Effective properties of three-phase electro-magneto-elastic composites

Coupling between the electric field, magnetic field, and strain of composite materials is achieved when electro-elastic (piezoelectric) and magneto-elastic (piezomagnetic) particles are joined by an elastic matrix. Although the matrix is neither piezoelectric nor piezomagnetic, the strain field in the matrix couples the electric field of the piezoelectric phase to the magnetic field of the piezomagnetic phase. This three-phase electro-magneto-elastic composite should have greater ductility and formability than a two-phase composite in which the electric field and the magnetic field are coupled by directly bonding two brittle materials. A finite element analysis (FEA) and micromechanics based averaging of a representative volume element (RVE) are performed in this work to determine the effective dielectric, magnetic, mechanical, and coupled-field properties of an elastic matrix reinforced with piezoelectric and piezomagnetic fibers as functions of the phase volume fractions, the fiber arrangements in the RVE, and the fiber material properties with special emphasis on the poling directions of the piezoelectric and piezomagnetic fibers. The effective magneto-electric moduli of this three-phase composite are found to be less than the effective magneto-electric moduli of a two-phase piezoelectric/piezomagnetic composite, because the elastic matrix is not stiff enough to transfer significant strains between the piezomagnetic and piezoelectric fibers.     

高体积含量颗粒增强复合材料的一个细观力学模型Ⅰ: 弹性分析与等效模量

提出了一种高体积含量颗粒增强复合材料的细观力学模型。该模型将颗粒简化为同质、同尺寸的弹性圆球, 两颗粒之间的粘接材料(基体) 简化为连接颗粒的一段圆柱体, 假设了圆柱形基体中的细观位移分布形式, 在此基础上分析了一对颗粒之间弹性的细观应力场和细观弹性系数, 将颗粒对的细观弹性系数在空间各个方向上平均, 得到材料的宏观弹性常数, 并建立了宏、细观分析之间的联系。最后用本模型分析了一种实际材料(两种体积含量) , 弹性常数的预测与实验吻合良好, 研究还发现颗粒的空间分布方式对材料宏观弹性常数的影响不大, 而对细观应力的影响显著。      

基于周期性边界条件的炭黑填充橡胶复合材料力学行为的预测

在分析炭黑填充橡胶复合材料的宏观与细观特征之间联系的基础上,提出了具有随机分布形态的代表性体积单元,推导并应用了周期性细观结构的边界约束条件,建立了三维多颗粒夹杂代表性体积单元的数值模型,对炭黑填充橡胶复合材料的宏观力学行为进行了模拟仿真。研究表明,该模型通过周期性边界条件的约束保证了宏观结构变形场和应力场的协调性;计算得到的炭黑填充橡胶复合材料的弹性模量明显高于未填充橡胶材料,并随着炭黑颗粒所占体积分数的增加而增大;该模型对复合材料有效弹性模量的预测结果与实验结果吻合较好,而且比Bergstrom三维模型的预测结果更好,证实了该模型能够用于炭黑颗粒增强橡胶基复合材料有效性能的模拟分析。     

Automatic generation of 2D micromechanical finite element model of silicon–carbide/aluminum metal matrix composites: Effects of the boundary conditions

Two-dimensional finite element (FE) simulations of the deformation and damage evolution of Silicon–Carbide (SiC) particle reinforced aluminum alloy composite including interphase are carried out for different microstructures and particle volume fractions of the composites. A program is developed for the automatic generation of 2D micromechanical FE-models with randomly distributed SiC particles. In order to simulate the damage process in aluminum alloy matrix and SiC particles, a damage parameter based on the stress triaxial indicator and the maximum principal stress criterion based elastic brittle damage model are developed within Abaqus/Standard Subroutine USDFLD, respectively. An Abaqus/Standard Subroutine MPC, which allows defining multi-point constraints, is developed to realize the symmetric boundary condition (SBC) and periodic boundary condition (PBC). A series of computational experiments are performed to study the influence of boundary condition, particle number and volume fraction of the representative volume element (RVE) on composite stiffness and strength properties.     

Elastic Constants of Particle and Fiber Reinforced Metal Matrix Composites

Abstract

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropics between directions of the fiber rich plane and that normal to the plane could be predicted by the model.     

