Finite element cohesive fracture modeling of airport pavements at low temperatures

Cold Regions Science and Technology 57 (2009) 123–130

Contents lists available at ScienceDirect

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

Finite element cohesive fracture modeling of airport pavements at low temperatures Hyunwook Kim ⁎, William G. Buttlar Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

a r t i c l e

i n f o

Article history: Received 15 September 2008 Accepted 9 February 2009 Keywords: Asphalt pavements Fracture Cohesive zone model Finite element method Viscoelastic

a b s t r a c t Low temperature cracking induced by seasonal and daily thermal cyclic loads is one of the main critical distresses in asphalt pavements. The safety of aircraft departure and landing becomes a crucial issue in runways when thermal cracks occur in airport pavements. The low-temperature fracture behavior of airport pavements was investigated using a bilinear cohesive zone model (CZM) implemented in the finite element method (FEM). Nonlinear temperature gradients of pavement structures were estimated based on national weather data and an integrated climate prediction model. Experimental tests were conducted to obtain the numerical model inputs such as viscoelastic and fracture properties of asphalt concrete using creep compliance tests, indirect tensile strength tests (IDT), and disk-shaped compact tension (DC(T)) tests. The finite element pavement fracture models could successfully predict the progressive crack behavior of asphalt pavements under the critical temperature and heavy aircraft gear loading conditions. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Cracks in highways are significant causes of structural degrading and they increase the maintenance and rehabilitation cost of pavement. The cracking phenomenon can occur through daily thermal changes and heavy traffic loads, i.e., trucks in highways and aircrafts in airports (Kim and Buttlar, 2002). Cyclic temperature variations are believed to be one of the primary causes of pavement cracking, especially in very cold climates (Rigo, 1993). It was described that both seasonal and daily temperature changes cause the thermal contraction of pavements and produce the critical tensile stress in pavement materials (Mukhtar and Dempsey, 1996). In cold climatic areas, cracks often occur upon the completion level of pavement rehabilitation in traditional pavement structures after the first or second winter (Buttlar and Bozkurt, 2002). When heavy traffic loads are applied onto the pavement surface at cold temperature, the tensile stress in pavement materials becomes critical due to the high stress concentration at the weakest or discontinuing points of pavement (Lytton, 1989). The thermal cracks are more significant in the airport pavements of cold regions due to the thicker pavement layers and the impact loads by heavy aircraft gears. Furthermore, the frequent maintenance and rehabilitation of airport pavements has caused many aircraft delays in international airports. In this study, a cohesive zone model approach was applied to represent the crack initiation and propagation of asphalt pavement in the finite element method (FEM). Using an enhanced integrated climate model (EICM) developed by the American Association of State Highway and Transportation Officials (AASHTO), nonlinear tempera-

⁎ Corresponding author. Tel.: +41 44 823 4474; fax: +41 44 821 6244. E-mail address: [email protected] (H. Kim). 0165-232X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2009.02.004

ture gradients through pavement layers could be obtained based on the US National Weather Database (NCHRP 1-37A, 2004). The determined temperature gradients were projected into pavement fracture models to investigate the thermal crack behavior in airport pavements. Also, heavy aircraft loading conditions were applied into FE pavement fracture models to understand the fracture behavior of pavements in thermo-mechanical loading conditions. Viscoelastic properties and fracture parameters of asphalt materials were obtained by a series of experimental tests and they were used as the input values of FE fracture models to characterize the fracture behavior of airport pavements.

