Transverse Electric Displacement Research Articles

Homogenization of size-dependent multiphysics behavior of nanostructured piezoelectric composites with energetic surfaces

Surface piezoelectricity considering the extended Gurtin--Murdoch coherent interface model has been incorporated into the composite cylinder assemblage (CCA), generalized self-consistent method (GSCM), as well as the multiphysics finite-element micromechanics (MFEM), for simulating the size-dependent multiphysics response of nanoporous materials wherein interface stress and electric displacement prevail. In the case of the CCA/GSCM model, the coherent interface model is implemented through the generalized Young--Laplace equations that govern the variation of the surface stress and the surface electric displacement. Three loading modes are utilized to identify the closed-form solutions for a complete set of Hill's moduli, and piezoelectric and dielectric constants. In the case of the MFEM, surface piezoelectricity is incorporated directly through additional surface energies associated with the elements that stretch along the interface. In order to assess the accuracy of the developed computational approaches, the generalized Kirsch problem under far-field transverse electric displacement loading is developed for recovering electric displacement concentration in the vicinity of the pore boundary. Homogenized properties are generated and critically examined for a broad variety of parameters and dimensions, predicted by the CCA/GSCM and MFEM methods. It is shown that all the predicted effective properties of these two families of homogenization techniques are similar except for the transverse shear moduli where they show marked differences that are reminiscent of what has been observed in the absence of surface electricity.

Improved analytical homogenization of the piezoelectric macro-fiber composite: active layer embedded among passive layers

Predicting the effective electromechanical properties for piezoelectric fiber composite patches is crucial for analysis of smart structures equipped with such patches. In particular, analytical homogenization models are attractive but are limited by the complexity of the microstructure and the multiphysics involved. The macro-fiber composite (MFC) has emerged as the most popular piezoelectric fiber composite. But typical reports of its effective properties provide layerwise homogenization with low-order mechanical plate theories. This approach discards most of the out-of-plane properties and may introduce significant error from the assumptions about the plate kinematics. Moreover, the effective permittivities of the transducer are not available analytically. In an attempt to overcome these limitations, a new analytical scheme for homogenizing all layers in the MFC patch is proposed. Based on the newly discovered mechanics of structure genome, we avoid all kinematical assumptions and obtain an exact analytical solution for a periodic, multi-layered, linear-piezoelectric material. In doing so, we prove that in any such multi-layered composite, the in-plane strains and the transverse stresses are equal in each layer and the in-plane electric fields and transverse electric displacement are constant between the electrodes. Using this knowledge, a hybrid rule of mixtures is developed to homogenize all the MFC layers to obtain the complete set of properties of the full MFC. In doing the new analytical framework can homogenize any layered piezoelectric medium. Finally, since we were able to avoid the plate-kinematics assumptions, we present studies on their effect on the properties they obtain. We find that the in-plane elastic properties are not affected but the transverse shear moduli are overpredicted from 100% to 300%. What is more, studies are presented on how using the ubiquitous plane-stress-like assumptions to homogenize the fiber layers causes an 8% underprediction of the lateral elastic modulus of the MFC.

2D semi-analytical solutions for single layer piezoelectric laminate subjected to electro-mechanical loading

Analysis of a piezoelectric laminate under plane strain condition of elasticity (cylindrical bending) has been performed with mixed semi-analytical model developed by Kant et al. (2007). The in-plane displacement, transverse displacement, transverse normal and shear stresses, electric potential and transverse electric displacement have been considered as fundamental dependent variables. The mathematical model consists of defining a two-point boundary value problem (BVP) governed by a set of coupled first order ordinary differential equations (ODEs). The accuracy and efficiency of the proposed model are assessed by comparing the numerical results from the present investigation with available elasticity solutions.

Layerwise mixed models for analysis of multilayered piezoelectric composite plates using least-squares formulation

This work provides an assessment of layerwise mixed models using least-squares formulation for the coupled electromechanical static analysis of multilayered plates. In agreement with three-dimensional (3D) exact solutions, due to compatibility and equilibrium conditions at the layers interfaces, certain mechanical and electrical variables must fulfill interlaminar C0 continuity, namely: displacements, in-plane strains, transverse stresses, electric potential, in-plane electric field components and transverse electric displacement (if no potential is imposed between layers). Hence, two layerwise mixed least-squares models are here investigated, with two different sets of chosen independent variables: Model A, developed earlier, fulfills a priori the interlaminar C0 continuity of all those aforementioned variables, taken as independent variables; Model B, here newly developed, rather reduces the number of independent variables, but also fulfills a priori the interlaminar C0 continuity of displacements, transverse stresses, electric potential and transverse electric displacement, taken as independent variables. The predictive capabilities of both models are assessed by comparison with 3D exact solutions, considering multilayered piezoelectric composite plates of different aspect ratios, under an applied transverse load or surface potential. It is shown that both models are able to predict an accurate quasi-3D description of the static electromechanical analysis of multilayered plates for all aspect ratios.