Elastic constants of particle and fiber reinforced metal matrix composites

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropies between directions of the fiber rich plane and that normal to the plane could be predicted by the model.This article is dedicated to Professor Dr. Paul Höller on the occasion of his 65th birthday.     

双随机分布细观分析模型与复合材料性能预报

发展了一种细观力学有限元分析方法——拟真实的参数化双随机分布模型, 该模型综合考虑了纤维增强树脂基复合材料的真实微结构特点和纤维单丝综合力学性能测试结果的离散性特征, 模拟了复合材料中纤维排列和强度分布的随机性。借助移动窗口法研究了该参数化双随机分布模型的可靠性, 确定了其代表性体积单元的尺寸。基于能量法原理推导了单向复合材料的弹性模量预测公式, 结合能量法和渐进失效分析方法, 利用该细观力学有限元方法分别预测了单向纤维增强树脂基复合材料T300/5228的弹性模量和强度性能。数值模拟结果和大部分试验结果吻合良好, 表明发展的细观力学有限元方法能够较好地预测复合材料的力学性能。      

Effective in situ material properties of micron-sized SiO2 particles in SiO2 particulate polymer composites

Several analytical models exist for determination of the Young’s modulus and coefficient of thermal expansion (CTE) of particulate composites. However, it is necessary to provide accurate material properties of the particles as input data to such analytical models in order to precisely predict the composite’s properties, particularly at high particle loading fractions. In fact, the constituent’s size scale often presents a technical challenge to accurately measure the particles’ properties such as Young’s modulus or CTE. Moreover, the in situ material properties of particles may not be the same as the corresponding bulk properties when the particles are embedded in a polymer matrix. To have a better understanding of the material properties and provide useful insight and design guidelines for particulate composites, the concept of “effective in situ constituent properties” and an indirect method were employed in this study. This approach allows for the indirect determination of the particle’s in situ material properties by combining the experimentally determined composite and matrix properties and finite element (FE) models for predicting the corresponding composite properties, then backing out the effective in situ particle properties. The proposed approach was demonstrated with micron-size SiO₂ particle reinforced epoxy composites over a range of particle loading fractions up to 35 vol.% by indirectly determining both the effective Young’s modulus and the effective CTE of the particles. To the best of our knowledge, this study is the first published report on the indirect determination of both the Young’s modulus and the CTE of micron size particles in particulate composites. Similar results on Young’s modulus of micron-size SiO₂ particles measured from nano-indentation testing are encouraging.     

A study on fiber orientation influence on the mechanical response of a short fiber composite structure

This paper deals with the structural analysis of composite materials with non-homogenous orientation of the reinforcement. During this research, a short fiber-reinforced polymer matrix composite is studied. In this case, inhomogeneity of the reinforcement orientation caused by injection molding manufacturing process is analyzed. The main objective of the paper is the investigation of an influence of process-induced orientation of the reinforcement on mechanical properties of the material in comparison with unidirectional and random reinforcement orientation. In particular, natural frequencies and transient response of an exemplary composite component are investigated. To specify effective properties of the composite, Mori–Tanaka’s micromechanical model is assumed. Orientation distribution of the reinforcement is determined by injection molding simulation. To determine elastic material properties dependent on non-homogenous orientation of the reinforcement, an orientation averaging procedure is taken into account. Therefore, during this study, effectiveness of the orientation averaging procedure and different closure approximations influence on the results are studied. Orientation averaging results are compared with numerical results obtained by finite element-based homogenization of composites with prescribed second-order orientation tensor. Finally, the obtained material parameters were applied into a macroscale finite element model, and numerical simulation with different boundary conditions was conducted.     

A micro-mechanics model for composites reinforced by regularly distributed particles with an inhomogeneous interphase

Based on the Mori–Tanaka method, a micro-mechanics model is developed to study the effective elastic properties of composites reinforced by regularly distributed particles. The spatial distribution of particles is supposed to be cube symmetric in the three-dimensional space, and the corresponding finite element method (FEM) computation has been performed through a unit cell model. Additionally, particle interaction and distribution are simultaneously taken into account by using the strain Green’s function, and the specified strain Green’s function is determined by utilizing the necessary conditions of geometric symmetry. In order to analyze particle size effect on the effective properties of composites, the Double-inclusion configuration and related theory are introduced to describe the role of the interphase between the matrix and particles. Finally, the overall elastic properties of the composite with regularly distributed particles are described by three independent elastic constants expressed in the explicit form, and the accuracy of the developed model is verified by comparing with FEM results.     