2. Constitutive laws 2.1. Cohesive zone model The cohesive zone model is an efficient tool to predict the damage evolution in the fracture process zone located ahead of a crack tip in materials. This model, which involves nonlinear constitutive laws described by displacement jump and the corresponding traction along the interfaces, provides a proper phenomenological approach to simulate various fracture behaviors such as crack nucleation, crack initiation, and crack propagation. Fig. 1 schematically illustrates the fracture process zone, defined as the distance between a cohesive crack tip where the traction is maximum and a material crack tip where a traction-free region develops (traction is assumed to be zero). Therefore, the process zone describes the region between the points of no damage (full load carrying capacity) and the points of complete failure (no load-bearing capacity). Along this zone, crack nucleation, crack bridging and crack propagation occur. As the displacement jump increases due to an increase of external force or compliance in the

124

H. Kim, W.G. Buttlar / Cold Regions Science and Technology 57 (2009) 123–130

Fig. 1. Concepts of fracturing in asphalt concrete.

structure, the traction first increases, reaches a maximum, and decays to zero. Various fracture mechanisms depend on the material strength (σc), critical displacement (δc) and fracture energy (Gf), which represent the constitutive cohesive zone model parameters. The potential based exponential cohesive law proposed introduces undesirable artificial compliance due to initial slope of the exponential cohesive law (Xu and Needleman, 1994). As the number of cohesive elements increases, the initial contribution of each cohesive element increases and, as a consequence, the compliance induced is significant. To alleviate such problems, a bilinear model is introduced to reduce the compliance by adjusting the initial slope of cohesive law (Espinosa and Zavattieri, 2003; Rahulkumar et al., 2000). Notice that the parameter λcr is a non-dimensional constant in which the traction is a maximum, and is incorporated to reduce the elastic compliance by adjusting the pre-peak slope of the cohesive law. λcr can be determined by replacing the denominators in Eq. (1) with displacements (δ1n for normal and δ1s for shear) where the traction becomes the maximum. In other words, as the value of λcr decreases, the prepeak slope of the cohesive law increases and as a result, artificial compliance is reduced. Herein, λcr of 0.01 was selected for asphalt materials based on verification results of DC(T) FE models and used to reduce the unnecessary compliance (Song, 2006). Non-dimensional effective displacement (λe) and effective traction (te) are defined as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 qffiffiffiffiffiffiffiffiffiffiffiffi δn δs 2 + = tn2 + ts and t λe = e δcn δcs

separation, i.e. zero traction, occurs; and tn and ts denote normal and shear effective tractions. For λe b λcr, the normal and shear tractions are given as (Dwivedi and Espinosa, 2003) tn = σ c

    1 δn 1 δs and ts = σ c : c λcr δn λcr δcs

ð2Þ

For λe N λcr, the tractions are described by tn = σ c

    1 − λe 1 δn 1 − λe 1 δs and t = σ s c 1 − λcr λe δcn 1 − λcr λe δcs

ð3Þ

where, σc represents material strength. The schematic representation of a bilinear cohesive zone model is shown in Fig. 2 and the shape of CZM depends on the selected value of λcr. Both traction and displacement were normalized by critical values where complement separation occurs. The initial slope of CZM can represent the initial compliance in the failure of materials. If λcr increases, then the initial slope of CZM decreases. The cohesive fracture energy (Gc) is computed by equating the area under the displacement–traction curve (Fig. 2) without normalizing, namely

Gc =

c Z δ

T ðδÞdδ =

1 c σ δ 2 c

ð4Þ

0

ð1Þ

where, δn and δs denote normal opening and shear sliding displacements, respectively; δcn and δsc are critical values where complete

where, T(δ) is the traction function and δ is the opening displacement. A bilinear cohesive model was used in the form of a user element (UEL) subroutine in ABAQUS™ (HKS Inc., 2007). The material parameters used in the cohesive fracture model are fracture energy and tensile strength. In the current cohesive fracture approach, cracking in the pavement is simulated in the FE model by means of a specialized cohesive zone element. These elements are embedded in the mesh along the interface of regular finite elements.

Table 1 Creep compliance data tested by IDT at low temperatures. Creep compliance (1 / Ga) at different temperatures

Fig. 2. Normalized parameters of the bilinear cohesive zone model.

