单元的驱动因子和形成机制,围绕单元系统、单元区划、风貌特色,揭示区域人居单元的基本特征,该概念可以为区域国土空间设计明确具体的实践对象,在区域尺度上为解决“千城一面”问题提供思路。