In the talk I present my Master thesis, which was supervised by Prof. Maria Giovanna Mora (Universita' di Pavia).
Dislocation Theory is a branch of Material Science that studies defects in the crystal lattice of a metal.
After a brief introduction to edge-type dislocations and to the mechanical model, I describe the problem of finding equilibrium configurations of dislocations. In our setting, a configuration of $N$ dislocations is a discrete probability measure on $\mathbb{R}^2$ of the form $\frac{1}{N}\sum_{i=1}^N \delta_{x_i}$. A configuration is said to be an equilibrium if it is a minimizer of a proper energy.
First of all, I state a $\Gamma$-convergence result concerning the discrete energies and I describe the main properties of the $\Gamma$-limit problem (existence, uniqueness and characterization of the minimizer). I explain that the minimizer of the $\Gamma$-limit energy is an approximation of the equilibrium configurations of $N$ dislocations, when $N$ is large.
In the second part, I present two particular problems (corresponding to different choices of the confinement potential) in which the $\Gamma$-limit energy admits an explicit minimizer. I explain that the solution of the first problem ("quadratic confinement" case) suits perfectly the mechanical intuition, while in the second problem ("physical confinement" case) the equilibrium configuration is quite unexpected.
I also describe the numerical algorithms which played a fundamental role in the solution of these problems.