Influence of residual surface stress on the fracture of nanoscale piezoelectric materials with conducting cracks

In this paper,we analyze the stress and electric field intensity factors affected by residual surface stress for conducting cracks in piezoelectric nanomaterials.The problem is reduced to a system of non-linear singular integral equations,whose solution is determined by iteration technique.Numerical results indicate that the residual surface stress can significantly alter the crack tip fields at nanometer length scales.Due to the residual surface stress,281he electric field can produce stress around crack tip.This suggests a strong electromechanical coupling crack tip field for nanoscale piezoelectric materials.Such a finding is considerably different from the classical fracture mechanics results.A transit electric field to stress load ratio is identified,for which influences of residual surface stresses vanish.The research is useful for the applications of nanoscale piezoelectric devices.     

A Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core

The electro-elastic interaction between a piezoelectric screw dislocation and an elliptical piezoelectric inhomogeneity, which contains an electrically conductive confocal elliptical rigid core under remote anti-plane shear stresses and in-plane electrical load is dealt with. The analytical solutions to the elastic field and the electric field, the interfacial stress fields of inhomogeneity and matrix under longitudinal shear and the image force acting on the dislocation are derived by means of complex method. The effect of material properties and geometric configurations of the rigid core on interfacial stresses generated by a remote uniform load, rigid core and material electroelastic properties on the image force is discussed.     

Electroelastic behaviour of an annular interfacial crack between dissimilar piezoelectric layers

An annular interfacial crack between dissimilar piezoelectric layers subjected to electroelastic loadings was investigated under an electrically impermeable boundary condition on the crack surface by using the Hankel transform technique and the Cauchy singular integral equation method. The stress intensity factors and energy release rates were determined. Numerical results reveal the effects of crack configuration, electric loads and material parameters on crack propagation and growth. The results should be useful for the design of piezoelectric composite structures and devices of high performance.     

The effect of initial stress on the propagation behavior of SH waves in piezoelectric coupled plates

This study analytically investigates the propagation of shear waves (SH waves) in a coupled plate consisting of a piezoelectric layer and an elastic layer with initial stress. The piezoelectric material is polarized in z-axis direction and perfectly bonded to an elastic layer. The mechanical displacement and electrical potential function are derived for the piezoelectric coupled plates by solving the electromechanical field equations. The effects of the thickness ratio and the initial stress on the dispersion relations and the phase and group velocities are obtained for electrically open and mechanically free situations. The numerical examples are provided to illustrate graphically the variations of the phase and group velocities versus the wave number for the different layers comparatively. It is seen that the phase velocity of SH waves decreases with the increase of the magnitude of the initial compression stress, while it increases with the increase of the magnitude of the initial tensile stress. The initial stress has a great effect on the propagation of SH waves with the decrease of the thickness ratio. This research is theoretically useful for the design of surface acoustic wave (SAW) devices with high performance.     

Love wave propagation in functionally graded piezoelectric material layer

An exact approach is used to investigate Love waves in functionally graded piezoelectric material (FGPM) layer bonded to a semi-infinite homogeneous solid. The piezoelectric material is polarized in z-axis direction and the material properties change gradually with the thickness of the layer. We here assume that all material properties of the piezoelectric layer have the same exponential function distribution along the x-axis direction. The analytical solutions of dispersion relations are obtained for electrically open or short circuit conditions. The effects of the gradient variation of material constants on the phase velocity, the group velocity, and the coupled electromechanical factor are discussed in detail. The displacement, electric potential, and stress distributions along thickness of the graded layer are calculated and plotted. Numerical examples indicate that appropriate gradient distributing of the material properties make Love waves to propagate along the surface of the piezoelectric layer, or a bigger electromechanical coupling factor can be obtained, which is in favor of acquiring a better performance in surface acoustic wave (SAW) devices.     

A numerical study of residual stress induced in machined silicon surfaces by molecular dynamics simulation

Residual stresses in machined surface are regarded as a critical factor affecting the quality and service life of components. However, little research has been conducted to reveal the formation of residual stresses as well as the relation between machining conditions and residual stresses at the nanometric scale. In this study, residual stresses in machined surfaces of monocrystalline silicon are computed based on molecular dynamics simulation. An orthogonal machining configuration is adopted, and diamond cutting tools are used. The numerical approach developed is able to reveal stress evolution during and after machining, as well as in-depth residual stress distributions. The results indicate that the material stresses are stabilized within a manageable amount of computation time, and the in-depth normal stress along the tool moving direction has a more dynamical and significant pattern compared with other stress components. Meanwhile, the effects of depth of cut and tool rake angle are investigated. It is found that the increase of depth of cut results in the decrease of maximum tensile residual stress on the machined surfaces and the increase of maximum compressive residual stress underneath the surface. Similar observations are observed when the tool rake angle changes from positive to negative. It is believed that the more negative tool rake angles or the larger depths of cut induce a more drastic phase transformation to the machined surfaces, and this makes the in-depth residual stress distributions more compressive.     