Assessment of a layerwise mixed least-squares model for analysis of multilayered piezoelectric composite plates

A layerwise mixed finite element model is developed based on the least-squares formulation for the coupled electromechanical static analysis of multilayered plates with piezoelectric and composite layers. The model assumes a layerwise variable description for displacements, transverse stresses and in-plane strains, along with the electrostatic potential, transverse electric displacement and in-plane electric field components, taken as independent variables. This original choice for the layerwise mixed formulation is intended to ensure the a priori and complete fulfilment of the interlaminar C0 continuity of both mechanical and electrical variables, which is due to compatibility and equilibrium conditions. Numerical applications are shown for assessment of the model predictive capabilities by comparison with available exact three-dimensional solutions, considering multilayered piezoelectric composite plates of various side-to-thickness ratios, under an applied load or surface potential.

Hamiltonian partial mixed finite element-state space symplectic semi-analytical approach for the piezoelectric smart composites and FGM analysis

A partial mixed finite element (FE)–state space method (SSM) semi-analytical approach is presented for the static analysis of piezoelectric smart laminate composite and functionally graded material (FGM) plates. Hence, using the Hamiltonian formalism, the three-dimensional piezoelectricity equations are first worked so that a partial mixed variational formulation, which retains the translational displacements, electric potential, transverse stresses, and transverse electric displacement as primary variables, is obtained; this allows, in particular, straightforward fulfillment of the electromechanical continuity constraints at the laminate interfaces. After an in-plane FE discretization only, the problem is first reduced, for a single layer, to a Hamiltonian eigenvalue problem that is solved using the symplectic approach; then, the multilayer solution is reached via the SSM propagator matrix. The proposed methodology is finally applied to the static analysis of piezoelectric-cross-ply hybrid laminated composite and FGM plates. In a comparison with open literature, available tabulated results show good agreements, thus validating the proposed approach.

Variable kinematics and advanced variational statements for free vibrations analysis of piezoelectric plates and shells

This paper investigates the problem of free vibrations of multilayered plates and shells embedding anisotropic and thickness polarized piezoelectric layers. Carrera’s Unified Formulation (CUF) has been employed to implement a large variety of electro-mechanical plate/shell theories. So-called Equivalent Single Layer and Layer Wise variable descriptions are employed for mechanical and electrical variables; linear to fourth order expansions are used in the thickness direction z in terms of power of z or Legendre polynomials. Various forms are considered for the Principle of Virtual Displacements (PVD) and Reissner’s Mixed Variational Theorem (RMVT) to derive consistent differential electro-mechanical governing equations. The effect of electro-mechanical stiffness has been evaluated in both PVD and RMVT frameworks, while the effect of continuity of transverse variables (transverse shear and normal stresses and transverse normal electric displacement) has been addressed by comparing various forms of RMVT. According to CUF, governing equations related to a given variational statement have been written in terms of fundamental nuclei whose form is independent of the order of expansion and of the adopted variable description. The numerical results have been restricted to simply supported orthotropic plates and shells, for which exact three-dimensional solutions are available. A large numerical investigation has been conducted to compute fundamental and higher vibrations modes. An exhaustive numerical evaluation of assumptions, related to the various PVD and RMVT forms, is given. Classical, higher-order, layer-wise and mixed assumptions have been compared to available three-dimensional solutions. The convenience of hierarchical approaches based on CUF is shown, along with the suitability of the implemented RMVT forms to accurately trace the free vibration response of piezoelectric plates and shells. RMVT applications permit the vibration modes of transverse electro-mechanical variables 1 Corresponding author: Salvatore Brischetto, Department of Aeronautics and Space Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy. Tel: +39.011.564.6869, Fax: +39.011.564.6899, Email: salvatore.brischetto@polito.it. 2 Department of Aeronautics and Space Engineering, Politecnico di Torino, Italy 260 Copyright © 2010 Tech Science Press CMES, vol.65, no.3, pp.259-341, 2010 to be accurately evaluated in the thickness plate/shell direction .

