Stochastic perturbation study for 2D brittle fractures: hybrid phase-field method and variational approach.

The study of fracture propagation is an essential topic for several disciplines in engineering and material sciences. Different mathematical approaches and numerical methods have been applied to simulate brittle fractures. Materials, naturally, present random properties that contribute its physical properties, durability, and resistance, for this reason, stochastic modeling is critical to obtain realistic simulations for fractures. In this article, we propose applying a Gaussian random field with a Mat\'ern covariance function to simulate a non-homogeneous energy release rate ($G_c$) of a material. We propose a surrogate mathematical model based on a weighted-variational model to reduce numerical complexity and execution times for simulations in the hybrid phase-field model. The FEniCS open-source software is used to obtain numerical solutions to the variational and hybrid phase-field models with Gaussian random fields on the parameter $G_c$. Results have shown that the weighted-variational model as a surrogate model is a competitive tool to emulate brittle fractures for real structures, reducing execution times by 90\%.