Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions

Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes. They can also be regarded as fundamental invariances existing in a given physical context. Classic examples include Hill's formulae for the effective bulk modulus of a two‐phase mixture when the phases have equal shear moduli, Levin's formulae linking the effective thermal expansion coefficient and effective bulk modulus of two‐phase mixtures, and Dykhne's result for the effective conductivity of an isotropic two‐dimensional polycrystalline material. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for matrix subspaces having a structure of special Jordan algebras. One of many new exact relations is for the effective shear modulus of a class of three‐dimensional polycrystalline materials. We present complete lists of exact relations for three‐dimensional thermoelectricity and for three‐dimensional thermopiezoelectric composites that include all exact relations for elasticity, thermoelasticity, and piezoelectricity as particular cases. © 2000 John Wiley & Sons, Inc.     

Methods of analysing ropes. The extension–torsion method

Two new approaches are used for calculating the stress–strain state of a rope and its stiffnesses. The first approach relies on the theory of fibrous composites and Saint-Venant's solution for a cylinder with helical anisotropy. The second approach is based on the solution by the finite element method of the three-dimensional problem of elasticity theory for a solid inhomogeneous cylinder formed by a finite number of elastic fibres arranged in helical lines and connected by a weak filler (in the sense that its Young's modulus is several orders of magnitude less than the Young's modulus of the fibre). The behaviour of the stiffness when the modulus of elasticity of the filler tends to zero is analysed, and the results of the limiting transition are discussed. The numerical results obtained are compared with calculations by other well-known applied theories.     

On the complexity and analysis of auxetic systems

A challenge in creating a model for an auxetic system based on a formalism that is fully computable, is the aim of this paper. Two major levels of complexity are discussed in a way of understanding the structure and processes that define an auxetic system. The auxeticity and structural complexity are interpreted in the light of Cosserat elasticity which admits degrees of freedom not present in classical elasticity, i.e. the rotation of points in the material, and a couple per unit area or the couple stress. The Young'modulus computing for a laminated periodic system made up of alternating aluminum and an auxetic material is an example of computing complexity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

预测纳米纤维复合材料有效弹性性能的界面模型和界面相模型北大核心CSCD

基于广义自洽法,同时采用Gurtin-Murdoch界面模型和界面相模型研究了纳米纤维复合材料的有效弹性性能,获得了两种模型下有效体积模量的封闭解析解和计算有效面内剪切模量数值解的全部公式.基于界面模型的解答,讨论了有效体积模量和有效面内剪切模量的界面效应.证明了界面模型的解答可由界面相模型的解答退化得到,其中有效体积模量可以实现解析退化,有效面内剪切模量则可以数值退化.以含纳米孔洞的金属铝为例,比较了两种模型计算结果的差异.结果表明,当纳米孔洞半径较小时,两个模型的结果存在很大差异,而当半径较大时两个模型的结果差别不大.     

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.     

Poroelastodynamic Boundary Element Method in Time Domain: Numerical Aspects

Based on Biot's theory the governing equations for a poroelastic continuum are given as a coupled set of partial differential equations (PDEs) for the unknowns solid displacements and pore pressure. Using the Convolution Quadrature Method (CQM) proposed by Lubich a boundary time stepping procedure is established based only on the fundamental solutions in Laplace domain. To improve the numerical behavior of the CQM-based Boundary Element Method (BEM) dimensionless variables are introduced and different choices studied. This will be performed as a numerical study at the example of a poroelastic column. Summarizing the results, the normalization to time and spatial variable as well as on Young's modulus yields the best numerical behavior. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective parameters of cancellous bone

