Asymptotic evaluation of effective complex moduli of fibre‐reinforced viscoelastic composite materials

We propose an asymptotic approach for the evaluation of effective complex moduli of viscoelastic fibre‐reinforced composite materials. Our method is based on the homogenization technique. We start with a non‐trivial expansion of the input plane‐strain boundary value problem by ratios of visco‐elastic constants. This allows to simplify the governing equations to forms analogous to the complex transport problem. Then we apply the asymptotic homogenization method, coming from the original problem on multi‐connected domain to the cell problem, defined on a unit cell of the periodic structure. For the analytical solution of the cell problem we apply the boundary perturbation technique, the asymptotic expansion by a distance between two neighbouring fibres and the method of two‐point Padé approximants. As results we derive uniform analytical representations for effective complex moduli, valid for all values of the components volume fractions and properties.     

Numerical calculation of effective elastic moduli for incompressible porous material

Incompressible elastic material with a periodic system of pores is considered. Processes are studied with a typical length which is much more than the typical diameter of pores and the typical distance between pores. Porous material behaves as a certain “effective” material without pores in such processes. The method of calculation of effective moduli based on mathematical homogenization theory is described. The estimates for the effective moduli are proved. The results of numerical calculations of effective moduli for materials with spherical and cubic pores are presented. The dependence of the effective moduli on the volume fracture of pores is investigated. The explicit formulae for effective coefficients are deduced. Comparison with the effective moduli for compressible materials is performed. Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October, 1995. Lomonosov Moscow State University, Department of Mathematics and Mechanics, Moscow. Published in Mekhanika Kompozitnykh Materialov, No. 5, pp. 579–587, September–October, 1996.     

Determination of effective thermal expansion coefficients of unidirectional fibrous nanocomposites

We present an efficient numerical scheme (based on complex variable techniques) to calculate the effective thermal expansion coefficients of a composite containing unidirectional periodic fibers. Moreover, the mechanical behavior of the fibers incorporates interface effects allowing the ensuing analytical model of the composite to accommodate deformations at the nanoscale. The resulting ‘nanocomposite’ is subjected to a uniform temperature variation which leads to periodic deformations within the plane perpendicular to the fibers and uniform deformations along the direction of the fibers. These deformation fields are determined by analyzing a representative unit cell of the composite subsequently leading to the corresponding effective thermal expansion coefficients. Numerical results are illustrated via several physical examples. We find that the influence of interface effects on the effective thermal expansion coefficients (in particular that corresponding to the transverse direction in the plane perpendicular to the fibers) decays rapidly as the fibers become harder. In addition, by comparing the results obtained here with those from effective medium theories, we show that the latter may induce significant errors in the determination of the effective transverse thermal expansion coefficient when the fibers are much softer than the matrix and the fiber volume fraction is relatively high.     

On the coupled theory of thermo-magnetoelectroelasticity

Within the context of the coupled theory of thermo-magnetoelectroelasticity,we derive some variational principles which fully characterizethe solution of the boundary-initial-value problem. Then weestablish a reciprocity relation using a new method of proof,which involves two thermoelastic processes at different instants.We show that this relation can be used to obtain reciprocityand uniqueness theorems. The reciprocity theorem avoids boththe use of the Laplace transform and the incorporation of initialconditions into the equations of motion. The uniqueness theoremis derived without making restrictions on the positive definitenessof the elastic moduli or the conductivity tensor. There arealso no restrictions on piezoelectric moduli, piezomagneticmoduli and thermal coupling coefficients other than symmetryconditions. The results obtained are applicable for some specialcases which can be deduced from our model.     

On the integral representation formula for a two‐component elastic composite

The aim of this paper is to derive, in the Hilbert space setting, an integral representation formula for the effective elasticity tensor for a two‐component composite of elastic materials, not necessarily well‐ordered. This integral representation formula implies a relation which links the effective elastic moduli to the N‐point correlation functions of the microstructure. Such relation not only facilitates a powerful scheme for systematic incorporation of microstructural information into bounds on the effective elastic moduli but also provides a theoretical foundation for inverse‐homogenization. The analysis presented in this paper can be generalized to an n‐component composite of elastic materials. The relations developed here can be applied to the inverse‐homogenization for a special class of linear viscoelastic composites. The results will be presented in another paper. Copyright © 2005 John Wiley & Sons, Ltd.     

