Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem

The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.     

Eshelby’s tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory

The Eshelby tensor for a plane strain inclusion of arbitrary cross-sectional shape is first presented in a general form, which has 15 independent non-zero components (as opposed to 36 such components for a three-dimensional inclusion of arbitrary shape). It is based on a simplified strain gradient elasticity theory that involves one material length scale parameter. The Eshelby tensor for an infinitely long cylindrical inclusion is then derived using the general form, with its components obtained in explicit (closed-form) expressions for the two regions inside and outside the inclusion for the first time based on a higher-order elasticity theory. This Eshelby tensor is separated into a classical part and a gradient part. The latter depends on the position, the inclusion size, the length scale parameter, and Poisson’s ratio. As a result, the new Eshelby tensor is non-uniform even inside the cylindrical inclusion and captures the size effect. When the strain gradient effect is not considered, the gradient part vanishes and the newly obtained Eshelby tensor reduces to its counterpart based on classical elasticity. The numerical results quantitatively show that the components of the new Eshelby tensor vary with the position, the inclusion size, and the material length scale parameter, unlike their classical elasticity-based counterparts. When the inclusion radius is comparable to the material length scale parameter, it is found that the gradient part is too large to be ignored. In view of the need for homogenization analyses of fiber-reinforced composites, the volume average of the newly derived Eshelby tensor over the cylindrical inclusion is obtained in a closed form. The components of the average Eshelby tensor are observed to depend on the inclusion size: the smaller the inclusion radius, the smaller the components. However, as the inclusion size becomes sufficiently large, these components are seen to approach from below the values of their classical elasticity-based counterparts.     

Solution of the Eshelby-type anti-plane strain polygonal inclusion problem based on a simplified strain gradient elasticity theory

The Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion of arbitrary-shape polygonal cross-section is analytically solved using a simplified strain gradient elasticity theory that incorporates one material length scale parameter. The Eshelby tensor (with four nonzero components) is obtained in a general form in terms of two scalar-valued potential functions. These potential functions, as area integrals over the polygonal cross-section, are first converted to two line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former does not contain any elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle (inclusion) size effect. For homogenization applications, the area average of the new position-dependent Eshelby tensor over the polygonal cross-section is also provided in a general form. To illustrate the newly obtained Eshelby tensor, five types of regular polygonal inclusions (i.e., triangular, quadrate, hexagonal, octagonal, and tetrakaidecagonal) are quantitatively studied by directly using the general formulas derived. The components of the induced strain and the averaged Eshelby tensor inside the inclusion are evaluated. Numerical results reveal that the induced strain varies with both the position and the inclusion size. The values of the induced strain components in a polygonal inclusion approach from below those in a corresponding circular inclusion when the inclusion size or the number of sides of the polygonal inclusion increases. The results for the averaged Eshelby tensor components show that the size effect is significant when the inclusion size is small but may be neglected for large inclusions.     

On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites

The paper focuses on the main uncertainty involved in classical Maxwell’s (1873) homogenization method for elastic composites. Maxwell’s scheme that equates the far fields produced by a set of inhomogeneities and by a fictitious domain with unknown effective properties (“effective inclusion”) is re-written in terms of the compliance and stiffness contribution tensors. It is shown that the shape of the effective inclusion substantially affects the overall elastic properties. The choice of this shape in the case of anisotropic composite is a non-trivial problem that has never been discussed in literature. In this paper, we show that the problem appears due to incompleteness of the Maxwell scheme and show that the problem can be realized when the effective inclusion is of ellipsoidal shape. We discuss how the aspect ratios of the ellipsoid have to be chosen and illustrates the approach by two examples – material with cracks having orientation scatter and a three-phase transversely-isotropic composite. It is also shown that tensor of the effective elastic constants calculated in the framework of Maxwell’s scheme is always symmetric with respect to couples of indices.     

On Bhattacharyya relative entropy in a homogenization of composite materials

The main idea of this work is an application of relative entropy in the numerical analysis of probabilistic divergence between original material tensors of the composite constituents and its effective tensor in the presence of material uncertainties. The homogenization method is based upon the deformation energy of the representative volume elements for the fiber-reinforced and particulate composites and uncertainty propagation begins with elastic moduli of the fibers, particles, and composite matrices. Relative entropy follows a mathematical model originating from Bhattacharyya probabilistic divergence and has been applied here for Gaussian distributions. The semi-analytical probabilistic method based on analytical integration of polynomial bases obtained via the least squares method fittings enables for determination of the basic probabilistic characteristics of the effective tensor and the relative entropies. The methodology invented in this work may be extended toward other probability distributions and relative entropies, for homogenization of nonlinear composites and also accounting for some structural interface defects.     