Mixed Elements for the Analysis of Anisotropic Multilayered Piezoelectric Plates

Mixed plate elements for accurate evaluation of transverse mechanical stresses and electrical displacement are developed and compared in this work. Various forms of the Reissner mixed variational theorem are considered to a priori fulfill the interlaminar continuity of transverse electromechanical variables. Classical formulation based on the principle of virtual displacements are implemented for comparison purposes. 2D assumptions are introduced by means of the Carrera unified formulation (CUF). Layer-wise theories are developed by using Legendre type polynomials of various order in the plate-thickness expansion. The evaluation of the interlaminar continuity of transverse electric displacement and/or transverse stresses on the static response of piezoelectric actuated/sensing plate is provided and compared. Orthotropic and anisotropic simply supported plates are considered. Results are assessed by comparison with available 3D piezoelasticity solutions. The mixed formulations that a priori evaluate the transverse electric displacement appear of particular interest. These last, in fact, permit an easy, convenient and reliable calculation of the electric charge and make feasible the development of robust control algorithms.

Exact three-dimensional piezothermoelasticity solution for dynamics of rectangular cross-ply hybrid plates featuring interlaminar bonding imperfections

An exact three-dimensional (3D) piezothermoelasticity solution is presented for static, free vibration and steady state harmonic response of simply supported cross-ply piezoelectric (hybrid) laminated rectangular plates with interlaminar bonding imperfections. The bonding imperfection is modeled by considering the jump in the displacements, electric potential and temperature across the non-rigid interface proportional, respectively, to the associated tractions, transverse electric displacement and heat flux. The solution includes the case when electric potentials are prescribed at the interfaces for effective actuation. Numerical results are presented for hybrid composite and sandwich plates with varying imperfection compliance. The effect of location of imperfect bonding on the response is investigated for mechanical, electric potential and thermal load cases. The effect of weak bonding at elastic–piezoelectric interface on the actuation authority of the piezoelectric layer is also investigated. These results would serve as benchmark for assessing 2D plate theories incorporating interlaminar bonding imperfections.

Two‐dimensional benchmark solution for buckling and vibration of simply supported hybrid piezoelectric angle‐ply flat panels

AbstractA benchmark 2D piezothermoelasticity exact solution is presented, based on state space approach for buckling and free vibration response of simply supported laminated hybrid angle‐ply panels in cylindrical bending under initial in‐plane thermo‐electromechanical loads. A mixed formulation of governing equations for the incremental problem of buckling/vibration is presented in terms of eight primary variables: three displacements, electric potential, three transverse stresses, and transverse electric displacement. The entities are expanded in Fourier series that satisfy the end conditions. The governing equations reduce to a z‐invariant state equation with the state vector of eight variables. The state vector at the bottom of the laminate is related to that at its top through a global transfer matrix using continuity/jump conditions at the layer interfaces. The eight homogenous boundary conditions at the top and the bottom of the angle‐ply panel are used to set up four homogeneous equations for u, v, w, and ϕ/Dz at the bottom. The determinant of their coefficient matrix is set to zero to obtain the buckling parameter or natural frequency. Benchmark results are presented for hybrid highly inhomogeneous test panel, composite panel, and sandwich panel. The effect of exclusion of the pre‐buckling strains in the formulation on the buckling temperature is investigated. The effects of thickness parameter, and the electric boundary conditions on the buckling loads are illustrated. The enhancement of buckling temperature by the application of actuation potential is discussed. The present benchmark solution would help assess 1D theories for buckling of hybrid angle‐ply panel.

Modified boundary layer analysis of an electrode in an electrostrictive material

A thin electrode embedded in an electrostrictive material under electric loading is investigated. In order to obtain an asymptotic form of electric fields and elastic fields near the electrode edge, we consider a modified boundary layer problem of an electrode in an electrostrictive material under the small scale saturation condition. The exact electric solution for the electrode is obtained by using the complex function theory. It is found that the shape of the electric displacement saturation zone is sensitive to the transverse electric displacement. A perturbation solution of stress fields induced by incompatible electrostrictive strains for the small value of the transverse electric displacement is obtained. The influence of transverse electric displacement on a microcrack initiation from the electrode edge is also discussed.

