Coupling of the phase field approach to the Armstrong-Frederick model for the simulation of ductile damage under cyclic load

The present contribution proposes a thermodynamically consistent model for the simulation of the ductile damage. The model couples the phase field method of fracture to the Armstrong Frederick plasticity model with kinematic hardening. The latter is particularly suitable for simulating the material behavior under a cyclic load. The model relies on the minimum principle of the dissipation potential. However, the application of this approach is challenging since potentials of coupled methods are defined in different spaces: The dissipation potential of the phase field model is expressed in terms of rates of internal variables, whereas the Armstrong-Frederick model proposes a formulation depending on thermodynamic forces. For this reason, a unique formulation requires the Legendre transformation of one of the potentials. The present work performs the transformation of the Armstrong-Frederick potential, such that final formulation is only expressed in the space of rates of internal variables. With the assumption for the free energy and the joint dissipation potential at hand, the derivation of evolution equations is straightforward. The application of the model is illustrated by selected numerical examples studying the material response for different load cases and sample geometries. The paper provides a comparison with the experimental results as well.