Periodic Composite Materials Research Articles

Stochastic homogenization of the fibre-reinforced metallic composites via the Monte-Carlo simulation technique

The main aim of this work is to apply the probabilistic homogenization method for the periodic fibre-reinforced composite material, whose components are metallic and may be subjected to the stochastic ageing processes. This homogenization method is mathematically based on the effective modules method, computational implementation is done thanks to the plane strain Finite Element Method code extended with the Monte-Carlo simulation and estimation algorithms. Initial material parameters of the fibre and matrix are modelled as the Gaussian random variables with the specified first two probabilistic moments. The stochastic aging processes are modelled as the linear decay of the Young moduli for the fibre and the matrix from their initial expectations and standard deviations using the ageing velocity defined as the Gaussian random variable also. The computer system MCCEFF used here, consisting of all necessary stochastic homogenization algorithms, allows to compute all necessary probabilistic moments and coefficients for certain time moments within the ageing history. As it is clear from the preliminary studies, the expected values of all homogenized tensor components decrease linearly in time, while their variances – apparently nonlinearly increase within the ageing time interval.

Periodic nanocomposites: A simple path for the preferential self-assembly of nanoparticles in block-copolymers

The use of diblock copolymers as chemical templates for the self-assembly of patterned and periodic materials has been the focus of extensive research in recent years. In this current work we show how a “one-pot” solution synthesis, which is based solely on the differences in the chemical reactivity of the two blocks towards metallic moiety, can be extended to create 1-, 2- and 3-dimensional ordered arrays of nanoparticles, conforming to the microstructure of the diblock copolymer. Chromium oxide nanoparticles (Cr 2O 3) synthesized in a poly(styrene- b-methyl methacrylate) diblock copolymer solution form, after solvent evaporation and annealing, thin films consisting of a periodic composite material. We show that the preferential incorporation of the Cr 2O 3 nanoparticles into the diblock copolymer through the direct polymer block – metal interactions does not hinder the self-assembly mechanism or the thermodynamically-driven microstructure formation of the diblock copolymer. Results show that the preferential segregation that occurred as early as the mixing stage of the reaction components was maintained throughout the annealing stage and the formation of the bulk ordered composite material.

Second-order homogenisation of functionally graded materials

The homogenisation theory for periodic composites is generalised to the case of quasi-periodic composites. In quasi-periodic composites, the unit cell does not repeat throughout the medium but gradually changes along one or more directions of periodicity (grading directions). Quasi-periodic composites are thus to functionally graded materials (FGMs) what periodic composites are to statistically uniform composite materials. Contrarily to most of the homogenisation methods applied to FGMs, the proposed second-order homogenisation theory takes explicitly into account the grading at the micro-level. The derived equivalent material happens to be a particular second gradient material in which few components of the strain gradient (second gradient of the displacement) should be taken into account in addition to the classical strains (first gradient of displacement). The second gradient theory therefore appears as the natural framework to appropriately handle functionally graded materials at the macro-level. It is worth mentioning that the presented second-order homogenisation procedure is somehow analogous to the one developed for periodic composite materials submitted to rapidly varying macroscopic strain fields as in regions of high gradients. In fact, both are a generalisation of the first-order homogenisation theory for periodic media and lead to a second gradient equivalent material. However, besides their different domains of application, they exhibit further substantial differences, which are highlighted in the paper.

On a Method to Reduce Uncertainties in Bulk Property Measurements of Two-Component Composites

For two-component composites, we address the inverse problem of estimating the structural parameters and decrease measurement errors in bulk property measurements. A measurement of the effective permittivity at one frequency gives microstructural information about the composite that is used in cross-property bounds to estimate the effective permittivity at other frequencies. We use this information and inverse bounds on microstructural parameters to tighten error bars on permittivity measurements at microwave frequencies. The method can be used in the design of random and periodic composite materials for a large variety of applications. We apply the method to a composite material used in radar applications.

