Multi-scale Method for the Crack Problem in Microstructural Materials

Abstract The paper deals with the numerical computation of a crack problem posed on microstructural heterogeneous materials containing multiple phases in the microstructure. The failure of such materials is a natural multi-scale effect since cracks typically nucleate in regions of defects on the microscopic scale. The modeling strategy for solving the crack problem concerns simultaneously the macroscopic and microscopic models. Our approach is based on an efficient combination of the homogenization technique and the mesh superposition method (s-version of the finite element method). The homogenized model relies on a double-scale asymptotic expansion of the displacement field. The mesh superposition method uses two independent (global and local) finite element meshes and the concept of superposing the local mesh arbitrarily on the global continuous mesh. The crack is treated by the local mesh and the homogenized material model is considered on the global mesh. Numerical experiments for problems on biomorphic microcellular ceramic templates with porous microstructures of different materials constituents are presented.

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

[2]  Jacob Fish,et al.  On adaptive multilevel superposition of finite element meshes for linear elastostatics , 1994 .

[3]  Ted Belytschko,et al.  Combined extended and superimposed finite element method for cracks , 2004 .

[4]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[5]  Jacob Fish,et al.  THE S-VERSION OF FINITE ELEMENT METHOD FOR LAMINATED COMPOSITES , 1996 .

[6]  V. Kouznetsova,et al.  Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme , 2002 .

[7]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[8]  Naoki Takano,et al.  Multi-scale finite element analysis of porous materials and components by asymptotic homogenization theory and enhanced mesh superposition method , 2003 .

[9]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[10]  J. Fish,et al.  Hierarchical modelling of discontinuous fields , 1992 .

[11]  Toshitaka Ota,et al.  Biomimetic Process for Producing SiC “Wood” , 1995 .

[12]  M. Cross,et al.  A multi-scale atomistic-continuum modelling of crack propagation in a two-dimensional macroscopic plate , 1998 .

[13]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[14]  Ronald H. W. Hoppe,et al.  Optimal shape design in biomimetics based on homogenization and adaptivity , 2004, Math. Comput. Simul..

[15]  Ronald H. W. Hoppe,et al.  Mechanical Failure in Microstructural Heterogeneous Materials , 2006, Numerical Methods and Applications.

[16]  Stéphane Bordas,et al.  An extended finite element library , 2007 .

[17]  Peter Greil,et al.  Biomorphic Cellular Silicon Carbide Ceramics from Wood: I. Processing and Microstructure , 1998 .

[18]  Yuri V. Vassilevski,et al.  Adaptive Grid Refinement for Computation of the Homogenized Elasticity Tensor , 2003, LSSC.

[19]  M. Rashid.  The arbitrary local mesh replacement method: An alternative to remeshing for crack propagation analysis , 1998 .

[20]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[21]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[22]  Brian Moran,et al.  Crack tip and associated domain integrals from momentum and energy balance , 1987 .

[23]  M. Wolcott.  Cellular solids: Structure and properties , 1990 .

[24]  Somnath Ghosh,et al.  Multiple scale computational model for damage in composite materials , 1999 .

[25]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[26]  J. Fish.  The s-version of the finite element method , 1992 .

[27]  R. Hoppe,et al.  Shape Optimization of Biomorphic Ceramics with Microstructures by Homogenization Modeling , 2006 .