Finite element cohesive fracture modeling of asphalt mixture based on the semi-circular bending (SCB) test and self-affine fractal cracks at low temperatures

Journal Pre-proof Finite element cohesive fracture modeling of asphalt mixture based on the semi-circular bending (SCB) test and self-affine fractal cracks at low temperatures

Ahmad Al-Qudsi, Augusto Cannone Falchetto, Di Wang, Stephan Büchler, Yun Su Kim, Michael P. Wistuba PII:

S0165-232X(19)30243-5

DOI:

https://doi.org/10.1016/j.coldregions.2019.102916

Reference:

COLTEC 102916

To appear in:

Cold Regions Science and Technology

Received date:

16 April 2019

Revised date:

29 July 2019

Accepted date:

4 October 2019

Please cite this article as: A. Al-Qudsi, A.C. Falchetto, D. Wang, et al., Finite element cohesive fracture modeling of asphalt mixture based on the semi-circular bending (SCB) test and self-affine fractal cracks at low temperatures, Cold Regions Science and Technology(2018), https://doi.org/10.1016/j.coldregions.2019.102916

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© 2018 Published by Elsevier.

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Finite Element Cohesive Fracture Modeling of Asphalt Mixture based on the Semi-Circular Bending (SCB) Test and Self-affine Fractal Cracks at Low Temperatures

Ahmad Al-Qudsi1, Augusto Cannone Falchetto1,2*, Di Wang1, Stephan Büchler1, Yun Su Kim1

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Technische Universität Braunschweig, Department of Civil Engineering (ISBS), Braunschweig, 38106, Germany. Department of Civil & Environmental Engineering, University of Alaska Fairbanks, Alaska, PO Box 755960 Fairbanks, AK 99775-5960, USA

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and Michael P. Wistuba1

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* Corresponding author: Augusto Cannone Falchetto, Technische Universität Braunschweig, Department of Civil Engineering (ISBS), Braunschweig, 38106, Germany. +49 (0)531-391 62064. [email protected]

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Abstract

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Thermal cracking is one of the most common distresses for asphalt pavement constructed in

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the cold regions. In order to address this issue, the combined use of fracture mechanics-based tests and simulation is a solid option. First, asphalt mixture samples are prepared based on the German standard and the low temperature strength are measured by Semi-Circular Bending (SCB) test and the Uniaxial Creep (UC) test at three different temperatures: -6, -12 and -18 ºC. Next, experimental test results are used to perform the cohesive zone (CZ) modeling by using a two-dimensional finite element (FE) simulation. As a new approach, the CZ is modeled along a self-affine crack path, which allows performing a simulation closer to reality. The FE results provide a comprehensive understanding of the mechanism of crack initiation and propagation while keeping the computational time within a reasonable level.

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Journal Pre-proof Keywords: Asphalt pavement; fracture mechanics; Semi-Circular Bending (SCB); Uniaxial Creep (UC); cohesive zone model; finite element method; viscoelasticity; low temperature properties.

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Introduction

Thermal cracking is one of the most serious distresses for asphalt pavements built in cold regions, such as northern Europe, northern U.S., northern China, and South Korea (Li and

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Marasteanu 2009 and 2010, Wang et al. 2016; Moon et al. 2017). Normally two types of thermal cracking, thermal fatigue crack caused by the repeatable thermal stress and thermal

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contraction cracks induced by the extremely low temperatures, can be observed in asphalt

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pavement in aforementioned regions (Wang et al. 2016). Since the thermal cracking caused by

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repeatable thermal stress is more common, hence, this type of thermal crack is investigated and evaluated in this paper. This phenomenon manifests as a set of almost parallel surface-initiated

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transverse cracks of various lengths and widths. The possible concurrent effect associated with

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water penetrating through surface cracks may further accelerate the overall deterioration of the