Eshelby tensors for a spherical inclusion in microelongated elastic fields

Eshelby tensors are found for a spherical inclusion in a microelongated elastic field. Here, a special micromorphic model is introduced to describe the damaged material which defines the damage as the formation and the growth of microcracks and microvoids occurred in the material at the microstructural level. To determine the new material coefficients of the model, an analogy is established between the damaged body and the composite materials and then Mori–Tanaka homogenization technique is considered to obtain overall material moduli. Following this idea, the determination of the Eshelby tensors which establish the relation between the strains of the matrix material and of the inclusion becomes the first task. Introducing the concept of eigenstrain and microeigenstrain, the general constitutive theory is given for a homogeneous isotropic centrosymmetric microelongated media with defects. Then by the use of Green’s functions, micro and macro elastic fields are presented for the case of spherical inclusions embedded in an infinite microelongated material. Thus, the Eshelby tensors are obtained for a microelongated elastic field with a spherical inclusion and it is also shown that the classical Eshelby tensors can be obtained as a limit case of the microelongation.     

The Composite Eshelby Tensors and their applications to homogenization

Summary In recent studies, the exact solutions of the Eshelby tensors for a spherical inclusion in a finite, spherical domain have been obtained for both the Dirichlet- and Neumann boundary value problems, and they have been further applied to the homogenization of composite materials [15], [16]. The present work is an extension to a more general boundary condition, which allows for the continuity of both the displacement and traction field across the interface between RVE (representative volume element) and surrounding composite. A new class of Eshelby tensors is obtained, which depend explicitly on the material properties of the composite, and are therefore termed “the Composite Eshelby Tensors”. These include the Dirichlet- and the Neumann-Eshelby tensors as special cases. We apply the new Eshelby tensors to the homogenization of composite materials, and it is shown that several classical homogenization methods can be unified under a novel method termed the “Dual Eigenstrain Method”. We further propose a modified Hashin-Shtrikman variational principle, and show that the corresponding modified Hashin-Shtrikman bounds, like the Composite Eshelby Tensors, depend explicitly on the composite properties.     

Modeling the effective elastic properties of nanocomposites with circular straight CNT fibers reinforced in the epoxy matrix

In the present study, the consistent effective elastic properties of straight, circular carbon nanotube epoxy composites are derived using the micromechanics theory. The CNT composites are known to provide high stiffness and elastic properties when the shape of the fibers is cylindrical and straight. Accordingly, in the present work, the effective elastic moduli of composite are newly obtained for straight, circular CNTs aligned in the specified direction as well as distributed randomly in the matrix. In this direction, novel analytical expressions are proposed for four cases of fiber property. First, aligned, and straight CNTs are considered with transverse isotropy in fiber coordinates, and the composite properties are also transversely isotropic in global coordinates. The short comings in the earlier developments are effectively addressed by deriving the consistent form of the strain tensor and the stiffness tensor of the CNT nanocomposite. Subsequently, effective relations for composites reinforced with aligned, straight CNTs but fibers isotropic in local coordinates are newly developed under hydrostatic loading. The effect of the unsymmetric Eshelby tensor for cylindrical fibers on the overall properties of the nanocomposite is included by deriving the strain concentration tensors. Next, the random distribution of CNT fibers in the matrix is studied with fibers being transversely isotropic as well as isotropic when CNT nanocomposites are subjected to uniform loading. The corresponding relations for the effective elastic properties are newly derived. The modeling technique is validated with results reported, and the variations in the effective properties for different CNT volume fractions are presented.     

On the capability of micromechanics models to capture the auxetic behavior of fibers/particles reinforced composite materials

This work investigates the possibility to predict the auxetic behavior of composites consisting of non-auxetic phases by means of micromechanical models based on Eshelby’s inclusion concept. Two specific microstructures have been considered: (i) the three-layered hollow-cored fibers-reinforced composite and (ii) a microstructure imitating the re-entrant honeycomb micro-architecture. The micromechanical analysis is based on kinematic integral equations as a formal solution of the inhomogeneous material problem. The interaction tensors between the inhomogeneities are computed thanks to the Fourier’s transform. The material anisotropy due to the morphological and topological textures of the inhomogeneities was taken into account thanks to the multi-site approximation of these tensors. In both cases, the numerical results show that auxetic behavior cannot be captured by such models at least in the case of elastic and isotropic phases. This conclusion is supported by corresponding finite element investigations of the second microstructure that indicate that auxetic behavior can be recovered by introducing joints between inclusions. Otherwise, favorable issues are only expected with auxetic components.     

