Computing overall elastic constants of polydisperse particulate composites from microtomographic data

In this paper, we use the well-known Hashin-Shtrikman-Willis variational principle to obtain the overall mechanical properties of heterogeneous polydisperse particulate composites. The emphasis is placed on the efficient numerical integration of complex three-dimensional integrals and on aspects of the anisotropic material response of real tomographically characterized packs. For this purpose, we numerically calculate the complete statistics of real packs, which are numerically or tomographically generated. We use the parallel adaptive sparse Smolyak integration method with hierarchical basis to integrate complex singular integrals containing the product of probability functions and the second derivative of Green's function. Selected examples illustrate both the numerical and physical facets of our work. First, we show the reduction of integral points for integration in spherical coordinates. Then, we comment on the parallel scalability of our method and on the numerical accuracy associated with the integration of a singular function. Next, we validate the solver against the experimental data and verify the results by comparing it to a closed-form expression. To investigate the ability of our scheme to capture the anisotropic nature of packs, we study a lattice type system. Finally, we report on the elastic constants computed for the modeled anisotropic particulate system that is tomographically characterized.     

Mechanical properties of lattice grid composites

An equivalent continuum method only considering the stretching deformation of struts was used to study the in-plane stiffness and strength of planar lattice grid com- posite materials. The initial yield equations of lattices were deduced. Initial yield surfaces were depicted separately in different 3D and 2D stress spaces. The failure envelope is a polyhedron in 3D spaces and a polygon in 2D spaces. Each plane or line of the failure envelope is corresponding to the yield or buckling of a typical bar row. For lattices with more than three bar rows, subsequent yield of the other bar row after initial yield made the lattice achieve greater limit strength. The importance of the buckling strength of the grids was strengthened while the grids were relative sparse. The integration model of the method was used to study the nonlinear mechanical properties of strain hardening grids. It was shown that the integration equation could accurately model the complete stress-strain curves of the grids within small deformations.     

Thermomechanical properties of smart composites

The smart composite materials reinforced by SMA show a high performance and special deformation behavior. The thermomechanical constitutive formulas of the composites are derived by means of Eshelby's equivalent inclusion method and Mori-Tanaka's mean field concept. The interaction between the inclusion and crack and toughening mechanism are considered and the energy release rate of a crack in the smart composite is calculated. This work shows that there are the multiple mechanisms contributing to the toughening of the smart composite materials reinforced by SMA.This project is supported by the National Natural Science Foundation of China.     

A Model for Predicting Effective Properties of Piezocomposites with Non-piezoelectric Inclusions

Most piezocomposites, which have been widely used in engineering, consist of piezoelectric inclusions and a non-piezoelectric matrix. Due to the limits of fabrication technology, it is hard to avoid the matrix intermingling with other non-piezoelectric inclusions, such as cavities. The non-piezoelectric inclusions can substantially affect performance of piezocomposites. In this paper we study the electromechanical fields in piezocomposites which are composed of a non-piezoelectric matrix embedded with both piezoelectric and non-piezoelectric inclusions. Closed-form relations are obtained for the effective electroelastic moduli of a piezocomposite with both piezoelectric and non-piezoelectric inclusions. The effective properties of a 1-3 type piezocomposite with non-piezoelectric spherical inclusions are analyzed carefully and explicit formulae for the effective electroelastic properties of a 1-3-0 piezocomposite are also obtained. The analysis shows that the effect of non-piezoelectric inclusions on the electroelastic properties of piezocomposites is significant and should not be neglected. The model proposed in this paper is expected to be useful for predicting and analyzing the overall electromechanical properties of piezocomposites with a non-piezoelectric matrix containing both piezoelectric and non-piezoelectric inclusions.     

Antiplane magneto-electro-elastic effective properties of three-phase fiber composites

In the present work, applying the asymptotic homogenization method (AHM), the derivation of the antiplane effective properties for three-phase magneto-electro-elastic fiber unidirectional reinforced composite with parallelogram cell symmetry is reported. Closed analytical expressions for the antiplane local problems on the periodic cell and the corresponding effective coefficients are provided. Matrix and inclusions materials belong to symmetry class 6mm. Numerical results are reported and compared with the eigenfunction expansion-variational method (EEVM) and other theoretical models. Good agreements are found for these comparisons. In addition, with the herein implemented solution, it is possible to reproduce the effective properties of the reduced cases such as piezoelectric or elastic composites obtaining good agreements with previous reports.     

A stochastic homogenization approach to estimate bone elastic properties

The mechanical properties of bone tissue depend on its hierarchical structure spanning many length scales, from the organ down to the nanoscale. Multiscale models allow estimating bone mechanical properties at the macroscale based on information on bone organization and composition at the lower scales. However, the reliability of these estimates can be questioned in view of the many uncertainties affecting the information which they are based on. In this paper, a new methodology is proposed, coupling probabilistic modeling and micromechanical homogenization to estimate the material properties of bone while taking into account the uncertainties on the bone micro- and nanostructure. Elastic coefficients of bone solid matrix are computed using a three-scale micromechanical homogenization method. A probabilistic model of the uncertain parameters allows propagating the uncertainties affecting their actual values into the estimated material properties of bone. The probability density functions of the random variables are constructed using the Maximum Entropy principle. Numerical simulations are used to show the relevance of this approach.     

