a micromechanical ﬁnite element model for linear and ...pages.mtu.edu/~qingdai/ijnamg_2006.pdf ·...

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2006; 30:1135–1158Published online 12 May 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.520

A micromechanical finite element model for linearand damage-coupled viscoelastic behaviour of

asphalt mixture

Qingli Dai1,n,y, Martin H. Sadd2 and Zhanping You3

1Department of Mechanical Engineering – Engineering Mechanics, Michigan Technological University,

Houghton, MI 49931, U.S.A.2Department of Mechanical Engineering and Applied Mechanics, University of Rhode Island,

Kingston, RI 02881, U.S.A.3Department of Civil and Environmental Engineering, Michigan Technological University,

Houghton, MI 49931, U.S.A.

SUMMARY

This study presents a finite element (FE) micromechanical modelling approach for the simulation of linearand damage-coupled viscoelastic behaviour of asphalt mixture. Asphalt mixture is a composite material ofgraded aggregates bound with mastic (asphalt and fine aggregates). The microstructural model of asphaltmixture incorporates an equivalent lattice network structure whereby intergranular load transfer is sim-ulated through an effective asphalt mastic zone. The finite element model integrates the ABAQUS usermaterial subroutine with continuum elements for the effective asphalt mastic and rigid body elements foreach aggregate. A unified approach is proposed using Schapery non-linear viscoelastic model for the rate-independent and rate-dependent damage behaviour. A finite element incremental algorithm with arecursive relationship for three-dimensional (3D) linear and damage-coupled viscoelastic behaviour isdeveloped. This algorithm is used in a 3D user-defined material model for the asphalt mastic to predictglobal linear and damage-coupled viscoelastic behaviour of asphalt mixture.For linear viscoelastic study, the creep stiffnesses of mastic and asphalt mixture at different temperatures

are measured in laboratory. A regression-fitting method is employed to calibrate generalized Maxwellmodels with Prony series and generate master stiffness curves for mastic and asphalt mixture. A com-putational model is developed with image analysis of sectioned surface of a test specimen. The viscoelasticprediction of mixture creep stiffness with the calibrated mastic material parameters is compared withmixture master stiffness curve over a reduced time period.In regard to damage-coupled viscoelastic behaviour, cyclic loading responses of linear and rate-inde-

pendent damage-coupled viscoelastic materials are compared. Effects of particular microstructure param-eters on the rate-independent damage-coupled viscoelastic behaviour are also investigated with finite elementsimulations of asphalt numerical samples. Further study describes loading rate effects on the asphalt visco-elastic properties and rate-dependent damage behaviour. Copyright # 2006 John Wiley & Sons, Ltd.

KEY WORDS: micromechanical modelling; viscoelasticity; damage behaviour; finite elements; asphaltmixture

Received 20 May 2005Revised 9 November 2005Accepted 6 March 2006Copyright # 2006 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: Qingli Dai, Department of Mechanical Engineering – Engineering Mechanics, MichiganTechnological University, Houghton, MI 49931, U.S.A.

1. INTRODUCTION

Asphalt mixture is a very complex heterogeneous and time-dependent material, and generallycontains aggregates, mastic and void space. Thus, the macro load-carrying behaviour dependson many micro-phenomena that occur at the aggregate/mastic level. Some important microbehaviours are related to mastic properties including volume percentage, elastic/viscoelasticmoduli, damage/time-dependent response, aging hardening, microcracking, and debondingfrom aggregates. Other microstructural features include aggregate size, shape, texture andpacking geometry. Because of these issues it appears that a micromechanical model would bebest suited to properly simulate such a material.

Recently, many studies have been investigating the micromechanical behaviour of particulate,porous and heterogeneous materials. For example, studies on cemented particulate materials byDvorkin et al. [1] and Zhu et al. [2] provided information on the normal and tangential loadtransfer between cemented particles. Applications of such contact-based micromechanical anal-ysis for asphalt mixture behaviour have been reported by Chang and Gao [3], Cheung et al. [4]and Zhu and Nodes [5].

Numerical modelling of cemented particulate materials has generally used both finite (FEM)and discrete (DEM) element methods. The DEM analyses particulate systems by modelling thetranslational and rotational behaviour of each particle using Newton’s second law withappropriate interparticle contact forces. Normally, the scheme establishes an explicit, time-stepping procedure to determine each of the particle motions. DEM studies on cementedparticulate materials include the work by Rothenburg et al. [6], Chang and Meegoda [7],Trent and Margolin [8], Buttlar and You [9], Ullidtz [10], and Sadd and Gao [11].

In regard to finite element modelling, Sepehr et al. [12] used an idealized finite elementmicrostructural model to analyse the behaviour of an asphalt pavement layer. Soares et al. [13]used cohesive zone elements to develop a micromechanical fracture model of asphalt mixture. Aparticular finite element approach to simulate particulate materials has used an equivalentlattice network system to represent the interparticle load transfer behaviour. Guddati et al. [14]recently presented a random truss lattice model to simulate microdamage in asphalt mixture anddemonstrated some interesting failure patterns in an indirect tension test geometry. Sadd et al.[15,16] employed a micro-frame network model to investigate the damage behaviour of asphaltmixture, this model used a special purpose finite element that incorporates the mechanical load-carrying response between neighbouring particles. Bahia et al. [17] have also used finite elementsto model the aggregate-mastic response of asphalt mixture.