Numerical simulation of the guided Lamb wave propagation in particle reinforced composites

This paper deals with the investigation of the Lamb wave propagation in particle reinforced composites excited by piezoelectric patch actuators. A three-dimensional finite element method (FEM) modeling approach is set up to perform parameter studies in order to better understand how the Lamb wave propagation in particle reinforced composite plates is affected by change of central frequency of excitation signal, volume fraction of particles, size of particles and stiffness to density ratio of particles. Furthermore, the influence of different arrangements is investigated. Finally, the results of simplified models using material data obtained from numerical homogenization are compared to the results of models with heterogeneous build-up. The results show that the Lamb wave propagation properties are mainly affected by the volume fraction and ratio of stiffness to density of particles, whereas the particle size does not affect the Lamb wave propagation in the considered range. As the contribution of the stiffer material increases, the group velocity and the wave length also increase while the energy transmission reduces. Simplified models based on homogenization technique enabled a tremendous drop in computational costs and show reasonable agreement in terms of group velocity and wave length.     

Eshelby tensors for an ellipsoidal inclusion in a micropolar material

Micropolar Eshelby tensors for an ellipsoidal inclusion are derived in an analytical form, which involves only one-dimensional integral. The numerical evaluation of the Eshelby tensors are also performed, it is found that the micropolar Eshelby tensors are not uniform in the ellipsoidal inclusion, however, their variations over the ellipsoidal domain are not significant. When size of inclusion is large compared to the characteristic length of the micropolar material, the micropolar Eshelby tensor is reduced to the classical one. It is also demonstrated that for a general ellipsoidal inclusion a uniform eigenstrain or eigentorsion produces on average only nonzero strain or torsion, and the average Eshelby relations are uncoupled.     

A dual homogenization and finite element approach for material characterization of textile composites

A novel procedure for predicting the effective nonlinear elastic moduli of textile composites through a combined approach of the homogenization method and the finite element formulation is presented. The homogenization method is first applied to investigate the meso-microscopic material behavior of a single fiber yarn based on the properties of the constituent phases. The obtained results are compared to existing analytical and experimental results to validate the homogenization method. Very good agreements have been obtained. A unit cell is then built to enclose the characteristic periodic pattern in the textile composites. Various numerical tests such as uni-axial and bi-axial extension and trellising tests are performed by 3D finite element analysis on the unit cell. Characteristic behaviors of force versus displacement are obtained. Meanwhile, trial mechanical elastic constants are imposed on a four-node shell element with the same outer size as the unit cell to match the force–displacement curves. The effective nonlinear mechanical stiffness tensor is thus obtained numerically as functions of elemental strains. The procedure is exemplified on a plain weave glass composite and is validated by comparing to experimental data. Using the proposed approach, the nonlinear behavior of textile composites can be anticipated accurately and efficiently.     

Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particle interactions

A micromechanical framework is proposed to predict effective elastic moduli of particle-reinforced composites. First, the interacting eigenstrain is derived by making use of the exterior-point Eshelby tensor and the equivalence principle associated with the pairwise particle interactions. Then, the near-field particle interactions are accounted for in the effective elastic moduli of spherical-particle-reinforced composites. On the foundation of the proposed interacting solution, the consistent versus simplified micromechanical field equations are systematically presented and discussed. Specifically, the focus is upon the effective elastic moduli of two-phase composites containing randomly distributed isotropic spherical particles. To demonstrate the predictive capability of the proposed micromechanical framework, comparisons between the theoretical predictions and the available experimental data on effective elastic moduli are rendered. In contrast to higher-order formulations in the literature, the proposed micromechanical formulation can accommodate the anisotropy of reinforcing particles and can be readily extended to multi-phase composites.     

Stiffness of particulate reinforced metal matrix composites with damaged reinforcements

Abstract

During tensile plastic deformation particulate reinforced metal matrix composites (MMCs) undergo reinforcement damage and a parallel reduction in stiffness. An analytical model is developed to calculate this stiffness reduction using the equivalent inclusion technique proposed by Eshelby. The model considers both damaged and undamaged reinforcement particles as ellipsoidal inclusions but with different stiffness tensors. The effect of the aspect ratio of the reinforcing particles has been accounted for in the model. The model is very flexible and can meet different specific damage situations by designing a suitable stiffness tensor for the damaged reinforcements. Finite element analysis is used to modify a numerical stiffness tensor for cracked reinforcement particles. The model is compared with an earlier model of modulus reduction in MMC materials and with a few experimental measurements made on a 15 vol.-%SiC particulate reinforced aluminium alloy 2618 MMC.     

