UNIVERSITI PUTRA MALAYSIA

MOHAMMAD MAHDI ROOZBAHANI

FK 2012 137

DEVELOPMENT OF COMPUTATIONAL METHOD TO DETERMINE MAXIMUM VOID RATIO OF SAND-SOIL PARTICLES USING

GRAVITATIONAL SPHERE PACKING METHOD

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DEVELOPMENT OF COMPUTATIONAL METHOD TO

DETERMINE MAXIMUM VOID RATIO OF

SAND-SOIL PARTICLES USING GRAVITATIONAL

SPHERE PACKING METHOD

By

MOHAMMAD MAHDI ROOZBAHANI

Thesis Submitted to the School of Graduate Studies, Universiti PutraMalaysia, in Fulfilment of the Requirements for the Degree of Master

of Science

October 2012

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DEDICATIONS

I would like to dedicate this project to all those who have helped me to complete it.

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Abstract of thesis presented to the Senate of Universiti Putra Malaysia infulfilment of the requirement for the degree of Master of Science

DEVELOPMENT OF COMPUTATIONAL METHOD TODETERMINE MAXIMUM VOID RATIO OF SAND-SOIL

PARTICLES USING GRAVITATIONAL SPHERE PACKINGMETHOD

By

MOHAMMMAD MAHDI ROOZBAHANI

October 2012

Chairman: Professor Bujang Bin Kim Huat, PhD

Faculty: Engineering

Abstract: These days, there has been an increasing interest in virtual reality sim-

ulation of the geotechnical field. More recently, literature has emerged that of-

fers linear experimental Equation which positively correlates the maximum and

minimum void ratio of sandy-soil particles. However, far too little attention has

been paid to virtually simulate one of this parameter in order to rapidly compute

these main parameters of the relative density without any need to do experimen-

tal works. Gravitational sphere packing which arranges non-overlapping spheres

within a confined space has powerful capability to simulate the interaction of large

number of particles. The goal of this research is to employ gravitational sphere

packing to compute maximum void ratio of sandy-soil particles with particle size

distribution as input data.

In this study, pre-processing, processing, and post-processing codes of the op-

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timized gravitational sphere packing were developed in MATLAB software. In

pre-processing stage, the initial position of the spheres was determined in (x-y)

plane by applying different random number distributions. At the stage of process-

ing, three manners of rolling, collision, and stability conditions were considered

to pack particles for each incoming sphere within the cylindrical container. The

post-process code computes porosity of packed particles through the cylindrical

container for both with and without boundary effects. The simulated model was

validated by ASTM D 4254 standard method for measuring the maximum void

ratio in metal cylindrical container, in which soil particles with different types of

sorting are employed for comparison.

Results showed that maximum void ratio of dry sandy-soil particles was computed

by considering wall-sphere interaction condition in gravitational sphere packing.

However, deactivation of boundary effects led to calculate the porosity of the

unconsolidated wet sandy-soil particles. Both above mentioned results had close

values in agreement with the experimental works. The study also revealed that

log-normally particle size distribution provides the lowest value of the porosity. In

addition, particles had high potential to be packed in square container with the

lowest value of the porosity.

All in all, the new presented method of optimized gravitational sphere packing

rapidly simulates arrangement of the soil particles in dilute suspension. The rapid

code implementation of this method can compute the main parameters of the

relative density with high accuracy which was verified by ASTM D 4254 standard

test.

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Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagaimemenuhi keperluan untuk ijazah Master Sains

