Periodic Composite Materials Research Articles

Bandgap formation mechanisms in periodic materials and structures

Bandgaps are frequency intervals of no wave propagation for mechanical waves. Spatial periodicity provides one mechanism for the emergence of bandgaps in periodic composite materials such as lattice materials, phononic crystals, and acoustic metamaterials. Coupling a propagating wave in a periodic medium with a local resonator provides an alternate mechanism for band-gap emergence. This study examines these two band-gap formation mechanisms using a receptance coupling technique. The receptance coupling technique yields closed-form expressions for the location of bandgaps and their width. Numerical simulations of Blochwaves are presented for the Bragg and sub-Bragg bandgaps and compared with the bounding frequency predictions given by the receptance analysis of the unitcell dynamics. It is observed that the introduction of periodic local resonators narrows Bragg bandgaps above the local resonant bandgap. Introduction of two fold periodicity is shown to widen the Bragg bandgap, thus expanding the design space. The generality of the receptance technique presented here allows straightforward extension to higher dimensional systems with multiple degrees of freedom coupling. Implication of this study for nano-electro-mechanical systems (NEMS) based filters will be discussed.

The effective electromechanical properties of cellular piezoelectret film: finite element modeling

Cellular space-charge polymer film, also called cellular piezoelectret, has very large piezoelectric effect due to their unique microvoid structure. In this article, the cellular piezoelectret film is considered to be a periodic composite material with closed-cell microvoids aligned periodically. Three dimensional finite element modeling is carried out to obtain the effective elastic modulus and piezoelectric coefficients. Sensitivity analysis was presented by modeling the effective electromechanical properties with different individual variable, including material constants and void shape parameters. By assuming a relation between void shape and void volume fraction, the finite element model can simulate quite well the inflation experiments of the voided charged Polypropylene film published in literature. Finally, the finite element model is used to explore the voided charge polymer film with non-uniform distribution of the trapped charges on the internal surface of voids. It was found that the resultant overall piezoelectric coefficients will be more significant if charges are closely gathered in the central area, and sparse in the around area of the internal surface of voids.

Effective longitudinal shear moduli of periodic fibre-reinforced composites with functionally-graded fibre coatings

This paper presents a homogenization method for unidirectional periodic composite materials reinforced by circular fibres with functionally graded coating layers. 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 coating domain is solved in closed form by applying the theory of hypergeometric functions, for different choices of grading profiles. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. The influence of microgeometry and grading parameters on the shear stress concentration at the coating/matrix interface is addressed, aimed at the composite optimization in regards to fatigue and debonding phenomena.

Periodic Bicontinuous Composites for High Specific Energy Absorption

We report on the mechanical behavior of an interpenetrating carbon/epoxy periodic submicrometer-scale bicontinuous composite material fabricated following the design principles deduced from biological composites. Using microscopic uniaxial compressive tests, the specific energy absorption is quantitatively evaluated and compared with the epoxy/air and carbon/air precursors. The carbon/epoxy material demonstrates extremely high specific energy absorption up to 720 kJ/kg and shear-dominant interphase interactions from the interlocked hard (carbon) and soft (epoxy) phases. Such bicontinuous nanocomposites are a new type of structural metamaterial with designed cell topology and mechanical anisotropy. Their inherent small length scale can play a critical role in prohibiting segregated mechanical responses leading to flaw tolerance.

On design of multi-functional microstructural materials

The design of periodic microstructural composite materials to achieve specific properties has been a major area of interest in material research. Tailoring different physical properties by modifying the microstructural architecture in unit cells is one of the main concerns in exploring and developing novel multi-functional cellular composites and has led to the development of a large variety of mathematical models, theories and methodologies for improving the performance of such materials. This paper provides a critical review on the state-of-the-art advances in the design of periodic microstructures of multi-functional materials for a range of physical properties, such as elastic stiffness, Poisson’s ratio, thermal expansion coefficient, conductivity, fluidic permeability, particle diffusivity, electrical permittivity and magnetic permeability, etc.

