The Relationship between the Fractal Dimension and Shape Properties of Particles

KSCE Journal of Civil Engineering (2011) 15(7):1219-1225DOI 10.1007/s12205-011-1310-x

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www.springer.com/12205

Geotechnical Engineering

The Relationship between the Fractal Dimension and Shape Properties of Particles

Seracettin Arasan*, Suat Akbulut**, and A. Samet Hasiloglu***

Received September 2, 2010/Accepted January 9, 2011

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Abstract

Due to their irregularity, the shape of particles is not accurately described by Euclidian geometry. However, fractal geometry usesthe concept of fractal dimension, DR, as a way to describe the shape of particles. In this study, the fractal dimensions and shapeproperties of particles were determined using image analysis. Exponential relationships between the fractal dimension androundness, sphericity, angularity, convexity were described. A set of empirical correlations were also presented which clearlydemonstrated the link between fractal dimension and shape properties of particles. Additionally, a new classification chart proposedfor use in describing and comparing particle shape and fractal dimension.Keywords: fractal dimension, particle shape, roundness, image analysis

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1. Introduction

The importance of the shape of aggregate particles on theirmechanical behavior is well recognized. In asphalt concrete, theshape of aggregate particles affects the durability, workability,shear resistance, tensile strength, stiffness, fatigue response, andoptimum binder content of the mixture (Shklarsky and Livneh,1964; Li and Kett, 1967; Stephens and Sinha, 1978; Kalcheff andTunnicliff, 1982; Huber and Heiman, 1987; Krutz and Sebaaly,1993; Kuo et al., 1996; Odurph et al., 2000; Masad et al., 2001;Topal and Sengoz, 2005; Al-Rousan et al., 2007; Arasan et al.,2010a). Since up to approximately 80 percent of the total volumeof concrete consists of aggregate, aggregate characteristics signi-ficantly affect the performance of fresh and hardened concreteand have an impact on the cost effectiveness of concrete (Ozol,1978; Hudson, 1999; Kwan et al., 1999; Erdogan, 2005; Erdoganet al., 2006; Jamkar and Rao, 2004). On the other hand, manyfundamental studies were published mainly in the literature dealingwith effect of particle shape on the geotechnical properties ofsoils. Some studies have been devoted to developing proceduresto determine the particle shape properties (Masad and Buton,2000; Rao and Tutumluer, 2000; Alshibli and Alsaleh, 2004;Masad et al., 2005; Al-Rousan et al., 2007; Arasan et al. 2010b).Others have focused on the effect of particle shape on maximumand minimum void ratios (Holubec and D'Appolonia, 1973; Youd,1973; Miura, 1997; Lade et al., 1998; Cubrinovski and Ishihara,1999; Cubrinovski and Ishihara, 2002; Yilmaz, 2009; Arasan etal., 2010c) and internal friction angle of granular soils (Chan andPage, 1997; Santamarina and Cho, 2004; Cho et al., 2006. Gori

and Mari, 2001). Due to their irregularity, the shape of aggregates is not accu-

rately described by Euclidian geometry. Fractals are relatively newmathematical concept for describing the geometry of irregularlyshaped objects in terms of frictional numbers rather than integer.The concept of fractals introduced by Mandelbrot (1977), whichhas the shape formed in nature, has been usually analyzed usingEuclidian geometry. The key parameter for fractal analysis is thefractal dimension, which is a real non-integer number, differingfrom the more familiar Euclidean or topological dimension. Thefractal dimension for a line of any shape varies between one andtwo, and for a surface between two and three.

In recent years, fractal geometry techniques have found wide-spread applications in many disciplines, including medicine,biology, geography, meteorology, manufacturing, and materialscience. Relatively, there have been a few applications of fractalgeometry in civil engineering. Some studies have been devotedto developing procedures to determine the particle fractaldimensions (Kaye 1978; Kennedy and Lin 1992; Hoyez, 1994;Vallejo 1995; Vallejo and Zhou, 1995; Hyslip and Vallejo, 1997;Akbulut, 2002; Hasiloglu et al., 2010). Others have focused onthe effect of fractal dimension of particles on engineering pro-perties of soils (Gori and Mari, 2001; Xu and Sun, 2005; Arasanet al., 2010d, 2010e). However, there is no comprehensive studythat investigated the effect of particle shape to the fractal dimen-sion of particles. Consequently, the present study was undertakento investigate the relationship between the fractal dimension andshape properties.