Damage mechanics provides a viable framework for the description of asphalt stiffness deg-radation, microcrack initiation, growth and coalescence, and damage-induced anisotropy.Continuum damage mechanics is based on the thermodynamics of irreversible processes tocharacterize elastic-coupled damage behaviours. Chaboche [18], and Simo and Ju [19] developedstrain- and stress-based anisotropic continuum damage models, while Kachanov [20] proposed amicrocrack-related continuum damage model for rate-independent solids. For viscoelasticmaterials, Taylor et al. [21] developed a computational algorithm to analyse thermomechanicalbehaviour. Some researches such as Simo [22] have proposed viscoelastic continuum damagemodels (VCDM) to describe the damage behaviour of viscoelastic materials. Schapery [23]developed detailed viscoelastic damage models based on non-equilibrium thermodynamics, visco-elastic fracture mechanics and elastic–viscoelastic correspondence principles, and a later study [24]incorporated a work potential theory. Park and Schapery [25] proposed an explicit

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:1135–1158

Q. DAI, M. H. SADD AND Z. YOU1136

viscoelastic damage model for particulate composites, and Park et al. [26] applied this model touniaxial behaviour of asphalt mixture. Recently, Schapery [27] developed constitutive equationsthat account for effects of viscoelastic, viscoplastic, growing damage and aging. Some recentstudies were conducted on damage constitutive modelling of viscoelastic composite materials,such as Canga et al. [28], Kaliske et al. [29], Haj-Ali and Muliana [30], and Kumar and Talreja[31]. Wu and Harvey [32] recently applied a 3D continuum damage mechanics method to modelthe cracking behaviour of asphalt mixture.

This paper presents a micromechanical modelling scheme for the linear and damage-coupledviscoelastic behaviour of asphalt mixture by using finite element methods. The model firstincorporates an equivalent lattice network structure whereby intergranular load transfer issimulated through an effective asphalt mastic zone. The previous work Sadd et al. [15,16]employed a network of user-defined micro-frame elements with a special stiffness matrix de-veloped to predict the load transfer between cemented particles. This study incorporates theuser-defined material subroutine with continuum elements for the effective asphalt mastic andrigid body defined with rigid elements for each aggregate. A unified approach for the rate-independent and rate-dependent damage behaviour has been developed using Schapery’s non-linear viscoelastic model. Properties of the continuum elements are specified through a usermaterial subroutine within the ABAQUS code and this allows linear and damage-coupledviscoelastic constitutive behaviour of the mastic cement to be incorporated. Section 2 of thispaper introduces a finite element incremental algorithm with recursive relationships for visco-elastic behaviour. This algorithm is later used in the linear and damage viscoelastic modelling ofthe asphalt mastic in the proposed microstructure model. Section 3 presents microstructuralmodelling of heterogeneous asphalt mixture. For the viscoelastic simulation, Section 4 comparesthe linear viscoelastic prediction of mixture creep stiffness using calibrated mastic properties andan image computational sample with laboratory test data. In regard to damage-coupledbehaviour, the cyclic loading responses of viscoelastic asphalt mixture with linear and rate-independent damage-coupled viscoelastic models are compared in Section 5. Section 6 studiesthe micro-parametric effects on rate-independent damage viscoelastic behaviour using finiteelement simulation on numerical samples with controllable microstructure. Finally, Section 7investigates rate-dependent damage behaviour and the effect of loading rate on the viscoelasticproperties and rate-dependent damage behaviour.

2. DAMAGE-COUPLED VISCOELASTIC MODEL

2.1. One-dimensional linear viscoelastic model

A generalized Maxwell model is commonly used to simulate the constitutive behaviour of linearsolid viscoelastic material. This model consists of an elastic spring with constant E1 in parallelwith M Maxwell elements. The stress–strain relationship for this model can be expressed as ahereditary integral

s ¼ E1eþZ t

0

EtdeðtÞdt

dt ð1Þ

where Et is expressed with a Prony series

Et ¼XMm¼1

Em e�ðt�tÞ=rm and rm ¼ZmEm

ð2Þ


MICROMECHANICAL FINITE ELEMENT MODEL 1137

In these equations, E1 is the relaxed elastic modulus, Et is the transient modulus as a function ofthe time, Em, Zm and rm are the spring constant, dashpot viscosity and relaxation time,respectively, for the mth Maxwell element.

The reduced time (effective time) is defined by using time–temperature superposition principleas

xðtÞ ¼Z t

0

1

aTdt ð3Þ

where the term aT ¼ aT ðTðtÞÞ is a temperature-dependent time-scale shift factor.A displacement-based incremental finite element modelling scheme with constant strain rate

over each increment has been developed. An incremental numerical algorithm for the linearviscoelastic integral has been used from the work of Zocher and Groves [33]. This algorithm wasdeveloped in closed form and results in a recursive relationship. This method creates anincremental formulation of the current stress state from recursive variables stored in the pre-vious step, and current variables of time and strain increments.

Assuming the stress is known at the reduced time xn�1, the current stress at reduced time xn,according to (1), is given by

sðxnÞ ¼ E1eðxnÞ þZ x

0

Etðxn � x0Þdeðx0Þdx0

dx0 ð4Þ

Its incremental form can be written in two parts,

Ds ¼ E0 � Deþ DsR ð5Þ

The first part includes the integration from the previous step xn�1 to the current step xn. Thereduced time increment is defined as Dx ¼ xn � xn�1: This formulation assumes the incrementalstrain De ¼ en � en�1 is known, and the strain changes with a constant strain rate Re during eachinterval xn�14x04xn: So the incremental modulus E 0 and strain rate Re can be expressed by

E 0 ¼ E1 þXMm¼1

EmrmDxð1� e�Dx=rmÞ and Re ¼

DeDx

ð6Þ

Note that the incremental stiffness E 0 is not dependent on the time if Dx remains constant.The second part of relation (5) is formulated with integration from 0 to the previous step xn�1,

and this leads to a recursive relation with the history variables Sm.

DsR ¼XMm¼1

�ð1� e�Dx=rmÞSmðxnÞ and SmðxnÞ ¼ EmRermð1� e�Dx=rmÞ þ Smðxn�1Þ e�Dx=rm

ð7Þ

For the initial increment, the history variable Sm(x1) equals to EmRermð1� e�Dx=rmÞ and issimilar to the following formulations.

