Average stress in the matrix and effective moduli of randomly oriented composites

By considering the variation of average stress in the matrix, some elastic properties of randomly oriented composites are established as a function of aspect ratio. Both three- and two-dimensional random orientations, resulting, respectively, in a complete and transverse isotropy, are considered. As the shape of inclusions changes, the isotropic bulk and shear moduli are shown to vary within the Hashin-Shtrikman bounds. The aspect ratio dependence of the five in-plane and out-of-plane moduli with planar orientations are also explicitly given; these results suggest that the in-plane properties are most effectively reinforced by fibrous inclusions, whereas the out-of-plane ones are more responsive to the disc type. The accompanying variations of average stress in the matrix are seen to be closely related to the corresponding variations of the moduli. Comparisons with some limited experimental data also show a reasonable agreement.     

The joint effect of shape and size of reinforcement on progressive damage of random composites

The properties of composites made by placing inclusions in a matrix are often controlled by the shape and size of the particles. In order to study the joint effect of shape and size of inclusions, we characterize the random shape of particles in composite mathematically by applying Fourier series, then generating random mesostructure of composite for cases of inclusions with (1) same size and different shape, (2) different size and same shape, or (3) random size and shape. Crack paths and effective stress–strain curves of these cases are predicted using spring network method which is given in detail. The study shows we need more elaborate statistical evaluation due to the random nature of composites. This paper outlines an approach to study effect of inclusion geometry on the elastic properties and crack of random composites.     

有限变形下含非完美界面复合材料有效模量的界限

导出了有限变形下含非完美界面两相复合材料的上下界限。在微小变形下, 所导出的界限还原成已知的线性情况下的相应结果。数值上预示了非完美界面特性、有限变形对复合材料有效模量的影响规律, 给出了与实际相符合的结果。      

Bounds on the effective thermal conductivity of composites with imperfect interface

A new model is developed to bound the effective thermal conductivity of composites with thermal contact resistance between spherical inclusions and matrix. To construct the trial temperature and heat flux fields which satisfy the necessary interface conditions, the transition layer for each spherical inclusion is introduced. For the upper bound, the trial temperature field needs to satisfy the thermal contact resistance conditions between spherical inclusions and transition layers and the continuous interface conditions between transition layers and remnant matrix. For the lower bound, the trial heat flux field needs to satisfy the continuous interface conditions between different regions. It should be pointed out that the continuous interface conditions mentioned above are absolutely necessary for the application of variational principles, and the thermal contact resistance conditions between spherical inclusions and transition layers are suggested by the author. According to the principles of minimum potential energy and minimum complementary energy, the bounds of the effective thermal conductivity of composites with imperfect interfaces are rigorously derived. The effects of the size and distribution of spherical inclusions on the bounds of the effective thermal conductivity of composites are analyzed. It should be shown that the present method is simple and does not need to calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides an entirely different way to bound the effective thermal conductivity of composites with imperfect interface, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds are finite when the thermal conductivity of spherical inclusions tends to ∞ and 0, respectively.     

The mechanical properties of ternary composites of polypropylene with inorganic fillers and elastomer inclusions

The effect of elastomer volume fraction and phase morphology on the elastic modulus of ternary composites polypropylene (PP)/ethylene-propylene rubber (EPR)/inorganic filler containing 30 vol % of either spherical or lamellar filler has been investigated. Phase morphology was controlled using maleated polypropylene (MPP) and/or maleated ethylene-propylene elastomer (MEPR). As revealed by SEM observations, composites of MPP/EPR/filler exhibit separation of the filler and elastomer and good adhesion between MPP and the filler, whereas composites of PP/MEPR/filler exhibit encapsulation of the filler by MEPR. Composite models were utilized to estimate upper and lower bounds for the elastic modulus of these materials, which is strongly dependent on the morphology of the ternary composite. A model based on the Kerner equation for perfect separation of the soft inclusions and rigid fillers gives a good prediction of the upper limit for relative elastic modulus as a function of filler and elastomer volume fractions. The lower limit, achieved in the case of perfect encapsulation, depends significantly on the particle shape. Good agreement was found between experimental data and lower limits predicted using the Halpin-Tsai equation for lamellar filler and the Kerner-Nielsen equation for spherical filler. In order to calculate reinforcing efficiency of the core-shell inclusions, the finite element method (ANSYS 4.4A, GT STRUDL) has been used.     