PEMBANGUNAN SUATU KAEDAH PENGIRAAN BAGIMENENTUKAN LOMPANG MAKSIMUM PARTIKEL-PARTIKEL

TANAH BERPASIR DENGAN MENGGUNAKAN KAEDAHPEMBUNGKUSAN SFERA GRAVITI

Oleh

MOHAMMAD MAHDI ROOZBAHANI

Oktober 2012

Pengerusi: Professor Bujang Bin Kim Huat, PhD

Fakulti: Kejuruteraan

Pada masa kini, minat terhadap simulasi realiti maya dalam bidang geoteknik se-

makin meningkat. Baru baru ini, kesusasteraan menawarkan persaman eksperimen

linear yang mana menghubungkaitkan nisbah lompang maksimum dan minimum

partikel tanah pepejal secara positif. Walaubagaimanapun, perhatian yang amat

sedikit telah diberikan kepada kaedah simulasi maya untuk mengira salah satu

parameter tersebut bagi secara pantas mengira parameter-parameter penting bagi

ketumpatan relatif tanpa menjalankan sebarang eksperimen. Kaedah pembungku-

san sfera graviti yang menyusun sfera-sfera yang tidak bertindih di dalam ruang

yang terbatas mempunyai kebolehan untuk menjalankan simulasi bagi sebilangan

besar partikel-partikel. Matlamat penyelidikan ini adalah untuk menggunakan

pembungkusan sfera graviti untuk mengira nisbah lompang maksimum bagi tanah

zarah berpasir dengan menggunakan taburan saiz partikel sebagai data input.

Di dalam penyelidikan ini, pre-pemprosesan, pemprosesan serta pemprosesan pasca

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bagi pembungkusan sfera graviti yang mengoptimumkan telah diperolehi dengan

menggunakan perisian MATLAB. Pada peringkat pre-pemprosesan, kedudukan

awal sfera-sfera di plane (x-y) telah ditentukan dengan menggunakan nombor

taburan rawak yang berbeza. Pada peringkat pemprosesan, tiga kaedah jatuh,

perlanggaran dan keadaan stabil telah diambilkira bagi membungkus setiap sfera

yang masuk ke dalam silinder. Kod yang digunakan pada peringkat pemprosesan

pasca pula mengira keporosan partikel-partikel yang dibungkus melalui silinder

bagi keadaan kesan dengan sempadan serta keadaan tanpa kesan sempadan. Model

simulasi telah disahkan oleh piawai ASTM D 4254 yang digunakan bagi mengukur

nisbah lompang maksimum dalam bekas silinder , di mana partikel-partikel tanah

di dalam susunan yang berbeza telah dibandingkan.

Keputusan menunjukkan nisbah lompang maksimum bagi partikel tanah zarah

berpasir yang kering, telah diukur dengan mengambil kira interaksi dinding sfera

di dalam pembungkusan sfera graviti. Walaubagaimanapun, penyahaktifan ke-

san sempadan telah membawa kepada pengiraan keporosan partikel tanah zarah

berpasir yang basah dan tidak tepu. Kedua-dua keputusan yang dibincangkan,

mempunyai nilai-nilai yang hampir sama dengan keputusan yang diperolehi melalui

eksperimen. Penyelidikan ini juga mendedahkan taburan saiz partikel log-biasa

mempunyai nilai keporosan paling rendah. Partikel-partikel tersebut juga mem-

puyai potensi yang tinggi untuk dibungkus di dalam bekas segiempat sama dengan

nilai keporosan yang paling rendah.

Rumusannya, kaedah baru yang mengoptimumkan pembungkusan sfera graviti ini

mensimulasikan dengan pantas kedudukan partikel-partikel tanah di dalam peng-

gantungan yang cair. Kod pantas bagi kaedah ini boleh digunakan bagi men-

gira parameter-parameter penting bagi ketumpatan relatif dengan ketepatan yang

tinggi yang telah disahkan oleh Ujian Piawai ASTM D 4254.

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ACKNOWLEDGEMENTS

My utmost gratitude goes to God. I would like to express a special thanks to

my academic advisor Professor Bujang Bin Kim Huat. It was a well rewarding

experience to have been under his supervision. I deeply appreciate his patient

instructions and continuous support he has given me during my study and research.

I am also graceful to my supervisory committee member, Dr. Haslinda Nahazanan.

I warmly thank Dr. Afshin Asadi for his generous contributions and effective

advices to my research. I indeed appreciate Professor Hasanah Mohd Ghazali for

her kind supports during the difficult moments of my research.

On a personal note, I am thankful of Professor Lori Graham-Brady for her kind,

supportive, and valuable helps during my research.

I would especially like to thank my wife, Elnaz Younessian, for her continuing

support and patience. I owe my loving thanks to my parents Hoeseinreza and

Fatemeh Roozbahani.

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I certify that a Thesis Examination Committee has met on 11 October 2012 toconduct the final examination of Mohammad Mahdi Roozbahani on his thesis enti-tled “Development of Computational Method to Determine Maximum Void Ratioof Sand-soil Particles using Gravitational Sphere Packing Method” in accordancewith the Universities and University Colleges Act 1971 and the Constitution ofthe Universiti Putra Malaysia [P.U.(A) 106] 15 March 1998. The Committee rec-ommends that the student be awarded the Degree of Master of Science.