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pavement package, ultimately resulting in premature failure of the road infrastructure (Cannone Falchetto and Moon 2015). This prompt the need to determine new and robust solutions to limit the detrimental effects of low temperature cracking. For this purpose, materials characterization based on solid experimental testing combined with the theoretical framework of fracture mechanics offers the tools to better evaluate the failure properties of asphalt mixture eventually promoting the design pavement with extended durability that can withstand thermal distresses. Over the years, a number of tests were developed to incorporate fracture mechanics in the characterization of asphalt materials such as Disk-Shaped Compact Tension test (DCT), Semi-Circular Bending (SCB) test and Three-Point Bending (3PB) (Zeng et al. 2014; Chiangmai and Buttlar 2015; Saeidi and Aghayan 2016). Among others, SCB represents an

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Journal Pre-proof advantageous configuration for determining the strength and fracture properties of asphalt mixture as also recently demonstrated (Cannone Falchetto et al. 2018). This is because it presents simple specimen preparation combined with the use of linear elastic fracture mechanics (Li and Marasteanu 2010, Cannone Falchetto et al. 2017). Specimens are of a semicircular shape with 75 mm radius, 30 mm thickness and 1mm wide notch of 15 mm in length. In fracture mechanical, there are three common failure modes: mode I is named opening mode which the tensile stress normal to the plane of the crack; mode II can be defined as the sliding

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mode, the shear stress acting parallel to the plane of the crack and perpendicular to the crack front, and mode III is known as the tearing mode, the shear stress acting parallel to the plane

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of the crack and parallel to the crack front (Marasteanu et al. 2012; Cannone Falchetto et al.

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2017). Conventionally, SCB test is used to address mode I fracture, although mode II can be

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also evaluated by changing the angle of the notch. In addition, fracture energy GF, peak stress σN and fracture toughness KIc can be derived from the load-displacement curve (Li and

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Marasteanu 2004; Cannone Falchetto et al. 2017).

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The simple geometry of the SCB significantly facilitates the analysis of the

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experimental results through numerical simulations. In this sense, a number of research efforts were devoted to investigating this test and the material response by performing Finite Element (FE) modeling (Ban et al. 2015; Cannone Falchetto et al. 2017). Overall, two main categories can be distinguished in FE simulations of asphalt. The first includes models, which simulate the asphalt as isotropic, homogeneous and linear elastic. In this specific case, cohesive elements are placed along a vertical line starting from the notch tip to simulate crack propagation (Saeidi and Aghayan 2016; Cannone Falchetto et al. 2017). Advantageously, this method is very fast and easy to implement. However, this approach does not take into account the aggregate distribution in the mixture, which controls the path of crack propagation. The other approach is more realistic, as it incorporates the lithic skeleton as well as the adhesion/interaction

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Journal Pre-proof between asphalt binder and aggregates. Nevertheless, the latter modeling solution is more complex and significantly more computationally time demanding. An alternative method, is suggested in this work for the first time in asphalt mixture, where a self-affine fractal crack path is predefined according to the aggregate distribution in the asphalt mixture. This is expected to provide simulations that are more representative of the actual phenomena occurring in the asphalt mixture during cracking at a significantly reduced computational time. Research approach

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2.

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In this paper, asphalt mixture AC 11, which is commonly used for wearing course in Germany

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(TL Asphalt-StB, 2013), is prepared with a type of unmodified binder 50/70 (EN 12591 2015). Then, two different tests: Semi-Circular Bending (SCB) test and the Uniaxial Creep (UC) test

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are performed at three different temperatures: -6, -12 and -18 ºC to measure the low temperature

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fracture and strength properties. Next, the fitted rheological parameters are used to run the twodimensional FE simulation. Finally, the common parameters between the experimental tests

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and numerical simulation are used to compare and evaluate the results. Figure 1 presents the

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flow chart of the research approach.

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Materials

Testing method

AC 11

Uniaxial Creep (UC) Test

Asphalt binder 50/70

Semi Circular Bending (SCB) Test

Temperature

Materials

1×123°C -6 °C

And

1×143°C -12 °C 1×163°C -18 °C

Experimental Work

SCB Test Results

UC Test Results

σmax – maximum peak stress

E1 – Maxwell spring stiffness E2 – Kelvin-Voigt spring stiffness

KIC – fracture toughness

η2 – Kelvin-Voigt dashpot damping coefficient

GF – fracture energy

Numerical Simulation output: σmax , KIC

and

GF

Numerical Simulation and Comparison

GF)

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Comparison of share parameters (σmax , KIC

and

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η1 – Maxwell dashpot damping coefficient

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between SCB test and numerical simulation results

Conclusions

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Evaluation of Results and Discussion

Figure 1: Schematic of the research approach.

