Begell House
International Journal for Multiscale Computational Engineering
International Journal for Multiscale Computational Engineering
1543-1649
9
3
2011
PrefaceSPECIAL ISSUEMULTISCALE MODELING AND UNCERTAINTY QUANTIFICATION OF HETEROGENEOUS MATERIALS
The increasing research activity for micro- and nano-scales over the last decade has significantly shown the need to account for disparate levels of uncertainty from various sources and across scales. Even over-refined deterministic approaches cannot account for the issue; the integration of stochastic and multiscale methodologies is required to provide a rational framework for uncertainty quantification and reliability analysis of heterogeneous materials. Facing the emergence of new advanced engineered materials, accurate stochastic modeling across multiple length scales becomes imperative.
The papers for this special issue are included in Volume 9, issues 3 and 4, 2011 of International Journal for Multiscale Computational Engineering and will focus on multiscale modeling and uncertainty quantification of heterogeneous materials. Particular emphasis is given to advanced computational methods which can greatly assist in tackling complex problems of multiscale stochastic material modeling. The papers can be grouped into several thematic topics that include homogenization and computation of effective elastic properties of random composites, development of computational models for large-scale heterogeneous microstructures, stochastic analysis and design of heterogeneous materials, and multiscale models for the simulation of fracture mechanisms in polycrystalline materials.
Volume 9, Issue 3 for the special issue consists of five papers that are devoted to the homogenization and computation of effective elastic properties of random composites.
George
Stefanou
Institute of Structural Analysis and Antiseismic Research, National Technical University of Athens, Greece
vii-viii
HOMOGENIZATION OF FIBER-REINFORCED COMPOSITES WITH RANDOM PROPERTIES USING THE LEAST-SQUARES RESPONSE FUNCTION APPROACH
The main issue in this elaboration is computational study of the homogenized elasticity tensor for the periodic random composite using the improved stochastic generalized perturbation technique. The uncertainty of the composite appears at the component's material properties, treated here as the Gaussian random variables, while its micro- and macrogeometry remains perfectly periodic. The effective modules method consisting in the cell problem solution is enriched with the generalized stochastic perturbation method. This method is implemented without the necessity of a large number of increasing order equations. The response function between the homogenized tensor and the input random parameter is determined numerically using several deterministic solutions and the least-squares approximation technique. Since classical polynomial approximation techniques may result in some errors for the lower and upper bound of the input parameter variability set, the least-squares approximation is used, where the degree of an approximant is the additional input variable. This approach has hybrid computational implementation{partially in the homogenization-oriented finite element method code MCCEFF and in the symbolic environment of the MAPLE 13 system, giving a wide range of approximation techniques that can also be modified in a graphical mode.
Marcin
Kaminski
Faculty of Civil Engineering, Architecture and Environmental Engineering, Technical University of Lodz, Poland
257-270
LARGE-SCALE COMPUTATIONS OF EFFECTIVE ELASTIC PROPERTIES OF RUBBER WITH CARBON BLACK FILLERS
A general method, based on a multiscale approach, is proposed to derive the effective elastic shear modulus of a rubber with 14% carbon black fillers from finite element and fast Fourier transform methods. The complex multiscale microstructure of such material was generated numerically from a mathematical model of its morphology that was identified from statistical moments out of transmission electron microscopy images. For finite element computations, the simulated microstructures were meshed from three-dimensional reconstruction of the isosurface using the marching cubes algorithm with special attention to the quality of the topology and the geometry of the mesh. To compute the shear modulus and to determine the representative volume element, homogeneous boundary conditions were prescribed on meshes and combined with a domain decomposition method. Regarding parallel computing, specific difficulties related to the highly heterogeneous microstructures and complex geometry are pointed out. The experimental shear modulus (1.8 MPa) obtained from dynamic mechanical analysis was estimated by the Hashin-Shtrikman lower bound ( 1.4 MPa) and the computations on simulated microstructures ( 2.4 MPa). The shear modulus was determined for two materials with the same volume fraction but different distribution of fillers. The current model of microstructures is capable of estimating the relative effect of the mixing time in processing associated with change in morphology on the elastic behavior. The computations also provide the local fields of stress/strain in the elastomeric matrix.
Aurelie
Jean
MINES ParisTech, Centre de Morphologie Mathématique, Mathématiques et Systémes; and Centre des Matériaux, CNRS UMR, France
Francois
Willot
MINES-ParisTech, Centre de Morphologie Mathématique, Mathématiques et Systémes, France
S.
