Local effective thermoelastic properties of graded random structure matrix composites

Summary  We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of ellipsoidal uncoated or coated inclusions, where the concentration of the inclusions is a function of the coordinates (functionally graded material). Effective properties, such as compliance and thermal expansion coefficient, as well as first statistical moments of stresses in the components are estimated for the general case of inhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the Green function technique as well as on the generalization of the multiparticle effective field method (MEFM), previously proposed for the research of statistically homogeneous random structure composites. The hypothesis of effective field homogeneity near the inclusions is used; nonlocal effects of overall constitutive relations are not considered. Nonlocal dependences of local effective thermoelastic properties as well as those of conditional averages of the stresses in the components on the concentration of the inclusions are demonstrated. Received 11 November 1999; accepted for publication 4 May 2000     

基于迭代法的非线性弹性均质化研究

微观结构对复合材料的宏观力学性能具有至关重要的影响, 通过合理设计复合材料微观结构可以得到期望的宏观性能. 均质化方法作为一种有效的设计方法, 它从微观结构的角度出发, 利用均匀化的概念, 实现了对复合材料宏观力学性能的预测和设计. 而当考虑非线性因素, 均质化的实现就非常困难. 本文利用双渐近展开方法, 将位移按照宏观位移和微观位移展开, 推导了非线性弹性均质化方程. 通过直接迭代法, 对非线性弹性均质化方程进行了求解, 并给出了具体的迭代方法和实现步骤. 本文基于迭代步骤和非线性弹性均质化方程编写MATLAB 程序, 对3种典型本构关系的周期性多孔材料平面问题进行了计算, 对比细致模型的应变能、最大位移和等效泊松比, 对程序及迭代方法的准确性进行了验证. 之后对一种三元橡胶基复合材料进行多尺度均质化, 将其分为芯丝尺度和层间尺度. 用线弹性的均质化方法得到了芯丝尺度的等效弹性参数, 并将其作为层间尺度的材料参数. 在层间尺度应用非线性弹性均质化方法对结构进行计算, 得到材料的宏观等效性能, 并以实验结果为基准进行评价.      

Multiscale modeling of the thermoelastic behavior of braided fabric composites for cryogenic structures

A multiscale analysis is performed to estimate the thermomechanical behavior of a column-type support post used in particle accelerators to sustain cryomagnets. First, the effective thermoelastic properties of a unidirectional composite material are computed for temperatures ranging between 1.9 and 293 K using a periodic homogenization method. Next, computed curves are used to model the thermoelastic behavior of a global representative volume element of the braided fabric composite. Finally, the thermoelastic behavior of the support in real working conditions is estimated using results previously obtained. For both unidirectional composite and braided fabric composite, a good agreement is obtained between estimated and measured values at several temperatures.     

Anisotropic effective higher-order response of heterogeneous Cauchy elastic materials

The homogenization results obtained by Bacca et al. (2013a), to identify the effective second-gradient elastic materials from heterogeneous Cauchy elastic solids, are extended here to the case of phases having non-isotropic tensors of inertia. It is shown that the nonlocal constitutive tensor for the homogenized material depends on both the inertia properties of the RVE and the difference between the effective and the matrix local elastic tensors. Results show that: (i) orthotropic nonlocal effects follow from homogenization of a dilute distribution of aligned elliptical holes and, in the limit case, of cracks; (ii) even under the dilute assumption and isotropic local effective behaviour, homogenization may lead to effective nonlocal orthotropic properties.     

Reiterated homogenization of a laminate with imperfect contact: gain-enhancement of effective properties

A family of one-dimensional (1D) elliptic boundary-value problems with periodic and rapidly-oscillating piecewise-smooth coefficients is considered. The coefficients depend on the local or fast variables corresponding to two different structural scales. A finite number of imperfect contact conditions are analyzed at each of the scales. The reiterated homogenization method (RHM) is used to construct a formal asymptotic solution. The homogenized problem, the local problems, and the corresponding effective coefficients are obtained. A variational formulation is derived to obtain an estimate to prove the proximity between the solutions of the original problem and the homogenized problem. Numerical computations are used to illustrate both the convergence of the solutions and the gain of the effective properties of a three-scale heterogeneous 1D laminate with respect to their two-scale counterparts. The theoretical and practical ideas exposed here could be used to mathematically model multidimensional problems involving multiscale composite materials with imperfect contact at the interfaces.     

