Micromechanical finite element framework for predicting ... - CiteSeerX

Materials and Structures DOI 10.1617/s11527-007-9303-4

ORIGINAL ARTICLE

Micromechanical finite element framework for predicting viscoelastic properties of asphalt mixtures Qingli Dai Æ Zhanping You

Received: 4 December 2006 / Accepted: 27 August 2007 RILEM 2007

Abstract A micromechanical finite element (FE) framework was developed to predict the viscoelastic properties (complex modulus and creep stiffness) of the asphalt mixtures. The two-dimensional (2D) microstructure of an asphalt mixture was obtained from the scanned image. In the mixture microstructure, irregular aggregates and sand mastic were divided into different subdomains. The FE mesh was generated within each aggregate and mastic subdomain. The aggregate and mastic elements share nodes on the aggregate boundaries for deformation connectivity. Then the viscoelastic mastic with specified properties was incorporated with elastic aggregates to predict the viscoelastic properties of asphalt mixtures. The viscoelastic sand mastic and elastic aggregate properties were inputted into micromechanical FE models. The FE simulation was conducted on a computational sample to predict complex (dynamic) modulus and creep stiffness. The complex modulus predictions have good correlations

Q. Dai Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA e-mail: [email protected] Z. You (&) Department of Civil and Environmental Engineering, Michigan Technological University, Houghton, MI 49931, USA e-mail: [email protected]

with laboratory uniaxial compression test under a range of loading frequencies. The creep stiffness prediction over a period of reduced time yields favorable comparison with specimen test data. These comparison results indicate that this micromechanical model is capable of predicting the viscoelastic mixture behavior based on ingredient properties. Keywords Microstructure Micromechanical modeling Finite element method Asphalt mixture Viscoelasticity Complex modulus Creep stiffness

1 Introduction Heterogeneous asphalt mixtures comprise of graded aggregates bound with mastic (asphalt binder plus fine aggregates and fines). For such materials, the macro properties depend on the aggregate and mastic microstructure. Important micro behaviors related to mastic properties include: volume percentage, viscoelastic and viscoplastic responses, microcracking, and bonding strength. The microstructural features of aggregate include: mineralogy, elastic modulous, size, shape, texture, and packing geometry. Chen et al. [1] evaluated the internal structures of asphalt mixtures constituting of four different percentages of flat and elongated (F&E) aggregates, and studied how the engineering properties of these mixtures change in terms of rut depth, particle movement and

Materials and Structures

orientation, and strain. The experimental procedure used in their study relied on the two-dimensional (2D) images subjected to wheel loading. It was concluded that the low percentages of F&E aggregates result in a stable internal structure that could develop stone-on-stone contact and provide a better interlocking mechanism [1]. Because of the heterogeneous nature of asphalt mixtures, a micromechanical model for asphalt concrete mixtures is needed to study and characterize their properties. Micromechanical models can predict fundamental material properties based upon the properties of the individual constituents such as the mastic and aggregate. Micromechanical models have tremendous potential benefits in the field of asphalt technology, for reducing or eliminating costly tests to characterize asphalt-aggregate mixtures and design purposes. Since these models allow a more thorough examination of microstructural material behavior, such as strain distribution within the aggregate skeleton and asphalt matrix, they can ultimately provide a powerful tool for optimizing mixture design on the basis of mechanistic performance. The use of micromechanical models to predict properties of asphalt mixtures and mastics has drawn increasing attention over the past 10 years, and a number of approaches have been investigated. Asphalt mixture was investigated by non-interaction particle micromechanics models without specified geometry [2–6], as well as, with specified geometry [7–10]. Discrete element method (DEM) was employed on cemented particulate materials in recent years [11–18]. A 2D micro-fabric discrete element model (MDEM) concept was developed to predict the stiffness of asphalt mixtures [19–21]. The MDEM holds promise to balance the advantages of microstructural model resolution with all the benefits of the discrete element approach including the simulation of aggregate cracking, debonding between aggregates and mastic, and microcracking initiation and propagation. However complex contact laws need to be developed for predicting complex mastic constitutive behavior. In addition, it is a challenge to use circular element to simulate the irregular shaped aggregate. On the other hand, finite element modeling of asphalt concrete microstructure potentially allows

