Dynamic effective property of fibrous piezoelectric composites with spring- or membrane-type imperfect interfaces

The present paper studies the dynamic effective property of piezoelectric composites embedded with cylindrical piezoelectric fibers under anti-plane harmonic electro-elastic waves. By using the dynamic generalized self-consistent method (DGSM) of electro-elastic coupling wave, the problem of randomly distributed cylindrical fibers in a piezoelectric medium can be analyzed in terms of a representative volume element with a coated fiber embedded in an equivalent effective medium. The interfaces between the fibers and the matrix are assumed to be imperfect which are here modeled as spring- or membrane-type interfaces. Through wave function expansion method and an iterative method, the effective piezoelectrically stiffened shear modulus and the effective wave number are obtained. Examples are conducted to verify the present solutions and to illustrate the dependence of the effective piezoelectrically stiffened shear modulus on the wave number (frequency) as well as the interface properties. The special size effect related to interfacial imperfection is also discussed.     

Semi-analytical method for computing effective properties in elastic composite under imperfect contact

A parallel fiber-reinforced periodic elastic composite is considered with transversely iso-tropic constituents. Fibers with circular cross section are distributed with the same periodicity along the two perpendicular directions to the fiber orientation, i.e., the periodic cell of the composite is square. The composite exhibits imperfect contact, in particular, spring type at the interface between the fiber and matrix is modeled. Effective properties of this composite for in-plane and anti-plane local problems are calculated by means of a semi-analytic method, i.e. the differential equations that described the local problems obtained by asymptotic homogenization method are solved using the finite element method. Numerical computations are implemented and comparisons with exact solutions are presented.     

A micromechanical model for effective elastic properties of particulate composites with imperfect interfacial bonds

A micromechanical model for effective elastic properties of particle filled acrylic composites with imperfect interfacial bonds is proposed. The constituents are treated as three distinct phases, consisting of agglomerate of particles, bulk matrix and interfacial transition zone around the agglomerate. The influence of the interfacial transition zone on the overall mechanical behavior of composites is studies analytically and experimentally. Test data on particle filled acrylic composites with three different interfacial properties are also presented. The comparison of analytical simulation with experimental data demonstrated the validity of the proposed micromechanical model with imperfect interface. Both the experimental results and analytical prediction show that interfacial conditions have great influence on the elastic properties of particle filled acrylic composites.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

Effect of overlapping inclusions on effective elastic properties of composites

We are considering, in this study, to quantify the difference between two morphologies: heterogeneous materials with overlapping identical spherical inclusions and heterogeneous materials with identical hard one. Coupling with numerical simulations, the statistical analysis of microstructures morphology was used to evaluate the representativeness of results. The methodology, developed in Kanit et al. (2003), is used to determine exactly the integral range (IR), variance and covariance of each microstructure type. The obtained results show that the integral range of microstructures with hard spheres, is simply, the volume of one inclusion in the deterministic representative volume element, and for microstructures with overlapping spheres, is 8 times the integral range in the case of hard spheres. The obtained results suggest us to define a new concept what we propose to name the Equivalent Morphology Concept (EMC). The relationships between parameters of two microstructures are presented.     

Influences of imperfect interfaces on effective properties of multiferroic composites with coated inclusion

In order to predict the effective properties of multiferroic composite materials, the effective material constants of multiferroic composites with the coated inclusion and imperfect interface are investigated. Based on the generalized self-consistent theory, the closed-form solutions of the effective material constants are derived. For the composites with piezomagnetic inclusion, piezoelectric coating and polymer matrix, numerical calculations are performed to present the influences of the imperfect interface cooperating with the coating on the effective material constants. From the results, it can be observed that the effective constants can be enhanced by the coating but reduced by the imperfect interface. Moreover, the coating has the shielding effects on the imperfect interface for the composite structures with its higher filling ratio.     

