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

The present Part II of this two-part study is concerned with the average field approximation (AFA), and the effective medium approximation (EMA) in two-phase matrix-based dielectric composites through the use of an auxiliary configuration in which a particle of the inclusion phase is first surrounded by some matrix material, and then embedded in the effective medium. Those models will be referred as the generalized self-consistent scheme-average field approximation (GSCS-AFA), and the generalized self-consistent scheme-effective medium approximation (GSCS-EMA). We show that there are four types of the GSCS-AFA and a single type of the GSCS-EMA. In this paper the application of those models to dielectric composites with isotropic constituents and an inclusion phase that consists of randomly oriented ellipsoidal particles will be studied. The analytical solution of the auxiliary problem, which consists of an ellipsoidal particle confocally surrounded by a matrix shell and embedded in the effective medium, is achieved by means of ellipsoidal harmonics. Our results show that the effective property predictions of the GSCS-EMA and GSCS-AFA for the considered systems differ from each other, and more importantly, out of the four GSCS-AFA models, three of them violate the Hashin-Shtrikman bounds. The predictions of the GSCS-EMA obey the bounds. It is then shown that the version of the GSCS-AFA which obeys the Hashin-Shtrikman bounds for an inclusion phase with randomly oriented ellipsoids will violate them in the case of a particle shape which is not simply connected. Moreover, it turns out that the SCS-AFA studied in Part I also violates the Hashin-Shtrikman bounds in that case; the EMA, as expected, owing to its realizability property, continues to obey the bounds. Among the AFA and EMA in matrix-based composites, the GSCS-EMA therefore stands out as the method to be recommended.     

Revisiting the generalized self-consistent scheme in composites: Clarification of some aspects and a new formulation

The determination of an effective property in composite materials necessitates the knowledge of some averaged field quantities in the constituents (like the average heat intensity or average strain) of a composite sample, which is subjected to homogeneous boundary conditions. In the generalized self-consistent scheme (GSCS) which is today a classical micromechanics model suited for the determination of the effective properties of matrix-based composites, those average quantities are estimated by using an auxiliary configuration in which a particulate phase is first surrounded by some matrix material and then embedded in the effective medium. In the present study, we revisit the GSCS both for two- and multi-phase matrix-based composites containing spherical particles, and clarify aspects related to the volume fractions of the particle core and matrix shell within the composite element which is embedded in the effective medium. The contribution of this study is believed to be mainly on the conceptual side and resides in a new formulation of the method in which the embedding volume fractions are determined in the course of the analysis by means of some fundamental relations on the averaged fields. The study is carried out in thermal conduction and elasticity and contains new results on the effective shear modulus of multi-phase composites.     

Elastic properties of inhomogeneous transversely isotropic rocks

The problem to determine the effective elastic moduli and velocities of elastic wave propagation in transversely isotropic solid containing aligned spheroidal inhomogeneities (solid grains, vugs and micro-cracks) has been solved using the self-consistent scheme known as effective medium approximation (EMA). Since a solution of so-called one-particle problem is a base for each self-consistent method, we solved this problem as a first step for spheroidal inhomogeneity in a transversely isotropic medium. In contrast to the known solution of this problem by Lin and Mura we obtained the expressions for the strain field inside inclusion in the explicit form (without quadratures). The obtained solution was used then in the symmetric variant of the EMA where each component of the system was considered as spheroid with its own aspect ratio. This approach was applied to simulate the properties of the rocks containing isolated pores and micro-cracks. For connected fluid-filled pores we used the anisotropic variant of the Gassmann theory. The results of the calculations, obtained for the effective elastic moduli, have been compared with the experimental data and theoretical simulations of the other authors. Unlike many other rock mechanics theories, EMA approximation gives correct elastic moduli values even in the nondilute concentration of inhomogeneities. The comparison of the experimental data for oriented crack system with the EMA predictions indicates their good correspondence.     