accurately modeling of aggregate and mastic complex constitutive behaviors and microstructure geometries. Research work has been conducted using FE techniques [22–31]. In addition, some research have been reported on the three-dimensional (3D) microstructure of asphalt mixture [29, 32–34]. A displacement discontinuity boundary element approach was applied in the modeling of asphalt mixtures [35]. An equivalent lattice network approach was developed and applied, where the local interaction between neighboring particles was modeled with a special frame-type FE [36–39]. A mixed FE approach was developed to study asphalt mixtures by using continuum elements for the effective asphalt mastic and rigid body defined with rigid elements for each aggregate [40]. A unified approach for the rateindependent and rate-dependent damage behavior was developed using Schapery’s nonlinear viscoelastic model. Properties of the continuum elements were specified through a user material subroutine within the ABAQUS code and this allows linear and damage-coupled viscoelastic constitutive behavior of the mastic cement to be incorporated. In these models, the microstructure of real asphalt materials was simulated with idealized elliptical aggregates and polygonal effective mastic zones [41]. Using image processing and ellipse fitting methods, particle dimensions and locations were determined from digital photographs of the sample’s microstructure. These models have been promising in predicting the mixture behavior, however, the aggregate shape is idealized as ellipse. Even though many research studies have been conducted on micromechanical modeling of asphalt mixtures, the irregular shape of the aggregate in the mixture still has not been successfully modeled to capture the micromechanical aggregate-to-aggregate contact behavior.

2 Objectives The objectives of this study are to (1) develop a microstructure-based FE model for heterogeneous asphalt mixture, and; (2) predict viscoelastic mixture properties (e.g., complex modulus and creep stiffness) by using the FE model with the input of the viscoelastic properties of sand mastic and elastic modulus of the aggregates.


3 Scope The asphalt mixture in this study is modeled as irregular shaped aggregates bonded with sand mastic. The microstructure of the mixture was obtained from the 2D scanned image of a sawn asphalt concrete. Therefore, the image of the asphalt mixture includes the coarse aggregate and sand mastic. The sand mastic is a mixture of fine sand and asphalt binder. Comparing with the coarse aggregate, the sand particles are so small that they can distribute in the asphalt binder uniformly. Therefore, it is reasonable treat the sand mastic as a homogonous composite. The FE mesh was generated within each aggregate and mastic subdomains by sharing nodes on the aggregate boundaries. Viscoelastic properties of the mastic elements are calibrated with the laboratory experimental test data and inputted into a user material subroutine within the ABAQUS code. Then the viscoelastic mastic was combined with elastic aggregates to predict the properties of asphalt mixtures. It should be noted that the air void in the mixture is ignored in this 2D model due to the limitation of smooth sawn image. Ongoing simulation work is conducted on X-ray scanned images with the indication of air void distribution.

4 Microstructure of asphalt mixture In this study, the 2D microstructure of asphalt concrete was obtained by optically scanning Fig. 1 Microstructure of an asphalt mixture specimen surface. (a) An original image of the surface, (b) The aggregate skeleton (sieving size [1.18 mm)