Cantournet
MINES ParisTech, Centre des Matériaux, CNRS UMR, France
Samuel
Forest
MINES MINES ParisTech, Centre des Matériaux, CNRS UMR, France
Dominique
Jeulin
MINES-ParisTech, Centre de Morphologie Mathématique, Mathématiques et Systémes, France
271-303
ELASTIC AND ELECTRICAL BEHAVIOR OF SOME RANDOMMULTISCALE HIGHLY-CONTRASTED COMPOSITES
The role of a non uniform distribution of heterogeneities on the elastic as well as electrical properties of composites is studied numerically and compared with available theoretical results. Specifically, a random model made of embedded Boolean sets of spherical inclusions (see, e.g., Proc. Eur. Conf. on Constitutive Models for Rubber, ECCMR 2007, Paris, Sept. 4-7) serves as the basis for building simple two-scale microstructures of \granular" type. Materials with infinitely contrasted properties are considered, i.e., inclusions elastically behave as rigid particles or pores, or as perfectly insulating or highly conducting heterogeneities. The inclusion spatial dispersion is controlled by the ratio between the two characteristic lengths of the microstructure. The macroscopic behavior as well as the local response of composites are computed using full-field computations, carried out with the fast fourier transfor method (C. R. Acad. Sci. Paris II, 318: 1417-1423, 1994). The entire range of inclusion concentrations and dispersion ratios up to the separation of length scales are investigated. As expected, the non uniform dispersion of inhomogeneities in multi scale microstructures leads to increased reinforcing or softening effects compared to the corresponding one-scale model (Willot and Jeulin, 2009); these effects are, however, still significantly far apart from Hashin-Shtrikman bounds. Similar conclusions are drawn regarding the electrical conductivity.
Francois
Willot
MINES-ParisTech, Centre de Morphologie Mathématique, Mathématiques et Systémes, France
Dominique
Jeulin
MINES-ParisTech, Centre de Morphologie Mathématique, Mathématiques et Systémes, France
305-326
OVERALL ELASTIC PROPERTIES OF POLYSILICON FILMS: A STATISTICAL INVESTIGATION OF THE EFFECTS OF POLYCRYSTAL MORPHOLOGY
In this paper we investigate the effects of polycrystal morphology on the overall properties of polysilicon. Focusing on two-dimensional representative volume elements (RVEs) of textured films, we numerically generate digital polycrystal morphologies through Voronoi tessellations and assume the in-plane orientation of the crystal lattice of silicon grains to be randomly distributed. First, we show how a regularization provision for the Voronoi tessellations, adopted in order to better match the grain boundary (GB) geometry featured by actual polysilicon films, affects the statistics of an internal length-scale which naturally emerges because of the presence of GBs. Second, we provide a numerical homogenization technique to estimate the overall in-plane elastic moduli of the polysilicon film and compare the outcomes with standard Voigt and Reuss bounds. Through this comparison, we furnish a way to also estimate the size of the RVE to get effective results. Third, through Monte Carlo simulations we investigate the effect of microstructural fluctuations on the scattering of the overall elastic moduli of polysilicon. We show that even when the RVE appears to be representative for a single polycrystal realization, the RVE might not be representative if one looks at the extreme values of the aforementioned scattered elastic moduli.
Stefano
Mariani
Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Italy
Roberto
Martini
Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Italy
Aldo
Ghisi
Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Italy
Alberto
Corigliano
Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Italy
Marco
Beghi
Politecnico di Milano, Dipartimento di Energia, NEMAS-Center for NanoEngineered Materials and Surfaces, Italy
327-346
VARIATIONAL FORMULATION ON EFFECTIVE ELASTIC MODULI OF RANDOMLY CRACKED SOLIDS
Formulation of variational bounds for properties of inhomogeneous media constitutes one of the most fundamental parts of theoretical and applied mechanics. The merit of rigorously derived bounds lies in them not only providing verification for approximation methods, but more importantly, serving as the foundation for building up mechanics models. A direct application of classical micromechanics theories to random cracked media, however, faces a problem of singularity due to a zero volume fraction of cracks. In this study a morphological model of random cracks is first established. Based on the morphological model, a variational formulation of randomly cracked solids is developed by applying the stochastic Hashin-Shtrikman variational principle formulated by Xu (J. Eng. Mech., vol. 135, pp. 1180-1188, 2009) and the Green-function-based method by Xu et al. (Comput. Struct., vol. 87, pp. 1416-1426, 2009). The upper-bound expressions are explicitly given for penny-shaped and slit-like random cracks with parallel and random orientations. Unlike previous works, no special underlying morphology is assumed in the variational formulation, and the bounds obtained are applicable to many realistic non-self-similar morphologies.
X. Frank
Xu
Department of Civil, Environmental and Ocean Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA
George
Stefanou
Institute of Structural Analysis and Antiseismic Research, National Technical University of Athens, Greece
347-363