On boundary conditions for homogenization of volume elements undergoing localization

This paper presents an adaption of periodic boundary conditions (BC), which is termed tessellation BC. While periodic BC restrict strain localization zones to obey the periodicity of the microstructure, the proposed tessellation BC adjust the periodicity frame to meet the localization zone. Thereby, arbitrary developing localization zones are permitted. Still the formulation is intrinsically unsusceptible against spurious localization. Additionally, a modification of the Hough transformation is derived, which constitutes an unbiased criterion for the detection of the localization zone. The behavior of the derived formulation is demonstrated by various examples and compared with other BC. It is thereby shown that tessellation BC lead to a reasonable dependence of the effective stress on the localization direction. Furthermore, good convergence of stiffness values with increasing size of the representative volume element is shown as well as beneficial characteristics in use with strain softening material.     

A new multi‐scale scheme for modeling heterogeneous incompressible hyperelastic materials

A computational homogenization scheme is developed to model heterogeneous hyperelastic materials undergoing large deformations. The homogenization scheme is based on a so‐called computational continua formulation in which the macro‐scale model is assumed to consist of disjoint unit cells. This formulation adds no higher‐order boundary conditions and extra degrees of freedom to the problem. A computational procedure is presented to calculate the macroscopic quantities from the solution of the representative volume element boundary value problem. The proposed homogenization scheme is verified against a direct numerical simulation. It is also shown that the computational cost of the proposed model is lower than that of standard homogenization schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

玻纤增强注塑件的均匀化弹性力学参数研究

基于均匀化方法,根据长玻纤增强聚丙烯(LGFR-PP)的微观特征,建立了非连续长玻纤增强复合材料的代表性体积单元(RVE),通过有限元方法模拟预测了复合材料的宏观等效弹性力学参数,与注塑样条拉伸性能测试结果进行了比较。研究表明,通过在玻纤两侧增加聚丙烯(PP)分布,所采用的RVE较传统连续纤维的有限元模型更为合理;当玻纤成单一取向时,玻纤增强聚丙烯为一种横观各向同性材料;改变玻纤取向与拉伸方向之间的角度,拉伸方向的等效模量先微幅减小,再迅速降低,而后趋于稳定。利用均匀化方法预测非连续长玻纤增强注塑件的等效弹性力学性能具有较高的工程可行性,能进一步为玻纤增强注塑件的结构服役性能分析提供科学依据。     

Optical properties of deposit models for paints: full-fields FFT computations and representative volume element

Abstract

A 3D model of microstructure containing spherical and rhombi-shaped inclusions ‘falling’ along a deposit direction is used to simulate the distribution of nanoscale color pigments in paints. The microstructure’s anisotropy and length scales, characterized by their covariance functions and representative volume element, follow that of transversely isotropic or orthotropic media. Full-field computations by means of the fast Fourier method are undertaken to compute the local and effective permittivity function of the mixture, as a function of the wavelength in the visible spectrum. Transverse isotropy is numerically recovered for the effective permittivity of the deposit model of spheres. Furthermore, in the complex plane, the transverse and parallel components of the effective permittivity tensor are very close to the frontiers of the Hashin–Shtrikman’s domain, at all frequencies (or color) of the incident wave. The representative volume element for the optical properties of paint deposit models are studied. At fixed accuracy, it is much larger for the imaginary part of the permittivity than for the real part, an effect of the strong variations of the electric displacement field, exhibiting hot-spots, a feature previously described in the context of conductivity.     

Numerical homogenization of N‐component composites including stochastic interface defects

The purpose of this paper is to present a mathematical formulation and numerical analysis for a homogenization problem of random elastic composites with stochastic interface defects. The homogenization of composites so defined is carried out in two steps: (i) probabilistic averaging of stochastic discontinuities in the interphase region, (ii) probabilistic homogenization by extending the effective modules method to media random in the micro‐scale. To obtain such an approach the classical mathematical homogenization method is formulated for n‐component composite with random elastic components and implemented in the FEM‐based computer program. The article contains also numerous computational experiments illustrating stochastic sensitivity of the model to interface defects parameters and verifying statistical convergence of probabilistic simulation procedure. Copyright © 2000 John Wiley & Sons, Ltd.     

