Elastic–plastic transition in three-dimensional random materials: massively parallel simulations,fractal morphogenesis and scaling functions

Elastic–plastic transitions were investigated in three-dimensional (3D) macroscopically homogeneous materials, with microscale randomness in constitutive properties, subjected to monotonically increasing, macroscopically uniform loadings. The materials are cubic-shaped domains (of up to 100?×?100?×?100 grains), each grain being cubic-shaped, homogeneous, isotropic and exhibiting elastic–plastic hardening with a J ₂ flow rule. The spatial assignment of the grains’ elastic moduli and/or plastic properties is a strict-white-noise random field. Using massively parallel simulations, we find the set of plastic grains to grow in a partially space-filling fractal pattern with the fractal dimension reaching 3, whereby the sharp kink in the stress–strain curve of individual grains is replaced by a smooth transition in the macroscopically effective stress–strain curve. The randomness in material yield limits is found to have a stronger effect than that in elastic moduli. The elastic–plastic transitions in 3D simulations are observed to progress faster than those in 2D models. By analogy to the scaling analysis of phase transitions in condensed matter physics, we recognize the fully plastic state as a critical point and, upon defining three order parameters (the ‘reduced von-Mises stress’, ‘reduced plastic volume fraction’ and ‘reduced fractal dimension’), three scaling functions are introduced to unify the responses of different materials. The critical exponents are universal regardless of the randomness in various constitutive properties and their random noise levels.     

Fractal Character of Dynamic Light Scattering of Particles

Dynamic light scattering signals from particles, exhibit fractal characteristics. This feature can be used to determine the particle size. The use of the fractal dimension, as a quantitative method to analyze the properties of dynamic light scattering signals from submicron particles, is presented. The analysis is performed directly on the time‐resolved scattered intensity, and the Box Dimensions of light scattering signals of particles with diameters 100, 200, 500 and 1000 nm. The experimental results show that the fractal dimensions of light scattering signals correlate well with particle size. In the submicron size range, the smaller the particles, the larger their fractal dimensions. Compared with the PCS technique, only several hundreds of samples are required in the fractal method. Therefore, the data processing is easily accomplished. However, this method only provides the mean particle size, but not the particle size distribution.     

The Use of Small Angle Light Scattering to Study Structure of Flocs

In this paper, a new approach using small angle light scattering to study the structure of large micron sized flocs is described. The flocs were made up from uniformly sized, approximately spherical hematite particles, induced to aggregate using 250 mM KCl to ensure that growth was governed by diffusion limited aggregation alone. The fractal dimensions of the hematite aggregates changed from 1.73 to 2.23 ± 0.05 as the aggregation proceeded, indicating that restructuring occurred during aggregation. The fractal dimensions obtained from the small angle light scattering technique are compared with those obtained from the combination of volume fraction and floc size measurement. The steady state fractal dimensions (2.23 ± 0.05) obtained by small angle light scattering are comparable to the fractal dimensions obtained from determining the slope of the relationship between floc volume fraction and floc size (2.09 and 2.14 ± 0.05). The discrepancy between the fractal dimensions obtained by these techniques during the initial stages of aggregation is due to the fact that the latter method measures the average fractal dimensions, whilst the former method measures the fractal dimensions at any particular instant.     

Compressibility and Fractal Dimension of Fine Coal Particles in Relation to Pore Structure Characterisation Using Mercury Porosimetry

Mercury porosimetry has been applied to characterize the pore structure of fine coals particles. Interparticle voids and compressibility effects on the mercury intrusion data were examined. It is found that coal compressibility has a significant effect on mercury porosimetry data when pressure P>20 MPa. The compressibility of the two coals used was determined to be 3.13×10⁻¹⁰ m² N⁻¹ and 2.50×10⁻¹⁰ m² N⁻¹ for CA and GO coals, respectively. Fractal dimension analysis provides a “fingerprint” to distinguish the effect of coal compression from the pore filling process during mercury intrusion. It is shown that fractal dimension can be evaluated from the compressibility corrected pore volume data. Results from the present study suggest that statistic self-similarity of the fractal dimension perspective is limited by certain artificial effects, such as crushing and grinding. Different surface irregularities exist over different pore size ranges, and a single fractal dimension value can only be used to describe the surface irregularity within a limited pore size range. The average fractal dimensions in the pore size range of 6–60 nm were found to be 2.71 and 2.43 for CA and GO coals, respectively.     

