Stochastic micromechanical model of the deterioration of ... · A stochastic modeling technique based on ... the asphalt microstructures used in the simulation. Next, the basis of

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. (2010)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.943

Stochastic micromechanical model of the deterioration of asphaltmixtures subject to moisture diffusion processes

Silvia Caro1,∗,†,‡, Eyad Masad1, Mauricio Sánchez-Silva2 and Dallas Little1

1Zachry Department of Civil Engineering and Texas Transportation Institute, Texas A&M University,College Station, TX 77843, U.S.A.

2Department of Civil and Environmental Engineering, University of Los Andes, Bogota, Colombia

SUMMARY

The deleterious effect of moisture in the microstructure of asphalt mixtures, usually referred to as moisturedamage, has been recognized as a main cause of early deterioration of asphalt pavements. The initiationand evolution of moisture-related deterioration is strongly influenced by the internal air void structureof asphalt mixtures. Despite its importance, the majority of works conducted on the micromechanicalmodeling of asphalt mixtures overlook the role of the air void structure, mainly because of its highcomplexity and variability. This paper explores the influence of air void variability on the performanceof asphalt mixtures subjected to moisture diffusion processes. A stochastic modeling technique based onrandom field theory was used to generate internal distributions of physical and mechanical propertiesof the asphalt matrix of the mixture that depend on probable air voids distributions. The analysis wasconducted by means of a coupled numerical micromechanical model of moisture damage. The resultsshowed that the variability and distribution of air voids are decisive in determining the moisture-dependentperformance of asphalt mixtures. Furthermore, it was also shown that a stochastic characterization ofthe diverse air void configurations is a feasible and useful approach to better represent and understandmechanically related deterioration processes in asphalt mixtures. Copyright � 2010 John Wiley & Sons,Ltd.

Received 1 April 2010; Accepted 13 April 2010

KEY WORDS: asphalt mixtures; moisture damage; random fields; micromechanics; stochastic process;air voids; diffusion

1. INTRODUCTION

Asphalt mixtures are viscoelastic composite materials comprised of aggregates (i.e. crushed rockswith different sizes and proportions), asphalt binder, and air voids. This material is used in theconstruction of surface and base layers of the road infrastructure. The initiation and evolution ofdistresses in asphalt pavements result from the combination of traffic loading and environmentalchanges. Within this context, the deleterious effects of moisture on the structural integrity of asphaltmixtures have been recognized as a primary cause of early damage to asphalt pavements [1]. Thisphenomenon, usually referred to as moisture damage, is defined as the progressive loss of structural

∗Correspondence to: Silvia Caro, Universidad de Los Andes, Bogotá, Colombia.†E-mail: [email protected]‡Assistant Professor.

Contract/grant sponsor: National Science Foundation; contract/grant number: CMS-0315564Contract/grant sponsor: Federal Highway Administration

Copyright � 2010 John Wiley & Sons, Ltd.

S. CARO ET AL.

integrity of the mixture that is primarily caused by the presence of moisture—in liquid or vaporstate—within the microstructure.

The microstructure of an asphalt mixture can be represented as a phase of coarse aggregatesembedded in a bulk asphalt matrix. The bulk matrix is comprised of the remaining components ofthe mixture, i.e. asphalt binder, the fine portion of the aggregates and the air void phase, which isusually called Fine Aggregate Matrix (FAM). The presence of moisture within this microstructurehas two main effects: (1) degradation of the quality of adhesion between the coarse aggregatesand the bulk matrix and (2) modification of the viscoelastic material properties of the FAM.These two processes are responsible for adhesive and cohesive degradation, respectively. In theparticular case of adhesive deterioration, Hefer et al. [2] explain the existence of thermodynamicand chemically favorable processes that are responsible for water to naturally disrupt the bondbetween the aggregates and the asphalt binder. The existence of such processes suggests thatadhesive deterioration is the main contributor to the overall structural degradation of the mixturein the presence of moisture.

In the process of understanding the causes of moisture damage, several studies have focusedon the analysis of the role of the internal air void structure on the initiation and evolution ofdamage [3–5]. These studies have verified the significance of the void structure as the indispensablecomponent for moisture to permeate the microstructure of the mixture and initiate moisture-induced deterioration. Furthermore, some of those studies showed that the total air void contentof an asphalt mixture (i.e. the ratio between the volume of voids to the total volume of themixture expressed in percentage) is not a sufficient parameter to characterize this structure sinceother characteristics, such as the air void sizes and spatial distribution, their connectivity and thetortuosity of the air void paths, also influence the performance of the material under the presence ofmoisture [3, 4].

There have been two approaches to account for the air void phase in the numerical modelingof asphalt mixtures at the microscale level: (1) as part of the viscoelastic bulk matrix, FAM, or(2) as an independent or separate entity. The first approach is the most common way to accountfor the air void phase. In this case, the FAM is modeled as a continuum with effective materialproperties that result from the characteristics of its constitutive components [6, 7]. Thus, the airvoid phase is assumed to be uniformly distributed within the bulk matrix and the characteristics ofthe internal void structure and its variability within the matrix are neglected. The second approachis more desirable since it explicitly accounts for the actual internal distribution of this phase, but thecomplexity of the void structure makes its numerical implementation a difficult and time-consumingtask [8].

This paper presents a new approach to incorporate the effect of the air void phase in themicromechanical modeling of asphalt mixtures. This approach relies on dividing the bulk matrix ofthe mixture into different, equal-size sections and assigning each section an air void content valueaccording to probable internal void distributions within the mixture. This information is furtherused to assign physical and mechanical material properties to the individual FAM sections. In thisway, the FAM phase indirectly accounts for the influence of the presence and variability of theinternal void structure on the mechanical performance of the asphalt mixture. The main advantageof this new approach is that its numerical implementation is simpler than the explicit inclusion ofvoids within the microstructure. In addition, the internal distribution of air voids can be formulatedto represent various asphalt mixture types.

