Determination of the fatigue-cracking resistance of asphalt concrete mixtures at low temperatures

Cold Regions Science and Technology 61 (2010) 116–124

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Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / c o l d r e g i o n s

Determination of the fatigue-cracking resistance of asphalt concrete mixtures at low temperatures Sungho Mun a,⁎, Sangyum Lee b a b

Expressway & Transportation Research Institute, Korea Expressway Corporation, 50-5, Sancheok-ri, Dongtan-myeon, Hwaseong-si, Gyeonggi-do, 445-812, South Korea Road Management Division, Seoul Metropolitan Government, Deoksugung-gil 15, Jung-gu, Seoul, 100-110, South Korea

a r t i c l e

i n f o

Article history: Received 17 November 2009 Accepted 12 February 2010 Keywords: Viscoelastic continuum damage Hot-mix asphalt Dynamic modulus Direct tension test Lime-modified mixture

a b s t r a c t This paper describes how to determine the fatigue-cracking resistance of hot-mix asphalt (HMA) mixtures and formulate a viscoelastic continuum damage algorithm based on the comparison between measured and calculated strain vs. stress histories at low temperatures. The fatigue-cracking resistance characteristics evaluated in this study were based on the dynamic modulus test for material stiffness characterization and the constant-crosshead-rate tension test for fatigue-cracking characterization, using dense-graded and limemodified HMA mixtures. The main contributions of this paper are the methods for determining the optimum asphalt content of the HMA mixtures to improve fatigue-cracking resistance and for evaluating the fatiguecracking resistance of various HMA mixtures. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The Strategic Highway Research Program (SHRP) began developing a new system for specifying asphalt materials in 1987 that resulted in SUperior PERforming asphalt PAVEments (Superpave). Superpave represents an improved system for specifying asphalt binder and mineral aggregate, developing asphalt mixture design, and establishing and analyzing pavement performance prediction. The system includes an asphalt binder specification and a hot-mix asphalt (HMA) design and analysis system. The core of the Superpave system is a performancebased specification system that has a direct relationship to field performance. The Superpave design method for HMA consists of three phases: material selection for the asphalt binder and aggregate, aggregate blending, and volumetric analysis of specimens compacted using the Superpave gyratory compactor, which is discussed in detail by Witczak et al. (2000). Although performance information from the Superpave design method was used to define the relationships between material properties and pavement performance, the present study uses a more mechanical performance evaluation of a continuum damage approach. The fundamental engineering properties of the continuum damage approach can be linked to the advanced material characterization methods of the detailed distress prediction models, which include a fatigue-cracking model at low temperatures. Several binder content levels and conventional dense-graded HMA mixtures were used as reference materials to compare with lime⁎ Corresponding author. Tel.: +82 31 371 3360; fax: +82 31 371 3439. E-mail address: [email protected] (S. Mun). 0165-232X/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2010.02.003

modified HMA materials in evaluating the fatigue-cracking resistance effects of hydrated lime-modified HMA mixtures at low temperatures. Hydrated lime is acknowledged as a superior additive for asphalt concrete. The beneficial nature of this material in asphalt concrete is related to both the particular chemistry of the system and the mechanical nature of fine particles in an asphalt binder matrix. The use of mineral filler such as lime for improving the performance of asphalt concrete mixtures has been well demonstrated (Little and Petersen, 2005; Anderson and Goetz, 1973; Tayebali et al., 2000). Appropriately selected fillers with physico-chemical characteristics compatible with the asphalt binder and aggregate particles act as bond-strengthening and crack-arresting agents to increase fatigue performance (Little and Petersen, 2005). Similarly, compatible mineral fillers aid in reducing permanent deformation by increasing the viscosity of the mastic, thus improving the plastic and viscoplastic characteristics of mixtures (Mohammad et al., 2006). The observed benefits of hydrated lime range from very significant as an active filler to less significant but still useful, i.e., similar to more standard inert fillers such as baghouse fines (Little and Petersen, 2005). Even though the beneficial nature of hydrated lime in HMA mixtures has been repeatedly demonstrated, confusion still exists as to the best method for adding lime to asphalt mixtures. Selecting the appropriate method is complicated by the lack of consistent results in the literature. Some researchers suggest that the method for adding lime is not important (Collins, 1988), whereas others clearly demonstrate differences in material properties depending on the method by which lime is added (Sebaaly et al., 2003). The conflicting results from the literature indicate that further study of this topic is required.

