Concurrent multi-scale analysis of elastic composites by a multi-level computational model

Abstract In this paper, an adaptive multi-level computational model is developed to comprehensively address issues of discretization and physically based modeling error in structures consisting of heterogeneous micro-structures. The model consists of three essential levels, (i) a purely macroscopic domain (level-0) with homogenized material parameters; (ii) a macro–micro domain (level-1) with the micro-domain represented by the periodic repetition of a representative volume element (RVE) and (iii) purely microscopic domain (level-2) where the RVE ceases to exist and extended micro-structural regions need to be modeled. All micro-structural computations of arbitrary heterogeneous domains are conducted using the adaptive Voronoi cell finite element model. An intermediate layer is used between the macroscopic and microscopic regions for smooth transition. The level-1 domain is utilized to estimate criteria for switching from macroscopic to microscopic calculations. Physically based error indicators are developed for transition from macroscopic to microscopic domain. Statistical functions are used to assess the extent of the RVE domains for non-uniform micro-structures. For the macroscopic domains, h - and hp -adaptations are executed using conventional error indicators based on the energy norm. The numerical simulations are compared with results from pure micro-mechanics and other multi-level models in the literature. The model is also tested for a realistic problem with macroscopic stress risers.

[1]  J. Lions.  On Some Homogenization Problems , 1982, Mai 1982.

[2]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[3]  Kumar K. Tamma,et al.  WOVEN FABRIC COMPOSITES: DEVELOPMENTS IN ENGINEERING BOUNDS, HOMOGENIZATION AND APPLICATIONS , 1999 .

[4]  J. Tinsley Oden,et al.  Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: I. Error estimates and adaptive algorithms , 2000 .

[5]  J. Oden,et al.  Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: Part II: a computational environment for adaptive modeling of heterogeneous elastic solids , 2001 .

[6]  Sia Nemat-Nasser,et al.  Overall moduli of solids with microcracks: Load-induced anisotropy , 1983 .

[7]  I. Babuška,et al.  The h-p version of the finite element method , 1986 .

[8]  J. Oden,et al.  Analysis and adaptive modeling of highly heterogeneous elastic structures , 1997 .

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

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

[11]  R. Pyrz.  Quantitative description of the microstructure of composites. Part I: Morphology of unidirectional composite systems , 1994 .

[12]  Somnath Ghosh,et al.  Quantitative characterization and modeling of composite microstructures by voronoi cells , 1997 .

[13]  E. Rybicki,et al.  On the Significance of Effective Modulus Solutions for Fibrous Composites , 1974 .

[14]  M. Ainsworth,et al.  Aspects of an adaptive hp-finite element method : Adaptive strategy, conforming approximation and efficient solvers , 1997 .

[15]  Peter Wriggers,et al.  A domain decomposition method for bodies with heterogeneous microstructure basedon material regularization , 1999 .

[16]  A Study of the Influence of Microstructure on the Modified Effective Modulus Approach for Composite Laminates , 1994 .

[17]  J. Oden,et al.  A procedure for a posteriori error estimation for h-p finite element methods , 1992 .

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

[19]  I. Babuska,et al.  Rairo Modélisation Mathématique Et Analyse Numérique the H-p Version of the Finite Element Method with Quasiuniform Meshes (*) , 2009 .

[20]  N. Kikuchi,et al.  Simulation of the multi-scale convergence in computational homogenization approaches , 2000 .

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

[22]  R. Christensen,et al.  Solutions for effective shear properties in three phase sphere and cylinder models , 1979 .

[23]  Ivo Babuška,et al.  A posteriori estimation and adaptive control of the pollution error in the h‐version of the finite element method , 1995 .

[24]  Somnath Ghosh,et al.  Interfacial debonding analysis in multiple fiber reinforced composites , 2000 .

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

[26]  G. P. Tandon,et al.  A Theory of Particle-Reinforced Plasticity , 1988 .

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

[28]  B. Budiansky.  On the elastic moduli of some heterogeneous materials , 1965 .

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

[30]  R. C. Morgan,et al.  An approach for constructing families of homogenized equations for periodic media. II: properties of the kernel , 1991 .

[31]  Somnath Ghosh,et al.  A two-dimensional automatic mesh generator for finite element analysis for random composites , 1991 .

[32]  Leszek Demkowicz,et al.  Toward a universal h-p adaptive finite element strategy , 1989 .

[33]  Mohammad A. Aminpour,et al.  A coupled analysis method for structures with independently modelled finite element subdomains , 1995 .

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

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

[36]  Somnath Ghosh,et al.  Adaptivity and convergence in the Voronoi cell finite element model for analyzing heterogeneous materials , 2000 .

[37]  R. Pyrz.  Correlation of Microstructure Variability and Local Stress Field in Two-Phase Materials , 1994 .

[38]  Somnath Ghosh,et al.  A Voronoi Cell finite element model for particle cracking in elastic-plastic composite materials , 1998 .

[39]  Ivo Babuška,et al.  An approach for constructing families of homogenized equations for periodic media. I: an integral representation and its consequences , 1991 .

[40]  Yusheng Feng,et al.  Local and pollution error estimation for finite element approximations of elliptic boundary value problems , 1996 .

[41]  Z. Hashin.  Analysis of Composite Materials—A Survey , 1983 .

[42]  Thermal stresses in aluminum-to-composite double-lap bounded joints , 1998 .

[43]  Somnath Ghosh,et al.  Tessellation-based computational methods for the characterization and analysis of heterogeneous microstructures , 1997 .

[44]  Ivo Babuška,et al.  Pollution-error in the h -version of the finite-element method and the local quality of a-posteriori error estimators , 1994 .

[45]  Jacob Fish,et al.  Multiscale finite element method for a locally nonperiodic heterogeneous medium , 1993 .

[46]  Y. Benveniste.  On the Mori-Tanaka's method in cracked bodies , 1986 .

[47]  Z. Hashin,et al.  The Elastic Moduli of Fiber-Reinforced Materials , 1964 .

[48]  N. J. Pagano,et al.  Revisiting the composite laminate problem with an adaptive multi-level computational model , 2001 .

[49]  J. Fish,et al.  Multi-grid method for periodic heterogeneous media Part 2: Multiscale modeling and quality control in multidimensional case , 1995 .

[50]  Somnath Ghosh,et al.  A MODEL FOR ANALYSIS OF ARBITRARY COMPOSITE AND POROUS MICROSTRUCTURES WITH VORONOI CELL FINITE ELEMENTS , 1996 .

[51]  Noboru Kikuchi,et al.  Digital image-based modeling applied to the homogenization analysis of composite materials , 1997 .

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

[53]  Jacob Fish,et al.  Multiscale analysis of composite materials and structures , 2000 .

[54]  Somnath Ghosh,et al.  Particle fracture simulation in non-uniform microstructures of metal-matrix composites , 1998 .

[55]  N. J. Pagano,et al.  The significance of eective modulus theory (homogenization) in composite laminate mechanics , 2000 .

[56]  R. Hill.  A self-consistent mechanics of composite materials , 1965 .