smoothly sawn asphalt mixture specimens. A highresolution scanner was used to obtain grayscale images from the sections. Image processing technique was used to process and analyze images. Figure 1 demonstrates the image processing on an asphalt specimen (with the nominal maximum aggregate size of 19 mm). Figure 1a shows an optical scanning image of a mixture specimen with 86 mm width and 106 mm height. A scanner with 1600 · 3200 Dots Per Inch (DPI) optical resolution was used. After improving the image contrast, the outlines of aggregates were converted into manysided polygons using a custom developed macro program in Image Pro Plus to define the microstructure of asphalt mixture [15]. The average of the polygon diameter was chosen as a threshold to determine which aggregates would be ‘‘retained’’ on a given sieve (i.e., the rest of the aggregates would be ‘‘passed’’ the given sieve), although some other measurement parameters were also attempted in analyzing the gradation of the aggregates [42]. The polygons used in the micromechanical model were treated as coarse aggregates. Figure 1b is the coarse aggregate retained on 1.18 mm sieve (i.e., No. 16), where the fine aggregates passing 1.18 mm were filtered as sand mastic. Elastic properties of aggregates [20, 21, 42] were assigned to aggregate subdomains, and viscoelastic mastic properties were also evaluated with mastic creep tests. Finite element simulation was conducted to predict the mixture behavior by combining aggregate and mastic properties.


5 Mastic viscoelastic model

each increment was developed in the following format:

Generalized Maxwell model was widely used for viscoelastic solids such as asphalt mixture. The generalized Maxwell model was applied to simulate the linear and damage-coupled viscoelastic behavior of asphalt mixture [40]. The linear constitutive behavior for this Maxwell-type model can be expressed as a hereditary integral Z t deij ðsÞ rij ¼ E1 eij þ ds ð1Þ Et ds 0 where Et is expressed with a Prony series Et ¼

M X

ðtsÞ

Em e qm ;

and

m¼1

qm ¼

gm Em

ð2Þ

In these equations, E1 is the relaxed modulus, Et is the transient modulus as a function of the time, Em, gm and qm are the spring constant, dashpot viscosity and relaxation time respectively for the mth Maxwell element. The reduced time (effective time) is defined by using the time-temperature superposition principle as Z t 1 nðtÞ ¼ ds ð3Þ a T 0 where the term aT ¼ aT ðTðsÞÞ is a temperaturedependent time-scale shift factor (Fig. 2). Three-dimensional behavior can be formulated with uncoupled volumetric and deviatoric stress– strain relations. A displacement-based incremental FE modeling scheme with constant strain rate over

Dr ¼ K De þ DrR

ð4Þ

Where Dr and De are incremental stress and strain, K is the incremental stiffness and DrR is the residue stress vector. The volumetric constitutive relationship is expressed with the volumetric stress rkk and strain ekk in the general form Z n dekk ðn0 Þ 0 rkk ðnÞ ¼ 3K1 ekk ðnÞ þ 3Kt ðn n0 Þ dn dn0 0 ð5Þ relaxed bulk where K1 ¼ E1 =3ð1 P2mÞ is the ðnn0 Þ q m modulus, Kt ðn n0 Þ ¼ M K e is the tranm¼1 m sient bulk modulus, and Km = Em/3(1–2m) is the bulk constants for the spring in the mth Maxwell element. The incremental formulation of the volumetric behavior is obtained with constant volumetric strain kk rate Rkk ¼ De Dn ; " Drkk ¼ 3 K1 þ

N X Km q m

m¼1

Dn

qDn

1e

#

m

Dekk þ DrRkk ð6Þ

and the residual part DrRkk can be expressed in a recursive relation with the history variable Sm, DrRkk ¼

M X Dn 1 eqm Sm ðnn Þ;

and

m¼1

Dn Dn Sm ðnn Þ ¼ 3Km Rkk qm 1 eqm þ Sm ðnn1 Þeqm ð7Þ

E∞

E1

E2

E3

η1

η2

η3

E M −1

η M −1

EM

ηM

For the initial increment, theDn history variable Sm(n1) q m equals to 3Km Rkk qm 1 e and is similar to the following formulations. For the deviatoric behavior, the constitutive relationship is written using deviatoric stress îj ¼ rij 13 rkk Dij and strain êij ¼ eij 13 ekk Dij ; r îj ðnÞ ¼ 2G1êij ðnÞ þ r

Z 0

Fig. 2 The generalized Maxwell viscoelastic model for the sand mastic

n

2Gt ðn n0 Þ

dêijðn0 Þ dn0 dn0

ð8Þ

where G1 ¼ E1 =2ð1 relaxed shear modðnn0 Þ Pþ mÞ is the q m ulus, Gt ðn n0 Þ ¼ M G e is the transient m¼1 m shear modulus, and Gm = Em/2(1 + m) is the shear constants for the spring in the mth Maxwell element.


