The effective properties of piezocomposites, part II: The effective electroelastic moduli

Since piezoelectric ceramic/polymer composites have been widely used as smart materials and smart structures, it is more and more important to obtain the closed-from solutions of the effective properties of piezocomposites with piezoelectric ellipsoidal inclusions. Based on the closed-from solutions of the electroelastic Eshelby's tensors obtained in the part I of this paper and the generalized Budiansky's energy-equivalence framework, the closed-form general relations of effective electroelastic moduli of the piezocomposites with piezoelectric ellipsoidal inclusions are given. The relations can be applicable for several micromechanics models, such as the dilute solution and the Mori-Tanaka's method. The difference among the various models is shown to be the way in which the average strain and the average electric field of the inclusion phase are evaluated. Comparison between predicted and experimental results shows that the theoretical values in this paper agree quite well with the experimental results. These expression can be readily utilized in analysis and design of piezocomposites. The project supported by the National Natural Science Foundation of China     

An explicit expression of the effective moduli for composite materials filled with coated inclusions

The obvious shortcoming of the generalized self-consistent method (GSCM) is that the effective shear modulus of composite materials estimated by the method can not be expressed in an explicit form. This is inconvenient in engineering applications. In order to overcome that shortcoming of GSCM, a reformation of GSCM is made and a new micromechanical scheme is suggested in this paper. By means of this new scheme, both the effective bulk and shear moduli of an inclusion-matrix composite material can be obtained and be expressed in simple explicit forms. A comparison with the existing models and the rigorous Hashin-Shtrikman bounds demonstrates that the present scheme is accurate. By a two-step homogenization technique from the present new scheme, the effective moduli of the composite materials with coated spherical inclusions are obtained and can also be expressed in an explicit form. The comparison with the existing theoretical and experimental results shows that the present solutions are satisfactory. Moreover, a quantitative comparison of GSCM and the Mori-Tanaka method (MTM) is made based on a unified scheme. The project supported by the National Natural Science Foundation of China under the Contract NO. 19632030 and 19572008, and China Postdoctoral Science Foundation     

A micromechanics constitutive theory for forward transformation plasticity with shear and dilatation effect: I,nonproportional loading history

A micromechanics constitutive theory which takes into account both the dilatation and shear effects of the transformation is proposed to describe the macroscopic plastic behavior of structure ceramics during forward transformation under different temperatures. Under some basic assumptions, the analytic expressions of the Helmholtz and complementary free energy of the constitutive element are derived in a self-consistent manner by using the Mori-Tanaka's method which takes into account the interaction between the transformed inclusions. In the framework of Hill-Rice's internal variable constitutive theory, the forward transformation yield function and incremental stress strain relations, in analogy to the theory of metal plasticity, for non-proportional loading histories are obtained. The project supported by National Natural Science Foundation of China     

A UNIFIED ENERGY APPROACH TO A CLASS OF MICROMECHANICS MODELS FOR MICROCRACKED SOLIDS

Micromechanics models have been developed for the determination ofthe elastic moduli of microcracked solids based on different approaches andinterpretations,including the dilute or non-interacting solution,the Mori-Tanakamethod,the self-consistent method,and the generalized self-consistent method.It isshown in the present study that all these micromechanics models can be unified withinan energy-equivalence framework,and that they differ only in the way in which themicrocrack opening and sliding displacements are evaluated.Relevance to thedifferential methods and the verification of these models are discussed.     

水泥砂浆界面相弹性常数的反演计算

为了计算水泥砂浆界面过渡区的弹性常数,采用广义自洽方法(GSCM),根据水泥砂浆内部的微细观结构,建立了由水泥浆基体、岩石离散夹杂(骨料)、界面过渡区(ITZ)和有效弹性材料组成的四相复合材料模型.推导了界面相的体积模量和剪切模量方程.利用已知水泥砂浆材料的实验数据,计算了界面相的弹性常数.发现界面相剪切模量约为水泥浆基体剪切模量的50%.     

