Strong-contrast expansion correlation approximations for the effective elastic moduli of multiphase composites

Following the previous approach of Pham and Torquato (J Appl Phys 94:6591–6602, 2003) and Torquato (J Mech Phys Solids 45:1421–1448, 1997; Random heterogeneous media, Springer, Berlin, 2002), we derive the strong-contrast expansions for the effective elastic moduli K _e,G _e of d-dimensional multiphase composites. The series consists of a principal reference part and a fluctuation part (perturbation about a homogeneous reference or comparison material), which contains multi-point correlation functions that characterize the microstructure of the composite. We propose a three-point correlation approximation for the fluctuation part with an objective choice of the reference phase moduli, such that the fluctuation terms vanish. That results in the approximations for the effective elastic moduli of isotropic composites, which coincide with the well-known self-consistent and Maxwell approximations for two-phase composites having respective microstructures. Applications to some two-phase materials are given.     

The boundaries of the effective elastic moduli for inhomogeneous solids

The calculation of the effective elastic moduli of inhomogeneous solids, which connect the stresses and strains averaged for the material, is accompanied by certain mathematical difficulties owing to correlation relationships of arbitrary orders. Neglect of correlation relationships leads to average elastic moduli, where averaging according to Voigt and Reuss establishes boundaries containing the effective elastic moduli [1]. Approximate values of the latter can be found by taking into account the correlation relationships of the second order in both calculation schemes [2, 3]. Another method of evaluating the true moduli consists of narrowing the boundaries of Voigt and Reuss on the basis of model representations [4-6]. The approximate effective elastic moduli for a series of polycrystals with various common-angle values are presented in [7]. An analysis of the effect of the correlation relationships between the grains of a mechanical mixture of isotropic components on the effective elastic moduli is carried out in [8], although in all the papers just mentioned the use of correlative corrections to narrow the range of elastic moduli is not investigated. Below it is shown that the calculation of the correlation corrections in the second approximation allows the range for the effective moduli to be narrowed.     

Effective elastic properties of matrix composites with transversely-isotropic phases

The present work addresses the problem of calculation of the macroscopic effective elastic properties of composites containing transversely isotropic phases. As a first step, the contribution of a single inhomogeneity to the effective elastic properties is quantified. Relevant stiffness and compliance contribution tensors are derived for spheroidal inhomogeneities. The limiting cases of spherical, penny-shaped and cylindrical shapes are discussed in detail. The property contribution tensors are used to derive the effective elastic moduli of composite materials formed by transversely isotropic phases in two approximations: non-interaction approximation and effective field method. The results are compared with elastic moduli of quasi-random composites.     

Stochastic functional expansion in elasticity of heterogeneous solids

The Volterra-Wiener functional expansion is employed to the analysis of statistic properties for random heterogeneous solids. For simplicity, the technique is displayed on an elastic suspension of spheres. The basis function in the expansion is chosen as that corresponding to the so-called “perfect disorder” of spheres (PDS), recently introduced by the authors. An infinite hierarchy of equations for the kernels in the expansion is derived whose truncating after the nth equation is shown to yield results for the averaged statistical characteristics which are valid to order cⁿ_f, where c_f is the volume fraction of the spheres. The kernels for the first and the second approximations, n = 1, 2, are found and related to the displacement fields in an infinite elastic body containing, respectively, one and two spherical inhomogeneities. Within the frame of the so-called singular approximation the overall tensor of elastic moduli for a suspension of perfectly disordered spheres is shown to coincide to the order c²_f with a formula, earlier obtained by means of the method of the effective field.     