Members of the Thesis Examination Committee were as follows:

Ahmad Rodzi Mahmud, PhDAssociate ProfessorFaculty of EngineeringUniversiti Putra Malaysia(Chairperson)

Abang Abdullah Abang Ali, PhDProfessorFaculty of EngineeringUniversiti Putra Malaysia(Internal Examiner)

Zainuddin Md. Yusoff, PhDSenior LecturerFaculty of EngineeringUniversiti Putra Malaysia(Internal Examiner)

Lori Graham-Brady, PhDProfessorFaculty of EngineeringJohns Hopkins UniversityCountry USA(External Examiner)

SEOW HENG FONG, PhDProfessor and Deputy DeanSchool of Graduate StudiesUniversiti Putra Malaysia

Date:

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This thesis was submitted to the Senate of Universiti Putra Malaysia and has beenaccepted as fulfilment of the requirement for the degree of Master of Science.The members of the Supervisory Committee were as follows:

Bujang Bin Kim Huat, PhDProfessorFaculty of EngineeringUniversiti Putra Malaysia(Chairperson)

Haslinda Nahazanan, PhDSenior LecturerFaculty of EngineeringUniversiti Putra Malaysia(Member)

BUJANG BIN KIM HUAT, PhDProfessor and DeanSchool of Graduate StudiesUniversiti Putra Malaysia

Date:

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DECLARATION

I declare that the thesis is my original work except for quotations and citations

which have been duly acknowledged. I also declare that it has not been previ-

ously, and is not concurrently, submitted for any other degree at Universiti Putra

Malaysia or at any other institution.

MOHAMMAD MAHDI ROOZBAHANI

Date:

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TABLE OF CONTENTS

Page

ABSTRACT ii

ABSTRAK iv

ACKNOWLEDGMENTS vi

APPROVAL vii

DECLARATION ix

LIST OF TABLES xii

LIST OF FIGURES xiii

LIST OF ABBREVIATIONS xvii

CHAPTER

1 INTRODUCTION 11.1 Motivation of the Research 11.2 Research Objectives 31.3 Scope of the Research 3

2 LITERATURE REVIEW 52.1 Introduction 52.2 Soil Inner Structure 5

2.2.1 Fabric of the Soil Particles 52.2.2 Particle Associations in Soils 62.2.3 Porosity 7

2.3 Particle Packing 132.3.1 Different types of packing 132.3.2 Equal-sized Sphere Packing 142.3.3 Gravitational Sphere Packing 162.3.4 Close Sphere Packing 21

2.4 Conclusion 22

3 METHODOLOGY 233.1 New Gravitational Sphere Packing Method 23

3.1.1 Spatial Binning 233.1.2 Computational Method 24

3.2 Rectangular Container 283.2.1 Computational Method in Different Random Numbers Dis-

tribution 303.3 Cylindrical Container 39

3.3.1 Computational Method in Cylindrical Container 39

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3.4 Experimental Works 433.4.1 Test Sieves 443.4.2 Specific Gravity 443.4.3 Maximum Void Ratio 46

4 RESULTS AND DISCUSSIONS 524.1 Model Validation 52

4.1.1 Validating the elimination of the boundary effect 524.1.2 Mono-sized sphere packing 55

4.2 Effect of rectangular container’s sides on porosity for equal-sizedsphere packing 57

4.3 Effect of Different Random Numbers Distribution on Porosity ofSpherical particles 604.3.1 Mono-sized sphere packing with different distributions 604.3.2 Multi-sized sphere packing with different distributions 62

4.4 Total Porosity Calculation of Unconsolidated Sandy Soil with NewGravitational Sphere Packing Method 674.4.1 The occurrence of variations in porosity by getting spheres

away from walls edge 674.4.2 Porosity variation in packing of bio-sized spheres through

the height of container 804.4.3 Porosity of unconsolidated sandy soil in different sorting co-

efficient 884.5 Calculation of Maximum Void Ratio of dry sandy soils Based on

Developed Gravitational Sphere Packing Method 904.5.1 Results of experimental works 904.5.2 Maximum void ratio 97