Rigorous bounds on the effective thermal conductivity of composites with ellipsoidal inclusions

A new method is developed to derive the bounds of the effective thermal conductivity of composites with ellipsoidal inclusions. The transition layer for each ellipsoidal inclusion is introduced to make the trial temperature field for the upper bound and the trial heat flux 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 bounds of the effective thermal conductivity 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 thermal conductivity of composites are analyzed. It should be shown that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides a powerful way to bound the effective thermal conductivity of composites, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds still are finite when the thermal conductivity of ellipsoidal inclusions tends to ∞ and 0, respectively.     

Hybrid effective medium approximations for random elastic composites

Several popular effective medium approximations for elastic constants of random composites are reformulated in terms of a pair of canonical functions and their transform variables. This choice of reformulation enables easier comparisons of the results of all these methods with rigorous bounds. Furthermore, insight into the various methods gained by taking this point of view suggests a number of new effective medium approximations that, in some cases, are natural variants and/or combinations (i.e., hybrids) of the existing ones, and in other cases are new ones based in part on the bounds themselves. Numerical comparisons are given for several standard inclusion models — including spherical, needle, and penny-shaped inclusions — as well as the penetrable sphere model. Of the various alternatives considered, a new method called the split-step differential (SSD) scheme is one of the more useful ones, as it simplifies the differential scheme by replacing half of this scheme’s integration routines with a simple update formula for the bulk modulus.     

Bounds on effective magnetic permeability of three-phase composites with coated spherical inclusions

An effective model is developed to bound the effective magnetic permeability of three-phase composites with coated spherical inclusions. In the present model, the trial magnetic potential for the upper bound and the trial magnetic induction field for the lower bound are constructed to satisfy continuity interface conditions. According to the variational principle, the upper and lower bounds on the effective magnetic permeability of three-phase composites with coated spherical inclusions are derived. In this paper, trial magnetic potentials with different function forms are taken and the optimal upper bound is obtained for the trail magnetic potential corresponding to the third-order function. When the three-phase model degenerates into the composite spheres assemblage model [1], it is interesting that the optimal upper and lower bounds are the same. The effects of the volume fraction of coated spherical inclusions and the thickness and magnetic permeability of coated layers between the matrix and spherical inclusions on the effective magnetic permeability of composites are analyzed. The upper and lower bounds are finite non-zero values when the magnetic permeability of spherical inclusions tends to ∞$\infty$ and 0, respectively.     

Some elastic properties of reinforced solids,with special reference to isotropic ones containing spherical inclusions

Based on Mori and Tanaka's concept of “average stress” in the matrix and Eshelby's solutions of an ellipsoidal inclusion, an approximate theory is established to derive the stress and strain state of constituent phases, stress concentrations at the interface, and the elastic energy and overall moduli of the composite. Both “stress-free” strain (polarization strain) and “strain-free” stress (polarization stress) are employed in these derivations under the traction- and displacement-prescribed conditions. The theory was developed first for a general multiphase, anisotropic composite with arbitrarily oriented anisotropic inclusions; explicit results are then given for a suspension of uniformly distributed, multiphase isotropic spheres in an isotropic matrix. Numerical results for stress concentrations in the spherical inclusions and at the interface are given for a 2-phase composite. Further, it is shown that the derived moduli are related to the Hashin-Shtrikman bounds and that, when the shear moduli are equal, the overall bulk modulus of a 2-phase composite reduces to Hill's exact solution. As compared with experimental data, the theory also provides reasonably accurate estimates for the Young's modulus of some 2- and 3-phase composites.     

Scale and boundary conditions effects in elastic properties of random composites

Summary We study elastic anti-plane responses of unidirectional fiber-matrix composites. The fibers are of circular cylinder shape, aligned in the axial direction, and arranged randomly, with no overlap, in the transverse plane. We assume that both fibers and matrix are linear elastic and isotropic. In particular, we focus on the effects of scale of observation and boundary conditions on the overall anti-plane (axial shear) elastic moduli. We conduct this analysis numerically, using a two-dimensional square spring net-work, at the mesoscale level. More specifically, we consider finite windows of observation, which we increase in size. We subject these regions to several different boundary conditions: displacement-controlled, traction-controlled, periodic, and mixed (combination of any of the first three) to evaluate the mesoscale moduli. The first two boundary conditions give us scale-dependent bounds on the anti-plane elastic moduli. For each boundary condition case we consider many realizations of the random composite to obtain statistics. In this parametric study we cover a very wide range of stiffness ratios ranging from composites with very soft inclusions (approximating holes) to those with very stiff inclusions (approaching rigid fibers), all at several volume fractions.     