S. Mun, S. Lee / Cold Regions Science and Technology 61 (2010) 116–124

The primary objective of this paper was to investigate the effect of lime modification on the fundamental behavior of HMA mixtures based on continuum damage mechanics. This was achieved by comparing the fundamental behaviors of dense-graded HMA mixtures to predict the fatigue-cracking performance of lime-modified HMA mixtures at low temperatures. The outline of the remainder of this paper is as follows. Section 2 describes the correspondence principle and viscoelastic continuum damage used in this study. Section 3 presents the material tests conducted to characterize the material parameters of HMA mixtures, as well as the numerical viscoelastic continuum damage computation. Section 4 describes the evaluation of fatigue-cracking performance of HMA mixtures at low temperatures based on continuum damage mechanics. Section 5 presents the concluding remarks.

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modulus; t is the time of interest; and τ is an integration constant. Using the definition of pseudo strain in Eq. (3), Eq. (1) can be rewritten as R

σ = ER ε :

ð4Þ

A correspondence can be found between Eq. (4) and a linear elastic stress–strain relationship (Hooke's Law). Fig. 1 illustrates the power of the pseudo strain. Fig. 1(a) shows the cyclic strain input, Fig. 1(b) illustrates the physical stress–strain hysteresis behavior, and Fig. 1(c) shows the characterization of the pseudo strain vs. stress relationship. No damage occurs if the material is in its LVE range, and the hysteretic behavior and accumulating strain are due only to viscoelasticity. Fig. 1 (c) shows the same stress data plotted against the calculated pseudo strains in the viscoelastic continuum damage case due to stiffness

2. Correspondence principle and viscoelastic continuum damage The uniaxial stress–strain constitutive equation for a linear viscoelastic material can be represented by a Boltzmann superposition integral t

σ ðt Þ = ∫ Eðt−τÞ 0

dε dτ; dτ

ð1Þ

where σ (t), E(t), and ε (t) are the stress, relaxation modulus, and strain, respectively. Schapery (1984) proposed the extended elastic– viscoelastic correspondence principle to address the problem of crack propagation; this principle is applicable to both linear and nonlinear viscoelastic (non-LVE) materials. The relaxation modulus of Eq. (1), E(t), which is based on a generalized Maxwell model consisting of a series of springs and dashpots (Biot, 1954) in the form of a Prony series, can be expressed as M

Eðt Þ = E∞ + ∑ Em exp ð−t = ρm Þ; m=1

ð2Þ

where E∞, ρm, and Em are the infinite relaxation modulus, relaxation time, and Prony coefficients, respectively. Schapery (1984) suggested that constitutive equations for certain viscoelastic (VE) media are identical to constitutive relationships for elastic cases. Three Corresponding Principles (CPs) were developed depending upon the boundary condition: 1) CP I: Both pseudo stress and strain (general boundary conditions); 2) CP II: Physical stress and pseudo strain (as the crack increases, the traction boundary condition increases); and 3) CP III: Pseudo stress and physical strain (as the crack heals, the traction boundary condition decreases). Constitutive relationships cannot be expressed in the convolution integral form for a non-LVE material or a VE material with some type of damage. Stresses and strains are not necessarily physical quantities in the VE body. Rather, they are pseudo variables in the form of convolution integrals. Pseudo variables account for all the hereditary effects. Thus, the time-dependent effects from the stress–strain relationship can be separated, even in the case of non-LVE or damaged materials. In other words, VE problems can be solved with elastic solutions when physical strains are replaced by pseudo strains. According to CP II (Schapery, 1984), the uniaxial pseudo strain εR is defined as t

R

ε =

1 dε ∫ Eðt−τÞ dτ; ER 0 dτ

ð3Þ

where ε is the uniaxial strain; ER is a reference modulus set to an arbitrary constant, typically unity; E(t) is the uniaxial relaxation

Fig. 1. Concept of Corresponding Principle II: (a) strain input, (b) physical hysteresis loop, and (c) curve of pseudo strain vs. stress.

