A Constitutive Model for Soft Mud Considering the Initial State of Effective Stress

2017 ◽  
Author(s):  
Chunyang Xu ◽  
Yongping Chen ◽  
Lili Yu
Keyword(s):  
Fractal Dimension ◽  
Constitutive Equation ◽  
Effective Stress ◽  
Volume Fraction ◽  
Soft Clay ◽  
Data Sets ◽  
Initial State ◽  
Floc Size ◽  
Sediment Volume ◽  
Self Similar

Consolidation is one of the most important processes that are needed to be accounted for in the model of cohesive sediment transport. Previous studies have shown that the consolidation paths for soil elements are not unique at low effective stress (here low effective stress means the effective stress is smaller than a threshold effective stress, which is short as TES and denoted as σb) In this study, a new theoretical constitutive equation for the effective stress of soft mud is derived to describe the non-unique consolidation paths at low effective stress. Firstly, based on the concept of fractal dimension, the mud flocs are treated as self-similar fractal entities and the relationship between mud floc size and sediment volume fraction is established. Due to the fact that the fractal dimension decreases as the floc size increases, the variation of fractal dimension is accounted for. Based on the self-similar model, the theoretical constitutive equation for the effective stress of soft mud is derived. The constitutive equation is validated using two data sets obtained from consolidation tests in a settling column for soft clay. At low effective stress, the model results curves follow the data trends and the non-uniqueness of consolidation paths are captured. The model results are also compared with existing numerical results and better agreement is presented especially when η is less than 0.5.

Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning

2021 ◽  
Author(s):  
Nicholas Dudu ◽  
Arturo Rodriguez ◽  
Gael Moran ◽  
Jose Terrazas ◽  
Richard Adansi ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Atmospheric Turbulence ◽  
Isotropic Turbulence ◽  
Passive Scalar ◽  
Counting Method ◽  
Data Set ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Box Counting Method

Abstract Atmospheric turbulence studies indicate the presence of self-similar scaling structures over a range of scales from the inertial outer scale to the dissipative inner scale. A measure of this self-similar structure has been obtained by computing the fractal dimension of images visualizing the turbulence using the widely used box-counting method. If applied blindly, the box-counting method can lead to misleading results in which the edges of the scaling range, corresponding to the upper and lower length scales referred to above are incorporated in an incorrect way. Furthermore, certain structures arising in turbulent flows that are not self-similar can deliver spurious contributions to the box-counting dimension. An appropriately trained Convolutional Neural Network can take account of both the above features in an appropriate way, using as inputs more detailed information than just the number of boxes covering the putative fractal set. To give a particular example, how the shape of clusters of covering boxes covering the object changes with box size could be analyzed. We will create a data set of decaying isotropic turbulence scenarios for atmospheric turbulence using Large-Eddy Simulations (LES) and analyze characteristic structures arising from these. These could include contours of velocity magnitude, as well as of levels of a passive scalar introduced into the simulated flows. We will then identify features of the structures that can be used to train the networks to obtain the most appropriate fractal dimension describing the scaling range, even when this range is of limited extent, down to a minimum of one order of magnitude.

scholarly journals Thermal and Wet Comfort of Fabrics Based on Fractal Dimension of Silicone Coating

2018 ◽  
Vol 13 (1) ◽  
pp. 155892501801300
Author(s):  
Yunlong Shi ◽  
Liang Wang ◽  
Wenhuan Zhang ◽  
Xiaoming Qian
Keyword(s):  
Fractal Dimension ◽  
Thermal Insulation ◽  
Coating Layer ◽  
Fractal Theory ◽  
Barrier Effect ◽  
Evaporative Resistance ◽  
Silicone Coating ◽  
Self Similar ◽  
Warmth Retention ◽  
Coated Fabrics

In this paper, thermal and wet comforts of silicone coated windbreaker shell jacket fabrics were studied. Both thermal insulation and evaporative resistance of fabric increased with an increase in coating area due to the barrier effect of the silicone coating layer. Moreover, the coated fabrics with self-similar structures showed different thermal insulation and evaporative resistance under the same total coating area. Fractal theory was used to explain this phenomenon. Optimal thermal-wet comfort properties were obtained when the fractal dimension (D=1.599) was close to the Golden Mean (1.618). When the fractal dimension of coating was lower than 1.599, fabric warmth retention was not high enough. In contrast, fabric evaporative resistance was beyond the value at which people would feel comfortable when the fractal dimension was greater than 1.599.

scholarly journals The limb-brightened jet of M87 down to the 7 Schwarzschild radii scale

2018 ◽  
Vol 616 ◽  
pp. A188 ◽  
Author(s):  
J.-Y. Kim ◽  
T. P. Krichbaum ◽  
R.-S. Lu ◽  
E. Ros ◽  
U. Bach ◽  
...  
Keyword(s):  
Brightness Temperature ◽  
Angular Resolution ◽  
Magnetic Energy ◽  
Large Mass ◽  
High Angular Resolution ◽  
Data Sets ◽  
Jet Formation ◽  
Physical Conditions ◽  
Self Similar ◽  
Jet Structure

