A variational form of the equivalent inclusion method for numerical homogenization

Due to its relatively low computational cost, the equivalent inclusion method is an attractive alternative to traditional full-field computations of heterogeneous materials formed of simple inhomogeneities (spherical, ellipsoidal) embedded in a homogeneous matrix. The method can be seen as the discretization of the Lippmann–Schwinger equation with piecewise polynomials. Contrary to the original approach of Moschovidis and Mura, who discretized the strong form of the Lippmann–Schwinger equation through Taylor expansions, we propose in the present paper a Galerkin discretization of the weak form of this equation. Combined with the new, mixed boundary conditions recently introduced by the authors, the resulting method is particularly well-suited to homogenization. It is shown that this new, variational approach has a number of benefits: (i) the resulting linear system is well-posed, (ii) the numerical solution converges to the exact solution as the maximum degree of the polynomials tends to infinity and (iii) the method can provide rigorous bounds on the apparent properties of the statistical volume element, provided that the matrix is stiffer (or softer) than all inhomogeneities. This paper presents the formulation and implementation of the new, variational form of the equivalent inclusion method. Its efficiency is investigated through numerical applications in 2D and 3D elasticity.     

Material sensitivity analysis in homogenization of linear elastic composites

Summary  The main goal of the paper is to present theoretical aspects and the finite element method (FEM) implementation of the sensitivity analysis in homogenization of composite materials with linear elastic components, using effective modules approach. The deterministic sensitivity analysis of effective material properties is presented in a general form for an n-components periodic composite, and is illustrated by the examples of 1D as well as of 2D heterogeneous structures. The results of the sensitivity analysis presented in the paper confirm the usefulness of the homogenization method in computational analysis of composite materials the method may be applied to computational optimization of engineering composites, to the shape-sensitivity studies and, after some probabilistic extensions, to stochastic sensitivity analysis of random composites. Received 10 November 2000; accepted for publication 24 April 2001     

Three-dimensional stochastic analysis using a perturbation-based homogenization method for elastic properties of composite material considering microscopic uncertainty

This paper discusses evaluation of influence of microscopic uncertainty on a homogenized macroscopic elastic property of an inhomogeneous material. In order to analyze the influence, the perturbation-based homogenization method is used. A higher order perturbation-based analysis method for investigating stochastic characteristics of a homogenized elastic tensor and an equivalent elastic property of a composite material is formulated.As a numerical example, macroscopic stochastic characteristics such as an expected value or variance, which is caused by microscopic uncertainty in material properties, of a homogenized elastic tensor and homogenized equivalent elastic property of unidirectional fiber reinforced plastic are investigated. The macroscopic stochastic variation caused by microscopic uncertainty in component materials such as Young’s modulus or Poisson’s ratio variation is evaluated using the perturbation-based homogenization method. The numerical results are compared with the results of the Monte-Carlo simulation, validity, effectiveness and a limitation of the perturbation-based homogenization method is investigated. With comparing the results using the first-order perturbation-based method, effectiveness of a higher order perturbation is also investigated.     

Modeling elasto-plastic indentation on layered materials using the equivalent inclusion method

This paper develops a fast semi-analytical model for solving the three-dimensional elasto-plastic contact problems involving layered materials using the Equivalent Inclusion Method (EIM). The analytical elastic solutions of a half-space subjected to a unit surface pressure and a unit subsurface eigenstrain are employed in this model; the topmost layer is simulated by an equivalent inclusion with fictitious eigenstrain. Accumulative plastic deformation is determined by a procedure involving an iterative plasticity loop and an incremental loading process. Algorithms of the fast Fourier transform (FFT) and the Conjugate Gradient Method (CGM) are utilized to improve the computation efficiency. An analytical elastic solution of layered body contact (O’Sullivan and King, 1988) and an indentation experiment result involving a layered substrate (Michler et al., 1999) are used to examine the accuracy of this model. Comparisons between numerical results from the present model and a commercial FEM software (Abaqus) are also presented. Case studies of a rigid ball loaded against a layered elasto-plastic half-space are conducted to explore the effects of the modulus, yield strength, and thickness of the coating on the hardness, stiffness, and plastic deformation of the composite body.     

