Effective Elastic and Plastic Properties of Interpenetrating Multiphase Composites

In interpenetrating phase composites, there are at least two phases that are each interconnected in three dimensions, constructing a topologically continuous network throughout the microstructure. The dependence relation between the macroscopically effective properties and the microstructures of interpenetrating phase composites is investigated in this paper. The effective elastic moduli of such kind of composites cannot be calculated from conventional micromechanics methods based on Eshelby's tensor because an interpenetrating phase cannot be extracted as dispersed inclusions. Using the concept of connectivity, a micromechanical cell model is first presented to characterize the complex microstructure and stress transfer features and to estimate the effective elastic moduli of composites reinforced with either dispersed inclusions or interpenetrating networks. The Mori–Tanaka method and the iso-stress and iso-strain assumptions are adopted in an appropriate manner of combination by decomposing the unit cell into parallel and series sub-cells, rendering the calculation of effective moduli quite easy and accurate. This model is also used to determine the elastoplastic constitutive relation of interpenetrating phase composites. Several typical examples are given to illustrate the application of this method. The obtained analytical solutions for both effective elastic moduli and elastoplastic constitutive relations agree well with the finite element results and experimental data.     

Effective elastic moduli of three-phase composites with randomly located and interacting spherical particles of distinct properties

A micromechanical analytical framework is presented to predict effective elastic moduli of three-phase composites containing many randomly dispersed and pairwisely interacting spherical particles. Specifically, the two inhomogeneity phases feature distinct elastic properties. A higher-order structure is proposed based on the probabilistic spatial distribution of spherical particles, the pairwise particle interactions, and the ensemble-volume homogenization method. Two non-equivalent formulations are considered in detail to derive effective elastic moduli with heterogeneous inclusions. As a special case, the effective shear modulus for an incompressible matrix containing randomly dispersed and identical rigid spheres is derived. It is demonstrated that a significant improvement in the singular problem and accuracy is achieved by employing the proposed methodology. Comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered. Moreover, numerical examples are implemented to illustrate the potential of the present method.     

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.     

Micromechanics modeling of viscoelastic properties of hybrid composites with shunted and arbitrarily oriented piezoelectric inclusions

A comprehensive micromechanics model is developed to estimate the effective viscoelastic properties of hybrid composites containing polymer matrix, conductive inclusions and shunted piezoelectric inclusions. The model is derived using the viscoelastic correspondence principle in conjunction with the Mori-Tanaka approach and the orientation averaging scheme. Three dimensional complex moduli are explicitly presented for hybrid composites with any orientation distribution. The model is first validated by comparison with available experimental results. Then, the loss factors are examined for hybrid composites with inclusions of various volume fractions and of shapes ranging from thin disks to long fibers. It is seen that hybrid composites with randomly oriented inclusions exhibit shear loss factors which are not possible with monolithic piezoelectric plate. Furthermore, the numerical results indicate that composites with long spheroid inclusions provide the best damping performance. The results recommend that aligned inclusion composites are good for alleviating longitudinal oscillations. If oscillation energy needs to be dissipated in all directions and for all modes, three dimensional random composites should be used. It is also observed that spherical inclusion composites cannot improve shear damping irrespective of the orientation and the volume fraction. In general, to achieve a pronounced damping piezoelectric inclusions that lie in aspect ratio range 0.1?α?2 should be avoided.     

Elastic Constants of Particle and Fiber Reinforced Metal Matrix Composites

Abstract

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropics between directions of the fiber rich plane and that normal to the plane could be predicted by the model.     

Elastic constants of particle and fiber reinforced metal matrix composites

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropies between directions of the fiber rich plane and that normal to the plane could be predicted by the model.This article is dedicated to Professor Dr. Paul Höller on the occasion of his 65th birthday.     

Calculation of effective moduli of fibrous composites with micro-mechanical damage

A micro-mechanics model for continuous fibrous composites was developed in order to determine the effective moduli of composites based on the material properties of their constituents, i.e. fiber and matrix materials. The model can calculate elastic or nonelastic effective moduli of composites depending on their constituents' behavior. Furthermore, micro-mechanical damage can also be considered in the present model to determine effective moduli. Predicted effective moduli from the present model compared very well with experimental data available elsewhere for both undamaged and damaged composites.     

On composites with periodic structure

The overall moduli of a composite with an isotropic elastic matrix containing periodically distributed (anisotropic) inclusions or voids, can be expressed in terms of several infinite series which only depend on the geometry of the inclusions or voids, and hence can be computed once and for all for given geometries. For solids with periodic structures these infinite series play exactly the same role as does Eshelby's tensor for a single inclusion or void in an unbounded elastic medium.For spherical and circular-cylindrical geometries, the required infinite series are calculated and the results are tabulated. These are then used to estimate the overall elastic moduli when either the overall strains or the overall stresses are prescribed, obtaining the same results. These results are compared with other estimates and with experimental data. It is found that the model of composites with periodic structure yields estimates in excellent agreement with the experimental observations.     

