a comparison of micro-mechanical modeling of asphalt ...pages.mtu.edu/~qingdai/mom_2005.pdf ·...

A comparison of micro-mechanical modeling ofasphalt materials using finite elements and doublet mechanics

Martin H. Sadd *, Qingli Dai

Department of Mechanical Engineering and Applied Mechanics, University of Rhode Island, 92 Upper College Road,

Kingston, RI 02881, USA

Received 11 July 2003; received in revised form 27 May 2004

Abstract

A comparative study is given between two micro-mechanical models that have been developed to simulate the behav-

ior of cemented particulate materials. The first model is a discrete analytical approach called doublet mechanics that

represents a solid as an array of particles. This scheme develops analytical expressions for the micro-deformation

and stress fields between particle pairs (a doublet). The second approach is a numerical finite element method that

establishes a network of elements between neighboring cemented particles. Each element has been developed to model

the local load transfer between particles. While the two modeling schemes come from very different beginnings, they

have a fundamental similarity. However, they also have some basic differences. In order to pursue these similarities

and differences, three example problems are investigated using each modeling approach. Even with the differences,

the two model predictions of the micro-stress distributions for each example compared quite closely. These results also

indicated significant micro-structural effects that differ from continuum elasticity theory and could lead to better expla-

nations of observed failures of these types of materials.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Finite element modeling; Doublet mechanics; Asphalt concrete; Micro-mechanical modeling; Cemented particulate mate-

rials; Material micro-structure

1. Introduction

The mechanical behavior of heterogeneous sol-

ids is commonly approached from two general

viewpoints depending on whether the material

phases are distributed as either continuous or dis-crete. Under continuous distribution, theories are

0167-6636/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechmat.2004.06.004

* Corresponding author. Tel.: +1 401 874 2425; fax: +1 401

874 2355.

E-mail address: [email protected] (M.H. Sadd).

Mechanics of Materials 37 (2005) 641–662

www.elsevier.com/locate/mechmat

mailto:[email protected]

based on continuum mechanics and have provided

many useful solutions to problems of engineering

interest. However, these theories generally develop

models that do not contain scaling effects and this

is normally regarded as a limitation to predictingmicro-mechanical material behavior. In order to

overcome this limitation, various elegant modifica-

tions to continuum mechanics have been made to

incorporate scale and micro-structural features

into the theory. On the other hand, discrete mode-

ling develops particular micro-force–deformation

relations between the various material phases

within the solid. Thus the micro-forces are distrib-uted at specific points and along directions inher-

ently connected to the discrete micro-structural

geometry. From a fundamental point of view, such

distributions are not directly representable by

traditional tensor variables used in continuum

mechanics. Discrete models normally lead to theo-

ries with one or more length scales resulting in

non-local behavior where the stress at a point willdepend on the deformation in a neighborhood

about the point. These length scales may represent

the sizes and/or separations of particles, dimen-

sions of internal cells, characteristic ranges of

particle or phase interactions, etc.

The particular material of interest in the current

work is asphalt concrete, a complex heterogeneous

material generally composed of aggregates, binder/cement, and void space. Because of these features,

the material has particular micro-structures that

affect the load carrying behavior. Typically these

micro-structures are related to aggregate geometry

such as particle size or spacing and occur at length

scales in the range 0.1–20mm. Recently a consider-

able amount of research has been conducted on

developing micro-mechanics models to predictthe behavior of asphalt, concrete, ceramics, rock

and other cemented granular materials. This previ-

ous work can be divided into analytical and com-

putational modeling.

Past analytical work has incorporated several

types of micro-mechanical continuum mechanics

theories. Examples include Cosserat/micro-polar

theories which introduce an additional kinematicrotational degree of freedom; see review articles

by Eringen (1968, 1999) and Kunin (1983). These

theories have been applied to granular materials

by Chang and Liao (1990) and Chang and Ma

(1991, 1992). Continuum theories using higher

order displacement gradients have also been used

to develop micro-mechanical models; Bardenha-

gen and Trianfyllidis (1994) for elastic latticemodels, and Chang and Gao (1995) for granular

materials. A large volume of work has used fabric

tensor theories to characterize the material micro-

structure and relate particular fabric tensors to

the material�s constitutive stress–strain response;

e.g. Nemat-Nasser and Mehrabadi (1983), Konshi

and Naruse (1988) and Bathurst and Rothenburg

(1988). Another area of analytical modeling hasused distributed body theory whereby porous/

multiphase micro-structure is accounted for using

an additional independent volume distribution

function. The general theory for elastic materials

was established by Cowin and Nunziato (1983),

and this was followed by many application papers;

e.g. Cowin (1984). Some work has approached the

problem using statistical methods to developmodels with random variation in micro-mechani-

cal properties; e.g. Ostoja-Starzewski and Wang

(1989).

One particular theory that has recently been ap-

plied to granular and asphalt materials is the dou-

blet mechanics model. This approach originally

developed by Granik (1978), has been applied to

granular materials by Granik and Ferrari (1993)and Ferrari et al. (1997). Recently Wang et al.

(2003) presented a micro-mechanical study of

top–down cracking of asphalt materials using

some results from this theory. Doublet mechanics

is a micro-mechanical discrete model whereby sol-

ids are represented as arrays of points, particles or

nodes at finite distances. A particle pair is referred

to as a doublet, and the particle spacing introduceslength scales into the micro-structural theory. The

model develops micro-stress–strain constitutive

laws for the extensional (axial), shear and torsional

deformations between the particles in each dou-

blet. The theory has shown promise is predicting

observed behaviors that are not predictable using

continuum mechanics. These behaviors include

the so-called Flamant paradox (Ferrari et al.,1997), where in a half-space under compressive

boundary loading, continuum theory predicts a

completely compressive stress field but observa-

642 M.H. Sadd, Q. Dai / Mechanics of Materials 37 (2005) 641–662

tions indicate regions of tensile stress. Other

anomalous behaviors include dispersive wave

propagation.

Another general area of micro-mechanical

modeling has taken the numerical computationapproach using finite element (FEM), boundary

element (BEM) and discrete element (DEM) meth-

ods. Examples of finite element modeling include

Liao and Chang (1992) for granular materials,

Stankowski (1990) on cemented particulate com-

posites, and Sepehr et al. (1994) for a study on as-

phalt pavement layers. Soares et al. (2003) used

cohesive zone elements to develop micro-mechani-cal fracture model of asphalt materials. A common

FEM approach to simulate particulate and hetero-

geneous materials has used an equivalent lattice

network system to represent the inter-particle load

transfer behavior. This type of micro-structural

modeling has been used previously; Bazant et al.