The air void content in each section of the FAM was modeled using a stochastic technique thatis based on random field theory. A random field is a generalization of a stochastic process in whicha parameter (e.g. material properties in a medium or pixel data from an image) is mapped into aspace after taking into consideration its spatial correlation and variability [9]. Random field theoryhas been widely used in geotechnical engineering to model the variability of material propertiesin soils; the reader is referred to Kim [10] and Fenton and Griffiths [9] for a comprehensivereview. The principles of this theory have also been applied to characterize and model physicaland mechanical processes in multiphase materials at different length scales [11–13] and to conductcomplex statistical image analysis, including pattern recognition techniques, with applications in

Copyright � 2010 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2010)DOI: 10.1002/nag

STOCHASTIC MICROMECHANICAL MODEL

a wide range of areas [14]. In asphalt pavement engineering, the application of this theory hasbeen limited. Some examples include the detection and characterization of surface distresses basedon image analysis [15], the modeling of the surface roughness of pavements and its effects onstochastic pavement loads [16], and the modeling of the variability and distribution of tire-pavementcontact loads [17]. This probabilistic-based technique, however, has not been used in the past tomodel the variability and distribution of material properties within an asphalt mixture or to estimatethe impact of such variability on the mechanical performance of this material.

The main objective of this study is to employ random field theory to investigate the effect ofair voids variability on the mechanical performance of asphalt mixtures subjected to the combinedaction of moisture diffusion and mechanical loading. The analysis was conducted using a microme-chanical model of moisture-induced damage in asphalt mixtures. This model integrates the effectof moisture diffusion with the mechanical cohesive and adhesive degradation processes that occurwithin the mixture.

Part one of this paper presents the micromechanical model and the geometric characteristics ofthe asphalt microstructures used in the simulation. Next, the basis of random field theory and theassumptions applied to generate internal air void distributions are presented. This is followed bythe description and characterization of the material models and properties used for each componentof the asphalt mixture. In this section of the paper the methodology utilized to relate the air voidsdistributions to the material properties of the FAM phase is explained. Finally, the paper presentsthe approach used to study the mechanical performance of the microstructures (i.e. characteristicsof the moisture diffusion and mechanical loading processes) and discusses the results obtainedfrom the numerical simulations.

2. COUPLED MICROMECHANICAL MODEL

A two-dimensional coupled micromechanical model of moisture-induced damage in asphaltmixtures was used in this study to investigate the effect of air voids distributions and variabilityin asphalt mixtures. The model, which has been implemented in the finite element (FE) softwareAbaqus�, couples the effects of moisture diffusion processes taking place within the microstructureof the mixture with the mechanical response of the material. Cohesive deterioration in the asphaltmatrix is accounted for by making the viscoelastic mechanical properties at each point withinthe matrix dependent on the amount of moisture diffused into the system. Adhesive damageprocesses are modeled by including special interfacial zones between the coarse aggregates andthe FAM phase. These interfacial zones, herein named adhesive zones, are characterized bymoisture-dependent mechanical properties which are used to represent the adhesive bond betweenboth materials. The adhesive zones have the ability to simulate initiation and propagation ofinterfacial cracking, and were integrated in the model by means of the Cohesive Zone Modeling(CZM) technique.

The CZM is considered as an efficient numerical methodology to simulate gradual fractureprocesses taking place between any two surfaces. The principles of this technique were derivedfrom the concepts introduced by Baremblat [18] and Dugdale [19] who proposed the existenceof a small traction-resistant zone in front of the crack tip. This zone was named the cohesivezone and represents the traction forces that are required to cause bulk materials to separate [20].The CZM model has been widely applied to simulate cohesive and adhesive fracture processesin diverse types of materials. This technique was selected to simulate moisture deterioration ofadhesive zones in asphalt mixtures because it has proved to be successful in integrating fractureprocesses within adhesive systems that are subjected to moisture [21, 22].

The main component of the CZM technique is the traction–separation law. This law determinesthe response of the adhesive elements prior to and after initiation of damage. In order to do so,the traction–separation relationship contains four parameters: (1) the mechanical response of theelements before reaching a damage initiation criterion (from �0 to �init in Figure 1), (2) a damageinitiation criterion (at [�init,�max]), (3) a damage evolution criterion (from �init to �fin), and (4) a


S. CARO ET AL.

n,s

n,sin

max

Linear Elastic:

fin

Prior todamage initiation

Damageevolution

CrackPropagation

Adhesive Bond Strength

=0

Quadratic nominal strain condition

Released energy according to Gc

Damage initiation criterion:

Figure 1. Traction–separation law characterizing the mechanical response of the adhesive zones(for modes I and II of failure).

final damage condition criterion (�fin). Once the damage initiation criterion is reached, the fractureenergy of the element is released. A crack initiates (i.e. an adhesive element is removed) whenthe final damage criterion is achieved. The mechanical loading that this element was previouslyable to resist is then transmitted to the surrounding elements, promoting the propagation of thecrack through the interface. When the material is subjected to mixed-mode conditions, a traction–separation law is required to characterize each mode of fracture (i.e. mode I for opening andmode II for shear, in a two-dimensional problem). In these cases, the definition of a global criticalfracture energy released rate (GC) is also required in order to define the rate at which the fractureenergy of the individual modes should be released.

The traction–separation law illustrated in Figure 1 was selected for this study. In this Figure 1,tn and ts are the nominal normal and shear stresses, εn and εs are the nominal normal and shearstrains (i.e. relative displacement between the opposite faces of the adhesive element, �, dividedby the thickness of the adhesive layer), Knn and Kss are the components of the elastic stiffnessmatrix, ε0n and ε0s are the peak values of the nominal strains at which separation processes initiate,GC

n and GCs correspond to the normal and shear critical energy released rates, GC represents the

overall critical energy released rate of the mixed-mode condition, and the brackets in the firstterm of the damage initiation criterion symbolize the Macaulay condition, i.e. pure compressionconditions do not produce damage. For simplification, it was assumed that the traction–separationlaw in Figure 1 characterizes both failure modes (i.e. modes I and II). More details regarding thecharacteristics of the adhesive zones can be found elsewhere [23].