2.2. One-dimensional damage-coupled viscoelastic model

Rate-independent failure processes are characterized by the fact that damage growth is the onlydissipative mechanism and that the current state does not depend on rate effects [34]. Therate-independent damage variable is defined as a function of maximum equivalent strain [22].A second damage mechanism referred to as rate-dependent damage is caused by the viscoelastictime-dependent material property and loading rate effects, and its damage variable is based on



the equivalent strain rate. A unified approach is presented for both failure mechanisms by usingthe Schapery model.

The original Schapery non-linear viscoelastic model [35] was given as

sðxÞ ¼ heE1eðxÞ þZ x

0

h1Etðx� x0Þdðh2eðx

0ÞÞdx0

dx0 ð8Þ

This model incorporates three different non-linear parameters: he is the non-linear factor of therelaxed elastic modulus E1, h1 measures the non-linearity effect in the transient modulus Et, andh2 accounts for the loading rate effect.

Following form (8), a unified damage-coupled viscoelastic model for both failure mechanismsis proposed by replacing three non-linear parameters with damage variables which would beexpressed with damage evolution functions.

sðxÞ ¼ heðemaxÞE1eðxÞ þZ x

0

h1ðemaxÞEtðx� x0Þdðh2ð’eÞeðx

0ÞÞdx0

dx0 ð9Þ

where he and h1 are the elastic and viscoelastic damage variables for the rate-independent failurebehaviour, h2 is the rate-dependent damage variable dependent on the strain rate ’e: The rate-independent variables he and h1 are functions of the maximum strain emax, which is defined asthe maximum value over the past history up to the current time x,

emax ¼ maxðeðx0ÞÞ; x0 2 ½0; x� ð10Þ

The elastic damage variable he ¼ 1� O measures the relaxed elastic stiffness reduction, andcan be described by using the inelastic damage evolution law in Sadd et al. [15,16],

heðemaxÞ ¼ e�bðemax=e0Þ ð11Þ

where the material parameters e0 and b are related to the softening strain and damage evolutionrate, respectively.

The viscoelastic variable h1 measures the damage effect in the transient modulus, and ischosen with the following exponential form by Simo [22],

h1ðemaxÞ ¼ bþ ð1� bÞ1� e�ðemax=e0Þ

emax=e0; b 2 ½0; 1� ð12Þ

The variable h1 will reduce from 1 to b as the maximum strain emax increases, and e0 is also thesoftening strain.

The rate-dependent damage variable h2 is proposed as

h2 ¼ 1þ’e’eC� 1

� �� a) h2 ¼

’e’eC

� ��aif ’e5’eC

1 if ’e5’eC

8><>: ð13Þ

where ’eC is the threshold strain rate that determines the start of the rate-dependent damage, anda50 is related to the rate-dependent damage evolution rate. When the parameter a ¼ 0 orloading rate ’e5’eC; h2 ¼ 1 and this model has only rate-independent damage behaviour. Theinfluence of parameters a and ’eC on model rate-dependent damage behaviour will be illustratedin the following loading rate effect study.

The incremental formulation of the 1D damage-coupled viscoelastic behaviour can beestablished by using a similar procedure as used in the previous section. According to



relation (9), the current stress at reduced time xn is given by

sðxnÞ ¼ heðemaxÞE1eðxnÞ þZ xn

0

h1ðemaxÞEtðxn � x0Þdðh2ð’eÞeðx

0ÞÞdx0

dx0 ð14Þ

and its incremental form can also be formulated into two parts

Ds ¼ E0 � Deþ DsR ð15Þ

Assuming again that the strain changes with constant rate Re during each interval xn�14x04xn;the first term in Equation (15) gives the incremental modulus E0 as

E0 ¼ heðemaxÞE1 þ h1ðemaxÞh2ðReÞXNm¼1

EmrmDxð1� e�Dx=rm Þ

where

Re ¼DeDx

and h2ðReÞ ¼

Re

’eC

� ��aif Re5’eC

1 if Re5’eC

8><>: ð16Þ

This incremental modulus E0 includes both rate-independent and rate-dependent damage var-iables. The second term of (15) also leads to a recursive relation with the history variables Sm,which includes the viscoelastic damage variable h1 and rate-dependent damage variable h2,

DsR ¼XNm¼1

�ð1� e�Dx=rmÞSmðxnÞ

and

SmðxnÞ ¼ Emh1ðemaxÞh2ðReÞRermð1� e�Dx=rm Þ þ Smðxn�1Þe�Dx=rm ð17Þ

This model includes two damage mechanisms when the strain rate ’e5’eC; and has onlyrate-independent damage behaviour if the strain rate is less than this threshold value.

2.3. Three-dimensional damage-coupled viscoelastic model

The numerical formulations for 1D damage-coupled viscoelastic behaviour are now to begeneralized to a multiaxial (3D) constitutive formulation for isotropic matrix material. Asemployed by Simo [22], and Haj-Ali and Muliana [30], uncoupled volumetric and deviatoricstress–strain relations are assumed. The formulation also assumes the incremental strain tensoris known and the strains change linearly during each interval xn�14x04xn:

The stress and strain tensors can be decomposed into the usual sum of volumetric anddeviatoric parts,

sij ¼ 1=3skkdij þ #sij ; eij ¼ 1=3ekkdij þ #eij ð18Þ

The elastic stress–strain relations for the volumetric and deviatoric behaviour are expressed by

skk ¼ 3Kekk; #sij ¼ 2G#eij ð19Þ

where K ¼ E=3ð1� 2uÞ and G ¼ E=2ð1þ uÞ are the elastic bulk and shear modulus. The termsK1 and G1 are the relaxed bulk and shear moduli, and Km and Gm are the bulk and shearconstants for the spring in the mth Maxwell element.