*Assistant Professor, Dept. of Civil Engineering, Ataturk University, Erzurum 25240, Turkey (Corresponding Author, E-mail: [email protected])**Professor, Dept. of Civil Engineering, Ataturk University, Erzurum 25240, Turkey (E-mail: [email protected])

***Associate Professor, Dept. of Computer Engineering, Ataturk University, Erzurum 25240, Turkey (E-mail: [email protected])

Seracettin Arasan, Suat Akbulut, and A. Samet Hasiloglu

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2. Particle Shape Properties

Particle geometry can be fully expressed in terms of threeindependent properties: form, angularity (or roundness), andsurface texture. Fig. 1 shows a schematic diagram that illustratesthe differences between these properties. Also, form, roundnessand surface texture are essentially independent properties ofshape because one can vary widely without necessarily affectingthe other two properties (Barrett, 1980). Form, the first orderproperty, reflects variations in the proportions of a particle.Angularity, the second order property, reflects variations at thecorners, that is, variations superimposed on shape. Surfacetexture is used to describe the surface irregularity at a scale that istoo small to affect the overall shape (Barrett, 1980; Masad,2004). Different researchers are using different shape indexes todescribe the shape of aggregate particles and even differentdefinitions for the same shape index. Besides, many researchersdiscussed the image analysis techniques used by most of theavailable imaging systems that utilize different mathematicalprocedures for the analysis of particle shape characteristics (Kuoand Freeman, 2000; Masad et al., 2005; Al-Rousan et al., 2007).

In this study, for the first order of shape, form, the imagingindex proposed is: sphericity is measured for each particle. Forthe second order of shape, roundness, the imaging indexesproposed are: roundness, angularity, and convexity are measuredfor each particle.

2.1 Sphericity (S)Sphericity is among a number of indexes that have been

proposed for measuring the form of particles.

(1)

2.2 Roundness (R)This is a shape factor that has a maximum value of 1 for a

circle and smaller values for shapes having a lower ratio of area(A) to perimeter (P), longer or thinner shapes, or objects havingrough edges.

(Cox, 1927) (2)

2.3 Angularity (K)This is a shape factor that has a minimum value of 1 for circle

and higher values for shapes having angular edges.

(3)

2.4 Convexity (C)Convexity is a measure of the surface roughness of a particle.

Convexity is sensitive to changes insurface roughness but notoverall form.

(4)

where, A: projected area of particle (mm2); L: projected longest dimension of particle (mm); P: projected perimeter of particle (mm); Pellipse=(L+1)

(π/2)Pellipse: perimeter of equivalent ellipse (mm),

I: projected intermediate dimension of particleArectangle: area of bounding rectangle (mm2). Arectangle=L×1

3. Fractal Dimension

Fractal theory uses the concept of fractal dimension, DR, as away to describe the shape of particles. Fractal dimension of part-icles was calculated with divider, parallel-line, and area-peri-meter methods. The area-perimeter method is also the easiestmethod (Hyslip and Vallejo, 1997). In this study, the fractaldimension of the particles is evaluated using the area-perimetermethod introduced by Mandelbrot (1983) and later Hyslip andVallejo (1997). This method based on the “ratio of linear extents,”proposed by Mandelbrot (1983). The “ratio of linear extents” offractal patterns is in themselves fractal, and that:

(5)

Where, c is a constant for similar fractal shapes and DR is theroughness fractal dimension of the population. Again, taking thelogarithm of Eq. (5), c yields a linear relationship between area Aand perimeter P with DR related to the slope coefficient, m, by:

(6)

Evaluating the fractal dimensions (DR) using Eq. (6), Dr=2/mis referred to as the “area-perimeter” method which determinesDR by evaluating an entire population of related shapes as

S Aπ L2×

4-------------⎝ ⎠⎛ ⎞------------------=

R 4 ΠA⋅P2

--------------=

K PPellipse--------------=

C AArec gletan------------------=

c P1 D⁄ RA0.5

-------------=

DR 2 m⁄=

Fig. 1. The Hierarchical View of Form, Roundness, and SurfaceTexture (Barrett, 1980)


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opposed to individual particles (Hyslip and Vallejo, 1997).