3. GEOMETRY OF THE MICROSTRUCTURE MODEL

Figure 2 illustrates the geometry of the 50mm× 50mm representative volume element (RVE)of an asphalt mixture selected for this study. The size of the RVE was determined based on astudy conducted by Kim et al. [24], who found that a 50mm× 50mm RVE size can be usedto represent the overall mechanical response of typical dense-graded Superpave mixtures. Theselected microstructure (Figure 2) was captured from a Superpave dense-graded asphalt mixture.It contains a total of 241 coarse aggregates that constitute 49.4% of the total area.

The asphalt mixture microstructure was modeled as a two-phase material comprised of coarseaggregates and a bulk matrix (FAM). The coarse aggregates include all particles with a diameterlarger than 0.42mm (i.e. sieve number 40), whereas all particles passing sieve number 40 wereassumed to be part of the FAM.

The FE mesh used to represent the two components of the microstructure is illustrated in theright-hand side of Figure 2. The typical length sizes of the elements were 0.5mm for the coarse



Original image Simplified image

Detail of the finite elements mesh

5 cm

5 cm

Figure 2. Section of the asphalt mixture used for the microstructure model and finite element mesh.

aggregates and 0.3mm for the FAM. The FE representation of the aggregates–FAM interfacesconsisted of thin layers of 0.05mm thickness that were inserted between the coarse aggregatesand the asphalt matrix. These thin layers are comprised of rectangular adhesive elements withdimensions of 0.05mm× 0.2mm that were designed to perfectly fit the geometry of the surroundingmaterials. Normal and transverse direction definitions were also assigned to each adhesive elementduring the meshing process in order to guarantee a correct representation of the opening and shearfracture modes during the mechanical simulations [25].

4. STOCHASTIC GENERATION OF THE AIR VOIDS INTERNAL DISTRIBUTION

Existing measured field air void distributions and variability data were used to generate realiza-tions of arrangements at the interior of the mixture’s microstructure. In the generation of thesearrangements, random field theory was used to treat the air void content as a random vector insteadof a constant-deterministic value. Thus, the stochastic technique provided a collection of spatiallycorrelated random values of air void contents that were mapped into a physical space, i.e. theasphalt mixture. The main advantage of applying random field theory to generate probable air voiddistributions is that it efficiently captures the complexity associated with this phase by includingboth its variability and spatial correlation. Details regarding the specific stochastic methodologyadopted for this study, including a formal definition of the spatial correlation concept, are presentedin the following sections.

The stochastic methodology used to generate diverse air voids distributions was designed toreflect the characteristics of a compacted asphalt layer in the field. Although the compactionprocess always aims at a pre-specified target value of air void content (e.g. 7%), the percent ofair voids within the asphalt course is not constant [26]. The variability of the air voids withinthe asphaltic layer is mainly determined by the characteristics of the mixture (e.g. physical andvolumetric properties, morphological and gradation characteristics of the coarse aggregates, etc.)and by the compaction method [27]. Therefore, if several field cores are extracted at differentlocations within the asphalt layer, a variation of air void content is expected. If the compactionprocess was conducted under strict quality control procedures, the average value of the air voidscontent from the different cores should be close to the target total air void content. However, evenunder these circumstances, the internal distribution of the air voids within each core would varyaround the target value due to the inhomogeneous distributions of the mixture components andboundary conditions during compaction. An illustration of variability in air voids in the field isdemonstrated in Figure 3.

The stochastic technique used in this paper takes into consideration the two different types ofvariability of air voids illustrated in Figure 3:

1. the variability of the air voids within the asphalt layer (i.e. the difference among the total airvoid content within the asphalt layer) and

2. the internal distribution of the air voids within the microstructure of the mixture (i.e. internalair void distribution within a particular location).


S. CARO ET AL.

1 542

AV %

dept

h (m

m)

3

AV % AV %AV % AV %

6.7% 5.9% 6.7% 7.7% 7.9%

Realization number

1 2 3 4 5% AV in field core

Internalvoiddistribution

Naturalvoidvariabilityin the field

Mean

7.0%

Figure 3. Probable realizations of the air voids content and spatial distribution within asphaltmixtures (AV,air voids content).

The stochastic technique is capable of producing realistic realizations of the air void distributionswithin an asphalt layer—each one containing a unique internal void structure—similar to thosethat would be obtained from randomly selected field cores (Figure 3).

4.1. Random field generation

Discrete random field generation methods divide the space of interest into several sections. In thispaper, the FAM phase of the asphalt mixture was divided into 100 elements (Figure 4). A realizationof the stochastic process consists of assigning a randomly generated value of a parameter P (e.g.hydraulic conductivity, porosity, strength, etc.) to each section. As each phase of the mixture hashigher probability of extending over an area longer than one element, adjacent elements are morelikely to have similar values of the parameter P than the values of elements that are further apart.Therefore, the value of P assigned to a given section should be somehow correlated with thevalues of P in neighbor cells. The distance at which this similarity is expected (i.e. the length ofinfluence of the parameter P) is called the autocorrelation distance or correlation length of therandom field, LT. In addition to the correlation length, a random field is also characterized bythe mean value of P and a dispersion measure, e.g. standard deviation. These parameters (meanand dispersion measure) are determined experimentally. The average value of P and its variabilityamong different realizations always converge to the mean and standard deviation of the randomfield.

A commonly used random field generation approach was proposed by El-Kadi andWilliams [28],who applied the covariance matrix decomposition technique as part of a method to generatetwo-dimensional autocorrelated distributions of the selected parameter, P , within a fixed space.The method proposed by El-Kadi and Williams [28] is based on two main assumptions: (1) theparameter of interest, P , has a normal or lognormal distribution within the space of interest and(2) P is characterized by an exponential autocorrelation function. The first assumption defines theprocess of determining probable values of P for each element of the space, whereas the seconddefines how the values of P at different locations are correlated. Both assumptions are supposedto be valid in characterizing the variability and distributions of air voids within the FAM phase ofthe mixtures.



Figure 4. 100 divisions of the FAM to which random values of air voids will be assigned.

