## 1 Introduction  
Quasi-brittle materials are widely used in civil and underground engineering. Among them, there are artificial materials, such as concrete and masonry, and natural materials, such as rock and hard clay. Quasi-brittle materials generally exhibit the following mechanical properties: inhomogeneity, anisotropy, structural discreteness, and nonlinearity. The failure of quasi-brittle materials causes engineering accidents and geological disasters, such as collapses, landslides, and tunnel collapses, creating challenges to the construction and operation of engineering structures. Therefore, exploring the failure behaviors of quasi-brittle materials is of great significance. In the past decades, many research studies were carried out on material failure behaviors. Most of these studies are based on experimental methods, involving material splitting failure [1; 2], impact failure [3], static uniaxial and triaxial compression failure [4; 5; 6], and dynamic uniaxial and triaxial compression failure [7; 8]. This method is simple and easy to operate. However, it also has some disadvantages, such as complex sample preparation, long cycles, high cost, large dispersion of results, and unclear failure mechanism. With the development of computer technology and numerical computation, numerical simulations have been gradually used to study failures. A range of methods and models have been proposed to simulate the initiation and propagation of cracks, such as the traditional finite element method [9; 10; 11; 12; 13], discrete element method [14; 15; 16; 17; 18], extended finite element method [19; 20; 21], boundary element method [22; 23; 24], displacement discontinuity method [25; 26; 27; 28; 29], etc. The most popular ones are the finite element method and discrete element method. The discrete element method considers the material as a combination of particles and expresses cracks by bond-breaking between them. It is suitable to deal with discontinuities and large deformations in the joint rock mass. However, it encounters difficulties in simulating the deformation and failure of intact rocks. In addition, the discrete element method is relatively computationally intensive, which limits either the length of a simulation or the number of particles, and is difficult to apply to complex problems, such as multiscale and multifield problems. The finite element method is mostly based on the damage model [30; 31; 32; 33] or phase field model [34; 35; 36], and simulates the initiation and propagation of cracks through stiffness reduction. Compared with the discrete element method, this method has the advantages of a clear physical meaning of parameters, high calculation efficiency, and strong applicability. The phase field and damage modeling approaches have essential differences. The phase field method introduces an additional degree of freedom that greatly increases the dimension of the problem, and requires fine grids, thus requiring higher computing resources and being limited to small-scale problems. The damage method does not need to introduce an additional degree of freedom; thus, it is computationally efficient and suitable for large-scale engineering simulations. It assumes that materials may fail in tension or shear and introduce a damage variable to characterize the degradation of stiffness. The key to adopting the damage method to simulate material failure is to construct appropriate failure criteria, a stiffness degradation model, and the damage evolution equation. The most-used tension failure criterion is the maximum tensile stress criterion. The commonly used shear failure criteria include the Mohr-Coulomb strength criterion [31; 32; 37; 38] and the Drucker-Prager criterion [33; 39]. These criteria approximate the envelope of the Mohr circles as a straight line. The error of this approximation is small under low-stress conditions but could be large under high-stress conditions. In addition, the simulation results based on these criteria in the tension-shear composite zone are far from the experimental results.