Thin interphase/imperfect interface in elasticity with application to coated fiber composites

The imperfect interface conditions which are equivalent to the effect of a thin elastic interphase are derived by a Taylor expansion method in terms of interface displacement and traction jumps. Plane and cylindrical interfaces are analyzed as special cases. The effective elastic moduli of a unidirectional coated fiber composite are obtained on the basis of the derived imperfect interface conditions. High accuracy of the method is demonstrated by comparison of solutions of several problems in terms of the imperfect interface conditions or explicit presence of interphase as a third phase. The problems considered are transverse shear of a coated infinite fiber in infinite matrix and effective transverse bulk and shear moduli and effective axial shear modulus of a coated fiber composite. Unlike previous elastic imperfect interface conditions in the literature, the present ones are valid for the entire range of interphase stiffness, from very small to very large.     

Effective elastic moduli of composites reinforced by particle or fiber with an inhomogeneous interphase

The solution of the strain energy change of an infinite matrix due to the presence of one spherical particle or cylindrical fiber surrounded by an inhomogeneous interphase is the basis of solving effective elastic moduli of corresponding composites based on various micromechanics models. In order to find out the strain energy change, the composite sphere or cylinder, i.e., the spherical particle or cylindrical fiber together with its interphase, is replaced by an effective homogeneous particle or fiber. Independent governing differential equations for each modulus of the effective particle or fiber are derived by extending the replacement method [J. Mech. Phys. Solids 12 (1964) 199]. As far as the strain energy changes of the infinite matrix subjected to various far-field stress systems are concerned, the present model is simple. Meanwhile, FEM analysis is carried out for a verification, which shows that the model can lead to rather accurate results for most practical interphases. Besides, to check the validity of the model further when the interactions among composite cylinders exist, the two problems of an infinite matrix containing two composite cylinders and the effective moduli of composites with the equilateral triangular distribution of composite cylinders are analyzed using FEM. The FEM results show that the model is still rather accurate, especially for the case of interphase properties varying between those of fiber and matrix. Therefore, composite spheres or cylinders are assumed as the effective homogeneous particles or fibers and simple expressions of the effective moduli of composites containing the composite spheres or cylinders are obtained. Furthermore, the present model is compared with some existing models that are based on very complicated derivations.     

Rigorous Bounds on the Torsional Rigidity of Composite Shafts with Imperfect Interfaces

We derive upper and lower bounds for the torsional rigidity of cylindrical shafts with arbitrary cross-section containing a number of fibers with circular cross-section. Each fiber may have different constituent materials with different radius. At the interfaces between the fibers and the host matrix two kinds of imperfect interfaces are considered: one which models a thin interphase of low shear modulus and one which models a thin interphase of high shear modulus. Both types of interface will be characterized by an interface parameter which measures the stiffness of the interface. The exact expressions for the upper and lower bounds of the composite shaft depend on the constituent shear moduli, the absolute sizes and locations of the fibers, interface parameters, and the cross-sectional shape of the host shaft. Simplified expressions are also deduced for shafts with perfect bonding interfaces and for shafts with circular cross-section. The effects of the imperfect bonding are illustrated for a circular shaft containing a non-centered fiber. We find that when an additional constraint between the constituent properties of the phases is fulfilled for circular shafts, the upper and lower bounds will coincide. In the latter situation, the fibers are neutral inclusions under torsion and the bounds recover the previously known exact torsional rigidity.      

Effect of an inhomogeneous interphase zone on the bulk modulus and conductivity of a particulate composite

A model is presented of a particulate composite containing spherical inclusions, each of which are surrounded by a localized region in which the elastic moduli vary smoothly with radius. This region may represent an interphase zone in a composite, or the transition zone around an aggregate particle in concrete, for example. An exact solution is derived for the displacements and stresses around a single inclusion in an infinite matrix, subjected to a far-field hydrostatic compression, and is then used to derive an approximate expression for the effective bulk modulus of a material containing a random dispersion of these inclusions. The analogous conductivity (thermal, electrical, etc.) problem is then discussed, and it is shown that the expression for the normalized effective conductivity corresponds exactly to that for the normalized effective bulk modulus, if the Poisson ratios of both phases are set to zero.     