Mathematical Models and Methods of the Mechanics of Stochastic Composites

The basic approaches used in mathematical models and general methods for solution of the equations of the mechanics of stochastic composites are generalized. They can be reduced to the stochastic equations of the theory of elasticity of a structurally inhomogeneous medium, to the equations of the theory of effective elastic moduli, to the equations of the theory of elastic mixtures, or to more general equations of the fourth order. The solution of the stochastic equations of the elastic theory for an arbitrary domain involves substantial mathematical difficulties and may be implemented only rather approximately. The construction of the equations of the theory of effective moduli is associated with the problem on the effective moduli of a stochastically inhomogeneous medium, which can be solved by the perturbation method, by the method of moments, or by the method of conditional moments. The latter method is most appropriate. It permits one to determine the effective moduli in a two-point approximation and nonlinear deformation properties. In the structure of equations, the theory of elastic mixtures is more general than the theory of effective moduli; however, since the state equations have not been strictly substantiated and the constants have not been correctly determined, theoretically or experimentally, this theory cannot be used for systematic designing composite structures. A new model of the nonuniform deformation of composites is more promising. It is constructed by performing strict mathematical transformations and averaging the output stochastic equations, all the constants being determined. In the zero approximation, the equations of the theory of effective moduli follow from this model, and, in the first approximation, fourth-order equations, which are more general than those of the theory of mixtures, follow from it     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ?; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.     

On reflected interactions in elastic solids containing inhomogeneities

Interactions in linear elastic solids containing inhomogeneities are examined using integral equations. Direct and reflected interactions are identified. Direct interactions occur simply because elastic fields emitted by inhomogeneities affect each other. Reflected interactions occur because elastic fields emitted by inhomogeneities are reflected by the specimen boundary back to the individual inhomogeneities. It is shown that the reflected interactions are of critical importance to analysis of representative volume elements. Further, the reflected interactions are expressed in simple terms, so that one can obtain explicit approximate expressions for the effective stiffness tensor for linear elastic solids containing ellipsoidal and non-ellipsoidal inhomogeneities. For ellipsoidal inhomogeneities, the new approximation is closely related to that of Mori and Tanaka. In general, the new approximation can be used to recover Ponte Castañeda–Willis׳ and Kanaun–Levin׳s approximations. Connections with Maxwell׳s approximation are established.     

Effective elastic properties of periodic composite medium

A new method is presented for calculating the bulk effective elastic stiffness tensor of a two-component composite with a periodic microstructure. The basic features of this method are similar to the one introduced by Bergman and Dunn (1992) for the dielectric problem. It is based on a Fourier representation of an integro-differential equation for the displacement field, which is used to produce a continued-fraction expansion for the elastic moduli. The method enabled us to include a much larger number of Fourier components than some previously proposed Fourier methods. Consequently our method provides the possibility of performing reliable calculations of the effective elastic tensor of periodic composites that are neither dilute nor low contrast, and are not restricted to arrays of nonoverlapping inclusions. We present results for a cubic array of nonoverlapping spheres, intended to serve as a test of quality, as well as results for a cubic array of overlapping spheres and a two dimensional hexagonal array of circles (a model for a fiber reinforced material) for comparison with previous work.     

Bounds for effective elastic moduli of disordered materials

Recently P.H. Dederichs and R. Zeller (1973) have developed a formal theory of the bounds of odd order n for the effective elastic moduli of linearly elastic, disordered materials. The bounds are established by use of statistical information given in terms of correlation functions up to order n (= 1, 3, 5,…). This theory is extended to include the bounds of even order n. It is indicated how these bounds can be made optimum under the given statistical information. The results for bounds of even and odd order are obtained in forms which resemble Neumann series, containing multiple integrals up to order (n?1). These integrals can be calculated for certain materials which are classified in terms of a gradual statistical homogeneity, isotropy and disorder. Materials which possess these properties up to the correlation functions of nth order are called overall grade n materials. The optimum bounds for overall grade 2 and grade 3 materials are given explicitly. Optimum bounds for materials which are of grade ∞ in homogeneity and isotropy (i.e. (statistically) perfectly homogeneous and isotropic) and, at the same time, disordered of grade 2 or 3 are also derived. Those for grade 2 in disorder are the Z. Hashin and S. Shtrikman's (1963) bounds. Those for grade 3 are the narrowest, explicit bounds so far derived for random elastic materials. They contain within themselves the so-called self-consistent elastic moduli.     