The formulation of the deviatoric behavior is obtained with constant deviatoric strain rate Dê Rîj ¼ Dnij ; "

# N X Dn Gm qm 1 eqm Dêij þ D^ rRij D^ rij ¼ 2 G1 þ Dn m¼1 ð9Þ and the residual part D^ rRij can be expressed in the recursive relation D^ rRij ¼

N X Dn 1 eqm Sm ðnn Þ;

calculated and then the incremental 3D linear viscoelastic behavior was formulated as 3 2 32 3 2 K1 K2 K2 0 Dexx 0 0 Drxx 7 6 Dr 7 6 K K 6 0 0 0 7 1 2 76 Deyy 7 6 yy 7 6 7 6 76 7 6 6 Drzz 7 6 K1 0 0 0 76 Dezz 7 7¼6 76 7 6 7 6 Dr 7 6 6 K3 0 0 7 76 Dexy 7 6 xy 7 6 7 6 76 7 6 4 Dryz 5 4 K3 0 54 Deyz 5 Drxz

and

Dn Dn Sm ðnn Þ ¼ 2Gm Rîj qm 1 eqm þ Sm ðnn1 Þeqm

þ6 6 6 6 4

ð10Þ

Drxx ¼ 1=3Drkk þ D^ rxx " # N X Km q qDn m 1 e m Dekk ¼ K1 þ Dn m¼1 " # N X Gm qm qDn 1 e m Dêxx þ 2 G1 þ Dn m¼1 þ 1=3DrRkk þ D^ rRxx

ð11Þ

where Dekk and Drkk are the incremental volumetric strain and stress, Dêxx and D^ rxx are the incremental deviatoric strain and stress components, and DrRkk and D^ rRxx are the recursive part of the volumetric and deviatoric behavior given in Eqs. 6 and 9. Incremental stresses Dryy and Drzz are determined in the same manner. The incremental shear stress can be formulated by using only the deviatoric behavior. For example, Drxy ¼ D^ rxy " ¼ 2 G1 þ

M X Gm q m

m¼1

Dn

1e

qDn

m

R 3 þ D^ rxx þ D^ rRyy 7 7 7 R 7 þ D^ rzz 7 7 7 D^ rRxy 7 7 R D^ ryz 5

K3

DrRkk 6 DrR 6 kk 6 6 DrR 6 kk

m¼1

The incremental normal stresses can be then formulated by combining the volumetric and deviatoric behavior. For example,

2

Dexz

ð13Þ

D^ rRzx where "

# N X Dn Km qm K1 ¼ K1 þ 1 e q m Dn m¼1 " # N X 4 Gm qm qDn þ G1 þ 1e m 3 Dn m¼1 " # N X Km qm qDn K2 ¼ K1 þ 1e m Dn m¼1 " # N X 2 Gm qm qDn G1 þ 1e m 3 Dn m¼1 " # N X Gm qm qDn K3 ¼ 2 G1 þ 1e m Dn m¼1

ð14Þ

This viscoelastic model was defined in the ABAQUS user material subroutine for mastic subdomains. A displacement-based time-dependent FE analysis was conducted by integrating elastic aggregate and viscoelastic mastic subdomains to predict the global behavior of asphalt mixture.