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reduction. In the linear case, all of the cycles collapse to a single line with a slope of 1.0. The use of pseudo strain significantly simplifies the modeling approach by allowing for the separation of VE (timedependent) behavior from any accumulated damage such as stiffness reduction. The constitutive model developed by Lee and Kim (1998) uses the elastic–viscoelastic correspondence principle to eliminate the time dependence of the material. The work potential theory (Schapery, 1990) is then used to model the damage growth in the material. The term damage is defined as all structural changes in the constitution of the material that result in the reduction of stiffness or strength as the material undergoes loading. Schapery (1990) developed a theory using the thermodynamics of irreversible processes to describe the mechanical behavior of elastic composite materials with growing damage. Three fundamental elements comprise the work potential theory as illustrated in Table 1. The damage parameter S in Table 1 represents the internal state variables, and WS(S) is the dissipated energy due to structural changes. Using Schapery's elastic–viscoelastic correspondence principle and rate-type damage evolution law (Schapery, 1984, 1990; Park et al., 1996; Mun and Geem, 2009), the physical strains ε are replaced with pseudo strains εR to include the effects of viscoelasticity. The use of pseudo strain accounts for all the time-dependent effects of the material through the convolution integral. Thus, the strain energy density function, W = W(ε, S), transforms into the pseudo strain energy density function, WR = WR(εR, S), where S is a damage parameter. Schapery's (1984, 1990) correspondence principle cannot be used to transform the elastic damage evolution law for use with VE materials because both the available force for the growth of S and the resistance against the growth of S in the damage evolution law are rate-dependent for most VE materials (Park et al., 1996). Quite simply, this means that microcracking damage is rate-dependent in asphalt concrete. Therefore, a form similar to power-law crack growth is used to describe the damage evolution in a VE material, S˙ =

−

∂W R ∂S

!α ;

ð5Þ

where Ṡ = dS/dt is the damage evolution rate with respect to time t, WR is the pseudo strain energy density function, and α is the material constant. Lee and Kim (1998) developed a constitutive model that is independent of loading mode to describe the fatigue and microdamage of asphalt concrete under uniaxial tensile cyclic loading. The slope of the hysteretic stress–pseudo strain loop in damage-inducing testing decreases as loading continues in both controlled stress and controlled strain testing. The decrease in the slope of the loop indicates a reduction in the stiffness of the material as damage accumulates. The stiffness reduction is defined by the pseudo secant modulus (C, pseudo stiffness) as follows: C=

σ : εR

ð6Þ

Constitutive equation (stress–strain relationship) Damage evolution law

R

W =

 2 1 R C ðSÞ ε 2

ð7Þ

and R

σ = C ðSÞε ;

ð8Þ

where C(S) is a function of the internal state variable S that represents the changing stiffness of the material due to microstructure changes such as accumulating damage. This internal state variable quantifies any microstructural changes that result in the observed reduction of stiffness. The relationship between the damage S and the normalized pseudo secant modulus C is referred to as the damage characteristic relationship and is a material function independent of loading condition (Daniel, 2001). The damage evolution law and experimental data are used to characterize the function C in Eq. (8). Values of C can be determined with Eq. (6) using the ratio of measured stresses to calculated pseudo strains. The values of S must be obtained using the following equation to determine the characteristic relationship between C and S: dS = dt

−

∂W R ∂S

!α :

ð9Þ

Substituting Eq. (7) into Eq. (9) and simplifying gives   dS 1 dC  R 2 α = − ε : dt 2 dS

ð10Þ

As the interval between successive data points becomes smaller, the derivative in Eq. (10) can be interpreted as the changes in S and t values, resulting in   ΔS 1 ΔC  R 2 α = − ε ; Δt 2 ΔS

ð11Þ

where Δ denotes the small change. The S values are plotted with the pseudo stiffness values C to obtain the damage characteristic curve. This relationship is then fitted to some analytical form. In this study, this simple form was used:   b C ðSÞ = exp −aS :

ð12Þ

3. Testing program and numerical computation Performance-based tests are tests that measure material properties that can be used in a fundamental response model to predict mixture responses to various loadings and environmental conditions. In this study, two different performance-based test protocols were used to determine the material behavior: the dynamic modulus test and the constant-crosshead-rate tension test. 3.1. Dynamic modulus test

Table 1 Fundamental elements of the work potential theory.