Variational bounds for the effective electroelastic moduli of piezoelectric composites with electromechanical coupling spring-type interfaces

The present work focuses on variational bounds for the effective electroelastic moduli of multiphase piezoelectric composites with thin piezoelectric interphase. Both the inhomogeneities and the matrix are assumed to be piezoelectric and transversely isotropic. The piezoelectric interphase is modeled as the spring-type interface with electromechanical coupling. The inhomogeneities are assumed to be spheroidal so that the reinforcement geometry is able to range from thin flake to continuous fiber. The effective properties of the piezoelectric composite with interfacial imperfection are defined and the principles of minimum internal energy and enthalpy are derived. These principles are applied to analytically obtain the upper and lower bounds for the effective electroelastic moduli. Unlike the Voigt–Reuss-type bounds for perfect interface, the present bounds depend not only on the material properties and volume fraction, but also on the interface parameters, inhomogeneity shape and orientation. An example of a two-phase composite is given for detailed discussion, where dependence of the electroelastic moduli and their bounds on the inhomogeneity shapes and orientations as well as the interface properties is provided and discussed. To qualitatively account for the dependence, analysis based on two possible mechanisms, i.e., the simple mixture rule of composite and the weakening effect by imperfect interface, are also provided.     

Interaction between screw dislocations and inclusions with imperfect interfaces in fiber-reinforced composites

A three-phase composite cylinder model is utilized to study the elastic interaction between screw dislocations and embedded multiple circular cross-section inclusions (fibers) with imperfect interfaces in composites. By means of complex variable techniques, the explicit solutions of stress and displacement fields are obtained. With the aid of the Peach–Koehler formula, the explicit expressions of image forces exerted on screw dislocations are derived. The equilibrium positions of the appointed screw dislocation near one of the inclusions are discussed for variable parameters (interface imperfection, material mismatch and dislocation position) and the influence of the nearby inclusions and dislocations is also considered. The results show that, if the inclusion is stiffer than the matrix and the magnitude of the degree of interface imperfection reaches the certain value, a new equilibrium position for the screw dislocation in the matrix can always be produced in comparison with the previous solution (the perfect interface). The effect of elastic constants of the inclusion on the image force and the equilibrium position of the appointed screw dislocation is weak when the interface imperfection is strong. It is also seen that the magnitude of the image force exerted on the appointed dislocation caused by multiple inclusions is always smaller than that produced by a single inclusion. The impact of the closer dislocations on the mobility of the appointed dislocation is very significant.     

残余应力对复合材料弹2塑性变形的影响

从细观力学的角度给出了分析残余应力对一般复合材料塑性性能影响的一种解析方法, 该方法基于应力二阶矩的割线模量法及Ponte Castaneda 和W illis 给出的弹性细观模型。有残余应力时, 所提的细观解析模型能够同时考虑纤维形状, 体积百分比, 纤维取向及纤维的分布对复合材料变形的影响。计算结果表明, 残余应力的存在会引起复合材料拉压变形的不对称, 材料宏观的拉压硬化曲线又与复合材料的细观结构参数密切相关。对单向复合材料, 本文作者对其等效割线热膨胀系数, 拉压应力-应变曲线的有限元分析结果与给出的细观解析模型定量吻合。      

The effect of slightly weakened interfaces on the overall elastic properties of composite materials

The effect of imperfect interfaces on the overall elastic properties of composites is studied in this paper. The imperfect interface is modeled by a linear spring-layer of vanishing thickness. The Mori-Tanaka estimate and its modification are used to evaluate the effective moduli of composites having slightly weakened interfaces. An interface is said to be slightly weakened, if the compliance of the spring-layer is very small. As an example, a composite consisting of aligned ellipsoidal particles is considered in detail. Explicit expressions of the Mori-Tanaka estimates of the effective moduli are derived when the particles are spherical.

Based on classical minimum energy principles, upper and lower bounds of the effective moduli of composites with imperfect interfaces are also derived.     

Two-dimensional and axisymmetric unit cell models in the analysis of composite materials

Unit cell models have been widely used for investigating fracture mechanisms and mechanical properties of composite materials assuming periodically arrangement of inclusions in matrix. It is desirable to clarify the geometrical parameters controlling the mechanical properties of composites because they usually contain randomly distributed particulate. To begin with a tractable problem this paper focuses on the effective Young’s modulus E of heterogeneous materials. Then, the effect of shape and arrangement of inclusions on E is considered by the application of FEM through examining three types of unit cell models assuming 2D and 3D arrays of inclusions. It is found that the projected area fraction and volume fraction of inclusions are two major parameters controlling effective elastic modulus of inclusions.     

