Stochastic Mechanics of Materials and Structures


This course presents an array of methods to study mechanics of spatially random material microstructures involving several scales. The course begins with lectures on random geometry and stochastic processes and fields, including spatial point processes, mathematical morphology, geodesics, ergodicity, and entropy. Subsequent topics include: periodic versus disordered truss- and beam-type lattices, and a construction of corresponding classical and non-classical (Cosserat, non-local, strain-gradient, chiral …) continua; introduction to statistical continuum theories, including thermomechanics of random media; scaling to Representative Volume Element (RVE) in conductivity, linear or finite (thermo)elasticity, elasto-plasticity, permeability, and coupled field phenomena; methods for problems below the RVE (i.e., those lacking the separation of scales) via micromechanically-based stochastic finite elements; tensor random fields, effects of microscale material randomness on waves and wavefronts (e.g. shocks) in linear and non-linear elastic/dissipative media; introduction to stochastic damage mechanics; elements of fractional calculus and fractals.

Key Topics

  • To develop familiarity with a wide array of methods in mechanics of spatially random materials and structures, specifically:
  • Introduction to stochastic geometric models of microstructures
  • Scalar and tensor random fields; fractal and Hurst effects
  • Scale-dependent homogenization of random elastic/inelastic materials
  • Stochastic (micro)mechanics as a basis for stochastic finite elements (SFE)
  • Waves in random media


Who Should Attend

Researchers in (thermo)mechanics, dynamics and transport phenomena in spatially random materials and structures. These include research scientists in solid and structural mechanics, graduate students and postdocs, faculty.

General Course Information

  • Type of Course: Instructor-Led Short Course
  • Course Level: Advanced
  • Course Length: 2 days
  • AIAA CEU's available: yes
  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 Methods and Bounds for 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 and stochastic partial differential equations, introduction to fractional calculus and fractals]

Recommended texts are M. Ostoja-Starzewski, Microstructural Randomness and Scaling in Mechanics of Materials , Chapman & Hall/CRC Modern Mechanics and Mathematics, 2008; and M. Ostoja-Starzewski, S. Kale, P. Karimi, A. Malyarenko, B. Raghavan, S.I. Ranganathan, and J. Zhang, “Scaling to RVE in random media,” Advances in Applied Mechanics 49, 111-211, 2016.


Martin Ostoja-Starzewski
Prof. Martin Ostoja-Starzewski is in the Department of Mechanical Science and Engineering as well as the Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign. His main research interests are in (thermo)mechanics and transport phenomena in random/fractal media, waves, advanced continuum theories, and spontaneous violations of the second law of thermodynamics. He wrote 190+ journal papers and two books: 1. Microstructural Randomness and Scaling in Mechanics of Materials, CRC Press (2008); 2. Thermoelasticity with Finite Wave Speeds, Oxford University Press (2009). He (co-)edited 15 books/journal special issues and co-organized numerous symposia and congresses. He is/was on the editorial boards of over a dozen journals, is co-Editor of the Modern Mechanics and Mathematics book series at CRC Press and Chair Managing Editor of Mathematics and Mechanics of Complex Systems (MEMOCS) journal. He is Fellow of ASME, AAM, SES, WIF, and Assoc. Fellow of AIAA. In 2012 he was Timoshenko Distinguished Visitor at Stanford University. Since 2014 he is Site co-Director of the NSF Industry/University Cooperative Research Center for Novel High Voltage/Temperature Materials and Structures. .


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