Effective Electroelastic Moduli Research Articles

Nonlinear micromechanical modeling of fully coupled piezo-elastic composite under large deformation and high electric field

This paper presents mathematical modeling and predictions of effective nonlinear electro-mechanical behavior of piezoelectric materials under large deformation and high electric field. The heterogeneous inclusion problem is extended to investigate the fields dependent electro-elastic behaviors under high fields. The associated strain field dependent Green tensors are introduced. The field dependent interaction tensors, related to Eshelby’s tensors, are explicitly formulated and used to derive micromechanical models based on the Mori–Tanaka approach and the self-consistent model. Due to electric-strain field dependence nonlinear algebraic tensors equations are resulted. Iterative incremental schemes based on the Newton–Raphson algorithm are elaborated and explicit semi-analytical formulations of electric-strain field dependent effective electro-elastic moduli of piezoelectric multi-phase composites are obtained for various inclusion types. Strain and electric field’s effects on effective properties can be analyzed for various electro-elastic composites. Numerical results of field dependent effective electro-mechanical properties are given for several inclusion’s volume fraction and shapes as well as types of matrix and inclusions phases.

Electroelastic ellipsoidal inclusion with imperfect interface and its application to piezoelectric composite materials

In this work, a new micromechanical model is presented to determine the effective behavior of piezoelectric composite materials with ellipsoidal reinforcements and imperfect interfaces. The integral equation is obtained by the Green's function technique applied to the electroelastic heterogeneous problem. This integral equation is further solved in the case of an ellipsoidal inclusion embedded in an infinite matrix with imperfect interface. By introducing the concept of interior and exterior-point Eshelby tensors, the solution of the inclusion problem is given in the case of displacement and electric fields discontinuities across the interface between inclusion and matrix. The concentration equation is expressed in anisotropic electroelasticity for ellipsoidal inclusions with imperfect interface. This equation is exact in the case of linear spring interface model for spherical and cylindrical inclusions. We compare the concentration tensors to the results given by exact model for cylindrical inclusion and in-plane loading. The Mori-Tanaka homogenization scheme is used to determine the effective electroelastic moduli of piezoelectric composite materials. In the case of spherical and cylindrical inclusions, the effective electroelastic moduli are compared to other models and experimental results. The combined influence of shape of inclusions and interface parameters is analyzed on the effective electroelastic behavior.

An XFEM/level set strategy for simulating the piezoelectric spring-type interfaces with apparent physical background

The piezoelectric spring-type interface is widely applied to describe the physical phenomenon that the displacement vector and the electric potential suffer jumps across an interface in a certain intelligent material, while both the normal traction vector and the normal electric displacement stay continuous across the same interface. This work is dedicated to accurately depicting the effects of the physics-based piezoelectric spring-type interfaces in arbitrary shapes and to predicting the effective electroelastic moduli of composites containing such interfaces. To achieve this two-fold goal, a computational approach combining the extended finite element method (XFEM) and the level set method (LSM) is developed and interpreted in detail. The accuracy and convergence performance of the elaborated approach are assessed with a benchmark problem for which the exact analytic solution is derived. Eventually, the approach validated is further utilized to explore the size- and shape-dependent effects induced by these spring-type interfaces on the overall couple-field moduli of fibrous piezoelectric composites.

Efficient interpolation algorithm of electro-elastic Green's function for boundary integral equation method and Eshelby inclusion problem

This study proposes an efficient interpolation algorithm for electro-elastic Green's functions and its derivatives. The extended concept of the interior and exterior point Eshelby tensor and its application are also presented. The computation efficiency of Green's functions in the boundary integral equation method and Eshelby inhomogeneity problem are greatly improved by the efficient interpolation of Green's functions, where the relevant tables of Green's functions are preliminarily computed using spherical coordinates. The validity of the interpolated electro-elastic Green's functions is tested by comparing the results to those of the effective electro-elastic moduli deduced from the Mori–Tanaka theory, the boundary integral equation method, and the extended concept of interior and exterior Eshelby tensor.

