The Variational Approach to Fracture

By Blaise Bourdin, Gilles A. Francfort, Jean-Jacques Marigo. 2008

2,076 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/8b008e645ad5bb12bf8f667b810afcaa99571956

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In Mechanics of Materials

Citations in the corpus

When and how do cracks propagate? (2009, background): “Of course, energy conservation needs to be re-formulated in such a framework (see [11, 14, 4, 13]).”
Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy (2015, methodology): “This theory was subsequently regularized into a phase-field model, suitable for numerical calculations [27], and which converges to the sharp variational theory of brittle fracture [32].”
Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments (2009, methodology): “The regularized model adopted in Bourdin et al. (2000, 2008) is characterized by a symmetric behavior in traction and in compression and allows for negative displacement jumps, i.e. material interpenetration in cracks associated to compressive actions.”
An efficient augmented finite element method for arbitrary cracking and crack interaction in solids (2014, methodology): “One method is the phase-field method [30–32], which introduces one additional nodal DoF to approximate a fracture surface with a highly concentrated yet continuous phase-field.”
Peridynamic Theory of Solid Mechanics (2010, background): “It is worthwhile to compare the peridynamic approach to fracture against variational approaches [28, 12, 13] in which the growth of a crack is determined by energy minimization, including contributions from continuous parts of the body and from energy consumed by the crack.”
A phase‐field model for cohesive fracture (2013, methodology): “An extension of the variational formulation for brittle fracture to cohesive fracture has been considered in [19], but a phase-field approximation is not yet available.”
Effective toughness of heterogeneous media (2014, methodology): “We do so following a variant of the variational fracture field approach of Bourdin and collaborators [13, 14].”
Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: One-dimensional examples (2017, positive): “profile, the mesh size h must be smaller than the internal length ! (at least h ≤ !/4), see [24] , rapidly leading to the need of large scale computations in 2D and 3D settings.”
Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity (2016, background): “that is on the energy level not convex in both solution variables u and φ simultaneously and requires sophisticated solution algorithms [20,18,23,36] if one aims for finding a global minimum.”
Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids (2013, background): “[123], Maugin and Trimarco [124] or Rinaldi and Lai [125] where variational principles modeling dissipative phenomena occurring in damage and fracture are formulated.”
Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations (2010, methodology): “The regularized setting of their framework, considered in Bourdin, Francfort & Marigo [5, 6], is obtained by Γ–convergence inspired by the work of image segmentation by Mumford & Shah [27].”
A phase-field/gradient damage model for brittle fracture in elastic–plastic solids (2015, methodology): “[40]), which by its turn was based on the phase-field approximation of the Mumford-Shah potential introduced by Ambrosio and Tortorelli [41].”
Phase-field modeling of fracture in linear thin shells (2014, positive): “However, the computed surface energy will be slightly overestimated [28, 45].”
2D and 3D Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture (2017, methodology): “This theory led to the development of the phase-field model [11, 12].”
Phase-field models for brittle and cohesive fracture (2014, positive): “An extension of the variational formulation for brittle fracture to cohesive fracture has been considered in [7], and a phase-field approximation has been developed in [22], with a focus on the application to adhesive fracture, i.”