Strain energy density function

Because all the tests conducted in this paper for the purpose of VE behavior characterization are in strain control, particularly the constant-crosshead-rate tests, the constitutive equations reduce to

Elasticity with damage

Viscoelasticity with damage

W = W(ε, S) W = 12 C ðSÞε2 σ = C(S)ε

WR = WR(εR, S)  2 W R = 12 C ðSÞ εR σ = C(S)εR

  S˙ = − ∂W ∂S

  R S˙ = − ∂W ∂S

Understanding the principle of time–temperature superposition (or time–temperature equivalence) is important to facilitate the presentation of results. Quite simply, this means that the same modulus value of a material can be obtained both with lowtemperature tests and short times (fast frequencies) or with hightemperature tests and long times (slow frequencies). More generally, the behavior of a material at high temperatures is the same as that

S. Mun, S. Lee / Cold Regions Science and Technology 61 (2010) 116–124

under long loading times or slow loading rates/frequencies, and the material behavior at low temperatures is the same as that under short loading times or fast loading rates/frequencies. Materials that exhibit this type of behavior are called thermorheologically simple (TRS). The time–temperature superposition of a material can be verified by performing dynamic modulus (e.g., |E*|) tests at various temperatures and frequencies, as described in the AASHTO TP 62 (2003) protocol. Asphalt concrete in the LVE range is a TRS material, and the effects of time and temperature can thus be combined into a joint parameter, i.e., reduced time/frequency, through the time–temperature shift factor aT using fR = f × aT :

ð13Þ

Fig. 2(b) shows typical data for this scenario. As expected, the |E*| increases as the loading frequency increases and the temperature decreases. This behavior allows for the horizontal shift of the data to form a single master curve and accounts for the constitutive behavior of asphalt concrete over a wide range of reduced frequencies. The simplifying feature afforded by time–temperature superposition is that all of these curves can be superimposed to form a single continuous curve by means of horizontal translations only. Fig. 2(b) presents such master curves for the four replicate tests shown in Fig. 3 (a). The amount of shift is dependent on the temperature chosen as the reference and therefore varies according to temperature, as shown in Fig. 2(a). 3.2. Constant-crosshead-rate tension test The secant pseudo stiffness C and the damage parameter S are defined in terms of structural material integrity and the amount of damage in the material, respectively. According to Daniel and Kim

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(2002), a single characteristic curve can be found that describes the reduction in material integrity as damage grows in the specimen, independent of the applied loading conditions (cyclic vs. monotonic, amplitude/rate, frequency) and temperature. Thus, a single damage characteristic curve can be obtained from either a monotonic tension test or a cyclic loading test as long as the same damage parameter is used. Constant-crosshead-rate tests (Fig. 3(b)) were performed by applying a constant rate of deformation over the complete loading train. Because each component in the loading train (machine ram, load cell, etc.) deforms slightly, the on-specimen displacement rate or strain rate is not constant (Chehab, 2002). Because of this, the measured strains obtained from the linear variable differential transformers (LVDTs) attached to the cylinder specimens were used as the inputs to the model evaluation in this study. Tests were completed at a constant temperature of 5 °C and at multiple rates. The relationships between the pseudo stiffness and the damage parameter (e.g., Eq. (12)) based on the dynamic modulus and constant-crosshead-rate tests were determined as shown in Table 2 depending on the asphalt content and either dense-graded HMA or lime-modified HMA. 3.3. Numerical viscoelastic continuum damage computation The linear VE constitutive relationship is derived from the generalized Maxwell model without considering the continuum damage model (Biot, 1954). A computation method suggested by Mun (2009) can be used to calculate the nonlinear stress at a given strain input as follows, 