3-3 型压电复合材料的有限元模型及材料参数对其性能的影响

根据均匀场理论分析3-3 型压电复合材料代表体积胞元, 通过罚函数方法引入周期性边界条件, 建立了3-3 型压电复合材料的有限元模型。数值计算结果与已有实验结果基本一致, 验证了模型的合理性, 并与解析解进行了对比, 表明该有限元模型能更精确地描述3-3 型复合材料的有效性能常数。用所建模型分析了基体相体积分数、复合材料基体泊松比、弹性模量和复合材料基体分布形状等参数对静水压压电常数值和静水压灵敏值的影响。      

Numerical experiments with the Bloch–Floquet approach in homogenization

This paper deals with a numerical study of classical homogenization of elliptic linear operators with periodic oscillating coefficients (period εY). The importance of such problems in engineering applications is quite well‐known. A method introduced by Conca and Vanninathan [SIAM J. Appl. Math. 1997; 57 :1639–1659] based on Bloch waves that homogenize this kind of operators is used for the numerical approximation of their solution u^ε. The novelty of their approach consists of using the spectral decomposition of the operator on ?^N to obtain a new approximation of u^ε—the so‐called Bloch approximation θ^ε—which provides an alternative to the classical two‐scale expansion u^ε(x)=u*(x)+Σε^ku_k(x,x/ε), and therefore, θ^ε contains implicitly at least the homogenized solution u* and the first‐ and second‐order corrector terms. The Bloch approximation θ^ε is obtained by computing, for every value of the Bloch variable η in the reciprocal cell Y′ (Brillouin zone), the components of u* on the first Bloch mode associated with the periodic structure of the medium. Though theoretical basis of the method already exists, there is no evidence of its numerical performance. The main goal of this paper is to report on some numerical experiments including a comparative study between both the classical and Bloch approaches. The important conclusion emerging from the numerical results states that θ^ε is closer to u^ε, i.e. is a better approximation of u^ε than the first‐ and second‐order corrector terms, specifically in the case of high‐contrast materials. Copyright © 2005 John Wiley & Sons, Ltd.     

The concept of representative crack elements for phase-field fracture: Anisotropic elasticity and thermo-elasticity

The realistic representation of material degradation at a fully evolved crack is still one of the main challenges of the phase-field method for fracture. An approach with realistic degradation behavior is only available for isotropic elasticity in the small deformation framework. In this paper, a variational framework is presented for the standard phase-field formulation, which allows to derive the kinematically consistent material degradation. For this purpose, the concept of representative crack elements (RCE) is introduced to analyze the fully degraded material state. The realistic material degradation is further tested using the self-consistency condition, where the behavior of the phase-field model is compared to a discrete crack model. The framework is applied to isotropic elasticity, anisotropic elasticity and thermo-elasticity, but not restricted to these constitutive formulations.     

On semi‐analytical probabilistic finite element method for homogenization of the periodic fiber‐reinforced composites

The main aim of this paper is a development of the semi‐analytical probabilistic version of the finite element method (FEM) related to the homogenization problem. This approach is based on the global version of the response function method and symbolic integral calculation of basic probabilistic moments of the homogenized tensor and is applied in conjunction with the effective modules method. It originates from the generalized stochastic perturbation‐based FEM, where Taylor expansion with random parameters is not necessary now and is simply replaced with the integration of the response functions. The hybrid computational implementation of the system MAPLE with homogenization‐oriented FEM code MCCEFF is invented to provide probabilistic analysis of the homogenized elasticity tensor for the periodic fiber‐reinforced composites. Although numerical illustration deals with a homogenization of a composite with material properties defined as Gaussian random variables, other composite parameters as well as other probabilistic distributions may be taken into account. The methodology is independent of the boundary value problem considered and may be useful for general numerical solutions using finite or boundary elements, finite differences or volumes as well as for meshless numerical strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