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3.1 Material model

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3. Numerical modeling background

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The Burgers viscoelastic model can be easily used to describe the viscoelastic behavior of asphalt mixture. This is a 4-parameter rheological model, which consists of a Maxwell model in series with a Kelvin-Voigt model (Gutierrez-Lemini 2014). It is composed of two springs and two dashpot elements. Figure 2 shows a schematic of the assembly.

Figure 2: Scheme of Burgers viscoelastic model.

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Journal Pre-proof Commercially available FEM tools such as ABAQUS do not have a Burgers model implemented in their software (Simulia 2014). However, they have a generalized viscoelastic model known as Prony series (Wang et al. 2017), which can be used to model different materials. Using the constants obtained from the experimental data, one can implement the Burgers model into this FEM based software by tweaking Prony series constants appropriately (Wang et al. 2017). The implementation procedure is explained as follows, where the Prony Series is modified to behave like the Burgers model. This technique was used by many

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researchers for different applications (Bharadwaj et al. 2010; Kong and Yuan 2010).

  2   2  p2 2  q1  q2 2 t t t t

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 1 2 E1 E2

q1  1

 1 2 E2

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p2 

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E12  E21  E11 E1 E2

p1 

q2 

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where the four parameters are expressed as

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  p1

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The general equation that describes the Burgers model is written as:

(1)

(2)

(3)

(4) (5)

Applying Laplace transformation to equation (1) and taking into account that the test is performed under a constant strain rate, one can obtain:

ˆ  p1sˆ  p2 s2ˆ  q1 0  q2 s 0

(6)

Solving for ˆ gives:

ˆ 

q1s  q2 s 2 0 s(1  p1s  p2 s 2 )

(7)

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Journal Pre-proof Expanding the equation (7) by partial fraction and performing the inverse Laplace transformation gives the stress relaxation:

 (t ) 

0 

[(q1  q2 r1 )e r t  (q1  q2 r2 )e r t ] 1

(8)

2

with

  p12  4 p2

(9)

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2 p1   p1  p1  4 p2 r1   2 p2 2 p2

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2 p1   p1  p1  4 p2 r2   2 p2 2 p2

(10)

(11)

q1  q2 r1  r t q1  q2 r1  r t e  e   1

2

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E (t ) 

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Expressing equation (8) in terms of the relaxation modulus results in:

(12)

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Prony series consists of a number of Maxwell elements in series, assembled in parallel

N



t

i

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E (t )  E   Ei e

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with spring. This can be expressed by the following formula as:

i 1

(13)

where E is the steady state stiffness

E (t )  E  E1e



t

1

 E2 e



t

2

(14)

Matching between equations (14) and (12) yields: E  0

E1 

(15)

q1  q2 r1 

E2  

q1  q2 r2 

(16)

(17)

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1 1 and  2  r1 r2

(18)

3.2 Self-affine fractal cracks Mandelbrot (Mandelbrot et al. 1984) introduced the term self-affinity when he conducted the first experimental study in fractal fracture mechanics and found that the fracture surfaces of steel are fractals. Self-affine fracture surface is a surface that has the same statistical properties and morphology for each length scale (Mandelbrot et al. 1984). Since then, many investigations

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have been conducted. For example, Saouma and Barton (1994) experimentally showed that fracture surfaces of concrete are fractals. Experimentalists have observed fractality in the

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fracture surfaces of many engineering materials (Cox and Wang 1993, Power et al. 1987, Power

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and Durham 1997).

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A crack can be described by a self-affine fracture surface as this provides a good representation for different length scales at various orders of magnitude. The self-affinity of

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fracture surfaces is decoded by analysis of its autocorrelation function Γ or its Fourier transform,

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which is called the power spectral density C (Klüppel and Heinrich 2000). For many natural

(see Figure 3).