The value of air voids in each realization is obtained by [28]:

AVn =Sln+ln (1)

where AV is a [100×1] vector containing the air voids content of the 100 elements of the FAM,S is the [100×100] autocorrelation matrix, e is a [100×1] vector containing standard normallydistributed values (i.e. with mean 0 and standard deviation 1), l is a [100×1] vector containingthe mean values of the internal air voids distribution within a particular microstructure, and thesubscript n refers to the results for the n-th realization of the random field. Note that the expectedvalue of Equation (1) is, effectively, the mean value of the random field:

E[AVn]=SE[εn]+ln =ln (2)

The autocorrelation matrix, S, can be found from:

C=SST (3)

where C is the covariance matrix of the field, which has to be positive definite. The matrix Scan be determined from C by using the Cholesky decomposition matrix technique. The Choleskytechnique is useful in decomposing a symmetric and positive definite matrix, such as C, into alower triangular matrix. Details of this methodology can be found elsewhere [29]. The covariancematrix C is defined as:

Ci, j =�2�i, j (4)

where Ci, j is the correlation function between the spatial points i and j , � is the standard deviationof the air void content in the field, and �i, j is the autocorrelation function between points i and j .Although the correlation function depends upon the problem at hand, several common modelshave been used extensively. Most common models include the polynomial decaying and differentexponential correlation functions [9]. For the purpose of this paper the following exponential formof the correlation function was selected:

�i, j =e− |di, j |

LT (5)

where di, j is the distance between points i and j , and LT is the autocorrelation distance, definedas the total area under the correlation function. For exponential correlation functions, as the oneused in this study, the autocorrelation length has been shown to be close to the distance where the


S. CARO ET AL.

spatial autocorrelation decays by 1/e (i.e. about 37%). If anisotropy is considered, the correlationof the random field becomes:

Ci, j =�2 exp

⎡⎢⎣−

√√√√(dxi, jLx

)2

+(dzi, jLz

)2⎤⎥⎦ (6)

where Lx and Lz are the autocorrelation distances in the x (i.e. horizontal) and z (vertical)directions, respectively, and dxi, j and d

zi, j are the horizontal and vertical components of the distance

between the points i and j (i.e. rectangular components of di, j ). It is noteworthy that large valuesof the autocorrelation distances (e.g. larger than the size of the space) will produce values of Ci, japproximately equal to the variance of the parameter in the field, which in turn results in a stronglycorrelated random field.

In summary, the following steps are needed to generate a realization of the air voids randomfield: (1) divide the FAM space into several subdivisions, (2) determine a mean value vector (ln),a standard deviation (�) and the autocorrelation distances (Lx and Ly) of the random field, (3)compute the standard normally distributed vector ε, (4) compute the covariance matrix C, (5)apply the Cholesky decomposition technique to determine the matrix S (Equation (3)), and (6)compute the realization of the air void random field by means of Equation (1). Notice that thegeneration of the random field (i.e. Equations (1)–(6)) captures the variability of both the total airvoid content (determined by its standard deviation, i.e. � in Equation (4)) and its spatial distribution(represented by the correlation function, i.e. �i j in Equation (5)).

4.2. Parameters of the random field

Regarding the actual variability of air voids content in the field, most state transportation agencies inthe United States have designed quality control methodologies to maintain the maximum allowablestandard deviation of the air voids in the range of 1–1.5% [30, 31]. This standard deviation isdetermined by experimentally measuring the total air void content of field cores specimens obtainedat different locations in the asphalt layer. The coefficient of variation (COV) of this property thatresults from those quality control specifications can then vary in a range of 10–20%. Nevertheless,studies conducted in the field to assess the influence of diverse compaction patterns in the pavementhave shown that the variability of the air voids in a compacted asphalt layer can reach COV valuesof about 30% [27].

In order to analyze the impact of the air void content variability on the moisture-related mechan-ical performance of an asphalt mixture, three different cases were considered. In all cases, it wasassumed that the microstructure model belongs to an asphalt course characterized by a mean valueof air void content of 7%, which is a common target for dense-graded mixtures in the field. Thethree cases considered COVs of air voids equal to 10, 20, and 30%. It is expected that the vari-ability of air voids in most asphalt layers will be close to the first two cases, whereas the third caserepresents an extreme case of variability. It is noteworthy that the use of a Gaussian distributionfor the generation of the air void contents for the random field is a valid assumption since theprobability of having negative values of this parameter in the most critical condition of variability(i.e. a COV of 30%) is close to zero (i.e. 0.00042). Also, notice that each COV represents thevariability of the air void content on a pavement in the field and not the variability of the air voidswithin a particular microstructure (i.e. internal air void distribution within a field core).

Although the mean value of the air voids random field was selected to be 7%, the actual internalair void distribution within a sample of an asphalt mixture is not constant. Figure 5 shows theair void distributions within different asphalt field cores that were obtained by means of X-rayComputed Tomography (X-ray CT) and image analyses techniques [33, 32]. It can be observedthat, in general, the air voids content at the surface of the pavement is always larger than that in themiddle and bottom sections of the asphaltic layer. This information was used to generate randommean vectors (ln in Equation (1)) that will characterize the internal void distribution within asample.



Air Voids (%)

Dep

th (

mm

)

0 2 64 8 10 12 140

10

20

30

40

50

Air Voids (%)

)m

m(ht pe

D

0 2 64 8 10 12 14 16 18 200

102030405060(a) (b)

Figure 5. Examples of the air voids distribution at the interior of different asphalt mixtures in the field: (a)Air voids (%), modified after Tashman et al. [32] and (b) air voids (%), modified after Masad et al. [33].

2% 4% 6% 8% 10% 12%

Air Voids Content (%)

10

20

30

40

50

00% 7%

Air

Voi

ds (

%)

b=1.0b=0.25

AVz=25AVz=0

Figure 6. Characteristics of the random generation of vertical distributions of air voids. The distributionis used as the mean vector for each realization of the random field.

For simplicity, it was assumed that the distribution of the percentage air voids with respect todepth, AV(z), which defines the mean vector of a realization of the random field (ln), follows aparabolic function (i.e. a ‘C’ shape, Figure 6):

AV(z)=a∗(z−25)2+AVz=25 (7)

where z represents the depth of a particular location within the microstructure (in mm, z=0corresponds to the top of the microstructure), and a and AVz=25 are the two parameters that definethe shape of the vertical air void distribution. Note that AVz=25 in Figure 6 corresponds to the airvoids content at 25mm in depth (i.e. at the middle height of the microstructure and vertex of theparabola).