Flory theory of polymeric fractals - intersection,saturation and condensation

We consider swelling effects of polymeric fractals, recently introduced by Cates, by usual simple Flory arguments for the free energy. The Flory arguments can be formulated to give a unified view for all polymers, linear, branched, or percolation clusters, as long they are of fractal connectivity.If the size of solvent molecules, being fractals themselves, is comparable to the given cluster, new values of the fractal dimensions can be found. The upper critical dimension is reduced. This is due to usual screening of the excluded volume. By standing overlap and repulsive energies of fractals of different fractal dimensions we find condensation to non-fractal objects depending on the value of the fracton dimension. A melt of polymeric fractals of the same fractal dimension and the same size becomes compact if the spectral dimension exceeds a “critical” value.These considerations are of relevance concerning recent experiments, considering static and dynamic properties of mixtures of microgels and linear polymers of different or equal sizes.     

Limits to gelation in colloidal aggregation

We show that the dynamics of large fractal colloid aggregates are well described by a combination of translational and rotational diffusion and internal elastic fluctuations, allowing both the aggregate size and internal elasticity to be determined by dynamic light scattering. The comparison of results obtained in microgravity and on Earth demonstrates that cluster growth is limited by gravity-induced restructuring. In the absence of gravity, thermal fluctuations ultimately inhibit fractal growth and set the fundamental limitation to the lowest volume fraction which will gel.     

Dielectric constant of porous ultra low-k dielectrics by fractal-Monte Carlo simulations

The effective dielectric constant of porous ultra low-k dielectrics is simulated by applying the fractal geometry and Monte Carlo technique in this work. Based on the fractal character of pore size distribution in porous media, the probability models for pore diameter and for effective dielectric constant are derived. The proposed model for the effective dielectric constant is expressed as a function of the dielectric coefficient of base medium and the volume fractions of pores and base medium, fractal dimension for pores, the pore size, as well as random number. The Monte Carlo simulations combined with the fractal geometry are performed. The predictions by the present simulations are shown in good accord with the available experimental data. The proposed technique may have the potential in analyzing other properties such as electrical conductivity and thermal conductivity in porous ultra low-k dielectrics.     

FORMATIONS OF FRACTAL ISLANDS AGGREGATED WITH DISCS IN GAUSSIAN DISTRIBUTION ON NONLATTICE SUBSTRATES

The growth mechanism of fractal islands on a two-dimensional nonlattice substrate with periodic boundary conditions has been investigated by using Monte Carlo technique. Results show that the fractal dimension d_f of the final ramified islands is almost independent of the diffusion step length, mobility and rigid rotation of the islands. The characteristics of the size distribution of the discs in an island do not change the dimension d_f of the island. However, we find that d_f increases linearly with the surface coverage ρ of the system and its slope decreases with the increase of the mean diameter of the discs.     

A fractal study for nucleate pool boiling heat transfer of nanofluids

In this paper, a fractal model for nucleate pool boiling heat transfer of nanofluids is developed based on the fractal distribution of nanoparticles and nucleation sites on boiling surfaces. The model shows the dependences of the heat flux on nanoparticle size and the nanoparticle volume fraction of the suspension, the fractal dimension of the nanoparticle and nucleation site, temperature of nanofluids and properties of fluids. The fractal model predictions show that the natural convection stage continues r...     