Electroelastic behavior of doubly periodic piezoelectric fiber composites under antiplane shear

The piezoelectric composites with a doubly periodic parallelogrammic array of piezoelectric fibers are dealt with under antiplane shear coupled with inplane electrical load. A rigorous analytical method is developed by using the doubly quasi-periodic Riemann boundary value problem theory integrated with the eigenstrain and eigen-electrical-field concepts. The numerical results are presented and a comparison with finite element calculations, experimental data and micromechanical analysis is made to demonstrate the efficiency and accuracy of the present method. Subsequently, the present solutions are used to study two important topics in piezoelectric fiber composites, i.e., (1) stress and electrical field fluctuations in the microstructure, (2) the macroscopic effective electroelastic moduli. The relation between the macroscopic properties of the composites and their microstructural parameters is discussed and many interesting electroelastic interaction phenomena are revealed, which are useful to estimate and optimize the performance of piezoelectric composites.     

Effective elastic shear stiffness of a periodic fibrous composite with non-uniform imperfect contact between the matrix and the fibers

In this contribution, effective elastic moduli are obtained by means of the asymptotic homogenization method, for oblique two-phase fibrous periodic composites with non-uniform imperfect contact conditions at the interface. This work is an extension of previous reported results, where only the perfect contact for elastic or piezoelectric composites under imperfect spring model was considered. The constituents of the composites exhibit transversely isotropic properties. A doubly periodic parallelogram array of cylindrical inclusions under longitudinal shear is considered. The behavior of the shear elastic coefficient for different geometry arrays related to the angle of the cell is studied. As validation of the present method, some numerical examples and comparisons with theoretical results verified that the present model is efficient for the analysis of composites with presence of imperfect interface and parallelogram cell. The effect of the non uniform imperfection on the shear effective property is observed. The present method can provide benchmark results for other numerical and approximate methods.     

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.     

Closed-form solutions to the effective properties of fibrous magnetoelectric composites and their applications

Magnetoelectric coupling is of interest for a variety of applications, but is weak in natural materials. Strain-coupled fibrous composites of piezoelectric and piezomagnetic materials are an attractive way of obtaining enhanced effective magnetoelectricity. This paper studies the effective magnetoelectric behaviors of two-phase multiferroic composites with periodic array of inhomogeneities. For a class of microstructures called periodic E-inclusions, we obtain a rigorous closed-form formula of the effective magnetoelectric coupling coefficient in terms of the shape matrix and volume fraction of the periodic E-inclusion. Based on the closed-form formula, we find the optimal volume fractions of the fiber phase for maximum magnetoelectric coupling and correlate the maximum magnetoelectric coupling with the material properties of the constituent phases. Based on these results, useful design principles are proposed for engineering magnetoelectric composites.     

Asymptotic determination of effective elastic properties of composite materials with fibrous square-shaped inclusions

We propose an asymptotic approach for evaluating effective elastic properties of two-components periodic composite materials with fibrous inclusions. We start with a nontrivial expansion of the input elastic boundary value problem by ratios of elastic constants. This allows to simplify the governing equations to forms analogous to the transport problem. Then we apply an asymptotic homogenization method, coming from the original problem on a multi-connected domain to a so called cell problem, defined on a characterizing unit cell of the composite. If the inclusions' volume fraction tends to zero, the cell problem is solved by means of a boundary perturbation approach. When on the contrary the inclusions tend to touch each other we use an asymptotic expansion by non-dimensional distance between two neighbouring inclusions. Finally, the obtained “limiting” solutions are matched via two-point Padé approximants. As the results, we derive uniform analytical representations for effective elastic properties. Also local distributions of physical fields may be calculated. In some partial cases the proposed approach gives a possibility to establish a direct analogy between evaluations of effective elastic moduli and transport coefficients. As illustrative examples we consider transversally-orthotropic composite materials with fibres of square cross section and with square checkerboard structure. The obtained results are in good agreement with data of other authors.     