The vertical air void distribution was determined by randomly generating the parameter b(Figure 6), which was assumed to be uniformly distributed between 0.25 and 0.7, and by keepingthe mean value of the distribution in a constant value of 7%. The parameter b symbolizes theratio between the minimum air void content (AVz=25 in Figure 6) and the maximum air voidcontent (top and bottom of the sample, AVz=0 in Figure 6) in the microstructure, as expressed byEquation (8). The range of variation of this parameter was obtained based on the shapes of actualair void distributions within diverse field cores as those shown in Figure 5. Values of b close to1.0 are associated with more uniform vertical distributions of voids, whereas values close to 0are associated with larger dispersion of the internal void structure. The shape parameters a andAVz=25 are unique for each value of b, and are determined by:

b= AVz=25

AVz=0= AVz=25

a∗252+AVz=25(8)

It is important to note that although the vector of vertical mean values of voids, ln , was assumedto be parabolic and its mean value was forced to a constant value of 7%, the actual distribution that


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Figure 7. Example of the air voids content characterizing the 100 divisions of the FAM that resulted froma stochastic realization of the random field.

results from a stochastic realization (Equation (1)) will not necessarily have this shape or meanvalue. Finally, the correlation of the air voids within the FAM was assumed to be anisotropic.This assumption is motivated by the influence of field compaction methods in making the air voidcontent of elements located at the same depth to be more correlated than the air void content ofelements at different vertical locations. Therefore, it was assumed that the horizontal autocorrelationdistance of the air voids within the FAM (Lx in Equation (6)) was 35mm, whereas the verticalautocorrelation distance (Lz in Equation (5)) was set to 10mm.

The random field generation technique was implemented in Matlab� and 15 realizations wereproduced for each of the three cases considered (i.e. asphalt layer with a COV of air voids of 10,20, and 30%). A typical realization of the air voids random field within the FAM is illustrated inFigure 7.

5. MATERIAL PROPERTIES AND MODELS

5.1. Fine aggregate matrix—moisture diffusion coefficient

Moisture diffusion coefficients within the FAM for each realization were determined based onthe randomly generated internal air voids distribution (Figure 7). In order to do so, a relationshipbetween the air voids content of the FAM and its corresponding moisture diffusion coefficient isrequired. Moisture diffusion coefficient data for FAMs found in the literature varies widely amongdifferent sources [3, 34, 35]. This is caused by differences among specimens studied in regard tothe maximum size, gradation and mineralogical nature of the aggregates, and type and amount ofasphalt binder. Furthermore, in most cases the total amount of air voids within the specimens—amain factor in determining the diffusivity of the mixture—is not reported.

The effective moisture diffusion coefficient of FAMs as a function of the total percent of airvoids can be numerically estimated if the diffusion coefficients of the individual phases of thematrix are known. Therefore, a two-dimensional FE model in Abaqus� was constructed in whichthe FAM was represented as a two-phase composite comprised of a portion of the matrix that isfree of voids and another portion characterized only by air voids. The moisture diffusion coefficientof the air phase was adopted from Geankoplis [36]. The moisture diffusion coefficient of the FAMportion with no voids was backcalculated by iteratively modifying its value until the effectivediffusion coefficient of the composite matrix was the same as the values reported by Kassem et al.[34] for FAM specimens with 14% air voids. Air voids were represented as circles of random sizesand spatial distribution within the FAM sample. Then, the diffusion coefficient of the FAM phasefree of voids was estimated as the mean coefficient of a set of simulations with random air voiddistributions. Moisture diffusion coefficients of the two individual phases were used to numericallyestimate the moisture diffusivity of FAMs containing different air voids contents (DFAM). Theresults showed the following good fit:

DFAM=3.7×10−7AV+1.25×10−7 for AV∈ [0%,20%] (9)



where the moisture diffusion coefficient has units of mm2/s. This equation was used to transformthe internal air void distributions obtained from each stochastic realization into random values ofthe moisture diffusion coefficient of the FAM phase.

5.2. Fine aggregate matrix—mechanical response

The mechanical response of the FAM phase was assumed to be linear viscoelastic. Data collectedby Kim et al. [7] was adopted to characterize the viscoelastic response of a FAM materialcontaining 7% air voids. In terms of Prony Series, the uniaxial relaxation modulus characterizingthe viscoelastic response of the asphalt matrix can be expressed as:

E(t)= E0−9∑

i=1Ei [1−exp(−t/�i )] and E∞ = E0−

n∑i=1

Ei (10)

where E0 is 1.12×108Pa, E∞ is 1.22×102Pa, and the values of Ei and �i are the parametersobtained after fitting the uniaxial relaxation modulusmaster curve into a Prony Series. The Poisson’sratio was defined time-independent and equal to 0.4.

Two different cases were considered for the viscoelastic properties of the FAM:

• Case A: The instantaneous relaxation modulus, E0, is a function of the internal air voidsdistribution of the FAM.

• Case B: The viscoelastic material properties are constant within the FAM and are equal tothose of a FAM with 7% air voids [7].

Note that the only difference between Cases A and B is the assumption made on the distributionof the mechanical properties of this material. Both cases are characterized by the same FAMmoisture diffusivity distributions and will produce the same internal moisture profiles withinthe mixture. However, Case A provides a better representation of an actual mixture, since theinfluence of the air void structure on the mechanical resistance of asphalt mixtures has been widelyrecognized [8, 31, 37].

To the extent of our knowledge, there is no published information regarding the relationshipbetween air voids content and the viscoelastic response of FAM materials. Consequently, experi-mental and numerical data regarding the effect of air voids on the resistance of asphalt mixturesreported by Epps et al. [31], Kassem [37], and Caro et al. [8] were used to assume the shapeof the relationship between the initial viscoelastic instantaneous modulus and the percentage airvoids for the FAM phase:

E0=−5.0×1010AV3+8.0×109AV2−9.0×108AV+1.54×108 for AV ∈ [4%,15%] (11)

where the units of E0 are Pa. This relationship was used in the generation of microstructures inCase A. As an example, Figure 8 presents the average vertical air voids and instantaneous modulusdistributions for the realization illustrated in Figure 7.

