\(\mathbb{B}_{I}^{\mathbf{u}}\) |
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 |