The elastic strain energy of a coherent inclusion with deviatoric misfit strains

The elastic strain energy of a misfitted coherent inclusion is discussed using the Eshelby method for ellipsoidal inclusions. Deviatoric and tetragonal misfit strains in an elastically inhomogeneous spheroidal inclusion are particularly considered. The variation of the elastic strain energy is evaluated as a function of the shape and orientation of the inclusion. Results obtained are compared with those for a coherent inclusion with purely dilatational misfit strains. Under certain deviatoric misfit strains and elastic moduli of the inclusion, the least elastic strain energy is achieved by a spheroid with an intermediate shape between a plate and a sphere or between a sphere and a needle.     

Elastic Moduli of Composites with Rigid Elliptical Inclusions

Effective elastic moduli of 2-D solids with randomly located perfectly rigid elliptical inclusions are derived using non-interacting approximation. This approximation constitutes the basic building block for various approximate schemes in micromechanics of materials with interacting inhomogeneities. Anisotropy due to non-random orientation of inclusions is investigated. The results are obtained in closed form and compared against existing solutions.     

A wavelet method for reducing the computational cost of BE-based homogenization analysis

A wavelet BEM is applied to the evaluation of the effective elastic moduli of unidirectional composites, based on the homogenization theory. This attempt is devoted to the reduction of computational cost for the BE-based homogenization analysis. Truncation for matrix compression is carried out by the Beylkin-type algorithm. A thresholding value for the truncation is set such that the discretization error of BE solution is comparable to its truncation error. Besides, rearrangement of the BE equations is proposed to attain rapid convergence of iterative solutions. Through investigation of asymptotical convergence of the effective moduli, it is found that the BE-based homogenization analysis ensures the same rate of convergence for effective moduli as for characteristic functions. By applying the wavelet BEM to heterogeneous media which have microstructures with many voids, the effective moduli with agreement of 2–4 digits can be evaluated using 20–50% memory requirements of conventional BE approaches.     

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.     

Analysis of the single-fiber fragmentation test

An analysis of the single-fiber fragmentation test was investigated.An approximate solution for the stress fields of a fiber embedded in a polymer matrix of different elastic moduli was obtained by the Eshelby method. The fiber was modeled as a prolate spheroid. The axial stress of the fiber increases with increasing aspect ratio and fiber-matrix shear modulus ratio and decreases with increasing matrix and fiber Poisson's ratios. Using this analysis, the fracture stress of a single-fiber fragmentation specimen was derived. The applied stress at fiber fracture decreases monotonically with increasing aspect ratio of the fragmented fiber and increases with increasing fiber and matrix Poisson's ratios. This model is in qualitative agreement with published experimental data.     

Convergence problems in the theory of random elastic media

A main result of the rigorous theory of random, linearly elastic media consists in the representation of the tensor of effective elastic moduli as a Neumann type infinite series which contains the infinite set of correlation functions of the distribution of the local elastic moduli. Under the restriction to statistically homogeneous and isotropic finite media it is proved that convergent series can always be obtained provided the local elastic moduli remain finite everywhere in the medium. This means that the mentioned theory cannot be applied in the above mentioned form to media with pores and/or rigid inclusions. It also means that the theory is not restricted to media with small fluctuations of the elastic parameter.     

含有扁长椭球形增强物复合材料的弹性常数

研究了含有扁长椭球形增强物复合材料的弹性过程并得到了材料整体的弹性常数.当材料基体和增强材料均为各向同性材料,增强材料的体积相对于基体非常地小并与基体紧密相联时,含有扁长椭球形增强物的材料具有各向异性,提高这种材料的杨氏模量是以牺牲剪切模量为代价的.     

