含正交排列夹杂和缺陷材料的等效弹性模量和损伤

研究含正交排列夹杂和缺陷材料的等效弹性模量和损伤,推导了以Eshelby－Mori－Tanaka方法求解多相各向异性复合材料等效弹性模量的简便计算公式,针对含三相正交椭球状夹杂的正交各向异性材料,得到了由细观参量（夹杂的形状、方位和体积分数）表示的等效弹性模量的解析表达式．在此基础上,提出了一个宏细观结合的正交各向异性损伤模型,从而建立了以细观量为参量的含损伤材料的应力应变关系．最后,对影响材料损伤的细观结构参数进行了分析．     

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.     

The averaged mechanical properties of prismatic lattice formed by ellipsoidal inclusions

Summary The objective of this paper is to evaluate the averaged elastic properties of 3-D grained composites in which identical inclusions form a prismatic network interacting with the matrix material. The inclusions are of ellipsoidal shape with transverse circular sections located at the nodes of a doubly-periodic lattice with an orthogonal elementary cell. When the arrays of inclusions are set at equal spacings in normal directions through the thickness of the matrix, the material formed is an anisotropic composite with tetragonal symmetry at planes transverse to the fiber axis. The longitudinal and transverse elastic and shear moduli as well as the longitudinal Poisson's ratios of such composites are evaluated in this paper. The averaged properties are studied in terms of the aspect ratio and volume fraction of the inclusions as well as the relative rigidity of the constituent phases. Employing the Eshelby's theory for the stress field around a single ellipsoidal inhomogeneity, which is surrounded by the effective anisotropic material, and considering the Mori-Tanaka's concept for the mutual interaction of the neighboring inclusions, we may evaluate the averaged elastic properties of grained composites with aligned ellipsoidal inclusions at finite concentrations. The results provided in a closed-form solution concern the stiffness of 3-D grained composites with parallely dispersed ellipsoidal inclusions forming a prismatic network inside the principal material. It is shown that the stiffness is affected by both the geometry of the inclusions and their concentration. The use of different composite models in the analysis shows that intense variations of stiffness occur mainly in hard composites weakened by soft ellipsoidal inclusions. These findings come in full verification with experimental or theoretical results from the literature. Received 10 February 1998; accepted for publication 27 November 1998     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Contact analysis in the presence of an ellipsoidal inhomogeneity within a half space

Many materials contain inhomogeneities or inclusions that may greatly affect their mechanical properties. Such inhomogeneities are for example encountered in the case of composite materials or materials containing precipitates. This paper presents an analysis of contact pressure and subsurface stress field for contact problems in the presence of anisotropic elastic inhomogeneities of ellipsoidal shape. Accounting for any orientation and material properties of the inhomogeneities are the major novelties of this work. The semi-analytical method proposed to solve the contact problem is based on Eshelby’s formalism and uses 2D and 3D Fast Fourier Transforms to speed up the computation. The time and memory necessary are greatly reduced in comparison with the classical finite element method. The model can be seen as an enrichment technique where the enrichment fields from the heterogeneous solution are superimposed to the homogeneous problem. The definition of complex geometries made by combination of inclusions can easily be achieved. A parametric analysis on the effect of elastic properties and geometrical features of the inhomogeneity (size, depth and orientation) is proposed. The model allows to obtain the contact pressure distribution – disturbed by the presence of inhomogeneities – as well as subsurface and matrix/inhomogeneity interface stresses. It is shown that the presence of an inclusion below the contact surface affects significantly the contact pressure and subsurfaces stress distributions when located at a depth lower than 0.7 times the contact radius. The anisotropy directions and material data are also key elements that strongly affect the elastic contact solution. In the case of normal contact between a spherical indenter and an elastic half space containing a single inhomogeneity whose center is located straight below the contact center, the normal stress at the inhomogeneity/matrix interface is mostly compressive. Finally when the axes of the ellipsoidal inclusion do not coincide with the contact problem axes, the pressure distribution is not symmetrical.     

Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces

A micromechanical framework is proposed to predict the effective elastic behavior and weakened interface evolution of particulate composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu, J., 1993a. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. Journal of Applied Mechanics 60 (4), 1048–1050; Qu, J., 1993b. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mechanics of Materials 14, 269–281] is adopted to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. Based on the Eshelby’s micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to characterize the varying probability of evolution of weakened interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to assess the potential of the present framework.     

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.     

含柔性涂层的颗粒增强复合材料弹性模量估计

采用线弹簧型弱界面模型来模拟柔性涂层,研究柔性涂层对复合材料宏观弹性模量的影响。首先利用Mori-Tanaka方法和弱界面球形夹杂问题的弹性解,获得单夹杂内部的平均应力和平均应变,进而求得具有柔性涂层的复合材料的宏观弹性模量,并研究界面柔度对复合材料弹性模量的影响。     

A transversely isotropic composite with a statistical distribution of spheroidal inclusions: a geometrical approach to overall properties

In the literature, the determination of global elastic properties of composites with ellipsoidal inclusions is based on the averaged stress, strain and elastic-energy fields (e.g. Compos. Sci. Technol. 27 (1986) 111). These are related to the local fields of the inclusion, the matrix, and the inclusion-matrix interface. In this study, we propose a method to obtain the global elastic properties of any transversely isotropic composite directly from the elastic properties of the matrix and the inclusions. Thus, it is not necessary to refer to the stress and strain applied globally or generated locally. The inclusions can have any transversely isotropic probability distribution of orientation. The problem is entirely geometrized and is treated in terms of averages of Walpole's (Adv. Appl. Mech. 21 (1981) 169) components of the fourth-order tensors describing the problem. We give a general numerical solution for any transversely isotropic statistical distribution of orientation, and also provide a validation of our method by applying it to some known cases and by retrieving the known exact solutions from the literature.     

