Enforcement of the irreversibility condition in the phase field damage model: Influence of the history variable approximation

The phase field damage model has been widely used in recent decades, based on regularizing the local variational problem (Griffith criterion revisited). This original phase field damage model must satisfy the irreversibility condition of cracks, so a constrained variational inequality must be solved. Interest in this method within the mechanical community grew after the publication of Miehe et al., where the damage irreversibility condition is simplified through a history variable in the well-known PF-AT2 model. The damage is governed by this new variable, which prevents a decrease in the elastic potential. This simplification makes it possible to solve an unconstrained variational equality and consequently consider a simplified problem. For the PF-AT2 model, this simplification is consistent. Later, other phase field models (PF-AT1, PF-CZM, $\dots$) were developed, and the use of the history variable was adapted by other researchers to these newer models. In simulating structural damage with the phase field method, many authors have employed the simplified formulation involving the history variable. Therefore, this paper presents a critical comparison of the original (consider damage irreversibility condition) and the simplified (use history variable to approximate the damage irreversibility condition) phase field damage model, through many supported by extensive numerical tests.