Mesh variational r-adaptivity for sharp modeling of brittle fracture

Purpose: Brittle fracture modeling is framed as a coupled evolution of spatial and material configurations. The displacement field between these configurations drives the fracture process and the crack path alters the geometry. This interplay has been expressed in the variational formalism of Francfort and Marigo. However, numerical models based on this formalism generally introduce a regularization to circumvent the need for geometry modification. Here, we propose to combine explicit sharp crack modeling with a variational formalism. Methods: Both mesh and displacement field are tuned to minimize the functional. Consequently, the crack path appears naturally. In this work, we first present the minimization of the fracture functional in a \((r)\)-adaptative scheme and the advantage it presents. Afterwards, we present a staggered optimization procedure to construct the spatial and material configurations of a body undergoing brittle fracture. Results: Afterwards, the method is demonstrated through several numerical examples. Namely, we recover theoretical behavior for a manufactured solution and present the crack path of single edge notch specimen. Conclusion: This method is a direct implementation of the variational formalism of fracture mechanics. Therefore, it provides a straightforward way to compute the energy release rate and crack path. Moreover, it does not require any regularization and the crack is directly represented in the geometry of the specimen. However, it is computationally expensive as the mesh needs to be optimized at each step of the loading.