Numerical analysis of interface fatigue of fiber reinforced composites

A numerical analysis of the interface debonding process is carried out for fiber–matrix composites under cyclic loading. The results show that the interface friction plays an important role in resisting debonding under fatigue. The degradation coefficient of interface friction influences the debonding velocity. An approximate formula of interface friction is found for fatigue debonding.     

The symplectic system method in the stress analysis of 2D elasto-viscoelastic fiber reinforced composites

With the aid of the variational method and Laplace transformation, the symplectic system method is employed into two-dimensional elastic–viscoelastic fiber reinforced composites. The fundamental eigenfunctions of the governing equations are generalized to the time domain. Therefore the problem can be discussed directly in the time domain, and the iterative application of Laplace transformation is not needed. Using this method, all the stress components of the inner fiber and outer matrix, and hence the stress transfer in the interface between the fiber and matrix, are expressed analytically. The results obtained by the approach are accurate, because all the boundary conditions prescribed on the surfaces and ends of the composites can be satisfied. Numerical example demonstrate that both the shear stress and the normal stress decrease with time due to the viscoelastic property of the matrix, and that stress concentration occurs near the end.     

The homogenization method for the study of variation of Poisson's ratio in fiber composites

Summary Materials with specific microstructural characteristics and composite structures are able to exhibit negative Poisson's ratio. This fact has been shown to be valid for certain mechanisms, composites with voids and frameworks and has recently been verified for microstructures optimally designed by the homogenization approach. For microstructures composed of beams, it has been postulated that nonconvex shapes (with reentrant corners) are responsible for this effect. In this paper, it is numerically shown that mainly the shape, but also the ratio of shear-to-bending rigidity of the beams do influence the apparent (phenomenological) Poisson's ratio. The same is valid for continua with voids, or for composites with irregular shapes of inclusions, even if the constituents are quite usual materials, provided that their porosity is strongly manifested. Elements of the numerical homogenization theory and first attempts towards an optimal design theory are presented in this paper and applied for a numerical investigation of such types of materials. Received 11 March 1997; accepted for publication 12 September 1997     

Numerical homogenization of elastic and thermal material properties for metal matrix composites (MMC)

A two-scale material modeling approach is adopted in order to determine macroscopic thermal and elastic constitutive laws and the respective parameters for metal matrix composite (MMC). Since the common homogenization framework violates the thermodynamical consistency for non-constant temperature fields, i.e., the dissipation is not conserved through the scale transition, the respective error is calculated numerically in order to prove the applicability of the homogenization method. The thermomechanical homogenization is applied to compute the macroscopic mass density, thermal expansion, elasticity, heat capacity and thermal conductivity for two specific MMCs, i.e., aluminum alloy Al2024 reinforced with 17 or 30 % silicon carbide particles. The temperature dependency of the material properties has been considered in the range from 0 to \(500{\,}^\circ \mathrm {C}\), the melting temperature of the alloy. The numerically determined material properties are validated with experimental data from the literature as far as possible.     

On the equivalent inclusion method and impotent eigenstrains

The equivalency equations and the nature of the solution are investigated when an inhomogeneity under applied stresses is simulated by an inclusion with eigenstrains. The equivalency equations by which the equivalent eigenstrain is obtained becomes singular when the inhomogeneity is void and the applied stress has a form of polynomials of coordinates of degree one. The solutions of the system of the equivalency equations are not uniquely determined. Explicit expressions are given for the impotent eigenstrains which do not generate any stress field throughout a material.This research was supported by the U.S. Army Research Office under Grant No. DAAG 29-77-G-0042 to Northwestern University.On leave of absence of Meiji University, Tokyo.     

Effective elastic moduli of composites reinforced by particle or fiber with an inhomogeneous interphase