The moisture dependency of the mechanical response of the FMA was accounted for by linearlydecreasing the relaxation modulus as a function of moisture content. The relaxation modulus of thebulk matrix at any time t was set to change from its original value—dry condition—to 90% of thisvalue under fully saturated conditions. This moisture-dependent relationship was selected in thisstudy because it describes what has been observed to be the effect of moisture on the viscoelasticresponse of some polymer composites [38]. The Poisson’s ratio of the FAM was assumed to bemoisture independent.

5.3. Coarse aggregates and adhesive zones

The moisture diffusion coefficient of the aggregates was assumed to be 1.28×10−3mm2/s basedon data reported for rocks [39]. Aggregates were modeled as a linear elastic material with an elasticmodulus, EA, of 5×104MPa, and a Poisson’s ratio, �A, of 0.3. The properties of the aggregateswere assumed to be independent of the moisture content.


S. CARO ET AL.

0

5

10

15

20

25

30

35

40

45

50

4 6 8 10 12 14 16

Air Voids (% )

Dep

th (m

m)

0

5

10

15

20

25

30

35

40

45

50

4 44 84 124

E0of FAM (MPa)

Dep

th (m

m)

Figure 8. Vertical average distribution of the air voids and instantaneous modulus of the FAMfor the realization shown in Figure 7.

Table I. Original mechanical properties of the microstructure’s components.

Aggregates Adhesive zones∗

Response Linear elastic Linear elastic prior to damage initiationExponential stiffness softening after damage initiation

Properties EA=5.0×1010 Pa Knn and Kss=4.48×107 Pa�=0.3 ε0n =ε0s =0.1mm/mm

Gcn=Gc

s =Gc=4.49×10−2 N/mm

∗The parameters of the traction–separation law for the adhesive zones listed here correspond to those presentedin Figure 1.

The moisture diffusion coefficient of the FAM was used to characterize moisture diffusionwithin the adhesive zones. The stiffness of the adhesive zones before the initiation of damagewas made comparable to that of the viscoelastic bulk by making the elastic stiffness of theadhesive elements equal to the average value of the relaxation modulus of the FAM duringthe time period of mechanical loading application. The peak values of the nominal strains (i.e.εn and εs in Figure 1) were estimated to be 0.1mm/mm. This value was adopted based onpreliminary experiments on the adhesive fracture in systems composed of thin asphalt binderlayers (i.e. 5�m) on metal substrate [40]. The fracture energy corresponding to both modesof failure was estimated to be 15% of the area within the elastic region of the traction–separation law.

Studies conducted on polymer–metal adhesive joints have demonstrated that moisture has twomain effects on the mechanical response of those interfaces: (1) it reduces the overall strength ofthe adhesive bond and (2) it reduces the fracture toughness of the adhesive system. Similar effectsare expected to occur on aggregate–FAM interfaces that are exposed to moisture. Therefore, themoisture dependency of the traction–separation law was accounted for by exponentially reducingthe stiffness of the elements prior to damage initiation (Knn and Kss in Figure 1) from its originalvalue, dry condition, to zero, saturated condition. It was also assumed that the peak values of thenominal normal and shear strain (εon,s in Figure 1) decrease linearly as a function of the moisturecontent from their original value, in the absence of moisture, to 80% of that value under saturatedconditions.

Table I summarizes the responses and mechanical material properties of the coarse aggregatesand adhesive zones.



6. FINITE ELEMENT MODELING

The modeling methodology consisted of subjecting the 45 microstructures (i.e. 15 for each case ofair voids variability in the field) to a 10-day moisture diffusion conditioning process. Afterwards,the microstructures were subjected to a force-controlled test that consisted of loading and unloadingthe microstructure for a total of 30 s with a monotonic load applied at a rate of 0.28N/s; Figure 9illustrates this process.

The methodology illustrated in Figure 9 was implemented in Abaqus� by means of a two-stepsequentially coupled technique. In the first step, transient moisture diffusion within the microstruc-ture was simulated and no mechanical loading was applied. The moisture profiles that were obtainedthrough time from this stage were used as prescribed conditions in the second step of the simula-tion technique. In this subsequent step, the mechanical response of the system was modeled aftermaking the mechanical properties of each FE to be dependent on the amount of diffused moisture.

Moisture diffusion was modeled using Fick’s second law with an implicit numerical scheme.The boundary conditions were fixed and consisted of a normalized moisture concentration of 1.0that was uniformly distributed along the perimeter of the microstructure. The normalized moistureconcentration is defined as the moisture concentration within the material divided by the maximummoisture concentration that the asphalt matrix can absorb (i.e. values of 0 and 1 indicate dryand full saturation conditions, respectively). Diffusion-based elements type DC2D4 in Abaqus�

were used to model the adhesive zones, and elements type DC2D3 were used for all the othercomponents of the model.

After the simulation of moisture diffusion was completed, the application of mechanical loadingwas modeled by means of an implicit scheme. Although the same FE mesh was used in both thediffusion and mechanical simulations, the diffusion-based elements were replaced in this secondstage by mechanical-based elements (i.e. elements type COH2D4 in Abaqus� for the adhesivezones and elements type CPE3 for all the other components of the model).

7. RESULTS AND DISCUSSION

7.1. Air void total content and distribution within the microstructures

Table II lists the mean values of the air voids content corresponding to the 15 stochastic realizationsobtained for each one of the three cases considered (i.e. field air voids with COV of 10, 20, and30%). Data in Table II verifies that a consequence of the stochastic methodology is that althoughthere is dispersion in the total air void content of the individual realizations, the average air voidscontent among all realizations is very close to the target value of the asphalt layer (i.e. 7%).

It should be noted that the mechanical performance of two microstructures containing the samevalues of total air voids content is not necessarily the same. This is due to the fact that theinternal distribution of the air voids within the FAM is not identical in any two microstructures.

Moisture Vapor

10-day moisture conditioning

F

Fracture Test

F

Time15 s 30 s

4.2 N

Figure 9. Representation of the modeling methodology.