Effective permittivity of periodic composite materials: Numerical modeling by the finite element method

The effective properties of composite materials are closely related to the composition and arrangement of its constituents. Many studies and articles are actively studying the dielectric properties of heterogeneous structures with random and periodic arrangement. In the quasistatic limit, we use the finite element method as a numerical tool to evaluate the effective permittivity of two and three component composites. Two heterostructures are investigated; the first is formed by crossed dielectric cylinders in permanent contact and arranged in three layers. The cylinders are immersed in a dielectric host medium. The second structure is similar to the first except that the tubes are covered by an interphase layer. The numerical tool used to extract the exact value of the effective permittivity takes into account all internal multipolar interactions which contribute to the polarization of the material medium. The impacts of the relative permittivity and volume fraction of cylinders, the thickness of interphase and its dielectric constant are reported. The Maxwell–Garnett theory fails to predict the effective permittivity of the studied structures for high volume fraction and permittivity contrast. To overcome this problem, an amendment was made to the McLachlan equation McQ also termed the Two Exponent Single Percolation Equation TESPE. The first exponent t is held equal to 1 and the other exponent s is depending on the volume fraction. s is calculated so that the whole values of the effective permittivity obtained by the McQ rule are exactly the same values obtained by the simulations. Finally, we obtained a chart and a model to find the values of s, a fast way that is very useful for practitioners and design engineers of composite materials.

Kinematic Shakedown Analysis of Anisotropic Heterogeneous Materials: A Homogenization Approach

A nonlinear mathematical programming approach together with the finite element method and homogenization technique is developed to implement kinematic shakedown analysis for a microstructure under cyclic/repeated loading. The macroscopic shakedown limit of a heterogeneous material with anisotropic constituents is directly calculated. First, by means of the homogenization theory, the classical kinematic theorem of shakedown analysis is generalized to incorporate the microstructure representative volume element (RVE) chosen from a periodic heterogeneous or composite material. Then, a general yield function is directly introduced into shakedown analysis and a purely kinematic formulation is obtained for determination of the plastic dissipation power. Based on the mathematical programming technique, kinematic shakedown analysis of an anisotropic microstructure is finally formulated as a nonlinear programming problem subject to only a few equality constraints, which is solved by a generalized direct iterative algorithm. Both anisotropy and pressure dependence of material yielding behavior are considered in the general form of kinematic shakedown analysis. The purely kinematic approach based on the kinematic shakedown analysis has the advantage of less computational effort on field variables and more convenience for displacement-based finite element implementation. The developed method provides a direct approach for determining the reduced macroscopic strength domain of anisotropic heterogeneous or composite materials due to cyclic or repeated loading.

Analysis of Electrical Property Variations for Composite Medium Using a Stochastic Collocation Method

In this paper, an efficient approach is used to evaluate the stochastic variations of the bulk permittivity for complex periodic composite material. This method is based on the stochastic collation method together with a periodic 3-D finite-difference approach. Based on the proposed method, the influence of the input uncertainties (e.g., mixture inclusion permittivity, conductivity, volume fraction, and host cell periodicity) on the complex bulk permittivity is discussed.

Thermo-mechanical properties of 5-harness satin fabric composites

The use of woven textile reinforcements in composite structures increased significantly in the past decades due to their interesting properties over unidirectional fibres. Therefore, the prediction of the thermo-mechanical properties of woven fabric composites is essential from a design and manufacturing standpoint. A micromechanical approach based on finite element method that utilizes three-dimensional unit cell was applied to predict the effective properties of a periodic woven fabric composite material. Using the resin processing properties models such as cure kinetics, shrinkage, glass transition temperature and elastic modulus models, the development of the periodic woven fabric composite material thermo-mechanical properties, as the cure progresses was predicted. The residual strains and stresses generated in the composite unit cell during the cure were also predicted and linked with the development of the material properties. The effective properties of the cured woven fabric composite material were compared to the one of an equivalent cross-ply composite material to verify the validity of neglecting the fibre waviness while modelling woven fabric composite.

Finite element modeling of temporal evolution of the quasi-piezoelectric d33 coefficient of cellular piezoelectret Polypropylene film

Voided charged polymer film, also called cellular piezoelectret, shows quasi-piezoelectric response macroscopically. The piezoelectric d33 coefficient can be as large as that of Lead Zirconate Titanate ceramics. Because of the inherent viscosity of polymer, the piezoelectric d33 coefficient of cellular piezoelectret is time-dependent. In this article, the effective piezoelectric d33 coefficient of cellular piezoelectret is simulated by means of the finite element method. The cellular piezoelectret film is regarded as the periodic composite material, where the closed flat voids are scattered periodically. The hollow oblate tetrakaidecahedron is used to model the void in a representative volume element. The polymer is taken to be isotropic and viscoelastic. The electrostatic field and the deformation are numerically solved. The piezoelectric d33 coefficient is obtained in terms of the converse piezoelectric effect. Qualitative analysis of temporal evolution of the effective piezoelectric d33 coefficient is presented with respect to various parameters, including the microstructural void parameters, charge density as well as the viscoelastic parameters. Finally, this finite element model is used to simulate the experimental results of the effective piezoelectric d33 coefficients of cellular piezoelectret Polypropylene film published in literature.

