## 3 Constitutive Model  
### Multiaxial Constitutive Model with Damage  
The \(J_{2}\) plastic model with damage variable \(D\), known as the Lemaitre's damage model, is considered in this work [51] for describing ductile and brittle fracture behaviors. In this model, the damage variable \(D=1\) represents the total fracture of materials and \(D=0\) represents intact materials. Moreover, the relationship between effective stress \(\mathbf{\sigma}(D)\) and real stress \(\mathbf{\sigma}\) in the theory of continuum damage mechanics (CDM) is given by [51]:  
\[\mathbf{\sigma}(D)=\frac{\mathbf{\sigma}}{(1-D)}. \tag{11}\]  
For ductile fracture problems, the yield function is given by [51]  
\[f=\overline{\sigma}(D)-\sigma_{\mathrm{Y}}(\overline{\epsilon}^{\mathrm{p}})=0, \tag{12}\]  
where \(\overline{\sigma}\) represents the von Mises equivalent stress, \(\sigma_{\mathrm{Y}}\) denotes the yield stress, and \(\overline{\epsilon}^{\mathrm{p}}\) is the accumulated plastic strain.  
The hardening law considered in this work is given by \(\sigma_{\mathrm{Y}}=\sigma_{\mathrm{Y}_{0}}+A\overline{\epsilon}^{\mathrm{p}}\), where \(\sigma_{\mathrm{Y}_{0}}\) is the initial yield stress and \(A\) is the hardening parameter.  
Combining the maximum dissipation principle [52] and [PERSON]'s work [53], the evolution of the incremental plastic strain, accumulated plastic strain, and damage can be expressed as [53]  
\[\Delta\epsilon^{\mathrm{p}}=\Delta\gamma\frac{\partial f}{\partial\mathbf{ \sigma}}, \tag{13}\]  
\[\Delta\overline{\epsilon}^{\mathrm{p}}=\frac{\Delta\gamma}{1-D}, \tag{14}\]  
\[\Delta D=\left\{\begin{array}{ll}\overline{D}\frac{\Delta\gamma^{\mathrm{p}} }{\overline{\epsilon}^{\mathrm{p}}_{0}-\overline{\epsilon}^{\mathrm{p}}_{0}} \,R_{\mathrm{v}},&\mathrm{if}\epsilon^{\mathrm{p}}\geq\overline{\epsilon}^{ \mathrm{p}}_{0}\text{ and }\sigma_{\mathrm{m}}\geq 0\\ 0,&\text{if not}\end{array}\right., \tag{15}\]  
where \(\Delta\gamma\) is the plastic multiplier, \(R_{\mathrm{v}}=\frac{2}{3}(1+\
u)+3(1-2\
u)\big{(}\frac{\sigma_{\mathrm{m}}}{ \overline{\sigma}}\big{)}^{2}\) is the triaxiality function, \(\sigma_{\mathrm{m}}\) is the hydrostatic pressure, \(\sigma_{\mathrm{m}}\geq 0\) ensures that the material does not fracture under compression, \(\overline{D}\) is the critical damage value, \(\overline{\epsilon}^{\mathrm{p}}_{\mathrm{f}}\) is the fracture strain, and \(\overline{\epsilon}^{\mathrm{p}}_{0}\) is the corresponding strain at the beginning of damage.  
For brittle fracture problems, the aforementioned damage model can be simplified as a subtle modification of Equation (15), i.e., adjusting \(\overline{\epsilon}^{\mathrm{p}}_{0}\) to \(0\), \(\overline{\epsilon}^{\mathrm{p}}_{\mathrm{f}}\) to a tiny value \(\overline{\epsilon}^{\mathrm{p}}_{\mathrm{f}}\), and \(\sigma_{\mathrm{Y}_{0}}\) to the tensile strength \(\sigma_{\mathrm{t}}\). The main idea is constructed by considering the material as an ideal plastic material and reducing the deformation in the plastic stage to simulate brittle fracture. Based on this approach, it is not only easier to use the tensile strength \(\sigma_{\mathrm{t}}\) as a failure criterion but also to spontaneously describe the stress degradation behavior of the material to avoid numerical oscillations caused by the sudden loss of bond force. Figure 2 gives a schematic diagram of the damage evolution during ductile and brittle fracture problems, where \(E\) is [PERSON]'s modulus.