On Bhattacharyya relative entropy in a homogenization of composite materials

The main idea of this work is an application of relative entropy in the numerical analysis of probabilistic divergence between original material tensors of the composite constituents and its effective tensor in the presence of material uncertainties. The homogenization method is based upon the deformation energy of the representative volume elements for the fiber-reinforced and particulate composites and uncertainty propagation begins with elastic moduli of the fibers, particles, and composite matrices. Relative entropy follows a mathematical model originating from Bhattacharyya probabilistic divergence and has been applied here for Gaussian distributions. The semi-analytical probabilistic method based on analytical integration of polynomial bases obtained via the least squares method fittings enables for determination of the basic probabilistic characteristics of the effective tensor and the relative entropies. The methodology invented in this work may be extended toward other probability distributions and relative entropies, for homogenization of nonlinear composites and also accounting for some structural interface defects.     

A nonlocal computational homogenization of softening quasi-brittle materials

In this paper, a computational counterpart of the experimental investigation is presented based on a nonlocal computational homogenization technique for extracting damage model parameters in quasi-brittle materials with softening behavior. The technique is illustrated by introducing the macroscopic nonlocal strain to eliminate the mesh sensitivity in the macroscale level as well as the size dependence of the representative volume element (RVE) in the first-order continuous homogenization. The macroscopic nonlocal strains are computed at each direction, and both the local and nonlocal strains are transferred to the microscale level. Two RVEs with similar geometries and material properties are introduced for each macroscopic Gauss point, in which the microscopic damage variables and the macroscale consistent tangent modulus and its derivatives are obtained by imposing the macroscopic nonlocal strain on the first RVE, and the macroscopic stress is computed by employing the microscopic damage variables and imposing the macroscopic local strain over the second RVE. Finally, numerical examples are solved to illustrate the performance of the proposed nonlocal computational homogenization technique for softening quasi-brittle materials.     

On the treatments of heterogeneities and periodic boundary conditions for isogeometric homogenization analysis

The treatments of heterogeneities and periodic boundary conditions are explored to properly perform isogeometric analysis (IGA) based on NURBS basis functions in solving homogenization problems for heterogeneous media with omni‐directional periodicity and composite plates with in‐plane periodicity. Because the treatment of the combination of different materials in IGA models is not trivial especially for periodicity constraints, the first priority is to clearly specify points at issue in the numerical modeling, or equivalently mesh generation, for IG homogenization analysis (IGHA). The most awkward, but important issue is how to generate patches for NURBS representation of the geometry of a rectangular parallelepiped unit cell to realize appropriate deformations in consideration of the convex‐hull property of IGA. The issue arises from the introduction of overlapped control points located at angular points in the heterogeneous unit cell, which must satisfy multiple point constraint (MPC) conditions associated with periodic boundary conditions (PBCs). Although two measures may be conceivable, we suggest the use of multiple patches along with double MPC that imposes PBCs and the continuity conditions between different patches simultaneously. Several numerical examples of numerical material and plate tests are presented to demonstrate the validity of the proposed strategy of IG modeling for IGHA. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermo‐mechanical analysis of periodic multiphase materials by a multiscale asymptotic homogenization approach

A spatial and temporal multiscale asymptotic homogenization method used to simulate thermo‐dynamic wave propagation in periodic multiphase materials is systematically studied. A general field governing equation of thermo‐dynamic wave propagation is expressed in a unified form with both inertia and velocity terms. Amplified spatial and reduced temporal scales are, respectively, introduced to account for spatial and temporal fluctuations and non‐local effects in the homogenized solution due to material heterogeneity and diverse time scales. The model is derived from the higher‐order homogenization theory with multiple spatial and temporal scales. It is also shown that the modified higher‐order terms bring in a non‐local dispersion effect of the microstructure of multiphase materials. One‐dimensional non‐Fourier heat conduction and dynamic problems under a thermal shock are computed to demonstrate the efficiency and validity of the developed procedure. The results indicate the disadvantages of classical spatial homogenization. Copyright © 2006 John Wiley & Sons, Ltd.     