Monte Carlo studies of two measures of polymer chain size as a function of temperature

Monte Carlo simulations of single polymer chains with both excluded volume and nearest-neighbor interaction energies are discussed. Two measures of chain size are obtained in the simulation, the radius of gyration of the polymer chain and the inverse radius of the polymer chain. Both of these are reported as a function of temperature, or interaction energy, and chain length,N. The possibility of estimating the fractal dimensions of these measures from the Monte Carlo data is discussed in the context of two different interpolation functions for the temperature dependence of the fractal dimensions. The approach to the fractal dimension as a function of chain length,N, is studied. It is suggested that the approach to fractal dimension of the measures of chain size of polymers is slow, perhaps a fractional power itself.     

分形聚集逾渗性质的计算机模拟

提出3种模型——小尺寸随机逐次成核生长模型和二维及三维代代聚集生长模型,在不同的近邻条件下和不同尺寸的网格中,通过蒙特卡罗模拟,系统地研究了一维、二维和三维分形聚集的逾渗性质.计算结果显示,分形聚集的逾渗阈值仅取决于空间维数和近邻条件,与模型的网格大小无关,是分形系统固有的临界属性;生长概率等于逾渗阈值时,聚集体可以无限生长并保持分形维数恒定,此时的分形维数只是空间维数的线性函数.     

Influence of complex component and particle polydispersity on radiative properties of soot aggregate in atmosphere

The effects of morphological structure, water coating, dust mixing and primary particle size distribution on the radiative properties of soot fractal aggregates in atmosphere are investigated using T-matrix method. These fractal aggregates are numerically generated using a combination of the particle-cluster and cluster-cluster aggregation algorithms with fractal parameters representing soot aggregate in atmosphere. The radiative properties of compact aggregate notably deviate from that of the branched one, and the effect of morphology changes on the radiative properties in wet air cannot be neglected. However it is reasonable to use realization-averaged radiative properties to represent that of the aggregates with certain morphology. In wet air, the scattering, absorption and extinction cross-section and symmetry parameter of soot aggregates coated with water notably increase with water shell thickness. The mixing structures of dust have little effect on radiative properties of aggregates, but the volume fraction of dust has an obvious effect on extinction, scattering and absorption cross-section of aggregates when the size parameters are above the Rayleigh limit. Although the primary particle size distribution of soot aggregate has mild effect on the scattering albedo and asymmetry parameter, the deviations of the extinction, scattering, absorption cross-section among the three size distributions are significant in this study. The size distribution has a significant effect on forward scattering of phase function, while the effect can be neglected as the size parameter approaches to the Rayleigh limit.     

Temperature Properties in Polydisperse Granular Mixtures

Numerical simulations are employed to consider the problem of determining the granular temperatures of the species of a homogeneous heated granular mixture with a power-law size distribution. The partial granular temperature ratios are studied as functions of the fractal dimension D, the restitution coefficient e, the rescaled viscosity time, the average occupied area fraction φ, the total particle number N and the number fraction. Different species of particles in a power-law system typically do not have the same mean kinetic energy, namely the granular temperature. It is found that the extent of nonequipartition of kinetic energy is determined by the fractal dimension D, the restitution coefficient e and the rescaled viscosity time, while is insensitive to the total particle number N , the area fraction φ and the number fraction.     

Three-angle scattering/extinction versus TEM measurements on soot in premixed ethylene/air flame

In this work, a three-angle scattering and extinction technique has been applied in order to study soot formation and growth in a rich ethylene/air premixed flame (Φ=2.34). The Rayleigh–Debye–Gans theory together with the fractal-like approach has been applied to derive soot parameters, in terms of volume concentration and morphology. A mathematical procedure is presented to obtain the radius of gyration by considering scattering signals collected at two supplementary angles. TEM measurements, carried out at different locations on the flame axis, are used to derive some parameters, such as fractal prefactor, fractal dimension and size distribution, to be entered in the mathematical treatment of optical data. The radius of gyration and the primary particle size as obtained by TEM and by optical measurements are compared. Good agreement has been found in the upper part of the flame investigated. Discrepancies observed low in the flame are discussed.     

