Improved bounds for the elastic moduli of perfectly-random composites

New upper and lower bounds are constructed for the elastic moduli of a class of isotropic composites with perfectly-random microgeometries ([1–3]), which improve upon the bounds on the elastic shear modulus given in [1].     

Calculation of effective elastic moduli of composite materials with multiphase interactions taken into consideration

The effective tensors of elastic moduli and pliability were calculated by renormalizing the equations of equilibrium and incompatibility. All multiphase interactions were taken into consideration, but the calculation was done approximately by using only the singular parts of derivatives of Green's functions, which is equivalent to the assumption of uniformity of the random components of stress and strain fields within grain boundaries. Solutions derived by two methods are reconciled, making it possible to determine the position of the elastic moduli of shear and of bulk K between the Hashin-Shtrikman bounds. Particular cases in which only the bulk modulus is not uniform, as well as those of composite materials with all of their constituents having the same ratio K=4/3 are considered.     

The effective elastic moduli of a random packing of spheres

The effective incremental elastic moduli of a random packing of identical elastic spheres are derived. Although the method of derivation is suitable for any initial deformed configuration that induces compressive forces between any spheres in contact, specific results are given only for the cases in which this corresponds to that of a hydrostatic compression or a uniaxial compression. For additional simplicity it is assumed that the spheres are either infinitely rough or perfectly smooth.     

Effective elastic moduli of third-order composite materials

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 118–123, November–December, 1990.     

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.     

On the overall moduli of non-linear elastic composite materials

From the work of R. Hill on constitutive macro-variables it is known that for an inhomogeneous elastic solid under finite strain an overall elastic constitutive law may be defined. In particular, the volume average of the strain energy of the solid is a function only of the volume-averaged deformation gradient. In view of the importance of this result it is re-derived in this paper as a prelude to a discussion of composite materials. A composite material consisting of a dilute suspension of initially spherical inclusions embedded in a matrix of different material is considered. For second-order isotropic elasticity theory an expression for the overall bulk modulus of the composite material is obtained in terms of the moduli of the constituents. When the inclusions are vacuous a known result for the bulk modulus of porous materials is recovered. In certain situations the strengthening/ weakening effects of the inclusions are less pronounced in the second-order theory than in the linear theory.     

Frequency dependence of the effective elastic moduli of cavernous bodies

The dependence of the effective parameters of microheterogeneous media on the frequency and structure of the pore space is obtained using the boundary integral equation method. The potential method is first used to solve dynamic three-dimensional elastic problems in multiply connected domains in the case of stationary oscillations. It is shown that if the wavelength corresponds to a finite number of blocks, the effective elastic moduli decrease.     

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.     

Asymptotics of the homogenized moduli for the elastic chess-board composite

We find the asymptotic behavior of the homogenized coefficients of elasticity for the chess-board structure. In the chess board white and black cells are isotropic and have Lamé constants (, ,) and (, ) respectively. We assume that the black cells are soft, so 0. It turns out that the Poisson ratio for this composite tends to zero with .     

On the quasistatic effective elastic moduli for elastic waves in three-dimensional phononic crystals

Effective elastic moduli for 3D solid–solid phononic crystals of arbitrary anisotropy and oblique lattice structure are formulated analytically using the plane-wave expansion (PWE) method and the recently proposed monodromy-matrix (MM) method. The latter approach employs Fourier series in two dimensions with direct numerical integration along the third direction. As a result, the MM method converges much quicker to the exact moduli in comparison with the PWE as the number of Fourier coefficients increases. The MM method yields a more explicit formula than previous results, enabling a closed-form upper bound on the effective Christoffel tensor. The MM approach significantly improves the efficiency and accuracy of evaluating effective wave speeds for high-contrast composites and for configurations of closely spaced inclusions, as demonstrated by three-dimensional examples.     

Estimate of effective elastic moduli with microcrack interaction effects

A method is developed for estimating the effects of microdefect interaction on the effective elastic properties of heterogeneous solids. An effective medium is defined to calculate the global effective elastic moduli of brittle materials weakened by distributed microcracks. Each microcrack is assumed to be embedded in an effective medium, the compliance of which is obtained from the dilute concentration method without accounting for interaction. The present scheme requires no iteration; it can account for microcrack interaction with sufficient accuracy. Analytical solutions are given for several two- and three-dimension problems with and without anisotropy.     

Bounds for effective elastic moduli of inhomogeneous solid bodies

In connection with the extensive use of various kinds of inhomogeneous materials (glass, carbon and boron reinforced plastics, cermets, concrete, reinforced materials, etc.) in technology, there arises a need to calculate the elastic properties of such systems. Here in each case it is necessary to work out specific methods for finding both elastic fields and effective moduli. Since, as a rule, such methods do not take into account the character of distribution of inhomogeneities in space, which is reflected on the form of the central moment functions [1], they can be referred to a single class and, consequently, can be obtained by a common method [2], In the given paper, by means of the method of solution of stochastic problems for microinhomogeneous solid bodies proposed in the work of the author [2], we find elastic fields and effective moduli in an arbitrary approximation. Depending on the choice of parameters, the latter form bounds within which there lie the exact values of the effective moduli. It is shown that the conditions used earlier for finding these parameters [3] are not the best ones. The effective elastic moduli of an inhomogeneous medium are calculated, and bounds, narrower than the bounds formed in [3], are found for them.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhniki, No. 5, pp. 144–150, September–October, 1973.     

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.     

Effective elastic moduli of triangular lattice material with defects

This paper presents an attempt to extend homogenization analysis for the effective elastic moduli of triangular lattice materials with microstructural defects. The proposed homogenization method adopts a process based on homogeneous strain boundary conditions, the micro-scale constitutive law and the micro-to-macro static operator to establish the relationship between the macroscopic properties of a given lattice material to its micro-discrete behaviors and structures. Further, the idea behind Eshelby’s equivalent eigenstrain principle is introduced to replace a defect distribution by an imagining displacement field (eigendisplacement) with the equivalent mechanical effect, and the triangular lattice Green's function technique is developed to solve the eigendisplacement field. The proposed method therefore allows handling of different types of microstructural defects as well as its arbitrary spatial distribution within a general and compact framework. Analytical closed-form estimations are derived, in the case of the dilute limit, for all the effective elastic moduli of stretch-dominated triangular lattices containing fractured cell walls and missing cells, respectively. Comparison with numerical results, the Hashin–Shtrikman upper bounds and uniform strain upper bounds are also presented to illustrate the predictive capability of the proposed method for lattice materials. Based on this work, we propose that not only the effective Young’s and shear moduli but also the effective Poisson’s ratio of triangular lattice materials depend on the number density of fractured cell walls and their spatial arrangements.     

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.     

Use of the parametrix method for estimating effective elastic moduli or randomly nonhomogeneous elastic bodies

The magnitude of the elasticity tensor of a comparison body remains unclarified if we use a singular approximation [1] to estimate the effective values of the elasticity tensor. Below we will use a parametrix method [2] to determine the first approximation of the random component of the deformation tensor and the effective values of the elasticity tensor, and will also compare the exact solution for one particular heterogeneous and a previously used approximation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 171–175, January–February, 1976.