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Catalogues >> Mathématiques et leurs applications >> Probabilités et statistiques
Responsables :

Dominique Jeulin
François Willot

Equipe Pédagogique :
Anthony Marais
Anthonin Steckmeyer
Xu Han

Niveau : Graduate

Langue du cours : Anglais

Période : Printemps

Nombre d'heures : 37

Crédits ECTS : 2
SGS_MP6124 ATHENS - MP08 - Physics and Mechanics of Random Media
Objectifs: Many solid media and materials (composites, granular media, metals, biomaterials, porous media, soils, rocks, etc.) encountered in materials sciences, geophysics, environmental sciences, energetics, hydrogeology,... display microstructures and structures of several length scales, showing often a non-deterministic disorder. A better understanding and prediction of the resulting multiscale and random nature of materials' mesoscopic and/or macroscopic properties requires a modeling approach based on a combination of probabilistic concepts with methods of physics and mechanics. The course, which aims to provide an introduction to this subject, is given in a self-contained series of lectures and training sessions on computers.

  • Motivated by a review of advanced experimental techniques for the microstructure description, and by typical results involving fluctuations present in plasticity, damage, fracture, and flows phenomena in porous media, basic tools of applied probability and random processes are recalled.
  • Probabilistic tools for the description random media and models together with their simulation are introduced.
  • Physics and mechanics of random media are first presented from the standpoint of approximate solutions of partial differential equations with random coefficients. For example, linear electrostatics problems in random media are studied by means of a perturbation expansion of the random electric and displacement fields, while bounds on the effective permittivity and of elastic moduli are derived from variational principles. This approach of homogenization, which can be applied to other physical properties like the composition of permeability, or of the thermal conductivity, is illustrated by third order bounds.
  • The use of numerical techniques (like Finite Elements), to provide an estimation of homogenized properties of random media from Monte Carlo type simulations is introduced. Bounds and numerical techniques are then extended to non linear behaviours, like the plasticity of polycrystals.
  • Given the importance of reliability problems in a multitude of engineering applications, several fracture statistics models (brittle, ductile, fatigue) are worked out from a probabilistic approach.
Structure of the course:
Single week. Lectures (80%) and practical training on computers (20%).

More information at:

Niveau requis : Prerequisites: basic knowledge of probability theory, physics and solids mechanics.

Modalités d'évaluation : The students prepare a written project from data processed during the training sessions.

Dernière mise à jour : jeudi 7 février 2013

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