(1990), Mora (1992), Sadd and Gao (1998) and

Budhu et al. (1997). Along similar lines, Guddatiet al. (2002) recently presented a random truss lat-

tice model to simulate micro-damage in asphalt

concrete, and Chang et al. (2002) used a lattice

micro-structure to develop a FEM model for

concrete failure. Bahia et al. (1999) have also used

finite elements to model the aggregate-binder re-

sponse of asphalt materials. Boundary element

applications for asphalt modeling have beenpresented by Birgisson et al. (2002). Discrete

element methods simulate particulate systems by

modeling the translational and rotational behav-

iors of each particle using Newton�s laws. DEM

studies on cemented particulate materials include

the work by Rothenburg et al. (1992), Chang

and Meegoda (1993), Trent and Margolin (1994),

Sadd et al. (1992), Sadd and Gao (1997), Buttlarand You (2001), and Ullidtz (2001).

Of special interest with respect to the current

study is the micro-mechanical finite element model

previously developed by the authors, Sadd et al.

(2004a,b). Similar to the doublet mechanics ap-

proach this computational model uses basic load

transfer mechanics between particles to establish

a numerical scheme to simulate asphalt materialbehavior. This is accomplished by first developing

a frame-type element to simulate the micro-load

carrying response between cemented aggregates.

The overall asphalt behavior is then modeled by

a network of these micro-finite elements. This

scheme has shown good success in several applica-

tions and has compared reasonably well with

experimental results.The purpose of this work is to present a com-

parison of modeling results from doublet mechan-

ics and the above mentioned finite element model.

We pursue such a comparison because of the inter-

esting similarities and differences between doublet

mechanics and the finite element approach. Each

model comes from very different beginnings, dou-

blet mechanics from a discrete analytical scheme,and finite elements based on the usual numerical

element equation model. Both schemes are based

on fundamental micro-mechanics between particle

pairs that make up the media, and each theory will

lead to the inclusion of one or more length scales.

However, there are also some important differ-

ences in each approach that warrant investigation.

After briefly reviewing the basics of the two mod-els, we apply each method to solve three basic

example problems. The first two examples will

incorporate a simplified doublet mechanics solu-

tion that has no length scale, while the final exam-

ple will use a more complete solution with a single

length scale. Each of these doublet mechanics solu-

tions is compared with a corresponding finite ele-

ment simulation of the equivalent problem.

2. Doublet mechanics

Originally developed by Granik (1978), doublet

mechanics (DM) is a micro-mechanical theory

based on a discrete material model whereby solids

are represented as arrays of points or nodes atfinite distances. A pair of such nodes is referred

to as a doublet, and the nodal spacing distances

introduce length scales into the micro-structural

theory. Current applications of the theory have

normally used regular arrays of nodal spacing thus

generating a Bravais lattice geometry. Each node

in the array is allowed to have a translation and

rotation, and increments of these variables are ex-panded in a Taylor series about the nodal point.

The order at which the series is truncated defines

the degree of approximation employed. The lowest

M.H. Sadd, Q. Dai / Mechanics of Materials 37 (2005) 641–662 643

order case using only a single term in the series will

not contain any length scales, while using more

than one term will produce a multilength scale the-

ory. This allowable kinematics develops micro-

strains of elongation, shear and torsion (aboutthe doublet axis). Through appropriate constitu-

tive assumptions, these micro-strains can be re-

lated to corresponding elongational, shear and

torsional micro-stresses.

Applications of this theory to geomechanics

problems have been given by Granik and Ferrari

(1993) and Ferrari et al. (1997). For these applica-

tions, a granular interpretation of doublet mechan-ics has been employed, in which the material is

viewed as an assembly of circular or spherical par-

ticles. A pair of such particles represents a doublet

as shown in Fig. 1. Corresponding to the doublet

(A,B) there exists a doublet or branch vector faconnecting the adjacent particle centers and defin-

ing the doublet axis a. The magnitude of this vec-

tor ga = jfaj is simply the particle diameter forparticles in contact. However, in general the parti-

cles need not be in contact, and for this case the

length scale ga could be used to represent a more

general micro-structural feature. As mentioned

the kinematics allow relative elongational, shear-

ing and torsional motions between the particles,

and this is used to develop an elongational

micro-stress pa, shear micro-stress ta, and torsionalmicro-stress ma as shown in Fig. 1. It should be

pointed out that these micro-stresses are not sec-

ond order tensors in the usual continuum mechan-

ics sense. Rather, they are vector quantities that

represent the elastic micro-forces and micro-cou-

ples of interaction between doublet particles. Their

directions are dependent on the doublet axes

which are determined by the material micro-struc-

ture. These micro-stresses are not continuously

distributed but rather exist only at particular

points in the medium being simulated by DM

theory.If u(x, t) is the displacement field coinciding

with a particle displacement, then the increment

function at x = xA (xA is the position vector of

the particle A) is written as

Dua ¼ uðxþ fa; tÞ � uðx; tÞ ð1ÞHere, a = 1, . . .,n, while n is referred to as the

valence of the Bravais lattice. Under the assump-

tion that the doublet interactions are symmetric,

the shear and torsional micro-deformations and

micro-stresses vanish, and thus only extensional

strains and stresses will exist. The extensional

micro-strain scalar measure ea, representing the

axial deformation of the doublet vector, is defined

ea ¼qa � Dua

gað2Þ

where qa = fa/ga is the unit vector in the a-direc-tion. The increment function (1) can be expanded

in a Taylor series as

Dua ¼XMm¼1

ðgaÞm

m!ðqa � rÞmuðx; tÞ þO jgaj

Mþ1� �

ð3Þ

Using this result into relation (2) develops theseries expansion for the extensional or axial

micro-strain

ea ¼ qaiXMm¼1

ðgaÞm�1

m!qak1 . . . qakm

omuioxk1 . . . oxkm

ð4Þ

where qak are the cosines of the angles between the

directions of micro-stress and the coordinates.

Fig. 1. Basic doublet geometry and micro-stress definitions.


As mentioned, the number of terms used in the

series expansion of the local deformation field

determines the order of the approximation in

DM theory. For the first order case (m = 1), it

has been shown that the scaling parameter ga willdrop from the formulation, and the axial micro-

strain is reduced to

ea ¼ qaiqajeij ð5Þ

where eij is the usual continuum strain tensor given

by eij = 1/2(ui,j + uj,i).

For this case, the elastic DM solution can be

calculated directly from the corresponding iso-

tropic continuum elasticity solution through therelation

rij ¼Xna¼1

qaiqajpa ð6Þ

and this can be expressed in matrix form

frg ¼ ½Q�fpg ) fpg ¼ ½Q��1frg ð7Þ

where {r} is the continuum elastic stress vector in

rectangular Cartesian coordinates, {p} is the

micro-stress vector, and [Q] is a transformation

matrix. For plane problems, this transformation

matrix can be written as

½Q� ¼ðq11Þ

2 ðq21Þ2 ðq31Þ

2

ðq12Þ2 ðq22Þ

2 ðq32Þ2

q11q12 q21q22 q31q32

264

375 ð8Þ

where qij are the cosines of the angles between the

micro-stresses and Cartesian coordinates. This re-

sult allows a straightforward development of firstorder DM solutions for many problems of engi-

neering interest, and will be used to generate DM

solutions for comparative example problems.