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and engineering surfaces, the power spectral density decays as a power-law of the wavenumber

For a self-affine path, Klüppel and Heinrich introduced the following equation for the power spectral density (Klüppel and Heinrich 2000):

C (q)  C0 (

q  (2 H 1) ) q0

(19)

Where H is called roughness- or Hurst-coefficient. This is a measure for surface irregularity. The values of H are located in the following range: (0 ≤ H ≤ 1). H=1 describes a smooth crack path, while H=0 indicates an extremely rough crack path. q is the wavevector.

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self-affine fracture crack

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Figure 3: Approximated power spectral density as a function of spatial frequency for a

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C0 is the constant determining the fracture amplitude. It can be described for self-affine

2 N  hrms

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C0 

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fracture paths as:

N

 i ( 2 H 1)

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i 1

(20)

N is the number of crack path points, hrms is the root mean square fluctuation of the

1 N

N

z i 1

2 i

zi  z ( xi )  z

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2 hrms 

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crack profile around the mean line. This parameter is described as:

(21) (22)

where is the mean height of the surface points z(xi). The Fourier-transform of the height-correlation function equals the power spectral density C(q). Therefore, the self-affine path is generated by MATLAB through inverse Fourier transform to the power spectral density with a normalizing pre-factor (see Figure 4).

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Figure 4: Profile shape for self-affine crack path generated in MATLAB.

3.3 Theory of cohesive zone model In order to simulate the crack propagation of semi-circular bending test, the cohesive zone

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model (CZM) was chosen (Li and Marasteanu 2010). The theoretical principle of CZM was introduced by Barenblatt (1962) and later further developed to quasi-brittle materials, such as

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asphalt mixture at low temperature (Tabaković et al. 2010, Cannone Falchetto et al. 2014 and

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2017). A first, a linear evolution phase of the stress up to the peak load is followed by the

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initiation of a cohesive crack. After reaching this peak, all the deformation is concentrated into the cohesive crack. In the CZM, the relationship between the transferred stress through

(23)

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  f (w) where f (0)  ft '

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cohesive crack σ and the corresponding crack opening w can be expressed as:

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where f(w)is a function of the material softening curve. The dissipated energy used to establish the unit surface, GF is calculated from the area of the stress-crack opening curve as: 



GF  0  dw  0 f (w)dw

(24)

The tensile strength, fracture energy, and the shape of the stress-crack softening curve must be calculated in order to characterize the fracture properties through the CZM. 4. Experimentation 4.1 Semi-circular bending fracture test SCB fracture tests are performed at low temperature according to the methodology suggested by Marasteanu et al. (2012) which has been further modified in recent researches (Cannone

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Journal Pre-proof Falchetto et al. 2017 and 2018). Asphalt mixture AC 11 is prepared based on the Germany standard (TL Asphalt-StB, 2013) by using unmodified binder 50/70 (EN 12591 2015). The SCB sample has a semi-circular shape with a diameter of 150 mm, a thickness of 30 mm and a straight vertical central notch of 15 mm in length and 1mm in width. The specimen is placed on a frame consisting of two fixed rollers and having a span of 120 mm. A load line displacement with a rate of 0.00025 mm/s is used with a conditioning time of 2 hours. The tests are performed at three different temperatures: -6, -12 and -18 ºC, at least three replicates are

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tested to obtain reliable experimental results. Figure 5 shows the test equipment for semi-

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circular bending test.

Mode I stress intensity factor KI can be calculated according to the following

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relationship (Li and Marasteanu 2009 and 2010) as:

s0 s s  B]   0    a  [YI (S0 / r )  ( a  0 )  B] r r r

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K I   0    a  [YI (S0 / r ) 

(25)

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where σ0 = P/ (2·r·t), r is the radius, t is thickness; YI is the normalized stress intensity

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factor (Li and Marasteanu 2004 and 2005), a is the notch length, sa/r is the actual span ratio,

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s0/r is the closest span ratio and B is a parameter depending on a and r (Li and Marasteanu 2004 and 2005). Fracture toughness, KIc, is obtained from the same formula when P reaches the peak load.

The fracture energy (i.e. energy release rate) GF is estimated accordingly as (Li and Marasteanu 2004 and 2005):

GF 

WF Alig

(26)

where WF is the work of fracture and Alig is the area of the ligament.

