Anti-plane time-harmonic Green's functions for a circular inhomogeneity in piezoelectric medium with a spring- or membrane-type interface

The present work focuses on the two-dimensional anti-plane time-harmonic dynamic Green's functions for a circular inhomogeneity immersed in an infinite matrix with an imperfect interface, where both the inhomogeneity and matrix are assumed to be piezoelectric and transversely isotropic. Two types of imperfect interface, the spring-type interface with electromechanical coupling and the membrane-type interface, are considered. The former type is often used to model the electromechanical damage along the interface while the latter is largely employed to simulate surface/interface effect of nano-sized inhomogeneity. By using the Bessel function expansions, explicit solutions for the electromechanical fields induced by a time-harmonic anti-plane line force and line charge located in an unbounded matrix as well as the circular inhomogeneity are respectively derived. The present solutions can recover the anti-plane Green's functions for some special cases, such as the dynamic or quasi-static Green's functions of piezoelectricity with perfect interface as well as the dynamic or quasi-static Green's functions of pure elasticity with imperfect interface. For detailed discussions, selected analytical results are presented. Dependence of the electromechanical fields on circular frequency as well as interface properties is examined. The size effect related to interfacial imperfection is also discussed.     

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.     

Transient response of piezoelectric composites caused by the sudden formation of localized defects

A continuum model is presented which is capable of generating the transient electroelastic field in piezoelectric composites of periodic microstructure, caused by the sudden appearance of localized defects. These defects are simulated by associating to every one of the ten piezoelectric parameters of the constituents a distinct damage variable. This procedure enables the modeling of localized cracks, soft and stiff inclusions and cavities. As a result, the constitutive equations of the piezoelectric phases appear in a specific form that includes eigen-electromechanical field variables which represent these defects. The method of solution is based on the combination of two distinct approach. In the first one, the representative cell method is employed according to which the periodic composite, which is discretized into several cells, is reduced to a problem of a single cell in the discrete Fourier transform domain. The resulting coupled elastodynamic and electric equations, initial, boundary and interfacial conditions in the transform domain are solved by employing a wave propagation in piezoelectric composite analysis which forms the second approach. The method of solution is verified by comparison with an analytical solution for the transient response of a piezoelectric material with a semi-infinite mode III-crack. Several applications are presented for the sudden formation of cracks in homogeneous and layered piezoelectric materials which are subjected to various types of electromechanical loading, and for the sudden appearance of a cavity. The effect of electromechanical coupling on the dynamic response is discussed.     

Dynamic focusing effect of piezoelectric fibers subjected to thermal shock

Based on finite Hankle transforms, this paper presents a theoretical method for analyzing the dynamic focusing effect of piezoelectric fibers subjected to the thermal shock of a transitory temperature change produced by a sudden electric current pulse. From analytical expressions and example calculations for two kinds of piezoelectric fibers, PZT-4 and BaTiO₃, it is found that the dynamic focusing effect is dependent on the material property of the piezoelectric fibers so that the maximum dynamic stress amplitude of the two kinds of piezoelectric fiber occur at different radial points. The mechanism of the dynamic focusing effect in piezoelectric fibers is relevant to the evaluation of the dynamic strength and electric signal of the piezoelectric fiber. The results carried out may be used as a reference to solve other transient coupled electrothermoelasticity problems in piezoelectric structures.     

Influence of imperfect elastic contact condition on the antiplane effective properties of piezoelectric fibrous composites

A fiber-reinforced periodic piezoelectric composite, where the constituents exhibit transverse isotropic properties, is considered. The fiber cross-section is circular and the periodicity is the same in two orthogonal directions. Imperfect mechanic contact conditions at the interphase between the matrix and fibers are represented in parametric form. In order to analyze the influence of the imperfect interface effect over the behavior of the composite, the effective axial piezoelectric moduli are obtained by means of the Asymptotic Homogenization Method. Some numerical examples are given.     