4. Materials and Methods

Basalt soils, calcareous soils, commercially available abrasiveball and bearing ball are used in this study. The specific gravityof materials is determined to be 2.90, 2.62, 7.0, and 6.5 accord-ing to ASTM D 854, respectively. 9 samples of Basalt soil (B), 9samples of Calcareous soils (C), 6 samples of Rounded Calcareoussoils (RC), 6 samples of Well-rounded Calcareous Soils (WRC),5 samples of Abrasive Ball (AB), and 7 samples of Bearing Ball(BB) were prepared with different graded and roundness forresearching all roundness class of soils. The grain size distribu-

tion of materials is also determined according to ASTM D 1140(Fig. 2 and Fig. 3). The rounded and well rounded calcareoussamples were prepared using Los Angeles Rattler Machinewithout steel balls as detailed in Arasan (2010). The numbers ofmachine revolution were selected 50.000 and 100.000 for round-ed and well rounded calcareous samples, respectively. Round-ness values, roundness classes and fractal dimensions of usedmaterials are given in Table 1. The roundness values and classesof samples were also determined by using Cox (1927) equationand Powers (1953) chart, respectively. Additionally, a typicalphotograph of used materials is given in Fig. 4.

The fractal dimensions (DR) of particles are examined from 2Dimages of at least 200 particles. The imaging system used by theauthors consists of a Nikon D80 Camera and Micro 60 mmobjective manufactured by Nikon. The other properties of usedmaterials test procedures, imaging system and digital image pro-cessing steps were also detailed in Arasan and Akbulut (2008)and Arasan et al. (2010a). The images were processed in ImageJthen output data of ImageJ was transferred to Excel. The outputdata also contain area and perimeter of each particle. Then thefractal dimensions of samples are determined by Eqs. (5) and (6).

5. Relationship between the Fractal Dimensionand Shape Properties

The correlation between fractal dimension of particles andparticle shape indexes (i.e., sphericity, roundness, angularity, andconvexity) of granular materials are presented in Figs. 5, 6, 7,and 8, respectively. It is seen that the sphericity (Fig. 5), round-ness (Fig. 6), and convexity (Fig. 8) decreased with the fractaldimension increased. However, the angularity increased as thefractal dimension increased (Fig. 7). It is an expected result sincehigher angularity and lower roundness values represent higherparticle surface irregularities (Russell and Taylor, 1937; Krumbein,1941; Rittenhouse, 1943; Pettijohn, 1949; Powers, 1953; Alshibliand Alsaleh, 2004), and it is well known that increasing particleirregularities increases fractal dimension (Vallejo, 1995; Vallejoand Zhou, 1995; Akbulut 2002; Kolay ve Kayaball, 2006).Similarly, Vallejo and Zhou (1995) indicated that the fractaldimension is significantly affected by the particle shape in a waythat fractal dimensions increase with increasing angularity ordecreasing roundness of the particles. Hence, it could be saidthat, since fractal dimension has a minimum value of 1 for a circleand larger values longer or thinner shapes, or particle having rough

Fig. 2. Grain-size Distrubition of the Basalt and Calcereous Soils(B: Basalt Soil, C: Calcereous Soil, RC: Rounden Calcere-ous Soil, WRC: Well Rounded Calcereous Soil) (Arasan etal., 2010e)

Table 1. Roundness and Fractal Properties of Materials used in Study

Sample Roundness Value Roundness Class Fractal Dimension Basalt (B) 0.609-0.737 Very Angular-Angular-Sub Angular 1.166-1.370Calcareous (C) 0.693-0.744 Angular-Sub Angular 1.121-1.289Rounded Calcareous (RC) 0.786-0.803 Rounded-Well Rounded 1.092-1.232Well Rounded Calcareous (WRC) 0.834-0.854 Well Rounded 1.076-1.159Abrasive Ball (AB) 0.855-0.872 Well Rounded 1.036-1.123Bearing Ball (BB) 0.902-0.904 Well Rounded 1.002-1.010

Fig. 3. Grain-size Distrubition of the Abrasive Ball (A.B.) and Bear-ing Ball (B.B.) (Arasan et al., 2010e)



Fig. 4. Typical Photographs of used Materials (Arasan et al., 2010e)

Fig. 5. Relationship between Fractal Dimension and Sphericity Fig. 6. Relationship between Fractal Dimension and Roundness


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edges, it can be concluded that approximation of the shape ofparticles to sphere or smooth particle surfaces resulted in lowerangularity and higher sphericity, roundness, convexity values.

