### **Models for material defects and grain boundaries**

**Prof. Adriana Garroni**
## Tentative program

The course will focus on some variational ''semi-discrete'' models for material defects in metals and their role in plastic deformation. The multiscale analysis of such models provides insight for the formulation of line tension energies for dislocations in crystals and their role in the modeling of grain boundaries.

## Tentative structure of the course:

PART 1

Introduction to Calculus of Variations. Classical Examples. Some hints of the final goal: elasticity/plasticity/material defects and grain boundaries. Partitions.

1) General scheme for the direct method. Application to the classical examples. Examples of non-existence.

2) Integral variational problems. Well poseness of variational integral problems: lower semicontinuity and coerciveness in Sobolev spaces (relaxation)

3) Gamma-convergence. Definition and main properties. Two classical examples: homogenisation and phase transitions

4) Gamma-convergence of Modica-Mortola functional in 1D. Ideas of the prof in general dimension

5) Introduction to BV and SBV (main results and characterisations). The partition problem

PART 2

1) Variational Framework for elasticiticity, plasticity and material defects. Models for elasticity: Korn’s inequality and rigidity.

2) Proof of the rigidity result. Application to problems in elasticity.

3) Models for dislocations. Some Gamma-convergence results with the core-radius approach.

4) Rigidity for incompatible fields. Some ideas for the asymptotic from which emerges a model for grain boundaries.

## Time and Location.

Tentative calendar (adjustments are possible depending on individual needs)

Lecture 1: Mo 30/05 - 16:00 - 17:30 (MI 03.04.011)

Lecture 2: Wed 1/06 - 16:00 - 17:30 (MI 03.08.022)

Lecture 3: Thu 2/06

Lecture 4: Fri 3/06 - 16:00 - 17:30 (MI 03.08.022)

Lecture 5: Tue 7/06 (remote)

Lecture 6: Wed 22/06 - 16:00 - 17:30. (MI 03.12.020)

Lecture 7: Thu 23/06 - 16:00 - 17:30

Lecture 8: Fri 24/06 - 16:00 - 17:30 (MI 03.08.022)

Lecture 9: Mo 27/06 - 16:00 - 17:30. (MI 03.08.022)

Lecture 10: Wed 29/06 - 16:00 - 17:30. (MI 03.08.022)

Lecture 11: Fri 1/07 - 14:00 - 15:30. (MI 03.08.022)

## Evaluation of the students

Depending on the student's background the exam may require a colloquium on some of the preliminary tools or a seminar given by the student focused on some of the technical results presented in the course or related results.

## Prerequisites

The results presented in the course will need a background in functional analysis and geometric measure theory that will be provided depending on the knowledge of the audience.

## References

* L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Oxford Mathematical Monographs) 1st edition (2000)

* S. Conti, A. Garroni, and M. Ortiz. The line-tension approximation
as the dilute limit of linear-elastic dislocations. Arch. Ration. Mech.
Anal., 218:699-755, 2015.

* Garroni, A., Leoni, G., Ponsiglione, M.: Gradient theory for plasticity via homogenization
of discrete dislocations. J. Eur. Math. Soc. 12(5), 1231–1266 (2010)

* G. Lauteri and S. Luckhaus.
An energy estimate for dislocation configurations and the emergence of Cosserat-type structures in metal plasticity. Preprint arXiv:1608.06155, 2016.

* S. Muller, L. Scardia, and C. I. Zeppieri. Geometric rigidity for
incompatible fields and an application to strain-gradient plasticity.
Indiana Univ. Math. J., 63:1365-1396, 2014.

* S. Muller, L. Scardia, and C. I. Zeppieri. Gradient theory for geometrically
nonlinear plasticity via the homogenization of dislocations.
In S. Conti and K. Hackl, editors, Analysis and Computation
of Microstructure in Finite Plasticity, pages 175{204. Springer,
2015.