Effective property of piezoelectric composites containing coated nano-elliptical fibers with interfacial debonding

Based on both the spring layer interface model and the Gurtin-Murdoch surface/interface model, the anti-plane shear problem is studied for piezoelectric composites containing coated nano-elliptical fibers with imperfect interfaces. By using the complex function method and the technique of conformal mapping, the exact solutions of the electroelastic fields in fiber, coating, and matrix of piezoelectric nanocomposites are derived under far-field anti-plane mechanical and in-plane electrical loads. Furthermore, the generalized self-consistent method is used to accurately predict the effective electroelastic moduli of the piezoelectric nanocomposites containing coated nano-elliptical fibers with imperfect interfaces. Numerical examples are illustrated to show the effects of the material constants of the imperfect interface layers, the aspect ratio of the fiber section, and the fiber volume fraction on the effective electroelastic moduli of the piezoelectric nanocomposites. The results indicate that the effective electroelastic moduli of the piezoelectric nanocomposites can be significantly reduced by the interfacial debonding, but it can be improved by the surface/interface stresses at the small scale, which provides important theoretical reference for the design and optimization of piezoelectric nanodevices and nanostructures.     

2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the SBFEM

The dynamic stress and electric displacement intensity factors of impermeable cracks in homogeneous piezoelectric materials and interface cracks in piezoelectric bimaterials are evaluated by extending the scaled boundary finite element method (SBFEM). In this method, a piezoelectric plate is divided into polygons. Each polygon is treated as a scaled boundary finite element subdomain. Only the boundaries of the subdomains need to be discretized with line elements. The dynamic properties of a subdomain are represented by the high order stiffness and mass matrices obtained from a continued fraction solution, which is able to represent the high frequency response with only 3–4 terms per wavelength. The semi-analytical solutions model singular stress and electric displacement fields in the vicinity of crack tips accurately and efficiently. The dynamic stress and electric displacement intensity factors are evaluated directly from the scaled boundary finite element solutions. No asymptotic solution, local mesh refinement or other special treatments around a crack tip are required. Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique.     

Pull-in voltage analysis of electrostatically actuated MEMS with piezoelectric layers: A size-dependent model

A size-dependent model for electrostatically actuated microbeam-based MEMS (micro-electro-mechanical systems) with piezoelectric layers attached is developed based on a modified couple stress theory. By using Hamilton's principle, the nonlinear differential governing equation and boundary conditions of the MEM structure are derived. In the newly developed model, the residual stresses, fringing-field and axial stress effects are considered for the fixed–fixed microbeam with piezoelectric layers. The results of the present model are compared with those from the classical model. The results show the size effect becomes prominent if the beam dimension is comparable to the material length scale parameter (MLSP). The effects of MLSP, the residual stresses and axial stress on the pull-in voltage are also studied. The study may be helpful to characterize the mechanical and electrostatic properties of small size MEMS, or guide the design of microbeam-based devices for a wide range of potential applications.     

The effective properties of composites with fiber-end cracks

The paper describes use of self-consistent finite element method (SCFEM) for predicting effective properties of fiber composite with partially debonded interface. The effective longitudinal Young's modulus and shear modulus for unidirectional fiber reinforced composites with fiber-end cracks are calculated. Numerical results show that the effective properties are considerably influenced by the fiber-end cracks. The effects of microstructural parameters, such as fiber volume fraction, modulus ratio of the constituents and fiber aspect, on the effective properties of the composites were discussed. The project supported by the National Natural Science Foundation of China     

Dynamic anti-plane shear of a cracked piezoelectric ceramic

The dynamic theory of antiplane piezoelectricity is applied to solve the problem of a line crack subjected to horizontally polarized shear waves in an arbitrary direction. The problem is formulated by means of integral transforms and reduced to the solution of a Fredholm integral equation of the second kind. The path-independent integral G is extended here to include piezoelectric effects, and is evaluated at the crack tip to obtain the dynamic energy release rate. Numerical calculations are carried out for the dynamic stress intensity factor and energy release rate. The material is piezoelectric ceramic.     

