A Unified Continuum Damage Mechanics Model for Predicting the Mechanical Response of Asphalt Mixtures and Pavements

Transcription

1 International Journal of Roads and Airports (IJRA) ISSN [on-line] - A Unified Continuum Damage Mechanics Model for Predicting the Mechanical Response of Asphalt Mixtures and Pavements Rashid K. Abu Al-Rub 1, Masoud K. Darabi 1, Taesun You 1, Eyad A. Masad 1,2, Dallas N. Little 1 1 Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843, USA 2 Mechanical Engineering Program, Texas A&M University at Qatar, Doha, Qatar Date received: 8 February 211 Abstract. Predicting the performance of asphalt pavements is a very challenging task due to the complex behavior of asphalt mixtures and the complex traffic and environmental loading conditions to which these pavements are subjected. In this paper, a unified continuum-based mechanistic model is presented and its effectiveness in predicting rutting and damage evolution in asphalt pavements is demonstrated. The model couples the characterization of nonlinear viscoelastic, viscoplastic, and viscodamage behavior of asphalt mixtures. Due to the very high computational cost of predicting the rutting of asphalt pavements through conducting threedimensional (3D) finite element (FE) simulations under many loading cycles, an effective and simple extrapolation technique is presented for predicting rutting based on conducting 3D FE simulations for a relatively small number of loading cycles and extending the analysis with twodimensional (2D) FE simulations for very many loading cycles. Further, it is shown that the unified continuum damage mechanics model presented herein can also be used in conducting novel 2D and 3D micromechanical simulations of asphalt mixtures where each constituent (e.g., aggregate, mastic) is modeled explicitly. Such micromechanical simulations can be used in guiding the material design of more rutting- and damage-resistant asphalt mixtures and pavements. Keywords. Damage; Finite Element Simulations; Rutting; Viscoelasticity; Viscoplasticity. This work is licensed under the current Creative Commons Attribution-Noncommercial License. Corresponding Author, D-Little@tamu.edu v. 1, n. 1 (211) 68-84

2 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 69 1 Introduction Asphalt mixtures are highly complex composite materials which can be viewed at three scales: (a) the micro-scale (mastic), where fine fillers are surrounded by the asphalt binder; (b) the meso-scale, fine aggregate mixture, where fine aggregates are surrounded by the mastic; and (c) the macro-scale which includes all the coarse aggregates surrounded by fine aggregate mixture. The material response of asphalt mixtures is time-, rate-, and temperature- dependent and exhibits both recoverable (viscoelastic) and irrecoverable (viscoplastic) deformations. The evolution of micro-cracks and micro-voids and rate-dependent plastic (viscoplastic) hardening offer other major sources of nonlinearity in the thermo-mechanical response of asphalt mixtures. There have been few attempts in the literature to development realistic mechanisticbased (not empirical-based) constitutive models for asphalt mixtures. Therefore, this area of research is still at a novel stage and more studies are needed for the development of more accurate constitutive models for asphalt binder and asphalt mixtures. Most of the work regarding modeling the viscoelastic response of asphalt mixtures has employed linear viscoelasticity theory as compared to nonlinear viscoelasticity theory (see the works of Kim and co-workers [1-5] and the references quoted therein). Nonlinear viscoelasticity as discussed in this paper implies that the viscoelastic strain at a specific stress level cannot be obtained from the viscoelastic response at a different stress level even in the absence of damage (i.e., one cannot employ the superposition principle). Recently, Masad and coworkers employed Schapery s nonlinear viscoelasticity model [6] to more accurately model the responses under loading of asphalt mixtures (see for example [7-1] and the references quoted therein). However, it is noteworthy that Kim and co-workers attributed the nonlinear viscoelastic response to damage evolution and modified the linear viscoelastic model accordingly to include nonlinearity. In terms of the viscoplastic behavior of asphalt mixes, the Perzyna s theory [7] has been extensively used to predict permanent deformation in asphalt mixtures (see for example, [8-12]). However, modeling the convolved viscoelastic and viscoplastic behavior alone of asphaltic mixtures is not sufficient to predict the performance of asphalt pavements. Microdamage (micro-cracks and micro-voids) evolution under different loading conditions must also be modeled for more accurate predictions of pavement performance. Models based on continuum damage mechanics [13] have been effectively used to model the degradation and damage evolution in asphalt mixtures (see for example [3-5, 9, 11, 14-15]). A more physically-sound approach based on cohesive zone modeling [16] has been recently employed for modeling crack nucleation and evolution in asphalt concrete [17-18]. The identification of the material constants associated with this approach is one of the most challenging tasks. Because of the complex behavior of asphalt mixtures, the necessity to couple the nonlinear thermo-viscoelasticity, thermo-viscoplasticity, and temperature- and rate-dependent damage (thermo-viscodamage) modeling seems inevitable. However, surprisingly, very limited work has focused on the development of such unified coupled continuum damage mechanics models. This paper employs the recently developed coupled nonlinear viscoelastic, viscoplastic, and viscodamage constitutive model by Darabi et al. [11] which has been derived consistently based on the laws of thermodynamics [19]. Therefore, the main objective of this paper is the use of this model to predict the microscopic and macroscopic behavior of asphalt mixtures.