Similarity of Stress-Singularity Problems for Materials with Linear and Bilinear Behaviors

An analogy is established between the solutions of the problems of singularities of stresses in linear and bilinear elastic isotropic media. It is shown that the distributions of stresses and displacements in the vicinity of singular points on the boundary of the body (characterized by the singularities of stresses) are described, in both cases, by the same functional dependences on the space coordinates but with different characteristics of the material. We deduce expressions for the effective moduli of elasticity and Poisson's ratio of the bielastic medium including the parameter of hardening of the material. The solution of the problem of singularities of stresses in bilinear materials is obtained from the solution of the corresponding problem for the linear elastic medium by replacing the elastic constants with the corresponding effective values depending on the parameter of hardening of the material. The cases of wedge-shaped notches (for various boundary conditions imposed on their edges), two-component wedges, plane wedge-shaped cracks, and circular conic notches or rigid inclusions in the bielastic space are studied in detail.     

Micromechanics determination of electroelastic properties of piezoelectric materials containing voids

This paper examines the electroelastic properties of piezoelectric materials that contain voids. A unified micromechanics approach is adopted for determining the properties. Voids are treated as spheroidal inclusions with zero elastic moduli. The surrounding material is assumed to be linearly piezoelastic and transversely isotropic. The electroelastic Eshelby tensors for spheroidal inclusions have been evaluated numerically for different aspect ratios. Utilizing these tensors and applying the Mori–Tanaka mean field theory that accounts for the interaction between inclusions and matrix, the effective electroelastic properties of the materials are obtained. Numerical examples are given based on PZT-5H and BaTiO₃. Influences of the volume fraction and aspect ratio of voids on the material properties have been studied. Emphasis has been placed on the piezoelastic coupling effect of the material. For both materials, the piezoelastic coupling provides a stiffening effect on the materials, and the influence is more pronounced when void volume increases and when the aspect ratio of voids becomes shorter.     

Average stress in the matrix and effective moduli of randomly oriented composites

By considering the variation of average stress in the matrix, some elastic properties of randomly oriented composites are established as a function of aspect ratio. Both three- and two-dimensional random orientations, resulting, respectively, in a complete and transverse isotropy, are considered. As the shape of inclusions changes, the isotropic bulk and shear moduli are shown to vary within the Hashin-Shtrikman bounds. The aspect ratio dependence of the five in-plane and out-of-plane moduli with planar orientations are also explicitly given; these results suggest that the in-plane properties are most effectively reinforced by fibrous inclusions, whereas the out-of-plane ones are more responsive to the disc type. The accompanying variations of average stress in the matrix are seen to be closely related to the corresponding variations of the moduli. Comparisons with some limited experimental data also show a reasonable agreement.     

Influence of imperfect interface on the elastic moduli of syntactic foams

Syntactic foams are manufactured by dispersing microspheres in a polymeric matrix, and the macroscale material properties of these foams are estimated by analyzing a periodic distribution of the inclusions. The analysis in the simplest form, further assume that the inclusions are perfectly bonded to the matrix material. It has been shown in a previous study [P.R. Marur, Mater. Lett. 59 (2005) 1954–1957.] that analytical model overestimated the experimentally determined elastic moduli, and that the morphology of particle distribution has negligible influence on the elastic moduli. In this paper, the assumption of perfect adhesion between the inclusion and the matrix is relaxed to allow for possible localized slip and separation at the particle interface. The analytical results obtained considering imperfect interface well agree with the measured elastic modulus reported in the literature.     

Effective thermoelastic moduli and stress concentrator factors in nanocomposites

Summary Nanocomposites are modeled as a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random field of spheroid nanofibers with prescribed random orientation. An estimation of the effective thermoelastic properties of NC was performed by the effective field method (see Buryachenko, [10]) taking into account the random orientation of nanofibers as well as justified selection of spatial correlations of fiber location. The independent justified choice of shapes of inclusions and correlation holes provides the matrix of effective moduli which is symmetric (in contrast to the Mori-Tanaka approach). One estimates also the effective tensor of thermal expansion and stress concentrator factors depending on the orientation of the fiber being considered as well as on the justified choice of the shape of correlation holes, concentration and orientation distribution functions of nanofibers.     

Prediction of elastic properties of precipitation-hardened aluminum cast alloys

A methodology is proposed for predicting the elastic properties of precipitation-hardened alloys by combining different modeling techniques: the CALPHAD method, first-principles calculation, and elasticity models. The proposed procedure was applied to conventional aluminum cast alloys to predict their elastic moduli. The predicted Young’s moduli are in reasonable agreement with values reported in the literature, which verifies the potential applicability of the methodology to the development of high-stiffness aluminum cast alloys.     