Tight bounds for three-dimensional nonlinear incompressible elastic composites

Tighter variational bounds, in the whole range of inclusion volume fraction, that is to say, even near percolation, for the effective energy of nonlinear composites, in the special case of 3D two-phase incompressible elastic composites with isotropic constituents are presented. Following the methodology of Talbot, Willis and Ponte Castañeda, a linear comparison material with the same microgeometry as the nonlinear composite is employed. The asymptotic homogenization method (AHM) combined with a finite element analysis (FEM), is used to find the displacement field as well as the effective properties for the comparison material. An elastic composite with periodically distributed spherical inclusions in a cubic array is considered as an example. Various numerical examples are performed. Comparisons with others theories (i.e. variational bounds, self-consistent estimates, etc.) are shown. Coincidence of the AHM-FEM results with the universal bounds of Nemat-Nasser, Yu and Hori serves as a useful check to the numerical calculation.     

含球夹杂复合材料的力学性能分析

对于复合材料的有效弹性模量,Eshelby[1]的等效夹杂法和Budians-ky和Wu[3]的自相似法仅仅考虑了夹杂的形状及基体和夹杂的力学性能,而忽略了夹杂的大小和相互作用。本文认为当复合材料的夹杂体积分数增大时,夹杂之间的相互作用影响是比较显着的。基于这一事实,本文在考虑夹杂的形状,大小,分布和相互作用前提下推导了材料的有效弹性模量。最后,本文给出了夹杂分布和基体泊松比对复合材料有效弹性模量的影响,并且部分结果与实验进行了比较。从比较的结果来看,本文的结果与实验值吻合的很好。      

Ellipsoidal bounds and percolation thresholds of transport properties of composites

Classical Hashin-Shtrikman bounds physically correspond to spherical inclusions distributed in a self-similar pattern. For general ellipsoidal inclusions, new bounds named as ellipsoidal bounds are theoretically derived in this study. As there remains a major theoretical question on rigorous determination of percolation thresholds, this study also fills this major gap between lack of theoretical prediction and the gigantic amount of experimental results produced each year on percolation of composites. Formulae of percolation thresholds are for the first time universally presented for electrical, thermal, magnetic, and hydraulic properties of a variety of composites. New bounds of transport properties and percolation thresholds estimated enable the geometry of fillers or cavities, the most direct and obvious microstructure information, to be explicitly taken into account for both engineering composites and natural media (rocks, soils, and sands) containing spheroidal particles/voids, fibers, cracks, nanotubes, etc.     

形状记忆合金的力学及界面参数和体积分数对大块金属玻璃基复合材料增韧的影响

采用考虑塑性的超弹性材料模型和基于损伤塑性的准脆性材料模型,建立了三维单胞有限元模型,模拟了形状记忆合金颗粒增韧大块金属玻璃基复合材料的单调拉伸行为。讨论了形状记忆合金的力学参数、体积分数、界面厚度和界面材料参数对金属玻璃增韧效果的影响。结果表明:提高形状记忆合金的相变应变和马氏体塑性屈服应力将显著提高形状记忆合金颗粒增韧大块金属玻璃基复合材料的拉伸失效应变;形状记忆合金弹性模量超过50.0GPa、马氏体塑性屈服应力超过1.8GPa后,复合材料的拉伸失效应变变化不大。能同时兼顾失效应变和失效应力的形状记忆合金体积分数为15%左右。复合材料界面弹性模量和界面屈服应力的增加将提高复合材料的失效应力,但对失效应变影响不大;复合材料界面厚度的增加在提高失效应变的同时,也降低了复合材料的失效应力。     

An irregular-shaped inclusion with imperfect interface in antiplane elasticity

Despite extensive studies of inclusions with simple shape, little effort has been devoted to inclusions of irregular shape. In this study, we consider an inclusion of irregular shape embedded within an infinite isotropic elastic matrix subject to antiplane shear deformations. The inclusion–matrix interface is assumed to be imperfect characterized by a single, non-negative, and constant interface parameter. Using complex variable techniques, the analytic function that is defined within the irregular-shaped inclusion is expanded into a Faber series, and in conjunction with the Fourier series, a set of linear algebraic equations for a finite number of unknown coefficients is determined. With this approach and without imposing any constraints on the stress distribution, a semi-analytical solution is derived for the elastic fields within the irregular-shaped inclusion and the surrounding matrix. The method is illustrated using three examples and verified, when possible, with existing solutions. The results from the calculations reveal that the stress distribution within the inclusion is highly non-uniform and depends on the inclusion shape and the weak mechanical contact at the inclusion/matrix boundary. In fact, the results illustrate that the imperfect interface parameter significantly influences the stress distribution.