Homogenization of fiber reinforced brittle materials: the intermediate case

We derive a cohesive fracture model by homogenizing a periodic composite material whose microstructure is characterized by the presence of brittle inclusions in a reticulated unbreakable elastic structure.

Effective conductivity of a composite material with non-ideal contact conditions

The effective conductivity of 2D doubly periodic composite materials with circular disjoint inclusions under non-ideal contact conditions on the boundary between material components is found. The obtained explicit formula for the effective conductivity contains all parameters of the considered model, such as the conductivities of matrix and inclusions, resistance coefficients, radii and centres of the inclusions and also the values of special Eisenstein functions. The method of functional equations is used to analyse the conjugation problem for analytic functions which is equivalently derived from the initial problem. Existence and uniqueness for the solution of the problem is obtained by using a reduction to a certain mixed boundary value problem for analytic functions in special functional spaces.

The vibrational properties of a periodic composite pipe in 3D space

Piping system vibrations exist in many fields. Vibration control is very important because it can limit possible damage to pipe systems caused by vibrations. Applying the idea of the Bragg scattering mechanism of phononic crystals (PCs), a pipe is designed as a periodic composite material structure. Using the transfer matrix method, the band structure of an infinite periodic straight pipe structure is calculated. With knowledge of the vibration band gaps, a 3D space pipe system is designed with a composite material periodic structure, and its various vibration modes’ frequency response functions are calculated. The accuracy of the transfer matrix method is verified with the commercial software MSC.Actran. The results show that the Bragg band gaps still exist in the 3D periodic pipe structure, and vibrations are strongly attenuated within the frequency ranges of the band gaps. Using the idea of PCs in piping systems provides a novel way to reduce pipe vibration.

Shape and topology optimization for periodic problems

In the present paper we deduce formulae for the shape and topological derivatives for elliptic problems in unbounded domains subject to periodicity conditions. Note that the known formulae of shape and topological derivatives for elliptic problems in bounded domains do not apply to the periodic framework. We consider a general notion of periodicity, allowing for an arbitrary parallelepiped as periodicity cell. Our cal- culations are useful for optimizing periodic composite materials by gradient type methods using the topo- logical derivative jointly with the shape derivative for periodic problems. Important particular cases of func- tionals to minimize/maximize are presented. A numer- ical algorithm for optimizing periodic composites using the topological and shape derivatives is the subject of a second paper (Barbarosie and Toader, Struct Multidis- cipl Optim, 2009).

207 ユニットセル配置にずれを有する周期材料への均質化理論の適用(OS6-(2) 複合材料の変形と破壊およびマルチスケール計算技術,オーガナイズドセッション)

In this study, a time-dependent homogenization theory is rebuilt for periodic composite materials in which their unit cells are arranged with misalignment. As a numerical example, the transverse elastic-viscoplastic properties of unidirectional carbon fiber-reinforced plastics (unidirectional CFRPs) with misaligned cell arrangements are analyzed using the present theory. It is thus shown that the transverse elastic-viscoplastic anisotropy of the unidirectional CFRPs decreases with the increase of misalignment of the cell arrangement.

Optimization of Periodic Composite Structures for Sub-Wavelength Focusing

Recently, there has been plenty of work in designing and fabricating materials with an effective negative refractive index. Veselago realized that a slab of material with a refractive index of −1 would act as a lens. Pendry suggested that the Veselago lens would act as a superlens, providing a perfect image of an object in contrast to conventional lenses which are only able to focus a point source to an image having a diameter of the order of the wavelength of the incident field. Recent work has shown that similar focusing effects can be obtained with certain slabs of “conventional” periodic composite materials: photonic crystals. The present work seeks to answer the question of what periodic dielectric composite medium (described by dielectric coefficient with positive real part) gives an optimal image of a point source. An optimization problem is formulated and it is shown that a solution exists provided the medium has small absorption. Solutions are characterized by an adjoint-state gradient condition, and several numerical examples illustrate both the plausibility of this design approach, and the possibility of obtaining smaller image spot sizes than with typical photonic crystals.