EFFECT OF FRACTIONAL ORDER PARAMETER ON THERMOELASTIC BEHAVIORS OF ELASTIC MEDIUM WITH VARIABLE PROPERTIES

This paper is concerned with the thermoelastic behaviors of an elastic medium with variable thermal material properties. The problem is in the context of fractional order heat conduction. The governing equations with variable thermal properties were established by means of the fractional order calculus. The problem of a half-space formed of an elastic medium with variable thermal material properties was solved, and asymptotic solutions induced by a sudden temperature rise on the boundary were obtained by applying an asymptotic approach. The propagations of thermoelastic wave and thermal wave, as well as the distributions of displacement,temperature and stresses were obtained and plotted. Variations in the distributions with different values of fractional order parameter were discussed. The results were compared with those obtained from the case of constant material properties to evaluate the effects of variable material properties on thermoelastic behaviors.     

Homogenization of a pressurized tube made of elastoplastic materials with discontinuous properties

In this paper we present the homogenization of a periodic multilayered pressurized tube made of very dissimilar elastoplastic materials. We focus on some aspects of technological importance, such as the effective properties, the behavior of the homogenized displacements and stresses, the discontinuities of hoop and longitudinal stresses, the homogenization-induced anisotropy. We conclude that the problem needs to be reformulated in order to be stable by homogenization and we define the effective elastic and incremental stress corrector matrices for the incremental stress–total strain matrix law. Finally, we present the numerical simulation for both the non-homogeneous and the homogenized material and two numerical examples confirming the theoretical results.     

复合材料中的渐近均匀化方法

本文将非均质弹性体的渐近均匀化方法应用于复合材料的宏观与细观分析之中。该方法基于平均化的思想，将复合材料视作由周期性的细观结构所构成，其场变量依赖于宏观和细观两个尺度的坐标变量而变化。通过建立位移和应力的渐近表达式，推导出关于周期性基元的细观平衡方程和细观本构关系，并与有限元数值方法相结合，得到材料的宏观等效性能和细观应力分布。对典型算例的分析，反映出该方法的有效性及准确性。     

Determining the effective thermoelastic characteristics of regularly laminated composite in the asymmetric theory of elasticity

The asymmetric theory of elasticity is used to model a hybrid laminated composite of regular structure with all phases isotropic. The effective thermoelastic characteristics of the composite are determined. It is shown that the equations derived can be used to determine stress–strain state in all the phases of the composite using the average components of the tensors of force stresses, couple stresses, strains, and wryness in a layered material, which is of fundamental importance for the design of composites based on structural theories of failure     

Three-dimensional stochastic analysis using a perturbation-based homogenization method for elastic properties of composite material considering microscopic uncertainty

This paper discusses evaluation of influence of microscopic uncertainty on a homogenized macroscopic elastic property of an inhomogeneous material. In order to analyze the influence, the perturbation-based homogenization method is used. A higher order perturbation-based analysis method for investigating stochastic characteristics of a homogenized elastic tensor and an equivalent elastic property of a composite material is formulated.As a numerical example, macroscopic stochastic characteristics such as an expected value or variance, which is caused by microscopic uncertainty in material properties, of a homogenized elastic tensor and homogenized equivalent elastic property of unidirectional fiber reinforced plastic are investigated. The macroscopic stochastic variation caused by microscopic uncertainty in component materials such as Young’s modulus or Poisson’s ratio variation is evaluated using the perturbation-based homogenization method. The numerical results are compared with the results of the Monte-Carlo simulation, validity, effectiveness and a limitation of the perturbation-based homogenization method is investigated. With comparing the results using the first-order perturbation-based method, effectiveness of a higher order perturbation is also investigated.     

Thermoelastic stresses in a cylinder or disk with cubic anisotropy

The thermoelastic stresses in a crystal in the shape of a circular cylinder or disk are considered. The crystal is a cubically-orthotropic linear elastic solid, with three independent elastic properties. The cubic anisotropy renders the problem asymmetric, despite the axisymmetry of the geometry and thermal loading. This problem is motivated by a thermoelastic model used for certain crystal growth processes. Two simplifying assumptions are made here: (a) the problem is two-dimensional with plane strain or plain stress conditions, and (b) the elastic properties do not depend on the temperature. A new Fourier-type perturbation method is devised and an analytic asymptotic solution of a closed form is obtained, based on the weak cubic anisotropy of the crystal as a perturbation parameter. A general solution technique is described which yields the asymptotic solution up to a desired order. Numerical results are presented for typical parameter values.     

