Variational damage model: A novel consistent approach to fracture

The computational modeling of fractures in solids using damage mechanics faces challenge when dealing with complex crack topologies. One effective approach to address this challenge is by reformulating damage mechanics within a variational framework. In this paper, we present a novel variational damage model that incorporates a threshold value to prevent damage initiation at low energy levels. The proposed model defines fracture energy density ($\tilde{\varphi }$) and damage field (s) based on the energy density (ϕ), crack energy release rate ($G_c$), and crack length scale (ℓ). Specifically, if $\varphi \le \frac{G_c}{2\ell }$, then $\tilde{\varphi }=\varphi$ and $s=0$; otherwise, $\tilde{\varphi }=-\frac{G_c^2}{4{\ell }^2}\frac{1}{\varphi }+\frac{G_c}{\ell }$ and $s=1-\frac{G_c}{2\ell }\frac{1}{\varphi }$. Furthermore, we extend the model with a threshold value to a higher-order version. Utilizing this functional, we derive the governing equation for fractures that evolve automatically with ease. The formulation can be seamlessly integrated into conventional finite element methods for elastic solids with minimal modifications. The proposed formulation offers sharper crack interfaces compared to phase field methods using the same mesh density. We demonstrate the capabilities of our approach through representative numerical examples in both 2D and 3D, including static fracture problems, cohesive fractures, and dynamic fractures. The open-source code is available on GitHub via the link https://github.com/hl-ren/vdm.