EVALUATION OF GENERALIZED CONTINUUM SUBSTITUTION MODELS FOR HETEROGENEOUS MATERIALS

Several extensions of standard homogenization methods for composite materials have been proposed in the literature that rely on the use of polynomial boundary conditions enhancing the classical affine conditions on the unit cell. Depending on the choice of the polynomial, overall Cosserat, second gradient, or micromorphic homogeneous substitution media are obtained. They can be used to compute the response of the composite when the characteristic length associated with the variation of the applied loading conditions becomes of the order of the size of the material inhomogeneities. A significant difference between the available methods is the nature of the fluctuation field added to the polynomial expansion of the displacement field in the unit cell, which results in different definitions of the overall stress and strain measures and higher order elastic moduli. The overall higher order elastic moduli obtained from some of these methods are compared in the present contribution in the case of a specific periodic two-phase composite material. The performance of the obtained overall substitution media is evaluated for a chosen boundary value problem at the macroscopic scale for which a reference finite element solution is available. Several unsatisfactory features of the available theories are pointed out, even though some model predictions turn out to be highly relevant. Improvement of the prediction can be obtained by a precise estimation of the fluctuation at the boundary of the unit cell.

Size effects on the hardening of channel-type microstructures: A field dislocation mechanics-based approach

A simple shear model based on the mechanical theory of dislocation fields is developed in order to predict in a straightforward way the effects of channel width on the kinematic hardening of a soft–hard periodic composite material with channel-type microstructures. The model uses a continuous description of the crystal incompatibility at channel–wall interfaces. A non-local tangential continuity condition on the plastic distortion rate is applied, accounting for the conservation of the Burgers vector at the interfaces. The constraints thus imposed on plasticity at the walls enhance kinematic hardening, which is found to be channel-size dependent for a fixed refined mesh size. The smaller the channel width becomes for a given volume fraction of walls (hard phase), the stronger the kinematic hardening and the Bauschinger effect. When the channel width is sufficiently large or when the continuous treatment of interfaces is overlooked, the model becomes equivalent to a size-independent isostrain mean field approach, which is only able to retrieve hard-phase volume fraction effects on composite hardening. Due to dislocation density transport, the incompatibility in the plastic distortion between phases is accommodated by the creation of a continuous layer of polar edge dislocation density through a gradual dynamic accumulation at the channel–wall interfaces. In contrast, the singular surface dislocations at the channel–wall interfaces commonly observed in Eshelby-type mean field approaches are shown to be unable to describe size-dependent hardening.

Non-linear macroscopic response of fiber-reinforced composite materials due to initiation and propagation of interface cracks

In the present work the effects of micro-crack initiation and evolution under mixed mode loading conditions on the macroscopic response of elastic periodic composite materials, are investigated. To this end an innovative micro-mechanical approach based on homogenization techniques and adopting stress based failure criteria in conjunction with fracture mechanics concepts is presented, taking into account also the influence of unilateral frictionless contact between crack faces. The crack propagation process, including also the competition between progressive interface cracking and kinking out of the interface, is driven by the maximum energy release rate criterion adopted to predict incremental changes in crack path, whereas a generalized coupled stress–energy criterion is adopted to predict crack initiation under mixed mode conditions. A novel strategy for quasi-automatic simulation of arbitrary crack initiation and propagation in 2D finite element models has been implemented, accounting for mixed mode loading conditions and taking advantage of a generalized J-integral formulation. Numerical applications are developed with reference to a 2D model of a fiber-reinforced composite material with an initially undamaged inclusion/matrix interface. The capability of the proposed approach is shown by computing the composite nonlinear macroscopic response for prescribed uniaxial and shear macro-strain paths, assuming a crack initiation at the fiber/matrix interface. The effects of loading in the transverse fiber direction, inclusion size and fiber volume fraction, are also examined.

Nonlinear homogenization techniques to solve masonry structures problems

The behaviour of masonry material subjected to different in-plane loading combination is studied in this work. The masonry is considered as a periodic composite material composed by a regular distribution of brick and mortar and it is analyzed using a homogenization technique. The mechanical properties of the masonry, as an orthotropic homogeneous material, depend on the geometrical and mechanical properties of the components based on the study of the equilibrium and compatibility of a basic cell. The masonry is a frictional material and its behaviour depends on the loading direction, for these reasons, a unilateral damage model is chosen for the analysis. This model describes the behaviour of brittle materials subjected to tension–compression cyclic loads based on the introduction of two damage variables and it assumes that the damage is due to the beginning and growth of cracks only in the mortar joints. It is considered that the bricks have a linear elastic constitutive relationship. Numerical applications are performed with a nonlinear finite element code in order to test the proposed procedure by comparing the results with those available in the literature and also with experimental data.