The effective thermoelectroelastic properties of microinhomogeneous materials

The paper is concerned with composite materials which consist of a homogeneous matrix phase with a set of inclusions uniformly distributed in the matrix. The components of these materials are considered to be ideally elastic and exhibit piezoelectric properties. One of the variants of the self-consistent scheme, the Effective Field Method (EFM) is applied to calculate effective dielectric, piezoelectric and thermoelastic properties of such materials, taking into account the coupled electroelastic effects. At first the coupled thermoelectroelastic problem for a homogeneous medium with an isolated inclusion is solved. For an ellipsoidal inclusion and constant external field the solution of this problem is found in a closed analytic form. This solution is then used in the EFM to derive the effective thermoelectroelastic operator for the composite containing a random array of ellipsoidal inclusions. Explicit formulae for the electrothermoelastic constants are given for composites, reinforced by spheroidal inclusions.     

Eshelby's problem of non-elliptical inclusions

The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.     

非理想界面对三相复合材料应力场的影响

对非理想界面的三相复合材料,提出了计算弹性应力场的微观力学模型,在适当的简化假设下,对带界相的颗粒增强和纤维增强复合材料,得到了应力场的计算公式。以剪切载荷为例给出了数值例子。给出的数值结果表明非理想界面对三相复合材料应力场的影响。     

Self-consistent field method in the problem about the effective properties of an elastic composite

This paper is devoted to the calculation of effective elastic properties of a medium containing a random field of ellipsoidal inhomogeneities. It is assumed that the centers of the inclusions (the inhomogeneities) form a random spatial lattice, i.e., the field of inhomogeneities considered is strongly correlated. The interaction between the inhomogeneities is taken into account within the frame-work of the self-consistent field approximation. It hence turns out that the symmetry of the tensor of the elastic properties of the medium is determined by the symmetry of the elastic properties of the inclusion matrix, as well as by the symmetry of the spatial lattice formed by the mathematical expectations of the centers of the inclusions.     

A generalized differential effective medium theory

A generalization of the Differential Effective Medium approximation (DEM) is discussed. The new scheme is applied to the estimation of the effective permittivity of a two phase dielectric composite. Ordinary DEM corresponds to a realizable microgeometry in which the composite is built up incrementally through a process of homogenization, with one phase always in dilute suspension and the other phase associated with the percolating backbone. The generalization of DEM assumes a third phase which acts as a backbone. The other two phases are progressively added to the backbone such that each addition is in an effectively homogeneous medium. A canonical ordinary differential equation is derived which describes the change in material properties as a function of the volume concentration φ of the added phases in the composite. As φ→ 1, the Effective Medium Approximation (EMA) is obtained. For φ < 1, the result depends upon the backbone and the mixture path that is followed. The approach to EMA for φ ? 1 is analysed and a generalization of Archie's law for conductor-insulator composites is described. The conductivity mimics EMA above the percolation threshold and DEM as the conducting phase vanishes.     

Solutions of inhomogeneity problems with graded shells and application to core-shell nanoparticles and composites

This paper first presents the Eshelby tensors and stress concentration tensors for a spherical inhomogeneity with a graded shell embedded in an alien infinite matrix. The solution is then specialized to inhomogeneous inclusions in finite spherical domains with fixed displacement or traction-free boundary conditions. The Eshelby tensors in the infinite and finite domains and the stress concentration tensors are especially useful for solving many problems in mechanics and materials science. This is demonstrated on two examples. In the first example, the strain distributions in core-shell nanoparticles with eigenstrains induced by lattice mismatches are calculated using the Eshelby tensors in the finite domains. In the second example, the Eshelby and stress concentration tensors in the three-phase configuration are used to formulate the generalized self-consistent prediction of the effective moduli of composites containing spherical particles within the framework of the equivalent inclusion method. The advantage of this micromechanical scheme is that, whilst its predictions are almost identical to the classical generalized self-consistent method and the third-order approximation, the expressions for the effective moduli have simple closed forms.     

Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity

Consider an infinite thermally conductive medium characterized by Fourier’s law, in which a subdomain, called an inclusion, is subjected to a prescribed uniform heat flux-free temperature gradient. The second-order tensor field relating the gradient of the resulting temperature field over the medium to the uniform heat flux-free temperature gradient is referred to as Eshelby’s tensor field for conduction. The present work aims at deriving the general properties of Eshelby’s tensor field for conduction. It is found that: (i) the trace of Eshelby’s tensor field is equal to the characteristic function of the inclusion, independently of the latter’s shape; (ii) the isotropic part of Eshelby’s tensor field over the inclusion of arbitrary shape is identical to Eshelby’s tensor field over a 2D circular or 3D spherical inclusion; (iii) when the medium is made of an isotropic material and when the inclusion has some specific rotational symmetries, the value of the Eshelby’s tensor field evaluated at the inclusion gravity center and the symmetric average of Eshelby’s tensor fields are both equal to Eshelby’s tensor field for a 2D circular or 3D spherical inclusion. These results are then extended, with the help of a linear transformation, to the general case where the medium consists of an anisotropic conductive material. The method elaborated and results obtained by the present work are directly transposable to the physically analogous transport phenomena of electric conduction, dielectrics, magnetism, diffusion and flow in porous media and to the mathematically identical phenomenon of anti-plane elasticity.     

Micro-mechanical behaviors of SMA composite materials under hygro-thermo-elastic strain fields

An analytical procedure to evaluate the behavior of shape memory alloy (SMA) composite under hygrothermal environment is presented. The SMA wires are considered as inclusions embedded in a homogeneous matrix medium of the composite. The inhomogeneity associated with the phase transformation and thermal strains in the SMA wire as well as the hygrothermal strain in the matrix is homogenized using Eshelby’s equivalent inclusion method. In the present work, a similar approach adopted for SMA composites by Marfia and Sacco [Marfia, S., Sacco, E., 2005. Micromechanics and homogenisation of SMA-wire-reinforced materials. J. Appl. Mech. 72 (2), 259–268.] is considered in order to validate the response of SMA composite subjected to thermo-elastic strain field. However, in the present approach, certain modifications and new derivations for the inelastic strain tensors is carried out. First, the constitutive laws for the SMA wire and matrix are expressed in terms of the average strain in the composite. The evolutionary equations used to characterize the pseudoelastic (PE) behavior of the SMA wire are redefined in terms of the eigen strains (phase transformation and thermal strains) occurring in the SMA wire, which are then expressed in terms of the average strain in the composite. Further, the SMA composite constitutive law under coupled hygro-thermo-elastic strain fields is proposed. The generic homogenized hygric and thermal inelastic composite tensors required for the proposed hygro-thermo-elastic constitutive law are derived. Finally, the SMA composite lamina is characterized using Eshelby’s equivalent inclusion method. Using the proposed modifications and derivations, the analytical results are validated for the case of thermo-elastic strain fields and the procedure is then extended to evaluate the SMA composite behavior under hygro-thermo-elastic strain fields. The results include the effect of thermo-elastic and hygro-thermo-elastic strains on the transformation stresses and the nature of hysteresis due to hygric and thermo-elastic strains.     

The application of the Eshelby equivalent inclusion method for unifying modulus and transformation toughening

When a crack is lodged in an inclusion, both difference between the modulus of the inclusion and matrix material and stress-free transformation strain of the inclusion will cause the near-tip stress intensity factor to be greater (amplification effect) or less (shielding or toughening effect) than that prevailing in a homogeneous material. In this paper, the inclusion may represent a second phase particle in composites and a transformation or microcracked process zone in brittle materials, which may undergo a stress-free transformation strain induced by phase transformation, microcracking, thermal expansion mismatch and so forth. A close form of solution is derived for predicting the toughening (or amplification) effect. The derivation is based on Eshelby equivalent inclusion approach that provides rigorous theoretical basis to unify the modulus and transformation contributions to the near-tip field. As validated by numerical examples, the developed formula has excellent accuracy for different application cases.     