3 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 7 It should be emphasized that continuum (either macroscopic or microscopic) mechanics models can be used to model the response of composite materials at different length scales. In this paper, the model developed by Darabi et al. [11] is used to investigate the response of asphalt mixtures at two different length scales: 1) at the macroscale level, where the asphalt mixture is modeled as a homogenous one-phase material with effective material properties that result from the combination of the individual properties of each phase; and 2) at the microscale level, where different phases of the mixture are considered as independent entities, each one with its own constitutive relationships. Therefore, first, 2D and 3D macroscopic finite element simulations are conducted to predict the rutting performance of asphalt pavements. However, due to the very high computational cost of conducting 3D performance simulations under many loading cycles and realistic loading conditions, a simplified extrapolation technique is proposed to predict rutting based on combined 2D and 3D finite element simulations. A comparison with rutting experimental data from a Wheel Tracking Test shows the robustness of the proposed extrapolation technique. Second, 2D and more realistic 3D micromechanical finite element simulations are conducted in order to show the effectiveness of the unified continuum damage mechanics model in predicting the overall macroscopic response of asphalt mixtures in terms of key microstructural features (e.g., aggregate properties and distribution, mastic properties, air voids distribution, etc). Such simulations are shown to be very useful in guiding the material design of distressresistant asphalt mixtures and pavements. 2 Unified Continuum Damage Mechanics Model The research team at Texas A&M University under the Asphalt Research Consortium (ARC) has been heavily involved in the development of a 3D unified continuum-based damage model for asphalt mixtures that couples nonlinear viscoelasticity, viscoplasticity, and viscodamage (i.e., rate- and time-dependent damage) [11, 14, 19-2]. The model relies on the decomposition of the total strain,ε, into a viscoelastic (recoverable) component, ve vp ε, and a viscoplastic (irrecoverable) component, ε, such that: ve vp ε = ε + ε The viscoelastic strain tensor ε ve is calculated based on Schapery s nonlinear viscoelastic model [6]: ( σ ) ψ τ 2 g : σ g1 ( ) d dτ ve t d g ε = D + ΔD ψ ψ τ (1) where D is the instantaneous compliance for undamaged material; N ΔD= D [1 exp( )] n 1 n λnt is the transient compliance for undamaged material with D and = n R tn = 1/ λ being the Prony series parameters and relaxation times, respectively; σ is the undamaged (effective) stress tensor; g n, g, and 1 g 2 are nonlinear parameters that can be a function of stress (when g = g 1 = g 2 = 1, a linear viscoelastic model is retrieved); and t t ( ) 1 ψ = aa d T e t is the reduced-time which is expressed in terms of time-temperature shift factor, a, and time-environment (e.g. aging, moisture) shift factor. Note that the superimposed bar indicates a quantity that is for the effective (undamaged or intact) T material.