反平面剪切作用下双材料滑动界面的细观力学模型

非理想粘结界面对多相材料力学性能具有重要影响。对于双材料间含众多随机分布微裂纹的界面，宏观上可以等效为连续损伤的弱界面，其两侧的面力连续而位移有间断。只有切线方向的位移间断，而法线方向位移连续的弱界面称之为滑动界面。在反平面剪切的作用下，我们证明了对于含有随机分布微裂纹的弹性双材料界面在宏观上等效为线弹簧型滑动界面，并获得了滑动界面柔度的一般表达式。利用Mori—Tanaka方法和广义自洽方法，我们研究了滑动界面柔度系数和微裂纹密度的关系。对这两种方法所得的结果进行比较发现，Mori—Tanaka方法得到的界面柔度比广义自洽方法得到的界面柔度大。当裂纹密度比较小时，这两种方法求得的界面柔度很接近。两种方法的结果都表明，界面柔度随裂纹密度的增加而增加。Mori—Tanaka方法比广义自治方法求解更为简便。     

Effective electroelastic constants for three-phase confocal elliptical cylinder model in piezoelectric quasicrystal composites

A three-phase confocal elliptical cylinder model is proposed to analyze micromechanics of one-dimensional hexagonal piezoelectric quasicrystal (PQC) composites. Exact solutions of the phonon, phason, and electric fields are obtained by using the conformal mapping combined with the Laurent expansion technique when the model is subject to far-field anti-plane mechanical and in-plane electric loadings. The effective electroelastic constants of several different composites made up of PQC, quasicrystal (QC), and piezoelectric (PE) materials are predicted by the generalized self-consistent method. Numerical examples are conducted to show the effects of the volume fraction and the cross-sectional shape of inclusion (or fiber) on the effective electroelastic constants of these composites. Compared with other micromechanical methods, the generalized selfconsistent and Mori-Tanaka methods can predict the effective electroelastic constants of the composites consistently.     

广义自洽Mori-Tanaka模型及涂层夹杂体复合材料的有效模量

经典广义自洽模型的最大不足是需要确定相材料的位移及应变场,这一过程十分繁杂,而且最后得到的有交剪切模量无法显式表达难以应用,该文摈弃这一经典做法,而从广义自洽模型的应变等价条件出发,在夹杂应变均匀的近似假定下,将Ｈｉｌｌ界面条件应用于整个二相体内,从而得到一种可以预报涂层夹杂体复合材料有效模量的广义自洽Ｍｏｒｉ－Ｔａｎａｋａ模型,与已有的实验及理论的比较表明,该模型准确可靠,而且有效体积和剪切模量     

Micromechanics predictions of the effective moduli of magnetoelectroelastic composite materials

In this paper, the self-consistent, generalized Mori–Tanaka and dilute micromechanics theories are extended to study the coupled magnetoelectroelastic composite materials. The heterogeneous inclusion problem of magnetoelectroelastic behavior is formulated in terms of five interaction tensors related to the Green's functions for an infinite three-dimensional transversely isotropic magnetoelectroelastic solid. These tensors are then used to predict the effective moduli of the magnetoelectroelastic solid based on the self-consistent, Mori–Tanaka and the dilute approaches. Numerical results are obtained for various types of inclusions. These results are employed to study the effects of the inclusion properties, such as moduli, volume fractions, shapes, etc., on the effective moduli of magnetoelectroelastic composites, in particular, the related magnetic properties. The results obtained using the self-consistent model, the generalized Mori–Tanaka's model and the dilute approach are compared with the existing experimental and theoretical results.     