SH waves of an arbitrary frequency in random fibrous composites via the K-K relations

The Kramers-Kronig relations method is shown to provide the dynamic response of a random fibrous composite for the full frequency interval, 0 < ω < ∞. The method yields a conceptually simple way of deriving the dynamic response of random composites if the approximation of an effective homogeneous medium is adopted.It is shown that some of the widely accepted theories may violate the causality and/or linearity of the effective medium. Extensive numerical data are given as well as comparison with other theories and experiments.     

Influence of the particle size and volume fraction on micro-damage of the composites

The bulk and shear modulus of metal matrix composites with various volume fractions of particles are modified based on the Eshelby’s equivalent inclusion method combined with self-consistent scheme. By introducing the modified modulus, a new model, which can predict the particle size effects on the stress–strain relation under interfacial debonding damage between matrix and particles, is established. The results obtained from the present investigation show a better agreement with the experimental data.     

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.     

Application of results on Eshelby tensor to the determination of effective poroelastic properties of anisotropic rocks-like composites

The present work is devoted to the determination of the macroscopic poroelastic properties of anisotropic elastic porous materials saturated by a fluid under pressure. It makes use of the theoretical results provided by Withers [Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59 (4), 759–781.] for the problem of an ellipsoidal inclusion embedded in a transversely isotropic elastic medium. The particular case of a spherical inclusion is very important for rock-like composites such as argillite and shales. The implementation of these results in a micromechanical theory of poroelasticity allows to quantify the effects of the solid matrix anisotropy and of pore space on the effective poromechanical properties. Closed form expressions of Biot tensor and of Biot modulus are presented as well as numerical applications for anisotropic shales.     

基于细观力学方法的混凝土热膨胀系数预测

建立混凝土材料的有效性质与微结构参数之间的关系,是混凝土材料优化设计的基础。本文用细观力学方法对复合材料宏观有效热膨胀系数进行研究,得到了含有一球形夹杂物的无限大介质在均匀变温作用下的应力场。假定混凝土为由骨料和砂浆基质组成的二相复合材料,根据混凝土宏观体积热膨胀量与组成混凝土的各相介质细观体积热膨胀量相等的原则,采用基于Mori-Tanaka方法的混凝土宏观有效剪切模量,推导出混凝土有效热膨胀系数的解答。对稀疏解法、自洽方法和有限单元数值试验结果的比较说明,本文提出的基于自洽方法的混凝土宏观有效热膨胀系数的理论公式能够较好的描述混凝土的热学特性,该方法可以推广到多相复合材料宏观有效热膨胀系数的预测中。     

三相多重铁性复合材料的有效磁电弹性模量预测

采用平均场Mori-Tanaka模型,计算了三相复合材料的有效磁电弹性模量,研究了复合材料磁电系数与微观结构之间关系;结果表明掺杂相的体积分数与颗粒形状系数对复合材料的有效磁电系数有很大的影响,这些结果可为复合材料的实验设计提供理论参考和指导.     

A one-parameter generalized self-consistent model for isotropic multiphase composites

The classical generalized self-consistent model (GSCM) is recognized to be suitable and efficient for estimating the effective moduli of an isotropic composite consisting of an isotropic host matrix and an isotropic inclusion phase. The present work aims to enlarge the scope of the GSCM so that it becomes applicable to a good number of important situations where the phases cannot be differentiated as the host matrix and inclusions. This objective is achieved first by inserting into the unknown effective medium a coated composite sphere whose core is made of the unknown effective medium and whose coatings are formed of the constituent phases and then by imposing an energy equivalency condition. The equations thus obtained to characterize the effective bulk and shear moduli involve a microstructural parameter which turns out to be capable of describing in some sense how far a microstructure is from the host matrix/inclusion morphology. The important case of two-phase composites is studied in detail to illustrate the salient features of the proposed model.