Prediction of the effective elastic moduli of materials with irregularly-shaped pores based on the pore projected areas

Statistical modeling is used to correlate geometric parameters of pores with their contributions to the overall Young’s moduli of linearly elastic solids. The statistical model is based on individual pore contribution parameters evaluated by finite element simulations for a small pore subset selected using the design of experiments approach, so there is no need to solve the elasticity problem for all pores in the material. A polynomial relating pore geometric parameters to the contribution parameters is then fitted to the results of the simulations. We found a good correlation between normalized projected areas of the pores on three coordinate planes and their contributions to the corresponding effective Young’s moduli. The model is applied and validated for two large sets of pore geometries obtained by X-ray microcomputed tomography of a carbon/carbon and a 3D woven carbon/epoxy composite specimens.     

An explicit expression of the effective moduli for composite materials filled with coated inclusions

The obvious shortcoming of the generalized self-consistent method (GSCM) is that the effective shear modulus of composite materials estimated by the method can not be expressed in an explicit form. This is inconvenient in engineering applications. In order to overcome that shortcoming of GSCM, a reformation of GSCM is made and a new micromechanical scheme is suggested in this paper. By means of this new scheme, both the effective bulk and shear moduli of an inclusion-matrix composite material can be obtained and be expressed in simple explicit forms. A comparison with the existing models and the rigorous Hashin-Shtrikman bounds demonstrates that the present scheme is accurate. By a two-step homogenization technique from the present new scheme, the effective moduli of the composite materials with coated spherical inclusions are obtained and can also be expressed in an explicit form. The comparison with the existing theoretical and experimental results shows that the present solutions are satisfactory. Moreover, a quantitative comparison of GSCM and the Mori-Tanaka method (MTM) is made based on a unified scheme. The project supported by the National Natural Science Foundation of China under the Contract NO. 19632030 and 19572008, and China Postdoctoral Science Foundation     

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.     

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.     

A generalized self-consistent Mori-Tanaka model for predicting the effective moduli of the coated inclusion based composite materials

The weak point of the generalized self-consistent method (GSCM) is that its solution for the effective shear moduli involves determining the complicated displacement and strain fields in constitutents. Furthermore, the effective moduli estimated by GSCM cannot be expressed in an explicit form. Instead of following the procedure of GSCM, in this paper a generalized self-consistent Mori-Tanaka method (GSCMTM) is developed by means of Hill's interface condition and the assumption that the strain in the inclusion is uniform. A comparison with the existing theoretical and experimental results shows that the present GSCMTM is sufficiently accurate to predict the effective moduli of the coated inclusion-based composite materials. Moreover, it is interesting to find that the application of Hill's interface condition in volumetric domain is equivalent to the Mori-Tanaka average field approximation. This project was supported by the National Natural Science Foundation of China and China Postdoctoral Science Foundation.     

Fracture of magnetoelectroelastic composite materials using boundary element method (BEM)

The behavior of cracked linear magnetoelectroelastic solids is analysed by means of the dual Boundary Element Method (BEM) approach. Media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. An explicit 2-D Green’s function in terms of the extended Stroh formalism for magnetoelectroelastic full-plane under static loading is implemented. Hypersingular integrals arising in the traction boundary integral equations are computed through a regularization technique. Evaluation of fracture parameters directly from computed nodal values is discussed. The stress intensity factors (SIF), the electric displacement intensity factor (EDIF), the magnetic induction intensity factor (MIIF) as well as the mechanical strain energy release rate (MSERR) are evaluated for different crack configurations in both finite and infinite solids subjected to in-plane combined magnetic–electric–mechanical loading conditions. The accuracy of the boundary element solution is confirmed by comparison with selected analytical solutions in the literature. The new results that can be of interest in the design and maintenance of novel magnetoelectroelastic devices are also discussed.     

Asymptotic determination of effective elastic properties of composite materials with fibrous square-shaped inclusions

We propose an asymptotic approach for evaluating effective elastic properties of two-components periodic composite materials with fibrous inclusions. We start with a nontrivial expansion of the input elastic boundary value problem by ratios of elastic constants. This allows to simplify the governing equations to forms analogous to the transport problem. Then we apply an asymptotic homogenization method, coming from the original problem on a multi-connected domain to a so called cell problem, defined on a characterizing unit cell of the composite. If the inclusions' volume fraction tends to zero, the cell problem is solved by means of a boundary perturbation approach. When on the contrary the inclusions tend to touch each other we use an asymptotic expansion by non-dimensional distance between two neighbouring inclusions. Finally, the obtained “limiting” solutions are matched via two-point Padé approximants. As the results, we derive uniform analytical representations for effective elastic properties. Also local distributions of physical fields may be calculated. In some partial cases the proposed approach gives a possibility to establish a direct analogy between evaluations of effective elastic moduli and transport coefficients. As illustrative examples we consider transversally-orthotropic composite materials with fibres of square cross section and with square checkerboard structure. The obtained results are in good agreement with data of other authors.     