Predictions of elastic property on 2.5D C/SiC composites based on numerical modeling and semi-analytical method

In this paper, the predictions of elastic constants of 2.5D (three-dimension angle-interlock woven) continue carbon fiber reinforced silicon carbide (C/SiC) composites are studied by means of theoretical model and numerical simulation. A semi-analytical method expressing elastic constants in terms of microstructure geometrical parameters and constitute properties has been proposed. First, both the geometrical model of the 2.5D composite and the representative volume element (RVE) in both micro- and meso-scale are proposed. Second, the effective elastic properties of the RVE in 2.5D C/SiC composites are obtained using finite element method (FEM) simulation based on energy equivalent principle. Finally, the remedied spatial stiffness average (RSSA) method is proposed to obtain more accurate elastic constants using nine correction factor functions determined by FEM simulations, also the effects of geometrical variables on mechanical properties in 2.5D C/SiC composites are analyzed. These results will play an important role in designing advanced C/SiC composites.     

碳化硅颗粒增强铝基复合材料颗粒表面改性技术研究现状

SiC颗粒增强铝基复合材料因具有高的比强度、比刚度、耐磨性及较好的高温稳定性而被广泛应用于航空航天、电子、医疗等领域,但由于SiC颗粒高熔点、高硬度的特点以及SiC颗粒与铝基体间存在界面反应,碳化硅铝基复合材料存在加工性差、界面结合力不足等问题,已无法满足航天等领域对材料性能更高的要求,因此开展如何改善基体与颗粒之间界面情况的研究对进一步提升复合材料综合性能具有重要的科学意义。结合国内外现有研究成果,总结了SiC颗粒与铝基体界面强化机制、界面反应特点、表面改性技术原理及数值建模的发展现状,结果表明,现有经单一表面改性方法处理后的增强颗粒对铝基复合材料性能的提升程度有限,因此如何采用新的手段使复合材料性能进一步提升将成为后续研究热点,且基于有限元数值模拟方法进行复合材料设计也是必然趋势。最后针对单一强化性能提升有限的问题,提出了基于表面改性的柔性颗粒多模式强化方法,同时针对现有的技术难点展望了后续的研究方向,以期为颗粒增强复合材料的制备提供理论参考。     

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.     

Numerical study of the influence of particle-cracking to the damage of MMC by the incremental damage theory

Based on the incremental damage theory, the influences of particle-cracking damage and its residue strengthening capacity on the stress–strain response of particle reinforced (metal matrix composite) MMC under uniaxial tension are carefully investigated in this paper. Two kinds of models are adopted in the numerical calculation to predict the damage evolution of MMC, one is modeling the broken particles as voids and the other is considering the remaining load carrying capacity of the damaged particles. Special emphasis is placed on the detailed comparison between the results predicted by the two models under different parameters such as the aspect ratio, volume fraction of particle and the elastic–plasticity properties of matrix. The damage process of MMC and the development of stress in the particles are predicted by two models and carefully analyzed.     

弹塑性多尺度分析的实现及其在颗粒增强复合材料中的应用

基于满足周期性假设和尺度分离假设的渐进展开均匀化原理,应用商业有限元软件ABAQUS实现了快速识别颗粒增强复合材料的等效弹性参数,及获取其宏-细观尺度下的非线性应力应变场信息。在细观尺度上,为了更好地逼近实际的复合材料结构,其增强颗粒采用不同直径和随机分布的球形进行近似。通过对不同颗粒含量的等效弹性参数的误差分析,证明了细观模型构造的合理性。此外,通过宏-细观尺度间的耦合机制,利用ABAQUS多个用户自定义子程序,实现了颗粒增强复合材料的非线性多尺度耦合分析,并提出了一套加速算法。最后据此研究了颗粒增强材料细观模型塑性演化过程对宏观力学性能的影响。由于编写的程序及分析的思路具有良好的通用性,这一工作为研究颗粒增强及其它复合材料的宏观力学性能提供了有益的参考。     

