A Model for the Quasi-Static Growth¶of Brittle Fractures:¶Existence and Approximation Results

By Gianni Dal Maso, Rodica Toader. 2001

334 citations recorded by the source corpus.

Abstract We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G. A. Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we cannot exclude the possibility that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time-discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.

Subfields: Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/fcfe2d03a5f08323ce626f3e0f8e9e9a438b3d89

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In Computational Mechanics

In Mechanics of Materials

Citations in the corpus

Existence and convergence for quasi‐static evolution in brittle fracture (2003, positive): “derivation is nearly identical but, since the setting is dierent — weak formulation in a SBV setting versus strong formulation in a two-dimensional setting with “connected” compact crack sites —, we establish them anew, while inviting the reader familiar with [ 11 ] to skip the derivation and go to ”
Quasistatic Crack Growth in Nonlinear Elasticity (2005, background): “Previous results on this subject have been obtained in [12] in the case n = 2 for a scalar-valued u and for W(ξ) = |ξ |2, which corresponds to the antiplane case in linear elasticity.”
Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures (2003, background): “[10] Cortesani G., Toader R.: A density result in SBV with respect to non-isotropic energies.”
Existence and regularity results for the Steiner problem (2012, background): “Our version of this theorem contains also the generalization of the semicontinuity result from [3] applicable to non-connected sets, which we will need to prove the existence result for the Steiner problem (ST ) and which is absent in that of [1].”
Existence results for a class of rate-independent material models with nonconvex elastic energies (2006, background): “In fact, for models in damage and fracture, see [9], [12], [15], [16], [22], it is desirable to avoid all di¤erentiable structures, i.”
Existence results for energetic models for rate-independent systems (2005, methodology): “Moreover, we show that the model of brittle fracture introduced in [FM93] and developed further in [FM98, DMT02 , Cha03] is a special case of our theory.”
The Variational Approach to Fracture (2008, background): “An adaptation of that result in [41] proves the existence of a minimizer to the strong discrete evolution at each time step under the restriction that the cracks have an a priori set number of connected components.”
Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments (2009, background): “Proofs of existence of solutions and convergence of the time-discrete quasi-static evolution are available for antiplane linear elasticity (Dal Maso and Toader, 2002; Francfort and Larsen, 2003) and for general linear (Chambolle, 2003) and non-linear (Dal Maso et al., 2005) elasticity without…”
An approximation result for special functions with bounded deformation (2004, background): “Does each minimization problem have a solution? Does there exist some limit evolution as δt ↓ 0? Some of these issues are addressed in [3, 18, 16, 21, 17], for variants of this problem (scalar versions, topological restrictions on the cracks, nonlinear elasticity).”
Compactness and lower semicontinuity in $GSBD$ (2018, background): “Another possible assumption is, in 2d, that the crack set K be connected [22, 10].”
Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation (2012, methodology): “…of the classical Griffith-type theory such as its non-ability to initiate cracks can be overcome by variational methods based on energy minimization as suggested in the pioneering works of Francfort and Marigo (1998) and Bourdin et al. (2008), see also Dal Maso and Toader (2002) and Buliga (1999).”
Quasi-static crack growth for a cohesive zone model with prescribed crack path (2005, methodology): “ends on the opening of the crack. For this reason we cannot apply directly the tools developed so far in the applications to fracture mechanics of the theory of free discontinuity problems (see [10], [5], [6], [1], [9], [3], [4]). To simplify the mathematical diﬃculties, we assume that the crack pat”