Theoretical study on dynamic effective electroelastic properties of random piezoelectric composites with aligned inhomogeneities

The closed-form solutions of the dynamic problem of heterogeneous piezoelectric materials are formulated by introducing polarizations into a reference medium and using the generalized reciprocity theorem. These solutions can be reduced to the ones of an elastodynamic problem. Based on the effective medium method, these closedform solutions can be used to establish the self-consistent equations about the frequencydependent effective parameters, which can be numerically solved by iteration. Theore...     

A self-consistent statistical mechanics approach for determining effective elastic properties of composites

A self-consistent statistical mechanics approach for determining the effective elastic properties of composites with random structure is developed. The problem is reduced to the model of a single inclusion with a non-homogeneous elastic neighbourhood in a medium with effective elastic properties. The inhomogeneous elastic properties and size of neighbourhood are defined by randomness of the geometry, random size of inclusions and random elastic properties of the inclusions. Numerical results are given for the effective elastic properties of a composite with hollow spherical inclusions.     

Predicting the properties of disoriented fibrous composites

International Applied Mechanics -     

Micromechanical analysis of fibrous piezoelectric composites with imperfectly bonded adherence

In this work, two-phase parallel fiber-reinforced periodic piezoelectric composites are considered wherein the constituents exhibit transverse isotropy and the cells have different configurations. Mechanical imperfect contact at the interface of the piezoelectric composites is studied via linear spring model. The statement of the problem for two-phase piezoelectric composites with mechanical imperfect contact is given. The local problems are formulated by means of the asymptotic homogenization method, and their solutions are found using complex variable theory. Analytical formulae are obtained for the effective properties of the composites with spring imperfect type of contact and different rhombic cells. Using the concept of a representative volume element (RVE), a finite element model is created, which combines the angular distribution of fibers and imperfect contact conditions (spring type) between the phases. Periodic boundary conditions are applied to the RVE, so that effective material properties can be derived. The fibers are distributed in such a way that the microstructure is characterized by a rhombic cell. The presented numerical homogenization technique is validated by comparing results with theoretical approach reported in the literature. Some studies of particular cases, numerical examples, and comparisons between the two aforementioned methods with other theoretical results illustrate that the model is efficient for the analysis of composites with presence of rhombic cells and the aforementioned imperfect contact.     

Estimation of critical loading parameters for composites with imperfect layer contact

Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 28, No. 5, pp. 22–27, May, 1992.     

Predicting the nonlinear strain properties of fibrous metal composites

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 9, pp. 45–51, September, 1989.     

New exact results for the effective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages

In this paper we present a unified treatment of composite ellipsoid assemblages in the setting of uncoupled phenomena like conductivity and elasticity and coupled phenomena like thermoelectricity and piezomagnetoelectricity. The building block of this microgeometry is a confocal ellipsoidal particle consisting of a (possibly void) core and a coating. All space is filled up with such units which have different sizes but possess the same aspect ratios. The confocal ellipsoids may have the same orientation in space or may be randomly oriented. The resulting microgeometry simulates two-phase composites in which the reinforcing components are short fibers or elongated particles. Our main interest is in obtaining information of an exact nature on the effective moduli of this microgeometry whose effective tensor symmetry structure depends on the packing mode of the coated ellipsoids. This information will sometimes be complete like the full effective thermoelectric tensor of an assemblage which contains aligned ellipsoids in which the coating is isotropic and the core is arbitrarily anisotropic. In the majority of the cases however the maximum achievable exact information will be only partial and will appear in the form of certain exact relations between the effective moduli of the microgeometry. These exact relations are obtained from exact solutions for the fields in the microstructure for a certain set of loading conditions. In all the considered cases an isotropic coating can be combined with a fully arbitrary core. This covers the most important physical case of anisotropic fibers in an isotropic matrix. Allowing anisotropy in the coating requires the fulfillment of certain constraint conditions between its moduli. Even though in this case the presence of such constraint conditions may render the anisotropic coating material hypothetical, the value of the derived solutions remains since they still provide benchmark comparisons for approximate and numerical treatments. The remarkable feature of the general analysis which covers all treated uncoupled and coupled phenomena is that it is developed solely on the basis of potential solutions of the conduction problem in the same microgeometry.