Homogenization problems of a hollow cylinder made of elastic materials with discontinuous properties

In this paper, we present the homogenization of an anisotropic hollow layered tube with discontinuous elastic coefficients. We focus on some aspects of technological importance, such as the effective coefficients of anisotropic materials, the behavior of the homogenized displacements and stresses, the discontinuities of in-plane shear, hoop and longitudinal stresses, the homogenization-induced anisotropy in the isotropic case. We conclude that the problem of cylindrically anisotropic tubes under extension, torsion, shearing and pressuring is stable by homogenization and we define the effective tensor of the material elastic coefficients. Some numerical examples confirm the theoretical results.     

Dual finite element methods in homogenization for elastic–plastic fibrous composite material

The problem of homogenization for a periodic, elastic–perfectly plastic, fiber reinforced, composite material is considered. The overall mechanical behavior of the material is described using the anisotropic model of elastic–plastic body with kinematic hardening. The appropriate initial–boundary value problem, set for one repeatable cell of the composite, is solved in order to find the effective constitutive relations. The cell problem is solved using the finite element method formulated in two dual forms: in displacements and in stresses. Stress functions are used in the latter formulation.     

On the limit and applicability of dynamic homogenization

Recent years have seen considerable research success in the field of dynamic homogenization which seeks to define frequency dependent effective properties for heterogeneous composites for the purpose of studying wave propagation. There is an approximation involved in replacing a heterogeneous composite with its homogenized equivalent. In this paper we propose a quantification to this approximation. We study the problem of reflection at the interface of a layered periodic composite and its dynamic homogenized equivalent. It is shown that if the homogenized parameters are to appropriately represent the layered composite in a finite setting and at a given frequency, then reflection at this special interface must be close to zero at that frequency. We show that a comprehensive homogenization scheme proposed in an earlier paper results in negligible reflection in the low frequency regime, thereby suggesting its applicability in a finite composite setting. In this paper we explicitly study a 2-phase composite and a 3-phase composite which exhibits negative effective properties over its second branch. We show that based upon the reflected energy profile of the two cases, there exist good arguments for considering the second branch of a 3-phase composite a true negative branch with negative group velocity. Through arguments of calculated reflected energy we note that infinite-domain homogenization is much more applicable to finite cases of the 3-phase composite than it is to the 2-phase composite. In fact, the applicability of dynamic homogenization extends to most of the first branch (negligible reflection) for the 3-phase composite. This is in contrast with a periodic composite without local resonance where the approximation of homogenization worsens with increasing frequency over the first branch and is demonstrably bad on the second branch. We also study the effect of the interface location on the applicability of homogenization. The results open intriguing questions regarding the effects of replacing a semi-infinite periodic composite with its Bloch-wave (infinite domain) dynamic properties on such phenomenon as negative refraction.     

Asymptotic homogenization of three-dimensional thermoelectric composites

Thermoelectric composites are promising for high efficiency energy conversion between thermal flows and electric conduction, though their effective behaviors remain poorly understood due to nonlinear thermoelectric coupling. In this paper, we develop an asymptotic homogenization theory to analyze the effective behavior of three-dimensional (3D) thermoelectric composites, built on the observation that the equations governing microscopic field fluctuations in the composite are actually linear instead of nonlinear after separation of length scales. A set of solutions similar to Green's function method are used to construct the unit cell problem, and appropriate interfacial continuity conditions and boundary conditions are derived. The homogenized governing equations are then developed for thermoelectric composites, and they are further reduced for a special case wherein the heat flow and electric conduction in the composite remains one-dimensional (1D) at macroscopic scale, even though the composite itself is 3D in general. The general homogenization theory is implemented using finite element method, and a key constant in the constructed solutions is determined using the reformulated eigenvalue problem. The algorithm is validated, and is applied for a number of case studies for the effective behavior of thermoelectric composites.     

Assessment of material uncertainties in solid foams based on local homogenization procedures

The present study is concerned with a numerical prediction and assessment of uncertainties in the macroscopic material properties of solid foams. The material properties are determined by means of a homogenization analysis considering a large scale representative volume element. The microstructure for the representative volume element is determined randomly using a Voronoï tesselation in Laguerre geometry with prescribed cell size distribution. For assessment of the scatter in the effective material response, the homogenization scheme is applied to subsets of the large scale representative volume element. By this means, an interrelation between the local microstructural characteristics and the local mesoscopic material response is established. In a first approach, the individual cells of the foam microstructure are employed as testing volume elements. As an alternative, a moving window technique is applied. The sets of homogenization results obtained by both approaches are evaluated by stochastic methods. For the local effective properties, a distinct scatter is observed. The results in both cases reveal that the local porosity is the most important influence parameter. For the microstructures investigated, only weak local correlations of the effective stiffnesses with a rapid spatial decay of the correlation is observed.     

