Numerical implementation of the variational formulation for quasi-static brittle fracture

Abstract

This paper presents the analysis and implementation of the variational formulation of quasi-static brittle fracture mechanics proposed by G. A. Francfort and J.-J. Marigo in 1998. We briefly present the model itself, and its variational approximation in the sense of Γ -convergence. We propose a numerical algorithm based on Alternate Minimizations and prove its convergence under restrictive assumptions. We establish a new necessary condition for optimality for the entire time evolution from which we derive the Backtracking algorithm. We give some elements of analysis of the Backtracking algorithm on a simple problem. We present realistic numerical simulations of a traction experiment on a fiber-reinforced matrix, and of the propagation of cracks in a perforated sample under mode-I loading.

A Derivation of Linearized Griffith Energies from Nonlinear Models

“…For the sake of a simplified mathematical description the investigation of fracture models in the realm of linearized elasticity is widely adopted (see for example [2,4,7,13,14,34]) and has led to a lot of realistic applications in engineering as well as to efficient numerical approximation schemes (we refer to [5,6,12,26,38,39,43] making no claim to be exhaustive). On the contrary, their nonlinear counterparts are usually significantly more difficult to treat since in the regime of finite elasticity the energy density of the elastic contributions is genuinely geometrically nonlinear due to frame indifference rendering the problem highly non-convex.…”

Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: One-dimensional examples

“…Renormalisation arguments allow us to set E 0 = 1 without loss of generality. The functions (21) are different from those considered in [4], where they were chosen to allow for explicit analytical results. The present model reduces to the ones commonly used in a phase-field regularisation of brittle fracture [55,47,24] when neglecting plasticity (σ p → ∞).…”

“…This can be considered as an extension of the consolidated alternate minimization scheme used in the regularised models of brittle fracture, [22,21,23,12,24] and fits the incremental energy minimization framework [54,59]. Similar algorithms have been used for coupled plasticity-damage problems also by [9,50,60].…”

“…The variational approach of brittle fracture and the associated regularised formulations [22,21,23,38] are nowadays widely adopted in computational mechanics. The mechanical interpretation of the regularised models as damage models added a further insight on the prediction of crack nucleation in the quasi-brittle setting, giving a precise connection between the regularising length-scale and the maximum allowable stress before failure [55,24].…”

Phase-field modeling of fracture in linear thin shells

“…The parameter 1 is introduced to avoid the singularity of disappearing internal energy density when the phase-field parameter is zero. In this model, cracks can propagate, branch and merge but can not reverse, whereas the last feature is reached by imposing υ i υ i−1 , such that υ i−1 and υ i are the phase-field parameters at step i − 1 and i [47].…”