Determination of sensitivity coefficients of the elastic effective properties for periodic fiber-reinforced composites using the response function method

Derivation and implementation of the homogenization method including determination of sensitivity gradients of the effective elasticity tensor using combined numerical-analytical approach are addressed in this paper. This is possible thanks to an application of the numerical response function together with the effective moduli method known from classical homogenization theory. Computational procedure is implemented using 4-noded quadrilateral plane strain finite elements (program MCCEFF) and the symbolic computations system MAPLE. The sensitivity coefficients are determined on the basis of partial derivatives of the homogenized elasticity tensor calculated using the response function method with respect to all composite components’ elastic characteristics. They are further separately subjected to normalization procedure for a final comparison with each other. Such an enriched homogenization procedure is tested on the periodic fiber-reinforced two component composites; the results of computational analysis are compared to the results of the central finite difference approach applied before. Computational methodology proposed here may be further successively applied not only in the context of homogenization method but also to extend various discrete computational techniques like boundary/finite element, finite difference and volumes together with various meshless methods.     

A differential approach to microstructure-dependent bounds for multiphase heterogeneous media

We present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. The microstructure is specified by the average Eshelby tensors. In the small contrast limit, we introduce and calculate the expansion coefficient tensors. We then show that the effective tensor satisfies a differential inequality with the initial condition given by the expansion coefficient tensors in the small contrast limit. Using the comparison theorem, we obtain rigorous bounds on the effective tensors of multiphase composites. These new bounds, taking into account the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent Hashin–Shtrikman bounds. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. We anticipate that this new approach will be useful for the modeling and optimal design of multiphase multifunctional composites.     

Effects of CNT waviness on the effective elastic responses of CNT-reinforced polymer composites

In this study, the effects of fiber waviness on the effective elastic responses of CNT–polymer composites are investigated based on the framework of micromechanics and homogenization. By taking advantage of an ad hoc Eshelby tensor, the load-transfer capability of wavy carbon nanotube (CNT) embedded in the polymer matrix is accounted for. Further, the effective elastic responses of composites are simulated by using the multi-phase Mori–Tanaka method to study the influence of randomly oriented wavy CNT. It is demonstrated that the proposed micromechanics-based closed form solution is effective to tackle the underlying problem. The present predictions and the comparisons with the available experimental data indicate that the CNT waviness leads to the degradation of effective responses of composites. Finally, in addition to the effect of CNT waviness, the significance of CNT interface is briefly discussed based on the experimental observations.     

Sensitivity studies for the effective characteristics of periodic multicomponent composites

The main goal is to present the application of design sensitivity analysis in homogenization of periodic multicomponent composites. The effective modulus approach is used to determine homogenized characteristics of the composite together with some upper and lower bounds estimators for these quantities. The approach related to homogenization problem is presented in a general form for a linear elastic n-component periodic composite and is implemented in the finite element method homogenization-oriented computer program MCCEFF. Sensitivity coefficients are determined numerically for various components of homogenized elasticity tensor and, using symbolic analysis, for their bounds, in both cases with respect to material parameters of the components. Structural response functional for such a composite is proposed as the strain energy in various strain states of the composite cell to detect the most influential parameters of the most popular fiber-reinforced structures as well as in the case of multicomponent superconducting composite device.     

基于改进PSO算法的非均匀材料区间均匀化分析

利用区间均匀化方法对有限弹性变形下的非均匀材料进行了研究,引入多尺度有限元机制,将非均匀材料等效为某个非局部的代表性体积单元(RVE).采用基于多尺度有限元与改进的粒子群(PSO)算法相结合的方式,对非均匀材料的有效参数(如弹性张量和第一 Piola-Kirchhoff应力以及应变能等)进行了区间分析,充分考虑了代表性...     

颗粒随机分布复合材料的细观构造对有效热传导系数的影响

结合建立的颗粒随机分布复合材料的微结构模型和基于均匀化理论统计的双尺度计算方法, 针对氮化硼颗粒增强高密度聚乙烯(BN/HDPE)复合材料, 研究了颗粒形状、体积分数和空间分布参数等微结构特征对复合材料有效热传导系数的影响。结果表明: 颗粒体积分数的增加将导致有效热传导系数升高; 球形颗粒的位置参数、长椭球颗粒的取向程度都对有效热传导系数有重要影响。数值试验表明, 材料的微结构特征对复合材料的有效热传导系数具有极大影响。