复合材料有效弹性性质分析方法eeeeee

建立复合材料的有效性质与微结构参数的关联,是复合材料优化设计的基础。本文具体针对有效弹性性质,重点介绍了建立有效性的基本思路 和主要分析方法。首先讨论了代表单元的概念,然后分别从复合材料有效性质的普适关系、界限理论和近似方法三个不同的视角较全面的介绍了建立非均质材料有效性质的方法、主要结果和最新进展。重点从构型的概念和微结构分布形式上分析了各种模型间及分析方法之间的联系与差别。最后还就建立非均质材料有效性质中存在的问题和研究热点做了简单的介绍。     

Micromechanics of particle-reinforced elasto-viscoplastic composites: Finite element simulations versus affine homogenization

The micromechanics of elasto-viscoplastic composites made up of a random and homogeneous dispersion of spherical inclusions in a continuous matrix was studied with two methods. The first one is an affine homogenization approach, which transforms the local constitutive laws into fictitious linear thermo-elastic relations in the Laplace–Carson domain so that corresponding homogenization schemes can apply, and the temporal response is computed after a numerical inversion of Laplace transform. The second method is the direct numerical simulation by finite elements of a three-dimensional representative volume element of the composite microstructure. The numerical simulations carried out over different realizations of the composite microstructure showed very little scatter and thus provided – for the first time – “exact” results in the elasto-viscoplastic regime that can be used as benchmarks to check the accuracy of other models. Overall, the predictions of the affine homogenization model were excellent, regardless of the volume fraction of spheres, of the loading paths (shear, uniaxial tension and biaxial tension as well as monotonic and cyclic deformation), particularly at low strain rates. It was found, however, that the accuracy decreased systematically as the strain rate increased. The detailed information of the stress and strain microfields given by the finite element simulations was used to analyze the source of this difference, so that better homogenization methods can be developed.     

The effective properties of composites with fiber-end cracks

The paper describes use of self-consistent finite element method (SCFEM) for predicting effective properties of fiber composite with partially debonded interface. The effective longitudinal Young's modulus and shear modulus for unidirectional fiber reinforced composites with fiber-end cracks are calculated. Numerical results show that the effective properties are considerably influenced by the fiber-end cracks. The effects of microstructural parameters, such as fiber volume fraction, modulus ratio of the constituents and fiber aspect, on the effective properties of the composites were discussed. The project supported by the National Natural Science Foundation of China     

Three-phase plane composites of minimal elastic stress energy: High-porosity structures

The paper establishes exact lower bound on the effective elastic energy of two-dimensional, three-material composite subjected to the homogeneous, anisotropic stress. It is assumed that the materials are mixed with given volume fractions and that one of the phases is degenerated to void, i.e., the effective composite is porous. Explicit formula for the energy bound is obtained using the translation method enhanced with additional inequality expressing certain property of stresses. Sufficient optimality conditions of the energy bound are used to set the requirements which have to be met by the stress fields in each phase of optimal effective material regardless of the complexity of its microstructural geometry. We show that these requirements are fulfilled in a special class of microgeometries, so-called laminates of a rank. Their optimality is elaborated in detail for structures with significant amount of void, also referred to as high-porosity structures. It is shown that geometrical parameters of optimal multi-rank, high-porosity laminates are different in various ranges of volume fractions and anisotropy level of external stress. Non-laminate, three-phase microstructures introduced by other authors and their optimality in high-porosity regions is also discussed by means of the sufficient conditions technique. Conjectures regarding low-porosity regions are presented, but full treatment of this issue is postponed to a separate publication. The corresponding “G-closure problem” of a three-phase isotropic composite is also addressed and exact bounds on effective isotropic properties are explicitly determined in these regions where the stress energy bound is optimal.     

PVC/纳米Al2O3复合材料的力学性能研究

通过熔融共混方法制备了PVC/纳米Al2O3复合材料,研究了纳米粒子对PVC的增强、增韧效果.采用细观力学方法理论上求解了纳米复合材料的有效弹性模量,比较并分析了试验值和理论计算值的偏差.     

多晶铁电材料的有效电弹性能预报

本文利用细观力学方法———Ｅｓｈｅｌｂｙ等效夹杂法和Ｍｏｒｉ Ｔａｎａｋａ的平均场理论,根据铁电材料的微结构特点,建立了一个细观统计模型对多晶铁电材料的有效电弹性能和模量进行分析预报．在本细观统计模型中,不仅考虑到单晶粒的形状影响,而且考虑铁电畴在外场作用下发生极化转动的影响．本文针对ＢａＴｉＯ３铁电陶瓷的有效电弹性能与系数的预报结果与实验观测结果相符     

In-situ mechanical properties identification of 3D particulate composites using the Virtual Fields Method

This paper presents an identification procedure based on the Virtual Fields Method (VFM) for identifying in situ mechanical properties of composite materials constitutive phases from 3D full-field measurements. The new procedure, called the Regularized Virtual Fields Method (RVFM), improves the accuracy of the VFM thanks to the imposition of mechanical constraints derived from an appropriate homogenization model. The developed algorithms were validated through virtual experiments on particulate composites. The robustness of both the VFM and the RVFM was assessed in the presence of noisy strain data for various microstructures. A study was also carried out to investigate the influence of the size of region of interests on the reliability of the identified parameters. Accordingly, the optimum size of region of interest was determined based on full-field measurement requirements and accuracy of the identified parameters. This study enables determining, a priori, the required magnification level of 3D images for composites of any mechanical and morphological characteristics.     

