Determining Effective Material Properties and Particles Size for Asphaltic Composites Using Microstructure Approach

1

DETERMINING EFFECTIVE MATERIAL PROPERTIES AND PARTICLES

SIZE FOR ASPHALTIC COMPOSITES USING MICROSTRUCTURE

APPROACH

Samer Dessouky1, M. ASCE, A.T. Papagiannakis2, F. ASCE 1Assistant Professor, email: [email protected]. 2Professor, email: [email protected].

Department of Civil and Environmental Engineering, University of Texas at San

Antonio, San Antonio, TX 78249, Ph. (210) 458-7072, Fax (210) 458-6475.

ABSTRACT

Asphaltic materials consist of wide range of distribution of particles. The

several order of magnitude differences in the stiffness between the constituents

(aggregate and binder) cause a wide range of stress and strain distribution within the

microstructure. This study aims to using two schemes to analyze the microstructure

of asphalt composites to identify the effective particle size and effective material

properties.

INTRODUCTION

Asphaltic materials consist of wide range of distribution of particles. The

several order of magnitude differences in the stiffness between the constituents

(aggregate and binder) cause a wide range of stress and strain distribution within the

microstructure. Constitutive models implemented by Finite Element methods have

been used to analyze the asphalt mix microstructure (e.g. Kose et al. 2000, Masad et

al. 2001). This approach is limited in capturing the details of the microstructure in a

representative volume element (RVE). This study aims to using two schemes to

analyze the microstructure of asphalt composites to identify the effective particle size.

An approach using micromechanics principal is also presented to determine and

effective material properties for the RVE. The purpose of these schemes is to

evaluate the effective length scale, average particles size, of the RVE. These schemes

can be used to introduce the RVE in a less heterogeneous representation. The two

methods used in the study are moving window technique and Auto correlation

function. The first is developed by Paley and Aboudi (1992). The moving window

divides a repeating rectangular representative element into any selected number of

rectangular subcells. The scheme is capable of predicting the behavior of periodic

microstructure from given properties of their constituents. The second scheme

describes the relative directional arrangement of the matrix constituents. The scheme

has a significant impact in engineering characteristics such as fluid flow in porous

media and strength anisotropy of granular materials.

EXPERIMENTAL CHARACTERIZATION

Six Superpave asphalt mix designs with different gradation are shown in

Table 1. The mixes were prepared with different nominal maximum aggregate sizes

2

and two aggregate types (limestone and gravel). Specimens were compacted using

the Superpave gyratory compactor. Each specimen had a diameter of 150 mm and a

height close to 100 mm. A diamond saw was used to cut these specimens vertically

in 50mm diametral parts (Dessouky et al. 2003).

Table 1. Description of Asphalt Mixes

Aggregate Gradation NMAS

(mm)

% Natural

sand

Design

AC, %

Mix label

Limestone Fine 9.5 0 5.3 A

Coarse 19 0 4.4 C

40 4.7 D

Gravel Fine 9.5 0 6.3 I

Coarse 19 0 5.4 K

40 4.8 L

* Nominal maximum aggregate size.

Gray scale images were captured using a digital camera. The original image

was first reduced to a rectangular image with dimensions of 400400 pixels and a

resolution of 0.3 mm/pixel. A pixel in a gray-scale image has intensity from 0,

representing black, to 255, representing white. Examples of images captured from

the mixes are shown in Figure 1. An image was thresholded and converted to a

black-and-white image, where black represented aggregate particles larger than the

image resolution of 0.3 mm and white represented the matrix, which consisted of

asphalt, air voids, and particles smaller than 0.3 mm.

Original true color mix Gray scale image (400X400

pixels)

Black/white

converted image

Fig. 1. Examples of Image conversion from color mode to black/white mode for mix

K

Effective Local Material Properties

3

The effective material properties are calculated using a micromechanics

solution that captures the influence of aggregate concentration within the moving

window on the effective material properties. Christensen (1990) developed a

differential dilute suspension method for effective shear and bulk moduli and ,

respectively to better represent nondilute composites. The concept of the differential

method is to add small percentages of particles (c) incrementally into the matrix.

