双周期圆柱形夹杂纵向剪切问题的精确解

研究无限介质中矩形排列双周期圆柱形夹杂的纵向剪切问题．利用Eshelby等效夹杂理论并结合双周期与双准周期解析函数工具，为这类考虑夹杂相互影响的问题提供了一个严格又实用的分析方法，求得了问题的全场级数解．作为退化情形得到单夹杂问题的经典解答，双周期孔洞、双周期刚性夹杂及单行(列)周期弹性夹杂等问题也可作为特殊情况被解决．数值结果揭示了这类非均匀材料力学性质随微结构参数变化的规律．     

Interfacial debonding of two-phase composite with periodic structure

A mixed analytical-numerical (boundary element method) procedure is presented for estimating the effective elastic moduli of a two-phase periodic composite by application of a unit cell. The two-phase composite consists of a metal/polymer matrix and one/three circular ceramic inclusions with adhesive and partial debonding of the interface. The results are displayed numerically with special attention given to development of plastic zones as debonding occurs. Dependence of load-time history is exhibited.     

Differential schemes for the elastic characterisation of dispersions of randomly oriented ellipsoids

The paper deals with the elastic characterisation of dispersions of randomly oriented ellipsoids: we start from the theory of strongly diluted mixtures and successively we generalise it with a differential scheme. The micro-mechanical averaging inside the composite material is carried out by means of explicit results which allows us to obtain closed-form expressions for the macroscopic or equivalent elastic moduli of the overall composite materials. This micromechanical technique has been explicitely developed for describing embeddings of randomly oriented not spherical objects. In particular, this study has been applied to characterise media with different shapes of the inclusions (spheres, cylinders and planar inhomogeneities) and for special media involved in the mixture definition (voids or rigid particles): an accurate analysis of all these cases has been studied yielding a set of relations describing several composite materials of great technological interest. The differential effective medium scheme (developed for generally shaped ellipsoids) extends such results to higher values of the volume fraction of the inhomogeneities embedded in the mixture. For instance, the analytical study of the differential scheme for porous materials (with ellipsoidal zero stiffness voids) reveals a universal behaviour of the effective Poisson ratio for high values of the porosity. This means that Poisson ratio at high porosity assumes characteristic values depending only on the shape of the inclusions and not on the elastic response of the matrix.     

Stress concentrations in the particulate composite with transversely isotropic phases

The accurate series solution have been obtained of the elasticity theory problem for a transversely isotropic solid containing a finite or infinite periodic array of anisotropic spherical inclusions. The method of solution has been developed based on the multipole expansion technique. The basic idea of method consists in expansion the displacement vector into a series over the set of vectorial functions satisfying the governing equations of elastic equilibrium. The re-expansion formulae derived for these functions provide exact satisfaction of the interfacial boundary conditions. As a result, the primary spatial boundary-value problem is reduced to an infinite set of linear algebraic equations. The method has been applied systematically to solve for three models of composite, namely a single inclusion, a finite array of inclusions and an infinite periodic array of inclusions, respectively, embedded in a transversely isotropic solid. The numerical results are presented demonstrating that elastic properties mismatch, anisotropy degree, orientation of the anisotropy axes and interactions between the inclusions can produce significant local stress concentration and, thus, affect greatly the overall elastic behavior of composite.     

Elastic-plastic behavior of elastically homogeneous materials with a random field of inclusions

A multiphase material is considered, which consists of a homogeneous elastic-plastic matrix containing a homogeneous statistically uniform random set of ellipsoidal elastic-plastic inclusions. The elastic properties of the matrix and the inclusions are the same, but the so-called “stress-free strains”, i.e. the strain contributions due to temperature loading, phase transformations, and the plastic strains, fluctuate. A general theory of the yielding for arbitrary loading (by the stress and by the temperature) is employed. The realization of an incremental plasticity scheme is based on averaging over each component of the nonlinear yield criterion. Usually, averaged stresses are used inside each component for this purpose. In distinction from this usual practice physically consistent assumptions about the dependence of these functions on the component's values of the second stress moments are applied. The application of the proposed theory to the prediction of the thermomechanical deformation behavior of a model material is shown.     

