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。     

On thermoelastic fields of a multi-phase inhomogeneity system with perfectly/imperfectly bonded interfaces

The stress fields of cylindrical and spherical multi-phase inhomogeneity systems with perfect or imperfect interfaces under uniform thermal and far-field mechanical loading conditions are investigated by use of the Boussinesq displacement potentials. The radius of the core inhomogeneity and the thickness of its surrounding coatings are arbitrary. The discontinuities in the tangential and normal components of the displacement at the imperfect interfaces are assumed to be proportional to the associated tractions. In this work, for the problems where the phases of the inhomogeneity system are homogeneous, the exact closed-form thermo-elastic solutions are presented. These solutions along with a systematic numerical methodology are utilized to solve various problems of physical importance, where the constituent phases of the inhomogeneity system may be made of a number of different functionally graded (FG) and homogeneous materials, and each interface may have a perfect or imperfect boundary condition, as desired. Also, the effect of the interfacial sliding and debonding on the stress field and elastic energy of an FG-coated inhomogeneity is examined.     

Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress

The fundamental framework of micromechanical procedure is generalized to take into account the surface/interface stress effect at the nano-scale. This framework is applied to the derivation of the effective moduli of solids containing nano-inhomogeneities in conjunction with the composite spheres assemblage model, the Mori-Tanaka method and the generalized self-consistent method. Closed-form expressions are given for the bulk and shear moduli, which are shown to be functions of the interface properties and the size of the inhomogeneities. The dependence of the elastic moduli on the size of the inhomogeneities highlights the importance of the surface/interface in analysing the deformation of nano-scale structures. The present results are applicable to analysis of the properties of nano-composites and foam structures.     

Size-dependent effective thermoelastic properties of nanocomposites with spherically anisotropic phases

Composites made of semi-crystalline polymers and nanoparticles have a spherulitic microstructure which can be reasonably represented by a spherically anisotropic volume element. Due to the high surface-to-volume ratio of a nanoparticle, the particle-matrix interface stress, usually neglected in determining the effective elastic moduli of particle-reinforced composites, may have a non-negligible effect. To account for the latter in estimating the effective thermoelastic properties of a composite consisting of nanoparticles embedded in a semi-crystalline polymeric matrix, this work adopts a coherent interface model for the nanoparticle-matrix interface and proposes an extended version of the classical generalized-self consistent method. In particular, Eshelby's formulae widely used to calculate the elastic energy change of a homogeneous medium due to the introduction of an inhomogeneity are extended to the thermoelastic case. The nanoparticle size effect on the effective thermoelastic moduli of the composite are theoretically shown and numerically illustrated.     

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.     

Size-dependent interaction of an edge dislocation with an elliptical nano-inhomogeneity incorporating interface effects

The elastic behavior of an edge dislocation, which is positioned outside of a nanoscale elliptical inhomogeneity, is studied within the interface elasticity approach incorporating the elastic moduli and surface tension of the interface. The complex potential function method is used. The dislocation stress field and the image force acting on the dislocation are found and analyzed in detail. The difference between the solutions obtained within the classical-elasticity and interface-elasticity approaches is discussed. It is shown that for the stress field, this difference can be significant in those points of the inhomogeneity-matrix interface, where the radius of curvature is smaller and which are closer to the dislocation. For the image force, this difference can be considerable or dispensable in dependence on the dislocation position, its Burgers vector orientation, and relations between the elastic moduli of the matrix, inhomogeneity and their interface. Under some special conditions, the dislocation can occupy a stable equilibrium position in atomically close vicinity of the interface. The size effect is demonstrated that the normalized image force strongly depends on the inhomogeneity size when it is in the range of several tens of nanometers, in contrast with the classical solution where this force is always constant. The general issue is that the interface elasticity effects become more evident when the characteristic sizes of the problem (inhomogeneity size, interface curvature radius and dislocation-interface spacing) reduce to the nanoscale.     

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.     

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.     

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

This work aims at estimating the size-dependent effective elastic moduli of particulate composites in which both the interfacial displacement and traction discontinuities occur. To this end, the interfacial discontinuity relations derived from the replacement of a thin uniform interphase layer between two dissimilar materials by an imperfect interface are reformulated so as to considerably simplify the characteristic expressions of a general elastic imperfect model which is adopted in the present work and include the widely used Gurtin–Murdoch and spring-layer interface models as particular cases. The elastic fields in an infinite body made of a matrix containing an imperfectly bonded spherical particle and subjected to arbitrary remote uniform strain boundary conditions are then provided in an exact, coordinate-free and compact way. With the aid of these results, the elastic properties of a perfectly bonded spherical particle energetically equivalent to an imperfectly bonded one in an infinite matrix are determined. The estimates for the effective bulk and shear moduli of isotropic particulate composites are finally obtained by using the generalized self-consistent scheme and discussed through numerical examples.     

