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Gilles A. Francfort

Publications and source records attributed to Gilles A. Francfort.

12 recordsLinked to original sources
arXiv2026

Periodic homogenization in scalar Hencky plasticity

This paper investigates the periodic homogenization of a two-phase elasto-plastic material in the framework of $Γ$-convergence. It establishes the link between the homogenized functional and that which was previously obtained in [12] in the context of two-scale convergence. Although there is no gap between the two formulations and that involving two-scale has the hallmark of a (two-scale) elasto-plastic problem, we are unable to positively ascertain the elasto-plastic character of the homogenized energy except in a one-dimensional setting.

math.AP↗
arXiv2026

Nucleation and propagation of brittle fracture as a constrained energy minimization problem

This paper presents a macroscopic, or continuum, theory aimed at describing when, where, and why cracks nucleate and propagate in nominally elastic brittle materials under monotonic, quasi-static, but otherwise arbitrary mechanical loads. Motivated by recent insights, the proposed sharp theory posits that: \emph{cracks nucleate and propagate exclusively in regions where the strength surface of the material is exceeded, with their evolution dictated by the minimization of the sum of the potential --- the elastic energy minus the work done by the externally applied forces --- and surface energies.} While the theory applies to materials with any elasticity (linear or nonlinear) and any material symmetry (isotropic or anisotropic), attention is restricted here to the most basic case of isotropic elastic brittle materials. For demonstration purposes, the theory is confronted with a set of nine tests that span the entire range of well-settled experimental knowledge on fracture nucleation and propagation --- the so-called ``Nine Circles of Elastic Brittle Fracture'' --- on both a hard material (a silicate glass) and a soft material (a synthetic rubber).

cond-mat.mtrl-sci↗
arXiv2025

Velocity optimization of self-equilibrated obstacles in a two-dimensional viscous flow

An obstacle is immersed in an externally driven 2D Stokes or Navier-Stokes fluid. We study the self-equilibration conditions for that obstacle under steady state assumptions on the flow. We then seek to optimize the translational and/or angular velocity of the obstacle by varying its shape. To allow general variations, we must consider a very large class of obstacles for which the notion of trace is meaningless. This forces us to revisit the notion of self-equilibration for both Stokes and Navier-Stokes in a measure theoretic environment.

math.AP↗
arXiv2024

Classical variational phase-field models cannot predict fracture nucleation

Notwithstanding the evidence against them, classical variational phase-field models continue to be used and pursued in an attempt to describe fracture nucleation in elastic brittle materials. In this context, the main objective of this paper is to provide a comprehensive review of the existing evidence against such a class of models as descriptors of fracture nucleation. To that end, a review is first given of the plethora of experimental observations of fracture nucleation in nominally elastic brittle materials under quasi-static loading conditions, as well as of classical variational phase-field models, without and with energy splits. These models are then confronted with the experimental observations. The conclusion is that they cannot possibly describe fracture nucleation in general. This because classical variational phase-field models cannot account for material strength as an independent macroscopic material property. The last part of the paper includes a brief summary of a class of phase-field models that can describe fracture nucleation. It also provides a discussion of how pervasively material strength has been overlooked in the analysis of fracture at large, as well as an outlook into the modeling of fracture nucleation beyond the basic setting of elastic brittle materials.

cond-mat.mtrl-sci↗
arXiv2023

Liquid Filled Elastomers: From Linearization to Elastic Enhancement

Surface tension at cavity walls can play havoc with the mechanical properties of perforated soft solids when the cavities are filled with a fluid. This study is an investigation of the macroscopic elastic properties of elastomers embedding spherical cavities filled with a pressurized liquid in the presence of surface tension, starting with the linearization of the fully nonlinear model and ending with the enhancement properties of the linearized model when many such liquid filled cavities are present.

math.AP↗
arXiv2021

Continuity equation and characteristic flow for scalar Hencky plasticity

We investigate uniqueness issues for a continuity equation arising out of the simplest model for plasticity, Hencky plasticity. The associated system is of the form $\rm{ curl\;}(μσ)=0$ where $μ$ is a nonnegative measure and $σ$ a two-dimensional divergence free unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic flow, as well as of the associated solutions.

math.AP↗
arXiv2021

Enhancement of elasto-dielectrics by homogenization of active charges

We investigate the PDE system resulting from even electromechanical coupling in elastomers. Assuming a periodic microstructure and a periodic distribution of micro-charges of a prescribed order, we derive the homogenized system. The results depend crucially on periodicity (or adequate randomness) and on the type of microstructure under consideration. We also show electric enhancement if the charges are carefully tailored to the homogenized electric field and explicit that enhancement, as well as the corresponding electrostrictive enhancement in a dilute regime.

math.AP↗
arXiv2018

A two-dimensional labile aether through homogenization

Homogenization in linear elliptic problems usually assumes coercivity of the accompanying Dirichlet form. In linear elasticity, coercivity is not ensured through mere (strong) ellipticity so that the usual estimates that render homogenization meaningful break down unless stronger assumptions, like very strong ellipticity, are put into place. Here, we demonstrate that a L^2-type homogenization process can still be performed, very strong ellipticity notwithstanding, for a specific two-phase two dimensional problem whose significance derives from prior work establishing that one can lose strong ellipticity in such a setting, provided that homogenization turns out to be meaningful.A striking consequence is that, in an elasto-dynamic setting, some two-phase homogenized laminate may support plane wave propagation in the direction of lamination on a bounded domain with Dirichlet boundary conditions, a possibility which does not exist for the associated two-phase microstructure at a fixed scale. Also, that material blocks longitudinal waves in the direction of lamination, thereby acting as a two-dimensional aether in the sense of e.g. Cauchy.

math.AP↗
arXiv2006

Spatial heterogeneity in 3D-2D dimensional reduction

A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret & Raoult. Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté, Fonseca & Mascarenhas, the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of $Γ$-convergence of the elastic energy, as the thickness tends to zero.

math.AP↗
arXiv2004

Quasistatic crack growth in finite elasticity

In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time.

math.AP↗
arXiv2004

Quasi-static evolution in brittle fracture: the case of bounded solutions

The main steps of the proof of the existence result for the quasi-static evolution of cracks in brittle materials, obtained in [7] in the vector case and for a general quasiconvex elastic energy, are presented here under the simplifying assumption that the minimizing sequences involved in the problem are uniformly bounded in $L^\infty$.

math.AP↗