Specific applications of doublet mechanics have

been developed for two-dimensional problems

with regular particle packing micro-structures.

One case that has been studied is the two-dimen-

sional hexagonal packing as shown in Fig. 2. Thisgeometrical micro-structure establishes three

Fig. 2. Two-dimensional hexagonal particle packing geometry.


doublet axes at 60� angles as shown. For this caseusing only the first order approximation, the shear

and torsional micro-stresses vanish, leaving only

the elongational micro-stress components

(p1,p2,p3) as shown. Positive elongational compo-nents correspond to tensile forces between

particles.

For the second order approximation (m = 2)

case, the axial micro-strain term is now given by

ea ¼ qaiqaj ui;j þga2!qakui;jk

� �ð9Þ

Choose the simplifying case that all doublets

originating from a common node have the same

magnitudes; i.e. ga = g (a = 1, . . .,n), and alsoassume that the interactions are purely axial

(no shear or torsional micro-stresses). For

homogeneous interaction, there will be only one

micro-modulus C0, and the constitutive

relationship between elongation micro-stress and

micro-strain is expressed by pa = C0ea. The

displacement field for the second order approxi-

mation can be written

u ¼ vþ gzþ g2w ð10Þ

where v represents the solution without the lengthscale, and corresponds to the classical elasticity

solution, z and w are first order and second order

displacement fields, respectively. By applying dis-

placement equilibrium equations and length-scale

independent boundary conditions, it follows that

z � 0. The governing equation for the second

order displacement field w has been previously

given as

Xna¼1

qaiqajqakqalwj;kl ¼1

12

Xna¼1

qaiqajqakqalqapqaqvj;klpq

ð11ÞThe solutions of w can be found by using Pap-

kovich–Neuber displacement potentials. Having

determined the displacements v and w, the micro-stress can then be evaluated by

pa ¼ C0qajqak vj;k þg2qalvj;kl þ g2wj;k

� �ð12Þ

This second order theory will be used later in a

particular application/comparison problem.

3. Micro-mechanical finite element model

The basics of our micro-mechanical finite ele-

ment model have been presented in previous stud-

ies (Sadd and Dai, 2001; Sadd et al., 2004a,b).

Here we will only briefly review some of the basic

model developments for the elastic case. Bitumi-

nous asphalt can be described as a multiphase

material containing aggregate, binder cement(including mastic and fine particles) and air voids

(see Fig. 3(a)). The load transfer between the

aggregates plays a primary role in determining

the load carrying capacity and failure of such

complex materials. In order to develop a micro-

mechanical model of this behavior, proper simula-

tion of the load transfer between the aggregates

Fig. 3. Micro-mechanical finite element modeling. (a) Asphalt material micro-structure, (b) resultant load transfer and (c) micro-frame

finite element.


must be accomplished. The aggregate material is

normally much stiffer than the binder, and thus

aggregates are taken as rigid particles. On the

other hand, the binder cement is a compliant mate-

rial with elastic, inelastic, and time-dependentbehaviors. In order to properly account for the

load transfer between aggregates, it is assumed

that there is an effective binder zone between

neighboring particles. It is through this zone that

the micro-mechanical load transfer occurs between

each aggregate pair. For the two-dimensional case,

this loading can be reduced to resultant normal

and tangential forces and a moment as shown inFig. 3(b). The similarities and differences in the

inter-particle load transfer for doublet mechanics

(Fig. 1) and the micro-finite element model (Fig.

3(b)) should be noted. While the normal and tan-

gential micro-forces are similar, the moment loa-

dings are not. DM theory formulates a torsional

loading about the in-plane doublet axis, while the

FEM model postulates a moment loading alongthe out-of-plane direction.

In order to model the inter-particle load trans-

fer behavior, some simplifying assumptions must

be made about allowable aggregate shape and bin-

der geometry. Aggregate geometry is commonly

quantified in terms of particle size, shape, angular-

ity and texture. However, for the present modeling

only size and shape are considered. In general, as-phalt concrete contains aggregate of very irregular

geometry as shown in Fig. 4(a). Our approach is to

allow variable size and shape using an idealized

elliptical aggregate model as represented in Fig.

4(b). Simplifying the shape allows a straight-for-

ward determination of binder geometry necessary

to calculate particular finite element properties.

The finite element model then uses an equivalent

lattice network approach, whereby the inter-parti-

cle load transfer is simulated by a network of spe-

cially created frame-type finite elements connected

at particle centers as shown in Fig. 4(c). From

granular materials research, the material micro-

structure or fabric can be characterized to someextent by the distribution of branch vectors which

are the line segments drawn from adjacent particle

mass centers. Note that the finite element network

coincides with the branch vector distribution.

Cementation between neighboring particles was

generated using a scheme shown in Fig. 3(c),

whereby the cementation was asymmetrically dis-

tributed parallel to the branch vector. The cemen-tation geometry parameters are shown in Fig. 3(c).

Since this scheme allows arbitrary non-symmetric

cementation, an eccentricity variable is defined

by e = (w2�w1)/2.

The current network model uses a specially

developed, two-dimensional frame-type finite ele-

ment to simulate the inter-particle load transfer.

These two-noded elements have the usual three de-grees-of-freedom (two displacements and a rota-

tion) at each node and the element equation can

thus be written in general form as

K11 K12 K13 K14 K15 K16

: K22 K23 K24 K25 K26

: : K33 K34 K35 K36

: : : K44 K45 K46

: : : : K55 K56

: : : : : K66

2666666664

3777777775

U 1

V 1

h1U 2

V 2

h2

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

¼

F n1

F t1

M1

F n2

F t2

M2

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

ð13Þ

where Ui, Vi and hi are the nodal displacements

and rotations, and F.. and M. are the nodal forces

and moments. The usual scheme of using bar and/

or beam elements to determine the stiffness terms is

not appropriate for the current applications, and

Fig. 4. Asphalt modeling concept. (a) Typical asphalt material, (b) model asphalt system and (c) network finite element model.


therefore these terms were determined using an

approximate elasticity solution from Dvorkin

et al. (1994) for the stress distribution in a cement

layer between two particles. The two-dimensional

model geometry (uniform thickness case) is shown

in Fig. 5.The stresses rx, rz and sxz within the cementa-

tion layer can be calculated for normal, tangential

and rotational particle motion cases. These stresses

can then be integrated to determine the total load

transfer within the cement binder, thus leading to

the calculation of the various stiffness terms

needed in the element equation. Details of this

process have been previously reported by Saddand Dai (2001), and the final result is given by

where Knn ¼ ðkþ 2lÞw=�h, Ktt ¼ lw=�h, k and lare the usual elastic moduli, w and �h are the

cementation width and average thickness, r1 and

r2 are the radial dimensions from each aggregate

center to the cementation boundary, w1 and w2

are left and right width of cementation. Each bin-

der element stiffness matrix will be different

depending on the two-particle layout and size,

and binder geometry. This procedure establishesthe elastic stiffness matrix, which is a function of

the material micro-structure and binder moduli.

