### Abstract

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 language | English (US) |
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Pages (from-to) | 3189-3199 |

Number of pages | 11 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 455 |

Issue number | 1989 |

State | Published - Jan 1 1999 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*455*(1989), 3189-3199.