Time (s)

− 20 °C

− 10 °C

0 °C

1 2 5 10 20 50 100 200 500 1000

0.059 0.066 0.075 0.085 0.096 0.116 0.137 0.163 0.209 0.257

0.109 0.123 0.146 0.171 0.2 0.249 0.304 0.372 0.511 0.674

0.214 0.258 0.331 0.408 0.498 0.669 0.821 1.025 1.413 1.192

H. Kim, W.G. Buttlar / Cold Regions Science and Technology 57 (2009) 123–130

125

Table 3 Aggregate gradation of asphalt mixture. Sieve size (mm) Passing % by mass

12.5 100

9.5 99

4.75 77

2.36 49

1.18 24

0.6 15

0.3 10

0.15 7

0.075 6

transformation, interconversion of the time dependent creep compliance function of Eq. (5) yields a relaxation function given as EðnÞ =

Fig. 3. Schematic representation of the generalized Maxwell model.

N X

Ei e

− n = τi

ð7Þ

i=1

2.2. Viscoelasticity An appropriate bulk material constitutive model is crucial to the accurate simulation of material behavior in the FE modeling technique. Asphalt concrete material is known to have time and temperature dependent behavior across most of the in-service temperature range. Creep tests on asphalt concrete materials have shown linear viscoelastic behavior at low and moderate temperatures. In order to obtain the relaxation modulus, a creep test was conducted with indirect tensile tests. The creep test was chosen for asphalt concrete because it is easier to perform than the relaxation test (Buttlar and Roque, 1996). The constant stress was imposed and a generalized Voigt–Kelvin model is used to describe creep compliance behavior, which is then used to obtain relaxation modulus using interconversion schemes (Hiltunen and Roque, 1994). The creep compliance function using a Voigt–Kelvin model is given as DðnÞ = Dð0Þ +

N X i=1

  n − n = τi + Di 1 − e ηv

ð5Þ

where ξ is reduced time, D(ξ) is creep compliance at the reduced time. D(0), Di, τi, and ηv are model constants of generalized Voigt– Kelvin model. N is the total number of Kelvin models. The reduced time ξ is obtained from t / aT where t is a real time and aT is a temperature shift factor. The creep compliance data at different temperatures are shown in Table 1. It is well known that the bulk material properties of polymer type materials satisfy the time–temperature superposition principle based on a shift factor, aT, that is defined by the Williams, Landel, Ferry (WLF) equation (Williams et al., 1955) as log10 ðaT Þ =

−C1 ðT − TR Þ C2 + ðT − TR Þ

ð6Þ

where TR, − 20 °C, is the reference temperature and T is the temperature at a test condition. The WLF shift factors (log10(aT)) used herein were 0 at −20 °C, 1.34 at − 10 °C, and 2.70 at 0 °C. Material constants, C1 and C2, can be determined by experimental data. Fig. 3 shows a generalized Maxwell model, which is a widely used constitutive model to describe the linear viscoelastic behavior of asphalt concrete. A customized shift function was used in ABAQUS™ to capture the temperature dependency of asphalt concrete properties by use of the time–temperature superposition principle. By using the inverse Laplace

Table 2 Relaxation parameters of the generalized Maxwell model. Relaxation modulus parameters i

Ei (GPa)

τi (s)

1 2 3 4 5

3.4 3.4 5.9 6.8 6.1

12 162 1852 17476 465460

where, E(ξ) is a relaxation modulus at the reduced time of ξ, and Ei and τi are model constants for the master relaxation modulus curve. The relaxation parameters are shown in Table 2. 3. Material properties and experimental tests Bulk and cohesive material properties were determined based on a typical surface mixture used in Illinois. Superpave 9.5 mm nominal maximum aggregate size (NMAS) mixture with a PG70-22 asphalt binder was used for laboratory tests. Binder contents in asphalt mixture were 5.5% by mass. Asphalt mixture was produced with a drum-type mix plant by the aggregate gradation curve shown in Table 3. Asphalt mixture was produced with a blend of three aggregate stockpiles, two of crushed dolomite limestone (FM02 and FM20) and one of natural sands (CM16). The produced asphalt mixture has the maximum theoretical density (Gmm) of 2.49 kg/m3 and the air void of 4% by volume. Indirect tensile (IDT) tests were conducted to obtain the tensile strength of asphalt concrete, which is used as the fracture parameter of asphalt material (AASHTO T322-03, 2004). The IDT test induces tension in a cylinder by applying a concentrated compressive load along a diametrical axis as shown in Fig. 4(a). The two halves of the circular specimen tend to arch away from each other, thereby inducing a tensile stress perpendicular to the direction of load. After a strength test, the indirect tension specimen is often found to have split in two halves, separated along the axis of loading. The unique aspect of IDT test is that displacement measurements are taken on the face of specimen as shown in Fig. 4(a). The determined tensile strength of asphalt concrete was 3.56 MPa at −20 °C. AASHTO T-322 standard test procedure was followed for both the test procedure and analysis. σc =

