## 2 Anisotropic Damage Model  
### Failure Criterion  
Two damage modes are typically assumed for quasi-brittle materials at the mesoscale: tension and shear. To unify the two types of damages with one criterion, the two-parameter parabolic-type yield criterion [40] was adopted as the failure criterion. It is assumed that the material will fail once the compressive stress \(\sigma\) normal to the failure plane and shear stress \(\tau\) on the failure plane satisfy the following parabolic relationship:  
\[\tau^{2}=n(\sigma+\sigma_{\text{t0}}), \tag{1}\]  
where \(n=\sigma_{\text{t0}}(\sqrt{m+1}-1)^{2}\), \(m=\sigma_{\text{c0}}/\sigma_{\text{t0}}\), \(\sigma_{\text{t0}}\) is the uniaxial tensile strength, and \(\sigma_{\text{c0}}\) is the uniaxial compressive strength. In the absence of compressive stress, this failure criterion is reduced to \(\tau=\sqrt{n\sigma_{\text{t0}}}\), where \(\sqrt{n\sigma_{\text{t0}}}\) is the shear strength of the failure plane without normal stress. Figure 1 shows the two-parameter parabolic-type yield envelope and the Mohr circles. It can be observed that the parabolic envelope shows strong nonlinear characteristics in both tensile and compressive stress regions, making up for the deficiency of the linear yield criterion. The left-end point of the envelope corresponds to the uniaxial tensile strength of the material. According to the relationship between the Mohr circles and strength envelope in the critical state, the shear and compressive stresses on the failure plane are as follows:  
\[\tau=\frac{\sigma_{1}-\sigma_{3}}{2}\cos\varphi, \tag{2}\]  
\[\sigma=\frac{\sigma_{1}+\sigma_{3}}{2}-\tau\tan\varphi, \tag{3}\]  
where \(\varphi\) is the internal friction angle; \(\sigma_{1}\) and \(\sigma_{3}\) are the first and third principal stresses, respectively. Moreover, the compressive stresses are positive. According to the failure condition Equation (1), tan \(\varphi=\partial\tau/\partial\sigma=n/2\tau\). Therefore, the stresses on the failure plane are as follows:  
\[\sigma=\frac{\sigma_{1}+\sigma_{3}}{2}-\frac{n}{2}, \tag{4}\]  
\[\tau=\frac{\sigma_{1}-\sigma_{3}}{2}\cdot\frac{\sqrt{2(\sigma_{1}+\sigma_{3})- 2n+4\sigma_{t}}}{\sqrt{2(\sigma_{1}+\sigma_{3})-n+4\sigma_{t}}}. \tag{5}\]  
Hence, the failure condition can be expressed in terms of the principal stresses as follows:  
\[\left\{ \begin{array}{cc}-\sigma_{3}\geq\sigma_{t0},&\sigma_{1}\leq n- \sigma_{t0}\\ (\sigma_{1}-\sigma_{3})^{2}-2n(\sigma_{1}+\sigma_{3})-4n\sigma_{t0}+n^{2}\geq 0,&\sigma_{1}>n-\sigma_{t0}\end{array} \right.. \tag{6}\]  
When \(\sigma_{1}\leq n-\sigma_{t0}\), the material fails in tension. When \(\sigma_{1}>n-\sigma_{t0}\), it fails in shear. Therefore, \(\sigma_{1}=n-\sigma_{t0}\) is the critical principal stress for switching the failure mode. In addition, for shear failure, \(\frac{\sigma_{1}+\sigma_{3}}{2}-\frac{n}{2}\geq 0\) corresponds to compression-shear failure, whereas \(\frac{\sigma_{1}+\sigma_{3}}{2}-\frac{n}{2}<0\) corresponds to the tension-shear failure. It is necessary to further establish the failure conditions expressed by strains for shear failure. According to the shear strength of the failure plane without normal stress and [PERSON]'s law for shear, the failure initiation shear strain on the failure plane is as follows:  
\[\gamma_{0}=\frac{2(1+\mu)}{E}\sqrt{n\sigma_{t}}. \tag{7}\]  
Hence, the shear failure condition can be expressed as follows:  
\[\gamma_{\text{eq}}=\gamma_{0}, \tag{8}\]  
where \(\gamma_{\text{eq}}\) is the equivalent shear strain on the failure plane, deduced using Equation (6), as follows:  
\[\gamma_{\text{eq}}=\sqrt{\left(\varepsilon_{1}-\varepsilon_{3}\right)^{2}+ \alpha(\varepsilon_{1}+\varepsilon_{3})+\beta}, \tag{9}\]  
where \(\alpha=-\frac{2n(1+\mu)^{2}}{(1-\mu)E}\) and \(\beta=\frac{n^{2}(1+\mu)^{2}}{E^{2}}\). As shown in Equation (8), the equivalent shear strain based on the two-parameter parabolic-type yield criterion is a nonlinear function of the strain, differing from the existing equivalent strain based on the linear yield criterion [41].