Effective elastic moduli and stress concentrator factors in random structure aligned fiber composites

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically uniform random set of aligned fibers. Effective elastic moduli as well as the stress concentrator factors in the components are estimated. The micromechanical approach is based on the Green’s function technique as well as on the generalization of the “multiparticle effective field method” (MEFM, see for references, Buryachenko [1]). The refined version of the MEFM takes into account the variation of the effective fields acting on each pair of fibers. The dependence of effective elastic moduli and stress concentrator factors on the radial distribution function of the fiber locations is analyzed. Received: October 20, 2004     

Polynomial approximations of a class of stochastic multiscale elasticity problems

We consider a class of elasticity equations in \({\mathbb{R}^d}\) whose elastic moduli depend on n separated microscopic scales. The moduli are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger–Reissner mixed problem that allows for computing the stress directly and the multiscale mixed problem with a penalty term for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real parameters which represent the probabilistic law of the stochastic equations. We study the multiscale homogenized problems that contain all the macroscopic and microscopic information. The solutions of these multiscale homogenized problems are written as generalized polynomial chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin approximating problems that project into the spaces of functions with only a finite number of N gpc modes. Assuming summability properties for the coefficients of the elastic moduli’s expansion, we deduce bounds and summability properties for the solutions’ gpc expansion coefficients. These bounds imply explicit rates of convergence in terms of N when the gpc modes used for the Galerkin approximation are chosen to correspond to the best N terms in the gpc expansion. For the mixed problem with a penalty term for nearly incompressible materials, we show that the rate of convergence for the best N term approximation is independent of the Lamé constants’ ratio when it goes to \({\infty}\). Correctors for the homogenization problem are deduced. From these we establish correctors for the solutions of the parametric multiscale problems in terms of the semidiscrete Galerkin approximations. For two-scale problems, an explicit homogenization error which is uniform with respect to the parameters is deduced. Together with the best N term approximation error, it provides an explicit convergence rate for the correctors of the parametric multiscale problems. For nearly incompressible materials, we obtain a homogenization error that is independent of the ratio of the Lamé constants, so that the error for the corrector is also independent of this ratio.     

非齐次小周期复合材料稳态热问题的多尺度渐近展开和收敛性分析

利用多尺度渐近展开和均匀化思想讨论了小周期复合材料的稳态热问题,得到了非齐次边界条件下二阶椭圆型方程的渐近解,并给出了原始解与渐近解之间的误差估计,数值结果表明了结论的正确性.     

Calculation of microstrains and microstresses in a thick non-symmetric heterogeneous plate by homogenization

We consider the thermoelastic behaviour of a thick heterogeneous plate containing in its thickness a large number of periodically distributed transverse holes or inclusions. We use the Reissner-Mindlin thermoelastic linear model of thick plates with a known temperature and we distinguish displacements in the upper and lower part of the plate with respect to the middle plane. Due to the structure of the plate, thermal and elastic coefficients are non-uniformly and rapidly oscillating functions of the space variable. Two-scale convergence, which is the state of the art in mathematical homogenization technics, is used and gives convergence results and formulae allowing to calculate the distribution of microstrains and microstresses inside the plate when a macroscopic behaviour is given.     

Calculation of deformative and thermal properties of multiphase composite materials filled with composite or hollow spherical inclusions by the effective medium method

A theoretical investigation was carried out to examine the possibilities of a structural approach to prediction of elastic constants, creep functions, and thermal properties of multiphase polymer composite materials filled with composite or hollow spherical Inclusions of several types. The problem of determining effective properties of the composite was solved by generalizing the effective medium method, a variant of the self-consistent method, for the case of a four-phase kernel-shell-matrix-equivalent homogeneous medium model. Exact analytical expressions for the bulk modulus thermal expansion coefficient, thermal conductivity coefficient, and specific heat were obtained. The solution for the shear modulus is given in the form of a nonlinear equation whose coefficients are the solution of a system of 12 linear equations.To be presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 462–472, July–August, 1995.     