Elastic Energies in Circular Inhomogeneities: Imperfect Versus Perfect Interfaces

The change in the total potential energy in a stressed elastic plane system, consisting of an unbounded matrix containing a cylindrical inhomogeneity of circular cross-section, is studied, when an imperfect bonding is formed across the interface. The imperfect bonding is simulated by linearly elastic springs distributed over the interface. Two loading cases are examined: an equilibrium system of fixed uniform tractions acting in the remote boundary of the matrix, and a phase transformation in the inhomogeneity prescribed by stress free uniform eigenstrains distributed in the inhomogeneity region. For both loadings, the fully elastic fields in explicit forms are derived involving the spring compliances and three new two-phase parameters depending on the elastic properties of the two materials. The elastic energies stored in the whole system and in its constituents are determined in simple and compact forms. It is shown that, in both loading cases, the total potential energy of the system is reduced. It is found that, in nanoscale, the ratio of the elastic energy stored in interface to the elastic energy stored in inhomogeneity increases rapidly for small values of the circular radius and tends to zero for large values. Also, this ratio increases as the matrix becomes softer compared to the inhomogeneity.     

Anti-plane time-harmonic Green's functions for a circular inhomogeneity in piezoelectric medium with a spring- or membrane-type interface

The present work focuses on the two-dimensional anti-plane time-harmonic dynamic Green's functions for a circular inhomogeneity immersed in an infinite matrix with an imperfect interface, where both the inhomogeneity and matrix are assumed to be piezoelectric and transversely isotropic. Two types of imperfect interface, the spring-type interface with electromechanical coupling and the membrane-type interface, are considered. The former type is often used to model the electromechanical damage along the interface while the latter is largely employed to simulate surface/interface effect of nano-sized inhomogeneity. By using the Bessel function expansions, explicit solutions for the electromechanical fields induced by a time-harmonic anti-plane line force and line charge located in an unbounded matrix as well as the circular inhomogeneity are respectively derived. The present solutions can recover the anti-plane Green's functions for some special cases, such as the dynamic or quasi-static Green's functions of piezoelectricity with perfect interface as well as the dynamic or quasi-static Green's functions of pure elasticity with imperfect interface. For detailed discussions, selected analytical results are presented. Dependence of the electromechanical fields on circular frequency as well as interface properties is examined. The size effect related to interfacial imperfection is also discussed.     

Effective elastic properties of matrix composites with transversely-isotropic phases

The present work addresses the problem of calculation of the macroscopic effective elastic properties of composites containing transversely isotropic phases. As a first step, the contribution of a single inhomogeneity to the effective elastic properties is quantified. Relevant stiffness and compliance contribution tensors are derived for spheroidal inhomogeneities. The limiting cases of spherical, penny-shaped and cylindrical shapes are discussed in detail. The property contribution tensors are used to derive the effective elastic moduli of composite materials formed by transversely isotropic phases in two approximations: non-interaction approximation and effective field method. The results are compared with elastic moduli of quasi-random composites.     

Estimation of anisotropic elastic properties of nanocomposites using atomistic-continuum interphase model

We have revised classical micromechanics by accounting for the effect of interface to predict the effective anisotropic elastic properties of heterogeneous materials containing nano-inhomogeneities. In contrast to sharp interface between the matrix and inhomogeneity, we introduce the concept of interphase to account for the interfacial-stress effect at the nano-scale. The interphase’s constitutive properties are derived from atomistic simulations and then incorporated in a micromechanics-based interphase model to compute effective properties of nanocomposites. This scale transition approach bridges the gap between discrete atomic level interactions and continuum mechanics. An advantage of this approach is that it combines atomistic with continuum models that consider inhomogeneity and interphase morphology. It thereby enables us to account simultaneously for both the shape and the anisotropy of a nano-inhomogeneity and interphase at the continuum level when we compute material’s overall properties. In so doing, it frees us from making any assumptions about the interface characteristics between matrix and the nano-inhomogeneity.     

Analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder

In this work we consider a cylindrical structure composed of a nonlinear core (inhomogeneity) surrounded by a different nonlinear shell (matrix). We elaborate a technique for determining its linear elastic moduli (second order elastic constants) and the nonlinear elastic moduli, which are called Landau coefficients (third order elastic constants). Firstly, we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems. Then, we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure. The following step consists in the exact solution of such a linear problem by means of the complex elastic potentials. As result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli, which are valid for any volume fraction of the core embedded in the external shell.     

