This course gives exposition of an array of methods developed over the past few decades, and necessary for reading the literature and doing research on mechanics of random and/or fractal material microstructures. This is the grand theme of contemporary mechanics of materials, including geomechanics and biomechanics. Besides (non)linear, (in)elastic responses, various coupled field phenomena or flow in porous media, can also be handled by techniques presented here.
Course Outline (6x2 hours)
1. Introduction to stochastic geometric models of microstructures
2. Lattice models (periodicity vs. randomness, rigidity, dynamics, and optimality)
3. Mesoscale bounds for random elastic media; size of representative volume element (RVE)
4. Mesoscale bounds for random nonlinear (in)elastic media
5. Scalar/tensor random fields; fractal and Hurst effects
6. Connection to stochastic partial differential equations and stochastic finite elements (SFE)
7. Wavefronts in random media
8. Mechanics of fractal media via dimensional regularization
9. Classical (Cauchy) versus generalized (Cosserat/micropolar or nonlocal) models
10. Elastic-plastic-brittle transitions and avalanches in disordered media
11. Generalized thermoelasticity theories
12. Continuum mechanics vis-à-vis violations of the second law of thermodynamics
Course Notes: to be distributed
Reference Texts (not required):
• M. Ostoja-Starzewski (2008), Microstructural Randomness and Scaling in Mechanics of Materials, CRC Press
• J. Ignaczak and M. Ostoja-Starzewski (2010), Thermoelasticity with Finite Wave Speeds, Oxford Mathematical Monographs, Oxford University Press.
• M. Ostoja-Starzewski, J. Li, H. Joumaa and P.N. Demmie (2013), “From fractal media to continuum mechanics,” ZAMM 93, 1-29
Who Will Benefit
Researchers in (thermo)mechanics and transport phenomena in heterogeneous random and/or fractal materials and stochastic multiscale problems.