Mixed piezoelectric plate elements with continuous transverse electric displacements

This paper proposes mixed finite elements, FEs, with an a priori continuous transverse electric displacement component Dz. The Reissner Mixed Variational Theorem (RMVT) and the Unified Formulation (UF) are applied to the analysis of multilayered anisotropic plates with embedded piezoelectric layers. Two forms of RMVT are compared. In a first, partial, form (P-RMVT), the field variables are displacements u, electric potential and transverse stresses n. The second, full, form (F-RMVT) adds Dz as an independent variable. F-RMVT allows the a priori and complete fulfillment of interlaminar continuity of both mechanical and electrical variables. We treat both equivalent single-layer models (ESLM), where the number of variables is kept independent of the number of layers, an layerwise models (LWM), in which the number of variables depends in each layer. According to the UF the order N of the expansions assumed for the u, , n and Dz fields in the plate thickness direction z as well as the number of the element nodes Nn have been taken as free parameters. In most cases the results of the classical formulation which are based on Principle of Virtual Displacements (PVD) are given for comparison purposes. The superiority of the F-RMVT results, with respect to the P-RMVT and to PVD ones, is shown by few examples for which three-dimensional solution is available. In particular, the F-RMVT results to be very effective for the evaluation of interlaminar continuous Dz fields.

Exact solutions of functionally graded piezoelectric shells under cylindrical bending

Within a framework of the three-dimensional (3D) piezoelectricity, we present asymptotic formulations of functionally graded (FG) piezoelectric cylindrical shells under cylindrical bending type of electromechanical loads using the method of perturbation. Without loss of generality, the material properties are regarded to be heterogeneous through the thickness coordinate. Afterwards, they are further specified to be constants in single-layer homogeneous shells and to obey an identical exponent-law in FG shells. The transverse normal load and normal electric displacement (or electric potential) are, respectively, applied on the lateral surfaces of the shells. The cylindrical shells are considered to be fully simple supports at the edges in the circumferential direction and with a large value of length in the axial direction. The present asymptotic formulations are applied to several benchmark problems. The coupled electro-elastic effect on the structural behavior of FG piezoelectric shells is evaluated. The influence of the material property gradient index on the variables of electric and mechanical fields is studied.

Asymptotic analysis of an impermeable crack in an electrostrictive material subjected to electric loading

The simple asymptotic problem of an impermeable crack in an electrostrictive ceramic under electric loading is analyzed. Closed form solutions of elastic fields are obtained by using the complex function theory. It is found that the K I-dominant region is very small compared to the electric saturation zone. A fracture parameter for an electrostrictive material subjected to electric loading is discussed. In order to investigate the influence of the transverse electric displacement on fracture behavior under the small-scale conditions, we also consider the modified boundary layer problem of a crack in an electrostrictive material. Analytic solutions of electric displacement fields for the asymptotic problem are obtained based on the nonlinear dielectric theory from a modified boundary layer analysis. The shape of the electric displacement saturation zone is shown to depend on the transverse electric displacement. Stress intensity factors induced by the electrostrictive strains are evaluated using the nonlinear solution of the electric displacements. It is found that the transverse electric displacement affects strongly the variation of the mode mixity.

Exact 3D piezothermoelasticity solution for buckling of hybrid cross‐ply plates using state space approach

AbstractA three dimensional exact piezothermoelasticity solution is presented for buckling of simply‐supported symmetrically laminated hybrid plates with elastic substrate and piezoelectric layers. Buckling is considered under thermal load in open as well as closed circuit conditions of the piezolayers. A mixed formulation is used to form the governing equations for buckling mode in terms of eight primary variables: Three displacements, electric potential, three transverse stresses and the transverse electric displacement. Using double Fourier series expansion of the entities, which satisfy the end conditions, the governing equations is reduced to a z‐invariant state equation with the state vector consisting of the the above eight variables. The state vector at the bottom of the laminate is related to that at its top through a global transfer matrix using continuity/jump conditions at the layer interfaces. The eight homogenous boundary conditions at the top and the bottom of the plate are used to set up four homogeneous equations for the three displacements and electric potential/electric displacement at the bottom. The determinant of their coefficient matrix is set to zero to obtain the buckling temperature/potential. Benchmark results are presented for hybrid highly inhomogeneous test plate, cross‐ply composite plate and sandwich plates. The effect of the exclusion of the prebuckling strains in the formulation on the buckling temperature is investigated. The effects of thickness parameter, aspect ratio and the electric boundary conditions on the buckling loads are illustrated. The enhancement of buckling temperature by the application of actuation potential is discussed.