Equivalent inhomogeneity method for evaluating the effective elastic properties of unidirectional multi-phase composites with surface/interface effects

A new technique is presented for evaluating the effective properties of linearly elastic, multi-phase unidirectional composites. Various effects on the fiber/matrix interfaces (perfect bond, homogeneously imperfect interfaces, uniform interphase layers) are allowed. The analysis of nano-composite materials based on the Gurtin and Murdoch model of material surface is also included. The basic idea of the approach is to construct a circular inhomogeneity in an infinite plane whose effects on the displacements and stresses at distant points are the same as those of a finite cluster of inhomogeneities (fibers of circular cross-section) arranged in a pattern representative of the composite material in question. The elastic properties of the equivalent inhomogeneity then define the effective elastic properties of the material. The volume ratio of the composite material is found after the size of the equivalent circular inhomogeneity is defined in the course of the solution procedure. This procedure is based on a semi-analytical solution of a problem of an infinite plane containing a cluster of non-overlapping circular inhomogeneities subjected to loading at infinity. The method works equally well for periodic and random composites and – importantly – eliminates the necessity for averaging either stresses or strains. New results for nano-composite materials are presented.     

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.      

Elastic fields generated by inhomogeneities: Far-field asymptotics,its shape dependence and relation to the effective elastic properties

We discuss connections between the effective elastic properties of a solid with inhomogeneities and the far-field asymptotics of the elastic fields generated by them. We focus on the dependence of the far-field asymptotics on the inhomogeneity shape. This shape dependence in the inhomogeneity problem is in contrast with shape independence of the far field in the eigenstrain problem. For the latter, the far field applies to inclusions of any shape. We show that the external fields in the eigenstrain – and the inhomogeneity problems are interrelated by the same tensor that characterizes the compliance contribution of an inhomogeneity.     

基于界面层模型的双周期纳米夹杂复合材料反平面问题研究

利用复变函数方法并结合双准周期Riemann边值问题理论，获得了含双周期分布非均匀相（夹杂/界面层）的复合材料在远场均匀反平面应力下弹性场的全场解答．该解答可用于对纳米夹杂复合材料的应力进行分析，结合平均场理论也用于预测纳米夹杂复合材料的有效性能．计算结果表明：当夹杂尺度在纳米量级时，应力和有效反平面剪切模量具有明显的尺度依赖性，并且随着夹杂尺寸的增加，趋近于不考虑界面效应时的结果；界面层厚度和性能对应力和有效反平面剪切模量明显变化时所对应的夹杂尺度范围和趋近于无界面效应结果的快慢有显著影响；当界面厚度足够薄时，界面层模型可用于模拟零厚度界面情况．     

A theory of plasticity for carbon nanotube reinforced composites

Carbon nanotubes (CNTs) possess exceptional mechanical properties, and when introduced into a metal matrix, it could significantly improve the elastic stiffness and plastic strength of the nanocomposite. But current processing techniques often lead to an agglomerated state for the CNTs, and the pristine CNT surface may not be able to fully transfer the load at the interface. These two conditions could have a significant impact on its strengthening capability. In this article we develop a two-scale micromechanical model to analyze the effect of CNT agglomeration and interface condition on the plastic strength of CNT/metal composites. The large scale involves the CNT-free matrix and the clustered CNT/matrix inclusions, and the small scale addresses the property of these clustered inclusions, each containing the randomly oriented, transversely isotropic CNTs and the matrix. In this development the concept of secant moduli and a field fluctuation technique have been adopted. The outcome is an explicit set of formulae that allows one to calculate the overall stress-strain relations of the CNT nanocomposite. It is shown that CNTs are indeed a very effective strengthening agent, but CNT agglomeration and imperfect interface condition can seriously reduce the effective stiffness and elastoplastic strength. The developed theory has also been applied to examine the size (diameter) effect of CNTs on the elastic and elastoplastic response of the composites, and it was found that, with a perfect interface contact, decreasing the CNT radius would enhance the overall stiffness and plastic strength, but with an imperfect interface the size effect is reversed. A comparison of the theory with some experiments on the CNT/Cu nanocomposite serves to verify the applicability of the theory, and it also points to the urgent need of eliminating all CNT agglomeration and improving the interface condition if the full potential of CNT reinforcement is to be realized.     

On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear

The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials.Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio μ₂/μ₁.