We now develop doublet mechanics and finite ele-

ment solutions to three example problems. Com-

parisons of the resulting micro-stress solutions

will be made.

4. Comparisons of DM analysis and FEM

simulation

4.1. Surface compression loading of a

semi-infinite mass

Surface loading of a semi-infinite body repre-

sents an important application problem in as-phalt concrete research related to roadway

performance. For example, top–down cracking

is a type of failure that initiates at or near the

pavement surface and is typically generated by

surface compression loading. As pointed out by

Wang et al. (2003), this problem is still not com-

pletely understood. Some of the conflicting issues

are related to the stress distribution under con-centrated and distributed surface loadings as

shown in Fig. 6(a) and (b). The elastic stress dis-

tribution in a semi-infinite solid under concen-

trated loading (Fig. 6(a)) is given by the

classical Flamant solution

½K� ¼

Knn 0 Knne �Knn 0 �Knne

� Ktt Kttr1 0 �Ktt Kttr2� � Kttr21 þ Knn

3w2

2 � w1w2 þ w21

� ��Knne �Kttr1 Kttr1r2 � Knn

3w2

2 � w1w2 þ w21

� �

� � � Knn 0 Knne

� � � � Ktt �Kttr2� � � � � Kttr22 þ Knn

3w2

2 � w1w2 þ w21

� �

2666666664

3777777775

ð14Þ

Fig. 5. Cementation between two adjacent particles.


rx ¼ �2Px2y=pðx2 þ y2Þ2

sxy ¼ �2Pxy2=pðx2 þ y2Þ2

ry ¼ �2Py3=pðx2 þ y2Þ2ð15Þ

This continuum mechanics solution specifies

that the stresses are everywhere compressive in the

region below the surface loading, and this wouldalso be true for the case shown in Fig. 6(b) where

the surface loading acts over a distributed area.

However as pointed out in the literature, there ex-

ists considerable experimental evidence that a

granular medium under similar surface loading

will exhibit tensile openings. Ferrari et al. (1997)

refer to this issue as Flamant�s paradox.It would thus appear that a micro-mechanical

model is needed to resolve this paradox, and Fer-

rari et al. (1997) and Wang et al. (2003) have ap-

plied doublet mechanics to investigate this issue.

For the Flamant problem using the transforma-tion given by (7) and (8), Ferrari et al. (1997) have

developed the micro-stresses for the first order,

non-scale case for a medium with hexagonal pack-

ing as shown in Fig. 2.

p1 ¼ �4Py2ffiffiffi3

pxþ y

� �=3pðx2 þ y2Þ2

p2 ¼ 4Py2ffiffiffi3

px� y

� �=3pðx2 þ y2Þ2

p3 ¼ �2Pyð3x2 � y2Þ=3pðx2 þ y2Þ2ð16Þ

These micro-stress directions are defined for a

granular micro-structure as shown in Fig. 6(c).These micro-stresses can be integrated to gener-

ate the more useful solution to the problem shown

in Fig. 6(b). For a uniformly distributed line load q

over the range �a 6 x 6 a, the micro-stresses at

Fig. 6. Surface compression problem. (a) Flamant problem, (b) integrated flamant problem and (c) computation model.


location (x,y) can be obtained by integration over

the coordinate origin variable u as

Although these DM micro-stresses actually ex-

ist only at discrete points and directions in the do-

main, we will use these results to make continuous

contour and x–y plots over the domain under

study. A similar statement would also apply for

the finite element micro-forces to be developed

next.

In order to generate a similar model for finiteelement simulation, a 2D-hexagonal Bravais lat-

tice structure was generated using a MATLAB

Material Generator Code. This model had 1296

circular particles and 3709 micro-frame elements

as shown in Fig. 6(c). Chosen model parameters

include: k = 0.58 MPa, l = 0.38 MPa, w1 = w2 =

4mm, h0 = 1mm, and particle size D = 10mm.

For the integrated Flamant problem, three centralparticles on the boundary had prescribed verti-

cal compressive loading and zero horizontal dis-

placement. Particles on the bottom layer were

supported with a very stiff vertical spring founda-

tion (compared with the asphalt binder stiffness).

Boundary particles on the vertical sides were un-

loaded and not constrained. To avoid boundary

effects present in the FEM model, only the centralportion of the domain (indicated by box in Fig.

6(c)) was used to compare with the doublet

mechanics results.

Micro-structural finite element simulation was

then conducted on this model, and the elastic nor-

mal axial force of each micro-frame element was

computed from the code. In order to compare fi-

nite element results with the corresponding dou-

blet mechanics predictions, the DM micro-

stresses from Eq. (17) were calculated at the

mid-point of each element. It should be noted

that the directions of these micro-stresses coincide

with the element axial forces. Fig. 7 shows com-

parisons between DM micro-stress contours and

FEM micro-force distributions for this problem.Plus and minus signs indicate regions of tensile

and compressive micro-stresses. It is observed that

the DM micro-stress and FEM micro-force con-

tours are quite similar. Comparable tensile zones

exist for each micro-stress–force distribution, thus

indicating possible regions of tensile or mode I

fracture behavior. Tensile zones for the P1 and

P2 distributions are located adjacent to the sur-face loading, and these have been observed re-

gions of surface or top–down cracking behavior.