Bounds for interpolating complex effective moduli of viscoelastic materials from measured data

The complex effective moduli of viscoelastic materials can be experimentally measured by a number of techniques giving relaxation data for materials at discrete frequencies. In this paper we present a method which allows one to find bounds on the moduli of the material at the unmeasured frequencies. Our bounds can be very tight, and are, therefore, good approximations to the data. Using our bounds, experimentalists can concentrate their efforts on gathering highly accurate data at only a few frequencies and numerically generate data at other frequencies that may be of interest. Received: 10 August 1999 Accepted: 21 December 2000     

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.     

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.     

Stress concentration and effective stiffness of aligned fiber reinforced composite with anisotropic constituents

The paper addresses the problem of calculation of the local stress field and effective elastic properties of a unidirectional fiber reinforced composite with anisotropic constituents. For this aim, the representative unit cell approach has been utilized. The micro geometry of the composite is modeled by a periodic structure with a unit cell containing multiple circular fibers. The number of fibers is sufficient to account for the micro structure statistics of composite. A new method based on the multipole expansion technique is developed to obtain the exact series solution for the micro stress field. The method combines the principle of superposition, technique of complex potentials and some new results in the theory of special functions. A proper choice of potentials and new results for their series expansions allow one to reduce the boundary-value problem for the multiple-connected domain to an ordinary, well-posed set of linear algebraic equations. This reduction provides high numerical efficiency of the developed method. Exact expressions for the components of the effective stiffness tensor have been obtained by analytical averaging of the strain and stress fields.     

Anisotropic effective higher-order response of heterogeneous Cauchy elastic materials

The homogenization results obtained by Bacca et al. (2013a), to identify the effective second-gradient elastic materials from heterogeneous Cauchy elastic solids, are extended here to the case of phases having non-isotropic tensors of inertia. It is shown that the nonlocal constitutive tensor for the homogenized material depends on both the inertia properties of the RVE and the difference between the effective and the matrix local elastic tensors. Results show that: (i) orthotropic nonlocal effects follow from homogenization of a dilute distribution of aligned elliptical holes and, in the limit case, of cracks; (ii) even under the dilute assumption and isotropic local effective behaviour, homogenization may lead to effective nonlocal orthotropic properties.     

Prediction of the overall moduli of a cylindrical short-fiber reinforced composite

With respect to obtaining the effective elastic moduli of the composite, the present theory differs from both Eshelby's equivalent inclusion method and Hill's self-consistent one, both of which only consider the mechanical properties of the matrix and inclusions (fibers). In fact, the inclusion-inclusion interaction is more pronounced when the volume fraction of inclusions of the composite increases. Hence, in this paper the effective elastic moduli of the composite are derived by taking into account the shapes, sizes and distribution of inclusions, and the interactions between inclusions. In addition, it is more convincing to assume short-fibers as cylindrical inclusions as in the present paper than as ellipsoidal ones as in others^[7,8]. Finally, numerical results are given.     

The effective properties of piezoelectric composite materials with transversely isotropic spherical inclusions

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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.     

Bounds on the effective conductivity of statistically isotropic multicomponent materials and random cell polycrystals

Upper and lower bounds on the effective conductivity of statistically isotropic multicomponent materials in d dimensions (d=2 or 3) are constructed from the minimum energy principles and appropriate trial fields. The trial fields, involving harmonic potentials and free parameters to be optimized, lead to the bounds containing up to three-point correlation information about the microgeometry of a composite. The bounds are applied to give estimates for the symmetric cell materials, which are optimal over some ranges of parameters, and asymmetric multicoated spheres, which yield the exact effective conductivity in certain cases. The results also agree with many known ones. New bounds for random cell polycrystals are obtained and illustrated on a number of polycrystalline aggregates.     

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.     

广义自洽Mori-Tanaka模型及涂层夹杂体复合材料的有效模量

经典广义自洽模型的最大不足是需要确定相材料的位移及应变场,这一过程十分繁杂,而且最后得到的有效剪切模量无法显式表达,难以应用.该文摈弃这一经典做法,而从广义自洽模型的应变等价条件出发,在夹杂应变均匀的近似假定下,将Hill界面条件应用于整个二相体内,从而得到一种可以预报涂层夹杂体复合材料有效模量的广义自洽Mori-Tanaka模型.与已有的实验及理论结果的比较表明,该模型准确可靠,而且有效体积和剪切模量均能显式表达.同时证实,以上的Hill界面条件应用于二相体内相当于Mori-Tanaka平均场近似.     

Lower bound of effective yield strength domain for random heterogeneous materials

A random heterogeneous material is represented by a finite family of microstructures, the environment of each microstructure being subjected to a perfect mix condition whose validity is justified both theoretically and numerically. Necessary conditions, satisfied by any permissible strain rate field on a representative and unlimited domain of the material, are highlighted. They enable one to obtain, by solving a discrete infmax problem, a lower bound of the effective yield strength domain of the material, that is rigorous and more predictive than the classical bound of Reuss. An analytical application on a porous medium illustrates the methodology.