Grayscale projection of finite size fractal clusters

Microscopy techniques are suitable to obtain structural information of colloidal clusters with high resolution, but yield only a two dimensional projection of the objects. When imaging finite size objects with fractal properties, such as clusters of colloidal particles, this projection process has to be taken into account for the calculation of the fractal dimension. In this paper we present a technique to calculate the fractal dimension of finite size clusters with fractal properties using grayscale projections such as images obtained by X-ray microscopy. The grayscales are interpreted as different occupation counts within a projection. It is shown, that the radial distribution of these occupation counts varies with the fractal dimension d of the cluster. Using the radius of maximum occupation probability the fractal dimension up to 2.2 of finite size clusters can be calculated. The theoretical predictions are verified by test calculations employing numerically generated clusters.     

Far-field intensity of electromagnetic waves scattered from random,self-affine fractal metal surfaces

Abstract

By means of the rigorous Green theorem integral equation formulation, we study the far-field intensity of linearly polarized, monochromatic electromagnetic waves scattered from a one-dimensionally rough silver surface characterized by a self-affine fractal structure. These surface fractal properties are ensured for the entire range of relevant length scales, from the illuminated spot size down to a sufficiently small (in terms of the wavelength) lower cut-off length. A peak in the specular direction is found in the angular distribution of the diffuse component of the mean scattered intensity, which becomes broader and smaller with increasing fractal dimension. For large fractal dimensions, enhanced backscattering in the case of p-polarization is observed owing to the roughness-induced excitation of surface plasmon polaritons. The interplay of different length scales of the fractal surface in the scattering process is analysed for an intermediate fractal dimension.     

Quantitative morphologic evaluation of white matter in survivors of childhood medulloblastoma

In survivors of pediatric brain tumors, cranial radiation therapy can cause a debilitating cognitive decline associated with decreased volume in normal-appearing white matter (NAWM). We applied fractal geometry to quantify white matter (WM) integrity in the brain of medulloblastoma survivors. Fractal features of WM were evaluated by indices of fractal dimensions (FDs) of WM intensity and boundary on T1-weighted magnetic resonance images. The FD index of WM intensity was calculated by using a fractional Brownian motion model, and the FD index of WM boundary was calculated by using a box-counting method. Fractal features of WM on 116 magnetic resonance images of 58 patients with medulloblastoma were investigated at the start of therapy (Start TX) and approximately 2 years later (After TX). Patients were assigned to one of two groups based on change in NAWM volumes. Fractal features in patients with decreased NAWM volume were significantly greater After TX, whereas those in patients with increased NAWM volumes were not. Multiple linear regression analysis showed that fractal features were strongly correlated with NAWM volumes After TX in patients with decreased NAWM volume. These results demonstrated significant deficit in NAWM integrity and WM density changes in children treated for medulloblastoma. Multiple regression analysis illustrated that deficits in NAWM integrity in these children may partly explain the decrease in NAWM volume. We conclude that fractal geometry can be used to monitor the morphologic effects of neurotoxicity in brain tumor survivors.     

Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

Two-dimensional random Lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap. In systems of finite spatial extension, the asymptotic decay of the survival probability is exponential and characterized by an escape rate γ, which can be related to the average positive Lyapunov exponent and to the dimension of the fractal repeller of the system. For infinite systems, the survival probability obeys a stretched exponential law of the form P(c,t)∼exp(−Ct^1/2). The transition between the two regimes is studied and we show that, for a given trap density, the non-integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space. Nevertheless, the repeller remains fractal. We determine the special scaling properties of this fractal.     

Collision Statistics of Driven Polydisperse Granular Gases

We present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the disks are subject to inelastic mutual collisions and driven by standard white noise. The inhomogeneity of the disk size distribution can be measured by a fractal dimension df. By Monte Carlo simulations, we have mainly investigated the effect of the inhomogeneity on the statistical properties of the system in the same inelasticity case. Some novel results are found that the average energy of the system decays exponentially with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state after a long evolution time. Furthermore, the inhomogeneity has great influence on the steady-state statistical properties. With the increase of the fractal dimension df, the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with df, but it is independent of time. Meanwhile, the velocity distribution deviates more strongly from the Gaussian one, but does not demonstrate any apparent universal behavior.