Cancellous bone is a two–component structure consisting of the bone frame and interstitial blood marrow. In the scope of this presentation, the multiscale finite element method is used for its modeling. This method results from a combination of homogenization theory and the theory of finite elements and is based on the calculation of effective material parameters by investigating representative volume elements (RVEs). For the particular kind of material considered here, a cubic two–phase RVE is assumed where the dry skeleton is modeled in different ways. Apart from the variations of the geometry, the influence of the usage of different types of finite elements is studied in this context. Note that the presence of a liquid phase requires dynamic investigation including the viscous phenomena. To this end, acoustic excitation and an analysis in the complex domain are chosen. The method permits calculation of the effective material parameters such as Young's modulus, bulk modulus and Poisson's ratio and furthermore the simulation of the behaviour of the complete bone or of its parts. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling viscoelastic response of particulate polymeric composites with high volume fractions of fillers

A generalized self-consistent method is extended to particulate viscoelastic composites with elastomeric matrices and high volume fractions of elastic inclusions. It is shown that the effective bulk modulus of a composite coincides with the bulk modulus of particles. A quadratic operator equation is derived for an analog of the effective shear relaxation kernel. This equation is explicitly solved using the Laplace transform method. The influence of material and geometrical parameters of a composite on its effective viscoelastic moduli is analyzed numerically.     

The governing equations of a thin elastic stressed beam with a periodic structure

The equilibrium equations, which govern the equations and boundary conditions for a thin elastic stressed beam with a periodic structure, are derived by the method of averaging. Unlike previous publications [1–3], initial stresses comparable with Young's modulus of the beam material are considered.     

具有非均匀界面相的颗粒和纤维增强复合材料弹性静力学问题的解析解

通过将以位移表示的平衡方程转化为黎卡提方程,得到了具有非均匀界面相的颗粒和纤维增强复合材料非均匀界面相内弹性场的解析解·　所得的解析解是弹性模量呈幂次方变化的非均匀界面相解的通用形式·　任意给定1个幂指数,可以得到具有非均匀界面相的颗粒和纤维增强复合材料体积模量的解析表达式·　通过改变幂指数及幂次方项的系数,此解析解可适用于具有多种不同性质的非均匀界面相·　结果表明:界面相模量和厚度对复合材料模量有很大的影响,当界面相存在时,粒子将出现一种"尺寸效应"·     

Analytical prediction of Young's modulus of carbon nanotubes using a variational method

Molecular mechanics and solid mechanics are linked to establish, a nanoscale analytical continuum theory for determination of stiffness and Young's modulus of carbon nanotubes. A space-frame structure consisted of representative unit cells has been introduced to describe the mechanical response of carbon nanotubes to the applied loading. According to this assumption a novel unit cell, given the name mechanical unit cell here is introduced to construct a graphene sheet or the wall of the carbon nanotubes. Incorporating the Morse potential function with the strain energy of the mechanical unit cells in a carbon nanotube is the key point of this study. The structural model of the carbon nanotube is solved to obtain its Young's modulus by using the principle of minimum total potential energy. It was found that the Young's modulus of the zigzag and armchair single-walled carbon nanotubes are 1.42 and 1.30 TPa, respectively. The results indicate sensitivity of the stiffness and Young's modulus of carbon nanotubes to chirality but show no dependence on its diameter. The presented analytical investigation provides a very simple approach to predict the Young's modulus of carbon nanotubes and the obtained results are in good agreement with the existing experimental and theoretical data.     

Vibration using for the longitudinal elasticity modulus determination

Longitudinal elasticity modulus, E, is a material specific feature, which,. in general, is establish on the pieces by longitudinal stress. This procedure is possible to apply to the compact material but not to the sintered power parts (or porous material test pieces). For sintered parts, the establishing of Young's modulus, in this paper, it is proposed by transmition of mechanical vibrations along to the test pieces. The test pieces of compact or porous material were strained at longitudinal vibrations. It was establish the linkage between vibration and density, respective between the density and the value of the longitudinal elasticity modulus. Using the test pieces of compact material we realized the methodology to obtain the longitudinal elasticity modulus regarding the compact material, and in this way can be establish the possibility to measure with a good result the longitudinal elasticity modulus for the pieces of sintered powders or of porous material. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation,Modelling and Analysis of stiffened GFRP-Samples

Glasfibre structures feature high potentials for optimization and substitution of conventional materials like steel and aluminum and their alloys. The paper deals with the insertion of glasfibre trusses into thin glasfibre structures to reinforce them. The effective material properties of the glasfibre structures were estimated by experiments and simulations. Furthermore the Young's modulus of the trusses was obtained by bending tests and tension tests. A comparison between bending experiments and bending simulations is given. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid

We undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ε. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size εtends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2‐scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2‐scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ρ_a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress–strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd.     

