Equation reference only.
1. Fields
| Symbol | Meaning | Convention |
|---|---|---|
| \(\boldsymbol{u}\) | displacement | primary field |
| \(p\) | pore pressure | primary field |
| \(\phi\) | phase field | \(0\): intact; \(1\): broken |
| \(H\) | history field | pointwise update |
| \(g(\phi)\) | tensile stiffness degradation | not the phase field |
| \(\ell_0\) | crack length scale | diffuse width |
| \(G_c\) | fracture energy | fracture parameter |
2. Phase field
\[ \left[1+\frac{2\ell_0(1-k)H}{G_c}\right]\phi -\ell_0^2\nabla^2\phi = \frac{2\ell_0(1-k)H}{G_c}. \]
\[ \mathbf{K}_{\phi}\boldsymbol{\Phi} = \mathbf{F}_{\phi}. \]
3. Driving energy and history
\[ Y = \psi_e^+(\boldsymbol{\varepsilon}) + \boldsymbol{\sigma}_0: \boldsymbol{\varepsilon}. \]
\[ H(\boldsymbol{x},t) = \max_{s\in[0,t]}Y(\boldsymbol{x},s), \qquad H_i^{n+1} = \max(H_i^n,Y_i^{n+1}). \]
4. Tensile energy and degradation
\[ \psi_e^+ = \frac{\lambda}{2} \left\langle\operatorname{tr}\boldsymbol{\varepsilon}\right\rangle_+^2 + \mu\,\operatorname{tr} \left[ (\boldsymbol{\varepsilon}^+)^2 \right]. \]
\[ \langle x\rangle_+=\max(x,0). \]
\[ g(\phi)=(1-k)(1-\phi)^2+k, \qquad g(0)=1, \qquad g(1)=k\approx0, \qquad g'(1)=0. \]
5. Solid mechanics
\[ \nabla\cdot\boldsymbol{\sigma} - \alpha\boldsymbol{I}\nabla p + \boldsymbol{b} = \boldsymbol{0}. \]
\[ \mathbf{K}_u\boldsymbol{U} = \boldsymbol{F}_u, \qquad \mathbf{K}_u^{(j)}\Delta\boldsymbol{U} = \boldsymbol{R}_u^{(j)}. \]
\[ \mathbb{D} = \frac{\partial\boldsymbol{\sigma}_e} {\partial\boldsymbol{\varepsilon}}, \qquad \mathbf{K}_u^e = \int_{\Omega_e} \mathbf{B}_u^{\mathsf T} \mathbf{D}_e \mathbf{B}_u\,\mathrm{d}\Omega. \]
\[ D_{ijkl} = \overline{D}_{ijkl} + \widetilde{D}_{ijkl}. \]
\[ \overline{D}_{ijkl} = \lambda \left[ g(\phi)H_e(\operatorname{tr}\boldsymbol{\varepsilon}) + H_e(-\operatorname{tr}\boldsymbol{\varepsilon}) \right] \delta_{ij}\delta_{kl}. \]
\[ \widetilde{D}_{ijkl} = 2\mu \left[ g(\phi)P^+_{ijkl} + P^-_{ijkl} \right]. \]
\(H_e\) is the Heaviside function; \(H\) is the history field.
6. Fluid transport
\[ \rho S\frac{\partial p}{\partial t} - \nabla\cdot \left( \frac{\rho\boldsymbol{K}_{\mathrm{eff}}}{\mu_{\mathrm{eff}}} \nabla p \right) = q_m - \rho\alpha\chi_r \frac{\partial\varepsilon_{\mathrm{vol}}}{\partial t}. \]
\[ \varepsilon_{\mathrm{vol}} = \nabla\cdot\boldsymbol{u}, \qquad S = \varepsilon_p c + \frac{(\alpha-\varepsilon_p)(1-\alpha)}{K_{Vr}}. \]
7. Property indicators
For \(c_1<c_2\):
\[ \chi_r(\phi) = \begin{cases} 1,&\phi\le c_1,\\ \dfrac{c_2-\phi}{c_2-c_1},&c_1<\phi<c_2,\\ 0,&\phi\ge c_2, \end{cases} \qquad \chi_f(\phi) = \begin{cases} 0,&\phi\le c_1,\\ \dfrac{\phi-c_1}{c_2-c_1},&c_1<\phi<c_2,\\ 1,&\phi\ge c_2. \end{cases} \]
\[ \chi_r+\chi_f=1, \qquad \alpha=\alpha_r\chi_r+\chi_f, \qquad \boldsymbol{K}_{\mathrm{eff}} = \boldsymbol{k}_r\chi_r+\boldsymbol{k}_f\chi_f. \]
8. Pressure discretization
\[ \left( \frac{\mathbf{M}_p}{\Delta t} + \mathbf{K}_p \right) \boldsymbol{P}^{n+1} = \boldsymbol{F}_p + \frac{\mathbf{M}_p}{\Delta t} \boldsymbol{P}^{n}. \]
\[ \left( \frac{\mathbf{M}_p}{\Delta t} + \mathbf{K}_p \right) \Delta\boldsymbol{P} = \boldsymbol{R}_p. \]
9. Staggered sequence
\[ \phi^n \rightarrow \{\chi_r,\chi_f,\alpha,\boldsymbol{K}_{\mathrm{eff}}\} \rightarrow \{\boldsymbol{u},p\} \rightarrow Y \rightarrow H^{n+1} \rightarrow \phi^{n+1}. \]
\[ \|\Delta\boldsymbol{u}\|, \quad \|\Delta p\|, \quad \|\Delta\phi\| < \text{tolerance}. \]