Finite volume simulations of dynamic brittle fracture using an exponential-based hyperbolic formulation of gradient damage models

This study provides a multidimensional extension of the hyperbolic formulation of gradient damage models introduced by Favrie et al. in one space dimension, and which allows an efficient explicit numerical solutionof dynamics problems. The proposed methodology is based on an “extended Lagrangian approach” developed by one of the authors for the nondissipative and dispersive shallow water equation. By using this strategy, the global minimization problem commonly derived for gradient damage models is recast as a hyperbolic one with purely local source terms. The numerical solution of the governing system of equations is then based on a fractional-step method consisting of a classical Godunov-type finite volume scheme to solve the homogeneous part of the system, followed by an implicit Ordinary Differential Equation solver for the local source terms. Stored-energy functions corresponding to well-known damage models (i.e. Ambrosio & Tortorelli or AT models) are used with, however, the introduction of a new relaxed damage variable to easily handle damage boundedness. The proposed model is illustrated on two-dimensional test cases: the Kalthoff-Winkler experiment and a crack branching test.