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．     

双周期平行四边形排列裂纹反平面问题的有限元分析

提出了双周期平行四边形排列裂纹反平面问题的有限元方法,通过对单位胞元引入周期边界条件,在裂纹尖端采用奇异单元,解决了有限元分析这类问题的效率和精度问题.利用Ansys软件计算,在各种有解析解对照的情形下,应力强度因子的相对误差都在0.2%以内.与现有通常限于对称阵列的双周期裂纹的研究相比,本文发展的方法适用于一般的非对称平行四边形裂纹阵列.算例揭示了行向裂纹间的相互干涉放大应力强度因子,而叠向裂纹间的作用相互屏蔽.对于平行四边形阵列的情形,这两种相反的干涉效应使应力强度因子与裂纹错动参数间呈现非单调依赖关系.     

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.     

Plane problem of elasticity for an anisotropic body with doubly periodic systems of thin inhomogeneities

A system of integral equations of the boundary element method for studying doubly periodic systems of thin inclusions in anisotropic bodies is constructed. Several dependences for determining the mean stresses and strains of a composite with regular systems of thin inhomogeneities are obtained. Numerical procedures of the proposed method are implemented, and generalized stress intensity factors are calculated together with the effective elasticity moduli of a composite with doubly periodic systems of thin elastic inclusions.     

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 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.     

EXACT SOLUTION FOR ORTHOTROPIC MATERIALS WEAKENED BY DOUBLY PERIODIC CRACKS OF UNEQUAL SIZE UNDER ANTIPLANE SHEAR

Orthotropic materials weakened by a doubly periodic array of cracks under far-field antiplane shear are investigated, where the fundamental cell contains four cracks of unequal size. By applying the mapping technique, the elliptical function theory and the theory of analytical function boundary value problems, a closed form solution of the whole-field stress is obtained. The exact formulae for the stress intensity factor at the crack tip and the effective antiplane shear modulus of the cracked orthotropic material are derived. A comparison with the finite element method shows the efficiency and accuracy of the present method. Several illustrative examples are provided, and an interesting phenomenon is observed, that is, the stress intensity factor and the dimensionless effective modulus are independent of the material property for a doubly periodic cracked isotropic material, but depend strongly on the material property for the doubly periodic cracked orthotropic material. Such a phenomenon for antiplane problems is similar to that for in-plane problems. The present solution can provide benchmark results for other numerical and approximate methods.     

涂层对复合材料电-磁-弹性性能的影响

研究带界面层的双周期排列电磁弹性纤维复合材料在远场反平面机械载荷和面内电磁载荷作用下的电-磁-弹性响应,结合本征广义应变(本征应变、本征电场、本征磁场)的概念和双准周期Rimann边值问题理论为其发展了一个严格的解析分析方法.解答可用于对胞元内电磁弹性场量进行精细分析,也可对复合材料有效电磁弹性模量做出准确预测.以纯弹性环氧树脂涂层为例,分析了涂层对应力集中系数和有效电磁弹性模量的影响.本文结果可作为一个"基准",用以评价有限元等数值与近似方法的精度.     

周期分布多边形夹杂奇异性应力干涉问题的研究

采用一种新型的杂交元模型和一种单胞模型来解决周期分布多边形夹杂角部的奇异性应力相互干涉的问题。新型杂交元模型是基于广义Hellinger-Reissner变分原理建立的,其中奇异性应力场分量和位移场分量是采用有限元特征分析法的数值特征解得到的。使用当前的新型杂交元模型,只需要在夹杂角部邻域的周界上划分一维单元,避免了像传统有限元模型那样需要划分高密度二维单元。文中给出了代表奇异性应力场强度的夹杂角部广义应力强度因子数值解,并考虑材料属性、夹杂尺寸和夹杂位置关系的影响。算例中,考虑了夹杂和基体完全接合的情况,并给出了考核例。结果表明:当前模型能得到高精度数值解,且收敛性好;与传统有限元法和积分方程方法相比,该模型更具有通用性,为非均质材料的细观力学分析打下了基础。     