Higher-order theory for periodic multiphase materials with inelastic phases

An extension of a recently-developed linear thermoelastic theory for multiphase periodic materials is presented which admits inelastic behavior of the constituent phases. The extended theory is capable of accurately estimating both the effective inelastic response of a periodic multiphase composite and the local stress and strain fields in the individual phases. The model is presently limited to materials characterized by constituent phases that are continuous in one direction, but arbitrarily distributed within the repeating unit cell which characterizes the material's periodic microstructure. The model's analytical framework is based on the homogenization technique for periodic media, but the method of solution for the local displacement and stress fields borrows concepts previously employed by the authors in constructing the higher-order theory for functionally graded materials, in contrast with the standard finite-element solution method typically used in conjunction with the homogenization technique. The present approach produces a closed-form macroscopic constitutive equation for a periodic multiphase material valid for both uniaxial and multiaxial loading. The model's predictive accuracy in generating both the effective inelastic stress-strain response and the local stress and inelastic strain fields is demonstrated by comparison with the results of an analytical inelastic solution for the axisymmetric and axial shear response of a unidirectional composite based on the concentric cylinder model and with finite-element results for transverse loading.     

Derivation of Micro/Macro Lithium Battery Models from Homogenization

In this article, we develop a micro–macroscopic coupled model aimed at studying the interplay between electrokinetics and transport in lithium ion batteries. The system studied consists of a solid (electrode material) and a liquid phase (electrolyte) with periodic microscopic features. In this work, homogenization of generalized Poisson–Nernst–Planck (PNP) equation set leads to a micro/macro formulation similar in nature to the one developed in Newman’s model for lithium batteries. Underlying conservation equations are derived for each phase using asymptotic expansions and mathematical tools from homogenization theory, starting from a PNP micromodel, and in particular Newman’s model is obtained as a corollary of the micro/macro approach developed here. The advantage of homogenization lies in the fact that effective parameters can be derived directly from the analysis of the periodic microstructure and from the application of the theory developed in this article. In addition, the advantages of using homogenization in Lithium ion battery modeling are outlined. Lastly, this work is a necessary step toward more general homogenized models and toward mathematical proofs, and it is also needed preliminary analysis for multiscale computational schemes.     

Strength homogenization of matrix-inclusion composites using the linear comparison composite approach

A homogenization procedure to estimate the macroscopic strength of nonlinear matrix-inclusion composites with different strength characteristics of the matrix and inclusions, respectively, is presented in this paper. The strength up-scaling is formulated within the framework of the yield design theory and the linear comparison composite (LCC) approach, introduced by Ponte Castaneda (2002) and extended to frictional models by Ortega et al. (2011), which estimates the macroscopic strength of composite materials in terms of an optimally chosen linear thermo–elastic comparison composite with a similar underlying microstructure. In the paper various combinations for the underlying material behavior for the individual phases of the composite are considered: The matrix phase can be a quasi frictional material characterized either by a Drucker–Prager-type (hyperbolic) or an elliptical strength criterion, which predicts a strength limit also in hydrostatic compression, while the inclusion phase either may represent empty pores, pore voids filled with a pore fluid, rigid inclusions, or solid inclusions, whose strength characteristics also maybe described by a Drucker–Prager-type or an elliptical strength criterion. For generating the homogenized strength criterion efficiently in such general cases of matrix-inclusion composites, a novel algorithm is proposed in the paper. The validation of the proposed strength homogenization procedure for selected combinations of strength characteristics of the matrix material and the inclusions is conducted by comparisons with experimental results and alternative existing strength homogenization models.     

Effective longitudinal shear moduli of periodic fibre-reinforced composites with radially-graded fibres

This paper presents a closed-form expression for the homogenized longitudinal shear moduli of a linear elastic composite material reinforced by long, parallel, radially-graded circular fibres with a periodic arrangement. An imperfect linear elastic fibre-matrix interface is allowed. The asymptotic homogenization method is adopted, and the relevant cell problem is addressed. Periodicity is enforced by resorting to the theory of Weierstrass elliptic functions. The equilibrium equation in the fibre domain is solved in closed form by applying the theory of hypergeometric functions, for new wide classes of grading profiles defined in terms of special functions. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. A parametric analysis investigating the influence of microstructural and material features on the effective moduli is presented. The feasibility of mitigating the shear stress concentration in the composite by tuning the fibre grading profile is shown.