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,