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.     

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.     

具有光滑界面的椭球夹杂的轴对称弹性场

本文在旋转椭球坐标系下,利用Papkovich—Neuber位移通解求解了具有光滑界面椭球夹杂由于均匀的特征应变引起的轴对称弹性场,与理想界面不同,在夹杂与基体界面不能经受剪应力而可自由滑动的情况下,解答只能是无穷级数形式,因此文中给出了数值算例。     

The axisymmetrical elastic field of an ellipsoidal inclusion with a slipping interface

This paper is concerned with the axisymmetrical elastic fields caused by an ellipsoidal inclusion with a slipping interface which undergoes a uniform eigenstrain. The problem is solved under a revolving ellipsoidal coordinate with the aid of Papkovich-Neuber general dipacement formula. In contrast to the perfectly bonded interface, when the interface between the inclusion and the matrix cannot sustain shear stress, and is free to slip, the solution cannot be expressed in closed form and involves infinite series. Therefore, the results are illustrated by numerical examples.     

Features of the stress field near tunnel inclusions in an inhomogeneous anisotropic space

This paper proposes a method to solve problems for interface tunnel defects in a piecewise-homogeneous elastic material that is under generalized plane strain and has no planes of elastic symmetry. The method is based on integral relations between the discontinuities and sums of the components of the displacement vector and stress tensor at the interface. Closed-form solutions are obtained for a system of interface tunnel inclusions with mixed contact conditions between the space and the inclusions. The dependences of the indices of singularity of the solutions on orthogonal coordinate transformation are established for different combinations of materials of monoclinic and orthorhombic systems. The effect of the antiplane component on the behavior of the solutions is revealed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 36–45, June 2008.     

线夹杂和线夹杂的相互作用

以短纤维复合材料为工程背景，本文利用线夹杂的工程计算模型以及无限平面中单夹杂的基本解，导出了线夹杂和线夹杂相互作用的平面问题的奇异积分方程。给出了夹杂端点的应力强度因子和夹杂界面应力的表达式，并作了具体的数值计算。     

多圆形刚性夹杂的反平面问题

构造任意分布且相互影响的多个圆形刚性夹杂模型的复应力函数，采用复变函数方法，达到满足各个夹杂的边界条件，利用坐标变换和围线积分将求解方程组化为线性代数方程组，推导出了圆形刚性夹杂任意分布的界面应力解析表达式，算例对多夹杂与单夹杂两种模型的界面应力最大值进行了对比，同时还给出了界面应力最大值随夹杂间距的变化规律，求出了刚性夹杂的合理间距。本文发展的分析方法为研究夹杂材料的细观机理探索了一条有效的分析途径。     

压电复合材料中的Eshelby夹杂问题

通过采用解析延拓和共形映射技术，获得了压电复合材料中有关Eshelby夹杂几个典型问题的精确弹性解答，即横观各向同性压电介质中任意形状的Eshelby夹杂与圆柱异相夹杂间相互作用；一般各向异性压电介质中任意形状的Eshelby夹杂与双压电材料所形成界面的相互作用．成功求解这些问题的关健在于构造一个辅助函数．与Ru所采用的方法不同，所引入的辅助函数在无穷远点不存在极点，从而使得所展开的分析更加自然合理．分析结果清楚地揭示出Eshelby夹杂的存在对压电复合材料机电耦合响应将产生不容被忽视的影响．很典型的一个例于是当一个Eshelby椭圆夹杂与圆柱异相夹杂相互作用时，每个夹杂体内部的应力场和电场都将是不均匀的；另一个例于是位于界面附近的Eshelby夹杂有可能是界面发生损伤的一个重要原因．     

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     

Interaction between elliptical and ellipsoidal inclusions under bending stress fields

Summary  This paper deals with interaction problems of elliptical and ellipsoidal inclusions under bending, using singular integral equations of the body force method. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x,y and r,θ,z directions in infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical and the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield the exact solutions for a single elliptical or spherical inclusion under a bending stress field. It yields rapidly converging numerical results for interface stresses in the interaction of inclusions. Received 9 September 1999; accepted for publication 15 January 2000     

纤维端部的界面裂纹分析

基于弹性力学空间轴对称问题的通解,研究了短纤维增强复合材料中纤维端部的轴对称币形和柱形界面裂纹尖端的应力奇异性,得到了裂纹尖端附近的奇异应力场．研究结果表明,这两种轴对称界面裂纹尖端的应力奇异性相同,并且与平面应变状态下相应模型的应力奇异性一致,材料性能对裂纹尖端附近奇异应力场的影响可用三个组合参数描述