Micromechanical modelling and two-scale simulation of epoxy/glass composites with interphases and interfaces

Composite systems consisting of glass fibres and epoxy matrix with interphases and interfaces will be considered in the modelling approach. The interphase forms the transition zone between the epoxy matrix and the glass fibre. The interface is the layer between the glass fibre and the surrounding interphase. The macroscopic strength of the composite material is intrinsically related to the bond strength of the polymeric/solid interface and the micromechanical characteristics of the three phases (epoxy, glass and interphase). Homogenization is an appropriate methodology to link these two scales to predict the overall physical behaviour of the composite. The nonlinear behaviour of amorphous polymers, cohesive interface elements and the elastic behaviour of glass fibres as part of the considered composite material are presented, as well as a representative example to show the necessity of taking interface influences into account. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media

The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically, waves of all periods (or quasi-momenta) are shown to persist and an appropriate extension of the notion of two-scale convergence is introduced. As a result, homogenised limit equations with none trivial quasi-momentum dependence are found as resolvent limits of the original operator family. This results in asymptotic spectral behaviour with a rich dependence on quasimomenta.     

Numerical Simulation for Convection-Diffusion Problem with Periodic Micro-Structure

In this article, the convection dominated convection-diffusion problems with the periodic micro-structure are discussed. A two-scale finite element scheme based on the homogenization technique for this kind of problems is provided. The error estimates between the exact solution and the approximation solution, of the homogenized equation or the two-scale finite element scheme are analyzed. It is shown that the scheme provided in this article is convergent for any fixed diffusion coefficient 5, and it may be convergent independent of δ under some conditions. The numerical results demonstrating the theoretical results are presented in this article.     

Multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations

This study develops a novel multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations. Firstly, the multiscale asymptotic analysis for these multiscale problems is given successfully, and the formal second-order two-scale approximate solutions for these multiscale problems are constructed based on the above-mentioned analysis. Then, the error estimates for the second-order two-scale (SOTS) solutions are obtained. Furthermore, the corresponding SOTS numerical algorithm based on finite element method (FEM) is brought forward in details. Finally, some numerical examples are presented to verify the feasibility and effectiveness of our multiscale computational method. Moreover, our multiscale computational method can accurately capture the local thermoelastic responses in composite block structure, plate, cylindrical and doubly-curved shallow shells.     

Homogenization of Incompressible Euler Equations

In this paper, we perform a nonlinear multiscale analysis for incompressible Euler equations with rapidly oscillating initial data. The initial condition for velocity field is assumed to have two scales. The fast scale velocity component is periodic and is of order one.One of the important questions is how the two-scale velocity structure propagates in time and whether nonlinear interaction will generate more scales dynamically. By using a Lagrangian framework to describe the propagation of small scale solution, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homogenized equation for the incompressible Euler equations. Preliminary numerical experiments are presented to demonstrate that the homogenized equation captures the correct averaged solution of the incompressible Euler equation.     

Loss of polyconvexity by homogenization: a new example

This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that polyconvex energies are not closed with respect to periodic homogenization. The counterexample is obtained through a rank-one laminated structure assembled by mixing two polyconvex functions with P-growth, where p ≥ 2 can be fixed arbitrarily.     

Homogenization for rate-independent systems

This paper is devoted to the homogenization for a class of rate-independent systems described by the energetic formulation. The associated nonlinear partial differential system has periodically oscillating coefficients, but has the form of a standard evolutionary variational inequality. Thus, the model applies to standard linearized elastoplasticity with hardening. Using the recently developed methods of two-scale convergence, periodic unfolding and the new introduced one, periodic folding, we show that the homogenized problem can be represented as a two-scale limit which is again an energetic formulation, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of problems of elasticity theory on composite structures with thin armoring

The paper considers the problems of elasticity theory on a flat slab armored by a periodic thin mesh or in a three-dimensional body armored by a periodic thin box structure. The composite medium depends on two small mutually related geometric parameters; one of them controls the periodicity cell and the other controls the thickness of the armoring structure. It is proved that the homogenization of the indicated problems is classical. In doing so, one applies V. V. Zhikov’s approach (“Zhikov measure approach”) together with the two-scale convergence method. Preliminarily, the paper studies the peculiarities of the two-scale convergence with the variable composite measure and also the Sobolev spaces of elasticity theory with variable composite measure. The obtained compactness principle (an analog of the Rellich theorem) in these spaces made it possible to prove the Hausdor. convergence of the spectrum of the problem studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

二元实值正交周期Box样条小波

In this paper, we construct a kind of bivariate real-valued orthogonal periodic box-spline wavelets. There are only 4 terms in the two-scale dilation equations. This implies that the corresponding decomposition and reconstruction algorithms involve only 4 terms respectively which are simple in practical computation. The relation between the periodic wavelets and Fourier series is also discussed.     

