## 1 Introduction
Notably, it is impossible to describe the continuous change of failure mode with stress state because the connection between tension and shear failure envelopes is not smooth. In terms of the stiffness degradation model, most studies use the isotropic damage model. That is, stiffness degradation is assumed to be the same for all directions. This can be reasonable for tension damage, but not for compression shear damage. When compression shear damage occurs, stiffness weakens mainly along the crack plane direction. The isotropic damage assumption will cause the interpenetration of crack faces. With respect to the damage evolution equation, conventional damage models generally adopt the tension damage evolution equation based on tensile strain and the shear damage evolution equation based on compressive strain. As the tension cracks are opened and the two crack faces are separated, the physical meaning of the tensile strain as the primary indicator to measure the degree of tension damage is clear. The shear cracks are mostly closed and only a relative slip occurs between the two crack faces. Therefore, the adoption of compressive strain as a basic indicator for measuring the degree of shear damage is unreasonable. Most studies use biaxial or triaxial compression tests of rock specimens to validate the damage models. A comparative analysis found that although these models simulate the general failure process, the following deficiencies still exist: (1) Tension-shear failure or compression-shear failure under low-stress conditions cannot be reflected. (2) The fracture morphology of the rock specimen exhibited a discontinuous mutation with confining pressure. (3) The fracture angle between the failure plane and the first principal stress plane did not change with the confining pressure. (4) There was no apparent regularity between the residual strength and confining pressure, which was inconsistent with the experimental results. (5) The shear crack faces interpenetrate each other and the relative sliding direction is unreasonable. The above problems existing in existing damage models cannot be ignored. The aim of this study is to propose a novel anisotropic damage model to more accurately simulate the meso- and macro-failure behaviors and strength characteristics of quasi-brittle materials. Compared to conventional damage models, the anisotropic damage model has the following three improvements. First, a two-parameter parabolic-type yield criterion was used as the failure criterion. It can reflect the nonlinear strength characteristics of quasi-brittle materials. Moreover, it describes the pure tension, tension-shear, and compression-shear damages using only one continuous criterion, enabling the continuous transition of different failure modes. Second, an anisotropic stiffness degradation model was used instead of an isotropic degradation model to reflect the anisotropic mechanical characteristics of quasi-brittle materials after shear damage. Third, the shear strain on the failure plane was selected as the independent variable rather than compression strain to establish the shear damage evolution equation, for which the physical meaning is clearer. The proposed anisotropic damage model was implemented using COMSOL Multiphysics and MATLAB and validated against two benchmark simulations: uniaxial and biaxial compression tests and Brazilian splitting tests. The effectiveness and performance of the proposed anisotropic damage model were verified by comparing the simulation results with experimental results and other numerical simulation results. The remainder of this paper is organized as follows: Section 2 presents a novel anisotropic damage model. Section 3 presents the implementation strategy of the numerical simulation. Section 4 evaluated the effectiveness of the proposed damage model. Section 5 presents the validation against the uniaxial and biaxial compression tests. Section 6 presents the validation against the Brazilian splitting test.