Effective properties of coated fibrous piezoelectric composites with spring-type interfaces under anti-plane mechanical and in-plane electrical loads

In this paper, we investigate the effective properties of three-phase (matrix/coating/fiber) cylindrical piezoelectric composites with imperfect interfaces under anti-plane mechanical and in-plane electrical loads. By using the electromechanically coupling spring-type interface model and the generalized self-consistent method (GSM), we analytically derived the effective electroelastic moduli. The present solutions include as special cases the three-phase cylindrical piezoelectric composites with perfect interfaces as well as the two-phase (matrix/fiber) case with imperfect or perfect interfaces. Selected calculations are graphically shown to demonstrate dependence of the effective moduli on the interfacial properties. The particular size-dependent characteristic due to the interfacial imperfection is also discussed.

Closed-form expressions for the effective moduli of heterogeneous piezoelectric materials

In this paper, analytical and semi-analytical expressions for the effective properties of transversely isotropic piezoelectric materials are derived based on the Mori–Tanaka model and Eshelby tensors using compact matrix formulations. Concise explicit relations are presented for the coupled and decoupled effective electroelastic moduli of piezoelectric materials with various inclusion types and shapes. In order to use these explicit relationships, some of the Eshelby tensor components are needed to be analytically or numerically computed. These closed-form relations are suitable for use in designing composite piezoelectric materials with optimized characteristics. Results of effective electroelastic moduli, obtained using the derived closed-form expressions, are presented for various types and shapes of piezoelectric inclusions and compared with self-consistent and incremental self-consistent numerical predictions.

Surface free energy effect on electro-mechanical behavior of piezoelectric thin film with square nanofibers under anti-plane shear

For a nano-sized piezoelectric film, the surface free energy at the surface/interface will exert great effect on the behavior of nano-sized piezoelectric structures. This paper concerns the effect of surface free energy at the surface/interface of square nanofibers on the electro-mechanical behavior of a piezoelectric thin film under anti-plane shear. Combining the electro-elastic surface/interface theory, the mapping method and the generalized self-consistent method, a closed-form solution of the effective electro-elastic moduli is obtained. Numerical examples are given to illustrate quantitatively the macro-effective electro-elastic moduli under different surfaces/interfaces when the size of the square nanofibers is on the order of nanometer. The effect of shape of square nanofibers on the effective electro-elastic moduli is also examined.

Variational bounds for the effective electroelastic moduli of piezoelectric composites with electromechanical coupling spring-type interfaces

The present work focuses on variational bounds for the effective electroelastic moduli of multiphase piezoelectric composites with thin piezoelectric interphase. Both the inhomogeneities and the matrix are assumed to be piezoelectric and transversely isotropic. The piezoelectric interphase is modeled as the spring-type interface with electromechanical coupling. The inhomogeneities are assumed to be spheroidal so that the reinforcement geometry is able to range from thin flake to continuous fiber. The effective properties of the piezoelectric composite with interfacial imperfection are defined and the principles of minimum internal energy and enthalpy are derived. These principles are applied to analytically obtain the upper and lower bounds for the effective electroelastic moduli. Unlike the Voigt–Reuss-type bounds for perfect interface, the present bounds depend not only on the material properties and volume fraction, but also on the interface parameters, inhomogeneity shape and orientation. An example of a two-phase composite is given for detailed discussion, where dependence of the electroelastic moduli and their bounds on the inhomogeneity shapes and orientations as well as the interface properties is provided and discussed. To qualitatively account for the dependence, analysis based on two possible mechanisms, i.e., the simple mixture rule of composite and the weakening effect by imperfect interface, are also provided.