Effect of inclusion shape on the effective elastic moduli for composites with imperfect interface

Summary Bounds on the effective elastic moduli for isotropic composites consisting of randomly oriented spheroidal inclusions with imperfect matrix-inclusion interface are proposed based on Hashin's extremum principle. Phenomenally, these bounds are the first-order ones for such composites, and contain the effect of the size and shape of inclusions, and the elastic properties of constituent phases and interfaces. In the limit cases, these bounds reduce to those known ones. The effect of inclusion shape and interface imperfection on the bounds is discussed with some numerical results for a WC/Co metal-matrix composite.     

Convergence problems in the theory of random elastic media

A main result of the rigorous theory of random, linearly elastic media consists in the representation of the tensor of effective elastic moduli as a Neumann type infinite series which contains the infinite set of correlation functions of the distribution of the local elastic moduli. Under the restriction to statistically homogeneous and isotropic finite media it is proved that convergent series can always be obtained provided the local elastic moduli remain finite everywhere in the medium. This means that the mentioned theory cannot be applied in the above mentioned form to media with pores and/or rigid inclusions. It also means that the theory is not restricted to media with small fluctuations of the elastic parameter.     

Overall elastoplastic property for micropolar composites with randomly oriented ellipsoidal inclusions

Overall linear and non-linear properties for micropolar composites containing 3D and in-plane randomly oriented inclusions are examined with an analytical micromechanical method. This method is based on Eshelby solution for a general ellipsoidal inclusion in a micropolar media and secant moduli method. The influence of inclusion’s shape, size and orientation on the classical effective moduli, yielding surface and non-linear stress and strain relation are examined. The results show that the effective moduli and non-linear stress–strain curves are always higher for micropolar composites than the corresponding classical composites. When the inclusion’s size is sufficiently large, the classical results can be recovered.     

Influence of parallelogram cells in the axial behaviour of fibrous composite

Effective longitudinal shear moduli closed-form analytical expressions of two-phase fibrous periodic composites are obtained by means of the asymptotic homogenization method (AHM) for a parallelogram array of circular cylinders. This work is an extension of previous reported results, where elastic, piezoelectric and magneto-electro-elastic composites for square and hexagonal arrays with perfect contact were considered. The constituents exhibit transversely isotropic properties. A doubly period-parallelogram array of cylindrical inclusions under longitudinal shear is studied. The behaviour of the anisotropic shear elastic coefficients is studied for several cell geometry arrays. Numerical examples and comparisons with other theoretical results demonstrate that the present model is efficient for the analysis of composites in which the periodic cell is rectangular, rhombic or a parallelogram. The effect of the arrangement of the cells on the shear effective property is discussed. The present method can provide benchmark results for other numerical and approximate methods.     

Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities

Summary Based on the general micromechanical framework proposed in a companion paper, effective elastic moduli of two-phase composites containing randomly dispersedspherical inhomogeneities are investigated in this paper. At variance with existing micromechanical pairwise interaction models (accurate up to the second-order in particle volume fraction ), the proposed approximate, probabilistic pairwise particle interaction formulationcoupled with the general ensemble-volume averaged field equations leads to a novel, higher-order (in ), and accurate method for the prediction of effective elastic moduli of two-phase composites containing randomly located spherical particles. The relevant ensemble integrals in the proposed formulation are absolutely convergent due to a renormalization procedure employed in a companion paper. In accordance with the analogy between the effective shear modulus of an incompressible elastic composite with randomly dispersed rigid spheres and the effective shear viscosity of a colloidal dispersion with randomly dispersed rigid spheres (at high shear rates), the proposed ensemble-micromechanical approach is extended to predict effective shear viscosities of colloidal dispersions at the high-shear limit. Comparisons with experimental data, classical variational bounds, improved three-point bounds, the second-order particle interaction model, and other micromechanical models are also presented. It is observed that significant improvement in predictive capability for two-phase composites with randomly dispersed spheres can be achieved by using the proposed method.     

残余应力对复合材料弹2塑性变形的影响

从细观力学的角度给出了分析残余应力对一般复合材料塑性性能影响的一种解析方法, 该方法基于应力二阶矩的割线模量法及Ponte Castaneda 和W illis 给出的弹性细观模型。有残余应力时, 所提的细观解析模型能够同时考虑纤维形状, 体积百分比, 纤维取向及纤维的分布对复合材料变形的影响。计算结果表明, 残余应力的存在会引起复合材料拉压变形的不对称, 材料宏观的拉压硬化曲线又与复合材料的细观结构参数密切相关。对单向复合材料, 本文作者对其等效割线热膨胀系数, 拉压应力-应变曲线的有限元分析结果与给出的细观解析模型定量吻合。      

Composite materials whose phases have partially equal elastic moduli

Hill [J. Mech. Phys. Solids 11 (1963) 357, 12 (1964) 199] discovered that, regardless of its microstructure, a linearly elastic composite of two isotropic phases with identical shear moduli is isotropic and has the effective shear modulus equal to the phase ones. The present work generalizes this result to anisotropic phase composites by showing and exploiting the fact that uniform strain and stress fields exist in every composite whose phases have certain common elastic moduli. Precisely, a coordinate-free condition is given to characterize this specific class of elastic composites; an efficient algebraic method is elaborated to find the uniform strain and stress fields of such a composite and to obtain the structure of the effective elastic moduli in terms of the phase ones; sufficient microstructure-independent conditions are deduced for the orthogonal group symmetry of the effective elastic moduli. These results are applied to elastic composites consisting of isotropic, transversely isotropic and orthotropic phases.     