2P πdh

ð8Þ

where, σc is the tensile strength, P is the applied force, d is the specimen diameter, and h is the specimen height (or thickness). A disk-shaped compact tension (DC(T)) test was proposed to investigate various fracture behavior of asphalt concrete (Wagoner et al., 2005; Kim et al., 2008). Fig. 4(b) illustrates a test set-up in temperature chamber and a DC(T) specimen, which has 150 mm diameter and 50 mm thickness. The length of mechanical notch was 27.5 mm. Loading pins are inserted in the holes and pulled apart with a closed-loop servo-hydraulic loading system to induce opening. Creep compliance of asphalt materials was measured at three temperatures (−20, −10, and 0 °C) with a creep load duration of 1000 s using the indirect tensile test (AASHTO T322-03, 2004). The time– temperature superposition principle was utilized to create a master creep compliance curve, shown in Fig. 5. The relaxation model parameters were obtained by the interconversion of creep testing data and were used in FE pavement fracture models. Fig. 6 shows the effect of loading rate at four temperatures on asphalt mixtures used in this study. Initially, four different temperatures (0, −10, −20, and −30 °C) were proposed with four different loading rates ranging from 0.1 to 10 mm/min. Three replicates of asphalt specimen were tested at each loading and temperature condition. The loading rate was controlled by

126

H. Kim, W.G. Buttlar / Cold Regions Science and Technology 57 (2009) 123–130

Fig. 4. Experimental tests.

the crack mouth opening displacement (CMOD), which is an opening displacement measured by the attached clip displacement gauge (Fig. 4(b)). The fracture energy increases by decreasing loading rate and by increasing temperature. For the fast loading rates at low temperature (−20 °C and −30 °C), asphalt materials exhibited a brittle failure, which is without a softening response after the peak load.

As early discussed in this paper, pavement cracks often occur in a single, critical cooling event from field observations. Thus, the coolest cooling events need to be searched from historical climate data before analyzing the thermal cracking behavior. Hourly climatic data can be obtained from US national oceanic and atmospheric administration (NOAA). The temperature gradients for different pavement sections were estimated from the pavement surface temperature using an enhanced integrated climatic model (EICM) developed by AASHTO.

Temperature gradients of pavements are transient and depend on factors including air temperature, solar energy, precipitation, etc. Fig. 7 provides the thickness information of selected pavement structures, 4L-22R and 9L-27R runways, for FE pavement fracture models from the O'Hare International Airport in US Northern region. The asphalt overlays were constructed in 1997 for 4L-22R and in 1999 for 9L-27R, respectively. The historical climate data in O'Hare airport region were searched to select the critical cooling event for the past 10 years. The temperature data were obtained from 1997 to 2005 and it was found that the coldest temperature occurred in January of 1999. The air temperature dropped down to −28 °C and the daily temperature change was −18 °C in January of 1999. Using the EICM, the critical temperature gradients and cooling cycles were estimated for both runway structures as shown in Fig. 8. The temperature change of pavement structures was very significant on the surface of asphalt materials. In the runway of 4L-22R, the critical cooling event occurred for 18 h from 1:00 pm on January 4th to 7:00 am on January 5th in 1999. On the other hand, the critical cooling event occurred for 14 h from 2:00 pm on January 4th to 4:00 am on January 5th in 1999 in the

Fig. 5. Creep compliance test results.

