Centre Emile Borel
Singularities in mechanics: formation, propagation and microscopic description January 7th - April 4th, 2008
Organizing committee: J. Eggers, C. Josserand, L. Saint-Raymond singular@ihp.jussieu.fr
Winter school – Fracture and Damage:
Formation and propagation of singularities in continuum mechanics
Program Wednesday, January 30th:
9h Tea/Coffee and registration
9h30 – 11h Jean-Jacques Marigo
11h – 11h30 Coffee Break
11h30 – 13h Gilles Francfort
Lunch
14h30 – 16h Jean-Jacques Marigo
16h – 16h30 Coffee Break
16h30 – 17h30 Short talk: Vincent Hakim
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Thursday, January 31st:
9h30 – 11h Guy David
11h – 11h30 Coffee Break
11h30 – 13h Gilles Francfort
Lunch
14h30 – 16h Guy David
16h–16h30 Coffee Break
Short talks:
16h30 – 16h50 Chiara Zanini 16h50 – 17h10 Chlo´e Arson 17h10 – 17h30 Lucia Scardia
Friday, February 1st:
9h30 – 11h Guy David
11h – 11h30 Coffee Break
11h30 – 13h Gilles Francfort
Lunch
14h30 – 16h Jean-Jacques Marigo
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Abstracts
Chlo´e Arson (CERMES, ENPC, Marne-la-Vall´ee): A mixed damage model for unsatured porous media.
A mixed damage model has been coupled to an unsaturated soil behaviour formulated in independent state variables (net stress and suction). The damage variable is a tensor, the eigenvectors and eigenvalues of which are the principle crack directions and densities, respectively. The Principle of Equivalent Elastic Energy is applied to compute the degraded mechanical and hydraulic stiffnesses. The behaviour laws of the state variables are derived from a postulated expression of the free energy. The irreversible strain increment turns out to be a function of the increment of damage. The damage evolution law is deduced from an associated flow rule. The relative permeability is supposed to depend only on hydraulic variables, and damage is taken into account in the intrinsic part of permeability. The intrinsic permeability is, in turn, split in a reversible and an irreversible component. The first one depends on an exponent of the void ratio. The second one is computed assuming that the flow is laminar in the homogenized cracks damaging the Representative Volume Element, and finally depends on the openings, radii and directions of the principal homogenized cracks. This damage model is being integrated in the finite element software Θ-Stock. Some discretization and programming issues will be discussed.
Guy David (Universit´e Paris-Sud): Regularity of the minimal segmentations for the Mumford-Shah functional.
The Mumford-Shah functional was introduced as a tool for image segmentation. Let Ω⊂Rn be a simple domain, g∈L∞(Ω) a bounded function on Ω, and set
(1) J(u, K) =Hn−1(K) + Z
Ω\K
|∇u|2+ Z
Ω\K
|u−g|2
forK⊂Ω closed in Ω andu∈W1,2(Ω\K), i.e., defined in Ω\K and with a derivative in L2(Ω\K). Minimal segmentations are pairs (u, K) as above, for which J(u, K) is minimal, and they should give reasonable segmentations of the initial image g.
We shall mostly care about the regularity properties of K when (u, K) is a minimal segmentation andN = 2 or 3. We shall prove some of them by hand (especially whenn= 2), mention some others, rapidly discuss the (known) existence of minimal segmentations and talk about blow-up limits, the monotonicity result of Bonnet, and global minimizers inRn. We should end with a few questions concerning minimizers in R3.
Gilles Francfort (Universit´e Paris-Nord): The variational approach to frac- ture.
In the first lecture, we will introduce and describe the variational approach to quasi- static fracture growth, and illustrate its behaviour with simple examples. In the second lecture, a proof of existence in the special case of global minimality will be described.
The last lecture will deal with numerical and computational aspects of this approach.
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Vincent Hakim (Ecole Normale Sup´erieure): Phase-field description of brittle fracture and laws of crack motion.
Jean-Jacques Marigo (Universit´e P. & M. Curie, Paris): Rupture avec
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energies de surfaces coh´esives : des crit`eres d’amor¸cage aux lois de fatigue, l’apport de l’approche variationnelle.
Cours 1: La th´eorie de Griffith: ses ingr´edients, sa formulation, ses forces et faiblesses, la n´ecessit´e de la compl´eter. La th´eorie de Barenblatt: ses ingr´edients et des premiers
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el´ements de comparaison avec la th´eorie de Griffith
Cours 2: La cr´eation de fissures dans les th´eories de Griffith et de Barenblatt. L’int´erˆet de l’approche variationnelle. Le cas 1D. Le cas 3D isotrope. Les crit`eres d’initiation en contraintes dans la th´eorie variationnelle de Barenblatt.
Cours 3: La mod´elisation de la fatigue,i.e.de la propagation de fissure sous chargement cyclique. La n´ecessit´e d’abandonner la th´eorie de Griffith au profit de celle de Barenblatt.
La question de l’irr´eversibilit´e dans la th´eorie de Barenblatt. Le principe variationnel incr´emental. Les lois d’´evolution limites obtenues par passage `a la limite lorsque la longueur caract´eristique du mat´eriau est petite devant celle de la structure. Liens entre les lois de Griffith et les lois de Paris.
Lucia Scardia (Max Planck Institute, Leipzig): Damage as Γ-limit of mi- crofractures in linearized elasticity.
A homogenization result is given for a material having brittle inclusions arranged in a periodic structure. According to the relation between the softness parameter and the size of the microstructure, three different limit models are deduced via Γ-convergence.
In particular, damage is obtained as limit of periodically distributed microfractures.
Chiara Zanini (Universit`a di Udine): A model of crack propagation based on the vanishing viscosity approach.
We consider the propagation of a single crack along a prescribed crack path and propose a notion of evolution of brittle fractures inspired by Griffith’s local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. In this talk we discuss existence of such an evolution using the vanishing viscosity method and compare this notion with the evolution introduced by Francfort and Marigo, based on a global stability criterion. This is joint work with Dorothee Knees and Alexander Mielke, WIAS, Berlin.
The winter school “Fracture and Damage: Formation and propagation of singularities in continuum mechanics” is organized with the support of the Centre Emile Borel and the Program “MICA” (Mouvements d’interfaces, calcul et applications) of the Agence Nationale de la Recherche.
Organizers: Antonin Chambolle (antonin.chambolle@polytechnique.edu), Olivier Pantz(olivier.pantz@polytechnique.edu),
CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France.
Secretary: MmeNassera Nacer (nassera.nacer@polytechnique.edu).
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