4 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 71 For simplicity, the following Arrhenius-type expression for the time-temperature shift factor, a, is assumed in order to incorporate the effect of temperature on the nonlinear T viscoelastic response of asphalt mixtures, such that: T at = exp δ 1 (2) T where T is the reference temperature and δ is the viscoelasticity temperature coupling material parameter. vp The viscoplastic strain rate & ε is given as a Perzyna-type viscoplastic model [7]: ( ) / vp vp N & ε =Γ T f σ y g σ (3) vp where Γ is the fluidity parameter, which controls the rate of viscoplastic strain evolution and is a function of temperature T ; f is the yield function which controls the onset of viscoplastic deformation; g is the viscoplastic potential, which is different than f for nonassociative viscoplasticity and which controls the direction of viscoplastic strain tensor; N characterizes the material s rate sensitivity σ is the compressive yield strength; and y is the Macaulay bracket defined by x = ( x+ x) /2. The functions f and g are defined as follows: f = τ αi 1 σ y, g = τ β I1, 3J 1 1 3J τ = vp vp d d 3 J2 where α and β are material parameters related to the material s internal friction; I 1 is the first invariant of σ ; J 2 and J 3 are the second and third deviatoric invariants of σ, respectively; and.78 d 1 is a material parameter which considers the distinction of asphalt vp mix behavior in compression and extension loading conditions. The flow stress σ y is assumed to have the following exponential form: vp κ = κ + κ 1 1 exp( κ2ε e ) (5) where κ, κ 1, and κ 2 are material parameters defining the initial yield stress, the saturated yield stress, and the strain hardening rate, respectively, at the reference temperature T. The viscoelastic and viscoplastic constitutive relations in Eqs. (1)-(4) are expressed in terms of the undamaged (effective) stress tensor σ since the stress in the intact (undamaged) material is the true stress that drives further viscoelastic and viscoplastic deformations as well as damage evolution. Based on continuum damage mechanics [21], the effective stress σ is related to the damaged (apparent) stress σ as follows: σ σ = (6) m 1 φ m where φ 1 is the damage density due to mechanical loading (representing the area of m m micro-cracks and micro-voids per unit area) such that φ = for no damage and φ = 1 when the material is completely damaged (i.e., fracture). The superscript in the above equations designates quantities for undamaged material. Darabi et al. [11] proposed the following evolution law for φ m which has been validated against a large set of experimental data under complex multi-axial loading conditions: (4)

5 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 72 q 2 ( T) Y Y ( k ) & m vd φ =Γ (1 φ) exp ε (7) e vd where Γ is the damage viscosity parameters that controls the rate of damage evolution and is a function of temperature T ; the material parameters Y, q, and k are obtained at a reference temperature T ; and ε e = ε : ε is the effective total strain. The term Y is the damage driving force which is assumed to have an analogous expression as τ in Eq. (4) (i.e., Y = τ ), such that: 3J 1 1 3J Y = vd vd d d 3 J2 vd where d > is the material parameter that is yields a distinct behavior between the asphalt mixture damage behavior under extension or compression loading conditions. For simplicity, the temperature-dependency of the viscoplastic and viscodamage behaviors can be considered by assuming that the viscosity parameters Γ vp and Γ vd are expressed in terms of time-temperature shift factor a in Eq. (2) such that: T vp vp / vd vd Γ =Γ a T, / a T Γ =Γ (9) vp vd where Γ and Γ are reference viscosities obtained at a reference temperature T. Abu Al-Rub et al. [2] and Darabi et al. [11] implemented the above unified continuum damage mechanics model into Abaqus [22] via the user material subroutine UMAT. This model has been validated extensively against a large set of experimental data under different multiaxial loading conditions and various temperatures. For more details, the reader should refer to [11-12, 14, 19-2]. It is noteworthy that the above continuum damage mechanics model can be used to simulate either the homogenized mechanical response of an asphalt mixture or the mechanical response of the mastic (i.e., binder with fine aggregates) in an asphalt mixture but with different material parameters. Therefore, this model can be used either to predict the fatigue damage and rutting performance of asphalt pavements or to predict the microstructural response of asphalt mixtures by assigning different properties of each phase in the asphalt mixture. (8) 3 Identification of the Material Parameters A detailed description of the procedure used to determine viscoelastic, viscoplastic, and viscodamage model parameters was presented by Abu Al-Rub et al. [2] and Darabi et al. [11, 19]. This procedure is summarized as follows. The viscoelastic and viscoplastic model parameters are determined using single creep-recovery tests in compression at low stress levels for which one can assume that the induced damage in the material is negligible. The first step of the calibration procedure is to separate the viscoelastic strain from the total strain in a creep-recovery compression test. Since the viscoplastic strain during the recovery process is constant, which is equal to its value at the end of the loading step (or equivalently at the end of the unloading step), the Prony series coefficients ( D and n λ n ) are determined first by analyzing the recovery strain at room temperature. Once the Prony series coefficients are obtained, the viscoplastic strain can be determined by subtracting the viscoelastic strain from the total strain. Consequently, the viscoplastic material parameters are