Influence of imperfect elastic contact condition on the antiplane effective properties of piezoelectric fibrous composites

A fiber-reinforced periodic piezoelectric composite, where the constituents exhibit transverse isotropic properties, is considered. The fiber cross-section is circular and the periodicity is the same in two orthogonal directions. Imperfect mechanic contact conditions at the interphase between the matrix and fibers are represented in parametric form. In order to analyze the influence of the imperfect interface effect over the behavior of the composite, the effective axial piezoelectric moduli are obtained by means of the Asymptotic Homogenization Method. Some numerical examples are given.     

Influence of fiber’s shape and size on overall elastoplastic property for micropolar composites

A general micromechanical method is developed for a micropolar composite with ellipsoidal fibers, where the matrix material is idealized as a micropolar material model. The method is based on a special micro–macro transition method, and the classical effective moduli for micropolar composites can be determined in an analytical way. The influence of both fiber’s shape and size can be analyzed by the proposed method. The effective moduli, initial yield surface and effective nonlinear stress and strain relation for a micropolar composite reinforced by ellipsoidal fibers are examined, it is found that the prediction on the effective moduli and effective nonlinear stress and strain curves are always higher than those based on classical Cauchy material model, especially for the case where the size of fiber approaches to the characteristic length of matrix material. As expected, when the size of fiber is sufficiently large, the classical results (size-independence) can be recovered.     

含非均匀界面相粒子填充复合材料的有效比热

基于细观力学复合球模型研究了含非均匀界面相粒子填充复合材料的有效热弹性性质,重点讨论了界面相性质的径向分布对有效比热的影响. 首先,将非均匀界面相沿径向离散为多个同心球壳,每个球壳内的材料性质假设是均匀的. 基于上述离散模型,利用含界面相的复合球模型,推导了复合材料的有效体积模量、有效热膨胀系数及有效比热的数值求解表达式;进一步,假设界面相的性质沿径向连续变化,建立了一组微分方程,上述有效性质依赖于该微分方程组的解. 特别地,当界面相杨氏模量为幂次分布时,通过求解该微分方程组得到了有效比热等热弹性性质的解析解. 算例结果表明,应用此方法预测的有效热膨胀系数与实验结果吻合良好;界面相热膨胀系数的径向分布对有效比热和有效热膨胀系数均有显著的影响,而界面相弹性模量的径向分布对有效比热有显著的影响,对有效热膨胀系数的影响相对较小.      

EFFECTIVE SPECIFIC HEATS OF SPHERICAL PARTICULATE COMPOSITES WITH INHOMOGENEOUS INTERPHASES

基于细观力学复合球模型研究了含非均匀界面相粒子填充复合材料的有效热弹性性质,重点讨论了界面相性质的径向分布对有效比热的影响. 首先,将非均匀界面相沿径向离散为多个同心球壳,每个球壳内的材料性质假设是均匀的. 基于上述离散模型,利用含界面相的复合球模型,推导了复合材料的有效体积模量、有效热膨胀系数及有效比热的数值求解表达式;进一步,假设界面相的性质沿径向连续变化,建立了一组微分方程,上述有效性质依赖于该微分方程组的解. 特别地,当界面相杨氏模量为幂次分布时,通过求解该微分方程组得到了有效比热等热弹性性质的解析解. 算例结果表明,应用此方法预测的有效热膨胀系数与实验结果吻合良好;界面相热膨胀系数的径向分布对有效比热和有效热膨胀系数均有显著的影响,而界面相弹性模量的径向分布对有效比热有显著的影响,对有效热膨胀系数的影响相对较小.     

