A unified variational damage model and an efficient length scale insensitive phase-field model

As an emerging method for simulating fracture in solids, the variational damage model is currently still mainly limited to the study of brittle fracture. To simulate the quasi-brittle failure of solids, this work proposes an efficient and unified variational damage model (vdczm) within a variational framework, together with its corresponding phase-field model (Tpfczm) that is insensitive to the length scale parameter. Specifically, a crack geometric function associated with the unified phase-field model and a purely geometric rational degradation function are introduced. The introduced constitutive functions are capable of recovering both the classical variational damage model and the phase-field models (including pfczm), thus ensuring the unification of the theoretical framework. This work also demonstrates the specific implementation of incorporating the cohesive zone model into the variational damage framework. The procedure includes deriving an analytical solution for quasi-brittle fracture in the one-dimensional case, based on which an equivalent cohesive zone model is constructed. This equivalent model can accurately reproduce exponential, hyperbolic, and Cornelissen softening laws, and typical constitutive parameters can be obtained by fitting these classical softening laws. Furthermore, this work proposes an efficient hybrid formulation of the unified variational damage model (vdczm), which provides greater advantages in energy decomposition. The effectiveness of the two proposed theories is verified through a series of numerical examples. The results show that both vdczm and Tpfczm are insensitive to mesh size, and Tpfczm is also insensitive to the length scale parameter when it is well resolved by the mesh. The comparison of computational efficiency indicates that vdczm is significantly more efficient than both Tpfczm and pfczm, while Tpfczm is also noticeably more efficient than pfczm.