5 CONCLUSIONS AND FUTURE WORK 985.1 Conclusions 985.2 Future Work 100

REFERENCES 101

BIODATA OF STUDENT 107

LIST OF PUBLICATIONS 109

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LIST OF TABLES

Table Page

2.1 Characteristics of different ideal packing structure. (Adapted fromDullien (1992); Mitchell and Soga (2005)) 15

3.1 Sorting specifications. (Adapted from Beard and Weyl (1973)) 51

4.1 Porosity values with the wall effects in the cylindrical container 56

4.2 Porosity values without the wall effects in the square container 57

4.3 Packing porosity for mono-sized spherical particles 62

4.4 Porosity for different distributions of multi-sized sphere packing 67

4.5 The statistical mode of filtered porosity results in different innercylinder containers for mono-sized sphere packing 72

4.6 The minimum porosity results through the height of cylindrical con-tainer for the biosized sphere packing 88

4.7 Porosity of sandy-soil particles in the different domains of the sort-ing coefficient 89

4.8 Sieve analysis 90

4.9 Specific gravity parameters 92

4.10 Characteristics of different particle size distributions 96

4.11 Comparison between the results of the simulated model and exper-imental works 97

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LIST OF FIGURES

Figure Page

2.1 Fabric of soil particle (Adapted from Mitchell and Soga (2005)) 6

2.2 Particle association (a) Individual clay platelet interaction, (b) in-dividual silt or sand particle interaction, (c) clay platelet group in-teraction, (d) clothed silt or sand particle interaction, and (e) partlyrecognizable particle interaction. (Adapted from Mitchell and Soga(2005)) 8

2.3 Different pore structure (Adapted from Korkut (2003)) 10

2.4 Structure of the MIP.(Adapted from John I. D’Souzae (2008)) 12

2.5 Ideal packings of uniform spheres: (a) simple cubic, (b) cubic tetra-hedral, (c) tetragonal sphenoidal, (d) pyramidal, and (e) tetrahe-dral. (Adapted from Dullien (1992); Mitchell and Soga (2005)) 16

2.6 Elimination of boundaries by periodic boundary condition for Cylin-drical Container (Adapted from Mueller (1992)). 20

2.7 Elimination of boundaries by periodic boundary condition for SquareContainer. (Adapted from Jodrey and Tory (1979)) 21

3.1 Spatial binning of the container 25

3.2 The geometrical method for making one sphere tangent to the topof three spheres 27

3.3 Spherical cap 30

3.4 Plan view of boundary effect elimination via square container 31

3.5 Side view of boundary effect elimination via square container 32

3.6 Program flow chart of rectangular container 33

3.7 Program flow chart accompanied by applying different random num-bers distribution 34

3.8 Normal distribution histogram 35

3.9 Exponential distribution histogram 36

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3.10 Log-normal distribution histogram 37

3.11 Arbitrary distribution of data in the square container 38

3.12 Arbitrary distribution histogram 38

3.13 Intersection volume with cylindrical container 41

3.14 Complete program flow chart within the cylindrical container 42

3.15 The main sample of soil provided in UPM Geotectonical Lab 45

3.16 The order of the sieves according to Table 4.8 in UPM GeotectonicalLab 46

3.17 Mechanical shaking the sieves 47

3.18 Evacuating the pyknometer 48

3.19 Pyknometer inside of desiccator containing anhydrous silica gel 48

3.20 Metal cylindrical container 49

3.21 Loosely pouring sandy soils in the standard mold 49

3.22 Carefully trimming the soil surface with a straightedge 50

4.1 100 square containers within the main cylindrical container 54

4.2 Average coordination number of square containers from Fig. 4.1 55

4.3 x-y plane of rectangular boundaries 57

4.4 Effect of container size on porosity 58

4.5 Increase proportion of porosity according to ratio of containers sides 59

4.6 x-y plane of the model 59

4.7 x-z plane of the model 60

4.8 x-y-z plane of the model 61

4.9 Mono-sized sphere packing with normally distributed initial (x− y)coordinates of spheres 63

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4.10 Mono-sized sphere packing with exponentially distributed initial(x− y) coordinates of spheres 64