6 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 73 determined by analyzing the viscoplastic strains for a uniaxial compression test. Subsequently, the viscodamage model parameters can be obtained using two creep tests in compression that include tertiary creep responses. The damage viscosity parameter at the refer- vd ence temperature, Γ, damage force at a reference condition, Y, and model parameter k can all be obtained from a creep test at the reference temperature and an arbitrary stress level that can be assumed as the reference stress. Once these damage parameters are determined, the stress-dependency parameter q will be determined from a creep test at the reference temperature but at a stress level different from the reference stress. Moreover, the two model parameters which are responsible for distinguishing the loading in tension and compression modes, d vp and d vd, are obtained from a creep-recovery tension tests. The vp value of d is obtained by adjusting model predictions with single creep-recovery experimental data in tension, while another creep test in tension that includes tertiary creep response is used to determine d. vd A large set of experimental data from University of Nottingham database was used to determine model material parameters. The asphalt mixture used in this study was a 1 mm dense bitumen Macadam which was a continuously graded mixture with an asphalt binder content of 5.5%. Granite aggregates and an asphalt binder with a penetration grade of 7/1 were used in preparing the asphalt mixtures. The identified material constants associated with this asphalt mixture are outlined in Tables 1-3 where the reference temperature is T = 2 C. These material parameters will be used in the subsequent numerical examples. Table 1. Viscoelastic model parameters. N λ (Sec -1 ) n C n (kpa -1 ) C (kpa -1 ) μ 1.73 δ Table 2. Viscoplastic model parameters. vp α β Γ (Sec -1 ) N κ (kpa) κ 1 (kpa) κ Table 3. Viscodamage model parameters. vd Γ (Sec -1 ) Y (kpa) q k Applications of the Unified Continuum Damage Mechanics Model 4.1 General layout Rutting is one of the most serious distresses in asphalt pavements affecting the pavement performance and service life. Therefore, the accurate simulation of rutting in asphalt pave-

7 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 74 ments is essential as a basis for understanding the limitations of the asphalt mixtures used in a specific pavement application and for improving the performance of a given mixture. The main mechanism of rutting is the accumulation of permanent deformation that increases progressively with increasing number of loading cycles. However, the complex nature and impact of accumulated damage over a very large number of loading cycles (millions of loading cycles), and complex constitutive behavior of asphaltic materials make the accurate prediction of rutting a very difficult and challenging task An extrapolation technique for predicting rutting at large loading cycles Recently, Abu Al-Rub et al. [23] showed that the finite element prediction of rutting in pavements using simplified assumptions such as 2D plane strain analysis instead of the 3D analysis will significantly overestimate rutting (almost doubling the rutting), but will significantly reduce the computational cost. However, 2D simulations give qualitative agreements with 3D simulations. Therefore, in this section a simple yet accurate method for extrapolating the results of 3D FE analysis based on the results of 2D FE analysis is proposed. The extrapolation equation is expressed as follows: u 3 DN, ref u 3 D, N rutting 2 D, N rutting u 2 DN, rutting ref urutting = (1) 3 DN, where u is the extrapolated rutting at N cycles from 3D simulations, 3 DN, ref rutting u and rutting 2 DN, ref u are the predicted rutting at a reference cycle N rutting ref from the 3D and 2D simulations, 2, respectively, and u DN rutting is the calculated rutting after N cycles from 2D simulations. Since permanent (viscoplastic) displacement is not considered as a degree of freedom at the element nodes in the classical finite element method, it is not possible to calculate the permanent surface deformation (i.e., rutting) directly. However, the magnitude of rutting can be calculated numerically by integrating the magnitude of the viscoplastic deformation through the pavement thickness. This can be achieved by dividing the thickness of the asphalt layer into a number of sub-layers, such that the rutting depth can be calculated as follows: u k vp() i () i rutting = ε h (11) i= 1 where u is the permanent displacement (rutting), vp( i ) rutting ε is the vertical viscoplastic strain at i th () i layer through the depth of the asphalt layer, and h is the i th layer thickness. In the following discussion and example, the rutting is only calculated at the center of the pavement slab for the purpose of conducting the numerical comparisons. The geometry of the Wheel Tracking Test simulated in this section and the corresponding 3D and 2D finite element meshes is illustrated in Figure 1. The wheel tracking test consists of an asphalt slab that is mm 3 in length, width, and depth, respectively. A wheel load is applied at the mid-width of the slab along and that moves back and forth along the length of the slab. The wheel moves with a speed of 4 passes per minute over a wheel path length of 23 mm, which is equivalent to a.55 km/hr speed. The wheel loading area is assumed as a rectangular shape with dimensions of 2 5 mm 2 in width and length, respectively. The employed asphalt layer, dimensions of the loading area, and the wheel speed are from the Wheel Tracking Test conducted by Hunter et al. [24]. The loading is applied as a step load within each loading cycle. Because of the symmetric nature of