It is also evident that a significant zone of hori-

zontal tensile micro-stress (P3), is located directly

below the loading. According to each theory, the

maximum value of this tensile field appears to be

located at a somewhat different location below the

loading surface. Clearly this location would bedependent on the fact that the DM solution is

for a semi-infinite half space, while the FEM

model has used particular dimensions and bound-

ary conditions for the asphalt domain. Since the

continuum elasticity case predicts only a compres-

P 1 ¼Z a

�ap1ðx� uÞdu ¼

Z a

�a�4qy2

ffiffiffi3

pðx� uÞ þ y

� �3p ðx� uÞ2 þ y2� �2� ��1

du

¼ 2qy3p

ffiffi3

py�x�a

ðxþaÞ2þy2�

ffiffi3

py�xþa

ðx�aÞ2þy2� 1

y tan�1 xþa

y

� �þ 1

y tan�1 x�a

y

� �h i

P 2 ¼Z a


Z a

�a4qy2

ffiffiffi3

pðx� uÞ � y

� �3p ðx� uÞ2 þ y2� �2� ��1

du

¼ � 2qy3p

ffiffi3

pyþxþa

ðxþaÞ2þy2�

ffiffi3

pyþx�a

ðx�aÞ2þy2þ 1

y tan�1 xþa

y

� �� 1

y tan�1 x�a

y

� �h i

P 3 ¼Z a


Z a

�a�2qyð3ðx� uÞ2 � y2Þ 3p ðx� uÞ2 þ y2

� �2� ��1

du

¼ 2qy3p

2xþ2aðxþaÞ2þy2

� 2x�2aðx�aÞ2þy2

� 1y tan

�1 xþay

� �þ 1

y tan�1 x�a

y

� �h i

ð17Þ


sive stress field, these tensile stress regions are

attributable to the micro-mechanical modeling

through the inter-particle mechanics based oneither doublet mechanics or finite element simula-

tion. Of course material failure could also occur

via shearing or mode II fracture at other locations

in the model.

4.2. Void under uniform compression

The existence of voids in asphalt pavementsresulting from improper compaction or entrap-

ment of spurious material is another important

issue related to roadway failure. Voids can raise

the local stress field and produce fatigue or frac-

ture failures emanating from the void boundary.

We wish to consider the problem of a circular

void in an asphalt material under uniform far-

field compression loading as shown in Fig. 8(a).The primary goal of this example is to investi-

gate the elevation of micro-stresses around the

void, and to look for zones of possible tensile

behavior.

The continuum elastic stress distribution

around a circular hole under uniform far-field

compression r0 as shown in Fig. 8(a) may be found

in any standard elasticity text and is given in polar

coordinates as

rr ¼ � r02

1� R2

r2

� �þ r0

21þ 3R4

r4 � 4R2

r2

� �cos 2h

rh ¼ � r02

1þ R2

r2

� �� r0

21þ 3R4

r4

� �cos 2h

srh ¼ � r02

1� 3R4

r4 þ 2R2

r2

� �sin 2h ð18Þ

where R is the radius of the hole and r and h are

the usual polar coordinates.

To obtain the DM micro-stresses from the con-tinuum field using transformation relation (7), the

polar coordinate stresses need to be first trans-

formed to Cartesian components. The micro-stress

directions for the problem (hexagonal Bravais lat-

tice) are shown in Fig. 8(a), and thus the transfor-

mation matrix Q can be expressed as

½Q� ¼cos2c cos2c 1

sin2c sin2c 0

� cos c sin c cos c sin c 0

264

375 ð19Þ

where the doublet structure angle c = 60� for this

case. Applying these appropriate transformations,

the DM micro-stresses become

Fig. 7. Comparison of DM analysis and FEM results for integrated flamant problem.


p1 ¼ � r0

3

" 1þ cos22hþ 1

2sin22hþ

ffiffiffi3

p

2sin 4h

�ffiffiffi3

psin 2hcos2h

!

þ�5 cos 2h� 4cos2h cos 2hþ sin22h

þffiffiffi3

psin 2h�

ffiffiffi3

psin 4h� 2

ffiffiffi3

pcos2h sin 2h

�R2

r2

þ 3cos22h� 3

2sin22hþ 3

ffiffiffi3

p

2sin 4h

þ3ffiffiffi3

pcos2h sin 2h

!R4

r4

#

p2 ¼ � r0

31þ cos22hþ 1

2sin22h�

ffiffiffi3

p

2sin 4h

"

þffiffiffi3

psin 2hcos2h

!

þ�5 cos 2h� 4cos2h cos 2hþ sin22h

�ffiffiffi3

psin 2hþ

ffiffiffi3

psin 4hþ 2

ffiffiffi3

pcos2h sin 2h

�R2

r2

þ 3cos22h� 3

2sin22h� 3

ffiffiffi3

p

2sin 4h

�3ffiffiffi3

pcos2h sin 2h

!R4

r4

#

p3 ¼r0

3

�cos22hþ ½4cos2hð1� 2 cos 2hÞ þ 2 cos 2h

ð1� cos 2hÞ�R2

r2þ ð6cos22h� 3sin22hÞR

4

r4

�

ð20Þ

The corresponding finite element model of this

problem was again generated using our MATLAB

code and the result is shown in Fig. 8(b). This

model has the required regular 2D hexagonal

Bravais lattice structure with 662 circular particlesand 1880 micro-frame elements. The circular void

was created in the central area of the model by

removing a single particle from the regular pack-

ing pattern. Model boundary conditions included

uniform compression loading on the top layer

while the bottom particles were supported with a

very stiff vertical spring foundation. The horizon-

tal x-displacements of the top and bottom layers

were fixed. Boundary particles on the vertical sides

were again unloaded and unconstrained. The over-

all sample dimensions were chosen to be more

than 10 times the size of the void diameter. As in

the previous problem to reduce boundary condi-tion effects, only the central subregion (indicated

by box in Fig. 8(b)) was used for comparisons with

the doublet mechanics results.

Finite element simulation was then conducted

on the model, and the axial element force distribu-

tions were obtained. As before, the corresponding

doublet micro-stresses from Eq. (20) were calcu-

Fig. 8. Void in granular asphalt materials under uniform

compression loading. (a) Void problem and (b) computation

model.


lated at the mid-point of each element. Fig. 9 illus-

trates comparisons between the correspondingDM and FEM distributions for the void example.

Again, plus and minus signs indicate regions of

tensile and compressive microstresses. As in the

previous example, results from each model for

the void case are very similar. Results for the p1and p2 micro-stresses indicate a general compres-

sion distribution in the domain except for particu-

lar portions on the boundary of the void. Thehorizontal micro-stress p3 is found to be primarily

tensile in the region except on the boundary of the

void. Fig. 10 shows each of the DM micro-stressesas a function of the angular coordinate both on the

void surface and in the far field (r ! 1). It can be

observed that the micro-stresses p1 and p2 have

small tensile zones on the void boundary and to-

tally compressive behavior for the far field. The

local maximum micro-stresses on the void surface

are elevated when compared with far field values.

A second void example was generated by rotat-ing the previous lattice through 90� (CCW). This

Fig. 9. Comparison of DM analysis and FEM results.

Fig. 10. Micro-stress concentration. (a) p1 micro-stress, (b) p2 micro-stress and (c) p3 micro-stress.


case and the new doublet axes are shown in Fig.

11(a). Notice that micro-stress p3 is now along

the compression loading direction. The new trans-

formation matrix for this case becomes

½Q� ¼sin2c sin2c 0

cos2c cos2c 1

sin c cos c � sin c cos c 0

264

375 ð21Þ

and the doublet structure angle is again c = 60�.Applying the transformations, the DM micro-

stresses now become

p1 ¼r0

3�1þ cos22hþ 1

2sin22hþ

ffiffiffi3

p

2sin 4h

"

þffiffiffi3

psin 2hsin2h

!

