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International Journal for Multiscale Computational Engineering

Published 6 issues per year

ISSN Print: 1543-1649

ISSN Online: 1940-4352

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Indexed in

AVERAGING PROPERTIES FOR PERIODIC HOMOGENIZATION AND LARGE DEFORMATION

Volume 10, Issue 3, 2012, pp. 281-293
DOI: 10.1615/IntJMultCompEng.2012002587
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ABSTRACT

The main motivation of this paper consists of using the periodic homogenization theory to derive several relations between macroscopic Lagrangian (e.g., deformation gradient, Piola−Kirchhoff tensor) and Eulerian (e.g., velocity gradient, Cauchy stress) quantities. These relations demonstrate that these macroscopic quantities behave formally in the same way as their microscopic counterparts. We say therefore that these relations are stable with respect to the periodic homogenization. We also demonstrate the equivalence between the two forms of the macroscopic power density expressed in the Lagrangian and Eulerian formulations. Two simple examples illustrate these results, and indicate also that the Green−Lagrange strain tensor and the second Piola−Kirchhoff stress tensor are not stable with respect to periodic homogenization.

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CITED BY
  1. Chatzigeorgiou George, Javili Ali, Steinmann Paul, Unified magnetomechanical homogenization framework with application to magnetorheological elastomers, Mathematics and Mechanics of Solids, 19, 2, 2014. Crossref

  2. Bibliography, in Thermomechanical Behavior of Dissipative Composite Materials, 2017. Crossref

  3. Zhu Jianchang, Bettaieb Mohamed Ben, Abed-Meraim Farid, Numerical investigation of necking in perforated sheets using the periodic homogenization approach, International Journal of Mechanical Sciences, 166, 2020. Crossref

  4. Zhu J.C., Ben Bettaieb M., Abed-Meraim F., Investigation of the competition between void coalescence and macroscopic strain localization using the periodic homogenization multiscale scheme, Journal of the Mechanics and Physics of Solids, 143, 2020. Crossref

  5. Zhu J. C., Bettaieb M. Ben, Abed-Meraim F., Comparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization scheme, Engineering with Computers, 38, 2, 2022. Crossref

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