TY - JOUR
T1 - Appropriate number of unit cells in a representative volume element for micro-structural bifurcation encountered in a multi-scale modeling
AU - Saiki, I.
AU - Terada, K.
AU - Ikeda, K.
AU - Hori, M.
N1 - Funding Information:
This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 12750416, 2000.
PY - 2002/3/29
Y1 - 2002/3/29
N2 - 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.
AB - 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.
KW - Block-diagonalization method
KW - Cellular solid
KW - Instability
KW - Multi-scale modeling
KW - Non-convex homogenization
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U2 - 10.1016/S0045-7825(01)00413-3
DO - 10.1016/S0045-7825(01)00413-3
M3 - Article
AN - SCOPUS:0037192734
VL - 191
SP - 2561
EP - 2585
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
SN - 0374-2830
IS - 23-24
ER -