8 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 75 the wheel loading condition and the slab geometry, the finite element model can be reduced to a half of the slab by constraining the horizontal direction on the vertical edge of the model to represent the middle of the slab. The boundary conditions in both 2D and 3D FE models are imposed such that the horizontal direction on the opposite side of the symmetric boundary is fixed; while the bottom of the slab is fixed in the vertical direction. The used element types in the 2D plane strain and 3D FE simulations in Abaqus are a plane strain four-node element with reduced integration (CPE4R) and a 3D eight-node element with reduced integration (C3D8R), respectively. The loading level is 77 kpa and is applied on the top of the asphalt layer. For simplicity, the shape of the applied load is assumed rectangular. Moreover, frictional and tangential loadings from the contact of the wheel with the asphalt top surface are neglected in this study. 1 mm 35 mm 14 mm Figure 1. Geometry and finite element mesh for (a) the 3D simulation and (b) the 2D simulation Figure 2 shows the extrapolated results at 2 C. Reference cycles of 2, 1, 2, and 3 repetitions were chosen for comparison. Figure 2 shows that using the reference rutting at a small number of cycles (2 th cycle) does not yield accurate extrapolation compared to the calculated rutting from 3D simulations. The extrapolation from a reference rutting at 1 th cycle or greater yields accurate predictions of rutting as compared with the rutting predictions from the 3D simulations. Hence, the proposed extrapolation technique based on the 2D rutting predictions gives an efficient method to predict and to extrapolate the rutting from the 3D simulations to a large number of loading cycles and, thus, significantly reducing the computational cost. The accuracy of the proposed extrapolation technique depends on N ref which is different from one mixture to another and varies with the temperature at which rutting is calculated. Therefore, one needs first to run the 3D simulations for a small number of cycles in order to identify N ref which yields an accurate extrapolation. As an example, the evolution of viscoplastic strain distribution at different loading cycles for the 3D FE simulation is plotted in Figure 3. Figure 3 shows that the maximum viscoplastic strain occurs at the top of the middle part of the asphalt layer which is consistent with previous studies [see e.g. [12]]. Moreover, it also shows that as the number of loading cycles increases, the compressive viscoplastic strain extends toward both top and bottom of the pavement which contributes to more permanent deformation. Also, damage distribution

9 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 76 contours are plotted for the same problem in Figure 4, which shows that the maximum damage occurs at the top of the middle part of asphalt layer which is exactly the region where the maximum viscoplastic strain occurs Rutting (mm) moving Loading (calculated) reference N=2 (extrapolation) reference N=1 (extrapolation) reference N=2 (extrapolation) reference N=3 (extrapolation) Cycles (N) Figure 2. The extrapolation of the 3D rutting predictions based on the 2D predictions Comparing with rutting experimental data In this section, the Wheel Tracking Test is modeled and the results are compared with experimental measurements at temperature 35 C for the 1 mm dense bitumen Macadam (DMB) asphalt mixtures. The material parameters associated with the nonlinear viscoelastic, viscoplastic, and viscodamage constitutive equations are listed in Tables 1, 2, and 3, respectively. The slabs of DMB materials with the dimensions of mm 3 were manufactured using a roller compactor. Materials were compacted in rigid molds using a roller compactor designed to simulate the action of the site compaction plant. The mold was moved back and forth under the rolling compactor to simulate a rolling action. The steel wheel applies 77 kpa moving load to the center of the slab with the frequency of 4 passes per minute. The total number of 96, loading cycles was applied to the slab and the rutting depth was measured every five minutes. Figure 5 shows the comparison between the experimental measurements and the 2D simulation results and the 3D extrapolation results. The 2D finite element simulates the rutting up to 96, cycles, while the 3D finite element only simulates the rutting up to 1, cycles (i.e. N ref = 1 ). Then, the extrapolation technique [Eq. (1)] was employed to predict the rutting in 3D up to 96, cycles. The results show that the rutting from the 2D simulation significantly overestimates the experimental measurements. However, the extrapolated results agree reasonably well with the experimental measurements where the error at loading cycle 96, is about 1%. Moreover, the rate of rutting from the 3D extrapolation is comparable to the experiment measurements. Therefore, the extrapolation method based on the 2D simulation results has the capability to predict the rutting from the 3D simulations.