A Piezoelectric Mixed Variational Theorem for Smart Multilayered Composites

This work derives, for piezoelectric media, the analog of Reissner's mixed variational theorem (RMVT), initially developed for elastic media, by adding the transverse electric field–potential relation as a constraint via a Lagrange multiplier. The latter is shown to be the transverse (normal) electric displacement, for which continuity can now be fulfilled in a natural way as is the case for transverse stresses in the RMVT. Hence, the newly proposed piezoelectric mixed variational theorem (PMVT) will be well suited for layered piezocomposites. Mixed piezoelectric constitutive equations to be used in conjunction with the PMVT are developed and guidelines for their numerical implementation are also given.

Electric assumptions for piezoelectric laminate analysis

In computational analysis of piezoelectric laminates which may comprise piezoelectric and non-piezoelectric plies, the transverse electric field is often assumed to be piecewise constant along the transverse direction. This conventional assumption may lead to significant errors unless the piezoelectric ply is considerably thinner than the overall thickness of the laminate. To this end, two alternate assumptions termed EL and DC on the spatial distributions of the electric variables are proposed. In EL, the transverse electric field is taken to be piecewise linear along the transverse direction. In the DC assumption, the transverse electric displacement is taken to be piecewise constant along the transverse direction. The rationales leading to the proposed assumptions are discussed. Numerical examples indicate that the proposed assumptions are markedly more accurate than the conventional one. However, EL becomes inaccurate when its piecewise function spans more than one ply. In this light, DC often consumes less electric potential d.o.f.s and is more efficient than EL.

Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle

A three-dimensional mixed variational principle is used toderive a Kth-order two-dimensional lineartheory for an anisotropichomogeneouspiezoelectric(PZT)plate.Themechanicaldisplacements, theelectricpotential,theinplane components of the stress tensor, and the in-plane components of the electric displacement are expressed as a e niteseries of orderK inthethickness coordinate bytakingLegendrepolynomialsas thebasisfunctions. However, the transverse shear stress, the transverse normal stress, and the transverse electric displacement are expressed as a e nite series of order (K +2) in the thickness coordinate. The formulation accounts for the double forces without moments that may change the thickness of the plate. Results obtained by using the plate theory are given for the bending of a cantilever thick plate loaded on the top and the bottom surfaces by uniformly distributed 1) normal tractions and 2) tangential tractions. Results are also computed for the bending of a cantilever thick PZT beam loaded by 1) a uniformly distributed charge density on the top and the bottom surfaces and 2) equal and opposite normal tractions distributed uniformly only on a part of the beam. The seventh-order plate theory captures well the boundary-layer effects near the clamped and the free edges and adjacent to the top and the bottom surfaces of a thick orthotropic cantilever beam with the span to the thickness ratio of two. Also, through-the-thickness variation of the transverse shear and the transverse normal stresses agree well with those computed from the analytical solution of the three-dimensional elasticity equations. The governing partial differential equations are second order, so that Lagrange basis functions can be used to solve the problem by the e nite element method.

New Thin Piezoelectric Plate Models

Early investigations on piezoelectric plates were based on a priori mechanical and experimental considerations. They assume plane stress and consider only transverse components of electric displacement and field. Beside, these were supposed constant in the plate thickness. Through an asymptotic analysis, this paper shows that mechanical hypotheses follow Kirchhoff-Love theory of thin plates. However, electric assumptions are found to be strongly dependent on the electric boundary conditions. That is, two regular problems should be distinguished: (1) the short circuited plate, for which only transverse electric displacement and field have to be considered-the electric potential is then found to be the sum of a known part, which depends on prescribed potentials, and an unknown part, which represents an induced potential and cannot be a priori neglected; the mechanical and electrical problems may be uncoupled; (2) the insulated plate, for which only in-plane electric displacement and field components are to be considered; the mechanical and electrical problems may be uncoupled for orthorhombic plates but not in general. Based on the above asymptotic analysis, two variational and local two-dimensional static models are presented for heterogeneous anisotropic plates. They are then applied to homogeneous and orthorhombic piezoelectric plates. For homogeneous orthorhombic piezoelectric plates, the electromechanical problem can be uncoupled. Hence, a mechanical problem is first solved for the mechanical displacement, then electric potentials are explicitly deduced from this displacement. Classical finite element codes having multilayer plate facilities can be used for solving the plate problems obtained.