Messier 87 (M 87) is one of the nearest radio galaxies with a prominent jet extending from sub-pc to kpc scales. Because of its proximity and the large mass of its central black hole (BH), it is one of the best radio sources for the study of jet formation. We study the physical conditions near the jet base at projected separations from the BH of ~7–100 Schwarzschild radii (Rsch). Global mm-VLBI Array (GMVA) observations at 86 GHz (λ = 3.5 mm) provide an angular resolution of ~50 μas, which corresponds to a spatial resolution of only 7 Rsch and reach the small spatial scale. We use five GMVA data sets of M 87 obtained from 2004 to 2015 and present new high angular resolution VLBI maps at 86 GHz. In particular, we focus on the analysis of the brightness temperature, the jet ridge lines, and the ratio of jet to counter-jet. The imaging reveals a parabolically expanding limb-brightened jet which emanates from a resolved VLBI core of ~(8–13) Rsch in size. The observed brightness temperature of the core at any epoch is ~(1–3) × 1010 K, which is below the equipartition brightness temperature and suggests magnetic energy dominance at the jet base. We estimate the diameter of the jet at its base to be ~5 Rsch assuming a self-similar jet structure. This suggests that the sheath of the jet may be anchored in the very inner portion of the accretion disk. The image stacking reveals faint emission at the center of the edge-brightened jet on sub-pc scales. We discuss its physical implication within the context of the spine-sheath structure of the jet.

scholarly journals On continuum damage mechanics

2019 ◽  
Vol 52 (3) ◽  
pp. 125-147
Author(s):  
Kari Juhani Santaoja
Keyword(s):  
Stress Tensor ◽  
Effective Stress ◽  
Elastic Strain ◽  
Damage Mechanics ◽  
Volume Fraction ◽  
Strain Tensor ◽  
Continuum Damage Mechanics ◽  
Continuum Damage ◽  
The Matrix ◽  
Elastic Strain Tensor

A material containing spherical microvoids with a Hookean matrix response was shown to take the appearance usually applied in continuum damage mechanics. However, the commonly used variable damage D was replaced with the void volume fraction f , which has a clear physical meaning, and the elastic strain tensor \Bold {ε}^e with the damage-elastic strain tensor \Bold {ε}^{de}. The postulate of strain equivalence with the effective stress concept was reformulated and applied to a case where the response of the matrix obeys Hooke’s law. In contrast to many other studies, in the derived relation between the effective stress tensor \Bold {\Tilde{σ}} and the stress tensor \Bold {σ}, the tensor \Bold {\Tilde{σ}} is symmetric. A uniaxial bar model was introduce for clarifying the derived results. Other candidates for damage were demonstrated by studying the effect of carbide coarsening on creep rate.

scholarly journals Dynamics of Subsiding Shells in Actively Growing Clouds with Vertical Updrafts

2020 ◽  
Vol 77 (4) ◽  
pp. 1353-1369 ◽  
Author(s):  
Vishnu Nair ◽  
Thijs Heus ◽  
Maarten van Reeuwijk
Keyword(s):  
Shell Thickness ◽  
Large Scale ◽  
Cloud Environment ◽  
Initial State ◽  
Cumulus Clouds ◽  
Mean Velocity ◽  
Flow Parameters ◽  
Shallow Cumulus ◽  
Order Of Magnitude ◽  
Self Similar

Abstract The dynamics of a subsiding shell at the edges of actively growing shallow cumulus clouds with updrafts is analyzed using direct numerical simulation. The actively growing clouds have a fixed in-cloud buoyancy and velocity. Turbulent mixing and evaporative cooling at the cloud edges generate a subsiding shell that grows with time. A self-similar regime is observed for first- and second-order moments when normalized with respective maximum values. Internal scales derived from integral properties of the flow problem are identified. A self-similarity analysis using these scales reveals that contrary to classical self-similar flows, the turbulent kinetic energy budget terms and velocity moments scale according to the buoyancy and not with the mean velocity. The shell thickness is observed to increase linearly with time. The buoyancy scale remains time invariant and is set by the initial cloud–environment thermodynamics. The shell accelerates ballistically with a magnitude set by the saturation value of the buoyancy of the cloud–environment mixture. In this regime, the shell is buoyancy driven and independent of the in-cloud velocity. Relations are obtained for predicting the shell thickness and minimum velocities by linking the internal scales with external flow parameters. The values thus calculated are consistent with the thickness and velocities observed in typical shallow cumulus clouds. The entrainment coefficient is a function of the initial state of the cloud and the environment, and is shown to be on the same order of magnitude as fractional entrainment rates calculated for large-scale models.