By applying the previous 1D damage-coupled viscoelastic model, the volumetric constitutiverelationship is expressed with the volumetric stress skk and strain ekk in the general form

skkðxÞ ¼ 3K1heðekkmaxÞekkðxÞ þZ x

0

3Ktðx� x0Þh1ðekkmaxÞdðh2ð’ekkÞekkðx

0ÞÞdx0

dx0 ð20Þ

where Ktðx� x0Þ ¼PM

m¼1 Km e�ðx�x0Þ=rm is the transient bulk modulus, ekkmax is the maxi-

mum volumetric strain over the past history, and ’ekk is the volumetric strain rate. It is assumedthat tension and compression damage behaviours are independent, so two different historyvariables ekkmax are used for the tension and compression volumetric behaviours.

Following the previous formulation procedures, the incremental formulation of thevolumetric behaviour is obtained with constant volumetric strain rate Rkk ¼ Dekk=Dx;

Dskk ¼ 3 K1heðekkmaxÞ þXNm¼1

h1ðekkmaxÞh2ðRkkÞKmrmDx

ð1� e�Dx=rmÞ

" #Dekk þ DsRkk

where

h2ðRkkÞ ¼

Rkk

’eC

� ��aif Rkk5’eC

1 if Rkk5’eC

8><>: ð21Þ

and the residual part DsRkk can be expressed in a recursive relation with the history variable Sm,

DsRkk ¼XMm¼1


and

SmðxnÞ ¼ 3Kmh1ðekkmaxÞh2ðRkkÞRkkrmð1� e�Dx=rmÞ þ Smðxn�1Þe�Dx=rm ð22Þ

For the deviatoric behaviour, the constitutive relationship is written using deviatoric stress #sijand strain #eij ;

#sijðxÞ ¼ 2G1heðeemaxÞ#eijðxÞ þZ x

0

2Gtðx� x0Þh1ðeemaxÞdðh2ð’eeÞ#eijðx

0ÞÞdx0

dx0 ð23Þ

where Gtðx� x0Þ ¼PN

m¼1 Gm e�ðx�x0Þ=rm is the transient shear modulus, eemax is the maximum

equivalent strain, and the equivalent strain ee and strain rate ’ee are defined as

ee ¼

ffiffiffiffiffiffiffiffiffiffiffiffi3

2#eij #eij

rand ’ee ¼

ffiffiffiffiffiffiffiffiffiffiffiffi3

2’#eij’#eij

rð24Þ

The formulation of the deviatoric behaviour is obtained with constant deviatoric strain rate#Rij ¼ D#eij=Dx;

D #sij ¼ 2 G1heðeemaxÞ þXNm¼1

h1ðeemaxÞh2ðReÞGmrmDx

ð1� e�Dx=rm Þ

" #D#eij þ D #sRij

where

Re ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2#Rij

#Rij

rand h2ðReÞ ¼

Re

’eC

� ��aif Re5’eC

1 if Re5’eC

8><>: ð25Þ



and the residual part D #sRij can be expressed in the recursive relation

D #sRij ¼XNm¼1


and

SmðxnÞ ¼ 2Gmh1ðeemaxÞh2ðReÞ #Rijrmð1� e�Dx=rm Þ þ Smðxn�1Þe�Dx=rm ð26Þ

The incremental normal stresses can be then formulated by combining the volumetric anddeviatoric behaviour, for example,

Dsxx ¼ 1=3Dskk þ D #sxx

¼ K1heðekkmaxÞ þXNm¼1

h1ðekkmaxÞh2ðRkkÞKmrmDx

ð1� e�Dx=rmÞ

" #Dekk þ 1=3DsRkk

þ 2 G1heðeemaxÞ þXNm¼1


ð1� e�Dx=rm Þ

" #D#exx þ D #sRxx ð27Þ

where Dekk and Dskk are the incremental volumetric strain and stress, D#exx and D #sxx are theincremental deviatoric strain and stress components, and DsRkk and D #sRxx are the recursive part ofthe volumetric and deviatoric behaviour given in Equations (22) and (26). Incremental stressesDsyy and Dszz are determined in the same manner.

The incremental shear stress can be formulated by using only the deviatoric behaviour. Forexample,

Dsxy ¼D #sxy

¼ 2 G1heðeemaxÞ þXNm¼1


ð1� e�Dx=rmÞ

" #D#exy þ D #sRxy ð28Þ

where D#exy and D #sxy are the incremental shear deviatoric strain and stress components, and therecursive term D #sRxy is also given in Equation (26).

Once the incremental stress components are developed, the incremental stiffness terms can becalculated as

@Dsxx@Dexx

¼ K1heðeemaxÞ þXNm¼1

h1ðeemaxÞh2ðRkkÞKmrmDx

ð1� e�Dx=rm Þ

" #

þ4

3G1heðeemaxÞ þ

XNm¼1


ð1� e�Dx=rmÞ

" #¼ Kd1

@Dsxx@Deyy

¼ K1heðeemaxÞ þXNm¼1

h1ðeemaxÞh2ðRkkÞKmrmDx

ð1� e�Dx=rm Þ

" #

�2

3G1heðeemaxÞ þ

XNm¼1


ð1� e�Dx=rmÞ

" #¼ Kd2

@Dsxy@Dexy

¼ 2 G1heðeemaxÞ þXNm¼1


ð1� e�Dx=rm Þ

" #¼ Kd3

ð29Þ



With these terms, the incremental 3D damage-coupled viscoelastic behaviour can be formulatedas

Dsxx

Dsyy

Dszz

Dsxy

Dsyz

Dsxz

2666666666664

3777777777775¼

Kd1 Kd2 Kd2 0 0 0

� Kd1 Kd2 0 0 0

� � Kd1 0 0 0

� � � Kd3 0 0

� � � � Kd3 0

� � � � � Kd3

2666666666664

3777777777775

Dexx

Deyy

Dezz

Dexy

Deyz

Dexz

2666666666664

3777777777775þ

DsRkk þ D #sRxx

DsRkk þ D #sRyy

DsRkk þ D #sRzz

D #sRxy

D #sRyz

D #sRzx

26666666666664

37777777777775

ð30Þ

This damage-coupled viscoelastic model can then be defined in the ABAQUS user materialsubroutine, and then combined with particular ABAQUS elements to implement a displace-ment-based non-linear finite element analysis.

