On the effective elastic properties of matrix composites: Combining the effective field method and numerical solutions for cell elements with multiple inhomogeneities

The work is devoted to calculation of effective elastic constants of homogeneous materials containing random or regular sets of isolated inclusions. Our approach combines the self-consistent effective field method with the numerical solution of the elasticity problem for a typical cell. The method also allows analysis of detailed elastic fields in the composites. By the numerical solution of the elasticity problem for a cell, integral equations for the stress field are used. Discretization of these equations is carried out by Gaussian approximating functions. For such functions, elements of the matrix of the discretized problem are calculated in explicit analytical forms. If the lattice of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure. The matrix-vector products with such matrices may be carried out by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Results are given for 2D-media with regular and random sets of circular inclusions, and compared with existing exact solutions.     

Calculation of electro and thermo static fields in matrix composite materials of regular or random microstructures

In the work, a numerical method for calculation of electro and thermo static fields in matrix composite materials is considered. Such materials consist of a regular or random set of isolated inclusions embedded in a homogeneous background medium (matrix). The proposed method is based on fast calculation of fields in a homogeneous medium containing a finite number of isolated inclusions. By the solution of this problem, the volume integral equations for the fields in heterogeneous media are used. Discretization of these equations is carried out by Gaussian approximating functions that allow calculating the elements of the matrix of the discretized problem in explicit analytical forms. If the grid of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure, and the matrix-vector product with such matrices can be calculated by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. In the case of an infinite medium containing a homogeneous in space random set of inclusions, our approach combines a self-consistent effective field method with the numerical solution of the conductivity problem for a typical cell. The method allows constructing detailed static (electric or temperature) fields in the composites with inclusions of arbitrary shapes and calculating effective conductivity coefficients of the composites. Results are given for 2D and 3D-composites and compared with the existing exact and numerical solutions.     

Effective conductive and dielectric properties of matrix composites with inclusions of arbitrary shapes

The effective field method is applied to the calculation of overall dielectric permittivities, and electro- or thermo-conductivities of composite materials consisting of a homogeneous matrix and a set of isolated inclusions. The problem is reduced to the solution of the one particle problem for a typical inclusion subjected to a constant external field. An original numerical method is proposed for the solution of the one particle problem for an inclusion of an arbitrary shape. As an example, the effective dielectric properties of composites with cylindrical inclusions of various sizes and properties are calculated.     

Fast solution of the elasticity problem for a planar crack of arbitrary shape in 3D-anisotropic medium

A planar crack of arbitrary shape in a 3D-anisotropic elastic medium subjected to an arbitrary external stress field is considered. An efficient numerical method of the solution of the problem is proposed. The problem is reduced to an integral equation for the crack opening vector on the crack surface. For discretization of this equation, Gaussian (radial) approximation functions centered at a system of nodes that covers the crack surface are used. For such functions, the elements of the matrix of the discretized problem are calculated in a quasi analytical form that involves standard non-singular integrals. If the node grid is regular, the matrix of the discretized system has Teoplitz’s structure, and the Fast Fourier Transform algorithm may be used for the calculation of matrix-vector products with such a matrix. It accelerate substantially the process of the iterative solution of the discretized system. Examples of the solutions for a circular crack in a transversally isotropic elastic medium are presented.     

Eigenstrain and Fourier series for evaluation of elastic local fields and effective properties of periodic composites

The elastic stress and strain fields and effective elasticity of periodic composite materials are determined by imposing a periodic eigenstrain on a homogeneous solid, which is constrained to be equivalent to the heterogeneous composite material through the imposition of a consistency condition. To this end, the variables of the problem are represented by Fourier series and the consistency condition is written in the Fourier space providing the system of equations to solve. The proposed method can be considered versatile as it allows determining stress and strain fields in micro-scale and overall properties of composites with different kinds of inclusions and defects. In the present work, the method is applied to multi-phase composites containing long fibers with circular transverse section. Numerical solutions provided by the proposed method are compared with finite element results for both unit cell containing a single fiber and unit cell with multiple fibers of different sizes.     

