The next session will be held on November 15-19, 2021.
Detailed preliminary program of the 2021 course
For registration or information contact firstname.lastname@example.org or email@example.com
Keywords: random structures, variability, simulations, homogenization, upscaling, elasticity, fracture statistics, reliability, computer aided design of materials, FFT methods
Organiser: François Willot (firstname.lastname@example.org, phone: +33 (0)1 64 69 48 07)
Time and location: November of each year, École des Mines de Paris (60 Bd Saint-Michel, Paris)
Participants: 30 maximum
Language of the courses: English
Subscription: 0€ (Mines ParisTech members), 650€ (students), 1,300€ (other). For more information, please contact Anne-Marie de Castro (
Goal: 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. First, 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. Then, probabilistic tools for the description random media and models together with their simulation are introduced. At the second stage, physics and mechanics of random media are 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 perturbation expansion of the random electric and displacement fields, while bounds on the effective permittivitty 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 third area of focus concerns the use of numerical techniques and especially FFT-based computations to provide estimates of homogenized properties of random media from Monte Carlo type simulations. 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: Five full days in a single week. Lectures (70%) and practical training on computers (30%).
Prerequisites: Basic knowledge in probability theory, physics and mechanics of solids.
Examination: The students prepare a written project from data processed during the training sessions. The project is submitted 3 weeks after the course.