Influence of interfaces on effective properties of nanomaterials with stochastically distributed spherical inclusions

The method of conditional moments is generalized to include evaluation of the effective elastic properties of particulate nanomaterials and to investigate the size effect in those materials. Determining the effective constants necessitates finding a stochastically averaged solution to the fundamental equations of linear elasticity coupled with surface/interface conditions (Gurtin–Murdoch model). To obtain such a solution the system of governing stochastic differential equations is first transformed to an equivalent system of stochastic integral equations. Using statistical averaging, the boundary-value problem is then converted to an infinite system of linear algebraic equations. A two-point approximation is considered and the stress fluctuations within the inclusions are neglected in order to obtain a finite system of algebraic equations in terms of component-average strains. Closed-form expressions are derived for the effective moduli of a composite consisting of a matrix and randomly distributed spherical inhomogeneities. As a numerical example a nanoporous material is investigated assuming a model in which the interface effects influence only the bulk modulus of the material. In that model the resulting shear modulus is the same as for the material without surface effects. Dependence of the bulk moduli on the radius of nanopores and on the pore volume fraction is analyzed. The results are compared to, and discussed in the context of other theoretical predictions.     

The effective elastic moduli of columnar composites made of cylindrically anisotropic phases with rough interfaces

The present work aims to determine the effective elastic moduli of a composite having a columnar microstructure and made of two cylindrically anisotropic phases perfectly bonded at their interface oscillating quickly and periodically along the circular circumferential direction. To achieve this objective, a two-scale homogenization method is elaborated. First, the micro-to-meso upscaling is carried out by applying an asymptotic analysis, and the zone in which the interface oscillates is correspondingly homogenized as an equivalent interphase whose elastic properties are analytically and exactly determined. Second, the meso-to-macro upscaling is accomplished by using the composite cylinder assemblage model, and closed-form solutions are derived for the effective elastic moduli of the composite. Two important cases in which rough interfaces exhibit comb and saw-tooth profiles are studied in detail. The analytical results given by the two-scale homogenization procedure are shown to agree well with the numerical ones provided by the finite element method and to verify the universal relations existing between the effective elastic moduli of a two-phase columnar composite.     

Micromechanical modeling of packing and size effects in particulate composites

This paper is devoted to the introduction of packing and size effects in micromechanical predictions of the overall elastic moduli of particulate composite materials. Whereas micromechanical models derived from the classical ‘point approach’ are known to be unable to model such effects, it is shown that the so-called ‘morphologically representative pattern-based approach’ (MRP-based approach) offers new means of taking some geometrical parameters into account such as the mean distance between nearest-neighbor particles or their size, so as to predict the dependence of the overall moduli on these parameters, at least in a relative way. Moreover, when internal lengths, such as the thickness of interphase shells of coated particles, are introduced, absolute size effects can be predicted as well. Illustrative applications are reported in view of comparisons between such new treatments and the predictions of some classical models which are shown to coincide with the ones derived from MRP-based models in definite limiting cases only.     

Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces

A micromechanical framework is proposed to predict the effective elastic behavior and weakened interface evolution of particulate composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu, J., 1993a. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. Journal of Applied Mechanics 60 (4), 1048–1050; Qu, J., 1993b. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mechanics of Materials 14, 269–281] is adopted to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. Based on the Eshelby’s micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to characterize the varying probability of evolution of weakened interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to assess the potential of the present framework.     

The effective properties of piezoelectric composite materials with transversely isotropic spherical inclusions

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Propagation of longitudinal elastic waves in composites with a random set of spherical inclusions (effective field approach)

The work is devoted to the problem of plane monochromatic longitudinal wave propagation through a homogeneous elastic medium with a random set of spherical inclusions. The effective field method and quasicrystalline approximation are used for the calculation of the phase velocity and attenuation factor of the mean (coherent) wave field in the composite. The hypotheses of the method reduce the diffraction problem for many inclusions to a diffraction problem for one inclusion and, finally, allow for the derivation of the dispersion equation for the wave vector of the mean wave field in the composite. This dispersion equation serves for all frequencies of the incident field, properties and volume concentrations of inclusions. The long and short wave asymptotics of the solution of the dispersion equation are found in closed analytical forms. Numerical solutions of this equation are constructed in a wide region of frequencies of the incident field that covers long, middle, and short wave regions of propagating waves. The phase velocities and attenuation factors of the mean wave field are calculated for various elastic properties, density, and volume concentrations of the inclusions. Comparisons of the predictions of the method with some experimental data are presented; possible errors of the method are indicated and discussed.