þ �3 cos 2hþ 4sin2h cos 2hþ 2ffiffiffi3

psin2h sin 2h

�

þ sin22hþffiffiffi3

psin 2h�

ffiffiffi3

psin 4h

�R2

r2

þ 3 cos 2h� 6sin2h cos 2h� 3

ffiffiffi3

psin2h sin 2h

� 3

2sin22hþ 3

ffiffiffi3

p

2sin 4h

!R4

r4

#

p2 ¼r0

3�1þ cos22hþ 1

2sin22h�

ffiffiffi3

p

2sin 4h

"

�ffiffiffi3

psin 2hsin2h

!

þ �3 cos 2hþ 4sin2h cos 2h� 2ffiffiffi3

psin2h sin 2h

�

þsin22h�ffiffiffi3

psin 2hþ

ffiffiffi3

psin 4h

�R2

r2

þ 3 cos 2h� 6sin2h cos 2hþ 3

ffiffiffi3

psin2h sin 2h

� 3

2sin22h� 3

ffiffiffi3

p

2sin 4h

!R4

r4

#

p3 ¼ � r0

3

�ðcos22hþ 2Þ þ ð8sin2h cos 2hþ 2sin22hÞ

R2

r2þ ð6cos22h� 3sin22hÞR

4

r4

�ð22Þ

The finite element model for this second void

problem was generated in analogous fashion and

is shown in Fig. 11(b). This model had the rotatedhexagonal Bravais lattice structure with 594 circu-

lar particles and 1680 micro-frame elements.

Boundary conditions for the FEM modeling were

identical to the previous case, and boundary effects

Fig. 11. Rotated hexagonal Bravais lattice structure for void

problem. (a) Void problem and (b) computation model.


were minimized by only considering the domain

interior to the box shown in Fig. 11(b). Fig. 12

shows the comparisons between DM and FEM dis-

tributions for the rotated void example. As in the

previous cases, results from each model are similar.

It is observed from the p1 and/or p2 contours that

each model generates tensile micro-stress domainsunder the uniform compressional loading. These

tension domains are caused by the presence of the

void. If the void is removed, the tension zones dis-

appear and the p1 and p2 fields are totally compres-

sive. It appears that the zone of maximum tensile

micro-stress p1 or p2 lies approximately along the

direction of the corresponding doublet axis. Since

the micro-stress p3 lies along the compressional

loading direction, it has only compression action

within model domain. Along a horizontal line(h = 0), p3 increases significantly as one approaches

the void. As shown in Fig. 13, the micro-stress dis-

tributions were again investigated as a function of

Fig. 12. Comparison of DM and FEM analysis for rotated lattice structure.

Fig. 13. Micro-stresses concentration. (a) p1 micro-stress, (b) p2 micro-stress and (c) p3 micro-stress.


the angular coordinate on the void boundary and

at far field. It can be observed that the micro-stres-

ses p1 and p2 have tensile zones on the void bound-

ary while p3 exhibits totally compressive behavior

on the void and at far-field.For the void problem shown in Fig. 8, the p3

micro-stress distribution around the void shown

in Fig. 10(c) indicates a range of �r0/3 6 p3 6 r0,while the far-field variation is given by 0 6 p3 6 r0/3. Thus in the horizontal direction (h = 0), the

local value of this micro-stress is three times higher

than the far-field value. Likewise, for the void

problem of Fig. 11, the ranges of p3 are shown inFig. 13(c) and would be �3r0 6 p3 6 0 around

the void and �r0 6 p3 6 2r0/3 for the far-field.

For this case, the stress concentration factor for

the vertical p3 micro-stress is three which matches

with classical continuum elasticity result.

4.3. Internal line loading (second order DM model)

We now wish to conduct a final comparison

study between the FEM model and a second order

doublet mechanics solution that will include alength scale. Existing DM solutions that include

the second order approximation are very limited,

and only the Kelvin problem has been given by

Ferrari et al. (1997). We therefore will use this

basic problem as our final comparative example.

The classic Kelvin problem shown in Fig. 14(a)

will be superimposed (integrated) to generate the

case of an internal line loading shown in case (b).The two-dimensional continuum elastic dis-

placement field for Kelvin�s problem is given by

v1 ¼ �c1xyr2

; v2 ¼ c1 2 log r þ x2

r2

� ð23Þ

Fig. 14. Internal compression example. (a) Kelvin problem, (b) integral Kelvin problem, (c) computation model and (d) maximum

micro-stress locations shown in bold (solid-tension, dotted-compression).


where r2 = x2 + y2, c1 ¼ 4P=9Ep, and E is the

modulus of elasticity. The second order displace-

ment field was formulated by Ferrari et al. (1997)

w1 ¼ c19y 64 x7

r10 � 96 x5

r8 þ 32 x3

r6 þ xr4

� �

w2 ¼ � c118

128 x8

r10 � 352 x6

r8 þ 320 x4

r6 � 100 x2

r4 þ 5r2

� �

ð24Þ

Thus from relation (10) with z = 0, the second

order displacement field for the Kelvin problem

can be expressed as

u1 ¼ v1 þ g2w1; u2 ¼ v2 þ g2w2 ð25ÞThe micro-stresses for the second order case of

the structure angle c = 60� can be developed from

Eq. (12) with C0 = E,

p1 ¼P

81pðx2 þ y2Þ6hg2�12x8y þ 444x2y7 � 780x4y5

þ 52x6y3 � 504ffiffiffi3

px5y4 þ 84

ffiffiffi3

px7y2 � 102

ffiffiffi3

pxy8

þ 588ffiffiffi3

px3y6 � 2

ffiffiffi3

px9 � 16y9

�

þ g 108xy9 þ 288x3y7 þ 216x5y5 � 36x9y� �

� 9ffiffiffi3

px11 þ 45y11 þ 189x2y9 þ 306x4y7

�

þ 234x6y5 þ 81x8y3 þ 9x10y þ 306ffiffiffi3

px5y6

þ 189ffiffiffi3

px3y8 þ 45

ffiffiffi3

pxy10 þ 234

ffiffiffi3

px7y4

þ 81ffiffiffi3

px9y2

�ið26Þ

p2 ¼P

81pðx2 þ y2Þ6hg2�12x8y þ 444x2y7 � 780x4y5

þ 52x6y3 þ 504ffiffiffi3

px5y4 � 84

ffiffiffi3

px7y2 þ 102

ffiffiffi3

pxy8

� 588ffiffiffi3

px3y6 þ 2

ffiffiffi3

px9 � 16y9

�

þ g �108xy9 � 288x3y7 � 216x5y5 þ 36x9y� �

þ 9ffiffiffi3

px11 � 45y11 � 189x2y9 � 306x4y7

�

� 234x6y5 � 81x8y3 � 9x10y þ 306ffiffiffi3

px5y6

þ 189ffiffiffi3

px3y8 þ 45

ffiffiffi3

pxy10 þ 234

ffiffiffi3

px7y4

þ 81ffiffiffi3

px9y2

�ið27Þ

p3 ¼4Py

81pðx2 þ y2Þ6g2 3x8 � 152x6y2 þ 390x4y4�

�96x2y6 � y8�þ g �9x9 þ 54x5y4�

þ72x3y6 þ 27xy8�þ �9x10 � 27x8y2�

þ18x4y6 � 18x6y4 þ 27x2y8 þ 9y10��

ð28Þ

where the micro-stress directions are defined inFig. 14(c).