Simple algebraic approximations for the effective elastic moduli of cubic arrays of spheres

Recently, Cohen and Bergman (Phys. Rev. B 68 (2003a) 24104) applied the method of elastostatic resonances to the three-dimensional problem of nonoverlapping spherical isotropic inclusions arranged in a cubic array in order to calculate the effective elastic moduli. The leading order in this systematic perturbation expansion, which is related to the Clausius-Mossotti approximation of electrostatics, was obtained in the form of simple algebraic expressions for the elastic moduli. Explicit expressions were derived for the case of a simple cubic array of spheres, and comparison was made with some accurate results. Here, we present explicit expressions for the effective elastic moduli of base-centered and face-centered cubic arrays as well, and make a comparison with other estimates and with accurate numerical results. The simple algebraic expressions provide accurate results at low volume fractions of the inclusions and are good estimates at moderate volume fractions even when the contrast is high.     

Theoretical properties of cubic crystals at arbitrary pressure—I. Density and bulk modulus

The theoretical elastic behaviour of simple monatomic cubic crystals at arbitrary pressure is to be presented in a series of papers. In the present paper, general expressions are derived for calculating the pressure P and the bulk modulus κ as a function of the all-round stretch λ for crystals in which the interatomic interaction energies are modelled by pairwise functions φ. With the aid of a particular family of functions φ, calculations are carried out for the three cubic structures, and a detailed study is made of the influence of crystal structure and explicit nature of φ upon the theoretical elastic behaviour under pressure loading. Calculations are also made of higher order derivatives of P(λ) and gk(λ) in the reference state (i.e. at λ = 1) and the “higher order moduli” thus calculated are used to formulate series expansion approximations to the functions P(λ) and κ(λ). Values of P(λ) and κ(λ) in the series approximations, based upon successively higher order moduli (evaluated in the reference state), are compared with the corresponding “exact” values evaluated in the current state. The theoretical results are useful as empirical relationships modelling the elastic behaviour of crystals at arbitrary pressure.     

含正交排列夹杂和缺陷材料的等效弹性模量和损伤

研究含正交排列夹杂和缺陷材料的等效弹性模量和损伤,推导了以Eshelby－Mori－Tanaka方法求解多相各向异性复合材料等效弹性模量的简便计算公式,针对含三相正交椭球状夹杂的正交各向异性材料,得到了由细观参量（夹杂的形状、方位和体积分数）表示的等效弹性模量的解析表达式．在此基础上,提出了一个宏细观结合的正交各向异性损伤模型,从而建立了以细观量为参量的含损伤材料的应力应变关系．最后,对影响材料损伤的细观结构参数进行了分析．     

Determination of material response functions for prestrained rubbers

The mechanical response of two natural rubber compounds is examined in order to determine relevant material parameters by non-linear finite element analysis. The materials are subjected to (a) combined static torsion and extension, and (b) small, steady-state torsional oscillations superposed on a large static simple extension. The materials are assumed to be incompressible and isotropic in their undeformed state and a time-strain separable relaxation modulus tensor is employed in order to characterize the steady-state harmonic viscoelastic response. The combined static torsion and extension experiments are used to determine the basic delayed elastic response functions. A Rivlin-type strain energy expression of third-order accuracy is used for the purpose. The two-constant, Mooney-Rivlin form is found to be adequate for both materials in the relatively limited range of deformation magnitudes considered.The torsional storage and loss moduli are determined under quasistatic conditions as functions of frequency and axial static pre-strain. The time-strain separability is found to be a resonable approximation in a relatively limited range of static prestrain magnitudes and frequencies considered for the natural gum rubbers investigated. The experimental methodology is discussed in some detail.     

