An FE2-X1 approach for multiscale localization phenomena

Abstract In this paper, a new methodology based on the Hill–Mandel lemma in an FE 2 sense is proposed that is able to deal with localized deformations. This is achieved by decomposing the displacement field of the fine scale model into a homogeneous part, fluctuations, and a cracking part based on additional degrees of freedom (X 1 )—the crack opening in normal and tangential directions. Based on this decomposition, the Hill–Mandel lemma is extended to relate coarse and fine scale energies using the assumption of separation of scales such that the fine scale model is not required to have the same size as the corresponding macroscopic integration point. In addition, a procedure is introduced to mimic periodic boundary conditions in the linear elastic range by adding additional shape functions for the boundary nodes that represent the difference between periodic boundary conditions and pure displacement boundary conditions due to the same macroscopic strain. In order to decrease the computational effort, an adaptive strategy is proposed allowing different macroscopic integration points to be resolved in different levels on the fine scale.

[1]  Ted Belytschko,et al.  Conservation properties of the bridging domain method for coupled molecular/continuum dynamics , 2008 .

[2]  Jacques-Louis Lions,et al.  Nonlinear partial differential equations and their applications , 1998 .

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

[4]  V. Kouznetsova,et al.  Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy , 2004 .

[5]  Ted Belytschko,et al.  Concurrently coupled atomistic and XFEM models for dislocations and cracks , 2009 .

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

[7]  I. Gitman.  Representative volumes and multi-scale modelling of quasi-brittle materials , 2006 .

[8]  Marc Bonnet,et al.  Inverse problems in elasticity , 2005 .

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

[10]  Mgd Marc Geers,et al.  Novel boundary conditions for strain localization analyses in microstructural volume elements , 2012 .

[11]  Udo Nackenhorst,et al.  An adaptive FE–MD model coupling approach , 2010 .

[12]  Barbara I. Wohlmuth,et al.  Mortar Finite Elements for Interface Problems , 2004, Computing.

[13]  M. Bonnet,et al.  Overview of Identification Methods of Mechanical Parameters Based on Full-field Measurements , 2008 .

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

[15]  Lennart Ljung,et al.  Nonlinear Black Box Modeling in System Identification , 1995 .

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

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

[18]  Guillaume Rateau,et al.  The Arlequin method as a flexible engineering design tool , 2005 .

[19]  D. Rixen,et al.  FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method , 2001 .

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

[21]  V. Kouznetsova,et al.  Enabling microstructure-based damage and localization analyses and upscaling , 2011 .

[22]  S. Eckardt,et al.  A mesoscale model for concrete to simulate mechanical failure , 2011 .

[23]  Somnath Ghosh,et al.  A multi-level computational model for multi-scale damage analysis in composite and porous materials , 2001 .

[24]  S. Eckardt,et al.  Adaptive Damage Simulation of Concrete Using Heterogeneous Multiscale Models , 2008 .

[25]  Ted Belytschko,et al.  Multiscale aggregating discontinuities method for micro–macro failure of composites , 2009 .

[26]  Carsten Könke,et al.  Coupling of scales in a multiscale simulation using neural networks , 2008 .

[27]  John E. Mottershead,et al.  Finite Element Model Updating in Structural Dynamics , 1995 .

[28]  Barbara I. Wohlmuth,et al.  Discretization Methods and Iterative Solvers Based on Domain Decomposition , 2001, Lecture Notes in Computational Science and Engineering.

[29]  James G. Boyd,et al.  Micromechanics and homogenization of inelastic composite materials with growing cracks , 1996 .

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

[31]  T. Belytschko,et al.  A bridging domain method for coupling continua with molecular dynamics , 2004 .

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

[33]  René de Borst,et al.  Computational homogenization for adhesive and cohesive failure in quasi‐brittle solids , 2010 .

[34]  Zvi Hashin,et al.  The Elastic Moduli of Heterogeneous Materials , 1962 .

[35]  Carsten Könke,et al.  An inverse parameter identification procedure assessing the quality of the estimates using Bayesian neural networks , 2011, Appl. Soft Comput..

[36]  Carsten Könke,et al.  Neural networks as material models within a multiscale approach , 2009 .

[37]  Jörg F. Unger,et al.  Multiscale Modeling of Concrete , 2011 .

[38]  Christian Miehe,et al.  Homogenization and two‐scale simulations of granular materials for different microstructural constraints , 2010 .

[39]  Harm Askes,et al.  Representative volume: Existence and size determination , 2007 .

[40]  Ted Belytschko,et al.  Multiscale aggregating discontinuities: A method for circumventing loss of material stability , 2008 .

[41]  Lennart Ljung,et al.  Nonlinear black-box modeling in system identification: a unified overview , 1995, Autom..

[42]  Frédéric Feyel,et al.  Multiscale FE2 elastoviscoplastic analysis of composite structures , 1999 .