Fig. 6. Fracture energy determined by DC(T) tests.

4. Numerical modeling 4.1. Temperature gradients

H. Kim, W.G. Buttlar / Cold Regions Science and Technology 57 (2009) 123–130

Fig. 7. Different types of airport pavement structures (unit: mm).

runway of 9L-27R. These temperature gradients were projected into the nodal points of FE pavement mesh and the cooling cycles were simulated with hourly time steps. 4.2. Aircraft gears and pavement properties Heavy aircraft gear loading, Boeing 777-200, was applied to achieve the critical loading condition at cold temperature. Boeing 777-200 has two main landing gears with a main gear width of 11 m. The gross weight of Boeing 777-200 aircraft is 288 t and the contact tire pressure of dual-tandem gears is 1.48 MPa, which was applied to FE pavement fracture models (FAA, 1995). The distance between two tires under the same main gear is 1.4 m. Viscoelastic properties of asphalt concrete, which were determined from laboratory creep tests as shown in Fig. 5, were applied to the asphalt overlays and existing AC layers after being interconverted to relaxation forms. Portland cement concrete (PCC) was assumed to have a Young's modulus of 27.6 GPa, while the modulus of subbase layer was taken as 6.89 GPa. A Winkler-type subgrade was assumed to have a stiffness or k-value of 81.4 MN/m3 (Huang, 2004). Poisson's ratios for PCC and subbase layers were assumed to be 0.15 and 0.20, respectively. Fracture energy of 200 J/m2 (at −20 °C and with 5 mm/min from Fig. 6) was assumed because the coldest temperature in pavements was beyond −20 °C and the loading speed of aircraft can be varied but faster than 5 mm/min. The thermal coefficients of asphalt concrete and cement concrete were assumed as 2.501 × 10− 5 1/°C and 9.902 × 10− 6 1/°C, respectively. The interface between asphalt overlay and Portland cement concrete layer was assumed to be fully bonded.

softening region. Also, there is the cracking threshold for completely separated (cracked) region (Fig. 10). Fig. 10 shows the numerical responses induced by the thermal loading in 4L-22R. The crack opening displacement increased gradually as the temperature decreased. The softening and separation thresholds were 1.12E03mm and 1.12E-01mm, which were determined with a 0.01 displacement ratio (δ1n / δcn), 3.56 MPa tensile strength, and 200 J/ m2 fracture energy. The critical cooling event in the winter season has a tremendous impact on the cracking of airport pavement although it was not fully fractured through AC overlays. Fig. 11 represents the contribution of aircraft gear loading after the critical thermal loading. The heavy aircraft gear loading had a significant contribution to progress the crack opening in airport pavements. It was assumed that the gear loading right above the cracks or joints is the critical traffic loading condition. From the responses of 9L-27R, there was a compression zone on the top region of AC overlay due to the thin AC surface layer on the thick PCC layer. The critical tensile opening displacement occurred at the bottom of AC overlay. It is important to notice that this simulation was conducted only by the single aircraft loading although there must be a lot of repeated aircraft departures and landings during the cold weather. However, it was a difficult task to apply all repeated aircraft loads due to the limitation of computational capability. Fig. 12 shows the progressive stress contours during the crack has been initiated and propagated through pavement structures. The crack opening displacement in contours was magnified with a deformation scale factor of 30 to show the material separation along the line of cohesive elements clearly. There was a characteristic fracture process zone ahead of the crack tip, which is an important

4.3. FE pavement fracture model Fig. 9 shows a schematic of FE pavement fracture model and the details of mesh. All FE simulations in this study were performed with two-dimensional FE models created along the direction of traffic movement. As shown in Fig. 9(a), FE pavement models contain 5 joints in PCC and were assumed that the existing asphalt layer was fully cracked. The distance between neighboring joints was 5.7 m so the total size of FE models was 28.5 m. Fig. 9(b) shows the FE meshes in the vicinity of a potential thermal cracking region, which is constructed with small elements. The size of CZM elements was 0.5 mm and they were inserted through AC overlays along each potential crack path. The FE models for low-temperature cracking simulations were constructed using graded meshes, which are used to significantly reduce the computational time. Graded meshes typically have a finer element size close to the areas of high stress variations and potential non-linearity, whereas in the regions of low stress gradients, larger elements are used. 5. Numerical results The model indicates the softening threshold between two faces of material at which the linear material behavior is transformed to

127

Fig. 8. Estimated critical cooling cycles and temperature gradients.

