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．     

双周期圆截面纤维复合材料平面问题的解析法

结合双准周期Riemann边值问题理论与Eshelby等效夹杂原理，为双周期圆截面纤维复合材 料平面问题发展了一个实用有效的解析方法，获得了问题的全场级数解并与有限元结果进行 了比较. 该方法为非均匀材料的力学性质分析和复合材料等新材料的微结构设计提供了 一个有效的计算工具，也可用来评估有限元等数值与近似方法的精度.     

Effective elastic properties of doubly periodic array of inclusions of various shapes by the boundary element method

A rectangular cell of known boundary conditions is cut out from a medium containing the doubly periodic array of inclusions. The stress and strain relationship of the rectangular cell is obtained by using the classical boundary element methods. By matching the boundary condition requirements, the effective elastic properties of the doubly periodic array of inclusions can then be calculated. Numerical examples from the sub-domain boundary element method and the single domain boundary element method are compared and discussed. However, the present method cannot be readily extended to domains having circular or curved boundary parts.     

Effective thermoelastic properties of graded doublyperiodic particulate matrix composites in varying externalstress fields

We consider a linear elastic composite medium, which consists of a homogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in a doubly periodicarray and subjected to inhomogeneous boundary conditions. The hypothesis of effective fieldhomogeneity near the inclusions is used. The general integral equation obtained reduces theanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusions insome representative volume element (RVE) . The integral equation is solved by a modifiedversion of the Neumann series; the fast convergence of this method is demonstrated for concreteexamples. The nonlocal macroscopic constitutive equation relating the cell averages of stress andstrain is derived in explicit iterative form of an integral equation. A doubly periodic inclusion fieldin a finite ply subjected to a stress gradient along the functionally graded direction is considered.The stresses averaged over the cell are explicitly represented as functions of the boundaryconditions. Finally, the employed of proposed explicit relations for numerical simulations oftensors describing the local and nonlocal effective elastic properties of finite inclusion pliescontaining a simple cubic lattice of rigid inclusions and voids are considered. The local andnonlocal parts of average strains are estimated for inclusion plies of different thickness. Theboundary layers and scale effects for effective local and nonlocal effective properties as well as foraverage stresses will be revealed.     

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.     

Inverse doubly periodic problem of the theory of bending of a plate with elastic inclusions

Based on the balanced strength principle, a problem of determining the optimal interference for fitting elastic inclusions into holes of an isotropic elastic plate weakened by a doubly periodic system of circular holes is solved. A closed system of algebraic equations is derived, which allows solving this problem. The resultant interference increases the load-carrying capacity of the composite plate being bent. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 153–161, July–August, 2006.     

双周期圆柱形压电夹杂反平面问题的精确解及其应用

研究无限压电介质中双周期圆柱形压电夹杂的反平面问题.借鉴Eshelby等效夹杂原理,通过引入双周期非均匀本征应变和本征电场,构造了一个与原问题等价的均匀介质双周期本征应变和本征电场问题.利用双准周期Riemann边值问题理论,获得了夹杂内外严格的电弹性解.作为压电纤维复合材料的一个重要模型,预测了压电纤维复合材料的有效电弹性模量.     

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.     

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.     

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.     

Effective elastic shear stiffness of a periodic fibrous composite with non-uniform imperfect contact between the matrix and the fibers

In this contribution, effective elastic moduli are obtained by means of the asymptotic homogenization method, for oblique two-phase fibrous periodic composites with non-uniform imperfect contact conditions at the interface. This work is an extension of previous reported results, where only the perfect contact for elastic or piezoelectric composites under imperfect spring model was considered. The constituents of the composites exhibit transversely isotropic properties. A doubly periodic parallelogram array of cylindrical inclusions under longitudinal shear is considered. The behavior of the shear elastic coefficient for different geometry arrays related to the angle of the cell is studied. As validation of the present method, some numerical examples and comparisons with theoretical results verified that the present model is efficient for the analysis of composites with presence of imperfect interface and parallelogram cell. The effect of the non uniform imperfection on the shear effective property is observed. The present method can provide benchmark results for other numerical and approximate methods.     

Prediction of the effective elastic properties of spheroplastics by the generalized self-consistent method

The problem of predicting the effective elastic properties of composites with prescribed random location and radius variation in spherical inclusions is solved using the generalized self-consistent method. The problem is reduced to the solution of the averaged boundary-value problem of the theory of elasticity for a single inclusion with an inhomogeneous transition layer in a medium with desired effective elastic properties. A numerical analysis of the effective properties of a composite with rigid spherical inclusions and a composite with spherical pores is carried out. The results are compared with the known solution for the periodic structure and with the solutions obtained by the standard self-consistent methods. Perm’ State Technical University, Perm’ 614600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 186–190, May–June, 1999.     

周期裂纹和刚性线尖端场干涉效应研究

研究双周期裂纹和刚性线夹杂非均匀材料的反平面剪切问题。基于保角变换技术和椭圆函数理论,获得了问题应力场的全场精确解,给出了裂纹和刚性线尖端应力强度因子的封闭形式解答,讨论了裂纹和刚性线尖端场的干涉效应。数值结果表明：改变水平和垂直分布周期对裂纹和刚性线尖端场影响明显不同;裂纹长度2a逐渐增大时(0≤a/ω1≤0.5),裂纹尖端应力强度因子从1逐渐增大到无限大,而刚性线的尖端场变化不大;刚性线长度2d逐渐增大时(0≤d/ω2≤1),刚性线尖端应力强度因子逐渐减小,而裂纹的尖端场仅略微增大。     

The interaction of doubly periodic cracks

In this paper, an extremely accurate and efficient method for computing the interaction of a set of or multiple sets of general doubly periodic cracks has been presented on the basis of superposition principle, pseudo-traction method, and isolating analysis technique. A great number of typical examples are given in this paper. The stress intensity factors (SIF), the minimum strain energy density factors (SED) of crack tip and the critical stress (CRS) of crack growth are calculated with the accuracy of six significant digits for the rectangularly distributed periodic cracks and five significant digits for the general doubly periodic cracks. The relation of the interaction effect of the double periodic cracks with the periods and the ratio of crack length to crack spacing is analyzed. Also in this paper, the key technique problems for this method are discussed.     