10 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 77 N=1 Cycle N=2 Cycles N=4 Cycles N=6 Cycles Figure 3. Viscoplastic strain distribution contours at different loading cycles for the 3D FE analysis at T = 2 o C.

11 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 78 N=1 Cycles N=2 Cycles N=4 Cycles N=6 Cycles Figure 4. Damage distribution contours at different loading cycles for the 3D FE analysis at T = 2 o C.

12 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little Rutting (mm) Experimental Measurements 2D simulation results 3D extrapolation results Cycles (N) Figure 5. Comparing 2D FE rutting predictions and extrapolated results with experimental data from a Wheel Tracking Test at temperature of 35 C 4.2 2D Micromechanical modeling of asphalt concrete response The constitutive model presented herein can be used to conduct micromechanical simulations of representative volume elements (RVEs) with realistic microstructures (Fig. 6). Such simulations can be used to guide the material design of distress-resistant asphalt mixtures for improved pavement performance. Therefore, by varying the properties of constituents of the asphalt mixture and their spatial distributions, one can predict the overall macroscopic response of asphalt mixtures under different loading conditions. Figure 6 shows an example of such a simulation where the RVE is generated using X-ray computed tomography with a realistic 2D microstructure. However, for simplicity, the effect of initial air voids in not considered in the current simulation, but it can be included in a straightforward manner. The aggregates are assumed to behave linearly elastic with a 25 GPa Young s modulus and.25 Poisson s ratio. The mastic surrounding the aggregates is assumed to behave as a nonlinear viscoelastic, viscoplastic, and viscodamage material with the material constants presented in Tables 1-3. It can be seen from Figure 6 that the current constitutive model can predict the variation of the composite response under different tensile strain rates. Moreover, crack-patterns at failure are shown in Figure 6c for the simulated tensile strain rates. It is apparent that the crack density increases as the rate of loading increases which qualitatively agree with experimental observations. These simulations can be repeated for different loading conditions, temperatures, and asphalt microstructures. However, this is beyond the scope of this paper and will be presented in details elsewhere.

13 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 8 2 mm Stress (MPa) E-3/second 1E-4/second 1E-5/second 1E-6/second (a) 2 mm Strain (%) (b) (c) Rate 1-5 /second Rate 1-4 /second Rate 1-3 /second Figure 6. Micromechanical simulations under uniaxial tension showing (a) asphalt microstructure (5% aggregate volume fraction) and finite element mesh, (b) stress-strain responses, and (c) crackpattern evolution at various strain rates. The simulations are conducted at 2 C 4.3 3D Micromechanical modeling of asphalt concrete response In order to conduct more realistic micromechanical simulations that can be used to effectively and accurately guide the design of fracture-resistant asphalt mixtures and pavements, 3D micromechanical simulations are desirable. In fact, due to the high complexity and very expensive computational cost, there have been very few attempts to model the 3D microstructure of asphalt mixtures based on the discrete element method [25-29] or the finite element method [3]. However, those few 3D micromechanical studies were limited, such that: 1) idealized microstructural representations were generated that do not sufficiently represent the real microstructure (i.e., aggregate shape, gradations, orientations, and angularities) of asphalt mixtures; 2) the effect of air voids was not considered; and 3) the material was assumed to be elastic or linear viscoelastic. The 3D microstructure of the asphalt mixtures can be generated based on the real images obtained from the X-ray computed tomography. For example, 2D X-ray computed tomography images (Fig. 7) were used to construct a realistic 3D finite element model as shown in Figure 8. The diameter and height of the generated virtual specimen is 5 mm by 75 mm, respectively. The unified continuum damage mechanics framework described herein was used to simulate the mechanical response of the mastic surrounding the aggregate using the material constants in Table 1-3 at room temperature under uniaxial tension. The aggregates are assumed to behave linearly elastic with a 25 GPa Young s modulus and.25 Poisson s ratio. The corresponding viscoelastic strain distribution, viscoplastic strain distribution, and crack pattern evolution are shown in Figure 9. It is obvious that such novel 3D micromechanical simulations can be used more effectively for establishing the propertystructure relationship and for identifying the key microstructural parameters for enhanced performance.

