On the effective elastic properties of matrix composites: Combining the effective field method and numerical solutions for cell elements with multiple inhomogeneities

The work is devoted to calculation of effective elastic constants of homogeneous materials containing random or regular sets of isolated inclusions. Our approach combines the self-consistent effective field method with the numerical solution of the elasticity problem for a typical cell. The method also allows analysis of detailed elastic fields in the composites. By the numerical solution of the elasticity problem for a cell, integral equations for the stress field are used. Discretization of these equations is carried out by Gaussian approximating functions. For such functions, elements of the matrix of the discretized problem are calculated in explicit analytical forms. If the lattice of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure. The matrix-vector products with such matrices may be carried out by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Results are given for 2D-media with regular and random sets of circular inclusions, and compared with existing exact solutions.     

Hybrid effective medium approximations for random elastic composites

Several popular effective medium approximations for elastic constants of random composites are reformulated in terms of a pair of canonical functions and their transform variables. This choice of reformulation enables easier comparisons of the results of all these methods with rigorous bounds. Furthermore, insight into the various methods gained by taking this point of view suggests a number of new effective medium approximations that, in some cases, are natural variants and/or combinations (i.e., hybrids) of the existing ones, and in other cases are new ones based in part on the bounds themselves. Numerical comparisons are given for several standard inclusion models — including spherical, needle, and penny-shaped inclusions — as well as the penetrable sphere model. Of the various alternatives considered, a new method called the split-step differential (SSD) scheme is one of the more useful ones, as it simplifies the differential scheme by replacing half of this scheme’s integration routines with a simple update formula for the bulk modulus.     

预报纤维增强复合材料有效热导率的随机微结构胞元模型

发展了能预报复合材料有效性质的随机微结构胞元模型以预测单向纤维增强复合材料横向热导率。研究了能反映宏观有效性质的模型最小化问题, 探讨了微结构影响宏观有效热传导性能的机制。结果表明: 通过对模型指定周期边界条件并且以多个合适的小规模模型计算的平均值取代大模型计算, 可大大改进收敛性并提高计算效率, 从10×10个到30×30个子胞的模型, 所得有效热导率计算结果的最大相对变化量仅为0.6%。不同纤维排列引起热流穿过热阻大的基体的路径长度改变, 造成有效热导率不同; 纤维热导率远大于基体热导率时, 纤维随机分布造成纤维偏聚, 部分纤维接触形成"热流通道", 使得有效热导率增大, 揭示了某一体积分数下有效热导率急剧增加是由"热流通道"贯通引起。与实验结果的比较说明了微结构随机性研究的必要性和本文工作的实用价值。     

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.     

一种有效实现弱平稳随机信号完整功率谱估计的方法

为解决某些振动信号中含有微弱频率难分辨的问题,本文采用现代信号处理的小波分析和MUSIC(多重信号分类)谱估计,提出了一种新的实现弱平稳随机信号完整功率谱估计的方法。该方法将弱平稳随机信号利用小波分解与重构得到各频段的细化信号,再将经MUSIC谱估计后的细化功率谱融合,得到具有较高分辨率的完整弱平稳随机信号功率谱。     

Effective parameters of static coupled physicomechanical field in matrix composites

Conclusions It is shown that the problem of evaluating the effective parameters of composites for a wide range of stationary coupled fields is reduced to examining uncoupled fields.The multiparticle method of the effective field was generalized with special reference to the problems of examining the static coupled physicomechanical fields in composites.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 4, pp. 105–111, July–August, 1991.     

结构非平稳随机响应方差数值计算方法

对受非平稳随机激励的工程结构,建立了在任意调制的演变随机白噪声和有色噪声激励下,结构响应方差矩阵的计算方法.提出了矩阵微分方程的数值解法,首先将矩阵微分方程转换为普通的代数微分方程,再应用常规的ODE求解器进行求解.该方法可以处理任意调制的演变随机白噪声或有色噪声,原理简明,易于掌握和应用.文中介绍了相应的关键计算步骤.数值仿真算例显示了该方法的正确性和可行性.     

Effective thermoelastic properties of random structure composites reinforced by the clusters of deterministic structure (application to clay nanocomposites)

Summary Polymer/clay nanocomposites consisting of an epoxy matrix reinforced by silicate clay plates have been observed to exhibit enhanced mechanical properties at low volume fraction of clay. The matrix and embedded nanoelements are modeled in the framework of continuum mechanics with known mechanical properties previously evaluated by, e.g., molecular dynamic simulation. Nanoclay composite is modeled by the aligned, uniformly distributed in the matrix stacks of parallel clay sheets separated from one another by interlayer matrix galleries of nanometer scale. Interaction of a finite number of oblate spheroidal inclusions modeling an individual stack inside the infinite matrix is carried by the multipole expansion technique. The obtained accurate numerical solution was incorporated into the multiparticle effective field method [5] for the estimation of effective thermoelastic properties. Detailed parametric analyses demonstrate the influence on the effective elastic moduli and stress concentrator factors of such key factors as the shape of nanoelements, interlayer distance, and the number of nanoelements in the stacks of deterministic structure.     

