Fracture failure in crack interaction of asphalt binder by using a phase field approach

Abstract

Fracture failure in crack interaction of asphalt binder has always been one serious problem in the pavement industry. In the state of the art research of asphalt cracking, single mode cracking has been studied by many researchers but there lacks theoretical and experimental research on the crack interaction of asphalt binder, which is more reasonable and realistic. The traditional way is to use the Griffith’s theory which is complex and complicated. In this paper, the phase field method (PFM) is presented for modeling, which describes the whole cracking system using a phase-field variable that assumes negative one in the void region (crack) and positive one in the solid region (intact). The fracture toughness is then considered as a material property and modeled as the surface energy stored in the diffuse interface between the intact solid and crack void. The non-conserved Allen–Cahn equation is adopted as the system governing equation to evolve the phase field variable to account for the growth of cracks. The energy based formulation of the phase-field method handles the competition between the growth of surface energy and release of elastic energy of crack interaction in a natural way: the crack propagation is a result of the energy minimization in the direction of the steepest descent. Both the linear elasticity and phase-field equation are solved in a unified finite element frame work, which is implemented in the commercial software COMSOL. Two cracking experiments, namely, direct tension test and double edge notch tension test are then performed for validation. It is discovered that the critical load of crack interaction by PFM agrees very well with both experiment results.

Appendix 1

The classic fracture mechanics is the traditional method to calculate the critical load that causing failure in pure Mode I case. For a single mode I crack, the Griffith criterion reads [27]

$$K_{I} \ge K_{\text{Ic}}$$

(26)

where K _I is the mode I stress intensity factor and K _Ic is the mode I fracture toughness which is a material parameter of asphalt binder. For the direct tension test specimen, the stress intensity factor is related to the tensile load as [22]

$$K_{I} = \sigma \sqrt {\pi a} \sqrt {\frac{2b}{\pi a}{ \tan }\frac{\pi a}{2b}} D_{I} (\frac{a}{b})$$

(27)

where a is the crack length, b is the specimen width (c.f. Fig. 7), and

$$D_{I} = \frac{{0.752 + 2.02\frac{a}{b} + 0.37(1 - { \sin }\frac{\pi a}{2b})^{3} }}{{{ \cos }\frac{\pi a}{2b}}}$$

(28)