and \(\varepsilon_{\text{eq}\mathrm{,}\text{crit}}^{p}\ \)as a threshold
value. \(\varepsilon_{\text{eq}}^{p}\) is often called von Mises
equivalent plastic strain. The variable \(p\) represents the
accumulation and localization of plastic strains. By making dependency
on \(\varphi\), \(p\) and degradation function \(g\), the fracture
process will be the natural consequence of ductile damage accumulation.
The variational derivative of \(E_{\mathcal{l}}\) with respect to\(\mathbf{\varepsilon}^{e}\) bring into the equilibrium equation\(\text{div\ }\mathbf{\text{σ\ }}=\ \mathbf{0}\), where the stress
takes the form