Appropriate number of unit cells in a representative volume element for micro-structural bifurcation encountered in a multi-scale modeling

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Abstract

The paper proposes a method to determine the number of unit cells (basic structural elements) to be employed for a representative volume element (RVE) of the multi-scale modeling for a solid with periodic micro-structures undergoing bifurcation. Main difficulties for the multi-scale modeling implementing instability are twofold: loss of convexity of the total potential energy that should be homogenized and determination of a pertinent RVE that contains multiple unit cells. In order to resolve these difficulties, variational formulation is achieved with the help of -convergence theory within the framework of non-convex homogenization method, while the number of unit cells in an RVE is determined by the block-diagonalization method of group-theoretic bifurcation theory. The latter method enables us to identify the most critical bifurcation mode among possible bifurcation patterns for an assembly of arbitrary number of periodic micro-structures. Thus, the appropriate number of unit cells to be employed in the RVE can be determined in a systematic manner. Representative numerical examples for a cellular solid show the feasibility of the proposed method and illustrate material instability at a macroscopic point due to geometrical instability in a micro-scale.

Original languageEnglish
Pages (from-to)2561-2585
Number of pages25
JournalComputer Methods in Applied Mechanics and Engineering
Volume191
Issue number23-24
DOIs
Publication statusPublished - 2002 Mar 29

Keywords

  • Block-diagonalization method
  • Cellular solid
  • Instability
  • Multi-scale modeling
  • Non-convex homogenization

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications

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