σ tn +

 1

" C ðSÞ n E∞ ε = ER

+ 1

M

+ ∑

m=1

res σm

#   n + 1 n + ∑ Modm ε −ε ; M

m=1

ð14Þ where εn and εn + 1 are strains at time tn and tn + 1, respectively. The variable σres m is defined as res

n

σm = exp ð−Δt = ρm Þσm ;

ð15Þ

where Δt is the time increment, and σnm is the state variable at time tn. The final variable Modm is defined as Modm =

ηm ½1− expð−Δt = ρm Þ; Δt

ð16Þ

where ηm is the mth coefficient of viscosity defined by Emρm. The ρm, Em, and E∞ are the coefficients shown in Eq. (2). Due to the nonlinear nature of damage described above, Newtontype iteration methods can be used to solve the equilibrium equations. The tangent stiffness needed in the solution procedure is obtained using an infinitesimal increase in the strain. Using Eq. (14), the following tangent stiffness can be determined by !   M ∂σ tn + 1 C ðSÞ E∞ + ∑ Modm = ER ∂ε m=1

ð17Þ

( ) C ðSÞ ηm M ∑ ½1− expð−Δt = ρm Þ : = E∞ + ER Δt m = 1

Fig. 2. Dynamic modulus (a) before master curve and (b) after master curve at reduced frequency and 25 °C.

The damage parameter, S, is updated using an incremental form of the rate-type evolution law (Eq. (9)) and the definition of pseudo strain energy density (Eq. (7)). The resulting updated procedure is R assumed given as ΔSupdate ) − WR(Sn)]/ +ΔSassumed }aΔt; n + 1 = {−[W (Sn + ΔSn + 1 n+1 update and Sn + 1 = Sn + ΔSn + 1 . The updating procedure is continuously carried out until the stress residue, |σtupdate − σtprevious |, is within an n+1 n+1

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Fig. 3. Test setup for (a) dynamic modulus test and (b) constant-crosshead-rate tension test.

error tolerance. If it is satisfied with the tolerance, then the next step is performed. 4. Prediction vs. measurement and fatigue-damage resistance results 4.1. Prediction vs. measurement results

specimens were used to predict the stress output histories. The measured and predicted strain vs. stress histories matched closely in the cases of dense-graded HMA with asphalt contents of 5.1%, 5.6%, 6.1%, and 6.6% (Figs. 6 and 7) and lime-modified HMA with asphalt contents of 4.8%, 5.3%, 5.8%, and 6.3% (Figs. 8 and 9). Based on the stress history prediction at 5 °C, the viscoelastic continuum damage analysis of HMA concretes can be conducted at any low temperature using the characteristic material parameters and

To verify the algorithm, the dynamic modulus master curve consisting of the components shown in Fig. 2 was constructed by shifting the modulus curve at multiple temperatures along the frequency axis using the time–temperature superposition principle. Thus, a dynamic modulus master curve at any targeted temperature can be constructed using the horizontal shifting principle in Eq. (13). As shown in Fig. 4, the Prony series representation (e.g., Eq. (2)) can be constructed using the dynamic modulus and phase angle data for the various mixtures (Mun et al., 2007; Mun and Zi, 2009). In addition, a series of crosshead-rate tension tests was conducted on the cylindrical specimens at 5 °C and at different strain rates using a servo-hydraulic closed-loop testing machine. The values (e.g., α in Eq. (9)) were found as shown in Table 3 using optimization search techniques. These fitted the pseudo stiffness C vs. damage parameter S closely for many values of strain rate and temperature (Mun, 2009; Mun and Geem, 2009). A low temperature of 5 °C was used for the evaluation of the prediction vs. the measurement. As shown in Fig. 5, the time vs. strain histories measured by the LVDTs attached to cylinder asphalt concrete

Table 2 Characteristic material parameters used for fitting pseudo stiffness functions. Materials

a*

b*

Dense-graded HMA with 5.1% asphalt content Dense-graded HMA with 5.6% asphalt content Dense-graded HMA with 6.1% asphalt content Dense-graded HMA with 6.6% asphalt content Lime-modified HMA with 4.8% asphalt content Lime-modified HMA with 5.3% asphalt content Lime-modified HMA with 5.8% asphalt content Lime-modified HMA with 6.3% asphalt content

0.0015703 0.0017216 0.0016698 0.0022088 0.0011091 0.0007388 0.0020003 0.0022623

0.5252920 0.50582002 0.5206491 0.5079348 0.5411329 0.5808464 0.5033175 0.4983656

*C(S) = exp (− aSb) shown in Eq. (12).

