A two-dimensional theory for piezoelectric layers used in electro-mechanical transducers—I : Derivation

An Nth order two-dimensional theory is derived for strongly coupled piezoelectric layers that is suitable for use in analyzing electro-mechanical transducer response in circumstances where the field variations over the transducer face are important. Specific formulae are given for the widely used ceramic PZT-5.The first few branches of the dispersion curves for PZT-5 are obtained numerically for both the three-dimensional theory and the approximate two-dimensional theory. The correspondence of these curves at long wavelengths is used to determine a single correction factor. The face boundary conditions accommodate unknown traction and voltage so that the theory can be used in transducer applications.     

A new two-dimensional model for electro-mechanical response of thick laminated piezoelectric actuator

This paper investigates the electro-mechanical behaviour of a thick, laminated actuator with piezoelectric and isotropic lamina under externally applied electric loading using a new two-dimensional computational model. The elastic core is relatively thick and thus it is modelled by Timoshenko thick-beam theory. Although the piezoelectric lamina is a beam-like layer, it is formulated via a two-dimensional model because of not only the strong electro-mechanical coupling, but also of the presence of a two-dimensional electric field. It is shown in this paper that a one-dimensional model for the piezoelectric beam-like layer is inadequate. The piezoelectric model is constructed within the scope of linear piezoelectricity. The actuation response is induced through the application of external electric voltage. Under the strong coupling of elasticity and electricity, the strain energy and work of electric potential are presented. The electro-mechanical response of the laminated Timoshenko beam is formulated and determined via a variational energy principle. Numerical examples presented illustrate convincing comparison with finite element solutions and existing published data. New numerical solutions are also presented to investigate the geometric effect on the electro-mechanical bending behaviour.     

A two-dimensional continuum theory for angle-ply laminates

A two-dimensional continuum theory of microstructure is developed for stress analysis of angle-ply laminates under in-plane loading. An example problem is used to evaluate the results of the theory against a reference solution obtained by the finite element method. The results are in satisfactory agreement; they also show that the in-plane stresses reach somewhat higher peak values than reported in previous literature.The theory is also presented in a simplified version, which is found to be adequate for predicting interlaminar stresses and in-plane stress resultants, but does not give acceptable results for the variation of in-plane stresses through the thickness of the laminations.     

A unified constitutive theory for polymeric liquids II. Applications to basic problems

A unified constitutive theory for polymeric liquids has been recently proposed. Its derivation is based on a combination of continuum mechanical approach, transient-network concept and thermodynamics of irreversible processes. In the resulting model, many modes may be present for each of which there are two time scales, associated with the loss rate and the nonaffine motion of transient network junctions, respectively. A single effective relaxation time, constructed from the two time scales, governs the behavior in the linear regime of deformation. Two new parameters for each mode, in comparison with other models, are introduced: (i) the ratio r of the two time scales, and (ii) the index a distinguishing the rates of loss and creation of junctions. Both are important only for the nonlinear regime of deformation. In this paper, the theory is applied to predict the following cases: (i) stress growth at constant shear strain rate, (ii) steady shear-rate-dependent viscosity and first normal-stress difference and (iii) transient elongational viscosity at constant elongational strain rate. Determination of the model parameters based on usual characterization experiments is described. Comparison of calculated and observed behavior of low-density polyethylene at 150 °C available in the literature are presented. In general, the agreement of the predictions with experiment appear gratifying even with the simplest version of the new model.     

A mixed electric boundary value problem for a two-dimensional piezoelectric crack

In this paper, a mixed electric boundary value problem for a two-dimensional piezoelectric crack problem is presented, in the sense that the crack face is partly conducting and partly impermeable. By the analytical continuation method, the unknown electric charge distributions on the upper and lower conducting crack faces are reduced to two decoupled singular integral equations and then these two equations are converted into algebraic equations to find the full field solution. Though the results suggest that the stress intensity factors at the crack tip are identical to those of conventional piezoelectric materials, but the electric field and electric displacement are related to the electric boundary conditions on the crack faces. The electric field and electric displacement are singular not only at crack tips but also at the junctures between the impermeable part and conducting parts. Numerical results for the variations of the electric field, electric displacement field and J-integral with respect to the normalized impermeable crack length are shown. Some discussions for the energy release rate and the J-integral are made.     

A numerical method for calculating unsteady two-dimensional laminar boundary layers

Summary A method for calculating unsteady two dimensional boundary layers in laminar incompressible flow has been developed and tested. No restrictive assumptions are made regarding the time-dependent terms in the boundary-layer equation. The differential equations are solved with an implicit difference scheme similar to that employed for steady two-dimensional boundary layers. At each step, here, known conditions at three stations are used to calculate the conditions at a new (fourth) station. The entire field is covered by a succession of these steps.