The solution of the strain energy change of an infinite matrix due to the presence of one spherical particle or cylindrical fiber surrounded by an inhomogeneous interphase is the basis of solving effective elastic moduli of corresponding composites based on various micromechanics models. In order to find out the strain energy change, the composite sphere or cylinder, i.e., the spherical particle or cylindrical fiber together with its interphase, is replaced by an effective homogeneous particle or fiber. Independent governing differential equations for each modulus of the effective particle or fiber are derived by extending the replacement method [J. Mech. Phys. Solids 12 (1964) 199]. As far as the strain energy changes of the infinite matrix subjected to various far-field stress systems are concerned, the present model is simple. Meanwhile, FEM analysis is carried out for a verification, which shows that the model can lead to rather accurate results for most practical interphases. Besides, to check the validity of the model further when the interactions among composite cylinders exist, the two problems of an infinite matrix containing two composite cylinders and the effective moduli of composites with the equilateral triangular distribution of composite cylinders are analyzed using FEM. The FEM results show that the model is still rather accurate, especially for the case of interphase properties varying between those of fiber and matrix. Therefore, composite spheres or cylinders are assumed as the effective homogeneous particles or fibers and simple expressions of the effective moduli of composites containing the composite spheres or cylinders are obtained. Furthermore, the present model is compared with some existing models that are based on very complicated derivations.     

Homogenization of elasto-plastic composites coupled with a nonlinear finite element analysis of the equivalent inclusion problem

This paper deals with the micromechanical modeling of particle reinforced elasto-plastic composites under general non-monotonic loading histories. Incremental mean-field (MF) homogenization models offer an excellent cost-effective solution, however there are cases where their predictions are inaccurate. Here, we assess the applicability of the equivalent inclusion representation, which sustains many homogenization schemes. To this end, MF models are fully coupled with a finite element (FE) solution of the equivalent inclusion problem (EIP). Consequently, Eshelby’s tensor is not used and most (but not all) approximations involved in the generalization of MF models from linear elasticity to the nonlinear regime are avoided. The proposal is implemented for Mori-Tanaka (M-T) and dilute inclusion models and applied to several composite systems with elasto-plastic matrix and spherical or ellipsoidal particles, subjected to various loadings (tension, plane strain, cyclic tension/compression). The predictions are verified against reference full-field FE simulations of multiparticle cells. Results show that the M-T model coupled with the nonlinear FE solution of the EIP is very accurate at the macro level up to 25% volume fraction of reinforcement, while the phase averages remain accurate as long as the volume fraction does not exceed 15%. The strain concentration tensor computed almost exactly from single inclusion FE analysis is compared against approximate expressions assumed by classical MF models. Implications for the development of advanced MF homogenization models are discussed.     

Effect of interphase layers on the overall elastic and conductive properties of matrix composites. Applications to nanosize inclusion

For a composite with thin interface layers between inclusions and the matrix, the effective elastic properties and the effective conductivity (thermal or electric) are almost unaffected by the layers, provided (1) the layer thickness is much smaller than the inclusion sizes and (2) the contrast between the properties of the layers and either of the phases is not overly high. For composites with nanoparticles, the interface thickness may be comparable to the particle sizes. Therefore, the effect of interfaces on the overall properties may be substantial. The controlling parameters are (1) the ratio of the interface thickness to particle sizes and (2) variability of the properties across the interface thickness. Explicit expressions constructed in the present work show that the overall elastic/conductive properties are affected, mostly, by the interface thickness (normalized to the size of the core particle) and are much less sensitive to the extent of the variation and its exact character. Similarities and differences between the elasticity and the conductivity problems are discussed.     

Smart damping of laminated thin cylindrical panels using piezoelectric fiber reinforced composites

In this paper performance of a new piezoelectric fiber reinforced composite (PFRC) material has been investigated for active constrained layer damping (ACLD) of laminated thin simply supported composite cylindrical panels. The constraining layer of the ACLD treatment has been considered to be made of this PFRC material. A finite element model of smart composite panels integrated with the patches of such ACLD treatment has been developed to demonstrate the performance of these patches on enhancing the damping characteristics of thin cross-ply and angle-ply laminated composite cylindrical panels. Particular emphasis has been placed on studying the effect of variation of the piezoelectric fiber orientation in the constraining PFRC layer and the shallowness angle of the panels on the control authority of the patches.     

