A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework

Phase-field models based on the variational formulation for brittle fracture have recently been gaining popularity. These models have proven capable of accurately and robustly predicting complex crack behavior in both two and three dimensions. In this work we propose a fourth-order model for the phase-field approximation of the variational formulation for brittle fracture. We derive the thermodynamically consistent governing equations for the fourth-order phase-field model by way of a variational principle based on energy balance assumptions. The resulting model leads to higher regularity in the exact phase-field solution, which can be exploited by the smooth spline function spaces utilized in isogeometric analysis. This increased regularity improves the convergence rate of the numerical solution and opens the door to higher-order convergence rates for fracture problems. We present an analysis of our proposed theory and numerical examples that support this claim. We also demonstrate the robustness of the model in capturing complex three-dimensional crack behavior.