We now wish to develop the solution for a uni-

formly distributed line load q over the range

�a 6 x 6 a (at y = 0) as shown in Fig. 14(b). This

can be accomplished as before by integrating the

micro-stresses at location (x,y) over the coordinate

origin variable u as

P 1 ¼Z a

�ap1ðx� uÞdu

¼ q

324p ðxþ aÞ2 þ y2� �5 g2 12y8 þ 6ðxþ aÞ8

��

þ64ðxþ aÞy7 � 400ðxþ aÞ3y5 þ 64ðxþ aÞ5y3

�246ðxþ aÞ2y6 þ 390ðxþ aÞ4y4 � 114ðxþ aÞ6y2

þ16ðxþ aÞ7y�þ g �144ðxþ aÞ2y7�

þ144ðxþ aÞ6y3 þ 72ðxþ aÞ8y � 72y9�

þ 108y10 þ 108ðxþ aÞ8y2 þ 432ðxþ aÞ6y4

þ 648ðxþ aÞ4y6 þ 432ðxþ aÞ2y8 þ 72ðxþ aÞ9y

þ 288ðxþ aÞ7y3 þ 432ðxþ aÞ5y5 þ 288ðxþ aÞ3y7

þ 72ðxþ aÞy9 þ 108tan�1 xþ ay

� y10

� 27 ln ðxþ aÞ2 þ y2� �

ðxþ aÞ10

� 27 ln ðxþ aÞ2 þ y2� �

y10 þ 108tan�1 xþ ay

�

� ðxþ aÞ10 � 135 ln ðxþ aÞ2 þ y2� �

ðxþ aÞ2y8

þ 540tan�1 xþ ay

� ðxþ aÞ2y8

� 135 ln ðxþ aÞ2 þ y2� �

ðxþ aÞ8y2


� ðxþ aÞ6y4



� ðxþ aÞ8y2

� 270 ln ðxþ aÞ2 þ y2� �

ðxþ aÞ6y4

� 270 ln ðxþ aÞ2 þ y2� �

ðxþ aÞ4y6

þ1080tan�1 xþ ay

� ðxþ aÞ4y6

�

� q

324p ðx� aÞ2 þ y2� �5 g2 12y8 þ 6ðx� aÞ8

��

þ64ðx� aÞy7 � 400ðx� aÞ3y5 þ 64ðx� aÞ5y3

�246ðx� aÞ2y6 þ 390ðx� aÞ4y4 � 114ðx� aÞ6y2

þ16ðx� aÞ7y�þ g �144ðx� aÞ2y7�

þ144ðx� aÞ6y3 þ 72ðx� aÞ8y � 72y9�þ 108y10

þ108ðx� aÞ8y2 þ 432ðx� aÞ6y4 þ 648ðx� aÞ4y6

þ 432ðx� aÞ2y8 þ 72ðx� aÞ9y þ 288ðx� aÞ7y3

þ 432ðx� aÞ5y5 þ 288ðx� aÞ3y7 þ 72ðx� aÞy9

þ 108tan�1 x� ay

� y10 � 27 ln ðx� aÞ2 þ y2

� �

� ðx� aÞ10 � 27 ln ðx� aÞ2 þ y2� �

y10


� ðx� aÞ10 � 135 ln ðx� aÞ2

�

þy2�ðx� aÞ2y8 þ 540tan�1 x� a

y

� ðx� aÞ2y8

� 135 ln ðx� aÞ2 þ y2� �

ðx� aÞ8y2


� ðx� aÞ6y4


� ðx� aÞ8y2

� 270 ln ðx� aÞ2 þ y2� �

ðx� aÞ6y4

� 270 ln ðx� aÞ2 þ y2� �

ðx� aÞ4y6

þ1080tan�1 x� ay

� ðx� aÞ4y6

�ð29Þ

P 3 ¼Z a

�ap3ðx� uÞdu

¼ � 2qy

81p ðxþ aÞ2 þ y2� �5 g2 58ðxþ aÞ5y2

�h

�70ðxþ aÞ3y4 � 2ðxþ aÞy6 � 2ðxþ aÞ7�

þ g 9y8 þ 18ðxþ aÞ2y6 � 18ðxþ aÞ6y2�

�9ðxþ aÞ8�þ 18ðxþ aÞ9 þ 72ðxþ aÞ7y2

þ108ðxþ aÞ5y4 þ 72ðxþ aÞ3y6 þ 18ðxþ aÞy8i

þ 2qy

81p ðx� aÞ2 þ y2� �5 g2 58ðx� aÞ5y2

�h

�70ðx� aÞ3y4 � 2ðx� aÞy6 � 2ðx� aÞ7�

þ g 9y8 þ 18ðx� aÞ2y6 � 18ðx� aÞ6y2�

�9ðx� aÞ8�þ 18ðx� aÞ9 þ 72ðx� aÞ7y2

þ108ðx� aÞ5y4 þ 72ðx� aÞ3y6 þ 18ðx� aÞy8i

ð30Þ

Note that micro-stress P2 can be simply ob-

tained from P1 by replacing x with �x.

To compare a numerical simulation with these

DM results, a similar finite element model was

generated with a 2D-hexagonal Bravais lattice

structure shown in Fig. 14(c). The model had 946

particles and 2715 micro-frame elements, andneighboring particles all had the same separation

(diameter plus cement spacing) of 1.1cm. To sim-

ulate the integrated Kelvin problem, four central

particles on the mid-line had vertical loading and

zero horizontal displacement. Particles on the top

and bottom layers were connected to a stiff vertical

spring foundation. Boundary particles on the ver-

tical sides were left unconstrained. To avoidboundary effects, a central portion of the model

(indicated by box in Fig. 14(c)) was again used.

The box had a horizontal dimension of 22cm

and a vertical height of 19cm. The loading line size

2a = 3.3cm and this gives a dimensionless length

scale (particle separation/a) of 0.67.