Influence of micro-cracking and contact on the effective properties of composite materials

In the present work a novel micro-mechanical approach to analyze the influence of micro-crack evolution and contact on the effective properties of elastic composite materials is proposed, based on homogenization techniques, interface models and fracture mechanics concepts. By means of the finite element method, enhanced non-linear macroscopic constitutive laws are developed by taking into account changes in micro-structural configuration associated with the growth of micro-cracks and with contact between crack faces. Numerical simulations are carried out for the cases of a porous composite with edge cracks and of a debonded fibre reinforced composite, loaded along extension/compression uniaxial macro-strain paths. Micro-crack propagation is modelled by using an original methodology based on the J-integral technique in conjunction with an interface model taking into account the unilateral contact of crack faces. In the context of a micro-to-macro transition obtained by controlling the macro-deformation of the micro-structure, the effects of adopting three types of boundary conditions on the macroscopic constitutive law, namely linear deformation, uniform tractions and periodic deformations and anti-periodic tractions, are studied. As a consequence, the proposed method can be applied to a large class of problems including periodic, locally periodic and irregular composite materials. Micro-crack and contact evolution result in a progressive loss of stiffness and can lead to failure for homogeneous macro-deformations associated with unstable crack propagation.

Phononic crystals in the diffraction regime

Phononic crystals are periodic composite materials exhibiting amazing wave propagation properties. In many works, complete band gaps are being looked for, i.e. the materials constituting the phononic crystal and its lattice arrangement are chosen such that propagation for all waves within a prescribed frequency range is forbidden. In other studies, the phononic crystal is considered a metamaterial, the anisotropic spatial dispersion of which can be tuned, and in which negative refraction can even be observed under certain circumstances. Such effects are usually considered in the sub-diffraction regime, i.e. below some critical onset frequency. In this work, we specifically examine phononic crystals in the diffraction regime. Indeed, the boundaries of a finite size phononic crystal embedded in a host propagation medium can be viewed as diffraction gratings, as we show. We will specifically consider two cases: two-dimensional phononic crystals composed of steel rods in water, and two-dimensional phononic crystals for surface acoustic waves achieved by etching cylindrical holes in a solid substrate.

Higher order asymptotic homogenization and wave propagation in periodic composite materials

We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steady-state elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.

Homogenization of Fiber-Reinforced Composites under the Stochastic Aging Process

The main aim of this work is to develop a homogenization method for the periodic fiberreinforced composite materials whose components are subjected to the stochastic aging processes. This homogenization method is entirely based on the effective modules method and the stochastic aging processes are modeled as the linear decay of the Young’s moduli for the fiber and the matrix from their initial expectations and standard deviations using the aging velocity defined as the random variable; no stochastic structural or interface defects are considered. Both initial values of those parameters as well as their aging velocities are assumed to be the Gaussian random variables with the specified first two probabilistic moments. Computational implementation of the stochastic composite model is provided using the Monte Carlo simulation method implemented on the homogenization-oriented finite element method code MCCEFF, where the stochastic behavior of the homogenized tensor probabilistic moments are modeled in certain time moments through the aging history. The specific case of glass fibers embedded into epoxy resin shows that the governing influence on this composite effective characteristics has the aging process in the matrix; the fiber aging can be of secondary importance depending only on the particular data.