Long-Term Creep of Hybrid Aramid/Glass-Fiber-Reinforced Plastics

The results of experimental investigation of the long-term creep of SVM aramid fibers, EDT-10 epoxy resin, aramid-epoxy FRP (fiber-reinforced plastics), glass-epoxy FRP, and aramid/glass-epoxy hybrid FRP with different volume fractions of aramid and glass fibers are presented. The long-term tests were continued for 50,000 h (5.7 years). A structural approach for predicting the long-term creep from the properties and content of the components is considered. The effect of hybridization (partial replacement of the aramid fibers by glass fibers) on the inelastic deformation of hybrid FRP is discussed. The redistribution of stresses in the components during long-term creep of the hybrid composites is analyzed.     

具有小周期孔隙复合材料弹性结构的双尺度有限元分析

对于具有小周期孔隙复合材料弹性结构,在双尺度渐近分析理论结果的基础上提出了双尺度有限元计算格式,并给出了严格的误差估计.     

Homogenization of the Euler system in a 2D porous medium

We study the homogenization of the Euler system in a periodic porous medium (of period ɛ) by using the notion of two-scale convergence. At the limit, we recover a system which couples a cell problem with the macroscopic one.     

Homogenization of a pseudoparabolic system

Pseudoparabolic equations in periodic media are homogenized to obtain upscaled limits by asymptotic expansions and two-scale convergence. The limit is characterized and convergence is established in various linear cases for both the classical binary medium model and the highly heterogeneous case. The limit of vanishing time-delay parameter in either medium is included. The double-porosity limit of Richards' equation with dynamic capillary pressure is obtained.     

Homogenization of periodic optimal control problems via multi-scale convergence

The aim of this paper is to provide an alternate treatment of the homogenization of an optimal control problem in the framework of two-scale (multi-scale) convergence in the periodic case. The main advantage of this method is that we are able to show the convergence of cost functionals directly without going through the adjoint equation. We use a corrector result for the solution of the state equation to achieve this.     

Homogenization for heat transfer in polycrystals with interfacial resistances

Heat conduction is investigated in periodic (single- or multi-phase) microstructures having disconnected phases and resistances on the interfaces between the phases. After deriving uniform a priori estimates for the microsolutions the macroscopic equations are obtained rigorously by means of two-scale convergence. The required generalization of two scale convergence for surfaces is shown with the help of a Weyl decomposition in the context of Sobolev spaces with respect to measures.     

Homogenization of a Nonlinear Convection-Diffusion Equation with Rapidly Oscillating Coefficients and Strong Convection

A Cauchy problem for a nonlinear convection-diffusion equationwith periodic rapidly oscillating coefficients is studied. Underthe assumption that the convection term is large, it is provedthat the limit (homogenized) equation is a nonlinear diffusionequation which shows dispersion effects. The convergence ofthe homogenization procedure is justified by using a new versionof a two-scale convergence technique adapted to rapidly movingcoordinates.     

Optimality Conditions for Semilinear Hyperbolic Equations with Controls in Coefficients

An optimal control problem for semilinear hyperbolic partial differential equations is considered. The control variable appears in coefficients. Necessary conditions for optimal controls are established by method of two-scale convergence and homogenized spike variation. Results for problems with state constraints are also stated.     

On averaging of the non-periodic conductivity coefficient using two-scale extensions

We consider a second order elliptic equation with known rapidly oscillated coefficient. The equation appears to describe such problems as heat transfer in composite materials, flow in non-homogeneous porous media as well as in many others. For the periodic coefficient the averaging procedure is well known. The non-periodic case is still a challenging problem. We present a two-scale extension approach and apply it on one numerical example in 2D. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

周期结构区域中压电问题的双尺度有限元方法

近年来纤维压电复合材料的力电性能预测已发展为一个重要的研究领域．对力电耦合周期结构的复合材料问题,通过引入匹配的边界层得到了电势与位移解的新型双尺度有限元计算方法,建立了电势与位移的双尺度耦合关系,分析了双尺度有限元解的误差.数值算例验证了方法的有效性．