Symmetry conditions for third order elastic moduli and implications in nonlinear wave theory

Several results are presented concerning symmetry properties of the tensor of third order elastic moduli. It is proven that a set of conditions upon the components of the modulus tensor are both necessary and sufficient for a given direction to be normal to a plane of material symmetry. This leads to a systematic procedure by which the underlying symmetry of a material can be calculated from the 56 third order moduli. One implication of the symmetry conditions is that the nonlinearity parameter governing the evolution of acceleration waves and nonlinear wave phenomena is identically zero for all transverse waves associated with a plane of material symmetry.     

Ultrasonic attenuation in polycrystals using a self-consistent approach

The self-consistent method of averaging elastic moduli to define the effective medium of a polycrystal is used to investigate the dynamic problem of wave propagation. An alternative covariance tensor describing the elastic moduli fluctuations of the polycrystal containing self-consistent elastic properties is derived and found to be significantly smaller than the covariance tensor formed through traditional Voigt averaging. Attenuation curves are generated using the self-consistent elastic moduli and covariance tensors and these results are compared with previous Voigt-averaged estimates. The second-order polycrystalline dispersion relation for the self-consistent scheme is found for cases of low and high crystallite anisotropy. The attenuation coefficients and dispersion relations derived through the self-consistent scheme are considerably different than previous estimates. Experimentally measured longitudinal attenuation coefficients support the use of the self-consistent scheme for estimation of attenuation.     

Effective elastic properties of solids with defects of irregular shapes

Pores and defects in real materials often have very irregular shapes. Thus, micromechanical modeling based on the analytical solutions of elasticity becomes inapplicable. The objective of this paper is to present a computational procedure to calculate the contribution of the irregularly shaped defects into the effective moduli of two-dimensional elastic solids. In this procedure, the cavity compliance tensor is constructed numerically for an individual defect, and then used in the elastic potential-based approach to predict the effective moduli of porous solids. Two computational methods are used in this paper to calculate the components of a cavity compliance tensor: finite element analysis and numerical conformal mapping. Application of this procedure to the regular hole shapes produces results that are in good correspondence with analytical predictions.     

On the stiffness of materials containing a disordered array of microscopic holes or hard inclusions

We study the macroscopic mechanical behavior of materials with microscopic holes or hard inclusions. Specifically, we deal with the effective elastic moduli of composites whose microgeometry consists of either soft or hard isolated inclusions surrounded by an elastic matrix. We approach this problem by taking the stiffness of the inclusion phase to be a complex variable, which we eventually evaluate at the soft or hard limits. Our main result states that there is a certain class of non-physical, negative-definite values of the elastic moduli of the inclusion phase for which the effective tensor does not have infinities or become otherwise singular.We present applications of this result to the estimation of effective moduli and to homogenization theorems. The first application involves using complexanalytic methods to obtain rigorous and accurate bounds on the effective moduli of the high-contrast composites under consideration. We also discuss the variational estimates of Rubenfeld & Keller, which yield a complementary set of bounds on these moduli. The best bounds are given by a combination of the analytical and variational results. As a second application, we show that certain known theorems of homogenization for materials with holes are simple consequences of our main result, and in this connection we establish corresponding new theorems for materials with hard inclusions. While our rederivation of the homogenization theorems for materials with holes can be closely related to other known constructions, it appears that certain elements provided by our main result are essential in the proof of homogenization for the hard-inclusion case.     

On estimation methods for effective moduli of microcracked solids

Summary  Most of the conventional methods for estimating the overall elastic moduli of microcracked solids are defined based on the concept of effective medium or effective field. The formal similarity of these methods is examined in this paper. A one-to-one correspondence relation exists between the effective medium methods and the effective field methods in the sense that they yield identical results. In addition to the conventional estimation techniques, any other number of such approaches may be constructed by appropriately specifying the effective matrix compliance (or stiffness) tensor and the effective stress (or strain) field which a microcrack is assumed to be subjected to. To generate continuous spectra of new methods for estimating the effective elastic moduli, two simple and straightforward approaches are proposed, which contain one or two adjustable parameters in order to yield results of good accuracy. The discussion in this paper can be extended to other kinds of heterogeneous materials. Received 4 October 2000; accepted for publication 30 January 2001