3. MICROSTRUCTURAL MODELLING

Asphalt mixture can be described as a multi-phase material containing coarse aggregates, masticcement (including asphalt binder and fine aggregates) and air voids (see Figure 1). The loadtransfer between the aggregates plays a primary role in determining the load-carrying capacityand failure of such complex materials. In order to develop a micromechanical model of thisbehaviour, proper simulation of the load transfer between the aggregates must be accomplished.The aggregate material is normally much stiffer than the mastic, and thus aggregates are takenas rigid particles. On the other hand, the mastic cement is a compliant material with elastic,inelastic, and time-dependent behaviours. Additionally, in most cases, when asphalt mixture issubject to mechanical loading, it generates distributed microstructure damage in the formof microcrack nucleation and growth due to higher stress concentration along theaggregate}mastic interface within the material. This microstructural modelling effort is toaccount for viscoelasticity and damage evolution properties of asphalt mixture.

In order to properly account for the load transfer between aggregates, it is assumed that thereis an effective mastic zone between neighbouring particles. It is through this zone that themicromechanical load transfer occurs between each aggregate pair. This loading can be reducedto resultant normal and tangential forces and a moment as shown in Figure 1.

In general, asphalt mixture contains aggregate of highly irregular geometry as shown inFigure 2(a). The approach used in this study is to allow variable size and shape using anelliptical aggregate model as represented in Figure 2(b). The finite element model shown inFigure 2(c) uses a rectangular strip to simulate the effective asphalt mastic zone, and incor-porates the ABAQUS user material subroutine with continuum elements to model the effectiveasphalt mastic behaviour.

As shown in Figure 3, the general modelling scheme employed four-node quadrilateralelements to mesh the effective cement material, and a rigid body defined with two-node rigidelements sharing the particle centre to model each aggregate. The particle centre is referred to asthe master node of the aggregate rigid body, and thus each rigid body element would haveidentical translation and rotation as the represented aggregate. The rigid elements act to link the



mastic deformation with the aggregate rigid body motion, thereby establishing the microme-chanical deformation behaviour. Properties of the quadrilateral elements are specified through auser-defined material subroutine within the ABAQUS code and this allows incorporation of thedeveloped linear and damage-coupled viscoelastic mastic cement models.

A MATLAB code was developed to generate 2D numerical samples of asphalt mixture, andthe element geometry properties and connectivity were created and saved in input files forABAQUS modelling and analysis. A 2D indirect tension sample has been generated with

Aggregate

F

Mastic

Void t

Fn

M

Resultant Load Transfer Between Aggregates

Figure 1. Schematic of multi-phase asphalt materials.

(a) (b) (c)

Effective Mastic Zone

Figure 2. Asphalt modelling concept: (a) typical asphalt material; (b) model asphalt system;and (c) ABAQUS finite element model.

Two-Node AggregateRigid Element

Four-Node Mastic Quadrilaterial Element

Figure 3. ABAQUS modelling scheme for a typical particle pair.



MATLAB code as shown in Figure 4(a), and its mesh figure was modelled by using ABAQUSelements as shown in Figure 4(b). This particular model has 65 particles (in four particle sizegroupings), 195 effective mastic zones, 7.6% porosity and an approximate overall diameter of102mm (4 in). This microgeometry results in a total of 780 deforming mastic elements and 1170rigid aggregate elements with connectivity as shown.

4. COMPARISON BETWEEN LINEAR VISCOELASTIC PREDICTIONOF MIXTURE CREEP STIFFNESS AND TEST DATA

The purpose of this study is to compare the mixture creep stiffness of a viscoelastic simulationon a computer-generated image model with test results from laboratory asphalt specimens. Inorder to capture real microstructure of asphalt mixture, simulation material models were gen-erated using image analysis procedures from sectioned surface photographic data of actualasphalt specimens. These imaging techniques and simulation model generation were described inthe previous work [36].

The image model generation was conducted on a uniaxial compression asphalt specimenshown in Figure 5. The specimen is a 19mm (nominal maximum aggregate size) mixture pro-duced in the work done by You [37], and You and Buttlar [38]. Its image was obtained by high-resolution scanner and preliminarily processed with imaging technology. Figure 5(a) shows abinary image of the sectioned specimen, where the sample aggregates have passed through a2.36mm sieve and its asphalt content is about 14%. In this specimen, the mastic portionincludes aggregate finer than the 2.36mm (passing the sieve) and asphalt binder. For the imageprocessing, each sieved aggregate was labelled and selected as a separate image. The boundary

(a) (b)

Figure 4. Indirect tension numerical sample: (a) numerical sample; and (b) mesh sample.



pixels of each aggregate were extracted, and the co-ordinates of these pixels were stored in anarray for the ellipse fitting. A least-squares, ellipse-fitting algorithm was incorporated todetermine the ‘best’ ellipse to represent each irregular aggregate geometry. The fitted ellipsewas then generated for each sieved aggregate with the developed MATLAB Code as shown inFigure 5(b). Based on these fitted ellipse parameters (centre co-ordinates, size and orientation), acomputation model is generated with maximally filled cementation between neighbouring par-ticles using MATLAB as shown in Figure 5(c). For this compressive computational model, theporosity was approximately 2% and the aggregate percentage was about 62%.