Giant Static Dielectric Constant of Strained PbTiO3

First-principles density functional perturbation calculations are employed to study the dielectric and piezoelectric properties of strained tetragonal PbTiO3. Lattice distortion, static dielectric constant, Born effective charge, zone-centre phonons, and piezoelectric constant are obtained. For the strained tetragonal PbTiO3, we obtain a giant static dielectric constant （3600） under a strain 0. 77%. Moreover, the calculated piezoelectric constant e15 of strained PbTiO3 reaches about 203 C/m^2 which is about 20 times of that of unstrained system. The giant static dielectric constant is mainly due to the softening of the lowest-frequency phonon mode and the reduce of Ti-O bond length. This work demonstrates a route to a giant static dielectrics for electrically microwave and other devices.     

Love waves in functionally graded piezoelectric materials by stiffness matrix method

A numerical matrix method relative to the propagation of ultrasonic guided waves in functionally graded piezoelectric heterostructure is given in order to make a comparative study with the respective performances of analytical methods proposed in literature. The preliminary obtained results show a good agreement, however numerical approach has the advantage of conceptual simplicity and flexibility brought about by the stiffness matrix method. The propagation behaviour of Love waves in a functionally graded piezoelectric material (FGPM) is investigated in this article. It involves a thin FGPM layer bonded perfectly to an elastic substrate. The inhomogeneous FGPM heterostructure has been stratified along the depth direction, hence each state can be considered as homogeneous and the ordinary differential equation method is applied. The obtained solutions are used to study the effect of an exponential gradient applied to physical properties. Such numerical approach allows applying different gradient variation for mechanical and electrical properties. For this case, the obtained results reveal opposite effects. The dispersive curves and phase velocities of the Love wave propagation in the layered piezoelectric film are obtained for electrical open and short cases on the free surface, respectively. The effect of gradient coefficients on coupled electromechanical factor, on the stress fields, the electrical potential and the mechanical displacement are discussed, respectively. Illustration is achieved on the well known heterostructure PZT-5H/SiO₂, the obtained results are especially useful in the design of high-performance acoustic surface devices and accurately prediction of the Love wave propagation behaviour.     

Love waves in a layered functionally graded piezoelectric structure under initial stress

Abstract

To study the effect of initial stress on the propagation behavior of Love waves in a layered functionally graded piezoelectric structure, a mathematical model is established. The piezoelectric layer is taken as exponentially graded material where as half-space is taken as simply elastic substratum. The coupled electromechanical field equations are solved analytically to obtain the mechanical displacements and electrical potential functions for the piezoelectric layer and elastic substrate. The dispersion relations are obtained for electrically open and short cases. The higher mode Love wave propagation has been considered. For numerical interpretation of the results, four sets of piezoelectric layer and elastic substrate have been taken into consideration. Graphical representation reveals about the effect of initial stress and the effect of inhomogeneity parameter on the phase velocity against wave number for electrically open and electrically short cases, respectively.     

Propagation of Bleustein-Gulyaev waves in a prestressed layered piezoelectric structure

Based on the theories of nonlinear continuum mechanics, piezoelectricity and elastic waves in solids, theoretical analysis of Bleustein-Gulyaev surface acoustic wave propagation in a prestressed layered piezoelectric structure are described. Numerical calculations are performed for the case that the layer and the substrate are identical LiNbO(3) except that they are polarized in opposite directions. It is found that an almost linear behavior of the relative change in phase velocity versus the initial stress is obtained for both surface electrically free and shorted cases. Potential applications in the design of acoustic wave devices are suggested.     

Dynamic response of a pre-stressed cylindrical shell to a moving load

The work presented in this paper is concerned with the response of a pre-stressed, finite, thin circular cylindrical shell under a moving local load with a constant velocity. An analysis is carried out by a dynamic method, and the solutions which are bounded even at the critical velocity are obtained. The effects of the initial stresses on the dynamic responses of the displacement and the stresses are examined in connection with the velocity of the load.     

Theoretical validation on the existence of two transverse surface waves in piezoelectric/elastic layered structures

In this paper, we analytically study the dispersion behavior of transverse surface waves in a piezoelectric coupled solid consisting of a transversely isotropic piezoelectric ceramic layer and an isotropic metal or dielectric substrate. This study is a revisit to the stiffened Love wave propagation done previously. Closed-form dispersion equations are obtained in a very simple mathematical form for both electrically open and shorted cases. From the viewpoint of physical situation, two transverse surface waves (i.e., the stiffened Love wave and the FDLW-type wave) are separately found in a PZT-4/steel system and a PZT-4/zinc system. All the observed dispersion curves are theoretically validated through the discussion on the limit values of phase velocity using the obtained dispersion equations. Those validation and discussion give rise to a deeper understanding on the existence of transverse surface waves in such piezoelectric coupled structures. The results can be used as a benchmark for the study of the wave propagation in the piezoelectric coupled structures and are significant in the design of wave propagation in the piezoelectric coupled structures as well.     

