Multiscale domain decomposition analysis of quasi‐brittle heterogeneous materials

SUMMARY A hybrid multiscale framework is presented, which processes the material scales in a concurrent manner, borrowing features from hierarchical multiscale methods. The framework is used for the analysis of non-linear heterogeneous materials and is capable of tackling strain localization and failure phenomena. Domain decomposition techniques, such as the finite element tearing and interconnecting method, are used to partition the material in a number of non-overlapping domains and adaptive refinement is performed at those domains that are affected by damage processes. This refinement is performed in terms of material scale and finite element size. It is verified that the results are independent of the chosen domain decomposition. Moreover, the multiscale analyses are validated with reference solutions obtained with a full fine-scale solution procedure. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  Chris J. Pearce,et al.  Scale transition and enforcement of RVE boundary conditions in second‐order computational homogenization , 2008 .

[2]  Zvi Hashin.  Elasticity of Random Media , 1965 .

[3]  Julia Mergheim,et al.  A variational multiscale method to model crack propagation at finite strains , 2009 .

[4]  Daniel Rixen,et al.  Domain decomposition techniques for the efficient modeling of brittle heterogeneous materials , 2011 .

[5]  Gilles Pijaudier-Cabot,et al.  CONTINUUM DAMAGE THEORY - APPLICATION TO CONCRETE , 1989 .

[6]  Somnath Ghosh,et al.  Concurrent multi-level model for damage evolution in microstructurally debonding composites , 2007 .

[7]  Philippe H. Geubelle,et al.  Multiscale cohesive failure modeling of heterogeneous adhesives , 2008 .

[8]  D J Evans,et al.  Parallel processing , 1986 .

[9]  David Dureisseix,et al.  A micro–macro and parallel computational strategy for highly heterogeneous structures , 2001 .

[10]  J. Tinsley Oden,et al.  Hierarchical modeling of heterogeneous solids , 1996 .

[11]  W. Voigt.  Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper , 1889 .

[12]  Daniel J. Rixen,et al.  Theoretical Relations Between Domain Decomposition and Dynamic Substructuring , 2004, PARA.

[13]  Olivier Allix,et al.  Nonlinear localization strategies for domain decomposition methods: Application to post-buckling analyses , 2007 .

[14]  Laurent Champaney,et al.  A two-scale approach with homogenization for the computation of cracked structures , 2007 .

[15]  V. G. Kouznetsova,et al.  Multi-scale computational homogenization: Trends and challenges , 2010, J. Comput. Appl. Math..

[16]  Vinh Phu Nguyen,et al.  Homogenization-based multiscale crack modelling: from micro diffusive damage to macro cracks , 2011 .

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

[18]  Daniel Rixen,et al.  Extended preconditioners for the FETI method applied to constrained problems , 2002 .

[19]  Ekkehard Ramm,et al.  Locality constraints within multiscale model for non‐linear material behaviour , 2007 .

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

[21]  Hervé Moulinec,et al.  A numerical method for computing the overall response of nonlinear composites with complex microstructure , 1998, ArXiv.

[22]  J. Z. Zhu,et al.  The finite element method , 1977 .

[23]  R. Hill.  Elastic properties of reinforced solids: some theoretical principles , 1963 .

[24]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[25]  Rhj Ron Peerlings,et al.  Gradient enhanced damage for quasi-brittle materials , 1996 .

[26]  W. Greub.  Linear Algebra , 1981 .

[27]  K. Tanaka,et al.  Average stress in matrix and average elastic energy of materials with misfitting inclusions , 1973 .

[28]  Fusao Oka,et al.  Dispersion and wave propagation in discrete and continuous models for granular materials , 1996 .

[29]  Damijan Markovic,et al.  On micro–macro interface conditions for micro scale based FEM for inelastic behavior of heterogeneous materials , 2004 .

[30]  C. Farhat,et al.  The second generation FETI methods and their application to the parallel solution of large-scale linear and geometrically non-linear structural analysis problems , 2000 .

[31]  P. Gosselet,et al.  Non-overlapping domain decomposition methods in structural mechanics , 2006, 1208.4209.

[32]  Heh Han Meijer,et al.  Effective properties of a viscoplastic constitutive model obtained by homogenisation , 1999 .

[33]  Ted Belytschko,et al.  Coarse‐graining of multiscale crack propagation , 2010 .

[34]  David Dureisseix,et al.  Multi‐scale domain decomposition method for large‐scale structural analysis with a zooming technique: Application to plate assembly , 2009 .

[35]  A. Reuss,et al.  Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle . , 1929 .

[36]  M. Geers,et al.  An enhanced multi‐scale approach for masonry wall computations with localization of damage , 2007 .

[37]  Jacob Fish,et al.  Multiple scale eigendeformation-based reduced order homogenization , 2009 .

[38]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[39]  Cv Clemens Verhoosel,et al.  A partition of unity‐based multiscale approach for modelling fracture in piezoelectric ceramics , 2010 .

[40]  F. Feyel.  A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua , 2003 .

[41]  Damijan Markovic,et al.  Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures , 2003 .

[42]  Pierre Gosselet,et al.  A Nonlinear Dual-Domain Decomposition Method: Application to Structural Problems with Damage , 2008 .

[43]  Vinh Phu Nguyen,et al.  On the existence of representative volumes for softening quasi-brittle materials – A failure zone averaging scheme , 2010 .

[44]  C. Duarte,et al.  Analysis and applications of a generalized finite element method with global-local enrichment functions , 2008 .

[45]  Fpt Frank Baaijens,et al.  An approach to micro-macro modeling of heterogeneous materials , 2001 .

[46]  C. Farhat,et al.  A method of finite element tearing and interconnecting and its parallel solution algorithm , 1991 .

[47]  J. Chaboche,et al.  FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials , 2000 .

[48]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[49]  L. J. Sluys,et al.  Coupled-volume multi-scale modelling of quasi-brittle material , 2008 .

[50]  André Chrysochoos,et al.  Two field multibody method for periodic homogenization in fracture mechanics of nonlinear heterogeneous materials , 2008 .

[51]  C. Miehe,et al.  Computational micro-to-macro transitions of discretized microstructures undergoing small strains , 2002 .

[52]  Alfio Quarteroni,et al.  Domain Decomposition Methods for Partial Differential Equations , 1999 .

[53]  P. Tallec,et al.  Domain decomposition methods for large linearly elliptic three-dimensional problems , 1991 .

[54]  Fodil Meraghni,et al.  Generalised Mori-Tanaka Scheme to Model Anisotropic Damage Using Numerical Eshelby Tensor , 2001 .