# rRxy Dêxy þ D^ ð12Þ

rxy are the incremental shear where Dêxy and D^ deviatoric strain and stress components, and the recursive term D^ rRxy is also given in Eq. 9. Once the incremental stress components are developed, the incremental stiffness terms can be

6 Laboratory tests of aggregates, sand mastic, and compacted asphalt mixture The purpose of this section is to measure and evaluate the material properties of sand mastic, aggregate (rock), and compacted asphalt mixture through laboratory tests. The uniaxial compression laboratory tests of sand mastic and aggregate


(cylinder specimen) were conducted to provide material input parameters for the FE models. The goal of the mixture test is to provide a comparison with the model prediction in order to validate the FE model simulation. The input parameters for the FE models include not only the microstructure information but also the material properties of the aggregate and mastic at different loading conditions. In this study, a modulus of 55.5 GPa [20, 21, 42] for the limestone was used for different temperatures and loading frequencies. The uniaxial compression creep test was conducted on the sand mastic and mixture samples under different temperatures (0C, –10C, and –20C). The mastic contains aggregates passing sieve 1.18 mm and asphalt content is about 14% [20, 21, 42]. The sand mastic was comprised of the portion of the aggregate gradation finer than the 1.18 mm sieve combined with the volume of binder normally used in the entire asphalt concrete mixture. The creep stiffness at different loading time and temperatures were obtained from the inverse of creep compliance. A regression fitting method was employed to evaluate mastic viscoelastic properties with a generalized Maxwell model at the reference temperature of – 20C, and the time shift factors were calculated for 0C and –10C. Master stiffness curves were generated for mastic and asphalt mixture from creep tests [43]. The shifted creep stiffness and fitted master curve for sand mastic at the reference temperature – 20C are shown as Fig. 3. The model for sand mastic includes one spring and four Maxwell elements in parallel. The viscoelastic parameters of the sand mastic are: E1 ¼ 59:7 MPa; E1 = 5710.6 MPa,

Mastic Creep Stiifness (GPa)

1E+02 0 C Test Data -10 C Test Data -20 C Test Data Fitted Model

1E+01

1E+00

1E-01 1E+00

1E+01

1E+02

1E+03

1E+04

1E+05

Reduced Time (sec)

Fig. 3 The shifted creep stiffness and fitted master curve for the sand mastic

E2 = 2075.1 MPa, s2 = 311.9 s, s1 = 26.2 s, E3 = 1449 MPa, s3 = 1678.8 s, E4 = 734.9 MPa, and s4 = 19952.6 s. The relaxed and transient moduli are determined from the master curve of mastic creep stiffness. Uniaxial behavior was extended to multiaxial (3D) constitutive formulation with the uncoupled volumetric and deviatoric Eqs. 5 and 8. The E elastic bulk and shear moduli are K ¼ 3ð12t Þ and E G ¼ 2ð1þtÞ : The terms K1 and G1 are the relaxed bulk and shear moduli, and Km and Gm are the bulk and shear constants for the spring in the mth Maxwell element. The Poisson’s ratio m is assumed as 0.3 for asphalt mixture. Therefore, the multi-axial properties were connected with the uniaxial measurements. For the gyratory compacted mixture specimens, uniaxial compression tests were conducted to measure complex (dynamic) modulus and creep stiffness with a number of mixture specimens. The complex moduli were measured at different temperatures and loading frequencies. The creep stiffnesses were calibrated over a period of loading time at different temperatures. This database was used to validate FE model for prediction of mixture complex moduli and creep stiffness at different loading conditions.

7 Development of micromechanical finite element models for asphalt mixtures As mentioned previously, the microstructure of the asphalt mixture was divided into different aggregate and mastic subdomains. The FE mesh was generated within the subdomains of aggregates and mastic and along the subdomain boundaries. Due to the very irregular aggregate and complex mastic distribution, the three-node triangle elements were used in the FE mesh for the complex geometry. Figure 4 shows the FE meshes in the aggregate and mastic subdomains of a specimen surface. Finite elements in the neighboring subdomains share the nodes on irregular boundaries as shown in Fig. 4, and therefore the displacements of neighboring subdomains were connected through the shared nodes. For this 2D FE mesh, plane stress elements with a solid section thickness were applied for both aggregate and mastic subdomains. After the FE model has been developed, uniaxial compression test was simulated. For the compression simulation, the x- and y-displacements of the nodes

