An approximation result for special functions with bounded deformation

By Antonin Chambolle. 2004

167 citations recorded by the source corpus.

A “special displacement with bounded deformation” is a function u:Ω⊂RN→RN whose symmetrized gradient is a bounded measure which coincides, outside a (N−1)-dimensional rectifiable “jump set” Ju, with a summable function e(u). We show that in dimension N=2, when u and e(u) are square integrable, and the total length H1(Ju) is finite, then such a displacement is approximated with a sequence (un)n⩾1 of piecewise continuous displacements whose jump sets Jun are (relatively) closed, with un and e(un) converging strongly in L2, respectively to u and e(u), and the lengths H1(Jun) converging to H1(Ju). As an application, we approximate with a sequence of elliptic functionals a functional which appears in the theory of brittle fracture in linearized elasticity.

Subfields: Materials Chemistry, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/505e667a77a25a28b1f2586e671344fd47acffcc

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Materials Chemistry

In Mechanics of Materials

Citations in the corpus

A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-Convergence and stress oscillations (2015, background): “Chambolle(2004) has proven that for continuous media this is the case, so that the functionalΠℓ for the smeared crack surfaceΓ-converges to that for the discrete crack surface for a vanishing length scale parameter, that isℓ → 0.”
Phase field approximation of cohesive fracture models (2014, background): “This result was extended in many directions, for example to vector-valued functions [30,31], to linearized elasticity [23,24,39], to second-order problems [5], to vectorial problems [42], and to models with nonlinear injectivity constraints [37]; for numerical simulations we refer to [12,16,15,21,22”
Experimental validation of a phase-field model for fracture (2017, background): “…a volume integral and, the energy functional becomes: 0 = ∫ ψed + ∫ Gc [ (c − 1)2 4l0 + l0 ∂c ∂xi ∂c ∂xi ] d (3) While and 0 are not equal, it has been shown (Chambolle 2004, 2005), following the ideas ofAmbrosio and Tortorelli (1990, 1992), that 0 approaches in the sense of -convergence as l0 →…”
Traces of functions of bounded deformation (2013, methodology): “This model consists in a constrained minimization of a Mumford-Shah type energy functional for which the subspace SBD(Ω) of special functions of bounded deformation introduced in [1, 5] (see also [6]) has found relevant.”
Phase-field models for brittle and cohesive fracture (2014, unknown): “However, C-convergence has so far only been proved for the quadratic degradation function [10].”
The Variational Approach to Fracture Mechanics. A Practical Application to the French Panthéon in Paris (2009, methodology): “This convergence was proved in [2] for the Mumford-Shah functional [25], and in [ 10 ] for the functional (3.3) in the two-dimensional case.”
High-accuracy phase-field models for brittle fracture based on a new family of degradation functions (2017, background): “…+ κ known as the energy degradation function that penalizes the material stiffness according to the value of φ. Equation (4) is essentially a direct adaptation of the earlier functional of Ambrosio and Tortorelli (1992), for which a proof of Γ-convergence was subsequently given by Chambolle (2004).”
Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments (2009, background): “D antiplane shear, by Francfort and Larsen (2003) for the n-dimensional case including cracks with an arbitrary number of connected components, and by Chambolle (2003, 2004) for plane elasticity with vectorial displacement fields.”
A density result for GSBD and its application to the approximation of brittle fracture energies (2013, background): “The proof of (1′′) and of (2′′) work as in [12, 13] since the definition of vh and of the discrete energies are the same.”
On penalization in variational phase-field models of brittle fracture (2018, background): “The functions g and w are the major ingredients of the regularized elastic energy and fracture surface energy functionals, and their specific choice establishes the rigorous link between (1) and (2) via the notion of Γ-convergence, see e.g. Braides [70], Chambolle [71], also giving a meaning to the ”
A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework (2014, background): “Chambolle [15] has provided a Γ-convergence proof for the case of brittle fracture in linear elasticity that shows that as `0 approaches zero the second-order approximate potential energy”