In the laboratory, the creep stiffnesses of mastic and asphalt mixture were measured at threetemperatures and different loading time in the study by You [37], and You and Buttlar [39]. Asmentioned previously, the mastic comprised of the portion with the aggregate gradation finerthan 2.36mm sieve and the volume of binder normally used in the entire asphalt concretemixture. Thus, the mastic had around 14% asphalt binder by weight. A regression-fittingmethod was employed to calibrate a generalized Maxwell models with Prony series, and gen-erate master stiffness curves for mastic and asphalt mixture from test data. Due to lengthlimitations, shift factor computations and master curve generation procedures are not given herebut can be found in References [40,41].

The shifted creep stiffness at three temperatures and the master stiffness curve (referencetemperature �208C) for mastic are shown in Figure 6. The time–temperature shift factors aT(T)were calculated as 0.1 and 0.008 for the temperatures �10 and 08C. The calibrated Maxwellmodel parameters with Prony series for mastic materials are given in Table I, and its corre-sponding master stiffness curve is plotted in Figure 6. Similarly, the shifted creep stiffness atthree temperatures and master stiffness curve (reference temperature �208C) for mixture areshown in Figure 7. The time–temperature shift factors aT(T) were calculated as 0.159 and 0.002at the temperatures �10 and 08C.

After calibrating mastic material properties and generating mastic and mixture stiffnessmaster curves, a comparison study was conducted between viscoelastic simulation using cal-ibrated mastic properties and the image model (shown in Figure 5(c)), and test data of lab-oratory mixture specimen (shown in Figure 7). For the compression creep simulation, the x- and

Figure 5. Uniaxial compression image model generation: (a) smooth surface of asphalt specimen;(b) elliptical fitted aggregates; and (c) image model with aggregate and mastic.



y-displacements of the particles on the bottom layer and the x-displacements of the particles onthe top layer were constrained. The constant force loading was evenly divided and imposed onparticles of the top layer. In the simulations, axial strain was calculated by dividing the averagevertical displacement of top particles with the initial height of the undeformed specimen. Axialstress was obtained by dividing the constant loading force on the top layer with the specimeninitial cross-section area. The finite element model simulation results for creep stiffness of

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Figure 6. Creep stiffness master curve for mastic.

Table I. Fitted generalized Maxwell model parametersfor mastic behaviour.

E0 59.66MPaE1 5710.63MPaE2 2075.07MPaE3 1449.00MPaE4 734.90MPar1 26.24 sr2 311.89 sr3 1678.80 sr4 19952.62 s

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Figure 7. Creep stiffness master curve for mixture.



asphalt mixture were compared with mixture stiffness master curve over a reduced time period(104 s) as shown in Figure 8. The average relative error between model prediction and test datawas approximately 11.7%. These errors are caused by the microstructural model assumption ofrigid aggregates with infinite stiffness and elliptical shape, and also due to the approximation ofmaster curves for mastic and mixture. In spite of some differences in the beginning of the curve,these comparison curves of mixture creep stiffness are very close after some reduced time.Therefore, this proposed micromechanical finite element model with user material subroutine isapplicable to predict the global viscoelastic behaviour of asphalt mixture.

5. CYCLIC LOADING RESPONSES WITH LINEAR AND RATE-INDEPENDENTDAMAGE-COUPLED VISCOELASTIC MODELS

The cyclic loading responses of linear and rate-independent damage-coupled viscoelastic modelswere investigated with indirect tension simulations on the numerical sample (shown in Figure 4).The mastic elements had 3-parameter viscoelastic constitutive properties with relaxed elasticmoduli E1 ¼ 412:8MPa and one Maxwell element of spring constant E1 ¼ 1232MPa andrelaxation time r1 ¼ 6:5 s: These constitutive properties were selected from asphalt mixturecharacterization testing by Gibson et al. [42]. It was assumed that Poisson’s ratio u did notchange with time and was given as 0.3. Model rate-independent damage parameters were chosenas b ¼ 1; b ¼ 0:3; and softening strain e0 ¼ 0:2: The rate-dependent damage parameter a wastaken as zero, and thus this initial study compares only the rate-independent damage with linearviscoelastic behaviour.

For the following cyclic loading simulations, both displacement- and force-controlled bound-ary conditions were used on the numerical sample. For the displacement-controlled boundaryconditions, both horizontal and vertical displacements of the bottom pair of aggregates are

Figure 8. Comparison between model prediction of mixture creep stiffness with mixture master curve.



constrained, while the top particle pair accepts the applied vertical displacement loading. Underforce-control, the vertical force is imposed to the top particle pair, and the displacements of thebottom pair of aggregates are again constrained.

Linear and damage-coupled viscoelastic simulations were conducted and compared for bothdisplacement- and force-controlled boundary conditions. Viscoelastic behaviour is demonstrat-ed by the decreasing relaxation force and increasing creep displacement with the unreversedloading cycles. The results also show that viscoelastic damage behaviour reduces the sample’sload-carrying ability for the displacement-controlled boundary conditions, and increases thesample creep displacement for the force loading.

Numerical simulations were also conducted with an incrementally increasing reversed cyclicloading as shown in Figure 9. Figure 9(a) shows the linear viscoelastic responses of thenumerical sample under the displacement and force loading. The maximum points in each loopare approximately located along a straight line in these two figures. Figure 9(b) gives thedamage-coupled viscoelastic responses. Since the rate-independent damage variables in the

Figure 9. Incrementally increasing reversed cyclic loading: (a) linear viscoelastic response;and (b) damage-coupled viscoelastic response.



model are all defined as exponential functions, the maximum points are distributed along anexponential curve. These damage-coupled cyclic responses illustrate that rate-independentdamage increases with the maximum deformation. This occurs since the rate-independentdamage parameters are a function of maximum strain. These results also demonstrate that mostof the loss in stiffness takes place during the first cyclic deformation, and the steady-stateresponse is obtained after very few cycles with the same loading amplitude.