In addition, some exponential relationships are found betweenthe fractal dimensions and shape indexes (Table 2). The correla-tion coefficient between fractal dimension and shape indexeswere found to be equal to 0.65-0.81, which indicated a relativelyhigh correlation between the variables. Similarly, Vallejo and

Zhou (1995) were proposed an empirical correlation betweenparticle roundness and fractal dimension.

6. New Classification Chart for Particle Shapes

The definition of particle shape in terms of roundness is widelyaccepted. However, methods have not been standardized becauseof the tedious task of making numerous readings. Analyses ofroundness are often made visually (Alshibli and Alsaleh, 2004).Numerous researchers supplemented roundness classes withdetailed descriptions so that the particle can be classified visually(Russell and Taylor, 1937; Krumbein, 1941; Rittenhouse, 1943;Pettijohn, 1949; Powers, 1953). However, there is no compre-hensive classification chart that presented together the fractaldimension. Consequently, Powers (1953) chart was modified anda new classification chart (i.e., fractal dimension scale) wasintroduced for particles in this study (Fig. 9). Six classes of parti-cle shape were defined in Fig. 9 such as very angular, angular,sub-angular, sub-rounded, rounded, and well rounded. The chartshowed that fractal dimension of natural particles was changedfrom 1 to 1.70. The Fig. 9 also indicated that the fractal dimen-sion could be used as a shape descriptor instead of roundness.

7. Conclusions

The present study was undertaken to investigate the effect ofparticle shape to the fractal dimension of particles. The test

Fig. 7. Relationship between Fractal Dimension and Angularity

Fig. 8. Relationship between Fractal Dimension and Convexity

Table 2. Relationship between Fractal Dimensions and ShapeProperties

y x Relationship R2

DR S y = 1.8619e-0.635x 0.78DR R y = 2.5189e-0.981x 0.81DR K y = 0.2533e1.3639x 0.74DR C y = 4.9693e-1.989x 0.65

Particle Images UnnaturalParticles

Roundness Class Very Angular Angular Sub-Angular Sub-Rounded Rounded Well RoundedRoundness - Cox (1927)* < 0.50 0.5-0.65 0.65-0.70 0.70-0.75 0.75-0.77 0.77-0.80 0.80-1.00Roundness – Russell and Taylor (1937 ) - - 0.00-0.15 0.15-0.30 0.30-0.50 0.50-0.70 0.70-1.00Roundness – Krumbein (1941) < 0.10 0.10-0.20 0.20-0.30 0.30-0.50 0.50-0.60 0,60-0.80 0.80-1.00Roundness – Pettijohn (1949 ) - - 0.00-0.15 0.15-0.25 0.25-0.40 0.40-0.60 0.60-1.00Roundness - Powers (1953) 0.00-0.12 0.12-0.17 0.17-0.25 0.25-0.35 0.35-0.49 0.49-0.70 0.70-1.00Roundness - Al-Rousan (2004) > 2.00 1.54-2.00 1.43-1.54 1.33-1.43 1.29-1.33 1.25-1.29 1.00-1.25Roundness – Alshibli and Alsaleh (2004) - > 1.50 1.40-.1.50 1.30-.1.40 1.20-.1.30 1.10-.1.20 1.00-.1.10Fractal Dimension 1.70-2.00 1.55-1.70 1.40-1.55 1.30-1.40 1.25-1.30 1.15-1.25 1.00-1.15*R = (4·πA)/P2 (This equation was used in this study).

Fig. 9. New Classification Chart for Particle Shapes



results indicated that there are exponential relationships betweenthe particle shape indexes (i.e., sphericity, roundness, angularity,and convexity) and fractal dimension. It is observed from theresults that the sphericity, roundness and convexity were decreasedand the angularity was increased with increasing fractal dimen-sion of particles. Additionally, a new classification chart wasproposed for using fractal dimension as a shape descriptor. Itcould be said that, since fractal dimension has a minimum valueof 1 for a circle and larger values longer or thinner shapes, orparticle having rough edges, it can be concluded that approxima-tion of the shape of particles to sphere or smooth particle surfacesresulted in lower angularity and higher sphericity, roundness,convexity values. It should also be pointed out that further studieson the correlation between surface roughness of particles andfractal dimension are needed to make more reasonable judgments.

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The Relationship between the Fractal Dimension and Shape Properties of Particles

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