A generalized self-consistent method accounting for fiber section shape

A three-phase confocal elliptical cylinder model is proposed for fiber-reinforced composites, in terms of which a generalized self-consistent method is developed for fiber-reinforced composites accounting for variations in fiber section shapes and randomness in fiber section orientation. The reasonableness of the fiber distribution function in the present model is shown. The dilute, self-consistent, differential and Mori–Tanaka methods are also extended to consider randomness in fiber section orientation in a statistical sense. A full comparison is made between various micromechanics methods and with the Hashin and Shtrikman’s bounds. The present method provides convergent and reasonable results for a full range of variations in fiber section shapes (from circular fibers to ribbons), for a complete spectrum of the fiber volume fraction (from 0 to 1, and the latter limit shows the correct asymptotic behavior in the fully packed case) and for extreme types of the inclusion phases (from voids to rigid inclusions). A very different dependence of the five effective moduli on fiber section shapes is theoretically predicted, and it provides a reasonable explanation on the poor correlation between previous theory and experiment in the case of longitudinal shear modulus.     

The effective medium and the average field approximations vis-à-vis the Hashin-Shtrikman bounds. I. The self-consistent scheme in matrix-based composites

The effective medium approximation (EMA) and the average field approximation (AFA) are two classical micromechanics models for the determination of effective properties of heterogeneous media. They are also known in the literature as ‘self-consistent’ approximations. In the AFA, the basic idea is to estimate the actual average field existing in a phase through a configuration in which a typical particle of that phase is embedded in the homogenized medium. In the EMA, on the other hand, one or more representative microstructural elements of the composite is embedded in the homogenized effective medium subjected to a uniform field, and the demand is made that the dominant part of the far-field disturbance vanishes. Both parts of this study are concerned with two-phase, matrix-based, effectively isotropic composites with an inclusion phase consisting of randomly oriented particles of arbitrary shape in general, and ellipsoidal shape in particular. The constituent phases are assumed to be isotropic. It is shown that in those systems the AFA and EMA give different predictions, with the distinction between them becoming especially striking regarding their standing vis-à-vis the Hashin-Shtrikman (HS-bounds). While due to its realizability property the EMA will always obey the bounds, we show that there are circumstances in which the AFA may violate the bounds. In the AFA for two-phase matrix-based composites, the embedded inclusion is a particle of the inclusion phase. If the particle is directly embedded in the effective medium, the method is called here the self-consistent scheme-average field approximation (SCS-AFA), and will obey the HS-bounds for an inclusion shape that is simply connected. If the embedded entity is a matrix-coated particle, then the method is called the generalized self-consistent scheme-average field approximation (GSCS-AFA), and may violate the HS-bounds. On the other hand, in the EMA for matrix-based composites with well-separated inclusions, we indicate that in view of its premises the embedding with a matrix-coated particle generally becomes the appropriate one, and the method is thus called the generalized self-consistent scheme-effective medium approximation (GSCS-EMA). Part I of this study is concerned with SCS-AFA in dielectrics and elasticity, and Part II with the GSCS-AFA and GSCS-EMA in dielectrics.     

A generalized self-consistent Mori-Tanaka model for predicting the effective moduli of the coated inclusion based composite materials

The weak point of the generalized self-consistent method (GSCM) is that its solution for the effective shear moduli involves determining the complicated displacement and strain fields in constitutents. Furthermore, the effective moduli estimated by GSCM cannot be expressed in an explicit form. Instead of following the procedure of GSCM, in this paper a generalized self-consistent Mori-Tanaka method (GSCMTM) is developed by means of Hill's interface condition and the assumption that the strain in the inclusion is uniform. A comparison with the existing theoretical and experimental results shows that the present GSCMTM is sufficiently accurate to predict the effective moduli of the coated inclusion-based composite materials. Moreover, it is interesting to find that the application of Hill's interface condition in volumetric domain is equivalent to the Mori-Tanaka average field approximation. This project was supported by the National Natural Science Foundation of China and China Postdoctoral Science Foundation.     

