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.     

Analysis of effective elastic modulus for multiphased hybrid composites

Short fiber reinforced composites inherently have fiber length distribution (FLD) and fiber orientation distribution (FOD), which are important factors in determining mechanical properties of the composites. Since the internal structure has a direct effect on the mechanical properties of the composites, a Micro-CT was used to observe the three dimensional structure of fibers in the composites and to acquire FLD and FOD. It was successful to investigate FLD, FOD, and fiber orientation states and to predict the elastic modulus of the hybrid system. Since hybrid composites used in this study consist of three phases of particles, glass fibers, and matrix, theoretical hybrid modeling is required to consider reinforcing effects of both particles and glass fibers. Interaction between the particles and matrix was considered by using a perturbed stress–strain theory, the Tandon–Weng model. In addition, the laminating analogy approach (LAA) was used to predict the overall elastic modulus of the composite. Theoretical prediction of hybrid moduli indicated that there was a possibility of poor adhesion between glass fibers and matrix. The poor interfacial adhesion was confirmed by morphological experiments. This theoretical and experimental platform is expected to provide more insightful understanding on any kinds of multiphased hybrid composites.     

Micromechanical effective elastic moduli of continuous fiber-reinforced composites with near-field fiber interactions

A higher-order micromechanical framework is presented to predict the overall elastic deformation behavior of continuous fiber-reinforced composites with high-volume fractions and random-fiber distributions. By taking advantage of the probabilistic pair-wise near-field interaction solution, the interacting eigenstrain is analytically derived. Subsequently, by making use of the Eshelby equivalence principle, the perturbed strain within a continuous circular fiber is accounted for. Further, based on the general micromechanical field equations, effective elastic moduli of continuous fiber-reinforced composites are constructed. An advantage of the present framework is that the higher-order effective elastic moduli of composites can be analytically predicted with relative simplicity, requiring only material properties of the matrix and fibers, the fiber?Cvolume fraction and the microstructural parameter ??. Moreover, no Monte Carlo simulation is needed for the proposed methodology. A series of comparisons between the analytical predictions and the available experimental data for isotropic and anisotropic fiber reinforced composites illustrate the predictive capability of the proposed framework.     

Modeling of the influence of fibers on creep of fiber reinforced cementitious composite

An analytical model has been developed to study the influence of fibers on creep of fiber reinforced cementitious composites. The model is based on the assumption that shear stress is produced between fiber and surrounding matrix as the matrix deforms. This shear stress in turn influences the matrix creep behavior resulting in macroscopic creep strain lower than that of pure cement-based matrix. In the present paper, a creep strain expression in the form of matrix creep strain multiplying by a fiber influence factor, which reflects the influences of matrix and fiber properties as well as fiber orientation characteristics, is presented. A parametric study, including the influence of elastic moduli of fiber and matrix, fiber dimension and fiber content is carried out. The modeling results indicate that creep strain of fiber reinforced cement-based composite is significantly influenced by the elastic moduli of fiber and matrix as well as fiber length and thickness (i.e. diameter for fiber with circular cross-section). Model predictions compare favorably with experimental measurements of creep strain of fiber reinforced mortar and concrete under compressive load.     

基于细观力学的纤维沥青混凝土有效松弛模量

为了研究纤维沥青混凝土的本构模型,将其视为以沥青混合料为粘弹性基体,纤维为弹性夹杂的两相复合材料。对基于复合材料细观力学理论建立的有效模量表达式进行了修正,提出了纤维沥青混凝土的割线有效松弛模量。以聚酯纤维沥青混凝土为例进行了有效松弛模量的解析分析和模拟蠕变实验的有限元分析,分析结果与试验数据的比较表明,该文提出的割线有效松弛模量模型对于纤维沥青混凝土粘弹性力学行为具有很好的预测能力。应用该模型对路面弯沉变形进行了有限元分析,结果表明:纤维的加入有效的改善了沥青混凝土路面的粘弹性性能。     

A generalized plane strain technique for estimating effective properties of particulate metal matrix composites using FEM

A procedure to estimate the effective elastic moduli and coefficient of thermal expansion (CTE) of particulate-reinforced metal matrix composites (MMCs) using a two-dimensional finite element method is presented. The actual microstructural geometry of the composites with randomly distributed second-phase particles is incorporated in the model. A generalized plane strain technique, realistically to describe the three-dimensional behaviour, is also incorporated in the model. The elastic moduli and the CTE, estimated using this model, agree favourably with the experimental data. The technique is shown to be superior compared to the conventional two-dimensional plane stress and plane strain approximations. Also, the results indicate that the effect of the shape of the randomly distributed second-phase particles on the effective elastic moduli is insignificant. Although the procedure is demonstrated for particulate MMCs, it can be easily extended to many other materials as well.     