New effective properties are obtained for the composite in each increment and are

subsequently used as matrix material properties in the following increments until c =

100%. The differential equations that represent this method are as follows:

01

3

4

0126

435

cdc

d

cdc

d

(1)

However, the / ratio for HMA depends on the concentration of aggregates,

and consequently, these solutions are not representative of asphalt mixes (Kim and

Little 2004). Therefore, the following exponential relation between and is

proposed to solve Eq. (1):

c

1exp (2)

where and are material constants determined based on experimental

measurements. Solution of Eq. 1 is obtained by defining the lower and upper limit of

Eq. 2, such that the composite material properties are equal to those of the binder

when c = 0 and equal to aggregate at c = 1. A Gaussian quadrature numerical scheme

was used and following expressions are obtained:

*

0 *

*

*

0 **

*

)1(

1exp)1ln(lnln

21exp)1(

1exp

6

5)1ln(

3

5lnln

dcc

cc

dccc

cc

c

m

c

m

(3)

c* is the volume fraction varied from 0 to 1. The integration part can be

solved numerically, and and are then found as a function of volume fraction. The

Young’s modulus E and Poisson’s ratio are calculated using the elastic relationship

(4). Figure 2 indicates the ratio of k/u using Eq. 3 (calculated) and Eq. 2

(exponential).

4

)3(2

23

3

9

E

(4)

Fig 2. The bulk-shear modulus ratio versus volume fraction using Eqs. 2 and 3

Moving Window Technique

The average strain in asphalt mix microstructure can be obtained by the

Taylor’s series expansion limited to the second-order term of the function )( rx

around a point x in the RVE as follows (Zbib and Aifantis 1989):

22cl)x( (5)

where lc is a characteristic length scale (average particles size)of the material

microstructure. 2 is the strain gradient determined using the strain at x and

neighbors points in the RVE.

Baxter and Graham (2000) presented an approach using FE analysis of

composite material microstructure. The effective material properties are averaged

within a moving window of definite size over the macrostructure using the

Generalized Method of Cells proposed by Paley and Aboudi (1992). The moving

window technique was used to calculate the effective elastic material properties in

Eq. (3) and the characteristic length scale, lc, in Eqs. (5). In this method, an image

that represents the RVE of a mix is divided into square windows of equal sizes. Each

window is shifted in the horizontal direction of the RVE with one pixel at least as

overlap. A parametric analysis conducted by Dessouky et al. (2006) implied that the

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size of the window capable to distinguish between mix design is 4040 pixels (1212

mm).

Effective material properties are calculated in three steps. First, the volume

fraction of aggregates is determined within the moving window. Second, Eq. (3) is

used to calculate the effective shear and bulk moduli. Finally, the corresponding

Young’s modulus and Poisson’s ratio are calculated using elasticity theory. These

effective properties are assigned to the element in the center of the moving window.

The coefficients and in Eq. (3) are calculated using the material properties for

aggregate and binder used by idealized microstructure. Consequently, the coefficients

and are found to be 3.40 and 1.67, respectively. Example of a medium of

effective properties is shown in Figure 3. It is evident that a more uniform field of

material properties is obtained using the effective material properties. This

distribution is less affected by small changes that might occur in the microstructure

distribution during image capturing and processing.

(a) Individual Constituent Properties (b) Effective Properties

Fig. 3. Young’s Modulus Distribution for Microstructures with Individual

Constituent Properties and Effective Properties (Dessouky et al. 2006)

Using the effective properties has a numerical advantage over using the

individual constituent properties. The several orders of magnitude difference in the

moduli of the aggregate and binder can cause a numerical instability (Somadevan

2000). The effective material properties reduce the difference in material properties

between constituents and consequently and eliminate this numerical limitation.

The moving window technique is also used to calculate lc. The method starts

by converting an image to a two-dimensional text array where 1 indicates a pixel that

belongs to the aggregate phase and 0 refers to a pixel that belongs to the matrix. The

average volume fraction over the RVE domain at position vector x is given as:

6

glxgg c

22)( (6)

where g is the percentage of particles at each window position and g is the average

percentage of particles in the entire image microstructure. )(xg is found by

averaging the volume fraction within each window in the RVE. g2 is calculated

between three adjacent windows positions in the horizontal direction. Example of lc is

shown in Figure 4.

Two types of RVE are examined to analyze the length scale, coarse and fine

with 19 and 9.5mm NMAS, respectively. Coarse RVE produces higher length scale

than fine microstructure. Measurements show that the effective length scale

distribution is larger for coarse particles (Figure 4). lc for the different mixes is

shown in Table 2. As the range of aggregate sizes becomes smaller, the probability

for the moving windows to have more uniform percentages of particles increases, and

consequently, lc becomes smaller.