128

H. Kim, W.G. Buttlar / Cold Regions Science and Technology 57 (2009) 123–130

Fig. 9. Schematic of the FE pavement fracture model.

parameter in the nonlinear fracture mechanism. The fracture process zone represents the softening zone area of materials and it was moved to the remaining ligament with a constant length and shape during the crack propagation.

6. Summary and conclusions A two-dimensional (2-D) finite element cohesive fracture model of a rehabilitated airport overlay system subjected to both thermal and aircraft gear loads was developed to study the fracture behavior of airport pavement structures. Material tests of asphalt concrete were performed to obtain viscoelastic and fracture properties using IDT creep tests, IDT strength tests, and DC(T) fracture tests. Critical temperature gradients of O'Hare International Airport were analyzed based on US

Fig. 10. Crack opening displacements due to critical cooling events (4L-22R).

Fig. 11. Crack opening displacements by both thermal and gear loads.

H. Kim, W.G. Buttlar / Cold Regions Science and Technology 57 (2009) 123–130

129

Fig. 12. Von Mises stress contour of progressive pavement crack propagation (9L-27R).

national weather data and the national climate model. Based on the past 10 years of US National Weather Service Database data, it was estimated that the critical temperature condition of asphalt runways in the O'Hare International Airport occurred in the winter of 1999. In January of 1999, the air temperature dropped down to −28 °C and the daily temperature change was −18 °C. The temperature gradients were projected into FE pavement fracture models to incorporate the critical thermal loading condition. Heavy aircraft gear loading were also applied for investigating the impact of thermo-mechanical loading to asphalt pavement structures. Based upon the findings of this study, the following conclusions can be drawn.

(1) Two dimensional FE pavement fracture models were developed by using the bilinear cohesive zone model and successfully integrated with experimental tests to investigate the fracture behavior of asphalt pavements. The developed pavement fracture models could simulate the crack initiation and propagation with a characteristic fracture process zone through the pavement structure. (2) Nonlinear temperature gradients, obtained from the national weather data, were successfully projected into all finite element nodal points of pavement fracture models. The FE simulation of critical thermal loading was conducted and the results showed that a single critical cooling event in a certain pavement structure could significantly contribute to the crack softening or the crack propagation. (3) When the heavy aircraft gear loading was coupled with the thermal loading, the fracture responses of selected pavement structures became more significant. Also, FE pavement fracture results indicated that the fracture behavior of asphalt pavements was different from two varied pavement structures although the same material parameters and loading conditions were applied. (4) It was known that the fracture energy of asphalt concrete depended on the loading rates and test temperatures based on the results of DC(T) tests. The faster loading rate with lower temperatures had much less fracture resistance.