Fig. 4. Prony series representations for (a) dense-graded HMA mixtures and (b) limemodified HMA mixtures. “AC” denotes the asphalt content.

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Table 3 Material parameters shown in Eq. (9). Materials

α

Dense-graded HMA with 5.1% asphalt content Dense-graded HMA with 5.6% asphalt content Dense-graded HMA with 6.1% asphalt content Dense-graded HMA with 6.6% asphalt content Lime-modified HMA with 4.8% asphalt content Lime-modified HMA with 5.3% asphalt content Lime-modified HMA with 5.8% asphalt content Lime-modified HMA with 6.3% asphalt content

2.9027576 3.0120482 2.9744199 2.8835063 2.9559563 2.7948575 3.0553010 3.1123561

by moving the dynamic modulus master curve horizontally based on a shift factor at a specific low temperature. 4.2. Fatigue-damage resistance results According to the viscoelastic continuum damage mechanics, fatigue-damage resistance at two low temperatures, 5° and −10 °C, was determined by analyzing the time vs. pseudo stiffness (e.g., C(S)) curves for the same crosshead strain rate of 0.001/s applied to all dense-graded and lime-modified HMA mixtures. In this comparison, the resistance to fatigue damage at the specified strain rate could be evaluated, depending on the asphalt content and either dense-graded or lime-modified HMA mixtures. The numerical viscoelastic continuum damage computation algorithm discussed in Section 3.3 was used to compute stress histories from strain histories. Figs. 10 and 11 illustrate the results of analyzing stress vs. strain and time vs. pseudo stiffness in the cases of dense-graded and limemodified HMA mixtures, respectively, at 5 °C. Figs. 12 and 13 show the similar results obtained at −10 °C. Generally, higher asphalt content resulted in relatively less stiffness, as shown in Figs. 10(a), 11(a), 12(a), and 13(a). In order to determine the optimum asphalt contents of

Fig. 5. Strain measured by LVDT for (a) dense-graded HMA mixtures and (b) limemodified HMA mixtures. “AC” denotes the asphalt content.

Fig. 6. Comparison between measured and predicted stress in dense-graded HMA mixtures for (a) 5.1% asphalt content and (b) 5.6% asphalt content.

Fig. 7. Comparison between measured and predicted stress in dense-graded HMA mixtures for (a) 6.1% asphalt content and (b) 6.6% asphalt content.

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Fig. 8. Comparison between measured and predicted stress in lime-modified HMA mixture for (a) 4.8% asphalt content and (b) 5.3% asphalt content.

Fig. 9. Comparison between measured and predicted stress in lime-modified HMA mixture for (a) 5.8% asphalt content and (b) 6.3% asphalt content.

Fig. 10. Stress and damage function calculation for the dense-graded HMA concrete at a given strain rate and 5 °C. (a) Stress vs. strain and (b) damage function vs. time. “AC” denotes asphalt content.

Fig. 11. Stress and damage function calculation of the lime-modified HMA concrete at a given strain rate and 5 °C. (a) Stress vs. strain and (b) damage function vs. time. “AC” denotes asphalt content.