S. CARO ET AL.

Table II. Total air voids content for the stochastic generated distributions for asphalt mixtures containingdifferent levels of spatial variability in the field.

Air voids Total air voids content (%) for each stochastic realization (i.e. microstructures)variabilityin the field 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

COV 10% 6.66 7.16 6.28 6.37 7.75 6.87 7.83 6.31 7.12 6.79 6.45 7.55 5.92 7.51 8.41Mean value: 7.00%COV: 10.03%



Figure 10. Internal air void distributions for two microstructures with similar total air voids contents(microstructures corresponding to field air void COV of 20%).

For example, realizations 1 and 9 in the second case of Table II (i.e. COV of air voids in the fieldequal to 20%) have very similar total air void content: 6.01 and 6.1%, respectively; however, theinternal distribution of the voids within the bulk matrix is dissimilar, as observed in Figure 10.Therefore, some differences can be expected in the mechanical response of both microstructuressince they will not produce the same moisture profiles or have the same mechanical resistance.

7.2. Mechanical performance of the microstructures

Figure 11 illustrates a typical force vs displacement curve that characterizes the mechanical responseof one of the microstructures after the mechanical test. Figure 11 also depicts the moisture profileat the interior of the microstructure after 10 days of moisture diffusion, the maximum deformationstate of the sample during the mechanical test, and some details of the adhesive failure occurring at



Displacement (mm)

)N(

ecroF

0 0.2 0.4 0.6 0.8 1.0 1.20

0.51

1.5

2.5

3.5

4.5

2

3

4

K

DE

1

1.0

0.0

0.5

NormalizedMoistureContent

(a) (b)

Figure 11. (a) Maximum deformation of the microstructure, moisture profile, and adhesive failures and(b) force vs displacement curve for one microstructure.

Table III. Statistical results of the total dissipated energy and overall stiffness for the three cases considered.

Variability in Mechanicalin the air voids properties of Standard

field FAM∗ Mean deviation (�) COV (%)

Dissipated COV 10% Case A 1.40 4.04×10−2 2.89energy Case B 1.41 1.94×10−2 1.37(DE, mJoules) COV 20% Case A 1.40 7.28×10−2 5.21

Case B 1.41 3.17×10−2 2.25COV 30% Case A 1.43 1.09×10−1 7.62

Case B 1.43 4.39×10−2 3.08

Overall COV 10% Case A 4.33 1.31×10−1 3.02Stiffness Case B 4.29 4.83×10−2 1.12(K , N/mm) COV 20% Case A 4.35 2.27×10−1 5.21

Case B 4.30 7.61×10−2 1.77COV 30% Case A 4.25 3.34×10−1 7.86

Case B 4.31 1.61×10−1 3.73

∗Case A: The mechanical properties of the FAM change according to the air voids distribution within the asphaltmatrix. Case B: The mechanical properties of the FAM are held constant.

the aggregate–FAM interfaces. The total energy dissipated by the composite, DE—defined as thearea within the force vs displacement curve in Figure 11—and the overall stiffness, K—definedas the maximum applied force divided by the maximum displacement in the same figure—wereselected as the two evaluation parameters by which the role of the air voids variability in themechanical performance of the microstructures was studied.

Table III summarizes the mean value, standard deviation, and COV of the total dissipated energyand stiffness corresponding to the 15 realizations in each one of the three cases considered. Thetable includes the results of the cases in which the mechanical properties of the FAM changedwithin the model according to the corresponding air voids content distributions (Case A), as wellas the cases in which the viscoelastic properties of the FAM were held constant and equal to theresponse of a viscoelastic matrix with 7% air voids (Case B).

As expected, an increase in the variability of the air void content within the asphalt mixtureincreases the variability of DE and K . Data in Table III show that the COV of DE and K in theasphalt mixture is between 25 and 30% of the magnitude of the overall variability of air voidcontents in the field for the microstructures in Case A. These variability ranges are in reasonableagreement with those obtained from applying an empirical stiffness performance model obtainedfrom regression analysis of experimental data [31]. The application of the regression model byEpps et al. [31] shows that the COV of the stiffness of an asphalt mixture is 40% of the COV ofthe field air voids.


S. CARO ET AL.

1.75

1.25

1.35

1.45

1.55

1.65

5.5 6.5 7.5 8.5 9.5 10.5 11.5

DE

(m

J)

Mean AV (%)

Case A

Case B

COV air voids in field 10%

1.25

1.35

1.45

1.55

1.65

1.75

4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5

DE

(m

J)

Mean AV (%)

Case A

Case B


1.25

1.35

1.45

1.55

1.65

1.75

3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5

DE

(m

J)

Mean AV (%)

Case A

Case B


(a) (b) (c)

Figure 12. Relationship between the average total value of air voids in the microstructure and the dissipatedenergy for asphalt layers with field air void content COV of (a) 10%, (b) 20%, and (c) 30%. Mean AV

refers to the mean value of air voids content within each microstructure or stochastic realization.

Data in Table III also demonstrate that the variability of the performance parameters (DE and K )of the microstructures in Case B (constant mechanical properties for the FAM) are always smallerthan the variability observed for the same parameters of the microstructures in Case A (variableFAM mechanical properties). The COVs of DE and K of the microstructures in Case B reflect thedispersion in the mechanical performance that is exclusively due to the effect of the air voids onthe moisture diffusion properties of the FAM. When the effects of the air void distributions on thestructural resistance of the mixture are also included (Case A), the variability of the mechanicalperformance rises by a factor of 2.

A careful analysis of the relationship that exists between the total air voids content of eachmicrostructure and their associated mechanical responses (i.e. DE and K ) for Cases A and B revealsinteresting information. Figure 12 illustrates such relationships from where two main observationscan be extracted:

1. The relationship between the total air voids content of the microstructures and the total energythat the material dissipates during the mechanical test follows a linear relationship. Thisconclusion is valid for microstructures tested using constant and variable internal viscoelasticmaterial properties of the FAM.

2. A critical air voids content exists that changes the relationship between Cases A and B. Belowthis threshold value, which is approximately 7.5% (vertical dashed lines in Figure 12), themicrostructures complying with the conditions of Case A demonstrated poorer mechanicalresponses than those having variable FAM viscoelastic properties. Above this threshold,however, mixtures of Case A performed better than those of Case B.

