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.     

Effective elastic moduli of inhomogeneous solids by embedded cell model

An embedded cell model is presented to obtain the effective elastic moduli for three-dimensional two-phase composites which is an exact analytic formula without any simplified approximation and can be expressed in an explicit form. For the different cells such as spherical inclusions and cracks surrounded by sphere and oblate ellipsoidal matrix, the effective elastic moduli are evaluated and the results are compared with those from various micromechanics models. These results show that the present model is direct, simple and efficient to deal with three-dimensional two-phase composites. The project supported by the National Natural Science Foundation of China (No. 19704100) and the National Natural Science Foundation of Chinese Academy of Sciences (No. KJ951-1-201)     

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     

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.     

Effective elastic moduli of an inhomogeneous medium with cracks

In the present paper, the effective elastic moduli of an inhomogeneous medium with cracks are derived and obtained by taking into account its microstructural properties which involve the shape, size and distribution of cracks and the interaction between cracks. Numerical results for the periodic microstructure of different dimensions are presented. From the results obtained, it can be found that the distribution of cracks has a significant effect on the effective elastic moduli of the material. The project supported by the National Education Committee for Doctor     

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.     

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.     

固体的弹性模量和内耗测量方法研究进展

弹性模量和内耗是固体材料的基本力学性质, 其测量的准确性和便捷性对工业生产和科学研究都很重要. 本文回顾了近一百年来固体材料弹性模量和内耗的测量方法, 主要分为四类: 准静态方法、低频法、共振法和波传播法. 首先对每类方法的测量原理进行了简单介绍及总体评价. 接着对几种共振方法, 包括自由梁共振法、脉冲激励法、超声共振谱方法和压电超声复合振动技术(PUCOT)进行了详细介绍和评价. 然后, 重点介绍了本课题组最新提出的基于机电阻抗的模量内耗测量方法(称之为M-PUCOT或Q-EMI), 它可以同时、准确、快速地测量杨氏/剪切模量及相应内耗. 最后, 对这种新型弹性模量/内耗测量方法的意义和应用前景进行了讨论和展望.      

One method of determining the elastic properties of inhomogeneous solids

This paper considers the inverse problem of reconstructing three inhomogeneous characteristics of a rod (Young’s modulus, shear modulus, and density) from amplitude-frequency characteristics for the regime of steady-state longitudinal, flexural, and torsional vibrations. The unknown characteristics are identified by constructing iterative processes based on the apparatus of the Fredholm integral equations of the first and the second kind. Results of numerical experiments are presented.     

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.     

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.     

Bounds for the effective conductivity and elastic moduli of fully-disordered multicomponent materials

Communicated by R. Kohn     

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.     

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.     

Effective elastic moduli of two dimensional solids with distributed cohesive microcracks

Effective elastic properties of a defected solid with distributed cohesive micro-cracks are estimated based on homogenization of the Dugdale–Bilby–Cottrell–Swinden (Dugdale–BCS) type micro-cracks in a two dimensional elastic representative volume element (RVE).Since the cohesive micro-crack model mimics various realistic bond forces at micro-scale, a statistical average of cohesive defects can effectively represent the overall properties of the material due to bond breaking or crack surface separation in small scale. The newly proposed model is distinctive in the fact that the resulting effective moduli are found to be pressure sensitive.     

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.     

Harmonic thickness vibrations of inhomogeneous elastic layers with curved boundaries

The thickness vibrations of elastic inhomogeneous bodies of different geometry under dynamic harmonic loading are studied. The dependence of the amplitude–frequency characteristics of homogeneous and inhomogeneous bodies on excitation frequency is analyzed in detail. The frequency spectra for plane, cylindrical, and spherical layers are determined     

On effective moduli of inhomogeneous media characterized by several small parameters

Composites and porous media of elongated structure, as well as materials with pores or inclusions having the shape of parallelepipeds or channels of rectangular cross-section, are considered under certain conditions on the inclusion-to-matrix modulus ratio and the volume fraction of inclusions (pores). The effective moduli are calculated by the method of mathematical homogenization theory. Numerical results on the dependence of the effective moduli on the prolateness of the structure, the shape of the inclusions (pores), the inclusion-to-matrix modulus ratio, and the volume fraction of inclusions (pores) are given. The effective moduli computed according to the algorithm of mathematical homogenization theory are compared with those given by the explicit approximate formulas earlier developed by the authors.