_Article_  
A Nonlinear Crack Model for Concrete Structure  
Based on an Extended Scaled Boundary Finite  
Element Method  
[PERSON] 1,* [PERSON] 1, [PERSON] 1, [PERSON] 2 and [PERSON] 1  
State Key Laboratory of Coastal and Offshore Engineering, Institute of Earthquake Engineering,  
Dalian University of Technology, Dalian 116024, China; [EMAIL_ADDRESS] (X.G. ); xiangfan  
Two main approaches are often taken to model the FPZ, which are: the smeared crack models and the discrete crack models. The smeared crack models proposed by [PERSON] [2] are based on the continuum approach, where the computational mesh of the FPZ remains constant while the fracture propagation is modeled by the growth of a number of parallel cracks smeared over the elements within the FPZ. Using this approach, [PERSON] et al. [3] and [PERSON] et al. [4] successfully carried out dynamic cracking analyses of the Koyna gravity dam under the influence of nonorthogonal cracks. [PERSON] et al. [5] also follow a similar approach incorporating the linear or bilinear softening dispersion crack models to predict the crack response of concrete gravity dams. The damage-based fracture mechanics was also developed sharing the similar concept by [PERSON] [6]. These methods have shown tremendous success in modeling concrete fractures under complex loading and boundary conditions. However, the strong mesh dependency often causes complications in determining the characteristic length scale, fracture strength, and fracture toughness to accurately describe the fracturing propagation process. The smeared crack models mainly combine with damage mechanics for crack propagation studies. On the other hand, the discrete crack model proposed by [PERSON] [7] and [PERSON] [8] utilizes a predefined fracture path as a part of the computational domain boundary, which reduces the mesh dependency comparing to the smeared crack model. [PERSON] et al. [9] firstly implemented the cohesive zone model (CZM) to describe the FPZ based on the discrete crack approach. Following this approach, numerous scholars have improved and implemented the cohesive zone method to model the fracture propagation process ([PERSON] et al. [10], [PERSON] et al. [11], [PERSON] et al. and [PERSON] et al. [12; 13; 14]). Among these improvements, the scaled boundary element method (SBFEM) was developed to model the complex fracture growth in terms of fracture branching and coalescing under complex loading conditions. However, the SBFEM fails to capture the nonlinearity brought by the cohesive interactions within the FPZ. Therefore, the mesoscale and atomic-scale-inserted CZMs were developed to model the FPZ [15; 16; 17; 18]. However, the predetermined fracture paths often cause computational complexities and inaccuracies. Recently, the extended scaled boundary finite element method (X-SBFEM) based on the level set method was developed on the basis of both the SBFEM [19; 20] and the extended finite elements (XFEM) [21; 22]. Capitalizing on the advantages of both methods, X-SBFEM [23; 24] can make full use of XFEM to describe the discontinuous displacement field and SBFEM to solve the problem of the stress singularity with higher precision. Simulating the crack body section using XFEM and the crack tip using SBFEM, the method finally establishes the total equilibrium equation of the crack body and solves the equation, thereby overcoming the disadvantages of XFEM, such as obtaining the analytical form of the displacement and the stress asymptotic fields of the crack tip in advance and constructing the complex enhanced functions, which can express nonsmooth behaviors near the crack tip. In some special circumstances, the enrichment functions are discontinuous or have nonpolynomial forms to specially address the issue when the stiffness matrix is constructed using numerical integration. Figure 1: Fracture process zone (FPZ) in concrete. Currently, the application of X-SBFEM in fractures [23] based on linear elastic fracture mechanics (LEFM) mainly focuses on the linear fracture.