We describe here formal analogies between the Darcy equations, that describe the flow of a viscous fluid in a porous medium, and some problems arising from the handing of congestion in crowd motion models.
At the microscopic level, individuals are identified to rigid discs, and the dual handling of the non overlapping constraint leads to discrete Darcy-like equations with a unilateral constraint that involves the velocities and interaction pressures, and that are set on the contact network. At the macroscopic level, a similar problem is obtained, that is set on the congested zone.
We emphasize the differences between the two settings: at the macroscopic level, a straight use of the maximum principle shows that congestion actually favors evacuation, which is in contradiction with experimental evidence. On the contrary, in the microscopic setting, the very particular structure of the discrete differential operators makes it possible to reproduce observed "Stop and Go waves", and the so called "Faster is Slower" effect.[-]

This talk revolves around two variational models in finite crystal plasticity where the possible deformations are restricted to plastic glide along one active slip system. In the first model for polycrystals, our focus lies on the set of attainable macroscopic strains, whose analysis is linked to the solvability of an inhomogeneous differential inclusion problem with affine boundary values. We discuss how to estimate this set by exploiting admissible boundary interaction, global compatibility, and the interplay between the slip mechanism and the polycrystalline texture. The second model describes high-contrast composites with periodically arranged layers and gives rise to energy functionals with non-convex differential constraints. We prove homogenization theorems via $\Gamma$-convergence in the Sobolev and BV setting and study the resulting limit models, addressing the uniqueness of minimizers and deriving necessary conditions. These results are joint work with Fabian Christowiak (University of Regensburg), Elisa Davoli (TU Vienna), Dominik Engl (KU Eichstätt-Ingolstadt), and Rita Ferreira (KAUST).[-]