4.11 Mono-sized sphere packing with log-normally distributed initial (x−y) coordinates of spheres 65

4.12 Mono-sized sphere packing with arbitrary distributed initial (x−y)coordinates of spheres 66

4.13 Multi-sized sphere packing with a normal distribution 68

4.14 Multi-sized sphere packing with an exponential distribution 69

4.15 Multi-sized sphere packing with a log-normal distribution 70

4.16 Multi-sized sphere packing with an arbitrary distribution 71

4.17 Mono-sized sphere packing for spheres with radius 1 73

4.18 Mono-sized sphere packing for spheres with radius 0.5 73

4.19 Mono-sized sphere packing for spheres with radius 0.25 74

4.20 Porosity variations of inner containers for un-filtered data of sphereswith radius 1 75

4.21 Porosity variations of inner containers for un-filtered data of sphereswith radius 0.5 76

4.22 Porosity variations of inner containers for un-filtered data of sphereswith radius 0.25 77

4.23 Porosity variations of inner containers for filtered data of sphereswith radius 1 78

4.24 Porosity variations of inner containers for filtered data of sphereswith radius 0.5 79

4.25 Porosity variations of inner containers for filtered data of sphereswith radius 0.25 80

4.26 Bio-sized sphere packing with radius 1 and 0.5 in (x-y-z) plane 81

4.27 Bio-sized sphere packing with radius 1 and 0.5 in (x-z) plane 82

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4.28 Bio-sized sphere packing with radius 1 and 0.25 in (x-y-z) plane 83

4.29 Bio-sized sphere packing with radius 1 and 0.25 in (x-z) plane 84

4.30 Porosity variations through the height of container for un-filtereddata of spheres with radiuses 1 and 0.5 85

4.31 Porosity variations through the height of container for un-filtereddata of spheres with radiuses 1 and 0.25 86

4.32 Porosity variations through the height of container for filtered dataof spheres with radiuses 1 and 0.5 87

4.33 Porosity variations through the height of container for filtered dataof spheres with radiuses 1 and 0.25 88

4.34 Particle size distribution 91

4.35 Particle size distribution for well sorted soil particles with Sc = 1.19 93

4.36 Particle size distribution for moderately sorted soil particles withSc = 1.68 94

4.37 Particle size distribution for poorly sorted soil particles with Sc =2.35 95

4.38 Particle size distribution for very poorly sorted soil particles withSc = 3.27 96

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LIST OF ABBREVIATIONS

VV Volume of voids

VS Volume of solids

VT Total volume

φ Porosity

Dr Relative density

emax Maximum void ratio based on a very loose state of the particles

emin Minimum void ratio based on a very dense state of the particles

e In situ void ratio

P Pressure

γ Surface tension of the liquid

θ Contact angle of the liquid

d Diameter of the capillary

VB Bulk volume of the sample

Va Volume of the vessel containing the sample

Vb Volume of the evacuated vessel

P2 Initial pressure

P1 Final pressure

N Coordination Number

C1, C2, C3 Pre-settled sphere centers

CS Coordinate of the center of the incoming sphere

r1, r2, r3 Radiuses of pre-settled sphere

rS Radius of incoming sphere

C1s Distance of the two touched spheres

h Height of spherical cap

a Radius of base circle

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σ Variance of data

µ The mean of data

r Sphere radius

R Container radius

b Smallest distance of the axis of the cylinder to the center of the sphere

Vi Volume of the intersection when b < R

Vc Volume of the intersection when b > R

A Largest root

B Second largest root

C Smallest root

K(k) First complete elliptic integral

E(k) Second complete elliptic integral

Π(k,−α2) Third complete elliptic integral

z The sine of the amplitude

α The modular angle

k The parameter

wi Residual of the ith data produced by regression smoothing method

ri Robust weight function

MAD Median(|ri|)

ρL Density of the liquid (Mg/m3)

m1 Mass of bottle(in g)

m2 Mass of bottle and dry soil (in g)

m3 Mass of bottle, soil and liquid (in g)

m4 Mass of bottle when full of liquid only (in g)

Gs Specific gravity

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CHAPTER 1

INTRODUCTION

1.1 Motivation of the Research

These days, simulations of the experimental works in the virtual environment

have been dramatically increased in the geotechnical field because of the rapid

data access and observing inner variations of the soil particles finely. To this end,

many methods have been extended such as sphere packing techniques (Tory et al.,

1973). Such these methods can simulate laboratory works in geotechnical field.

Sphere packing is an arrangement of non-overlapping spheres within a container.

