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