Modelling an simulating the mechanical behavior of ice floes

Keywords :

• Homogénéisation
• Fracture
• Gamma-Convergence
• Méthodes variationnelles
• Banquise

This doctoral dissertation is part of a research project on sea ice modeling, initiated by TOTAL S.A. and the Université Grenoble Alpes.This project lead to the development of a granular model for the evolution of sea ice, and in particular the mechanical behavior of ice floes in the marginal ice zone.The implementation of the model can simulate the collisions of one million ice floes, and their interaction with rigid structures.This PhD thesis improves the current granular model by adding an efficient model for ice floe fracture.Firstly, we present a fracture model for an ice floe subject to a boundary displacement.This model is a brittle fracture model, relying on the work of citeauthor{GRIFFITH1921}.It is written in a variational framework inspired from that of citeauthor{FM98}'s model: we minimize the total energy of the material.We show that, under some hypothesis, the total energy of the ice floe has a minimum.This variational model is efficient, and can be used in the collision model which simulates the behavior of a large number of floes.This efficiency relies on a strong geometric hypothesis, although mitigated by the use of a quasistatic loading : we restrict the space of admissible fractures to the set of segment lines.Secondly, we present a research strategy to obtain an expression of the boundary displacement during the percussion of two ice floes.The strategy is the following : we consider the ice floe as the limit of an isotropic mass-spring lattice.For a given lattice, we can write the differential equation verified by each mass, and thus we hope to derive an expression of the boundary displacement.We identify three mathematical limits which we deem necessary to the understanding of the percussion phenomenon, and we obtain two of them.Doing so, we prove two Gamma-convergence results of discrete functionals, defined on different lattices, to the classical elastic energy.In particular, we work with a stochastic isotropic lattice, built as the Delaunay triangulation of a stochastic point process.In that case, we will prove the almost-sure Gamma-convergence.

Directors:

• Mr Jérôme Weiss
• Mr Stéphane Labbé (Professeur des universités - Université Grenoble Alpes )

Raporteurs:

• Mr Hervé Le Dret (Professeur des universités - Université Pierre et Marie Curie )
• Mr Gregory Vial (Professeur des universités - École centrale Lyon )

Examinators:

• Mr Dorin Bucur (Professeur des universités - Université de Savoie Mont Blanc )
• Mr Vladimir Yastrebov (Chargé de recherche - Mines ParisTech )