## 1 Introduction  
Fiber-reinforced polymer (FRP) laminates are widely used in aerospace, automotive, medical components, and sports equipment, due to their advantages. The transverse loading and low-velocity impact of the fiber-reinforced polymer laminates are the common loading conditions of the composite structures. The complexity of failure processes involving intra- and inter-laminar damage in composite laminates needs reliable failure theories and a damage evolution mechanism. To determine the fracture behavior in engineering structures under quasi-static or impact load, the studies on a reasonable numerical analysis method are mainstream in computational fracture mechanics. Various intralaminar damage mechanisms have been proposed and demonstrated in the composite structures. The traditional fracture mechanics using crack propagation theory have been developed and matured well to solve these issues, but it is difficult to explain the complex failure process of the composites such as kinking and crushing. The first-ply failure model, in which all material properties are set to zero once a prescribed failure criterion is satisfied, are proposed in the numerical simulation, but the initiation of damage does not cause instantaneous failure of the entire structure, because of the complex fracture nature of laminated composites [1; 2]. The cohesive zone model (CZM) was recently used for simulating intralaminar failure [3; 4], but the CZM approach has a shortcoming; it is difficult for CZM to describe arbitrary matrix cracks in composites. Continuous damage mechanics (CDM) of composite materials stands on its own as a mature field solidly having a variety of analytical and computational methodologies associated with it [5; 6; 7; 8]. CDM theory was the first applied to study the creep rupture of materials [9], and the progressive damage failure analysis based on CDM has been widely employed to predict the damage initiation and evolution of composites. [PERSON] et al. [10; 11] proposed a continuum damage model for the prediction of the intralaminar failure mechanisms in composite laminates, and [PERSON] et al. [12] proposed a combined elastoplastic damage model for the progressive failure analysis of composite materials. [PERSON] et al. [13] adopted the shear lag model and defined transverse and shear damage as an internal state variable representing the lamina damage state directly. [PERSON] et al. [14] developed the improved shear damage evolution model for fiber-reinforced laminates and considered the effect of matrix cracks on transverse and shear damage evolution. [PERSON] et al. [15] proposed the CDM model with the introduction of the crack direction parameter, which has the capability to predict the initiation and direction of crack propagation. Interlaminar delamination has been widely studied, where agreement between models and experiment is common [16; 17; 18; 19; 20]. The delamination simulation of composites using the finite element method (FEM) has been generally performed by the CZM approach. The CZM approach does not require remeshing for crack propagation and can predict both the initiation and progression of delamination. [PERSON] [21] first proposed CZM, and [PERSON] [22] introduced an idea of cohesive forces on a molecular scale. [PERSON] et al. [23] proposed the exponential cohesive zone model, and [PERSON] et al. [24] introduced the bilinear cohesive law. The bilinear hardening traction-separation relationship is still a basic element in several widely-used cohesive laws for the delamination analysis [25; 26; 27; 28; 29; 30; 31; 32]. Composite failure is characterized by a strong interaction of intralaminar failure and interlaminar delamination. [PERSON] [33] presented the damage model considering the relation between matrix crack density and delamination in order to predict the propagation of matrix cracking and delamination in laminated composites.