14 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 81 (a) (b) (c) Figure 7. CT image: (a) gray-scale image, (b) contrasted image, and (c) image with well-separated aggregates. + + = (a) (b) (c) (d) Figure 8. 3D finite element mesh shown in (d) and constructed by adding (a) large aggregates, (b) mastic (i.e., binder with fine aggregates), and (c) air voids (a) (b) (c) Figure 9. Mechanical response at room temperature under uniaxial compressive loading showing (a) viscoelastic and (b) viscoplastic strain distributions, and (c) crack patterns. Only half of the specimen is shown and in (a) and (b) aggregates are not shown

15 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 82 5 Summary In this paper, a unified continuum damage mechanics constitutive model is presented that can be used effectively in predicting the complex nonlinear behavior of asphalt mixtures and pavements. It is shown that this unified model can be used for conducting two types of very useful computer simulations that can aid in the design of more distress-resistant and long-lasting pavements: a. Performance simulations of an asphalt pavement structure to predict fatigue damage and rutting due to different traffic and environmental loading conditions; b. Micro-mechanical simulations that can be used for virtual testing to guide the design of the asphalt mixture s microstructure for enhanced fatigue resistance and rutting due to different loading, moisture, and temperature effects. Moreover, since the 2D simulations are computationally inexpensive compared to the 3D simulations, an extrapolation technique is proposed to extrapolate the results from 2D simulations to 3D. Therefore, one can establish the extrapolation relation by conducting the 2D simulation for a large number of loading cycles and the 3D simulation under moving loading for a small number of cycles. Using the proposed extrapolation expression, the rutting depth for a large number of loading cycles can be predicted with reasonable accuracy. This approach is used to compare the experimental measurements of the Wheel Tracking Test and to the model predictions using the extrapolation technique. The comparison shows that the proposed extrapolation technique is capable of predicting the experimental data, and that applying this methodology saves offers great savings in computational time. The effects of more realistic wheel-pavement contact stresses and the wheel speed is the focus of future work by the authors. Finally, 2D and more realistic 3D micromechanical FE models have been constructed based on X-ray computed tomography images. The unified constitutive model presented herein is being successfully used to predict the overall macroscopic response of the asphalt mixture based on the distinct mechanical response of its constituents. Such micromechanical simulations can be used to conduct virtual testing of the behavior of asphalt mixtures behavior under different loading conditions and considering key microstructural features. Therefore, these simulations allow one to save significant time and cost in experimentally examining the behavior of an asphalt mix, and in exploring the key microstructural features and mechanistic behavior that leads to design of asphalt mixtures for long-lasting pavements. The authors research is directed toward continuing validation of this approach. Acknowledgment Authors acknowledge the financial support provided by the Federal Highway Administration through the Asphalt Research Consortium, and the partial financial support by the Qatar National Research Fund. References 1. Kim, Y.R. and D.N. Little, One-Dimensional Constitutive Modeling of Asphalt Concrete. Journal of Engineering Mechanics-Asce, (4): p Kim, Y.R. and D.N. Little, One-dimensional constitutive modeling of asphalt concrete. Journal of