Analysis of effective properties of electroelastic composites using the self-consistent and asymptotic homogenization methods

The problem of the calculation of overall properties of a binary electroelastic fibre-reinforced composite is studied here, whose constituents are an elastic matrix and piezoelectric fibres that have transversely isotropic properties. Randomly positioned fibres as well as periodic distributed fibres are considered. The former case is dealt with by means of the self-consistent method and the latter one by asymptotic homogenization. Closed-form expressions are given for two variants of the self-consistent method, one in explicit form (effective field) and the other implicitly (effective medium). The former agree with the Mori-Tanaka equations. The equations derived using the asymptotic homogenization method are also explicit. It is shown that the three sets of effective coefficients satisfy analytically Schulgasser’s universal relations; the Milgrom-Shtrikman determinant is also explicitly satisfied by the effective field method variant. Overall properties are computed as a function of the fibre concentration. It is generally found that the properties calculated using the effective field self-consistent and homogenization methods are very close to each other for at least concentrations up to or near 0.3. In many cases the agreement is beyond that. Also the case when the constituents have either the same or opposite poling directions can be studied with the exact formulae. The antiplane strain related properties display interesting larger effects with opposite polings.     

Evaluation of effective elastic properties of 3D printable interpenetrating phase composites using the meshfree radial point interpolation method

ABSTRACT

Interpenetrating phase composites (IPCs) have recently been fabricated using three-dimensional (3D) printing methods. In a two-phase IPC, the two phases are topologically interconnected and mutually reinforced in the three dimensions. As a result, such IPCs exhibit higher stiffness, strength, and toughness than particle- or fiber-reinforced composites. In the current study, three unit cell models for the IPCs with the simple cubic (SC), face-centered cubic (FCC), and body-centered cubic (BCC) microstructures are developed using the meshfree radial point interpolation method. Radial basis functions with polynomial reproduction are applied to construct shape functions, and the Galerkin method is employed to formulate discretized equations. These unit cell-based meshfree models are used to evaluate effective elastic properties of 3D printable IPCs. The simulation results are compared with those based on the finite element (FE) method and various analytical bounding techniques in micromechanics, including the Voigt–Reuss, Hashin–Shtrikman, and Tuchinskii bounds. It is found that all of the simulation results for the effective Young's modulus and shear modulus fall between the Voigt–Reuss upper and lower bounds for each IPC considered, with the FE models predicting higher values than the meshfree models. In addition, it is seen that the SC microstructure leads to higher effective Young's modulus than the BCC and FCC microstructures. Furthermore, the numerical results reveal that the IPCs with the SC, BCC, and FCC microstructures can be approximated as isotropic materials (with the Zener anisotropic ratio varying between 0.9 and 1.0), with the BCC IPC being the most isotropic one, and the SC IPC being the least isotropic one among the three types of IPCs.     

A polarization‐based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast

It is recognized that the convergence of FFT‐based iterative schemes used for computing the effective properties of elastic composite materials drastically depends on the contrast between the phases. Particularly, the rate of convergence of the strain‐based iterative scheme strongly decreases when the composites contain very stiff inclusions and the method diverges in the case of rigid inclusions. Reversely, the stress‐based iterative scheme converges rapidly in the case of composites with very stiff or rigid inclusions but leads to low convergence rates when soft inclusions are considered and to divergence for composites containing voids. It follows that the computation of effective properties is costly when the heterogeneous medium contains simultaneously soft and stiff phases. Particularly, the problem of composites containing voids and rigid inclusions cannot be solved by the strain or the stress‐based approaches. In this paper, we propose a new polarization‐based iterative scheme for computing the macroscopic properties of elastic composites with an arbitrary contrast which is nearly as simple as the basic schemes (strain and stress‐based) but which has the ability to compute the overall properties of multiphase composites with arbitrary elastic moduli, as illustrated through several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical investigation of the connection between state of dispersion and percolation and its effect on the elastic properties of 2D random microstructures

The present work is dedicated to a numerical investigation of the connection between state of dispersion and percolation and its effect on the elastic properties of 2D random microstructures. The main objective consists in checking out the link between percolation and mechanical response in the context of a heterogeneous medium the reinforcements of which are not homogeneously dispersed. Besides, the influence of the stiffness of inclusions is also investigated since this could impact on the percolation effects. For these purposes, large samples of volume elements are generated according to the Monte Carlo method. We consider the low cost framework of 2D random grids which enables large and in-depth investigations. Besides, the spatial distribution of heterogeneities is simulated with the help of the 2-scale Boolean scheme of disks which is a powerful tool for modelling and studying several states of dispersion. The numerical results highlight beneficial mechanical reinforcements for a heterogeneous dispersion when the percolation phenomenon is enhanced. This improvement is highly sensitive to the stiffness of heterogeneities.