# An Improved Cohesive Zone Model for Interface Mixed-Mode Fractures of Railway Slab Tracks
## 1 Introduction  
Chinese railway track systems (CRTS) have successfully served for more than 10 years in China's high-speed railway (CRH) and have performed well during the period. However, with increasing operation time and the influence of complex temperature and environmental conditions, hundreds of interfacial cracks (as shown in Figure 1) between track slab and cement asphalt mortar (CA mortar) have appeared on the high-speed railway tracks [1, 2]. Under extremely high temperatures in summer, defects of slab arching also occur. Typical interlayer defects, such as slab arch [3], are closely related to the interfacial cracks between the track slab and the under-layer. During operation, the track directly undertakes the effects of the cyclic loads from the high-speed train and environmental temperature, which increases the possibility of interface cracking. As a vertical multilayer and longitudinally heterogeneous structure, the slab ballastless track has weak parts between the new and old concrete interface and composite connection surface. Therefore, a reasonable simulation of interlayer interactions is the key to studying the defects of track structures. Cohesive zone model (CZM), an effective and favored crack model in interface fracture mechanics, has been widely used to simulate crack initiation and propagation in various materials, such as metals [4, 5, 6], polymers [7], ceramics [8], concrete [9, 10, 11], and fiber-reinforcedcomposites [12]. In CZM, material failure is characterized by a traction-separation law, which relates the traction across the crack to the corresponding separation [13]. The approach ensures that CZM maintains the continuity conditions mathematically and removes the singularity present in Linear Elastic Fracture Mechanics (LEFM) [14; 15; 16]. With the development of CZM over the past six decades since it was proposed by [PERSON] [17] and [PERSON] [18], a large variety of traction-separation laws have been established. The most popular are bilinear [19; 20; 21; 22], trapezoidal [23; 24; 25], exponential [26; 27; 28], polynomial [29; 30; 31], and so on. Various cohesive zone models may have different applicable conditions due to different initial assumptions. For instance, the trapezoidal cohesion zone model proposed by [PERSON] [23] could not consider the situation where the mode I fracture energy is not equal to the mode II fracture energy. The exponential cohesive zone model proposed by [PERSON] and [PERSON] [27] could consider the difference values of normal and tangential fracture energies, but when the two fracture energies are different, there is a \"self-repairing\" problem at the crack tip under mixed-mode loading and unloading. The Park-Paulino-Roesler (PPR) model is a kind of polynomial traction-separation law for mixed-mode fractures that was proposed by [PERSON] et al. [32] in 2009. This model is versatile because it can consider different fracture energies with respect to fracture modes and can be applied to represent various material softening responses, i.e., ductile, brittle, and quasi-brittle, due to the controllable softening given by the shape parameters [13; 32]. More significantly, the model guarantees the consistency of the cohesive constitutive relationship under mixed-mode conditions [33; 34; 30]. Due to the above advantages and the convenient implementation in commercial software ABAQUS as a user subroutine [35; 36; 34], the PPR model has been utilized to investigate a wide range of failure phenomena and cited in many papers. The model was found to still have limitations that need to be improved. [PERSON] et al. [37] indicated that due to the different cohesive interaction regions between the normal and tangential tractions when fracture energies are different, one traction component might become zero while the other traction component had not yet vanished. This situation does not conform to reality in which normal and tangential tractions typically fail simultaneously when a fracture happens. In addition, [PERSON] et al.