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Journal Pre-proof In order to determine the material properties of the cohesive zone, SCB fracture tests were performed at low temperature using three replicates for each temperature (Cannone

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Falchetto et al. 2017).

Figure 5: Test equipment for semi-circular bending test.

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4.2 Uniaxial creep test

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For the uniaxial creep test, the same materials and test temperatures are used compared to the

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SCB test, at least three replicates are tested to obtain reliable experimental results. The Burgers model parameters were determined using uniaxial creep test on asphalt mixture following an internal testing procedure (Büchler et al. 2010) (see Figure 6). A total of three replicates per temperature were prepared and tested.

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Figure 6: Test equipment for uniaxial creep test for low temperature behavior.

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The material properties for bulk and cohesive zone (CZM) are listed in Tables 1 and 2. Table1: Burgers model parameters for AC 11 mixture at different temperatures. E2 [MPa] 4719.8 11738.9 20970.7

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E1 [MPa] 13604 18056.7 25181

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T [C°] -6 -12 -18

η1 [MPa.s] 80551.7 230143.9 577199.4

η2 [MPa.s] 1056.8 1791 2829

Table2: Cohesive zone model parameters for AC 11 mixture at different temperatures. T [C°] -6 -12 -18

σmax [MPa] 1.065 1.104 1.218

GF [mJ/mm2] 1.689 0.885 0.435

KIc [MPa/mm] 2.222 3.548 6.333

5. Finite elements simulation and results 5.1 Finite element model Commercial software for finite element modeling – ABAQUS - was selected to simulate SCB test (Simulia 2014). The geometry of the SCB (Figure 7) was meshed with 2D planar shell deformable elements having a thickness of 30mm. (see Figure 8).

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Figure 7: The geometry of the semi-circular bending test (Cannone Falchetto et al. 2017).

Figure 8: The mesh of the semi-circular bending test specimen. The asphalt mixture in the bulk part of the specimen was modeled as isotropic, homogeneous and viscoelastic (see subsection 2.1). Cohesive elements were placed along a self-affine fractal line starting from the notch tip to simulate the crack propagation. The cohesive elements were stopped 1mm away from the top of SCB specimen. The identified geometry was adopted to prevent singularities in crack propagation which is considered as a simulation error as shown in previous studies (Zegeye 2012; Cannone Falchetto et al. 2017).

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Journal Pre-proof During the FEM simulation, reaction forces, displacements, and maximum stress were recorded. In order to investigate the effectiveness of the FE model, load versus load-line displacement (LLD) curves of the experimental and the simulation results were compared. 5.2 Simulation results of SCB fracture test Figure 9 shows how the crack propagated during the semi-circular bending test at a temperature of -6 °C, while Figures 10 and 11 illustrate the result of the finite element simulation of the

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SCB test and the crack propagation during the test, respectively. From the comparison between

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Figures 9 and 11, one can notice similar trends of crack propagation.

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Figure 9: Crack path of the semi-circular bending test specimen at T=-6 °C.

Figure 10: Simulation results of SCB fracture test at T=-6 °C.

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Figure 11: Crack propagation in SCB test specimen using FE simulation at T=-6 °C.

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Figures 12 to 14 compare FE simulation and experimental results using the load versus

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LLD curves at different temperatures. It can be observed that the calculated force-displacement curve of FE simulation has the same slope of the loading part of the curve determined in the

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laboratory tests, which mean that the loading phase of the bilinear force-displacement curve

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can be described very well with the cohesive zone model.

Figure 12: Comparison of FE simulation and experimental SCB results at T=-6 °C.

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Figure 13: Comparison of FE simulation and experimental SCB results at T=-12 °C.

Figure 14: Comparison of FE simulation and experimental SCB results at T=-18 °C.