An exact solution for the three-phase thermo-electro-magneto-elastic cylinder model and its application to piezoelectric–magnetic fiber composites

A three-phase cylindrical model for analyzing fiber composite subject to in-plane mechanical load under the coupling effects of multiple physical fields (thermo, electric, magnetic and elastic) is presented. By introducing an eigenstrain corresponding to the thermo-electro-magnetic-elastic effect, the complex multi-field coupling problem can be reduced to a formal in-plane elasticity problem for which an exact closed form solution is available. The present three-phase model can be applied to fiber/interphase/matrix composites, such that a lot of interesting thermo-electro-magnetism and stress coupling phenomena induced by the interphase layer are revealed. The present model can also be applied to fiber/matrix composites, in terms of which a generalized self-consistent method (GSCM) is developed for predicting the effective properties of piezoelectric–magnetic fiber reinforced composites. The effective piezoelectric, piezomagnetic, thermoelectric and magnetoelectric moduli can be expressed in compact explicit formulae for direct references and applications. A comparison of the predictions by the GSCM with available experimental data is presented, and interesting magnification effects and peculiar product properties are discussed. As a theoretical basis for the GSCM, the equivalence of the three sets of different average field equations in predicting the effective properties are proved, and this fact provides a strong evidence of mathematical rigor and physical realism in the formulation.     

Residual strength of a fiber reinforced metal matrix composite with a crack

The residual strength of a cracked unidirectional fiver reinforced metal matrix composite is studied. We propose a bridging model based on the Dugdale strip yielding zones in the matrix ahead of the crack tips that accounts for ductile deformations of the matrix and fiber debonding and pull-out in the strip yielding zone. The bridging model is used to study the fracture of an anisotropic material and its residual strength is calculated numerically. The predicted results for a SiC/titanium composite agree well with the existing experimental data. It is found that a higher fiber bridging stress and a larger fiber pull-out length significantly contribute to the composite's residual strength. The composite's strength may be more notch-insensitive than the corresponding matrix material's strength depending on several factors such as fiber-matrix interface properties and the ratio of the matrix modulus to an ‘effective modulus’ of the composite.     

含界面相效应的纤维增强复合材料桥联增韧力学分析

本文对纤维增强复合材料桥联增韧进行了详细的断裂力学分析，基于Ｃａｓｔｉｇｌｉａｎｏ＇ｓ定理和界面剪滞模型，得到了含界丰效应的复合材料桥联增专访和裂纹线开位移的控制议程；并按照第二类Ｆｒｅｄｈｏｌｍ积分方程的迭代解法，给出其数值结果，为例题于分析界面相参数对增韧效果的影响，寻求了该控制方程的近似解解析表达式，对近似解进行了误差估计，证明了解的可行性，在此基础上得到了界面剪切模量，裂纹长度。界面厚度，     

End-shaped copper fibers in an epoxy matrix—predicted versus actual fracture toughening

End-shaped copper fibers are placed in a brittle thermoset epoxy matrix at 10 vol% and tested in four-point bending to determine the fracture toughness of the composite. Results from four-point bend tests agree well with the theoretical predictions of the fracture toughness increment ‘ΔG’ of a metal fiber/brittle thermoset matrix composite based on single fiber pullout (SFP) tests. This close agreement demonstrates that SFP testing, along with the theoretical model, can be used as an effective end-shape screening tool for ductile fibers before full scale composite testing. The model predicts that the composite’s fracture toughness will be 46% higher with flat end-impacted fibers and 4% lower with rippled fibers compared to straight fibers at a 0° orientation. Four-point bend results show the actual composite’s fracture toughness is 49% higher with flat end-impacted fibers and 5% lower with rippled fibers compared to straight fibers. Further, four-point bend results show that end-shaped copper fibers improve both the flexural strength and modulus of the composite, demonstrating that end-shaped ductile fibers provide a good stress transfer to the fibers by anchoring the fibers into the matrix. Lastly, experimental validation of the model also indicates that at low fiber volume fractions, fiber–fiber interaction has only a minor influence on the fracture toughness for the tested ductile fiber/brittle matrix composite.     

Effective moduli for micropolar composite with interface effect

Size-dependence is well observed for metal matrix composites, however the classical micromechanical model fails to describe this phenomenon. There are two different ways to consider this size-dependency: the first approach is to include the nonlocal effect by idealizing the matrix material as a high order continuum (e.g., micropolar or strain gradient); the second is to take into account the interface effect. In this work, we combine these two approaches together by introducing the interface effect into a micropolar micromechanical model. The interface constitutive relations and the generalized Young–Laplace equation for micropolar material model are firstly presented. Then they are incorporated into the micropolar micromechanical model to predict the effective bulk and shear moduli of a fiber-reinforced composite. Two intrinsic length scales appear: one is related to the microstructure of the matrix material, the other comes from the interface effect. The size-dependent effective moduli due to the nonlocal effect and interface effect can be synchronized or desynchronized for nanosize fibers, depending on the nature of the interface. For the relatively large fiber size, the size-dependence is dominated by the nonlocal effect. As expected, when the fiber size tends to infinity, classical result can be recovered.     