含随机分布缺陷的多晶铁电材料的有效电弹性能

针对铁电材料含有随机分布缺陷以及微结构在外场作用下发生变化的特点,建立起一个细观统计力学模型,考虑到缺陷和酶极化转动之间的相互影响,利用细观力学方法－Eshelby等效夹杂法和Mori-Tanaka的平均理论,具体分析了所含缺陷以及外场对铁电材料有效电弹性能和模量的影响。针对BaTiO3铁电陶瓷的有效电弹性能与常数的预报结果表明了缺陷的存在将增强材料的压电性能。     

基于自洽法的电化学沉积修复饱和混凝土细观描述

针对当前电化学沉积修复混凝土缺乏细观层次上的理论描述,首先,以饱和混凝土的细观结构和电化学沉积修复的主要机理为基础, 提出了含电化学沉积产物、水和混凝土基体在内的三相复合材料细观力学模型. 其次,为了从细观层次定量描述电化学沉积修复对混凝土宏观性能的影响,基于自洽法对上述细观力学模型进行多层次均匀化处理,获取电化学沉积修复饱和混凝土的有效性能,其中,第一层均匀化是采用广义自洽模型获取等效夹杂的有效性能,第二层均匀化是采用自洽方法和沃伊特(Voigt) 上限相结合的方式获取修复混凝土的有效性能. 最后,为了验证该文所提出模型和方法的有效性,对比了该文的预测结果、试验数据和已有模型,结果表明了该文模型和方法是合理的.      

基于细观力学分析的砂土弹塑性本构模型

利用细观力学分析中的自洽理论建立了在固结排水条件下的砂土弹塑性本构模型,在该模型中考虑了部分砂土颗粒因剪切作用在颗粒接触面滑移而引起的整体剪切模量的降低,给出了求解整体剪切模量的解析方法,并将模型预测与不同应力路径的试验结果作了比较,证明该模型是合理的.     

Effective moduli of piezoelectric material with microcavities

The effective electroelastic moduli of microcavity-weakened piezoelectric plates are investigated by the dilute, self-consistent, Mori-Tanaka and differential micromechanics theories. The results of perturbed heat intensity, strain and electric field (SEF) due to the presence of voids are obtained for two-dimensional (2-D) piezoelectric plates with microcavities of various shapes, and then the above four micromechanics models can be established with the results. These models can be applicable to a wide range of microcavities such as ellipse, circle, crack, triangle, square and pentagon. Some numerical results are presented to illustrate the applicability of these models.     

Consistency between independence theorems and generalized self-consistent method

Recently Zheng & Hwang established a series of independence theorems concerning with planar effective elastic properties. It is manifested that the estimation of the effective elastic properties of microcracked solids through the generalized self-consistent method (GSCM) contradicts with these independence theorems. In this paper it is shown that such contradiction is actually caused by the approximate algorithm adopted, while the exact solution of GSCM is consistent with these rigorously established independence theorems. Since only an approximate algorithm in GCSM is available in dealing with problems involving non-circular inclusions or holes, an intrinsic GSCM is proposed, which can be performed based on an approximate algorithm and the corresponding estimations are consistent with the independence theorems.     

确定裂纹体等效弹性模量的边界元方法

采用边界元方法计算含有序分布裂纹的裂纹体在压缩载荷作用下的等效弹性模量，利用一种能适当考虑裂纹有间相作用的自洽理论，建立了相应的迭代格式，通过算例研究了裂纹方向，裂纹面间摩擦系数对裂纹体等效弹性模量的影响。     

Equivalent inclusions in micromechanics with interface energy effect

In order to apply classical micromechanics in predicting the effective properties of nanocomposites incorporating interface energy, a concept of equivalent inclusion(EI) is usually adopted. The properties of EI are obtained by embedding a single inclusion with the interface into an infinite matrix. However, whether such an EI is universal for different micromechanics-based methods is rarely discussed in the literature. In this paper, the interface energy theory is used to study the applicability of the above mentioned EI. It is found that some elastic properties of the EI are related only to the properties of the inclusion and the interface, whereas others are also related to the properties of the matrix. The former properties of the EI can be applied to both the classical Mori-Tanaka method(MTM) and the generalized self-consistent method(GSCM). However, the latter can be applied only to the MTM. Two kinds of new EIs are proposed for the GSCM and used to estimate the effective mechanical properties of nanocomposites.