Stability and bound-exceeding elastic and viscoelastic response of composite materials having a negative-stiffness phase

We show that a composite material with one phase having negative bulk and Young's moduli (and therefore being unstable by itself) can be stable overall under dead traction boundary conditions if the encapsulating phase is sufficiently stiff. We also show experimental evidence that appropriate tuning of the negative-stiffness phase can lead to bound-exceeding elastic and viscoelastic overall material response. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of Unidirectional and Arbitrarily Oriented Fiber-Reinforced Materials by the Method of Conditional Moments

In this paper the method of conditional moments is developed for the case of a two–component matrix composite with randomly distributed unidirectional and arbitrarily oriented ellipsoidal inclusions. The algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters and orientation of inclusions is discussed. It is assumed that the components of the composite show orthotropic symmetry of thermoelastic properties. As a numerical example arbolite (straw particle inclusions in a cement matrix) is considered. The dependencies of Young's moduli, Poisson's ratios and shear moduli from the concentration of inclusions and for certain orientations of the inclusions are predicted and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mechanical behavior of a number of polymers at helium temperatures

An experimental investigation of the mechanical behavior of a number of polymers in the range 4.2–240°K has been made. It has been shown that at helium temperature the Poisson ratio is governed by the free volume. It has been established that the dynamic Young's modulus and shear modulus of these polymers at 4.2°K depend on the chemical structure: their values are determined by the mean distance between neighboring macromolecule chains.     

纳米晶体弹性模量的模拟研究

通过分子动力学(MD)方法模拟纳米晶体(1～3nm)的结构,并对模拟的结果进行了X射线衍射的点阵常数、结合能及弹性模量等模拟计算.结果表明纳米晶体无论是晶界和晶粒都与传统的粗晶粒晶体材料没有根本的区别,只是由于晶粒尺寸变小以及晶界的体积分数等的作用而导致诸如弹性模量大幅度减少等一系列不同性能.     

Dual‐primal FETI methods for linear elasticity

Dual‐primal FETI methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by Lagrange multipliers, as in one‐level FETI methods. These methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems. The purpose of this article is to develop strategies for selecting these constraints, which are enforced throughout the iterations, such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them. The algorithms are described in terms of a change of basis that has proven to be quite robust in practice. A theoretical analysis is provided for the case of linear elasticity, and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the Young's modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain. The strategies have already proven quite successful in large‐scale implementations of these iterative methods. © 2006 Wiley Periodicals, Inc.     

Micromechanical analysis of the stress transfer in single-fiber composite: The influence of the uniform and graded interphase with finite-thickness

A theoretical model is developed to analyze the stress transfer between fiber and matrix through the interphase with finite thickness. The Young's modulus of interphase is assumed to be homogeneous uniform or power-graded along radial direction while other material parameters are constants. The bonds between fiber and interphase as well as between interphase and matrix are perfect. The geometrical equations are strictly satisfied except that the radial displacement gradient with respect to the axial direction is neglected, as its magnitude is much smaller than that of the axial displacement gradient with respect to the radial direction. The equilibrium equations along radial direction are strictly satisfied, while the equilibrium equations along axial direction are satisfied in the integral forms. In addition, both the interfacial displacement and stress continuity conditions as well as stress boundary conditions are enforced exactly. Two coupled 2nd-order ordinary differential equations can be obtained in terms of average axial stresses in fiber and matrix. Finite element analysis (FEA) with refined mesh for single-fiber composite containing uniform interphase with finite thickness is developed to validate the present model. Series of parameter studies are performed to investigate the influence of interphase properties and thickness as well as the fiber volume content and model length on the stress distribution in composites.