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Fig. 12. Stress and damage function calculation of the dense-graded HMA concrete at a given strain rate and − 10 °C temperature. (a) Stress vs. strain and (b) damage function vs. time. “AC” denotes asphalt content. Fig. 14. Comparison between dense-graded (DG) and lime-modified (LM) HMA mixtures at (a) 5 °C and (b) − 10 °C.

dense-graded and lime-modified HMA mixtures, the histories of time vs. pseudo stiffness are investigated, based on the fact that a large value of pseudo stiffness can be considered as higher material fatigue resistance. The optimum asphalt content, such as 5.6% and 5.3%, corresponded to dense-graded and lime-modified HMA mixture, respectively, can be determined by observing Figs. 11(b), 12(b), and 13(b) because the pseudo stiffness of the optimum asphalt content is relatively higher than others, except that the dense-graded HMA mixture with 6.6% asphalt content resulted in more fatigue resistance at 5 °C, as shown in Fig. 10(b). The case of dense-graded HMA mixtures shows the different optimum asphalt contents of 5.6% and 6.6% at −10 °C and 5 °C, respectively. Finally, as illustrated in Fig. 14, a comparison of the fatigue-damage resistance of dense-graded and lime-modified HMA mixtures shows the relative advantage of the limemodified HMA mixture because its pseudo stiffness value is equivalent or better than that in dense-graded HMA before the peak stress in which the mixture starts to fail. This failure phenomenon is discussed by Mun et al. (2004). 5. Conclusions

Fig. 13. Stress and damage function calculation of the lime-modified HMA concrete at a given strain rate and − 10 °C temperature. (a) Stress vs. strain and (b) damage function vs. time. “AC” denotes asphalt content.

This paper has presented an advanced fatigue-damage method for determining fatigue-damage resistance characteristics at low temperatures. This method is based on a numerical computation of the viscoelastic continuum damage mechanics of HMA mixtures, using the material stiffness of the dynamic modulus and the material parameters of damage mechanics obtained from lab tests; however, excepting a significant impact of aging and healing properties on the performance of a mixture in the field. As shown in Section 4, the results of stress vs. strain prediction at 5 °C agreed closely with those of the measured history. This paper also presented a computational

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viscoelastic continuum damage method, which is recently developed as well as provides a fatigue-damage resistance tool for determining the optimum asphalt contents in the cases of dense-graded and limemodified HMA mixtures, based on investigating the histories of time vs. pseudo stiffness. This provides the advantage of fatigue resistance in the case of the lime-modified HMA mixture, based on the pseudo stiffness comparison between HMA mixtures before the peak stress. References American Association of State Highway and Transportation Officials (AASHTO), 2003. TP-62 standard method of test for determining dynamic modulus of hot-mix asphalt concrete mixtures. Washington D.C. Anderson, D.A., Goetz, W.H., 1973. Mechanical behavior and reinforcement of mineral filler-asphalt mixtures. Association of Asphalt Paving Technologists 42, 37–66. Biot, M.A., 1954. Theory of stress–strain relations in anisotropic viscoelasticity and relaxation phenomena. Journal of Applied Physics 25, 1385–1391. Chehab, G., 2002. Characterization of asphalt concrete in tension using a viscoelastoplastic model. Ph.D. Dissertation, North Carolina State University, Raleigh, NC. Collins, R., 1988. Status Report on the Use of Hydrated Lime in Asphaltic Concrete Mixtures in Georgia. Georgia DOT, Materials and Research, Georgia. Daniel, J.S., 2001. Development of a simplified fatigue test and analysis procedure using a viscoelastic, continuum damage model and its implementation to WesTrack mixtures. Ph.D. Dissertation, North Carolina State University, Raleigh, NC. Daniel, J.S., Kim, Y.R., 2002. Development of a simplified fatigue test and analysis procedure using a viscoelastic continuum damage model. Association of Asphalt Paving Technologists 71, 619–650. Lee, H.J., Kim, Y.R., 1998. A uniaxial viscoelastic constitutive model for asphalt concrete under cyclic loading. Journal of Engineering Mechanics 124, 32–40. Little, D.N., Petersen, J.C., 2005. Unique effects of hydrated lime filler on the performance-related properties of asphalt cements: physical and chemical interactions revisited. Journal of Materials in Civil Engineering 17, 207–218. Mohammad, L.N., Abadie, C.D., Dranda, C., Wu, Z., Zhongie, Z., 2006. Permanent deformation analysis of hot-mix asphalt mixtures using simple performance tests

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