The first observation is important because it demonstrates the significance of air void contentsin affecting the mechanical performance of asphalt mixtures subjected to moist environments. Itis noteworthy that although these relationships provide a broad idea of the expected performanceof the mixture, the actual mechanical response also depends on the internal distribution of theair voids and not only on its total percentage of air voids. The second observation is interestingbecause it suggests that the validity of the assumptions about the internal viscoelastic propertiesdistributions within the bulk matrix (Cases A or B) depends on the amount of air voids present inthe mixture.

Microstructures in Case A are a better representation of a real asphalt mixture. However, itsnumerical implementation is complex, since the geometrical section representing the FAM must beseparated into several regions, each one having different moisture-dependent material properties.Furthermore, it was observed that the adhesive failure at the aggregate–FAM interfaces withinmicrostructures in Case A was more severe than that found in Case B. The initiation and propagationof adhesive cracks within the model are associated with numerical instability difficulties duringthe simulation. Therefore, a constant distribution of the mechanical properties of the bulk matrix(Case B) is desirable in order to minimize the difficulties associated with the numerical simulations.This assumption seems to be valid when the spatial variability of the field air voids is small



DE (mJ)

FD

C

COV 10%

COV 20%

COV 30%

K (N/mm)

FD

C

COV 10%

COV 20%COV 30%

(a) (b)

Figure 13. Cumulative probabilistic distribution of (a) the total dissipated energy and (b) overallstiffness (mixtures in Case B). The COV values correspond to the variability of the air voids content

of the asphalt mixture in the field.

(e.g. Figure 12(a)) and/or when the total air void content is closer to the mean value of the randomfield (between 7 and 8%). However, when there is a considerable variability of the field air voids(e.g. COV of the air voids in the asphalt layer larger than 20%) and the total air voids content of themixture is greater than 7.5%, this assumption can underestimate the actual structural susceptibilityof the mixture to develop damage in the presence of moisture (particularly adhesive damage).Similar results were found for the inverse linear relationship that exists between the total air voidcontent of the microstructures and the mixture’s stiffness (K ).

The probabilistic distribution characterizing the mechanical response of the microstructureswas explored based on the 15 realizations conducted for each of the three spatial field air voidsvariability cases. A best fit analysis of the three sets of data of DE and K was performed usingthe Crystall Ball� software. The results suggest that the dissipated energy and the stiffness followa three-parameter Weibull probabilistic distribution:

P(k)=��−1[(k−c)

�

]−exp

{−[(k−c)

�

]}(12)

where k represents either DE or K and �, �, and c are the parameters of the distribution, usuallydenominated shape, scale and locator, respectively. Figure 13 presents the cumulative probabilisticdistribution (CDF) for DE and K for the three field air voids considered and for those microstruc-tures with variable FAM viscoelastic properties. The shape of the three cumulative distributionsreflects the increase in variability of the air void content within the mixture (Table III). The impor-tance of knowing the probabilistic distribution is that it can be easily used to estimate the expectedresponse of an asphalt mixture when subjected to moisture diffusion and mechanical loading. Forexample, information from Figure 13(b) can be used to conclude that the probability that thestiffness of an asphalt mixture is smaller or equal to 4.4N/mm after being subjected to a 10-daydiffusion conditioning process is between 50 and 60%, independent of the air voids variabilitypresent in the field.

8. SUMMARY AND CONCLUSIONS

A stochastic-based methodology was successfully applied to generate diverse internal air voidsdistributions within the viscoelastic bulk matrix of asphalt mixtures (i.e. FAM). The internal voiddistributions were used to estimate physical and mechanical material properties within the FAMphase. This information was further used as an input in a micromechanical model of moisture-induced damage to evaluate the impact of such distributions on the mechanical performance ofasphalt mixtures subjected to moisture diffusion processes.

The results from the simulations showed that the spatial variability of the total air voids contentof an asphalt mixture in the field, as well as the internal distribution of those voids, plays animportant role in the mechanical resistance of the material. In fact, the results suggest that theCOVs of the energy dissipated by the mixture and its overall stiffness are 25–30% of the magnitude


S. CARO ET AL.

of the COV characterizing the air void content of the mixture in the field. This range was foundto be close to reported data on the relationship between the air voids variability and the expectedmechanical response of the mixtures in the field. However, it is important to note that this rangeonly provides a general idea of the expected performance of the material since more accurateestimates require collecting experimental data regarding the actual influence of air voids on thematerial properties of the FAM phase.

It can also be concluded that a direct relationship exists between the total air voids content ofthe microstructures and their mechanical performance. Besides, the use of FAMs with constantmechanical properties (independent of the internal air voids distribution of the mixture) wasobserved to be a fair simplification of the problem in those cases in which the dispersion of thespatial field air voids is low (e.g. COV below 15%) and/or when the total air void content of themicrostructure is in a range of 7–8%.

Finally, it was observed that Weibull distributions provide a good representation of the mechan-ical performance of the moisture-conditioned mixtures. This information is very useful to conductprobabilistic analysis on the expected response of an asphalt mixture layer that is characterized bycertain dispersion of the of air void phase.

The main advantage of the proposed stochastic methodology is that although it does not explicitlyinclude the actual air void structure within the microstructure, it considers the role of the air voidsdistribution and its variability on the susceptibility of the mixture to experience moisture-induceddamage. Thus, it constitutes an efficient way to evaluate the effects of compaction processes on themechanical performance of asphalt courses. Information about the impact of air void variability onthe performance of the mixtures can be further used as an input parameter in life-cycle analysesof pavement structures.

ACKNOWLEDGEMENTS

This research was performed while the first author was working as a graduate research assistant at TexasTransportation Institute (TTI)—Texas A&M University. The authors thank the National Science Foundationthrough grant CMS-0315564 for the funding provided to develop the initial part of the numerical modelused in this study. Funding provided by Federal Highway Administration through the Asphalt ResearchConsortium (ARC) to develop the stochastic model presented in this paper is highly appreciated.

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