The Interaction between a Crack and a Particle Cluster

A numerical method has been developed to study the interaction between a crack and second phase particles in a discontinuously reinforced composite material. The simulation is achieved using a ‘dual’ boundary integral method, coupled with a maximum energy release rate criterion for determining the direction of crack propagation. The method has been applied to a composite material composed of components having the elastic properties of Aluminium (matrix) and Silicon Carbide (reinforcement). In particular, the method is used to investigate the crack trajectory and energetics as it interacts with a single particle and with clusters consisting of two particles or a random distribution of ten particles. It is found that although the energy release rate is affected by the particle(s) at relatively large distances, the crack trajectory is not substantially altered until the crack is very close to the particle(s). A pre-existing interface flaw is observed to attract the crack and substantially increase the energy release rate. This revised version was published online in August 2006 with corrections to the Cover Date.     

The effect of interfacial imperfections on the micromechanical stress and strain distribution in fibre reinforced composites

A mathematical model for the determination of micromechanical stress and strain distribution in a unidirectional fibre reinforced composite is developed. The model consists of three phases represented as concentric cylinders, including the existence of an interphase. Both fibre and matrix have well defined elastic properties, while the interphase properties follow an exponential law of variation. The effect of an abrupt variation of elastic properties at the fibre—interphase boundary on the micromechanical state of stress is also presented. The degree of adhesion between fibre and matrix is described by means of adhesion parameters introduced, and a parametric study is performed wherein the stress and strain distribution around the fibre are determined as a function of adhesion efficiency and fibre volume fraction. Analytical results were confirmed by means of a finite element technique introduced and applied to the model.     

An energetic homogenisation procedure for the elastic properties of general cellular sandwich cores

The present study provides a general procedure for the determination of the effective elastic properties of two-dimensional cellular sandwich cores with arbitrary cell topology and geometry. The scheme uses a strain energy-based representative volume element procedure assuming that macroscopically equivalent strain states have to cause the same strain energy in a representative volume element whether the real microstructure or the “effective” homogenised medium is considered. The strain energy can be evaluated either by analytical or pure numerical methods. Both approaches agree well in a number of examples considering different sandwich core geometries.     

Mechanical analysis of 3-D braided composites by the finite multiphase element method

A finite multiphase element method (FMEM), in which the element comprises more than one kind of material, has been proposed to predict the effective elastic properties of 3-D braided composites. This method is based on the variational principle and our previous geometric model that assumes the existence of different types of unit cells in the three regions in a 3-D braided composite, i.e. the interior, surface and corner. The numerical procedure involved two steps. First, a fine local mesh at the unit cell level is used to analyze the stress/strain of each unit cell. Then, a relatively coarse global mesh is used to obtain the overall responses of the composite at macroscopic level. By using the stress volume averaging method, the effective elastic properties of the composite can be calculated under the prescribed uniform strain boundary conditions. Finally, the predicted stress/strain curves are compared with experimental results, demonstrating the applicability of the FME method.     

Heterogeneous and homogenized models for predicting the indentation response of particle reinforced metal matrix composites

In this study, three-dimensional heterogeneous and homogenized finite element models are used to predict the indentation response of particle reinforced metal matrix composites (PRMMCs). The matrix is assumed to have elasto-plastic behavior whereas the particles (uniform in size and spherical in shape) are assumed to be harder than the matrix, and possess linear elastic behavior. The particles (25 % by volume) are randomly distributed in the metal matrix. Two modeling approaches are used. In the first approach, the PRMMC is fully replaced by an equivalent homogenous material, and its material properties are obtained through homogenization using representative volume element approach under periodic boundary conditions. In second approach, a small cubical volume under the indenter is modeled as heterogeneous material with randomly distributed particles, whereas the remaining domain is assigned equivalent material properties obtained through homogenization. The elastic material properties obtained through simulations are found within Hashin–Shtrikman bounds. A suitable size cubical volume consisting of heterogeneities under the indenter is established by considering different cubical volumes so as to capture the actual indentation response. The simulations are also carried out for different particle sizes to establish a suitable particle size. These simulations show that the second modeling approach yields harder indentation response as compared to first modeling approach due to the local particle concentration under the indenter.