This method is widely employed in two different structures which are statistical

geometric and numerical. The former one is used in models with a small number

of the elements, whereas the latter one is applied for a large number of elements.

The random loose sphere packing was exercised in this research with the statistical

geometric procedure which is founded by Tory and Church (Tory et al., 1973).

This method has been updated to be utilized in different fields such as powder

and nano-technology, chemical, and geological engineering.

The content of this research is concerned with loosely packing of the sandy-soil

particles with gravitational sphere packing method. However, different applica-

tions of this method have been used to compute an ideal system with low value of

the porosity. Thus, a sever optimization is needed to get close this method with

reality conditions in the experimental works of the geotechnical field.

Maximum void ratio, emax, of the sandy-soil particles is defined by this method.

emax is one of the main parameters of the relative density, Dr, based on the

Equation 1.2. Void ratio, e, is the ratio of the volume of voids to the volume

of solids according to Equation 1.1. More recently, literature has emerged that

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offers linear experimental Equation which positively correlates the maximum and

minimum void ratio, emin, of sandy-soil particles (Yilmaz and Mollamahmutoglu,

2009; Cubernovski and Ishihara, 2002). However, far too little attention has been

paid to virtually simulate emax in order to rapidly compute these main parameters

of the relative density (emax and emin) without any need to do experimental works.

Therefore, this project comes to this end to simulate one of these parameters, emax,

with particle size distribution as input data in order to rapid data access to these

parameters in a virtual environment.

e =VVVS

=VV

VT − VV=

φ

1− φ;φ =

VVVT

(1.1)

where;

VV = volume of voids

VS = volume of solids

VT = total volume

φ = porosity

Dr =emax − e

emax − emin(1.2)

where;

emax = maximum void ratio in correspondence with a very loose state of the

particles

emin = minimum void ratio in correspondence with a very dense state of the

particles

e = in situ void ratio

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In the gravitational or loosely sphere packing, very little research has been reported

on the geotechnical behavior of the elements. This research is going to relate this

method to the geotechnical field.

1.2 Research Objectives

Two main objectives of current research are categorized as follows:

• To find minimum porosity of the unconsolidated sandy soils after mechanical

rearrangement of the sand grains.

• To compute maximum void ratio of dry sandy-soil particles based on the

statistical geometric approach of the gravitational or loosely sphere packing

method.

It should be noted that above objectives are achieved according to the prelimi-

nary objectives mentioned below which are directly related to the fundamental

concepts of the random loose sphere packing (gravitational sphere packing) de-

veloping this method for different applications such as geotechnical field. For

example, better perception to choose an efficient container or random numbers for

the post-processing computations in porosity.

• To investigate the different porosity values of the identical particles in dif-

ferent rectangular containers.

• To find the lowest value of the porosity by examining different distributions

of random numbers into loosely packed multi-sized spheres.

1.3 Scope of the Research

The main scope of this research is divided into two parts which are as follows:

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Simulation part: Vital necessities of optimization in the sphere packing method

would play crucial role to provide logical results in agreement with experi-

mental works. In Chapter 2, the different methods of the sphere packing are

briefly introduced, and exclusive literature review of the gravitational sphere

packing has also been described in mentioned Chapter.

Sphere packing method has been optimized in the rapid collision detection

and container interaction with soil particles which are explained in Chapter

3. K Nearest Neighbors based on the KD-tress method and binary search

are applied to find the neighbors of every incoming sphere in the system,

and also 3D Apollonius is applied to break 2 orders equations in simple ones

so as to rapid making tangent of spheres. Two conventional containers are

considered for this method which are rectangular and cylindrical container.

Interactions of the spheres with cylindrical container were solved by applying

complete elliptic integrals. These intersections were resolved by spherical caps

for rectangular container.

It should be noted that some outlier data of the porosity values through the

height and diameter of the container are mathematically filtered by Lowes

Local Regression Smoothing procedure explained in Chapters 3 and 4.

In Chapter 4, The results of the simulated model according to the mentioned

objectives are verified by last simulation models of the literatures and current

experimental work.

Experimental works: Experimental works were done to find the maximum void

ratio for sandy-soil particles with different sorting coefficients which are clas-

sified from very well sorted to the very poorly sorted. In addition, specific

gravity, and sieve analysis are the Lab works which are discussed in Chapters

3 and 4.

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