Bond stresses for a composite material reinforced with various arrays of fibers

The problem treated here is that of an isotropic body having a doubly periodic rectangular or triangular array of perfectly bonded circular elastic inclusions. The body is in tension or compression. This simulates a composite material wherein a relatively weak matrix is reinforced by stronger (and more rigid) fibers. Bond stresses for both rectangular and triangular arrays have been calculated using either boundary point matching or boundary point least squares techniques. Numerical results based on a plane strain analysis are given in graphical form.     

Interaction of Anisotropic Ellipsoidal Inclusions in an Isotropic Medium under Mechanical Loads and Uniform Heating

Eshelby's equivalent-inclusion method is extended to a finite number of arbitrarily oriented anisotropic ellipsoidal inclusions in an elastic isotropic matrix under polynomial mechanical loading and heating. The interaction of two identical and two different triaxial ellipsoidal inclusions in an elastic medium is studied as numerical examples. In special cases, the results are compared with those obtained by other authors     

A micromechanically based couple-stress model of an elastic orthotropic two-phase composite

We determine couple-stress moduli and characteristic lengths of a two-dimensional matrix-inclusion composite, with inclusions arranged in a periodic square array and both constituents linear elastic of Cauchy type. In the analysis we replace this composite by a homogeneous planar, orthotropic, couple-stress continuum. A generalization of the original Mindlin's (1963) derivation of field equations for such a continuum results in two (not just one!) characteristic lengths. We evaluate the couple-stress properties from the response of a unit cell under several types of boundary conditions: displacement, displacement-periodic, periodic and mixed, and traction controlled. In the parametric study we vary the stiffness ratio of both phases to cover a range of different media from nearly porous materials through composites with very stiff inclusions. We find that the aforementioned boundary conditions result in hierarchies of orthotropic couple-stress moduli, whereas both characteristic lengths are fairly insensitive to boundary conditions, and fall between 0.12% and 0.22% of the unit cell size for the inclusions' volume fraction of 18%.     

Stokes flow through microstructural model of fibrous media

Viscous flow through a two-dimensional model of fibrous media is analyzed. The periodic model is based on array of cylindrical fibres on a triangular lattice. The numerical approach consists in satisfying boundary conditions by using a least squares method applied to the analytical solution of Stokes motion around a cylinder. Our interest is in the drag force exerted on the array, and hence the permeability of such arrays. Results are given over the full porosity range. Furthermore, the flow structure analysis is carried out as a function of porosity. For low porosity values, recirculation motion composed of Stokes eddies are obtained that play an important role in pollution phenomena.     

On collinear microcrack–inclusion arrays interaction and related problems

Investigated is a crack problem for an array of collinear microcracks in composite matrix. Inclusions are situated in between the neighbouring microcracks tips and exhibit different elastic properties than matrix. The problem is solved using the technique of distributed dislocations. A developed approximate fundamental solution for a single dislocation lying in a general point between inclusions is employed in the distribution of continuously distributed dislocation to cracks modelling. Stress intensity factor is calculated for various cracks/inclusions geometries and elastic moduli mismatches. Stability and/or instability of the straight microcrack paths is investigated for slowly growing microcracks with inclusions located in between the neighbouring microcracks tips. Applications to periodic microcrack tunnelling and microcracks weakening ahead of the main crack are discussed.     

Heterogeneous linearly piezoelectric patches bonded on a linearly elastic body

In [1], we studied the response of a thin homogeneous piezoelectric patch perfectly bonded to an elastic body. Here we extend this study to the case of a very thin heterogeneous patch made of a periodic distribution of piezoelectric inclusions embedded in a linearly elastic matrix and perfectly bonded to an elastic body. Through a rigorous mathematical analysis, we show that various asymptotic models arise, depending on the electromechanical loading together with the relative behavior between the thickness of the patch and the size of the piezoelectric inclusions.     

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.     

Elastodynamic analysis of inhomogeneous anisotropic bodies

The integral equation method is presented for elastodynamic problems of inhomogeneous anisotropic bodies. Since fundamental solutions are not available for general inhomogeneous anisotropic media, we employ the fundamental solution for homogeneous elastostatics. The terms induced by material inhomogeneity and inertia force are regarded as body forces in elastostatics, and evaluated in the form of volume integrals. The scattering problems of elastic waves by inhomogeneous anisotropic inclusions are investigated for some test cases. Numerical results show the significant effects of inhomogeneity and anisotropy of materials on wave propagations.     

