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.     

A numerical simulation for effective elastic moduli of plates with various distributions and sizes of cracks

A fast convergent numerical model is developed to calculate the effective moduli of plates with various distributions and sizes of cracks, in which the crack line is divided into M parts to obtain the unknown traction on the crack line. When M=1, the model reduces to Kachanov's approximation method [Int. J. Solids Struct. 23 (1987) 23]. Six types of crack distributions and three kinds of crack sizes are considered, which are four regular (equilateral triangle, equilateral hexagon, rectangle, and diamond) and two random distributions (random location and orientation, and parallel orientation and random location), and one, two and random crack sizes. Some typical examples are also analyzed using the finite element method (FEM) to validate the present model. Then, the effective moduli associated with the crack distributions and sizes are calculated in detail. The present results for the regular distributions show some very interesting phenomena that have not been revealed before. And for the two random distributions, as the effective moduli depend on samples due to the randomness, the effect of the sample size and number are analyzed first. Then, effective moduli for plates with the three sizes of cracks are calculated. It is found that the effect of crack sizes on the effective moduli is significant for high crack densities, and small for low crack densities, and the random crack size leads to the lowest effective moduli. The present numerical results are compared with several popular micromechanics models to determine which one can provide the optimum estimation of the effective moduli of cracked plates with general crack densities. Furthermore, some existing numerical results are analyzed and discussed.     

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.     

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

This work aims at estimating the size-dependent effective elastic moduli of particulate composites in which both the interfacial displacement and traction discontinuities occur. To this end, the interfacial discontinuity relations derived from the replacement of a thin uniform interphase layer between two dissimilar materials by an imperfect interface are reformulated so as to considerably simplify the characteristic expressions of a general elastic imperfect model which is adopted in the present work and include the widely used Gurtin–Murdoch and spring-layer interface models as particular cases. The elastic fields in an infinite body made of a matrix containing an imperfectly bonded spherical particle and subjected to arbitrary remote uniform strain boundary conditions are then provided in an exact, coordinate-free and compact way. With the aid of these results, the elastic properties of a perfectly bonded spherical particle energetically equivalent to an imperfectly bonded one in an infinite matrix are determined. The estimates for the effective bulk and shear moduli of isotropic particulate composites are finally obtained by using the generalized self-consistent scheme and discussed through numerical examples.     

Estimates for the elastic moduli of random trigonal polycrystals and their macroscopic uncertainty

Explicit expressions of the upper and lower estimates on the macroscopic elastic moduli of random trigonal polycrystals are derived and calculated for a number of aggregates, which correct our earlier results given in Pham [Pham, D.C., 2003. Asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals. Int. J. Solids Struct. 40, 4911–4924]. The estimates are expected to predict the scatter ranges for the elastic moduli of the polycrystalline materials. The concept of effective moduli is reconsidered regarding the macroscopic uncertainty of the moduli of randomly inhomogeneous materials.     

The effective elastic moduli of columnar composites made of cylindrically anisotropic phases with rough interfaces

The present work aims to determine the effective elastic moduli of a composite having a columnar microstructure and made of two cylindrically anisotropic phases perfectly bonded at their interface oscillating quickly and periodically along the circular circumferential direction. To achieve this objective, a two-scale homogenization method is elaborated. First, the micro-to-meso upscaling is carried out by applying an asymptotic analysis, and the zone in which the interface oscillates is correspondingly homogenized as an equivalent interphase whose elastic properties are analytically and exactly determined. Second, the meso-to-macro upscaling is accomplished by using the composite cylinder assemblage model, and closed-form solutions are derived for the effective elastic moduli of the composite. Two important cases in which rough interfaces exhibit comb and saw-tooth profiles are studied in detail. The analytical results given by the two-scale homogenization procedure are shown to agree well with the numerical ones provided by the finite element method and to verify the universal relations existing between the effective elastic moduli of a two-phase columnar composite.     

Evaluation of elastic moduli of bilaminate filament-wound composites

This paper investigates the elastic behavior of bilaminate-composite coupons segmented from a filament-wound rocket case and of a laboratory prepared flat panel for direct tension-compression and flexural loading under static and dynamic conditions. New methods of testing have been developed which are primarily applicable to composite constructions. The dynamic test consists of exciting the primary free-free resonant mode of a specimen. Flexure tests utilizing a unique pure-bend system are employed for the static evaluations. The composite moduli determined from the static and dynamic test are compared with analytic values. The analytic values for the composite are derived from tension-compression and flexure analytical models using the material properties of the constituents. The measured elastic moduli compared favorably with analytical prediction and are indicative of the history of loading effects as well as the crazed condition of the composite constructions. The moduli determined by the dynamic test showed the closest agreement with analytic values, with a difference of 0 to 16 percent.     

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.     

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].     

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.     

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.     

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.     

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.     

Canonical ensemble for static elastic structures with random microstructures

A constant-(N,V,T,θ) ensemble is proposed to describe the elastostatics of random solid structures. Within the harmonic approximation, the energy of such a solid structure is the sum of a thermal and a strain component without mutual interaction. Systems in this ensemble thus draw energies from two separate baths: one thermal and the other mechanical. A mechanical entropy and an effective temperature (θ) can then be defined on the same rigorous basis as the thermal entropy and the Kelvin temperature (T). This ensemble approach can be used to calculate the properties of solid structures sharing similar microstructural randomness.     

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.