Microstructural disorder, mesoscale finite elements and macroscopic response

Martin Ostoja, S. Tarzewski

Research output: Contribution to journalArticlepeer-review


Consideration of boundary-value problems in mechanics of materials with disordered microstructures leads to the introduction of an intermediate scale, a mesoscale, which specifies the resolution of a finite-element mesh relative to the microscale. The effective elastic mesoscale response is bounded by the Dirichlet and Neumann boundary-value problems. The two estimates, separately, provide inputs to two finiteelement schemes-based on minimum potential and complementary energy principles, respectively-for bounding the global response. While in the classical case of a homogeneous material, these bounds are convergent with the finite elements becoming infinitesimal, the presence of a disordered non-periodic microstructure prevents such a convergence and leads to a possibility of an optimal mesoscale. The method is demonstrated through an example of torsion of a bar having a percolating two-phase microstructure of over 100 000 grains. By passing to an ensemble setting, we arrive at a hierarchy of two random continuum fields, which provides inputs to a stochastic finite-element method.

Original languageEnglish (US)
Pages (from-to)3189-3199
Number of pages11
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number1989
StatePublished - Jan 1 1999
Externally publishedYes


  • Computational mechanics
  • Finite elements
  • Mesoscale
  • Microstructural disorder
  • Random media
  • Stochastic finite elements

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)


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