Course code MP08
Course title Physics and Mechanics of Random Media
Institution Mines ParisTech
Course address 60 boulevard Saint-Michel, 75272 Paris cedex 06
City Paris
Minimum year of study 4th year
Minimum level of English Good
Minimum level of French Poor
Key words   random structures, variability, simulations, homogenization, upscaling, elasticity, fracture statistics, reliability, computer aided design of materials
Language English
Professor responsible François Willot, Centre de Morphologie Mathématique, ENSMP
Participating professors   M. Bornert, B. Figliuzzi, C. Lantuejoul, B. Noetinger, Yves-Patrick Pellegrini, Anne-Françoise Gourgues-Lorenzon, F. Willot
Number of places Minimum: 5, Maximum: 25, Reserved for local students: 0
Objectives   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, will be given in a self-contained series of lectures and training sessions on computers.
Programme to be followed

The main topics of the course are as follows :

-  Introduction and basic concepts (material variability of mechanical properties at different scales, introduction to applied probability and probabilistic models, morphological characterization of random sets and of random functions, examples of models and simulations of random structures)
-  Homogenization of random media (linear and non linear properties): bounds and numerical techniques  (numerical homogenization by Fast Fourier Transforms)

-      - Transport in random media. Fracture Statistics models.

The courses detail the construction of models, their main properties, and their use from experimental data by means of examples of application.

A large part of the course is based on training by means of softwares Micromorph and Morph'Hom developed in CMM. 
Structure of the course : Five full days in a single week. Lectures (70 %) and practical training on PC computers (30 %).

The daily course programme can be consulted some ten days prior to the course, please see : (under the link , Ingénieurs civils)


Prerequisites  Basic knowledge in probability theory. Physics and Mechanics of solids
Course exam The students prepare a written project from data processed durig the training session