Grad Course on the Mechanics of Random Media Offered in Spring 2016

MECHANICS OF RANDOM MEDIA (TAM 598)

Spring 2016 - 4 hours/week

Prof. Martin Ostoja-Starzewski Department of Mechanical Science & Engineering

Objective: This course is intended to expose students to an array of methods developed over the past few decades, and necessary for reading the literature and doing research on mechanics of random material microstructures involving several scales. This is the grand theme of contemporary mechanics of materials, including biomechanics. Besides (in)elastic responses, various coupled field phenomena (thermoelastic, flow in porous media, electromagnetic…) can also be handled by techniques presented here.

COURSE CONTENTS

1. Planar and Spatial Random Processes and Models of Material Microstructures

[point processes, directional data, random line fields, random tessellations (models of polycrystal, granular, porous media,…), elements of mathematical morphology]

2. Elements of Random Processes and Fields

[types of random fields (Markov/Gibbs, evolutionary versus stationary, anisotropic, ergodic, tensor,…), classes of correlation functions, maximum entropy method]

3. Lattice Models

[periodic versus non-periodic (central-force, beam…) models and generalized continuum models, rigidity, randomness, dynamics, and optimality]

4. Statistical Continuum and Bounding Methods for Effective Transport and Elastic Properties

[statistical hierarchy of functions, non-local effect, homogenization method, percolation problems, maximum entropy approach]

5. Mesoscale Bounds for Random Media

[Hill condition, separation of scales and Representative Volume Element (RVE), hierarchies of mesoscale bounds in (non)linear elastic and/or inelastic microstructures]

6. Random Field Models and Stochastic Finite Elements (SFE)

[mechanics restrictions imposed on tensor-valued random fields, SFE, random plastic media, damage and fracture in random microstructures]

7. Advanced Stochastic Solid Mechanics

[introduction to stochastic ordinary & partial differential equations, introduction to fractional calculus, fractals in mechanics]

Basic text: M. Ostoja-Starzewski, Microstructural Randomness and Scaling in Mechanics of Materials, Chapman & Hall/CRC Press, 2008.

Recommended background: 400-level (or higher) continuum/solid mechanics, standard probability theory and statistics

Grading: Homeworks (30), Project (40), Tests-Exam (30)

Broad grade equivalents: 90-100, A; 80-90, B; 70-80, C; 60-70, D, 0-60, F