16 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 83 engineering mechanics, : p Lee, H.J., J.S. Daniel, and Y.R. Kim, Continuum damage mechanics-based fatigue model of asphalt concrete. Journal of Materials in Civil Engineering, 2. 12: p Lee, H.J. and Y.R. Kim, Viscoelastic continuum damage model of asphalt concrete with healing. Journal of Engineering Mechanics-Asce, (11): p Park, S.W., Y.R. Kim, and R.A. Schapery, A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete. Mechanics of Materials, (4): p Schapery, R.A., On the characterization of nonlinear viscoelastic materials. Polymer Engineering & Science, (4): p Perzyna, P., Thermodynaic Theory of Viscoplastcity. Advances in Applied Mechanics, : p Masad, E., et al., Viscoplastic modeling of asphalt mixes with the effects of anisotropy, damage and aggregate characteristics. Mechanics of Materials, (12): p Saadeh, S., E. Masad, and D. Little, Characterization of hot mix asphalt using anisotropic damage viscoelastic-viscoplastic model and repeated loading. ASCE Journal of Materials in Civil Engineering, : p Masad, E., S. Dessouky, and D. Little, Development of an elastoviscoplastic microstructural-based continuum model to predict permanent deformation in hot mix asphalt. International Journal of Geomechanics, 27. 7: p Darabi, M.K., et al., A thermo-viscoelastic-viscoplastic-viscodamage constitutive model for asphaltic materials. International Journal of Solids and Structures, (1): p Huang, C.W., et al., Three dimensional simulations of asphalt pavement performance using a nonlinear viscoelastic-viscoplastic model. ASCE Journal of Materials in Civil Engineering, (1): p Kachanov, L.M., Introduction to continuum damage mechanics. Mechanics of elastic stability , Dordrecht ; Boston: M. Nijhoff. x, 135 p. 14. Abu Al-Rub, R.K., et al., A micro-damage healing model that improves prediction of fatigue life of asphalt mixes. International Journal of Engineering Science, (11): p Uzan, J., Viscoelastic-viscoplastic model with damage for asphalt concrete. Journal of Materials in Civil Engineering, (5): p Ortiz, M. and A. Pandolfi, A class of cohesive elements for the simulation of three-dimensional crack propagation. International Journal for Numerical Methods in Engineering, : p Kim, Y.R., D.H. Allen, and D.N. Little, Damage-Induced Modeling of Asphalt Mixtures through Computational Micromechanics and Cohesive Zone Fracture (ASCE). Journal of Materials in Civil Engineering (5): p Kim, Y.R., D.H. Allen, and D.N. Little, Computational constitutive model for predicting nonlinear viscoelastic damage and fracture failure of asphalt concrete mixtures. International Journal of Geomechanics, 27. 7: p Darabi, M.K., et al., Thermodynamic based model for coupling viscoelastic, viscoplastic, and viscodamage constitutive behavior of asphalt mixtures. International Journal for Numerical and Analytical Methods in Geomechanics (in press), Abu Al-Rub, R.K., E.A. Masad, and C.W. Huang, Improving the Sustainability of Asphalt Pavements through Developing a Predictive Model with Fundamental Material Properties. Final Report submitted to Southwest University Transportation Center, Report # SWUTC/8/ , 29: p Kachanov, L.M., On time to rupture in creep conditions (in Russian). Izviestia Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, : p Abaqus, Version 6.8. Habbit, Karlsson and Sorensen, Inc, Providence, RI, Abu Al-Rub, R.K., et al., Comparing finite element and constitutive modeling techniques for predicting rutting of asphalt pavements. International Journal of Pavement Engineering (in press), Hunter, A.E., G.D. Airey, and O. Harireche, Numerical modeling of asphalt mixture wheel tracking experiments. International Journal of Pavement Engineering & Asphalt Technology, 27. 8: p Adhikari, S. and Z.P. You, 3D discrete element models of the hollow cylindrical asphalt concrete specimens

17 Rashid K. Abu Al-Rub, Masoud K. Darabi, Taesun You, Eyad A. Masad, Dallas N. Little 84 subject to the internal pressure. International Journal of Pavement Engineering, (5): p Collop, A.C., G.R. McDowell, and Y.W. Lee, Modelling dilation in an idealised asphalt mixture using discrete element modelling. Granular Matter, 26. 8(3-4): p You, Z.P., S. Adhikari, and Q.L. Dai, Three-Dimensional Discrete Element Models for Asphalt Mixtures. Journal of Engineering Mechanics-Asce, (12): p You, Z.P. and Y. Liu, Three-Dimensional Discrete Element Simulation of Asphalt Concrete Subjected to Haversine Loading An Application of the Frequency-Temperature Superposition Technique. Road Materials and Pavement Design, (2): p Collop, A.C., G.R. McDowell, and Y.W. Lee, On the use of discrete element modelling to simulate the viscoelastic deformation behaviour of an idealized asphalt mixture. Geomechanics and Geoengineering, 27. 2(2): p Mo, L.T., et al., 2D and 3D meso-scale finite element models for ravelling analysis of porous asphalt concrete. Finite Elements in Analysis and Design, (4): p