Symbol Name
\(\mathbb{B}_{I}^{\mathbf{u}}\) Strain-displacement Tensor
\(\mathbb{B}_{I}^{\varphi}\) Strain-Phase field Tensor
\(E_{\mathcal{l}}\) Total energy
\(\mathbf{F}_{\text{I\ ext}}^{\mathbf{u}}\) External force
\(\mathbf{F}_{\text{I\ int}}^{\mathbf{u}}\) Internal force
\(g_{p}\) Plastic phase field function
\(H_{e}\) History variable
\(h\) Hardening modulus
\(\mathbb{K}\) Stiffness matrix
\(\mathcal{l}\) Length scale
\(N_{I}^{u}\) Displacement shape function
\(N_{I}^{\varphi}\) Phase field shape function
\(n\) Calibration parameter
\(p\) Accumulated plastic strains
\(q\) Calibration parameter(Stabilization)
\(\mathbf{r}_{I}^{\mathbf{u}}\) Displacement field Residual
\(\mathbf{r}_{I}^{\varphi}\) Phase field Residual
\(\mathbf{u}\) Displacement field
\(\alpha\) Internal hardening variable
\(\mathbf{\varepsilon}\) Strain
\(\varepsilon_{\text{eq},\text{crit}}^{p}\) Critical plastic strain
\(\varepsilon_{\text{eq}}^{p}\) Equivalent plastic strain
\(\Gamma\) crack set
\(\eta\) Stabilization parameter
\(\mu,\ \lambda\) Lame constants
\(g\left(\varphi\right)\) Phase field function
\(g\left(\varphi\mathrm{,}p\right)\) Plasticity dependent phase field function
\(\varphi\) Phase-Field parameter
\(\varphi_{c}\) Maximum value of damage
\(\psi^{e}\) Total elastic energy
\(\psi_{p}\) Plastic energy
\({\psi_{e}}^{+}\) Elastic energy(tension)
\({\psi_{e}}^{-}\) Elastic energy(compression)
\(\Omega\) Reference configuration
\(\mathcal{G}_{c}\) Fracture toughness