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.     

The dynamic response of random composites by a causal differential method

An extension of the method of causal differential media is given. The present approach makes use of the concept of the generalized extinction cross section of inclusion in a lossy medium to evaluate the total effect of losses, and then resorts to the Kramers—Kronig relations to find the phase velocity. Numerical results and comparison with alternative theories are given.     

随机结构复合材料等效导热系数的理论预测

基于随机生成结构法创建了纤维随机排布的复合材料纤维体,采用链表数据结构实现了一种物理直观、不依赖网格划分的纤维体等效导热系数理论预测方法。将该方法用于酚醛浸渍碳烧蚀材料,研究了影响等效导热系数的相关因素。结论表明:复合材料的等效导热系数并非材料固有属性,纤维长度与试件尺度相近时,试件尺度会影响材料导热系数;单位空间的纤维根数与等效导热系数呈非线性正相关关系;但等效导热系数并非体积分数的单变量函数,还取决于纤维的连通性,有效长度愈小,则表明连通性愈好,等效导热系数愈大。      

Successive iteration method applied to composites containing sliding inclusions: effective modulus and anelasticity

The effective shear modulus and Poisson's ratio of a body containing spherical sliding inclusions are calculated. First, the amount of sliding on the surface of a single spherical sliding inclusion is determined so that the tangential traction of the stresses due to the sliding, the Somigliana dislocations, cancels that of the external stresses on the surface of the inclusion. Next, the influence of other inclusions is accounted for by using a successive iteration method based on the average field theory. The successive iteration converges into closed forms, leading to analytical forms of the effective elastic constants. It is shown that the sliding occurs in a first order kinetics, the relaxation time of which is proportional to the radius of the inclusions with a constant depending on the volume fraction of the inclusions. The two-dimensional problem of a body containing aligned cylindrical fibers is also solved.     

An isogeometric collocation method for efficient random field discretization

This paper presents an isogeometric collocation method for a computationally expedient random field discretization by means of the Karhunen-Loève expansion. The method involves a collocation projection onto a finite-dimensional subspace of continuous functions over a bounded domain, basis splines (B-splines) and nonuniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves an exact geometrical representation of many commonly used physical or computational domains and exploits the regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, in the collocation method, the construction of the system matrices for a d-dimensional eigenvalue problem asks for at most d-dimensional domain integrations, as compared with 2d-dimensional integrations required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a huge computational advantage over the existing Galerkin method. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the collocation method for obtaining eigensolutions.     

Influence of testing method on mechanical properties of ceramic matrix composites

Mechanical properties of a monolithic zircon and zircon-matrix composites reinforced with silicon carbide monofilaments and/or whiskers were measured in three-point flexure and uniaxial tension modes to study the influence of testing methods on mechanical behaviour. A number of composite characteristics, such as the first-matrix cracking stress and strain, the ultimate composite strength and strain, and the modulus were obtained from the load-deflection behaviour in flexure and tension tests. The results indicated that the modulus values and the qualitative dependence of mechanical properties on composite parameters were similar in flexure and tension tests. In contrast, all of the other mechanical properties of the monolithic and composites were different in tests performed in flexure and tension modes. Typically, the first-matrix cracking stress and strain were higher in flexure tests than in tension tests, and these stress and strain values were independent of the filament-matrix interfacial properties. Similarly, the ultimate strengths of the monolithic and composites were higher in flexure than in tension, and these strengths were independent of interfacial properties. Therefore, the mechanical properties of composites obtained in flexure should not be used for a quantitative comparison with the predictions of micromechanical models, which are derived under the assumption of a uniform tensile stress. However, the flexure data are perfectly valid in demonstrating the qualitative dependence of mechanical properties on composite parameters.     

Micromechanical modelling and experimental investigation of random discontinuous glass fiber polymer–matrix composites

Videomeasurements were used to estimate the damage in chopped random glass fiber polymer–matrix composites. In order to predict the overall mechanical behaviour, voiding evolution induced by fiber debonding is incorporated into a micromechanics-based constitutive model. The comparison between the experimental data and the numerical predictions shows a very good agreement.