Finite element analysis of thermoelastic field in a rotating FGM circular disk

This study focuses on the finite element analysis of thermoelastic field in a thin circular functionally graded material (FGM) disk subjected to a thermal load and an inertia force due to rotation of the disk. Due to symmetry, the FGM disk is assumed to have exponential variation of material properties in radial direction only. As a result of nonuniform coefficient of thermal expansion (CTE) and nonuniform temperature distribution, the disk experiences an incompatible eigenstrain which is taken into account. Based on the two dimensional thermoelastic theories, the axisymmetric problem is formulated in terms of a second order ordinary differential equation which is solved by finite element method. Some numerical results of thermoelastic field are presented and discussed for an Al₂O₃/Al FGM disk. The analysis of the numerical results reveals that the thermoelastic field in an FGM disk is significantly influenced by temperature distribution profile, radial thickness of the disk, angular speed of the disk, and the inner and outer surface temperature difference, and can be controlled by controlling these parameters.     

Temperature and displacement fields in brake and clutch systems with thermoelastic instabilities

In brake and clutch systems kinetic energy is converted into thermal energy. Experiments show that the corresponding temperature field can develop unstable periodic structures. The temperature field couples to the displacement field by thermal expansion. Local pressure maxima in the frictional plane and the corresponding maxima in heat generated cause thermoelastic instabilities (TEI). A model describing both effects covers layers of thermoelastic materials for all necessary mechanical components of the system. The set of field equations of each layer can analytically be solved by separation of constants. These solutions must fulfill the boundary conditions e.g. in the sliding plane. A stability discussion yields whether TEI appear or not. As a study a brake system is analyzed comprising two pads pressed against a rotating disk with cooling channels inside. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of a Thick Plate Model with Inclusions or Openings Periodically Distributed in the Thickness

AMS (MOS): 65 M-N

Using Hencky-Mindlin thermoelastic linear model for plates with a large number of small identical inclusions or openings in their thickness, we obtain equilibrium equations in which coefficients depend on x and periodically on x/ε where ε is the diameter of the cell of the periodic structure.

Formal asymptotic expansions give “homogenized” equations with coefficients independant of ε corresponding to an equivalent homogeneous plate.

Solution of these homogenized equations is proved to be (in a weak sense) the limit, when ε tends to zero, of original equations solution.

Moreover this result leads to a method giving explicit computation of thermal and elastic coefficients of the equivalent homogeneous plate in the case of right cylindrical openings. Numerical results for different shapes of openings are given.     

Homogenization of dynamic laminates

This paper addresses the study of the homogenization problem associated with propagation of long wave disturbances in materials whose properties exhibit not only spacial but also temporal inhomogeneities (called dynamic materials). The study was initiated by Lurie in his pioneering work of 1997. Homogenization theory is employed to replace an equation with oscillating coefficients by a homogenized equation. Two typical examples of periodic homogenization are considered: the wave equation and Maxwell's system coefficients oscillating rapidly not only in space but also in time. Conditions that generate applicability of the homogenization procedure to dynamic materials composites are developed. In particular, we examine a cell problem for periodic composites as well as the laminate formulae. The effective tensors of rank-one laminates for one-dimensional wave equation and the full Maxwell's system are computed explicitly. We also note some dramatic differences between the hyperbolic and the elliptic cases.     