---

Übersicht Es wird ein Verfahren zur Berechnung von zeitabhängigen zweidimensionalen Grenzschichten in laminarer inkompressibler Strömung entwickelt und auf seine Genauigkeit geprüft. Dabei werden keine einschränkenden Annahmen über die zeitabhängigen Glieder in der Grenzschichtgleichung gemacht. Die Differentialgleichungen werden durch ein implizites Differenzen-Verfahren gelöst, das dem gewöhnlich für stetige zweidimensionale Grenzschichten verwendeten ähnlich ist. Hier werden für jeden Schritt schon bekannte Werte an drei Stellen benutzt, um die Werte an einer neuen (vierten) Stelle zu berechnen. Das ganze Feld wird auf diese Weise schrittweise überdeckt.

---

---

This work was done while the author was at The John Hopkins University, on leave from the Royal Aircraft Establishment, Farnborough. It was supported in part by a grant from the National Science Foundation (Geophysical Fluid Mechanics, GA-641X). British Crown Copyright, reproduced with the permission of the Controller, Her Britannic Majesty's Stationery Office.     

A two-dimensional closed-form solution for the free-vibrations analysis of piezoelectric sandwich plates

This work presents a two-dimensional (2D) closed-form solution for the free-vibrations analysis of simply-supported piezoelectric sandwich plates. It has the originality to consider all components of the electric field and displacement, thus satisfying exactly the electric equilibrium equation. Besides, the formulation considers full layerwise first-order shear deformation theory and through-thickness quadratic electric potential. Its independent mechanical and electric variables are decomposed using Fourier series expansions, then substituted in the derived mechanical and electric 2D equations of motion. The resulting eigenvalue system is then condensed so that only nine mechanical unknowns are retained. After its validation on single- and three-layer piezoelectric, and hybrid sandwich plates, the present approach was then used to analyze thickness modes of a square sandwich plate with piezoceramic faces and elastic cross-ply composite core. It was found that only the first three thickness modes are global, thus can be modeled by the mixed equivalent single-layer/layerwise approach, often retained in the literature; the remaining higher thickness modes being characteristic of sandwich behavior; i.e., dominated by the deformations of either the core or the faces. These results, together with presented through-thickness variations of the mechanical and electric variables clearly recommend full layerwise modeling. Several numerical results are provided for future reference for validation of 2D approximate analytical or numerical approaches; in particular, of 2D piezoelectric adaptive finite elements.     

A generalized JKR-model for two-dimensional adhesive contact of transversely isotropic piezoelectric half-space

The aim of the present paper is to investigate the adhesive behavior between a transversely isotropic piezoelectric half-space and a cylinder punch subjected to combined mechanical and electric loads under plane-strain condition. The effect of adhesion is described by using a generalized JKR-model which can account for the non-slip condition in the contact regions. Analytical function theory is employed to find the solution of the resulting singular integral equations. Our analysis shows that the adhesive contact behavior for different types of piezoelectric materials may be quite different. The results obtained in this paper may be helpful to understand the contact mechanics of piezoelectric materials at micro-scale.     

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.     

A new approximate solution technique for randomly excited non-linear oscillators—II

The method of weighted residuals is applied to the reduced Fokker-Planck equation associated with a non-linear oscillator, which is subjected to both additive and multiplicative Gaussian white noise excitations. A set of constraints are deduced for obtaining an approximate stationary probability density for the system response. One of the constraints coincides with the previously proposed criterion of dissipation energy balancing, and the others are useful for calculating the equivalent conservative force. It is shown that these constraints imply certain relationships among certain statistical moments; their imposition guarantees that such moments computed from the approximate probability density satisfy the corresponding exact equations derived from the original equation of motion. Moreover, the well-known procedure of stochastic linearization and its improved version of partial linearization are shown to be special cases of this scheme, and they are less accurate since the approximations are not chosen from the entire set of the solution pool of generalized stationary potential. Applications of the scheme are illustrated by examples, and its accuracy is substantiated by Monte Carlo simulation results.     

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

A recently developed coupled third-order zigzag theory for the statics of piezoelectric hybrid cross-ply plates is extended to dynamics. The theory combines a third-order zigzag approximation for the in-plane displacements and a sub-layerwise linear approximation for the electric potential, considering all components of the electric field. The nonuniform variation of the transverse displacement due to the piezoelectric field is accounted for. The conditions for the absence of shear traction at the top and bottom surfaces and continuity of transverse shear stresses in the presence of electromechanical loading are satisfied exactly, thereby reducing the number of displacement variables to five, which is the same as in a first- or third-order equivalent single-layer theory. The governing equations of motion are derived from the extended Hamilton's principle. The theory is assessed by comparing the Navier solutions for the free and forced harmonic vibration response of simply supported plates with the exact three-dimensional piezoelasticity solutions. Comparisons for hybrid test, composite and sandwich plates establish that the present theory is quite accurate for the dynamic response of moderately thick plates.     