Coupled Flexural-Longitudinal Vibration in a Curved Periodic Pipe Conveying Fluid

A longitudinal vibration coupling with a flexural vibration in a curved fluid-filled periodic pipe is studied. The pipe is fabricated by a periodic composite material structure based on the Bragg scattering mechanism of Phononic Crystals. Using the transfer matrix method, the band structure of an infinite periodic straight pipe is calculated, and the elastic wave propagation characteristics of the periodic pipe are discussed. Furthermore, the vibrational frequency response functions of a finite curved pipe are performed and the coupled vibration is studied. Finally, the transmission properties of longitudinal vibration, flexural vibration and their coupling vibration in the curved periodic fluid-filled pipe are investigated.

Homogenisation of fibrous composites under stochastic ageing

The main aim of this paper is to present an application of the generalised stochastic perturbation technique to model the stochastic ageing processes of fibre-reinforced periodic composite materials in terms of their effective properties. The response surface method is used to detect, for various time moments of the ageing process, the analytical interrelations between the effective elasticity tensor components and the composite parameters exposed to the ageing phenomena. A numerical procedure based on the finite-element code MCCEFF and the system MAPLE is used to model ageing of glass–epoxy composites, but it may find general application in the reliability analysis of other composite structures.

Homogenization of metallic fiber-reinforced composites under stochastic ageing

The main aim of this paper is to present an application of the generalized stochastic perturbation technique to model stochastic ageing processes of the metallic fibre-reinforced periodic composite materials in terms of their effective properties. Those ageing processes are modelled here as two-parametric time series having Gaussian random initial values and time rate, both defined uniquely by their expectations and standard deviations. Computational homogenization procedure is discrete and based on the Finite Element Method program MCCEFF as well as the computer algebra system MAPLE, where the Response Function Method and the stochastic analysis are entirely implemented. This numerical strategy is used to analyze probabilistic moments of the effective elastic tensor of the few metal matrix composites as well as to simulate stochastic ageing of two representative composites – MoSio 2–SiC and Ti–SiC. The approach proposed and results of computations may be further applied in the reliability analysis of metallic or the other composites.

Multi‐objective topology optimisation for microstructure of periodic composite material

By using the homogenisation method and finite element method (FEM) jointly, a multi‐objective optimisation model for the microstructure of a periodic composite material was developed in which the objective was to maximise the weighted sum of the main diagonal elements of the equivalent elasticity modular matrix and to restrain the amount of material in the microstructure. An optimisation of material distribution in the microstructure was carried out. In the optimisation, the design objective and material distribution were controlled by the optimisation criteria method and the chequerboard influence was eliminated using the sensitivity filter method. Influences of volume fraction and finite element (FE) mesh sizes on the optimal microstructures are discussed. Numerical examples indicate that the present optimisation model and method are effective.

Theoretical and Experimental Investigation of Flexural Wave Propagating in a Periodic Pipe with Fluid-Filled Loading

Based on the Bragg scattering mechanism of phononic crystals (PCs), a periodic composite material pipe with fluid loading is designed and studied. The band structure of the flexural wave in the periodic pipe is calculated with the transfer matrix (TM) method. A periodic piping experimental system is designed, and the vibration experiment is performed to validate the attenuation ability of the periodic pipe structure. Finally, a finite-element pipe model is constructed using the MSC-Actran software, and the calculated results match well with the vibration experiment. The errors between the theoretical calculation results and the vibration experimental results are analyzed.

Wave Propagation in Periodic Composites: Higher-Order Asymptotic Analysis Versus Plane-Wave Expansions Method

Abstract This work is devoted to a comparison of different methods determining stop-bands in 1D and 2D periodic heterogeneous media. For a 1D case, the well-known dispersion equation is studied via asymptotic approach. In particular, we show how homogenized solutions can be obtained by elementary series used up to any higher-order. We illustrate and discuss a possible application of asymptotic series regarding parameters other than wavelength and frequency. In addition, we study antiplane elastic shear waves propagating in the plane through a spatially infinite periodic composite material consisting of an infinite matrix and a square lattice of circular inclusions. In order to solve the problem, a homogenization method matched with asymptotic solution on the cell with inclusion of the large volume fracture is proposed and successfully applied. First and second approximation terms of the averaging method provide the estimation of the first stop-band. For validity and comparison with other approaches, we have also applied the Fourier method. The Fourier method is shown to work well for relatively small inclusions, i.e., when the inclusion-associated parameters and matrices slightly differ from each other. However, for evidently contrasting structures and for large inclusions, a higher-order homogenization method is advantageous. Therefore, a higher-order homogenization method and the Fourier analysis can be treated as mutually complementary.