Localization of elastic deformation in strongly anisotropic, porous, linear materials with periodic microstructures: Exact solutions and dilute expansions

Exact solutions are derived for the problem of a two-dimensional, infinitely anisotropic, linear-elastic medium containing a periodic lattice of voids. The matrix material possesses either one infinitely soft, or one infinitely hard loading direction, which induces localized (singular) field configurations. The effective elastic moduli are computed as functions of the porosity in each case. Their dilute expansions feature half-integer powers of the porosity, which can be correlated to the localized field patterns. Statistical characterizations of the fields, such as their first moments and their histograms are provided, with particular emphasis on the singularities of the latter. The behavior of the system near the void close-packing fraction is also investigated. The results of this work shed light on corresponding results for strongly non-linear porous media, which have been obtained recently by means of the “second-order” homogenization method, and where the dilute estimates also exhibit fractional powers of the porosity.     

ELASTIC SH WAVE PROPAGATION IN PERIODIC LAYERED COMPOSITES WITH A PERIODIC ARRAY OF INTERFACE CRACKS

The interaction of anti-plane elastic SH waves with a periodic array of interface cracks in a multi-layered periodic medium is analyzed in this paper. A perfect periodic structure without interface cracks is first studied and the transmission displacement coefficient is obtained based on the transfer matrix method in conjunction with the Bloch-Floquet theorem. This is then generalized to a single and periodic distribution of cracks at the center interface and the result is compared with that of perfect periodic cases without interface cracks. The dependence of the transmission displacement coefficient on the frequency of the incident wave, the influences of material combination, crack configuration and incident angle are discussed in detail. Compared with the corresponding perfect periodic structure without interface cracks, a new phenomenon is found in the periodic layered system with a single and periodic array of interface cracks.     

Asymptotic Analysis of High-Contrast Phononic Crystals and a Criterion for the Band-Gap Opening

We investigate the band-gap structure of the frequency spectrum for elastic waves in a high-contrast, two-component periodic elastic medium. We consider two-dimensional phononic crystals consisting of a background medium which is perforated by an array of holes periodic along each of the two orthogonal coordinate axes. In this paper we establish a full asymptotic formula for dispersion relations of phononic band structures as the contrast of the shear modulus and that of the density become large. The main ingredients are integral equation formulations of the solutions to the harmonic oscillatory linear elastic equation and several theorems concerning the characteristic values of meromorphic operator-valued functions in the complex plane, such as the generalized Rouché’s theorem. We establish a connection between the band structures and the Dirichlet eigenvalue problem on the elementary hole. We also provide a criterion for exhibiting gaps in the band structure which shows that smaller the density of the matrix is, the wider the band-gap is, provided that the criterion is fulfilled. This phenomenon was reported by Economou and Sigalas (J Acoust Soc Am 95:1734–1740, 1994) who observed that periodic elastic composites whose matrix has lower density and higher shear modulus compared to those of inclusions yield better open gaps. Our analysis in this paper agrees with this experimental finding.     

Influences of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture

Local mechanical properties in aluminum cast components are inhomogeneous as a consequence of spatial distribution of microstructure,e.g.,porosity,inclusions,grain size and arm spacing of secondary dendrites.In this work,the effect of porosity is investigated.Cast components contain voids with different sizes,forms,orientations and distributions.This is approximated by a porosity distribution in the following.The aim of this paper is to investigate the influence of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture.A micromechanical model with a spherical void located at the center of the matrix material,called the representative volume element(RVE),is developed.Fully periodic boundary conditions are applied to the RVE and the values of stress triaxiality and Lode parameter are kept constant during the entire course of loading.For this purpose,a multi-point constraint(MPC)user subroutine is developed to prescribe the loading.The results of the RVE model are used to establish the constitutive equations and to further investigate the influences of initial porosity,stress triaxiality and Lode parameter on elastic constant,plastic deformation and ductile fracture of an aluminum die casting alloy.     