6. MICROSTRUCTURAL EFFECTS ON RATE-INDEPENDENT VISCOELASTICDAMAGE BEHAVIOUR OF ASPHALT NUMERICAL SAMPLES

Micro-parameter effects on inelastic damage behaviour of asphalt numerical samples wereinvestigated in detail by Dai and Sadd [43]. Following this previous work, the relationshipbetween microstructure parameters and damage-coupled viscoelastic behaviour of particularasphalt model samples is studied. The microstructure parameters of asphalt mixture can becategorized by: proportions of asphalt mixture components such as model porosity (area ratioof aid voids to entire sample) and aggregate percentage (area percentage of aggregate over thesample model); particle/aggregate measures including particle orientation, shape and size; andpacking fabric descriptions such as aggregate gradation (aggregate area percentage passingdifferent sieving sizes) and branch vector distribution. Particle orientation is commonly rep-resented by an angular measure (with respect to a reference direction) of the particle’s longestaxis. Particle shape ratio (aspect ratio) is defined as the ratio of the longest axis dimension to theshortest axis dimension. Particle size is the longest axis dimension while the branch vector runsfrom one particle centre to another neighbouring particle centre. Packing fabric can be cat-egorized as the branch vector distribution and aggregate gradation. Figure 10 illustrates particleorientation and branch vectors for a model sample of cemented particulate material. In order toquantify the distributions of these vectors for an entire sample, an angular distribution plot(Rose diagram) is normally constructed. Such a plot indicates the frequency of the vectors lyingin a particular direction as a function of angular measure. These will be used to correlate variousnumerically generated asphalt samples for finite element simulation.

Branch Vectors Orientation Vectors

Figure 10. Vector microstructure measures in particulate materials.



Microstructure effects were investigated through finite element simulation of a series ofasphalt numerical samples. Using a specially developed MATLAB code, numerical sampleswere generated with controllable microstructure variation in an effort to determine the effect ofa particular microstructural variable on the material damage viscoelastic response. A series ofthe finite element simulations were conducted on indirect tension samples, and displacement-controlled boundary conditions were used (as shown in Figure 4). The viscoelastic materialconstants include E1 ¼ 412:8MPa; u ¼ 0:3; and six Maxwell elements from asphalt mixturecharacterization testing by Gibson et al. [42]. Damage parameters are same as previous section,and rate-independent damage behaviour was also considered in this microstructure study.

6.1. Model porosity

Air voids provide an important indication of the mixture’s pavement service performances.Model porosity, which is defined as area ratio of air voids over entire sample, affects the

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Figure 11. Damage-coupled viscoelastic simulations with different model porosity.



viscoelastic damage response of the model. Figure 11 illustrates three indirect tension damage-coupled viscoelastic simulations on models with porosities of 1.1, 4.6 and 6.7%. Each numericalsample had the same number of particles (181) and elements (574), and had identical particlelocations, shape (aspect ratio), and particle area percentage. Particle size and number percentageare also same as: 2mm (44.8%), 3mm (14.9%), 4mm (8.3%), 7mm (16%), 9mm (10.5%) and

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11mm (5.5%). All samples had the same branch and orientation vector distributions as shownin the figure. The model porosity was modified by only changing the effective mastic zone widthwhile other model parameters were kept the same. The damage-coupled viscoelastic simulationsof these numerical samples were conducted under the same unreversed saw-toothed displace-ment loading and the results (shown in Figure 11) indicated that lower-porosity models hadstiffer behaviour.

6.2. Aggregate gradation

Aggregate gradation curves are very helpful in making necessary adjustments to asphalt mixturedesign. In order to investigate the effect of aggregate gradation, a pair of numerical models wasgenerated with variation in the smaller-sized particles. The two indirect tension models, shownin Figure 12, were created from five different particle size groupings: 2, 4, 7, 11 and 14mm.Model B-1 was composed of a mix of 96 particles, and the particle distribution for different sizesare 2mm (area percentage of 2% for the 23 particles), 4mm (area percentage of 4% for the 15particles), 7mm (area percentage of 24% for the 29 particles), 11mm (area percentage of 38%for the19 particles) and 14mm (area percentage of 33% for the 10 particles). This resulted in amodel with 292 finite elements and a particle percentage of 54.6%. Model B-2 includedadditional small aggregates of 2 and 4mm size, and the particle distribution changed to 2mm(area percentage of 7% for the 108 particles), 4mm (area percentage of 4% for the 15 particles),7mm (area percentage of 22% for the 29 particles), 11mm (area percentage of 36% for the 19particles) and 14mm (area percentage of 31% for the 10 particles). The added small aggregateslead to a model with 181 total particles, 574 finite elements and a particle percentage of 64.1%.The gradation curves for each of the models are plotted as particle area passing percentage withdifferent particle sizes shown in Figure 12. These curves are compared with typical gradations ofsome actual asphalt mixture in the same plot with 0.45 power of aggregate sieving sizes. Allother micro-parameters are identical in the two models. The damage-coupled viscoelastic sim-ulation results under the unreversed saw-toothed displacement loading illustrate that the addedsmall aggregates increase the model stiffness and load-carrying ability.

7. LOADING RATE EFFECT ON RATE-DEPENDENT VISCOELASTICDAMAGE BEHAVIOUR

The purpose of this section is to describe rate-dependent damage behaviour and loading rateeffects on sample damage by assuming the strain rate ’e is larger than the threshold strain rate ’eCand the parameter a has a positive value. For this study, indirect tension simulations were againconducted on the same numerical sample (shown in Figure 4) with displacement-controlledboundary conditions. These simulations used the same 3-parameter viscoelastic constitutiveproperties as previous cyclic loading study, and the rate-independent damage parameters werechosen as b ¼ 1; b ¼ 0:5; and softening strain e0 ¼ 0:1:

Effects of the two model parameters ’eC and a used in rate-dependent damage variable h2 areillustrated in Figure 13. The loading displacement increases linearly from zero to 6mm with aloading rate 0.6mm/s. Figure 13(a) shows the simulation results for three different thresholdstrain rates ’eC with constant a. These results indicate that more rate-dependent damage isgenerated as the threshold strain rate decreases. Figure 13(b) shows simulation results for the



cases of a ¼ 0:1; 0.3 and 0.5 with constant threshold strain rate. Note that the sample has morerate-dependent damage behaviour as a is increased.