A generalized plane strain technique for estimating effective properties of particulate metal matrix composites using FEM

A procedure to estimate the effective elastic moduli and coefficient of thermal expansion (CTE) of particulate-reinforced metal matrix composites (MMCs) using a two-dimensional finite element method is presented. The actual microstructural geometry of the composites with randomly distributed second-phase particles is incorporated in the model. A generalized plane strain technique, realistically to describe the three-dimensional behaviour, is also incorporated in the model. The elastic moduli and the CTE, estimated using this model, agree favourably with the experimental data. The technique is shown to be superior compared to the conventional two-dimensional plane stress and plane strain approximations. Also, the results indicate that the effect of the shape of the randomly distributed second-phase particles on the effective elastic moduli is insignificant. Although the procedure is demonstrated for particulate MMCs, it can be easily extended to many other materials as well.     

Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particle interactions

A micromechanical framework is proposed to predict effective elastic moduli of particle-reinforced composites. First, the interacting eigenstrain is derived by making use of the exterior-point Eshelby tensor and the equivalence principle associated with the pairwise particle interactions. Then, the near-field particle interactions are accounted for in the effective elastic moduli of spherical-particle-reinforced composites. On the foundation of the proposed interacting solution, the consistent versus simplified micromechanical field equations are systematically presented and discussed. Specifically, the focus is upon the effective elastic moduli of two-phase composites containing randomly distributed isotropic spherical particles. To demonstrate the predictive capability of the proposed micromechanical framework, comparisons between the theoretical predictions and the available experimental data on effective elastic moduli are rendered. In contrast to higher-order formulations in the literature, the proposed micromechanical formulation can accommodate the anisotropy of reinforcing particles and can be readily extended to multi-phase composites.     

Modeling the elastic properties of concrete composites: Experiment,differential effective medium theory,and numerical simulation

Concrete is a mixture of cement, water and aggregates. In terms of microstructure, besides the cement paste matrix and aggregate inclusions, there is a third phase, which is called the interfacial transition zone (ITZ), which forms due to the wall effect and can be thought of as a thin shell that randomly forms around each aggregate. Thus, concrete can be viewed as a bulk paste matrix containing composite inclusions. To compute the elastic properties of a concrete composite, a differential effective medium theory (D-EMT) is used in this study by assigning elastic moduli to corresponding bulk paste matrix, ITZ and aggregate. In this special D-EMT, each aggregate particle, surrounded by a shell of ITZ of uniform thickness and properties, is mapped onto an effective particle with uniform elastic moduli. The resulting simpler composite, with a bulk paste matrix, is then treated by the usual D-EMT. This study shows that to assure the accuracy of the D-EMT calculation, it is important to consider the increase in the water:cement mass ratio (w/c) of the ITZ and the corresponding decrease in w/c ratio of the bulk matrix. Because of this difference in w/c ratio, the contrast of elastic moduli between the ITZ and the bulk paste matrix needs to be considered as a function of hydration age. The Virtual Cement and Concrete Testing Laboratory (VCCTL) cement hydration module is used to simulate the microstructure of cement paste both inside and outside the ITZ. The redistribution of calcium hydroxide between ITZ and bulk paste regions can further affect the elastic contrast between ITZ and bulk paste. The elastic properties of these two regions are computed with a finite element technique and used as input into the D-EMT calculation. The D-EMT predictions of the elastic properties of concrete composites are compared with the results measured directly with a resonant frequency method on corresponding composites. This comparison shows that the D-EMT predictions agree well with experimental measurements of the elastic properties of a variety of concrete mixtures.     

Fourier series expansion in non-orthogonal coordinate system for the homogenization of linear viscoelastic periodic composites

The Eshelby method and the Fourier series are used in order to determine the linear elastic and viscoelastic properties of composites with periodically distributed inclusions in a non-orthogonal coordinate system. The relaxation moduli provided by the proposed method are compared with the moduli obtained via FEM for a unidirectional composite. This comparison gives good results. The procedure results very useful for periodic composites with hexagonal symmetry, such as some transversely isotropic composites.     

含球夹杂复合材料的力学性能分析

对于复合材料的有效弹性模量,Eshelby[1]的等效夹杂法和Budians-ky和Wu[3]的自相似法仅仅考虑了夹杂的形状及基体和夹杂的力学性能,而忽略了夹杂的大小和相互作用。本文认为当复合材料的夹杂体积分数增大时,夹杂之间的相互作用影响是比较显着的。基于这一事实,本文在考虑夹杂的形状,大小,分布和相互作用前提下推导了材料的有效弹性模量。最后,本文给出了夹杂分布和基体泊松比对复合材料有效弹性模量的影响,并且部分结果与实验进行了比较。从比较的结果来看,本文的结果与实验值吻合的很好。