A numerical study of vorticity layers in a two-dimensional stratified shear flow

A numerical study is conducted to find out the conditions of occurrence of a secondary Kelvin-Helmholtz instability in the thin layers (referred to as baroclinic layers) that form in a stably-stratified shear layer. For this purpose, three high resolution calculations of a moderately stratified shear layer have been carried out, at a fixed Reynolds number. The wavelength of the initial perturbation is progressively increased, starting from the fundamental wavelength predicted by linear stability theory up to twice this fundamental wavelength. The baroclinic layer of the flow is shown to lengthen and destabilize progressively from one calculation to the other, eventually bearing a secondary Kelvin-Helmholtz instability. The structure and dynamics of the baroclinic layers of the three calculations are examined in the frame of a theoretical model proposed by Corcos and Sherman ([1]). An excellent agreement with the predictions of this model have been found. We next show that the stability of the layer is controlled by the large-scale Kelvin-Helmholtz vortex, via the strain field that it induces in the stagnation point region of the layer. A consequence of this study is that secondary Kelvin-Helmholtz instabilities are fostered by the pairing of primary Kelvin-Helmholtz vortices in a strongly-stratified shear layer.

---

Sommario E stato condotto uno studio numerico per trovare le condizioni in cui insorge una instabilità secondaria di Kelvin-Helmholtz negli strati sottili che si formano in uno strato di scorrimento stabilmente stratificato. A questo scopo sono state effettuate tre simulazioni ad alta risoluzione a fissato numero di Reynolds e stratificazione bassa. La lunghezza d'onda della perturbazione iniziale è stata progressivamente aumentata dalla lunghezza fondamentale predetta dalla teoria lineare della stabilità fino a due volte questa stessa lunghezza. È stato osservato che da una simulazione all'altra lo strato baroclino del flusso si allunga e si destabilizza progressivamente, generando eventualmente un'instabilità di Kelvin-Helmholtz secondaria. Utilizzando il modello teorico proposto da Corcos e Sherman (1976), per le tre simulazioni sono state analizzate la struttura e la dinamica dello strato baroclino. È stato trovato un accordo eccellente con le predizioni di questo modello. È stato in seguito mostrato che la stabilità dello strato è controllato dai vortici di Kelvin-Helmholtz di larga scala attraverso il campo di deformazione che inducono nella regione del punto di ristagno dello strato. Una conseguenza di questo studio è che le instabilità secondarie di Kelvin-Helmholtz sono forzate dall'accoppiamento dei vortici primari in uno strato di scorrimento fortemente stratificato.

     

A MODE Ⅱ CRACK IN A TWO-DIMENSIONAL OCTAGONAL QUASICRYSTALS

The present study develops the fracture theory for a two-dimensional octagonal quasicrystals. The exact analytic solution of a Mode Ⅱ Griffith crack in the material was obtained by using the Fourier transform and dual integral equations theory, then the displacement and stress fields, stress intensity factor and strain energy release rate were determined, the physical sense of the results relative to phason and the difference between mechanical behaviors of the crack problem in crystal and quasicrystal were figured out. These provide important information for studying the deformation and fracture of the new solid phase.     

Non-local theory solution for a Mode I crack in piezoelectric materials

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials subjected to a uniform tension loading. The permittivity of the air in the crack is considered. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the materials constants, the electric boundary conditions and the lattice parameter on the stress and the electric displacement fields near the crack tips. It can be obtained that the effects of the electric boundary conditions on the electric displacement fields are large. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allowing us to use the maximum stress as a fracture criterion.     

Hypoplasticity theory for granular materials—II: Three-dimensional axially symmetric exact solutions

In part I of this paper, we consider the governing equations of hypoplasticity theory for two-dimensional steady quasi-static plane strain compressible gravity flow and determine some exact analytical solutions applying for certain special cases. Similarly, for the three-dimensional situation considered here in part II, we undertake a similar mathematical investigation to determine some simple solutions of the governing equations for three-dimensional steady quasi-static axially symmetric compressible gravity flow for hypoplastic granular materials. We again find that for certain special cases, we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solutions is not a trivial matter, and that the solutions determined here might be exploited as benchmarks for full numerical schemes.