The calculated maximum stress resulting from FE simulation nearly coincides with the force determined with the laboratory experiments, which means that the parameter of the maximum stress needs to be slightly adjusted. However, it can be seen that the calculated

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Journal Pre-proof force-displacement curves resulting from FEM deviate from the force-displacement curves determined in laboratory tests. This can be justified due to the dissipated energy resulting from the viscoelastic nature of asphalt, which causes an additional dissipation of energy in addition to the deformation energy resulting from the fracture. The fracture energy was calculated without considering this additional component of the dissipated energy. Since there is currently no method for the exact calculation of the dissipated fracture energy as a function of the crack propagation, a calibration based on the algorithms for iterative solution strategies is necessary.

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The fracture energy must be calibrated until the area under the calculated force-displacement curve equals the area under the laboratory-determined curve. One can observe that, with

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decreasing temperature, the effect of the viscoelastic dissipation becomes higher and larger

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deviations between FE simulation and experimental results occur. Another justification of the

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differences between FE simulation and the experimental results may be due to the specific cohesive zone model for which ABAQUS assumes a homogeneous material, while this is not

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cohesive zone.

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an entirely realistic assumption due to the inhomogeneity of the asphalt material also in the

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6. Summary and conclusions

In this paper, a simple FE analysis was developed to simulate SCB fracture test with the aim of understanding the crack initiation and propagation mechanism of asphalt mixture at low temperatures. A conventional asphalt mixture for wearing course was taken into account and evaluated at three different temperatures. In order to get closer to a realistic simulation of asphalt cracking with a reduced computational cost, a random self-affine fractal crack path was predefined according to the aggregate distribution in the asphalt mixture. Finally, the results of the FE simulation were compared to the experimental SCB results. From the comparison, it can be concluded that based on cohesive zone modeling, it is possible to determine maximum strength and fracture toughness. On the other hand, fracture energy cannot be correctly 18

Journal Pre-proof estimated. This may have a two-fold reason. Asphalt mixture behaves in a viscoelastic manner, and therefore, additional energy dissipation occurs because of viscoelasticity beside the energy dissipation resulting from the fracture. Therefore, based on CZM and SCB experimental data, the FE simulation must be calibrated in terms of fracture energy. The second reason is that the selected cohesive zone model is based on the assumption of homogeneous material, which does not fully take into account the inhomogeneous nature of the asphalt mixture. Future directions for their research in this area include phase-field modeling and by

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taking into account the asphalt as a porous media. This study is addressing the behavior of asphalt mixture prepared only with plain binder. In this sense the use of modified binder might

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overall affect the parameters of the model. However, this should be addressed in a follow-up

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study to concrete determine the effectiveness of the Burgers model in representing the bulk

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part of the material away from the cohesive zone. Acknowledgments

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Financial support of the German Federal Ministry of Transportation for project No. FE

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References

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07.0290/2016/ERB (POTEA) is gratefully acknowledged.

Ban, H., Im, S., and Kim, Y. R. 2015. Mixed-mode fracture characterization of fine aggregate mixtures using semicircular bend fracture test and extended finite element modeling. Construction and Building Materials, 101, 721-729. doi: 10.1016/j.conbuildmat.2015.10.083 Barenblatt, G. I. 1962. The mathematical theory of equilibrium cracks in brittle fracture. In Advances in applied mechanics, Elsevier, 7, 55-129. Bharadwaj, M., Claramunt, S. and Srinivasan, S. 2010. Modeling Creep Relaxation of Polytetrafluorethylene Gaskets for Finite Element Analysis. International Journal of Materials, Mechanics and Manufacturing 5(2), 123-126, 2017. doi: 10.18178/ijmmm.2017.5.2.302 Büchler, S. 2010. Rheologisches Modell zur Beschreibung des Kälteverhaltens von Asphalten. (Ph.D. thesis), Braunschweig University of Technology, Germany (In German). Cannone Falchetto, A., and Moon, K. H. 2015. Micromechanical-analogical modelling of asphalt binder and asphalt mixture creep stiffness properties at low temperature. Road Materials and Pavement Design, 16(1), 111-137. doi: 10.1080/14680629.2015.1029708 Cannone Falchetto, A., Marasteanu, M. O., Balmurugan, S., and Negulescu, I. I. 2014. Investigation of asphalt mixture strength at low temperatures with the bending beam