The influence of porosity on the elastic response of homogeneous and structural composite media

Continuum theories of composites are employed to analyze the influence of inclusions and porosity on the elastic response of both homogeneous and laminated composite media. The general model analyzed consists of a periodic array of two perfectly bonded laminates; one of which consists of an elastic homogeneous material while the other is made up of a periodic array of cylindrical elastic inclusions that are distributed in another elastic matrix material. Several specific models are deduced as special cases. In all cases, porosity is simulated in the limit as the properties of the inclusions identically vanish. It is demonstrated that porosity plays a major role in the geometric dispersion of such media; in particular, it increases the arrival and rise times (spreading) of a propagating transient pulse. For the special case of elastic inclusions in a homogeneous matrix media, the present results correlate very well with existing experimental data and other approximate analyses.     

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.     

Long wavelength inner-resonance cut-off frequencies in elastic composite materials

We revisit an ancient paper (Auriault and Bonnet, 1985) which points out the existence of cut-off frequencies for long acoustic wavelength in high-contrast elastic composite materials, i.e. when the wavelength is large with respect to the characteristic heterogeneity length. The separation of scales enables the use of the method of multiple scale expansions for periodic structures, a powerful upscaling technique from the heterogeneity scale to the wavelength scale. However, the results remain valid for non-periodic composite materials which show a Representative Elementary Volume (REV). The paper extends the previous investigations to three-component composite materials made of hard inclusions, coated with a soft material, both of arbitrary geometry, and embedded in a connected stiff material. The equivalent macroscopic models are rigorously established as well as their domains of validity. Provided that the stiffness contrast within the soft and the connected stiff materials is of the order of the squared separation of scales parameter, it is demonstrated (i) that the propagation of long wave may coincide with the resonance frequencies of the hard inclusions/soft material system and (ii) that the macroscopic model presents a series of cut-off frequencies given by an eigenvalue problem for the resonating domain in the cell. These results are illustrated in the case of stratified composites and the possible microstructures of heterogeneous media in which the inner dynamics phenomena may occur are discussed.     

Subharmonic Melnikov method of six-dimensional nonlinear systems and application to a laminated composite piezoelectric rectangular plate

In this paper, the bifurcations of subharmonic orbits are investigated for six-dimensional non-autonomous nonlinear systems using the improved subharmonic Melnikov method. The unperturbed system is composed of three independent planar Hamiltonian systems such that the unperturbed system has a family of periodic orbits. The key problem at hand is the determination of the sufficient conditions on some of the periodic orbits for the unperturbed system to generate the subharmonic orbits after the periodic perturbations. Using the periodic transformations and the Poincaré map, an improved subharmonic Melnikov method is presented. Two theorems are obtained and can be used to analyze the subharmonic dynamic responses of six-dimensional non-autonomous nonlinear systems. The subharmonic Melnikov method is directly utilized to investigate the subharmonic orbits of the six-dimensional non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Using the subharmonic Melnikov method, the bifurcation function of the subharmonic orbit is obtained. Numerical simulations are used to verify the analytical predictions. The results of the numerical simulation also indicate the existence of the subharmonic orbits for the laminated composite piezoelectric rectangular plate.     

A local finite element implementation for imposing periodic boundary conditions on composite micromechanical models

Recent advances in computational speed have resulted in the ability to model composite materials using larger representative volume elements (RVEs) with greater numbers of inclusions than have been previously studied. Imposing periodic boundary conditions on very large RVEs can mean enforcing thousands of constraint equations. In addition, a periodic mesh is essential for enforcing the constraints. The present study investigates a method that uses a local implementation of the constraints that does not adversely affect the computational speed. The present study demonstrates the method for two-dimensional triangular and square RVEs of periodically-spaced regular hexagonal and square arrays of composite material containing fibers of equal radii. To impose the boundary conditions along the edges, this study utilizes a cubic interpolant to model the displacement field along the matrix edges and a linear interpolant to model the field along the fiber edges. It is shown that the method eliminates the need for the conventional node-coupling scheme for imposing periodic boundary conditions, consequently reducing the number of unknowns to the interior degrees of freedom of the RVE along with a small number of global parameters. The method is demonstrated for periodic and non-periodic mesh designs.     

Effective properties of elastic periodic composite media with fibers

A series solution to obtain the effective properties of some elastic composites media having periodically located heterogeneities is described. The method uses the classical expansion along Neuman series of the solution of the periodic elasticity problem in Fourier space, based on the Green's tensor, and exact expressions of factors depending on the shape of the inclusions. Some properties of convergence of the solution are presented, more specifically concerning the elasticity tensor of the reference medium, showing that the convergence occurs even for empty fibers. The solution is extended for rigid inclusions. A comparison is made with previous exact solutions for a fiber composite made of cylindrical fibers with circular cross-sections and with previous estimates. Different examples are presented for new situations concerning the study of fiber composites: composites with elliptic cross-sections and multi-phase fibrous composites.