We are planning an excursion to the San Canziano caves on Wednesday afternoon.
More information about the excursion and the social dinner will be available soon.

Schedule

	Monday, June 6th	Tuesday, June 7th	Wednesday, June 8th	Thursday, June 9th	Friday, June 10th
9.30-10.15	registration	Chambolle	Ortiz	Braides	McMeeking
10.30-11.15	Peletier	Larsen	Toader	Cicalese	Hackl
11.30-12.15	coffee break	coffee break	coffee break	coffee break	coffee break
12.15-13.00	Le Bris	Savaré	Conti	Francfort	Mielke
13.15-15.00	lunch	lunch	lunch	lunch	lunch
15.00-15.45	Marigo	Geers	excursion	Ravi-Chandar
16.00.16.30	coffee break	coffee break	excursion	coffee break
16.30-17.15	Walton	Lazzaroni	excursion	Acharya
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20.00-			social dinner

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Contributions

Amit Acharya: Line Defect dynamics and solid mechanics.


Andrea Braides: Homogenization of spin systems.
We address the problem of the description of the macroscopic behaviour of spin systems with inhomogeneous interactions. In particular
1) we completely describe the effective surface tension of periodic mixtures of two types of ferromagnetic interactions. This problem is linked to optimal design of networks and their metric properties (joint work with L.Kreutz);
2) we study conditions ensuring the possibility of describing systems of mixtures of ferromagnetic and antiferromagnetic interactions via an effective surface tension. Examples of frustrated systems are given when this can or cannot be done (joint work with M.Cicalese);
3) we study the case when antiferromagnetic interactions appear with low (but finite) probability, and show that generically such system behave as ferromagnetic systems (joint work with A.Causin, A.Piatnitski, and M. Solci).


Antonin Chambolle: Numerical approximation of a fracture model with non interpenetration.
This talk (based on joint work with S. Conti, G. Francfort) will recall how the non interpenetration of the fracture can be practically enforced in variational models for brittle fracture in linearized elasticity. We will discuss in particular the validity of phase-field approximations of this model.


Marco Cicalese: On global and local minimizers of prestrained thin elastic rods.
We study the stable configurations of a thin 3-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By Gamma-convergence we derive a limit 1-dimensional theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the 3-dimensional model. In the case of isotropic materials and for two-layers prestrained 3-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam.  In this case we study global and local stability of straight and helical configurations. Some simple simulations help us to compare our results with real experiments. 


Sergio Conti: Variational modeling of dislocation microstructures.
Dislocations are topological singularities of the strain field, which play a crucial role in the plastic deformation of crystals. I shall discuss the derivation of variational models for dislocations and dislocation densities starting from three-dimensional linear elasticity, within the framework of Gamma convergence. The limiting process leads to relaxation and formation of microstructures, in the form of oscillations in the orientation of the dislocations and of patterning in situations with multiple dislocations present.


Gilles A. Francfort: Periodic and stochastic homogenization in linear elasticity.
In 1993, Giuseppe Geymonat, Stefan Müller and Nicolas Triantafyllidis demonstrated that, in the setting of linearized elasticity, a Gamma-convergence result holds for highly oscillating sequences of elastic energies whose functional coercivity constant over the whole space is zero while the corresponding coercivity constant on the torus remains positive. We find sufficient conditions for such a situation to occur through a rigorous revisiting of a laminate construction given by Gutierrez in 1999. We further demonstrate that isotropy prohibits such an occurrence.

The results apply to both the periodic and the stochastic setting. They were obtained in part with Marc Briane (Rennes), and in part with Antoine Gloria (Brussels) and Scott Armstrong (Paris).


Marc G. D. Geers: Interfaces in copper-rubber based stretchable electronics.


Klaus Hackl: Modeling of quasi-brittle damage in Fourier space.


Christopher J. Larsen: Limits of minimizers of cohesive fracture energy with history.
For proving existence of (energy minimizing) quasi-static Griffith fracture evolutions, the biggest hurdle was proving that limits of unilateral minimizers for a Griffith energy were themselves unilateral minimizers.  Existence for corresponding quasi-static cohesive models remains open, largely due to the failure to prove the analogous minimality (minimality with history) for limits of minimizers of cohesive energies. 
I will explain why in general this minimality is false, and what the best minimality is that we can hope for.  Finally, I will show that this minimality does hold, a least for a class of cohesive energies.


Giuliano Lazzaroni: Dynamic evolutions for a peeling test in dimension one.
We present a simplified model of dynamic crack propagation, where the equation of elastodynamics is coupled with Griffith's principle. In recent years there has been an increasing interest in studying systems where second-order equations for displacements are coupled with first-order flow rules for internal variables. Despite a number of papers devoted to regularised models, only partial results are available for dynamic fracture and heavy mathematical difficulties have to be overcome. In our work we deal with a problem of debonding propagation for a one-dimensional thin film, partially glued on a substrate and subject to oscillations in the debonded part. We provide existence and uniqueness results for dynamic evolutions and study the limit as the speed of external loading tends to zero. We establish the properties of the limit solution and see that in general it does not coincide with the expected quasistatic limit. Joint collaboration with Gianni Dal Maso and Lorenzo Nardini (SISSA).