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rheometer. Road Materials and Pavement Design, 15(1), 28-44. doi: 10.1080/14680629.2014.926618 Cannone Falchetto, A., Moon, K. H., Lee, C. B., and Wistuba, M. P. 2017. Correlation of low temperature fracture and strength properties between SCB and IDT tests using a simple 2D FEM approach. Road Materials and Pavement Design, 18(2), 329-338. doi: 10.1080/14680629.2017.1304258 Cannone Falchetto, A., Moon, K. H., Wang, D., Riccardi, C., and Wistuba, M. P. 2018. Comparison of low-temperature fracture and strength properties of asphalt mixture obtained from IDT and SCB under different testing configurations. Road Materials and Pavement Design, 19(3), 591-604.doi: 10.1080/14680629.2018.1418722 Chiangmai, C. N., and Buttlar, W. G. 2015. Cyclic loading behavior of asphalt concrete mixture using disk-shaped compact tension (DC (T)) test and released energy approach. Journal of the Association of Asphalt Paving Technologists, (84). Cox, B. L., Wang, J. S. Y. 1993. Fractal surfaces: measurement and applications in the earth sciences. Fractals, 1(01), 87-115. EN 12591. 2015. Bitumen and bituminous binders. Specifications for paving grade bitumens. European Committee for Standardization. Gutierrez-Lemini, D. 2014. Engineering viscoelasticity. New York: Springer. Hillerborg, A. 1985. The theoretical basis of a method to determine the fracture energy GF of concrete. 18(4), 291-296. Klüppel, M., and Heinrich, G. 2000. Rubber friction on self-affine road tracks. Rubber chemistry and technology, 73(4), 578-606. doi: 10.5254/1.3547607 Kong, J., and Yuan, J. Y. 2010. Application of linear viscoelastic differential constitutive equation in ABAQUS. In Computer Design and Applications (ICCDA), 2010 International Conference on (5), V5-152. IEEE. doi: 10.1109/ICCDA.2010.5541456 Li, X. J., and Marasteanu, M. O. 2009. Using semi circular bending test to evaluate low temperature fracture resistance for asphalt concrete. Experimental Mechanics, 50(7), 867876. doi:10.1007/s11340-009-9303-0 Li, X. J., and Marasteanu, M. O. 2010. The fracture process zone in asphalt mixture at low temperature. Journal of Engineering Fracture Mechanics, 77(7), 1175-1190. doi: 10.1016/j.engfracmech.2010.02.018 Li, X., and Marasteanu, M. O. 2004. Evaluation of the low temperature fracture resistance of asphalt mixtures using the semi-circular bend test. Journal of the Association of Asphalt Paving Technology, 73, 401426. Li, X., and Marasteanu, M. O. 2005. Cohesive modeling of fracture in asphalt mixtures at low temperatures. International Journal of Fracture, 136(1-4), 285-308. doi: 10.1007/s10704005-6035-8 Mandelbrot, B. B., Passoja, D. E., and Paullay, A. J. 1984. Fractal character of fracture surfaces of metals. Nature, 308(5961), 721. Marasteanu, M., et al. 2012. Investigation of low temperature cracking in asphalt pavements national pooled fund study–phase II. (Report No. MnDOT 2012-23). Saint Paul, MN: Minnesota Department of Transportation. Moon, K. H., Cannone Falchetto, A., and Marasteanu, M. O. 2013. Rheological modeling of asphalt material properties at low temperatures: From time to frequency domain. Road Material and Pavement Designs, 14(4), 810-830. doi: 10.1080/14680629.2013.817351 Moon, K. H., Cannone Falchetto, A., Wang, D., Wistuba, M. P., and Tebaldi, G. 2017. Lowtemperature performance of recycled asphalt mixtures under static and oscillatory loading. Road Materials and Pavement Design, 18(2), 297-314. doi: 10.1080/14680629.2016.1213500

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Journal Pre-proof Highlights 

Fracture properties of asphalt mixtures are investigated at three different low temperatures

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Finite Elements (FE) are used to model Semi Circular Bending (SCB) fracture test

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A self-affine fractal crack path is adopted for a more realistic simulation of crack propagation Strength and fracture toughness are determined by using cohesive zone modeling

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Fractal crack provides a reasonable compromise in terms of computation time

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