Materials and Structures Fig. 4 The asphalt mixture image and the three-node triangle FEM meshes for the aggregate and mastic subdomains. (a) A scanned image of the specimen surface, (b) The finite element meshes of the specimen surface, (c) Enlarged meshes for the aggregates and mastic

on the bottom layer and the x-displacements of the nodes on the top layer were constrained. The constant or dynamic force loading was evenly divided and imposed on nodes of the top layer. The generalized Maxwell model parameters for mastic and the elastic modulus of aggregates were inputted to the FE model. Simulation was conducted to predict the global viscoelastic properties of the asphalt mixture. In the simulation, axial strain was calculated by dividing the average vertical displacement of top particles with the initial height of the undeformed specimen, and axial stress was obtained by dividing the constant loading force on the top layer with the specimen initial cross-section area.

Fig. 5 The compression strain contour in a portion of the digital specimen

One of the benefits of the micromechanical model is to present the detailed stress and strain distributions within the microstructure of the mixture sample. Figure 5 shows the compression strain distribution contour in a portion of the digital specimen under a uniaxial constant compression force loading. The aggregate skeletons are indicated in the figure with the skeleton curves. The high compression strains were generated in the vertical mastic gap between neighboring aggregates. The computational results show that the highest local strain is about eight times of average strain for this portion of the computational sample. Figure 6 shows the shear strain distribution contours in the same portion of the digital specimen

Materials and Structures Fig. 6 The shear strain contour in the same portion of the digital specimen

under a constant compression force loading. This figure indicates that high shear strain zones are distributed in horizontal narrow mastic gap along the large-size aggregates. From our previous study, it was found that under the mechanical loading, the cracks initiate in the narrow gap between coarse aggregates of asphalt mixture specimen. These strain distributions agree with the laboratory observations. The strain contours help examine micro material behavior, such as strain intensities within the asphalt mastic and aggregate phases. They also can provide useful information in analyzing crack initiation and optimizing mixture design on the basis of mechanistic performance for further study.

8 Complex modulus simulation and results Sinusoidal cyclic loading was imposed to the simulation specimen for calculating the complex modulus under the different loading frequencies (10 Hz, 5 Hz, 1 Hz and 0.1 Hz) as shown in Fig. 7. In the simulation, the loading cycles were taken as 30 for 0.1 Hz, 50 for 1 Hz, 5 Hz, and 10 Hz. In these figures, the constant cyclic curve is the imposed stress load with the right-side axial scale, and the other curve indicates the strain response with the left-side axial scale. For better illustration, the final several

cycles were magnified in the right-side figures for each frequency. The computation points are indicated in the right-side figures. The magnitude of the dynamic modulus was calculated using the last ten cycles for each frequency. As indicated in these figures, the strain increases with the loading time for each frequency. Comparing different frequency responses, it was found that the strain value deceases with the loading frequencies. It also indicates the complex modulus value increases with the loading frequencies due to decreasing relaxation time. Figure 8 shows the complex modulus comparisons between FE simulation results and test data for different loading frequencies at a test temperature of –20C. As mentioned previously, the mastic master curve was obtained at the reference temperature – 20C. Therefore the FE simulation with the input mastic properties generated the mixture behavior at this selected temperature. The comparisons indicate that the simulation results are reasonable and applicable for complex modulus prediction although it slightly under predicted the mixture modulus. The differences between the predictions and measurements may cause by the following reasons: (1) the aggregates sieved from the 2D image may reduce the real aggregate content/percentage, and (2) the 2D mixture microstructure model may underestimate the


0.35

12

0.30

10

0.25

8

0.20

6

0.15

4

0.10

2

0.05

0

0.00 3.0

4.0

0.35

12

0.30

10

0.25

8

0.20

6

0.15

4

0.10

2

0.05 0.00

0 4.6

5.0

4.7

Strain 0.40

12

0.35 0.30

10

0.25 0.20

8 6

0.15 0.10

4 2

0.05 0.00

0 0

2

4

6

8

Com pression S tress (GP a)