单向短纤维增强复合材料的复模量

本文提出了单向短纤维增强复合材材纤维方向模量的一个计算模型.这个模型考虑了横向相邻纤维间以及纵向相接纤维间的相互作用.籍助于这一模型,根据粘弹性对应原理,本文用解析方法计算了单向短纤维增强复合材料的纤维方向复模量,着重分析了端部间隙对复模量的实部和虚部的影响.计算结果和文献中已有的计算结果和试验结果作了比较.      

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.     

Elastic response of a carbon nanotube fiber reinforced polymeric composite: A numerical and experimental study

Premature failure due to low mechanical properties in the transverse direction to the fiber constitutes a fundamental weakness of fiber reinforced polymeric composites. A solution to this problem is being addressed through the creation of nanoreinforced laminated composites where carbon nanotubes are grown on the surface of fiber filaments to improve the matrix-dominated composite properties. The carbon nanotubes increase the effective diameter of the fiber and provide a larger interface area for the polymeric matrix to wet the fiber. A study was conducted to numerically predict the elastic properties of the nanoreinforced composites. A multiscale modeling approach and the Finite Element Method were used to evaluate the effective mechanical properties of the nanoreinforced laminated composite. The cohesive zone approach was used to model the interface between the nanotubes and the polymer matrix. The elastic properties of the nanoreinforced laminated composites including the elastic moduli, the shear modulus, and the Poisson’s ratios were predicted and correlated with iso-strain and iso-stress models. An experimental program was also conducted to determine the elastic moduli of the nanoreinforced laminated composite and correlate them with the numerical values.     

Cellulose fibers as Reinforcement in Composites: Determination of the stiffness of cellulose fibers

A new method for the determination of the elastic modulus of cellulose fibers is presented. Cellulose fibers separated by different pulping processes were fractionated in order to get the same aspect ratio. Composites were prepared under well-controlled conditions, by impregnating unoriented, laboratory-made handsheets with liquid unsaturated polyester resin. The tensile properties of the composites were determined. The elastic moduli of the cellulose fibers were then calculated, using micromechanical relations for short-fiber composites, and compared with values obtained from measurements on the unbonded fiber systems. Good correlation between the fiber moduli obtained by these different methods was found.     

Effective Moduli Evaluation of Carbon Nanotube Reinforced Polymers Using Micromechanics

A multistep homogenization method is adopted to compute the effective moduli of carbon nanotube reinforced composites. The composite is assumed to be reinforced with isolated individual fibers and clustered fibers. A uniform agglomeration model is introduced assuming constant carbon nanotube cluster size throughout the matrix. Agglomeration volume fraction—a critical parameter in the simulation—is considered to be an explicit function of inter-particle distance and quality of dispersion of fibers. The micromechanics model also incorporates random fiber orientation using a statistical approach. It is seen that these parameters reduce the stiffening effect of carbon nanotubes significantly in the composite.     

New higher-order bounds on effective transverse elastic moduli of three-phase fiber-reinforced composites with randomly located and interacting aligned circular fibers

A higher-order structure for three-phase composites containing randomly located yet unidirectionally aligned circular fibers is proposed to predict effective transverse elastic moduli based on the probabilistic spatial distribution of circular fibers, the pairwise fiber interactions, and the ensemble-area homogenization method. Specifically, the two inhomogeneity phases feature distinct elastic properties and sizes. In the special event, two-phase composites with same elastic properties and sizes of fibers are studied. Two non-equivalent formulations are considered in detail to derive effective transverse elastic moduli of two-phase composites leading to new higher-order bounds. Furthermore, the effective transverse elastic moduli for an incompressible matrix containing randomly located and identical circular rigid fibers and voids are derived. It is demonstrated that significant improvements in the singular problems and accuracy are achieved by the proposed methodology. Numerical examples and comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered to illustrate the potential of the present method.     

Dynamic Effective Properties of Particle-Reinforced Composites with Viscoelastic Matrix

The frequency-dependent dynamic effective properties of the particle-reinforced composites with the viscoelastic matrix are studied. Several equations to predict the effective wavenumber of the coherent plane waves propagating through particle-reinforced composites are discussed and the equation given by Gubernatis, J.E., [‘Effects of microstructure on speed and attenuation of elastic waves in porous materials’, Wave Motion, 6, 1984, 579–589] based on the independent scattering approximation is used in this paper. The effective phase velocity, the effective attenuation and the effective elastic moduli are evaluated. Numerical calculations are carried out for two kinds of composites, namely, Lead-Epoxy and Glass-Epoxy and the numerical results show that the frequency-dependent dynamic effective properties are related to both the multiple scattering effects among the distributed particles and the viscous dissipative effects of the viscoelastic matrix. However, these effects in the composites with distributed heavy particles (lead) and light particles (glass) are of evidently different features.     

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.     