Electroelastic behavior of doubly periodic piezoelectric fiber composites under antiplane shear

The piezoelectric composites with a doubly periodic parallelogrammic array of piezoelectric fibers are dealt with under antiplane shear coupled with inplane electrical load. A rigorous analytical method is developed by using the doubly quasi-periodic Riemann boundary value problem theory integrated with the eigenstrain and eigen-electrical-field concepts. The numerical results are presented and a comparison with finite element calculations, experimental data and micromechanical analysis is made to demonstrate the efficiency and accuracy of the present method. Subsequently, the present solutions are used to study two important topics in piezoelectric fiber composites, i.e., (1) stress and electrical field fluctuations in the microstructure, (2) the macroscopic effective electroelastic moduli. The relation between the macroscopic properties of the composites and their microstructural parameters is discussed and many interesting electroelastic interaction phenomena are revealed, which are useful to estimate and optimize the performance of piezoelectric composites.     

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.     

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.     

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%.     

Generalized stress intensity factors in the interaction within a rectangular array of rectangular inclusions

Summary To evaluate the mechanical strength of fiber-reinforced composites, it is necessary to consider singular stresses at the end of fibers because they cause crack initiation, propagation, and final failure. A square array of rectangular inclusions under longitudinal tension is considered in this paper. The body-force method is applied to a unit cell region. Then, the problem is formulated as a system of singular integral equations, where the unknown functions are the densities of body forces distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. The unknown functions are expressed as piecewise-smooth functions using power series and two types of fundamental densities which express singular stresses. Generalized stress intensity factors at the corners of inclusions are systematically calculated with varying the shape and spacing of a square array of square and rectangular inclusions.     

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.     

Groundwater flow in a medium with periodic inclusions

A study is made of the problem of two-dimensional confined steady flow through a porous reservoir whose percolation coefficient is a step function, while the homogeneity zones are rectangular inclusions (blocks) and a matrix (fractures). The velocity distributions, streamlines, isochrones of the motion of marked particles, and their time of passage through the elementary cell are found for this doubly periodic structure under conditions of continuity of the fluid head and the normal component of the flow on the phase contact boundary. Conclusions concerning the convective component of the longitudinal and transverse dispersion are drawn on the basis of these essentially two-dimensional hydrodynamic characteristics. The exact solutions obtained are compared with the results of numerical calculations carried out by the finite-difference method.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 139–148, September–October, 1995.     

形状记忆合金纤维复合材料的等效力学行为

在Aboudi提出的胞元模型以及Liu等建立的形状记忆合金的本构模型的基础上，由Legendre多项式，假设每个子胞元的位移场、应变场和应力场，再由子胞元间交界面的应力连续条件和外荷载边界条件推导出基体为弹塑性材料的形状记忆合金纤维复合材料的胞元模型；模拟了呈周期对称的形状记忆合金纤维复合材料受轴向单向拉伸、横向拉伸和横向剪切荷载作用下的等效力学行为，与有限元解进行了比较，结果基本一致。与有限元法比较起来，本文推导出的形状记忆合金纤维复合材料的胞元模型更具高效性。     

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。     

An embedding method for modeling micromechanical behavior and macroscopic properties of composite materials

This paper presents a numerical method for modeling the micromechanical behavior and macroscopic properties of fiber-reinforced composites and perforated materials. The material is modeled by a finite rectangular domain containing multiple circular holes and elastic inclusions. The rectangular domain is assumed to be embedded within a larger circular domain with fictitious boundary loading represented by truncated Fourier series. The analytical solution for the complementary problem of a circular domain containing holes and inclusions is obtained by using a combination of the series expansion technique with a direct boundary integral method. The boundary conditions on the physical external boundary are satisfied by adopting an overspecification technique based on a least squares approximation. All of the integrals arising in the method can be evaluated analytically. As a result, the elastic fields and effective properties can be expressed explicitly in terms of the coefficients in the series expansions. Several numerical experiments are conducted to verify the accuracy and efficiency of the numerical method and to demonstrate its application in determination of the macroscopic properties of composite materials.