Results of loading rate effects on asphalt viscoelastic damage behaviour are demonstrated inFigure 14. For these indirect tension simulations, rate-dependent damage parameters werechosen as ’eC ¼ 0:001 and a ¼ 0:1: Figure 14(a) shows the simulation results under the mono-tonic linear displacement loading to the same compression displacement 5mm at loading ratesof 5, 1, 0.5 and 0.25mm/s. Solid lines indicate the linear viscoelastic behaviour, dashed linesindicate the damage-coupled viscoelastic behaviour. Higher loading rates generate larger samplereaction forces since the sample has less relaxation time. Comparing the linear and damage-coupled viscoelastic behaviour, the higher loading rate also leads to more rate-dependent

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Figure 14. Loading rate effect on the asphalt viscoelastic damage behaviour: (a) simulations undermonotonic linear loading; and (b) simulations under reversed saw-toothed loading.



damage behaviour at the end of the displacement loading. Park et al. [26], and Lee and Kim [44]conducted uniaxial tension experiments with asphalt mixture specimens at different strain rates,and their test data also show such a damage increase with loading rate. Figure 14(b) illustratesthe damage simulations under cyclic linear displacement loading (� 5mm) but with differentloading rates 1, 0.5 and 0.33mm/s. Results indicate that the higher loading rate also leads to astiffer sample under cyclic loading. These simulation results under constant loading rate alsodemonstrate most of the stiffness reduction occurs during the first cycle deformation and thedamage responses reach steady state after very few cycles.

Simulations were also conducted with different material properties for a loading history withthree loading rate increments as shown in Figure 15. A threshold strain rate was chosen as’eC ¼ 0:001: Figure 15(a) shows the compression displacement input indicating the three loadingrate increments. Figure 15(b) shows the simulation results with four types of mastic materialproperties: linear viscoelastic, rate-independent damage with a ¼ 0; and rate-dependent damagewith a ¼ 0:1 and 0.3. Comparison between linear and rate-independent damage behaviourshows that the rate-independent damage again increases with the loading displacement. Com-parisons between two rate-dependent damage cases (a ¼ 0:1 and 0.3) and rate-independentdamage ða ¼ 0Þ behaviour, indicate additional rate-dependent damage increases with the load-ing rate. Also more rate-dependent damage is generated with a larger a parameter.

8. CONCLUSIONS

A micromechanical constitutive model was formulated and used to simulate the 2D linear anddamage-coupled viscoelastic behaviour of asphalt mixture. The aggregate-mastic microstructurewas simulated by incorporating the user material subroutine with continuum elements forasphalt mastic and rigid body elements for each aggregate. Rate-independent and rate-depend-ent damage mechanisms are defined and combined within the Schapery non-linear viscoelasticmodel. A finite element incremental algorithm with recursive relationships for the linear anddamage-coupled viscoelastic behaviour was developed. The numerical implementation of this

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Figure 15. Simulations under three loading rate increments: (a) compression displacement; and(b) simulations with different material properties.



method was also addressed. This behaviour was then defined in the ABAQUS user materialsubroutine for the asphalt mastic to predict global linear and damage-coupled viscoelasticbehaviour of asphalt mixture.

The linear viscoelastic predictions of mixture creep stiffness using calibrated mastic propertiesand an image computational model were compared with specimen test data. The average rel-ative error between model prediction and mixture master curve is approximately 11.7%. Theerrors are caused by the model assumption of rigid aggregates with infinite stiffness and ide-alized elliptical shape, and the approximate material master curves. In spite of some differencein the beginning, the comparison curves showed very close response of mixture creep stiffnessafter some reduced time. Therefore, this proposed micromechanical finite element model isapplicable to predict the global viscoelastic behaviour of asphalt mixture.

The cyclic loading response of linear and rate-independent damage viscoelastic asphalt mix-ture was compared for both displacement- and force-controlled boundary conditions. Thesecyclic loading responses show rate-independent damage increases with the maximum deforma-tion. The results also demonstrate that most of the loss in material stiffness occurs during thefirst cycle of deformation, and both linear and damage viscoelastic behaviour reaches steadystate after very few cycles.

Specific microstructure parameters of asphalt samples were identified and categorized fornumerical analysis. A series of indirect tension asphalt samples were then generated with sys-tematic variation of particular microstructures. Numerical experiments included investigationson sample porosity and aggregate gradation. Model porosity studies indicated that the sampleswith lower porosity resulted in higher load-carrying behaviour. Simulation investigations onaggregate gradation showed the sample had higher stiffness with added small aggregates.

The study of loading rate effects showed that higher loading rate leads to a stiffer asphaltsample response and also generates more rate-dependent damage behaviour. By using the uni-fied damage model, this study also compared the viscoelastic response of linear, rate-independ-ent damage, and rate-dependent damage material properties under different loading rate andamplitude.

The linear viscoelastic model verification study was conducted by comparing the mixturecreep stiffness with test data, and the simulation results on 2D models had satisfactory pre-diction. As future modelling efforts are extended to 3D, it is expected that model predictions willbe closer to test data. The damage-coupled viscoelastic simulations were in qualitative agree-ment with observed and expected behaviour of asphalt mixture. More detailed experimental testdata are needed to further verify the quantitative results of such viscoelastic damage modellingeffort.

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