Computational homogenization of nonlinear elastic materials using neural networks

In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three‐dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two‐scale example of heterogeneous structure with graded nonlinear properties. Copyright © 2015 John Wiley & Sons, Ltd.     

Mechanical Properties of Open Cell Foams: Simulations by Laguerre Tesselation Procedure

The Laguerre tessellation procedure is used for simulation of microstructures of open-cell foams. In contrast with the conventional Voronoii tessellation, the Laguerre one permits to simulate the foam microstructures with a given law of distribution of cell diameters. An original finite element method is developed for calculating the elastic properties: the ligaments are modelled as Timoshenko beams and each ligament is treated as one finite element. The size of the representative volume element for reliable calculations of the effective elastic properties is evaluated by computational experiments. Dependence of the properties on the cell size distributions and ligament shapes are analyzed.     

Computational damage mechanics for composite materials based on mathematical homogenization

This paper is aimed at developing a non‐local theory for obtaining numerical approximation to a boundary value problem describing damage phenomena in a brittle composite material. The mathematical homogenization method based on double‐scale asymptotic expansion is generalized to account for damage effects in heterogeneous media. A closed‐form expression relating local fields to the overall strain and damage is derived. Non‐local damage theory is developed by introducing the concept of non‐local phase fields (stress, strain, free energy density, damage release rate, etc.) in a manner analogous to that currently practiced in concrete [1, 2], with the only exception being that the weight functions are taken to be C⁰ continuous over a single phase and zero elsewhere. Numerical results of our model were found to be in good agreement with experimental data of 4‐point bend test conducted on composite beam made of Blackglas™/Nextel 5‐harness satin weave. Copyright © 1999 John Wiley & Sons, Ltd.     

Composite material design of two‐dimensional structures using the homogenization design method

Composite materials of two‐dimensional structures are designed using the homogenization design method. The composite material is made of two or three different material phases. Designing the composite material consists of finding a distribution of material phases that minimizes the mean compliance of the macrostructure subject to volume fraction constraints of the constituent phases, within a unit cell of periodic microstructures. At the start of the computational solution, the material distribution of the microstructure is represented as a pure mixture of the constituent phases. As the iteration procedure unfolds, the component phases separate themselves out to form distinctive interfaces. The effective material properties of the artificially mixed materials are defined by the interpolation of the constituents. The optimization problem is solved using the sequential linear programming method. Both the macrostructure and the microstructures are analysed using the finite element method in each iteration step. Several examples of optimal topology design of composite material are presented to demonstrate the validity of the present numerical algorithm. Copyright © 2001 John Wiley & Sons, Ltd.     

周期性复合材料等效传热系数预测的渐进均匀化新算法及其实现

渐进均匀化方法具有严格的数学基础,预测周期性复合材料的等效热传导系数具有较高的计算精度。本文提出了基于渐进均匀化方法预测周期性复合材料等效热传导系数的新算法。相比原有的算法具有两个优点：它的实现与代表体元方法一样简单,新的算法以现有的有限元商业软件为黑箱,通过简单的几个分析步骤,即可以获得复合材料的等效热传导系数;可以利用商业软件提供的多种单元类型去离散同一个单胞,在处理复杂几何单胞结构时,可以节约大量的计算费用。通过几个典型的算例,验证了方法的有效性。该工作对于推广渐进均匀化方法在预测复合材料等效热传导系数的广泛应用具有积极作用。