respectively.
To enhance the efficiency of phase-field computations, the higher-order and hybrid formulations were recommended in Miehe8and Ambati32, respectively. Extension of the quasi-static formulations Eq.(5) and Eq.(9) to the dynamic setting has been presented in numerous contributions.12,13,33,34
2.2 Phase-field model of ductile fracture:
For the energy functional presented in Eq. (1) crack growth is driven by elastic strain energy. To extend this theory to ductile materials and provide a mechanism for plastic yielding to contribute to crack growth plastic energy density function will add,\(\psi_{p}\left(\alpha\right)\), to the stored energy. The stored energy functional proposed in Ambati35 is,