## 1 Introduction
[PERSON] [34] proposed the variational approach reflecting the initiation of local delamination near the tips of transverse cracks in \(90^{\circ}\) layers and longitudinal cracks in \(0^{\circ}\) layers parallel to the loading. Conventionally, these intra- and inter-laminar failure mechanisms have been treated separately in fracture finite element analysis of composites, but recently, these interactions of multiple damage modes have been studied by several researchers [35]. [PERSON] et al. [36] simulated the interaction of intra- and inter-laminar damage for composite laminates using CZM and CDM, respectively, and compared with the experiment. It was concluded that CZM cannot be widely applied to a variety of problems because of prescribing the damage-evolution path and CDM delays the delamination initiation significantly and underpredicts the final delamination crack length. An approach, which combines the floating node method with the VCCT, is proposed to simulate delamination migration in cross-ply tape laminates [37]. [PERSON] et al. [3; 38] developed 2D FE models to simulate some modes of inter- and intra-laminar cracks and their interaction by employing cohesive zone elements (CZEs) in the intralaminar failure; this has the disadvantage that they are only available for a pre-known damage-evolution path. The phantom node method for matrix cracking and CZM for delamination are used for failure analysis of composite laminates [39]. CZM in conjunction with the extended finite element method (XFEM) cannot capture the load transfer between cohesive interfaces and the solid elements when crack bifurcation or coalescence occurs [40]. An extended cohesive damage model is introduced by combining XFEM and CZM for simulating multicrack propagation in fiber composites [41; 42]; XFEM is very effective for treating cracks to even multiple levels, but it has some shortcomings, such as difficulty with 3D crack propagation and high computational cost [43]. The phase field model combined with CZM [44] and the XFEM approach coupled with CDM [45; 46]have also been proposed to simulate the interaction of delamination and intralaminar crack. CZM and CDM are used together to predict the damage caused by low velocity impact in carbon-epoxy \([0_{4}904]_{s}\) laminates [47]. The method of using CDM for the intralaminar failure and using CZM for the interlaminar delamination of composites is still extensively used in the laminated composite damage analysis [48], but it has led to the loss of coupling information for simulating multiple damage modes and also results in inaccurate prediction of crack paths [3]. CZM in conjunction with CDM cannot capture the high stresses at the tip of the transverse crack in the numerical simulation, for the elements where the transverse crack is predicted soften without being able to capture the stress field accurately at the interface, and this approach also does not explicitly represent the kinematics of the cracks [37; 49]. The interaction between matrix cracks and delaminations in composite laminates has been investigated experimentally or theoretically in the previous studies, but the drawbacks of not being able to reflect interaction between several failure mechanisms, such as delamination due to matrix cracking, have not yet been remedied in the approaches of using CDM and CZM together. A few studies have been performed to solve the issues concerning the damage interaction for failure simulation of FRP laminates. In a word, there is no effective progressive damage model both capturing all multi-damage modes together and reflecting the interaction between the complex failure mechanisms for the FRP composites. This work presents a novel approach for modeling the interaction of delamination and intralaminar crack in composite materials with a polymeric matrix reinforced with carbon fibers subject to quasi-static load.