Stress

14

Com pression S train


Stress

0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

10 8 6 4 2 0 9.0

10 8

0.3

6

0.2

4 0.1

2 0

0.0

Stress

40

Straint 0.5 0.4

10 8

0.3

6

0.2

4 0.1

2

0.0

0 47

48

Stress 0.7

10

0.6 0.5

8

0.4 6

0.3

4

0.2

2

0.1

0

0.0 150

10

12

Strain

12

100

9.8

49

50

Loading time (s)

0.8

50

Stress

46

14

0

9.6

14

50

200

250

300

Loading Time (s)

Com pression Stress (GPa)

C om pr e s s ion Str e s s (GPa )

20 30 Loading time (s)


0.4


12

(d)

9.4

Strain 0.5

10

9.2

Loading Time (s)

14

0

Strain

12

10


Com pression Stress (Gpa)

Stress

5.0

14

Loading Time (s)

(c)

4.9

Loading Time (s)

Loading Time (s)

(b)

4.8

Com pression Strain

2.0

14

Com pression Strain

1.0

Strain

Strain

14

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

12 10 8 6 4 2 0 260

270

280

290

Com pression Strain

0.0

Stress

Com pression Strain

Strain


Stress

14

Com pression Strain


(a)

300

Loading Time (s)

Fig. 7 The FEM simulation results under sinusoidal loading. (a) loading frequency = 10 Hz, (b) loading frequency = 5 Hz, (c) loading frequency = 1 Hz, (d) loading frequency = 0.1 Hz

aggregate-aggregate contact or aggregate interlock effects. With the development of 3D modeling and the additional calibration of mastic and aggregates, the model prediction perhaps will be closer to the test data with more capacity to describe mixture complex behavior.

In order to compare the FE prediction, the authors also conducted the discrete element modeling to predict the mixture complex modulus with the input of the measured mastic complex moduli at different loading frequencies and test temperatures, and aggregate modulus [19–21, 43]. It was found that the

Materials and Structures 100.0 Relaxation Modulus (GPa)

Mixture Complex Modulus (GPa) .

100

10

100.0

1000.0

10000.0

Reduced Time (s)

1 1

10.0

1.0 10.0

Test Data (-20C) FEM Simulation 0.1

FEM DEM (c=0.62), Calibrated DEM (c=0.55), Calibrated Test Data

10

Loading Frequency (Hz)

Fig. 9 Prediction from the FEM and the calibrated DEM simulation, and the laboratory measurements of the asphalt mixture creep stiffness for a reduced time up to 104 s

Fig. 8 The complex modulus comparison with the FEM simulation and the test data at a temperature –20C

complex modulus predictions from the FE models had a good agreement with the DEM and the lab measurements.

9 Creep stiffness simulation and results In this section, the creep stiffness simulation by using the micromechanical FE model was discussed. In addition, the predictions from the micromechanical FE model, a microstructure based discrete element model (DEM) [21] and a microstructure based FE network model (FENM) were compared. Micromechanical FE simulation was conducted with a constant force loading condition to predict the creep stiffness of asphalt mixture. When the constant force was applied to the microstructure-based FE model, the creep displacement responses were captured over a period of time. Then the macro creep strain and creep stiffness of the asphalt mixture were computed. The creep stiffness (reverse of the creep compliance) varies with time. FE model simulation was compared with the measurements of the master curve of creep stiffness of the mixture across a reduced time up to 104 s as shown in Fig. 9. The FE model had a good comparison with test measurements, and slightly underpredicted the creep stiffness (perhaps due to the limitation of 2D model). In this case study, the authors consider that the major contributions to creep deformation are viscoelastic properties. It also exits slight unrecoverable deformation caused by the viscoplastic behavior for real asphalt mixture. In the ongoing work, the authors study the viscoplastic-viscoelastic behavior by