Electro-mechanical coupling properties of piezoelectric nanocomposites with coated elliptical nano-fibers under anti-plane shear

Surfaces/interfaces in nanostructures play a significant role in deciding the electro-mechanical coupling properties of piezoelectric nanocomposites. A comprehensive electro-mechanical model of nano-size is developed to predict the effective coupling properties of piezoelectric nanocomposites with coated elliptical nano-fibers. The effects of surfaces/interfaces around the coating layer and the shape of the nano-fibers on the effective elastic modulus and piezoelectric coefficient are considered. Combining the generalized self-consistent method, the electro-elastic surface/interface model, and the mapping method, a closed-form solution of the effective electro-mechanical coupling properties is given. Through numerical calculations, it is found that the elastic modulus and dielectric constant increase with the semi-major axes, while piezoelectric constant expresses different behaviors. The interface effect on the effective electro-elastic moduli is also examined.

Evaluation of effective electroelastic properties of piezoelectric coated nano-inclusion composites with interface effect under antiplane shear

A theoretical study on piezoelectric coated nano-inclusion composites with interface effect under antiplane shear is reported. Based on the theory of Gurtin–Murdoch surface/interface theory and the generalized self-consistent method, a closed-form solution of the effective electroelastic moduli are obtained. The numerical results reveal the size dependence of the effective electroelastic moduli when the size of the coating and inclusion are on the order of nanometer. The effects of the coating thickness and inclusion radius on the effective electroelastic moduli of the composites are discussed.

Interaction between periodic cracks and periodic rigid-line inclusions in piezoelectric materials

Piezoelectric materials with a doubly periodic array of cracks and rigid-line inclusions under far-field antiplane mechanical load and inplane electric load are investigated. By employing the conformal mapping technique and the elliptical function theory, an exact solution of the whole-field stress and electrical displacement is obtained. The closed-form formulae for the stress and electrical displacement intensity factors at the tips of cracks and rigid-line inclusions and the effective electroelastic moduli of the composites are presented. Some existing solutions can be regarded as degenerated cases of the present results. Numerical examples are provided to show the interesting electroelastic interaction between multiple cracks and multiple rigid-line inclusions.

Size-dependent effective electroelastic moduli of piezoelectric nanocomposites with interface effect

The problem of nanocomposite materials under far-field antiplane mechanical load and inplane electric load is investigated. Based on the theory of Gurtin–Murdoch surface/interface model, an exact solution is obtained for the inhomogeneity/matrix/equivalent medium model, in terms of which a generalized self-consistent approach is proposed for predicting the effective electroelastic moduli of nanocomposites. A closed-form solution of the effective electroelastic moduli is presented. The numerical results reveal that the effective electroelastic moduli are size dependent when the size of the inhomogeneity is on the order of nanometer. With the increase in the size of the inhomogeneity, the present solution approaches to the classical results obtained in the classical theory.

An eigenfunction expansion-variational method for the anti-plane electroelastic behavior of three-phase fiber composites

The anti-plane electroelastic behavior of three-phase piezoelectric composites (fiber/interphase/matrix) with doubly periodic microstructures is dealt with. A new variational functional for a unit cell is constructed by incorporating the periodic boundary conditions into the energy functional. Then, by combining with the eigenfunction expansions of the complex potentials satisfying the fiber–interphase–matrix interfacial conditions, an eigenfunction expansion-variational method based on a unit cell is developed. The numerical results of the effective electroelastic moduli show a rapid convergence of the present method. A unified first-order approximation formula is also provided, where an equivalent parameter matrix reflecting the overall influence of the electroelastic properties of the fiber and interphase on the effective properties, is found. The equivalent parameter matrix can greatly simplify the complicated relation of the effective electroelastic properties to the internal structure of a three-phase fiber composite. Though the equivalent parameter matrix is extracted in the first-order approximation formula, its validity is also verified in the high-order numerical results.

The exact connections of the effective thermal properties of ferroelectric crystals with domain configurations

Exact connections between the effective thermal expansion and pyroelectric coefficients and the effective elastic and piezoelectric coefficients have been established for trigonal, tetragonal, and hexagonal ferroelectric crystals with compatible domain configurations, taking advantage of compatible but not necessarily uniform internal field resulted from a hydrostatic stress and uniform temperature change. These exact connections reduce the number of independent material constants, and thus the number of necessary experimental measurements, and allow the determination of the effective thermal moduli of a ferroelectric crystal from its effective electroelastic moduli. Numerical results are also presented to demonstrate the application of such exact connections.

Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects

We consider the macroscopic behavior of two-phase fibrous piezoelectric composites. The fibers are of circular cross-section with the same radius. Along the interfaces between the fibers and the matrix we consider the effects of surface stress and surface electric displacement. The constituents are transversely isotropic and exhibit pyroelectricity. We find that the overall thermoelectroelastic moduli of these solids must comply with two sets of exact connections. The first set, similar to Hill’s universal connections, provides five constraints between the six axisymmetric overall electroelastic moduli. The second set relates the effective coefficients of thermal stress and pyroelectric coefficients to the effective electroelastic moduli, in analogy with Levin’s formula. In contrast to their conventional counterparts, i.e., without surface effects, the presence of surface effects makes both sets of connections dependent on the absolute size of the nanoinclusions.

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.

An analytical study on prediction of effective properties in porous and non-porous piezoelectric composites

Eshelby type micromechanics model with a newly developed piezoelectric Eshelby tensor is proposed for predicting the effective electroelastic properties of the piezoelectric composite. The model is applied for piezoelectric solids containing both porosities and metal inhomogeneities. The effective electroelastic moduli of the composites such as stiffness, piezoelectric constants, and dielectric constants are predicted by the present model, which are extensively compared with the existing experimental results from the literatures. The validity of Eshelby type model for predicting the effective properties of the composite is thoroughly examined. It can be concluded from this study that a new mechanism is needed to compute correctly the dielectric constants among the effective properties of the composites.

The effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics

We present a micromechanical analysis on the effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics in terms of their microstructural information. The overall behaviors of ferroelectric ceramics are profoundly influenced by the microstructural phenomena, where the macroscopic pyroelectric effect can be induced in an originally isotropic, thus non-pyroelectric ceramic composed of randomly oriented ferroelectric grains through poling, during which the polar axes of grains are switched by the applied electric field or mechanical stress. To analyze these complicated phenomena, we will first establish an exact connection between the effective thermal moduli and the effective electroelastic moduli of ferroelectric ceramics, and then combine the exact connection with the effective medium approximation to provide an estimate on the effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics in terms of the orientation distribution of grains and poling conditions, where the texture evolution as a result of domain switching during poling has been taken into account. Numerical results are presented and good agreements with known theoretical results and some experimental data are observed.

Constitutive model development and micro-structural topology optimisation for Nafion hydrogel membranes with ionic clustering

The deployment of electroactive ionic polymer hydrogel-metal composites in artificial muscle and BioMEMS applications has recently been intensively investigated. In order to analyse their electromechanical responses to externally applied electrical fields, it is critical to develop a constitutive model linking the macro-mechanical moduli with the micro-mechanical characteristics, and to determine the geometric size and shape of the micro-structural cluster and investigate the effect of cluster morphology on the effective electro-elastic moduli of the polymer hydrogels. As a typical ionic polymer-based hydrogel, the Nafion membrane is studied in this work. Based on the Biot poroelasticity theory, a multi-scale constitutive model which includes both macro and micro characteristics is developed using an asymptotic homogenisation method. The effect of water-volume fraction on the effective elastic moduli of the hydrogel membrane is examined for different equivalent weights. Numerical investigations show that the simulated effective constitutive moduli agree well with experimental data. The presently developed constitutive model is thus validated. In order to determine the micro-structural shape of the polymer skeleton subject to fluid pressure, a representative volume element (RVE) is designed by topology optimisation of the periodic microstructures of the Nafion hydrogels, through the minimisation of the electro-elastic interaction energy between the polymer-based fluorocarbon matrix and the surrounding fluid. This optimal RVE correctly predicts the geometric shapes of the clusters.

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.