Micromechanics modeling of viscoelastic properties of hybrid composites with shunted and arbitrarily oriented piezoelectric inclusions

A comprehensive micromechanics model is developed to estimate the effective viscoelastic properties of hybrid composites containing polymer matrix, conductive inclusions and shunted piezoelectric inclusions. The model is derived using the viscoelastic correspondence principle in conjunction with the Mori-Tanaka approach and the orientation averaging scheme. Three dimensional complex moduli are explicitly presented for hybrid composites with any orientation distribution. The model is first validated by comparison with available experimental results. Then, the loss factors are examined for hybrid composites with inclusions of various volume fractions and of shapes ranging from thin disks to long fibers. It is seen that hybrid composites with randomly oriented inclusions exhibit shear loss factors which are not possible with monolithic piezoelectric plate. Furthermore, the numerical results indicate that composites with long spheroid inclusions provide the best damping performance. The results recommend that aligned inclusion composites are good for alleviating longitudinal oscillations. If oscillation energy needs to be dissipated in all directions and for all modes, three dimensional random composites should be used. It is also observed that spherical inclusion composites cannot improve shear damping irrespective of the orientation and the volume fraction. In general, to achieve a pronounced damping piezoelectric inclusions that lie in aspect ratio range 0.1?α?2 should be avoided.     

Fourier series expansion in non-orthogonal coordinate system for the homogenization of linear viscoelastic periodic composites

The Eshelby method and the Fourier series are used in order to determine the linear elastic and viscoelastic properties of composites with periodically distributed inclusions in a non-orthogonal coordinate system. The relaxation moduli provided by the proposed method are compared with the moduli obtained via FEM for a unidirectional composite. This comparison gives good results. The procedure results very useful for periodic composites with hexagonal symmetry, such as some transversely isotropic composites.     

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.     

Elastic moduli of Al-Li alloys treated at a high pressure of 5.4 GPa

The elastic moduli of high pressure treated supersaturated Al-Li solid solutions were measured. An interesting elastic behaviour was observed, in that, the bulk modulus decreased with an increase in the lithium content, whereas the Young's modulus and shear modulus increased. In order to clarify this property, we investigated the compressibility of the Al-Li supersaturated solid solutions prepared by high-pressure solid-solidification by high-pressure Synchrotron X-ray diffraction. The obtained pressure-volume relations were fitted to Birch's equation of state. The calculated bulk moduli were lower than those of pure Al at a reduction of about 0.6 GPa per mol% Li. The temperature dependence of the elastic modulus for those supersaturated solid solutions was also measured, and it was found that the trend in the variation of the elastic modulus against the lithium concentration was maintained in the temperature range of 5–290 K. Therefore the attractive relationship between the bulk modulus and Young's modulus was demonstrated to be intrinsic.     

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.     

Correlation of elastic properties of melt infiltrated SiC/SiC composites to in situ properties of constituent phases

The ability to correlate the elastic properties of melt infiltrated SiC/SiC composites to properties of constituent phases using a hybrid Finite Element approach is examined and the influence of material internal features, such as the fabric architecture and intra-tow voids, on such correlation is elucidated. Tensile testing was carried out in air at room temperature and 1204 °C. Through-thickness compressive elastic modulus utilizing the stacked disk method was measured at room temperature. In situ moduli of constituent materials were experimentally evaluated using nano-indentation techniques at room temperature. A consistent relationship is observed between constituent properties and composite properties for in-plane normal and shear moduli and Poisson’s ratio at room temperature. However, experimental data for through-thickness compressive elastic modulus is lower than the calculated value. It is hypothesized that the existence of voids inside the fiber tows and their collapse under compressive loads is the cause of such discrepancy. Estimates for the change in elastic moduli of constituent phases with temperature were obtained from literature and used to calculate the elastic properties of the composites at 1204 °C. A reasonable correlation between the in-plane elastic moduli of the composite and the in situ elastic properties of constituent phases is observed.