Homogenization of microstructures in dynamics using periodic boundary conditions

A homogenization method is used to calculate effective material properties of periodic microstructures subjected to small, time-harmonic strain fields. The method is based on the equivalence of averaged micro-field quantities in dispersive heterogeneous media and those in effective homogeneous media with frequency dependent elastic constants. Solutions to the elastodynamic boundary value problem (BVP) are obtained from a boundary element method where periodic boundary conditions on the unit-cell are incorporated using Lagrangian multipliers. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the representation formula for well‐ordered elastic composites: a convergence of measure approach

The aim of this paper is to derive an integral representation formula for the effective elasticity tensor for a two‐component well‐ordered composite of elastic materials without using a third reference medium and without assuming the completeness of the eigenspace of the operator ? defined in (2.16) in (J. Mech. Phys. Solids 1984; 32 (1):41–62). As shown in (J. Mech. Phys. Solids 1984; 32 (1):41–62) and (Math. Meth. Appl. Sci. 2006; 29 (6):655–664), this integral representation formula implies a relation which links the effective elastic moduli to the N‐point correlation functions of the microstructure. Such relation not only facilitates a powerful scheme for systematic incorporation of microstructural information into bounds on the effective elastic moduli but also provides a theoretical foundation for de‐homogenization. The analysis presented in this paper can be generalized to an n‐component composite of elastic materials. The relations developed here can be applied to the de‐homogenization for a special class of linear viscoelastic composites. The results will be presented in another paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Thermally Stressed State of Bodies with Two Ellipsoidal Inclusions

Solutions are obtained for the interaction of two ellipsoidal inclusions in an elastic isotropic matrix with polynomial external athermal and temperature fields. Perfect mechanical and temperature contact is assumed at the phase interface. A solution to the problem is constructed. When the perturbations in the temperature field and stresses in the matrix owing to one inclusion are re-expanded in a Taylor series about the center of the second inclusion, and vice versa, and a finite number of expansion terms is retained, one obtains a finite system of linear algebraic equations in the unknown constants. The effect of a force free boundary of the half space on the stressed state of a material with a triaxial ellipsoidal inhomogeneity (inclusion) is investigated for uniform heating. Here it was assumed that the elastic properties of the inclusions and matrix are the same, but the coefficients of thermal expansion of the phases differ. Studies are made of the way the stress perturbations in the matrix increase and the of the deviation from a uniform stressed state inside an inclusion as it approaches the force free boundary.     

Numerical estimation of the thermal expansion of lamellar single domains in metal/ceramic composites

A porous ceramic preform, the pore structure of which is created via a freeze-casting technique, is infiltrated with a metal melt via sqeeze-casting. The microstructure of the obtained metal/ceramic composites has lamellar domains with geometrical characteristics which are dependent on the manufacturing parameters. The aim of our study is to predict the coefficients of thermal expansion of single domains of parallel Al₂O₃ platelets embedded in Al. For this purpose, a homogenization procedure was employed for microstructure based finite element and micromechanical modeling. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

空心球复合材料热弹性性质的一些精确结果

本文基于所提出的基体均匀场方法研究了空心球增强复合材料的热弹性性质·导出了均匀边界条件激发的局部热场和力学场量的关系，并进而得到了复合材料等效热弹性性质之间的精确关系·对于具有某种特定内外径比的空心球所构成的宏观各向同性复合材料，如果基体和空心球的热膨胀系数相同，可以证明其等效体积模量和线膨胀系可以精确地确定·     

Homogenization of magneto-electro-elastic multilaminated materials

In this work, based on the periodic unfolding homogenizationtechnique, the limiting equations modelling the behaviour ofthree-dimensional magneto-electro-elastic periodic structuresare rigorously established. The local problems and the correspondinghomogenized coefficients of the elastic, dielectric, magneticpermittivity, piezoelectric, piezomagnetic and magneto-electric(ME) tensors are explicitly described. The homogenization modelis exemplified for laminated composites and a unified generalformula for all effective properties of periodic multilaminatedmagneto-electro-elastic composites is obtained. This formulais applied to investigate the global behaviour for the importantcase of transversely isotropic constituents and any finite numberof layers in each periodic cell. Examples that provide theoreticalevidence of the presence of both a product property and theME effect are given.