A micromechanical model for interpenetrating multiphase composites

The dependence relation between the macroscopic effective property and the microstructure of interpenetrating multiphase composites is investigated in this paper. The effective elastic moduli of such composites cannot be calculated from conventional micromechanics methods based on Eshelby’s tensor because an interpenetrating phase cannot be extracted as dispersed inclusions. Employing the concept of connectivity, a micromechanical cell model is presented for estimating the effective elastic moduli of composites reinforced with either dispersed inclusions or interpenetrating networks. The model includes the main features of stress transfer of interpenetrating microstructures. The Mori–Tanaka method and the iso-stress and iso-strain assumptions are adopted in an appropriate manner of combination, rendering the calculation of effective moduli quite easy and accurate.     

Calculation of effective moduli of fibrous composites with micro-mechanical damage

A micro-mechanics model for continuous fibrous composites was developed in order to determine the effective moduli of composites based on the material properties of their constituents, i.e. fiber and matrix materials. The model can calculate elastic or nonelastic effective moduli of composites depending on their constituents' behavior. Furthermore, micro-mechanical damage can also be considered in the present model to determine effective moduli. Predicted effective moduli from the present model compared very well with experimental data available elsewhere for both undamaged and damaged composites.     

粘弹性复合材料中的渐近均匀化方法

主要研究了由各向同性线弹性加强体和各向同性线粘弹性基体组成的复合材料的问题。在已有的线弹性多层材料的渐近均匀化方法的基础上,应用弹性-粘弹性对应原理,在Carson域中求解粘弹性问题,通过两次运用均匀化方法,得到一类单向强化复合材料的有效模量的表达式。反演可得到单向强化复合材料的有效松弛模量在时间域中的表达式,并且与其它结果进行了比较。     

Effective Elastic and Plastic Properties of Interpenetrating Multiphase Composites

In interpenetrating phase composites, there are at least two phases that are each interconnected in three dimensions, constructing a topologically continuous network throughout the microstructure. The dependence relation between the macroscopically effective properties and the microstructures of interpenetrating phase composites is investigated in this paper. The effective elastic moduli of such kind of composites cannot be calculated from conventional micromechanics methods based on Eshelby's tensor because an interpenetrating phase cannot be extracted as dispersed inclusions. Using the concept of connectivity, a micromechanical cell model is first presented to characterize the complex microstructure and stress transfer features and to estimate the effective elastic moduli of composites reinforced with either dispersed inclusions or interpenetrating networks. The Mori–Tanaka method and the iso-stress and iso-strain assumptions are adopted in an appropriate manner of combination by decomposing the unit cell into parallel and series sub-cells, rendering the calculation of effective moduli quite easy and accurate. This model is also used to determine the elastoplastic constitutive relation of interpenetrating phase composites. Several typical examples are given to illustrate the application of this method. The obtained analytical solutions for both effective elastic moduli and elastoplastic constitutive relations agree well with the finite element results and experimental data.     

Elastic field and effective moduli of periodic composites with arbitrary inhomogeneity distribution

This paper develops a semi-analytic model for periodically structured composites, of which each period contains an arbitrary distribution of particles/fibers or inhomogeneities in a three-dimensional space. The inhomogeneities can be of arbitrary shape and have multiple phases. The model is developed using the Equivalent Inclusion Method in conjunction with a fast Fourier Transform algorithm and the Conjugate Gradient Method. The interactions among inhomogeneities within one computational period are fully taken into account. An accurate knowledge of the stress field of the composite is obtained by setting the computational period to contain one or more structural periods of the composite. The effective moduli of the composite are calculated from average stresses and elastic strains. The model is used to analyze the stress field and effective moduli of anisotropic composites that have cubic symmetry. It shows that the bulk and shear moduli predicted by the present model are well located within the Hashin-Shtrikman bounds. The study also shows that the stress field of the composite can be significantly affected by the distribution of inhomogeneities even though the effective moduli are not affected much.     

Tensile properties of as fabricated,aged, and stretched SiC whisker and particle reinforced 2009 aluminium alloy

Abstract

Tensile tests were carried out using specimens of 2009 aluminium alloy reinforced by either SiC whiskers or particles. The size distributions of the whiskers and particles in the matrix were obtained by image analysis. It was found that failure was a result of uniform void nucleation and coalescence in the as fabricated composites, or a result of fast crack propagation initiated by a flaw developed at clusters of SiC in the aged or stretched and aged composites. The strengths of the as fabricated composites were estimated based on the results of image analysis using continuum mechanics and dislocation theories. The estimation indicated that the tensile strengths are largely contributed to by composite strengthening, supplemented by residual dislocation strengthening and work hardening. Owing to the flaw controlled failure, the tensile strengths of the aged or stretched and aged composites were independent of aging time, aging temperature, and the amount of stretching. The elastic moduli of the composites were estimated using the Halpin–Tsai model and a good correlation was found between the measured and estimated moduli.

MST/3438