Interacting cracks and ellipsoidal inhomogeneities by the equivalent inclusion method

Based on the Eshelby's equivalent inclusion method (EIM) and Hill's theorem on discontinuities of elastic fields across the interfaces, a theory for the determination of the stress intensity factors (SIFs) of arbitrarily oriented interacting cracks under non-uniform far-field applied stress (strain) is developed. As shown in this investigation the EIM proposed by Moschovidis and Mura can be extended for treatment of such problems, but their formulations are quite cumbersome and computationally inefficient. An alternative analytical approach is proposed that is computationally more efficient, and unlike the method of Moschovidis and Mura can easily handle complex problems of interacting inhomogeneities and cracks. It is seen that as the interaction between the inhomogeneities becomes stronger, this method yields results that are closer to the solutions reported in the literature than the solutions obtained using the extended EIM of Moschovidis and Mura, which is developed herein. Problems involving combinations of interacting elliptic and penny shape cracks and inhomogeneities are excellent candidates for demonstration of the accuracy and robustness of the present theory, for which the previous EIM produces less accurate results. Due to the limitations imposed on the existing methods, every reported treatment has been tailored for a certain category of problems, and only uniform far-field loadings have been remedied. In contrast, the present theory is more general than the previously reported theories and it encompasses interacting cracks having a variety of geometries subjected to non-uniform far-field applied stress (strain); moreover, it is applicable to modes, I, II, III, and mixed mode fracture.     

Numerical evaluation of the equivalent properties of Macro Fiber Composite (MFC) transducers using periodic homogenization

This paper focuses on the evaluation of the homogeneous properties of the active layer in Macro Fiber Composite (MFC) transducers using finite element periodic homogenization. The proposed method is applied to both d₃₁ and d₃₃-MFCs and the results are compared to previously published analytical mixing rules, showing a good agreement. The main advantages of the finite element homogenization is the possibility to take into account local details in the representative volume element such as complicated electrode patterns or local variations of the poling direction due to curved electric field lines. Although these influences have been found to be rather small in the present study, the method presented is useful for a better understanding of the behavior of piezocomposite transducers.     

Bounds on the elastic and transport properties of two-component composites

Weshow in detail how the McCoy bounds on the effective shear modulus of a statistically isotropic composite, can be simplified and expressed in terms of the volume fraction, f₁, and two geometric parameters. ζ₁ and η₁. We simplify Silnutzer's bounds on the effective elastic moduli of fibre-reinforced composites and find they can be expressed in terms f₁and two geometric parameters, ζ'₁ and η'₁. The parameter ζ'₁ also determines bounds on the transport and optical constants of such composites. Also, the Elsayed-McCoy bounds on the transport properties of fibre-reinforced, symmetric-cell materials are shown to depend on three geometric parameters.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

Effective medium method in the problem of axial elastic shear wave propagation through fiber composites

The effective medium method (EMM) is applied to the solution of the problem of monochromatic elastic shear wave propagation through matrix composite materials reinforced with cylindrical unidirected fibers. The dispersion equations for the wave numbers of the mean wave field in such composites are derived using two different versions of the EMM. Asymptotic solutions of these equations in the long and short wave regions are found in closed analytical forms. Numerical solutions of the dispersion equations are constructed in a wide region of frequencies of the incident field that covers long, middle and short wave regions of the mean wave field. Velocities and attenuation factors of the mean wave fields in the composites obtained by different versions of the EMM are compared for various volume concentrations and properties of the inclusions. The main discrepancies in the predictions of different versions of the EMM are indicated, analyzed and discussed.     

The nonlinear analysis of elastic wave of piezoelectric crystal plate with perturbation method

When investigating or designing acoustic wave sensors, the behavior of piezoelectric devices is supposed to be linear. However, if the sensors are subjected to a strong elastic field, the amplitude of the elastic strain induced in the piezoelectric material is so large that the nonlinearity, which affects the stability and performance of the piezoelectric sensors, can no longer be ignored. In this paper, we perform a theoretical analysis on nonlinear anti-symmetric thickness vibration of thin-film acoustic wave resonators made from quartz. Using Green’s identity, under the usual approximation of neglecting higher time harmonics, a perturbation analysis is performed from which the resonator frequency–amplitude (A–F) relation is obtained. Numerical calculations are made. Furthermore, the validity of the method is examined.     

Analysis of temperature and impact damage of fiber reinforced composites

Fracture resistance of fiber reinforced composites with polymer matrix is examined owing to impact loading and heat treatment over a wide range of temperatures. Test data are presented for the nonhomogeneous peeling behavior of layered composite specimen damaged by impact. Analytical results are also given for the peeling growth rate under the action of edge peel off and transverse shear.