The effective thermoelectroelastic properties of microinhomogeneous materials

The paper is concerned with composite materials which consist of a homogeneous matrix phase with a set of inclusions uniformly distributed in the matrix. The components of these materials are considered to be ideally elastic and exhibit piezoelectric properties. One of the variants of the self-consistent scheme, the Effective Field Method (EFM) is applied to calculate effective dielectric, piezoelectric and thermoelastic properties of such materials, taking into account the coupled electroelastic effects. At first the coupled thermoelectroelastic problem for a homogeneous medium with an isolated inclusion is solved. For an ellipsoidal inclusion and constant external field the solution of this problem is found in a closed analytic form. This solution is then used in the EFM to derive the effective thermoelectroelastic operator for the composite containing a random array of ellipsoidal inclusions. Explicit formulae for the electrothermoelastic constants are given for composites, reinforced by spheroidal inclusions.     

Effective elastic moduli of triply periodic particulate matrix composites with imperfect unit cells

In many problems the material may possess a periodic microstructure formed by the spatial repetition of small microstructures, or unit cells. Such a perfectly regular distribution, of course, does not exist in actual cases, although the periodic modeling can be quite useful, since it provides rigorous estimations with a priori prescribed accuracy for various material properties. Triply periodic particulate matrix composites with imperfect unit cells are analyzed in this paper. The multiparticle effective field method (MEFM) is used for the analysis of the perfect and imperfect periodic structure composites. The MEFM is originally based on the homogeneity hypothesis (H1) (see for details [Buryachenko, V.A., 2001. Multiparticle effective field and related methods in micromechanics of composite materials. Appl. Mech. Rev. 54, 1–47]) of effective field acting on the inclusions. In this way the pair interaction of different inclusions is taken directly into account by the use of analytical approximate solution. For perfect periodic structures the hypothesis (H1) is enough for estimation of effective properties. Imperfection of packing necessitates exploring some additional assumption called a closing hypothesis. The next imperfections are analyzed. (A) The probability of location of an inclusion in the center of a unit cell below one (missing inclusion). (B) Some hard inclusions are randomly replaced by the porous (modeling the complete debonding) with some probability. At first, one obtains general explicit integral representations of the effective elastic moduli and strain concentrator factors depending on three numerical solutions: for the perfect periodic structure, for the infinite periodic structure with one imperfection, and for the infinite periodic structure with two arbitrary located imperfections. The method proposed is general; it is not limited by concrete numerical scheme. No restrictions were assumed on both the concrete microstructure and inhomogeneity of stress fields in the inclusions. The inclusions of one kind are assumed to be aligned. The problem (A) is solved at the level of numerical results obtained in the framework of the hypothesis (H1). For the problem (B) the numerical results are obtained if the elastic inclusions (for example hard inclusions) are randomly replaced by another inclusion (for example by the voids modeling the complete debonding). The mentioned problems are solved by three methods. The first one is a Monte Carlo simulation exploring an analytical approximate solution for the binary interacting inclusions obtained in the framework of the hypothesis (H1). The second one is a generalization of the version of the MEFM proposed for the analysis of the perfect periodic particulate composites and based on the choice of a comparison medium coinciding with the matrix. The third method uses a decomposition of the desired solution on the solution for the perfect periodic structure and on the perturbation produced by the imperfections in the perfect periodic structure. All three methods lead to close results in the considered examples; however, the CPU times expended for the solution estimation by Monte Carlo simulation differ by a factor of 1000.     

Local effective thermoelastic properties of graded random structure matrix composites

Summary  We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of ellipsoidal uncoated or coated inclusions, where the concentration of the inclusions is a function of the coordinates (functionally graded material). Effective properties, such as compliance and thermal expansion coefficient, as well as first statistical moments of stresses in the components are estimated for the general case of inhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the Green function technique as well as on the generalization of the multiparticle effective field method (MEFM), previously proposed for the research of statistically homogeneous random structure composites. The hypothesis of effective field homogeneity near the inclusions is used; nonlocal effects of overall constitutive relations are not considered. Nonlocal dependences of local effective thermoelastic properties as well as those of conditional averages of the stresses in the components on the concentration of the inclusions are demonstrated. Received 11 November 1999; accepted for publication 4 May 2000