Fundamental understandings on the bitumen fracture mechanism are vital to improve the mixture design of asphalt concrete. In this paper, a diffuse interface model, namely, phase-field method is used for modeling the quasi-brittle fracture in bitumen. This method describes the microstructure using a phase-field variable which assumes one in the intact solid and negative one in the crack region. Only the elastic energy will directly contribute to cracking. To account for the growth of cracks, a nonconserved Allen-Cahn equation is adopted to evolve the phase-field variable. Numerical simulations of fracture are performed in bituminous materials with the consideration of quasi-brittle properties. It is found that the simulation results agree well with classic fracture mechanics.
One of the most serious distresses in pavement structure is brittle cracking in bitumen [
The phase-field method (PFM) was originally proposed by Cahn and Hilliard [
In the simulation of preflawed bitumen cracking, the phase-field variable is set as
Figure
Local free energy density with respect to different
Figure
Local free energy density with respect to different
Figure
Local free energy density with respect to different
Actually, for bitumen and bituminous materials, the materials are quasi-brittle since the visco- and plastic properties cannot be neglected no matter how small their magnitudes are. The cracking process is always initiated by the local plastic deformation at the crack tip. The plastic work needs to be done first before crack can advance to create new surfaces. The total stress is thus considered as the sum of the elastic part, the viscous part and, the plastic part as
The elastic part can be obtained according to Hooke’s law as
The viscous part could similarly be obtained as [
The plastic stress is obtained as
The evolution of the stress fields is determined by the principle of momentum conservation:
Note that the viscous part and plastic part almost have no contributions to quasi-brittle cracking. Allen-Cahn equation is employed as the governing equation:
Quasi-brittle cracking calculation flowchart.
The derivations of our previous theoretical analysis based on PFM lies in that bituminous material can be considered isotropic and homogenous. However, in reality, there will exist large coarse aggregates in asphalt mixture, as shown in Figure
Asphalt mixture with coarse aggregates.
The cracking simulation using the diffuse interface approach used in this paper could be not only good for preflawed bitumen quasi-brittle cracking but also good for bituminous mixture using fine aggregates (Figure
(a) Fine aggregates and (b) coarse aggregates.
The authors have previously analyzed the asphalt binder cracking using phase-field method at low temperature [
Consider the preflaw in bitumen specimen as a rectangular crack with depth
Cracking contour.
Based on Hooke’s law, we further have elastic strain energy density as
Since there is no traction on the fracture surface, we thus have
Also, note that, during the linear elastic cracking process, the elastic strain energy will be totally transformed to the fracture energy
Equation (
A two-dimensional finite element model is established in COMSOL to simulate the quasi-brittle cracking process of bitumen under tension loading. A fixed Eulerian mesh is used to describe the internal interfaces between the intact solid and crack void, as shown in Figure
Fixed Eulerian mesh in preflawed bitumen model.
Actually the overall cracking process in preflawed bitumen should be simulated in three dimensions. However, it will be very costly to do such simulation. Previous researches [
Pure tension loading is applied on top and bottom boundaries, which is set as
To study the quasi-brittle cracking behavior and viscoelastic cracking behavior on bitumen, two simulations are conducted simultaneously. The first is to consider the bitumen as linear elastic, while the second is to consider bitumen as viscoelastic plastic.
Quasi-brittle cracking: the following material properties are used:
Viscoelastic cracking is used to reflect the viscoelastic property of bitumen.
In order to study the differences between quasi-brittle cracking and viscoelastic cracking in our phase-field model on bitumen, the fracture results are first calculated and compared to classic fracture mechanics, where the Griffith criterion is
For bitumen at low temperature,
The critical load calculation based on the classic fracture mechanics without considering viscous property can be calculated as 313310.96 Pa. In phase-field method, the load value at the time instant when
For Case
For Case
The reason why the critical load in quasi-brittle phase-field model is much smaller than that in the viscoelastic model is that only the elastic stress part will contribute to fracture rather than the total stress (including elastic stress, viscous stress, and plastic stress) during brittle-cracking. It means bitumen has more resistance to fracture when considering viscoelastic properties than pure brittle linear elastic.
For infinitesimal deformation during bitumen brittle cracking, Cauchy stress is the same as Piola-Kirchhoff stress, which indicates the pure brittle cracking. However, considering the “quasi-brittle” property, finite deformations will occur at crack tip, and the second Piola-Kirchhoff stress is used to express the stress with Gauss-point evaluation. For the
In this paper, we present a phase-field model to analyze the quasi-brittle fracture in bitumen subject to tension loading. In phase-field model, only the elastic stress will contribute to the fracture. Overall, compared with the classic fracture mechanics, the phase-field model does not need to explicitly treat the crack surface and can easily solve the quasi-brittle cracking of preflawed bitumen specimen.
The authors declare that they have no competing interests.
The research performed in this paper is supported by National Natural Science Foundation of China (no. 51308042 and no. 41372320), Natural Science Foundation of Shandong Province (ZR2015EQ009), and the Fundamental Research Funds for the Central Universities (06500036).