The effective elastic moduli of columnar composites made of cylindrically anisotropic phases with rough interfaces

The present work aims to determine the effective elastic moduli of a composite having a columnar microstructure and made of two cylindrically anisotropic phases perfectly bonded at their interface oscillating quickly and periodically along the circular circumferential direction. To achieve this objective, a two-scale homogenization method is elaborated. First, the micro-to-meso upscaling is carried out by applying an asymptotic analysis, and the zone in which the interface oscillates is correspondingly homogenized as an equivalent interphase whose elastic properties are analytically and exactly determined. Second, the meso-to-macro upscaling is accomplished by using the composite cylinder assemblage model, and closed-form solutions are derived for the effective elastic moduli of the composite. Two important cases in which rough interfaces exhibit comb and saw-tooth profiles are studied in detail. The analytical results given by the two-scale homogenization procedure are shown to agree well with the numerical ones provided by the finite element method and to verify the universal relations existing between the effective elastic moduli of a two-phase columnar composite.     

Influences of reinforcing particle and interface bonding strength on material properties of Mg/nano-particle composites

In the present study, an effective model is proposed to predict the effective elastic behavior of the three-phase composite containing spherical inclusions, each of which is surrounded by an interphase layer. The constitutive equations are derived for the stress and strain of each phase of the composite subjected to a far-field tension. Based on these constitutive laws, the effective bulk, shear and Young’s modulus are obtained. A statistical debonding criterion is adopted to characterize the varying probability of the evolution of interphase debonding. Influences of debonding damage, particle volume fraction, interphase properties and bonding strength on overall mechanical behavior of composites are also discussed. Numerical analyses are carried out on particle-reinforced composites and the predictions have a good agreement with the experimental results.     

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.     

Optimal strength of a compound column

A column of fixed length and variable cross section consists of two homogeneous and isotropic components. The components are joined along their side surfaces and have different Young's moduli, but the same Poisson's ratio. One of the components encloses the other that has the smaller Young's modulus. For different values of the ratio of the moduli, the shape of the column, which has the largest critical buckling load under axial thrust, is determined, assuming that the volumes of the components are prescribed. The problem is solved for the case of pinned ends.It appears that the solution of the most general problem, in which each of the areas of the component cross sections may be varied, is a combination of the solutions of some more elementary problems. Therefore, two types of problems are discussed: the compound bar with an inner component of fixed cross section and the general compound bar.The method of solution may be extended to other boundary conditions.     

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.     

Failure instability of a debonded interface in the presence of a main crack

The problem of fracture initiating from an edge crack in a nonhomogeneous beam made of two dissimilar linear elastic materials that are partially bonded along a common interface is studied by the strain energy density theory. The beam is subjected to three-point bending and the unbonded part of the interface is symmetrically located with regard to the applied loading. The applied load acts on the stiffer material, while the edge crack lies in the softer material. Fracture initiation from the tip of the edge crack and global instability of the composite beam are studied by considering both the local and global stationary values of the strain energy density function, dW/dV. A length parameter l defined by the relative distance between the maximum of the local and global minima of dW/dV is determined for evaluating the stability of failure initiation by fracture. Predictions on critical loads for fracture initiation from the tip of the edge crack, crack trajectories and fracture instability are made. In the analysis the load, the length of the edge crack and the length and position of the interfacial crack remained unchanged. The influence of the ratio of the moduli of elasticity of the two materials, the position of the edge crack and the width of the stiffer material on the local and global instability of the beam was examined. A general trend is that the critical load for crack initiation and fracture instability is enhanced as the width and the modulus of elasticity of the stiffer material increase. Thus, the stiffer material acts as a barrier in load transfer.     

The effective moduli of composite reinforced by spherical coating particles

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