Effective elastic moduli of three-phase composites with randomly located and interacting spherical particles of distinct properties (original) (raw)
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International Journal of Engineering Science, 2011
The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix-vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.
Elastic property of multiphase composites with random microstructures
Journal of Computational Physics, 2009
We propose a computational method with no ad hoc empirical parameters to determine the elastic properties of multiphase composites of complex geometries by numerically solving the stress-strain relationships in heterogeneous materials. First the random microstructure of the multiphase composites is reproduced in our model by the random generation-growth method. Then a high-efficiency lattice Boltzmann method is employed to solve the governing equation on the multiphase microstructures. After validated against a few standard solutions for simple geometries, the present method is used to predict the effective elastic properties of real multiphase composites. The comparisons between the predictions and the existing experimental data have shown that the effects of pores/ voids in composites are not negligible despite their seemingly tiny amounts. Ignorance of such effects will lead to over-predictions of the effective elastic properties compared with the experimental measurements. When the pores are taken into account and treated as a separate phase, the predicted Young's modulus, shear modulus and Poisson's ratio agree well with the available experimental data. The present method provides an alternative tool for analysis, design and optimization of multiphase composite materials. Published by Elsevier Inc.
Composite Structures, 2007
The aim of presenting this paper is to evaluate the effective material properties of spherical particle reinforced composites for different volume fractions up to 60%. A numerical homogenization technique based on the finite element method (FEM) with representative volume element (RVE) was used to evaluate the effective material properties with periodic boundary conditions. The numerical approach is based on the FEM and it allows the extension of the composites with arbitrary geometrical inclusion configurations, providing a powerful tool for fast calculation of their effective material properties. Modified random sequential adsorption algorithm (RSA) was used to generate the three-dimensional RVE models of randomly distributed spherical particles. The effective material properties obtained using the numerical homogenization techniques were compared with different analytical methods and good agreement was achieved. Several investigations had been conducted to estimate the influence of the size of spherical particles and of the RVE on effective material properties of spherical particle reinforced composites.
Zeitschrift für angewandte Mathematik und Physik, 2006
We consider a transversal loading of a linearly elastic isotropic media containing the identical isotropic aligned circular fibers at non-dilute concentration c. By the use of solution obtained by the Kolosov-Muskhelishvili complex potential method for two interacting circles subjected to three different applied stresses at infinity, and exact integral representations for both the stress and strain distributions in a microinhomogeneous medium, one estimates the effective moduli of the composite accurately to order c 2 .
A Micropolar Homogenization Approach for Random Particle-Based Composites
2016
This study presents a multiscale procedure for determining the size of the Representative Volume Element (RVE) and the homogenized moduli of particle-based composite materials, modeled as micropolar continua. The homogenization, consistent with a generalized Hill-Mandel condition, is adopted in conjunction with a statistical procedure, by which two hierarchies of scale-dependent bounds on classical and micropolar constitutive moduli are obtained using Dirichlet and Neumann BCs. Two different types of inclusion, either stiffer or softer than the matrix, are considered in the numerical applications. The results highlight the importance of accounting for micropolar bending deformation modes, spatial randomness of the medium, and presence of inclusions crossing the edges of the test window used in the homogenization. Sommario. Questo studio presenta una procedura multiscala per la determinazione della dimensione dell’Elemento di Volume Rappresentativo e dei moduli omogeneizzati di mater...
Random Stiffness Tensor of Particulate Composites with Hyper-Elastic Matrix and Imperfect Interface
Materials
The main aim of this study is determination of the basic probabilistic characteristics of the effective stiffness for inelastic particulate composites with spherical reinforcement and an uncertain Gaussian volume fraction of the interphase defects. This is determined using a homogenization method with a cubic single-particle representative volume element (RVE) of such a composite and the finite element method solution. A reinforcing particle is spherical, located centrally in the RVE, surrounded by the thin interphase of constant thickness, and remains in an elastic reversible regime opposite to the matrix, which is hyper-elastic. The interphase defects are represented as semi-spherical voids, which are placed on the outer surface of this particle. The interphase is modeled as hyper-elastic and isotropic, whose effective stiffness is calculated by the spatial averaging of hyper-elastic parameters of the matrix and of the defects. A constitutive relation of the matrix is recovered ex...
Computational homogenization of elastic–plastic composites
International Journal of Solids and Structures, 2013
This work describes a computational homogenization methodology to estimate the effective elastic-plastic response of random two-phase composite media. It is based on finite element simulations using threedimensional cubic cells of different size but smaller than the deterministic representative volume element (DRVE) of the microstructure. We propose to extend the approach developed in the case of elastic heterogeneous media by Drugan and Willis (1996) and Kanit et al. (2003) to elastic-plastic composites. A specific polymer blend, made of two phases with highly contrasted properties, is selected to illustrate this approach; it consists of a random dispersion of elastic rubber spheres in an elastic-plastic glassy polymer matrix. It is found that the effective elastic-plastic response of this particulate composite can be accurately determined by computing a sufficient number of small subvolumes of fixed size extracted from the DRVE and containing different realizations of the random microstructure. In addition, the response of an individual subvolume is found anisotropic whereas the average of all subvolumes leads to recover the isotropic character of the DRVE. The necessary realization number to reach acceptable precision is given for two examples of particle volume fractions.
Journal of Mathematical Chemistry
This paper focuses on the development of a surrogate model to predict the macroscopic elastic properties of polymer composites doped with spherical particles. To this aim, a polynomial chaos expansion based Kriging metamodeling technique has been developed. The training experimental design is constructed through a dataset of numerical representative volume elements (RVEs) considering randomly dispersed spherical particles. The RVEs are discretized using finite elements, and the effective elastic properties are obtained by implementing periodic boundary conditions. Parametric analyses are reported to assess the convergence of the scale of the RVE and the mesh density. The accuracy of the proposed metamodelling approach to bypass the computationally expensive numerical homogenization has been evaluated through different metrics. Overall, the presented results evidence the efficiency of the proposed surrogate modelling, enabling the implementation of computationally intensive technique...
International Journal for Multiscale Computational Engineering, 2004
In this work, a 3D finite element model has been developed to compute the macroscopic elastic properties of polymer/clay nanocomposites (PCNs) from the microstructure morphologies and the elastic behaviour of each phase. Microstructural parameters of clay or clay stacks such as elastic properties, aspect ratio, interlayer spacing and clay volume fraction have been taken into account in the proposed models. A parametric study of the effect of these parameters on the macroscopic elastic properties of PCNs has been carefully investigated. The results show that the macroscopic rigidity of PCNs materials depends not only on the clay volume fraction but also on the dispersion state of clay platelets in the polymer matrix. An exfoliated structure may improve the macroscopic rigidity of PCNs much more efficiently than intercalated ones, particularly at high volume fraction of clays. The key role of interphase on the mechanical properties enhancement of PCNs has also been demonstrated. In addition, the partially exfoliated morphology, in which individual clay layers and intercalated blocks are simultaneously present in the polymer matrix, has been studied. The last morphology is commonly encountered in PCN processing, especially when high clay content is used. The comparison with the experimental and theoretical results extracted from the literature has been performed.