## 2 Phase-Field-Regularized Cohesive Zone Model
### Principle of Virtual Work and Governing Equations  
By using the results from the previous section, the total potential energy functional can be written as  
\[\Psi(\mathbf{u},\phi)=\int_{\Omega}\omega(\phi)\psi_{0}(\mathbf{\varepsilon}(\mathbf{u})) \mathrm{d}\Omega+\int_{\Omega}G_{c}\gamma(\phi,\
abla\phi)\mathrm{d}\Omega \tag{12}\]  
Here, the compact tensor is not used. For a linear-elastic material, \(\psi_{0}(\mathbf{\varepsilon})=\frac{1}{2}\lambda\varepsilon_{kk}\varepsilon_{ ll}+\mu\varepsilon_{ij}\varepsilon_{ij}\), with \(\lambda\) and \(\mu\) being the Lame constants; \(i,j,k,l=1,\ldots,d\), with \(d\) being dimensional numbers. For small strains, i.e., \(\varepsilon_{ij}=\frac{1}{2}\big{(}\partial u_{i}/\partial x_{j}+\partial u_{ j}/\partial x_{i}\big{)}\). The governing equations are obtained as follows: we first define the variation of the external work increment as \[\delta W_{\text{ext}}\ =\int_{\Omega}b_{j}^{*}\delta u_{j}\text{d}\Omega+\int_{ \partial\Omega_{t}}t_{j}^{*}\delta u_{j}\text{d}A \tag{13}\]  
On the other hand, the variation of the internal energy increment is given by  
\[\delta W_{\text{int}}\ =\delta\Psi=\frac{\partial\Psi}{\partial\epsilon_{ij}} \delta\epsilon_{ij}+\frac{\partial\Psi}{\partial\phi}\delta\phi \tag{14}\]  
which for the case of (12) yields  
\[ \begin{split}\delta\Psi&=\int_{\Omega}\omega(\phi) \sigma_{ij}\delta\epsilon_{ij}\text{d}\Omega+\int_{\Omega}\omega^{\prime}(\phi )\delta\phi\psi_{0}(\epsilon)\text{d}\Omega\\ &\quad+\int_{\Omega}G_{c}\frac{1}{c_{0}}\left(2\ell_{0}\frac{ \partial\phi}{\partial x_{i}}\frac{\partial\delta\phi}{\partial x_{i}}+\frac{ 1}{t_{0}}\alpha^{\prime}(\phi)\delta\phi\right)\text{d}\Omega\end{split}  \tag{15}\]  
By appropriate constant transformation with divergence theorem, it can be found that the above expression is equivalent to  
\[ \begin{split}&\delta\Psi=\int_{\partial\Omega}\omega(\phi)n_{i} \sigma_{ij}\delta u_{j}\text{d}A-\int_{\Omega}\omega(\phi)\frac{\omega_{ij}}{ \partial x_{i}}\delta u_{j}\text{d}\Omega\\ &+\int_{\Omega}\omega^{\prime}(\phi)\delta\phi\psi_{0}(\epsilon) \text{d}\Omega+\int_{\partial\Omega}G_{c}\frac{1}{c_{0}}2\ell_{0}\frac{ \partial\phi}{\partial x_{i}}n_{i}\delta\phi\text{d}A\\ &\quad+\int_{\Omega}G_{c}\frac{1}{c_{0}}\left(-2\ell_{0}\frac{ \partial^{2}\phi}{\partial x_{i}\partial x_{i}}\delta\phi+\frac{1}{t_{0}} \alpha^{\prime}(\phi)\delta\phi\right)\text{d}\Omega\end{split}  \tag{16}\]  
Combine the terms in (13) and (16), imposing that \(\delta W_{int}=\delta W_{ext}\) should hold for the arbitrary values of \(\delta u\) and \(\delta\phi\). This leads to the strong form of the governing equations:  
\[\omega(\phi)\frac{\partial\sigma_{ij}}{\partial x_{i}}+b_{j}^{*}=0\quad\text{ in }\Omega \tag{17}\]  
\[\omega(\phi)\sigma_{ij}n_{i}=t_{j}^{*}\quad\text{on}\ \partial\Omega_{t} \tag{18}\]  
\[u_{j}=u_{j}^{*}\quad\text{on}\ \partial\Omega_{u} \tag{19}\]  
\[-\frac{2\ell_{0}}{c_{0}}G_{c}\frac{\partial^{2}\phi}{\partial x_{i}\partial x _{i}}+\frac{G_{c}}{c_{0}\ell_{0}}\alpha^{\prime}(\phi)+\omega^{\prime}(\phi) \psi_{0}(\epsilon)=0\quad\text{in}\ \Omega \tag{20}\]  
\[\frac{\partial\phi}{\partial x_{i}}n_{i}=0\quad\text{on}\ \partial\Omega \tag{21}\]  
The above shows a coupled system consisting of the modified stress equilibrium (17) and the phase field evolution (20); (18), (19) and (21) are the boundary conditions for the modified stress equilibrium and the phase field evolution equations, respectively. For quasi-brittle materials, there is significant tensile-compression anisotropic behavior because their compressive strength is much greater than their tensile strength. To avoid the propagation of the crack under compression, [PERSON] [17], [PERSON] [18] and other scholars proposed a volumetric-deviatoric split, spectral decomposition and other methods to overcome the above problem in the classical phase field fracture model.