Claude Le Bris: Homogenization of problems with defects.
We present a general approach to approximate at the fine scale the solution to an elliptic equation with oscillatory coefficient when this coefficient consists of a "nice" (in the simplest possible case say periodic) function which is, in a to be made precise, perturbed. The approach is based on the determination of a local profile, solution to an equation similar to the corrector equation in classical homogenization. We prove that this equation has a unique solution, in various functional settings depending upon the perturbation: local perturbation, two different periodic structures separated by a common interface, etc. We then perform the homogenization of the problem and show the quality of the approximation, at different scales and in different norms.
The works are a series of joint works in collaboration with Xavier Blanc (University Denis Diderot, Paris), Pierre-Louis Lions (College de France, Paris), and also Marc Josien (Ecole des Ponts, Paris).


Jean-Jacques Marigo: Gradient damage models coupled with plasticity.


Robert McMeeking: Inelastic response of ceramics under impact: brittle damage, plasticity and comminution.


Alexander Mielke: Rate-independent microstructure evolution via relaxation of a two-phase model.
Based on the recent work [HeM2016] with Sebastian Heinz we revisit the quasistatic two-well model for phase transformation of a linearly elastic body that was introduced and studied in [MTL2002]. The relaxed energetic rate-independent model is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase as state variables. The evolution is driven by an energy functional and a dissipation distance.
Using an approach based on mutual recovery sequences enable us to justify the weak-limit passage from the pure two-phase model to the relaxed mixture model despite the fact that this model does not have any internal material length scale, and hence no obvious compactness. The new observation is that suitable laminate constructions allow for an explicit calculation of the associated H-measures because of a separation of length scales.

[HeM2016] S. Heinz, A. Mielke: Existence, numerical convergence, and evolutionary relaxation for a rate-independent phase-transformation model.  Phil. Trans. Royal Soc. A 374 (2016) 20150171.
[MTL2002] A. Mielke, F. Theil, V.I. Levitas: A variational formulation of rate-independent phase transformations using an extremum principle.  Arch. Rational Mech. Analysis 162 (2002) 137-177.
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Michael Ortiz: Atomistic-to-continuum modeling of grain boundaries.


Mark A. Peletier: Upscaling the dynamics of dislocations.
Plasticity, the permanent deformation that one observes in metals, is the net effect of the movement of a large number of microscopic defects in the atomic lattice. These defects, called dislocations, are curve-like topological mismatches, and migrate  through the metal under the influence of internal and external forces. Macroscopic, permanent, deformation arises through the concerted movement of a large number of these dislocations.

It is a major challenge to connect a microscopic description of dislocation movement on one hand with models of macroscopic plastic behaviour on the other hand. If this were possible, then much could be gained: metals could be designed at the workstation with tailor-made properties, design of hybrid materials would become much easier, and generally the holy grail of 'materials by design' would come a little closer. At this stage we are not able to do this; there is a major gap between the models at these different spatial and temporal scales. Part of the difficulty lies in the complex interactions between dislocations: they attract and repel each other, and form complex higher-level structures that appear to play an important role in determining the macroscopic behaviour.

Interestingly, the situation for the dynamics of the dislocations is significantly more complex than that of the energetics. 

I will outline some recent results in this field, describe some of our own recent results in two dimensions, and mention some open questions and one or two mysteries. 


Krishnaswamy Ravi-Chandar: On the deformation and failure of polycrystalline materials.


Giuseppe Savaré: Viscous approximations of rate independent evolutions.
We will discuss two kinds of viscous approximations of rate independent systems driven by an energy-dissipation mechanism.
A first situation (studied in collaboration with L. Minotti) concerns a viscous correction in the standard incremental minimization scheme leading to energetic solutions. The characterization of the corresponding Visco-Energetic solutions involves a refined description of their jump behaviour, which reflects the modified approximation scheme.
A second approximation (studied in collaboration with V. Agostiniani and R. Rossi) deals with the vanishing viscous dissipation limit: in this case the main difficulty relies on the lack of compactness in time and the existence of a limit can be proved under suitable assumptions on the critical points of the energy.
In both cases the variational approach gives an important insight concerning the jump behaviour of the solutions, which provides an essential contribution to recover the energy balance.


Rodica Toader​: Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case.
We study the asymptotic behavior of a variational model for damaged elasto-plastic materials when the coefficients of the problem depend on a small parameter which forces damage concentration on a set of codimension one. In the antiplane shear case we show that the limit energy functional contains a surface energy term depending on the crack opening. The talk is based on the results obtained in collaboration with G. Dal Maso and G. Orlando (SISSA).


Jay R. Walton: Interfacial Mechanics and the Modeling of Contact and Fracture.