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

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 Γ-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.

[1]  T. Healey.  A group-theoretic approach to computational bifurcation problems with symmetry , 1988 .

[2]  Muneo Hori,et al.  Analysis on evolution pattern of periodically distributed defects , 1997 .

[3]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[4]  Analysis of non-isothermal mold filling process in resin transfer molding (RTM) and structural reaction injection molding (SRIM) , 1995 .

[5]  M. Ashby,et al.  Cellular solids: Structure & properties , 1988 .

[6]  U. Hornung.  Homogenization and porous media , 1996 .

[7]  Somnath Ghosh,et al.  Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method , 1995 .

[8]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[9]  Noboru Kikuchi,et al.  Characterization of the mechanical behaviors of solid-fluid mixture by the homogenization method , 1998 .

[10]  P. Tallec.  Numerical methods for nonlinear three-dimensional elasticity , 1994 .

[11]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[12]  N. Ohno,et al.  Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation , 2002 .

[13]  Kenjiro Terada,et al.  Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain , 2003 .

[14]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[15]  S. D. Papka,et al.  In-plane compressive response and crushing of honeycomb , 1994 .

[16]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[17]  N. Kikuchi,et al.  Preprocessing and postprocessing for materials based on the homogenization method with adaptive fini , 1990 .

[18]  N. Triantafyllidis,et al.  Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity , 1993 .

[19]  Kazuo Murota,et al.  Imperfect Bifurcation in Structures and Materials: Engineering Use of Group-Theoretic Bifurcation Theory. Applied Mathematical Sciences, Vol 149 , 2002 .

[20]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[21]  Nicolas Triantafyllidis,et al.  An Investigation of Localization in a Porous Elastic Material Using Homogenization Theory , 1984 .

[22]  S. Nemat-Nasser,et al.  Micromechanics: Overall Properties of Heterogeneous Materials , 1993 .

[23]  J. Tinsley Oden,et al.  Qualitative methods in nonlinear mechanics , 1985 .

[24]  K. Matsui,et al.  Kinematically Correct Formulation for Two-scale Finite Deformation Problems , 2000 .

[25]  Grégoire Allaire,et al.  Mathematical approaches and methods , 1996 .

[26]  Kazuo Murota,et al.  Computational Use of Group Theory in Bifurcation Analysis of Symmetric Structures , 1991, SIAM J. Sci. Comput..

[27]  N. Kikuchi,et al.  A class of general algorithms for multi-scale analyses of heterogeneous media , 2001 .

[28]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[29]  Stefan Müller,et al.  Homogenization of nonconvex integral functionals and cellular elastic materials , 1987 .

[30]  F. Devries,et al.  Homogenization and damage for composite structures , 1989 .

[31]  Mark S. Shephard,et al.  Computational plasticity for composite structures based on mathematical homogenization: Theory and practice , 1997 .