As might have known, the fracture mechanism (i.e. the evolution of the phase-field) in the brittle fracture framework is carried out mainly by the elastic strains, thus the contribution of the plastic strains will be negligible.
From the above statement, it becomes clear that a basis of the ductile phase-field model evolution is the right choice of the degradation function and, particularly, that this function should rely not only on the phase-field variable \(\varphi\) but also to some extent of the plastic strain state.

2.3 The proposed model

In this contribution, the recommended free energy functional is,