Transversely isotropic sequentially laminated composites in finite elasticity

A general expression for the energy-density function of sequentially laminated composites is derived. For the class of neo-Hookean composites in the limit of small deformations well-known results for linear transversely isotropic composites are recovered. However, it is shown that under large deformations these composites are not isotropic. Transversely isotropic composites are obtained with sequentially-coated composites in which the next rank composite is constructed by lamination of the previous composite with thin layers of the matrix phase. The transverse behavior of this sequentially-coated composite is neo-Hookean with shear modulus in the form of the Hashin-Shtrikman bounds for the corresponding class of linear composites. Comparison of the behaviors of these composites with recent estimates for transversely isotropic composites reveals good agreement up to relatively large deformations and volume fractions of the inclusion phase.     

Bounds and self-consistent estimates for the overall properties of anisotropic composites

Bounds of Hashin-Shtrikman type and self-consistent estimates for the overall properties of composites, which may be anisotropic, are developed. Bodies containing aligned ellipsoidal inclusions are considered particularly, generalizing previously known results. The overall thermal conductivity of a body containing aligned spheroidal inclusions is discussed as an example including, as limiting cases, bodies containing highly-conducting aligned needles and bodies containing aligned pennyshaped cracks.     

Dispersion and attenuation in thermoelastic multisize particulate composites

This paper deals with the problem of multiple scattering by a random distribution of spherical solid particles in a solid. The material properties of both media are taken as thermoelastic. The radii of the inclusions may be different. The self-consistent method in its variant of the effective medium is used to find the dispersion and attenuation of quasi-elastic, quasi-thermal and shear waves. The single scattering problem required by this technique is solved approximately by means of the Galerkin method applied to an integral equation using the Green function. Numerical results display a characteristic resonance phenomena which appears in the interval where the results are approximately valid, that is, for very long waves down to wavelengths about twice the largest diameter of the spheres. Examples are shown, for composites with two sets of inclusions, which have either a very similar or dissimilar size. Comparisons are made with the elastic counterpart. Among the material properties, the mass density ratio, inclusion to matrix, seems to play an important and simple role. Frequency intervals are distinguished and shown to depend on that ratio, where the attenuation and dispersion of quasi-elastic and P-waves are either very close to each other or not at all. The same applies to shear waves in either composite. The mass density ratio also displays a simple monotonic decreasing behaviour as a function of the frequency at the first attenuation maximum and velocity minimum. These results may be of interest for the nondestructive testing characterization of particulate composites.     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Neo-Hookean fiber-reinforced composites in finite elasticity

The response of a transversely isotropic fiber-reinforced composite made out of two incompressible neo-Hookean phases undergoing finite deformations is considered. An expression for the effective energy-density function of the composite in terms of the properties of the phases and their spatial distribution is developed. For the out-of-plane shear and extension modes this expression is based on an exact solution for the class of composite cylinder assemblages. To account for the in-plane shear mode we incorporate an exact result that was recently obtained for a special class of transversely isotropic composites. In the limit of small deformation elasticity the expression for the effective behavior agrees with the well-known Hashin-Shtrikman bounds. The predictions of the proposed constitutive model are compared with corresponding numerical simulation of a composite with a hexagonal unit cell. It is demonstrated that the proposed model accurately captures the overall response of the periodic composite under any general loading modes.     

On the overall elastic moduli of composites with spherical coated fillers

A micromechanical approach is presented to estimate the overall linear elastic moduli of three phase composites consisting of two phase coated spherical particles randomly dispersed in a homogeneous isotropic matrix. The theoretical method is based on Eshelby’s equivalent inclusion method and its recent extension by Shodja and Sarvestani [J. Appl. Mech. 68 (2001) 3] to evaluate the local field variables in case of double (multi) inhomogeneities. Using Tanaka–Mori theorem [J. Elasticity 2 (1972) 199] and a decomposition of Green’s function integral equation, the pair-wise average phase values of stress and strain in two interacting coated particles are estimated. Following Ju and Chen [Acta Mech. 103 (1994) 103; Acta Mech. 103 (1994) 123] the ensemble phase volume average of stress and strain fields can be evaluated within a representative volume element containing a finite number of coated particles. Comparisons with classical bounds are presented to illustrate the accuracy of the proposed method.     

Effective magnetoelastic properties of laminated composites

Effective physical parameters of a fine-layered medium whose layers exhibit linearized magnetostriction as ferrites are determined. Ferromagnetic materials of cubic system with ferromagnetic resonance are considered __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 36–43, August 2006.