Fig. 15 shows the comparisons between DM and

FEM micro-stress distributions for the integratedKelvin problem. For the DM analysis, the length


scale is taken to be the same as the particle spacing

used in the FEM model; i.e. g = 1.1cm, and theloading-line size was selected to match the FEM

model. The thick line contour indicates the separa-

tion between compression and tension zones in the

DM distribution. Again, results from each model

show very similar distributions. P1 and P2 con-

tours are symmetric to each other about the cen-

tral vertical line, and their compression and

tension domains have similar distribution patterns.Maximum values of the P1 and P2 micro-stresses

occur at the edges of the loading line as shown

in Fig. 14(d), with maximum tensile and compres-

sive values occurring at the doublets indicated by

the solid and dotted bold lines. Contours for P3

indicate symmetry about the vertical centerline

and skew-symmetry about the horizontal. Results

indicate a significant P3 tension zone below theload line and a corresponding compression zone

above. It was found that the maximum tensile

and compressive doublets were on the loading cen-

terline as indicated in Fig. 14(d). These DM results

match very closely with corresponding predictions

from the FEM model.

Finally it was desired to investigate the effect of

model length scale on the maximum DM micro-

stresses. The model geometry and the loading line

size were kept fixed as before, and length scale was

varied from 0 to a. The loading magnitude q was

taken as unity, and this resulted in a constant total

loading for each analysis. Fig. 16 shows the trend

of the dimensionless maximum micro-stresses

versus the dimensionless length scale g/a. It is

Fig. 16. Effect of the length scale on maximum micro-stress

(unit compression loading).

Fig. 15. Comparison of DM analysis and FEM results for integrated Kelvin problem.


observed from the figure that the maximum micro-

stresses show sizeable increase with length scale,

thus indicating significant dependency on this mi-

cro-structural parameter. As the length scale

decreases, these micro-stresses reduce to the first-order elastic solution (without length scale). For

a unit compression loading, the maximum value

of P1 (or P2) for the elastic case was calculated

approximately as 0.5, while the maximum of P3

was about 0.14.

An FEM investigation on the effect of length

scale was also conducted on the computation

model shown in Fig. 14(c). Six simulations wereconducted on the same model geometry with a

loading line size 2a changed to impose the uniform

load to 3, 4, 5, 6, 7, 9 particles, respectively. For a

given simulation each particle had the same load-

ing force, and this was adjusted to keep the total

loading identical for all simulations. FEM results

are shown in Fig. 16, and these predictions match

reasonably well with those from doublet mechan-ics. Thus the effects of length scale within each

model appear to provide similar behaviors.

5. Summary and conclusions

A comparative study has been given between

two micro-mechanical models that have beendeveloped to simulate the behavior of cemented

particulate materials. Specific applications of these

models are made to asphalt concrete. The first

model came from a discrete analytical approach

called doublet mechanics that represents micro-

structural solids as an array of particles. This

scheme developed analytical expressions for the

micro-deformation and stress fields between parti-cle pairs that define a doublet. The second model

was based on a numerical finite element technique

that establishes a network of elements between

neighboring cemented particles. Elements within

this model were specially developed in order to

simulate the local load transfer between particles.

While the two modeling schemes come from very

different origins, they have a common similarityof constructing a theory based on local interaction

between neighboring particle pairs. Both theories

allow for normal and tangential micro-forces be-

tween adjacent particles. However, the local mo-

ment interactions are not the same in each

theory. Doublet mechanics incorporates an in-

plane moment load transfer as shown in Fig. 1,

while the finite element scheme used a more com-mon out-of-plane moment as illustrated in Fig. 3b.

In order to pursue the effects of such similarities

and differences, three example problems were

investigated using each approach. Comparisons

were made between the DM micro-stresses with

the FEM element micro-forces. In spite of the dif-

ferences, model predictions compared favorably

for each of the examples. Results also indicated sig-nificant micro-structural effects that differ from

continuum elasticity theory. For example, results

from the first problem dealing with compressive

loading of a half-space, indicated sizeable zones

of tensile micro-stress. This situation is completely

different from continuum elasticity and could lead

to material failure through delamination between

particles. The second example problem of an infi-nite medium under uniform compression with a

stress-free hole also showed tensile zones that differ

from the continuum solution field. The final exam-

ple investigated an integrated Kelvin problem for

the case where the DM solution yields a length

scale dependent solution. Micro-structural depend-

ent tensile zones were again found and compared

well with the corresponding FEM solution. It wasfound from both DM and FEM that maximum

micro-stresses increased with increasing relative

length scale. In general, these comparisons tend

to provide additional verification for each theory.

The good comparisons between doublet

mechanics and finite element modeling would tend

to indicate that the extensional response between

particle pairs must be the dominant micro-mechanical mechanism in the problems consid-

ered. It would thus be interesting to investigate

additional cases where the shear and tor-

sional micro-stresses are present. Unfortunately

such DM solutions have not been found in the lit-

erature, and it is expected that development of

such solutions might be a formidable task. It is

anticipated that for these cases, the comparisonswould not be as good.

The doublet mechanics model offers analytical

solutions to simulate micro-mechanical behavior


of particular mechanics problems. However, for

more realistic problems involving irregular particle

sizes and packing geometries, it is unlikely that

such analytical DM solutions can be found. Of

course the finite element model can be used forsuch problems, and would therefore be the recom-

mended approach.

The presented comparisons have been limited to

only the elastic response. However, our finite ele-

ment model has incorporated damage mechanics

concepts in order to simulate inelastic and failure

of cemented particulate materials (Sadd et al.,

2004a,b). Likewise, doublet mechanics theory hasalso been extended to describe inelastic behavior,

and some specific failure criteria have been devel-

oped (Ferrari and Granik, 1995). Thus compari-

sons of the inelastic predictions between the two

theories appear to be workable and would be an

interesting future study.

Since most real particulate materials are three-

dimensional in nature, two-dimensional modelingis always subject to concern. Two-dimensional

limitations on material micro-structure (fabric)

and on out-of-plane degrees of freedom will nor-

mally not capture all of the micro-mechanics in

the particulate system. The usual simple uniform

scaling through the thickness can only approxi-

mate the actual behavior. In regard to three-

dimensional modeling, the additional degrees offreedom will add considerable complexity to both

the doublet mechanics and micro-mechanical fi-

nite element models. It would be expected that

constructing three-dimensional DM solutions

would be very difficult. We have recently begun

research on a three-dimensional micro-finite ele-

ment modeling scheme. Within this new model

each particle is allowed to have 6 degrees of free-dom, and this generates a two-noded, micro-finite

element with 12 degrees of freedom. We hope to

report three-dimensional simulations in the near

future.

Acknowledgment

The authors would like to acknowledge support

from the Transportation Center at the University

of Rhode Island under Grants 01-64 and 02-86.

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