replacing the elastic spring with a viscoplastic-elastic element in the presented model. In order to compare the creep stiffness prediction of a microstructure based discrete element model (DEM) [21], the discrete element model predictions with the coarse aggregate volume concentration ratios (i.e, c = 0.55 and c = 0.62) were compared with the master curve of the creep stiffness of the mixture as shown in Fig. 9. A calibration method was applied to reduce the possibility of under-counting the aggregate particles in the mixture model, since the 2D modeling approach may count insufficient aggregate-aggregate contact or interlock. The calibration method was adding extra fine aggregate particles (between 0.6 mm and 1.18 mm). Therefore, the extra fine aggregate particles were part of the original aggregate skeleton, and therefore the mixture’s coarse aggregate volume concentration ratio increased. When comparing with the master curve of the creep stiffness from discrete element models, the model prediction was improved with the increasing aggregate volume concentration ratio. Finite element network model (FENM) using elliptical aggregates was developed to study asphalt mixture behavior including creep stiffness [41, 44, 45]. In the FENM model, the microstructure of asphalt materials is simulated with idealized elliptical aggregate and polygonal effective mastic zone. FENM integrates viscoelastic mastic elements with rigid elliptical aggregates to predict global mixture behavior. Figure 10 shows the comparison among mixture creep stiffness predictions from the FE and idealized FENM simulation, and the laboratory measurements. It was observed from the figure that the FENM slightly over-predicted mixture creep


Relaxation Modulus (GPa)

100.0 FEM

FENM

Test Data

10.0

1.0 10.0

100.0

1000.0

10000.0

Reduced Time (s)

Fig. 10 Comparison among mixture creep stiffness prediction from the FEM, and the idealized FENM simulation, and the laboratory measurements

stiffness especially in the beginning of the loading time. This occurs due to the model assumption of rigid aggregates with infinity stiffness and idealized particle and mastic zone shapes. The measurements of the creep stiffness are between the FE and FENM prediction bounds. Although some limitations within 2D micromechanical FE and FENM predictions, the FE model predictions in general, are reasonable by comparing the measurements of mixture creep tests.

10 Conclusions The microstructure-based FE model was developed and applied to predict viscoelastic properties (complex modulus and creep stiffness) of heterogeneous asphalt mixture. The 2D microstructure of asphalt mixture was obtained by optically scanning the smoothly sawn surface of asphalt specimens. In the microstructure, aggregates and sand mastic were divided into different subdomains. Finite element mesh was generated within each aggregate or sand mastic subdomain. Therefore the very irregular aggregate geometry and mastic domains are modeled using a number of FEs. Then the viscoelastic mastic with specified properties in an ABAQUS user subroutine was combined with elastic aggregates to predict the global viscoelastic properties of asphalt mixtures. An experimental program was developed to measure the properties of the aggregates, sand mastic, and asphalt mixture for FEM simulation and validation. The mastic viscoelastic properties and aggregate elastic modulus were inputted for FE simulation. The laboratory measurements of the complex

modulus and creep stiffness of the asphalt mixture were used to compare the model predictions. In general, micromechanical FE models provided reasonable predictions of the complex modulus over a range of frequencies, and creep stiffness across a period of reduced loading time. In order to show the different model predictions, comparisons have been conducted among the FE model, a DEM and a FE network model (FENM). FENM slightly overpredicted creep stiffness especially in the beginning of loading time due to the model assumption of rigid aggregates and idealized aggregate and mastic zone shapes. Based upon the predictions from FE and DE models, it was found that these models slightly underpredicted mixture modulus and creep stiffness, because of the limitation of aggregate-to-aggregate contact and interlock effects in the 2D models. As future modeling efforts are extended to three-dimensions, the prediction will be improved with larger amount of inter-particle contacts and more measurements of aggregates and mastic. Acknowledgement The authors acknowledge Dr. William Buttlar’s assistance in the laboratory tests at the University of Illinois at Urbana-Champaign.

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