ESTIMATING THE FRACTAL DIMENSION OF PLANTS USING THE TWO-SURFACE METHOD: AN ANALYSIS BASED ON 3D-DIGITIZED TREE FOLIAGE

Fractals ◽  
2006 ◽  
Vol 14 (03) ◽  
pp. 149-163 ◽  
Author(s):  
FRÉDÉRIC BOUDON ◽  
CHRISTOPHE GODIN ◽  
CHRISTOPHE PRADAL ◽  
OLIVIER PUECH ◽  
HERVÉ SINOQUET
Keyword(s):  
Fractal Dimension ◽  
Field Measurements ◽  
Three Dimensions ◽  
Total Leaf Area ◽  
Topological Information ◽  
Surface Method ◽  
Plant Foliage ◽  
Self Similar ◽  
Branching System ◽  
Peach Trees

In this paper, we present a method to estimate the fractal dimension of plant foliage in three dimensions (3D). This method is derived from the two-surface method introduced in the 90s to estimate the fractal dimension of tree species from field measurements on collections of trees. Here we adapted the method to individual plants. The multiscale topology and geometry of the plant must first be digitized in 3D. Then leafy branching systems of different sizes are constructed from the plant database, using the topological information. 3D convex envelops are then computed for each leafy branching system. The fractal dimension of the plant is finally estimated by comparing the total leaf area and the convex envelop area of these leafy modules. The method was assessed on a set of four peach trees entirely digitized at shoot scale. Results show that the peach trees have a marked self-similar foliage with fractal dimension close to 2.4.

scholarly journals A laboratory study of sediment deformation: stress heterogeneity and grain-size evolution

1996 ◽  
Vol 22 ◽  
pp. 167-175 ◽  
Author(s):  
Neal R. Iverson ◽  
Thomas S. Hooyer ◽  
Roger Leb. Hooke
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Normal Stress ◽  
Size Distribution ◽  
Grain Size Distribution ◽  
Local Stress ◽  
Large Strains ◽  
Self Similar ◽  
Deformation Stress ◽  
Stress Heterogeneity

In shearing sediment beneath glaciers, networks of grains may transiently support shear and normal stresses that are larger than spatial averages. Consistent with studies of fault-gouge genesis, we hypothesize that crushing of grains in such networks is responsible for surrounding larger grains with smaller grains. At sufficiently large strains, this should minimize stress heterogeneity, favor intergranular sliding and abrasion rather than crushing, and result in a self-similar grain-size distribution.This hypothesis is tested with a ring-shear device that slowly shears a large annular sediment sample to high strains. Shearing and comminution of weak equigranular (2.0–3.3 mm) sediment resulted in a self-similar grain-size distribution with a fractal dimension that increased with shear strain toward a steady value of 2.85. This value is significantly larger than that of gouges produced purely by crushing, 2.6, but it is comparable to values for tilts thought to be deforming beneath modern glaciers, 2.8 to nearly 3.0. At low strains, under a steady mean normal stress of 84 kPa, variations in normal stress measured locally ranged in amplitude from 50 to 300 kPa with wavelengths that were 100 times larger than the initial grain diameter. Crushing of grains, observed through the transparent walls of the device, apparently caused the failure of grain networks. At shearing displacements ranging from 0.7 to 1.0 m, the amplitude of local stress fluctuations decreased abruptly. This change is attributed to fine sediment that distributed stresses more uniformly and caused grain networks to fail primarily by intergranular sliding rather than by crushing of grains. Sliding between grains apparently produced silt by abrasion and resulted in a fractal dimension that was higher than if there had been only crushing.A size distribution with a fractal dimension greater than 2.6 is probably a necessary but not sufficient condition for determining whether a basal till has been highly deformed. Stress heterogeneity in subglacial sediment that is shearing through its full thickness should contribute to the erosion of underlying rock.

ON THE APPLICATION OF METHODS USED TO CALCULATE THE FRACTAL DIMENSION OF FRACTURE SURFACES

Fractals ◽  
2001 ◽  
Vol 09 (01) ◽  
pp. 105-128 ◽  
Author(s):  
TAYFUN BABADAGLI ◽  
KAYHAN DEVELI
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Natural Fracture ◽  
Data Sets ◽  
Data Set ◽  
Fracture Surfaces ◽  
Power Spectral ◽  
Data Points ◽  
2D Data ◽  
Fracturing Mechanism

This paper presents an evaluation of the methods applied to calculate the fractal dimension of fracture surfaces. Variogram (applicable to 1D self-affine sets) and power spectral density analyses (applicable to 2D self-affine sets) are selected to calculate the fractal dimension of synthetic 2D data sets generated using fractional Brownian motion (fBm). Then, the calculated values are compared with the actual fractal dimensions assigned in the generation of the synthetic surfaces. The main factor considered is the size of the 2D data set (number of data points). The critical sample size that yields the best agreement between the calculated and actual values is defined for each method. Limitations and the proper use of each method are clarified after an extensive analysis. The two methods are also applied to synthetically and naturally developed fracture surfaces of different types of rocks. The methods yield inconsistent fractal dimensions for natural fracture surfaces and the reasons of this are discussed. The anisotropic feature of fractal dimension that may lead to a correlation of fracturing mechanism and multifractality of the fracture surfaces is also addressed.

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