To solve the Eq.(41), extension of the return mapping algorithm made
taking into account a special treatment of the phase-field
development.41,42
Dependent on the small elastic strain tensor,\(\varepsilon^{\mathbf{e}}\), plastic strain tensor,\(\varepsilon^{\mathbf{p}}\), and the internal strain like scalar
hardening variable, \(\alpha\), the problem of the increment calculation
of \(\varepsilon^{\mathbf{e}}\)and \(\varepsilon^{\mathbf{p}}\)typically solved by an operator split into an elastic predictor and
plastic corrector.43,45 The calculation of the trial
elastic state \({(\ \ )}^{\text{tr}}\) according to the\(J_{2}\)-plasticity model with linear isotropic hardening and based on
freezing the plastic flow at the time \(t_{\ n+1}\) is given by,