Study of wave dispersion in periodic composites by higher‐order asymptotic homogenization method

AbstractWe present [1] an application of the higher‐order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable to the size of heterogeneities, successive reflections and refractions of the waves at the components interfaces lead to the formation of a complicated sequence of pass and stop frequency bands. The AHM provides a long‐wave approximation valid in the low frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch theorem by the plane‐wave (PW) expansions method. Anti‐plane shear waves in a fibre‐reinforced composite with a square lattice of cylindrical fibres are considered. The dispersion curves are obtained, the pass and stop bands are identified. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The relation of constant mean curvature surfaces to multiphase composites with extremal thermal conductivity

The behaviour of a periodic composite material depends not only on the properties of the constituent phases but also strongly on the microstructural texture of those phases such as spheres, lamellae and needles. This paper shows how to design the microstructure for a specific extremal bulk (effective) thermal conductivity in a three-phase composite medium. An inverse homogenization technique that is driven by the computational topology optimization algorithm is presented. Apart from benchmarking examples such as the Vigdergauz-type and sandwich-like architectures, a series of new single length-scale designs of microstructures are generated from this procedure. The topological design results are validated by comparing their conductivities against the empirical formulae in the two-phase composites. This study interestingly finds that the phase interfaces yielded from the topology optimization highly resemble the constant mean curvature surfaces. A comparison of their respective attainability with the Milton–Kohn physical bounds is made and the equivalence of these two sets of topologies is consequently justified.

Asymptotic simulation of imperfect bonding in periodic fibre-reinforced composite materials under axial shear

An asymptotic approach for simulation of the imperfect interfacial bonding in composite materials is proposed. We introduce between the matrix and inclusions a flexible bond layer of a volume fraction c (3) and of a non-dimensional rigidity λ (3), derive a solution for such three-component structure, and then set c (3)→0, λ (3)→0. In the asymptotic limit depending on the ratio λ (3)/ c (3) different degrees of the interface's response can be simulated. A problem of the axial shear of elastic fibre-reinforced composites with square and hexagonal arrays of cylindrical inclusions is considered. The performed analysis is based on the asymptotic homogenization method, the cell problem is solved using the underlying principles of the boundary shape perturbation technique. As a result, we obtain approximate analytical solutions for the effective shear modulus and for the stress field on micro level depending on the degree of the interfacial debonding. Developed solutions are valid for all values of the components’ volume fractions and properties. In particular, they work well in cases of rapid oscillations of local stresses (e.g., in the case of densely packed perfectly rigid inclusions), while many of other commonly used methods may face computational difficulties.

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.

Modeling of 3-D periodic multiphase composites by homogenization

An efficient technique is proposed for analyzing two- and three-dimensional lossy periodic composite materials, combining an asymptotic multiscale method with the unfolding method. The computed effective conductivities for square cylinders and cubes suspended in a host isotropic medium are compared to the Maxwell-Garnett mixing formula predictions. The electromagnetic field in a finite lattice of round cylinders is compared with the exact electromagnetic field calculated directly in the heterogeneous lattice

Interface defects in unidirectional composites by multiresolutional finite element analysis

The main goal of the paper is to use all the opportunities created by the wavelet analysis to model interface defects and some structural non-homogeneities in unidirectional periodic composite materials. The influence of these defects is detected by the comparison of a defected composite with the behaviour of additional composites with perfect interfaces. Some algebraic combination of the Haar wavelet, Mexican hat and harmonic wavelets is proposed to describe analytically the material properties, i.e. heat conductivity and capacity, Young modulus as well as mass density. The interface defects are simulated here as a significant decrease of any material property in the neighbourhood of a multi-material boundary inserted into the component having smaller material coefficients. The composites with and without interface defects with material characteristics so defined are compared with one another using some basic wavelet-based FEM tests of different discretisation order for transient heat transfer and free vibrations problems. The multiresolutional meshing process is completed thanks to various order meshes, where 1D two- or three-noded finite elements discretize a single representative volume element (RVE). Necessary numerical data pre-processing is carried out using the symbolic computations package MAPLE, which was developed for determination of the effective characteristics for analogous composites. Computational studies highlight the crucial role of the interface in the macro-scale behaviour of even unidirectional composites and should be extended to multiresolutional elastoplastic computational analysis, 2D and 3D analysis by the wavelet functions.