A surrogate model for studying random field energy release rates in 2D brittle fractures

This article proposes a weighted-variational model as an approximated surrogate model to lessen numerical complexity and lower the execution times of brittle fracture simulations. Consequently, Monte Carlo studies of brittle fractures become possible when energy release rates are modelled as a random field. In the weighed-variational model, 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. Numerical solutions to the weighed-variational model, along with the more standard but computationally demanding hybrid phase-field models, are obtained using the FEniCS open-source software. The results have indicated that the weighted-variational model is a competitive surrogate model of the hybrid phase-field method to mimic brittle fractures in real structures. This method reduces execution times by 90\%. We conducted a similar study and compared our results with an actual brittle fracture laboratory experiment. We present an example where a Monte Carlo study is carried out, modeling $G_c$ as a Gaussian Process, obtaining a distribution of possible fractures, and load-displacement curves