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

0% 20% 40% 60% 80% 100%

Moving windows %

Le

ng

th s

ca

le (

m2)

Fig. 4. Length scale calculated for mixes with different gradation

Autocorrelation Function

The autocorrelation function (ACF) describes the relative arrangement of

different phases in a composite material. It evaluates the probability of locating two

points of the same material within the RVE. It is assumed that the microstructure of

asphalt mixes is statistically homogeneous, and consequently, ACF depends on the

average difference in the coordinate values between two points rather than the

locations of these points. Consequently, the two-point ACF is given as (Berryman

and Blair 1986):

iM

x

jN

y jNiM

jyixfyxfjiS

1 1 ))((

),(),(),( (7)

Coarse

Fine

7

where 1, yxf if a pixel at point yx, is located within the aggregate phase, and

0, yxf otherwise. i and j are the distances between any two pixels in two

orthogonal coordinate axes. M and N are the number of pixels in the HMA

microstructure image in two orthogonal coordinate axes.

The ACF carries important information about the microstructure distribution.

Berryman and Blair (1986) have shown that the following microstructure features can

be estimated from the ACF (Fig. 5):

fS )0( , 2

)(lim frSr

, 4

)0(s

S , s

ffrc

)1(4 (8)

where f is the average volume fraction of aggregate particles, 22 jir is the

distance between two points in the microstructure, s is the specific surface area, and

rc is the effective distance between particles (mean free path). The effective particle

size, rg, can also be determined from the ACF as shown in Fig. 5. The figure shows

the trend of the correlation function for all RVEs. The initial values (r=0) correspond

to the volume fraction (c) for each image, and then it stabilizes to the square of this

value (c2) as suggested by Berryman and Blair (1986). The representative length scale

of the microstructure can be represented by the distance rg.

Fig 5. Illustration of the Autocorrelation Function

Figure 6 shows the trend of the correlation function for all mixes. A three-

dimensional plot of the ACF distribution is given in Figure 7. According to the

results in Table 2 mixes with 19 mm NMAS have higher rg values than the 9.5 mm

NMAS mixes. There are also differences between gravel and limestone mixes with

similar particle size distributions. Limestone mixes have higher values for rc and rg

and lower values for s than gravel mixes. These results can be interpreted by the fact

f

f

S(0) = -s / 4

rg rc

2

f

S(

r )

r = (i2+j2)0.5

8

that limestone aggregate are more elongated (or less spherical) and the particles tend

to be oriented more toward the horizontal in a mix than gravel aggregates. Therefore,

the ACF measured in the horizontal direction gives larger effective length for

limestone aggregates. This shows that the parameters from the ACF can be used to

reflect the length scales associated not only with the size of particles but with their

shape as well.

Fig. 6. Autocorrelation Function for Different Mixes

9

-40

-20

0

20

40

-40

-20

0

20

40-0.5

0

0.5

1

ij

f(i,j)

Fig. 7. Three-Dimensional Representation for the Autocorrelation Function for

aggregate volume fraction

Table 2. Results of Microstructure Analysis of Asphalt Mixes

Mix Characteristic

Length Scale

lc

(mm)

Specific

Surface

area “s”

Mean

Free

Path

“rc”

(mm)

Effective

Particle

Size

“rg”

(mm)

Average

Percentage

of

Particles

“ f ”

*

f2

Measured

**

f2

A 4.11 0.178 1.5 5.4 0.660 0.436 0.433 C 5.78 0.125 2.0 9.3 0.701 0.491 0.495 D 5.28 0.140 2.0 8.4 0.639 0.409 0.405 K 2.79 0.182 1.5 4.8 0.625 0.391 0.383 I 4.73 0.166 1.6 9 0.672 0.452 0.454 L 4.75 0.157 1.9 9.3 0.523 0.273 0.449

* Using direct measurement of percentage of particles.

** Evaluated using graphical analysis of Fig 6.

CONCLUSION

A methodology for microstructure analysis of asphalt composite based on

effective material properties that capture the influence of aggregate volume fraction

on the local microscopic response has been developed. Effective material properties

have been determined using differential method developed by Christensen at any

given volume fractions. Experimental procedures were developed to determine the

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material characteristic length scale using the moving window technique and the ACF.

Asphalt mixes with course gradation have shown larger lc values compared to fine

mix. The moving window utilizes the effective materials properties to develop a

uniform field of transitioned material properties. This field can lead to more

stabilized numerical solutions for microstructural analysis of asphalt mixes. The ACF

was also used to provide more insight into the mixtures microstructure and to extract

geometric information for the particles. Limestone particles have shown more

effective length due to its elongated shape comparing to the gravel particles.

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