The Variational Approach to Fracture

1 Introduction;2 Going variational; 2.1 Griffith's theory; 2.2 The 1-homogeneous case - A variational equivalence; 2.3 Smoothness - The soft belly of Griffith's formulation; 2.4 The non 1-homogeneous case - A discrete variational evolution; 2.5 Functional framework - A weak variational evolution; 2.6 Cohesiveness and the variational evolution;3 Stationarity versus local or global minimality - A comparison; 3.1 1d traction; 3.1.1 The Griffith case - Soft device; 3.1.2 The Griffith case - Hard device; 3.1.3 Cohesive case - Soft device; 3.1.4 Cohesive case - Hard device; 3.2 A tearing experiment;4 Initiation; 4.1 Initiation - The Griffith case; 4.1.1 Initiation - The Griffith case - Global minimality; 4.1.2 Initiation - The Griffith case - Local minimality; 4.2 Initiation - The cohesive case; 4.2.1 Initiation - The cohesive 1d case - Stationarity; 4.2.2 Initiation - The cohesive 3d case - Stationarity; 4.2.3 Initiation - The cohesive case - Global minimality;5 Irreversibility; 5.1 Irreversibility - The Griffith case - Well-posedness of the variational evolution; 5.1.1 Irreversibility - The Griffith case - Discrete evolution; 5.1.2 Irreversibility - The Griffith case - Global minimality in the limit; 5.1.3 Irreversibility - The Griffith case - Energy balance in the limit; 5.1.4 Irreversibility - The Griffith case - The time-continuous evolution; 5.2 Irreversibility - The cohesive case;6 Path;7 Griffith vs. Barenblatt;8 Numerics and Griffith; 8.1 Numerical approximation of the energy; 8.1.1 The first time step; 8.1.2 Quasi-static evolution; 8.2 Minimization algorithm; 8.2.1 The alternate minimization algorithm; 8.2.2 The backtracking algorithm; 8.3 Numerical experiments; 8.3.1 The 1D traction (hard device); 8.3.2 The Tearing experiment; 8.3.3 Revisiting the 2D traction experiment on a fiber reinforced matrix;9 Fatigue; 9.1 Peeling Evolution; 9.2 The limitfatigue law when d tends to 0; 9.3 A variational formulation for fatigue; 9.3.1 Peeling revisited; 9.3.2