Although the current numerical approach was a useful tool for investigating the fracture behavior of pavements, the field validation is much more needed to prove the benefits of FE pavement fracture models for future. Also, different mixture types in various pavement structures should be comprehensively investigated to achieve better understanding of complex pavement fracture behavior. Acknowledgements This paper was prepared from a study conducted at the FAA Center of Excellence for Airport Technology (CEAT) and funded by the O'Hare Modernization Program (OMP). The contents do not necessarily reflect the official views and policies of OMP and FAA. This paper does not constitute a standard, specification, or regulation. We are grateful for the support provided by the OMP. References AASHTO T322-03, 2004. Standard test method for determining the creep compliance and strength of hot mix asphalt (HMA) using the indirect tensile test device, Standard Specifications for Transportation Materials and Methods of Sampling and Testing, 24th Edition. USA. Buttlar, W.G., Bozkurt, D., 2002. Three-dimensional finite element modeling to evaluate benefits of interlayer stress absorbing composite for reflective crack mitigation. Federal Aviation Administration Technology Transfer Conference, Atlantic City, USA. Buttlar, W.G., Roque, R., 1996. Evaluation of empirical and theoretical models to determine asphalt mixture stiffnesses at low temperatures. Journal of the Association of Asphalt Paving Technologists 65, 99–141. Dwivedi, S.K., Espinosa, H.D., 2003. A modeling dynamic crack propagation in fiber reinforced composites including frictional effects. Mechanics of Materials 35, 481–509. Espinosa, H.D., Zavattieri, P.D., 2003. A grain level model for the study of failure initiation and evolution in polycrystalline brittle materials. Part I: theory and numerical implementation. Mechanics of Materials 35, 333–364. Federal Aviation Administration (FAA), 1995. Advisory Circular: Airport Pavement Design for the Boeing 777 Airplane, AC No.150/5320-16. Department of Transportation, Washington, D.C., USA. Hibbitt, Karlsson and Sorensen (HKS), Inc., 2007. ABAQUS™ Users Manual. Pawtucket, Rhode Island, USA. Hiltunen, D.R., Roque, R., 1994. The use of time–temperature superposition to fundamentally characterize asphaltic concrete mixtures at low temperatures. Engineering Properties of Asphalt Mixtures and the Relationship to Performance: ASTM STP 1265. American Society for Testing and Materials, Philadelphia, USA.

130

H. Kim, W.G. Buttlar / Cold Regions Science and Technology 57 (2009) 123–130

Huang, Y.H., 2004. Pavement Analysis and Design, Second edition. Prentice Hall, Inc., Englewood. Cliffs, New Jersey, USA. Kim, J., Buttlar, W.G., 2002. Analysis of reflective crack control system involving reinforcing grid over base-isolating interlayer mixture. Journal of Transportation Engineering, American Society of Civil Engineering 128 (4), 375–385. Kim, H., Wagoner, M.P., Buttlar, W.G., 2008. Simulation of fracture behavior in asphalt concrete using a heterogeneous cohesive zone discrete element model. ASCE Journal of Materials in Civil Engineering 20 (8), 552–563. Lytton, R.L., 1989. Use of geotextiles for reinforcement and strain relief in asphalt concrete. Journal of Geotextiles and Geomembranes 8. Mukhtar, M.T., Dempsey, B.J., 1996. Interlayer Stress Absorbing Composite (ISAC) for Mitigating Reflective Cracking in Asphalt Concrete Overlays. Transportation Engineering Series No. 94. University of Illinois at Urbana-Champaign, Urbana, IL, USA. NCHRP 1-37A, 2004. Development of the 2002 Guide for the Design of New and Rehabilitated Pavement Structures: Phase II. National Cooperative Highway Research Program (NCHRP), American Association of State Highway and Transportation Officials (AASHTO), USA.

Rahulkumar, P., Jagota, A., Bennison, S.J., Saigal, S., 2000. Cohesive element modeling of viscoelastic fracture: application to peel testing of polymers. International Journal of Solids and Structures 37, 1873–1897. Rigo, J.M., 1993. General introduction, main conclusion of the 1989 conference on reflection cracking in pavements, and future prospects. Proceedings of the 2nd RILEM Conference on Reflective Cracking in Pavement, Liege, Belgium. Song, S.H., 2006. Fracture of asphalt concrete: a cohesive zone modeling approach considering viscoelastic effects. Ph.D. Dissertation, University of Illinois at UrbanaChampaign, USA. Wagoner, M.P., Buttlar, W.G., Paulino, G.H., 2005. Disk-shaped compact tension test for asphalt concrete fracture. Experimental Mechanics 45 (3), 270–277. Williams, M.L., Landel, R.F., Ferry, J.D., 1955. Temperature dependence of relaxation mechanisms in amorphous polymers and other glass forming liquids. Journal of American Chemical Society 77, 3701–3707. Xu, X.P., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 42 (9), 1397–1434.