Propagation of SH-waves in two anisotropic layers bonded to an isotropic half-space under gravity

This study reports a theoretical investigation of the propagation of SH-wave in a piezoelectric layer superimposed on a self-reinforced layer overlying an isotropic gravitational half-space. The expressions of the dispersion relation of SH-wave have been established for electrically open and electrically short conditions in closed form. For the purpose of numerical computation, lithium niobate piezoelectric material has been considered. The dispersion curves have been depicted graphically and the prominent impacts of piezoelectric constant, dielectric constant, reinforced parameter, width ratio, and Biot’s gravity parameter on the phase velocity of SH-wave have been unraveled for both the electrical conditions. As a special case of the problem, it is found that the obtained dispersion relation concurs with classical Love wave equation for both the electrical conditions. Moreover, some important peculiarities have also been traced out through numerical computations for both the electrical cases.     

The Flow Patterns and Wall Stresses in a Louvered‐Wall Moving Granular Filter Bed of Quartz Sand

The flow patterns and wall stresses in a two‐dimensional louvered moving granular filter bed of quartz sand were investigated. The flow pattern histories of granular solids in the filter bed were recorded using a digital camcorder and a pressure gauge for simultaneously measuring the normal stresses as well as shear stresses of the granular solids. The static wall stress distributions produced by the granular solids were measured, and agreed with a theoretical prediction formed using the differential slice and Runge‐Kutta (order four) methods. The variations in the dynamic wall stresses with time in a moving granular filter bed were obtained and the effect of the louver angle upon the flow patterns and wall stresses was also investigated. Employing the results obtained by stress measurements and image processing, the wall pressure pulsation phenomena in a moving granular filter bed may be further understood. The results reported here provide fundamental information for the design of moving granular filter beds to act as high‐temperature flue gas cleanup filters.     

Comparison of several methods to characterise the high frequency behaviour of piezoelectric ceramics for transducer applications

Thickness mode resonances in commercial piezoelectric ceramics have been characterised as a function of frequency by two methods. The first is based on a fit on the electrical impedance for the fundamental and the overtones. This method has been applied to a large number of PZT ceramic samples and frequency dependence for all the parameters is investigated, in particular for the piezoelectric coefficient e33. The second is based on the measurement of the mechanical displacement at the centre of the surface of a PZT ceramic disk. With a modified KLM scheme, this displacement is modelled. The dielectric, elastic and piezoelectric parameters are extracted and compared for the fundamental and the third overtone. The results are found to be in good agreement.     

Transverse surface waves in a layered structure with a functionally graded piezoelectric substrate and a hard dielectric layer

As to an ideally layered structure with a functionally graded piezoelectric substrate (material parameters change continuously along the thickness direction) and a hard dielectric layer, the existence and propagation behavior of transverse surface waves is studied by analytical technique. The dispersion equations for the existence of the transverse surface waves with respect to phase velocity are obtained for electrically open and short circuit conditions, respectively. A detailed investigation of the effect of gradient coefficient on dispersion relation, electromechanical coupling factor and penetration depth is carried out. It is found by numerical examples that adjusting gradient coefficient makes the electromechanical coupling factor of the transverse surface waves achieve quite high values at some appropriate ratio values of the layer thickness to the wavelength, and at the same time, the penetration depth can be reduced to the same order as the wavelength.     

Features of the acoustic field of angle surface-wave transducers

The influence of process features in the fabrication of angle transducers (deviation from uniform distribution of mechanical stresses/strains over their transmitting/receiving surface) on the acoustic field of excited surface waves has been investigated. Calculation was carried out within the parabolic approximation by replacing a real angle transducer of surface waves with a linear source located on the surface of a waveguide (control object) with a nonuniform (cosine) distribution of stress amplitudes. The expressions for the stress (strains) at an arbitrary observation point for a point receiver (reflector) and for the stress averaged over the receiver plane are obtained in complex form. The coordinate dependences of the stress amplitudes and phases have been calculated for typical technological modes; the calculation results are presented as plots. It is shown that the effect of deviations from uniform distribution is strongest in the transition (from the near-to far-field) zone, where nonuniformity of displacements of 10–15% at the edges of a piezoelectric cell causes a change in their amplitude by 15–30% in comparison with uniform distribution; this must be taken into account when estimating diffraction corrections. The possibilities of experimentally verifying the degree of nonuniformity of the stress distribution over the angle-transducer surface are discussed.     

Identification of elastic-plastic and phase transition characteristics for relaxor ferroelectric PMN-PT anisotropic single crystals using nanoindentation technique

As one kind of important ferroelectric ceramics, relaxor ferroelectric PMN-PT single crystals have triggered a revolution in electromechanical devices owing to their giant piezoelectric properties and ultra-high electromechanical coupling factors. The present study focused on the mechanical responses of [100]- and [110]-oriented poled PMN-PT ferroelectric single crystals under an indenter loading. The hardness and Young’s modulus with different crystallographic orientations of the crystals were measured by using the continuous stiffness measurement (CSM) with nanoindentation technique. Using a spherical indenter pressured at different indentation depths, the typical quasi-static nanoindentation tests with displacement-controlled mode were performed on the PMN-PT single crystal samples. Load–displacement curves of indentations were recorded to reveal the yielding or inelasticity behaviour in [100]- and [110]-oriented PMN-PT through a pop-in event. It was further verified by the stress–strain curves evaluated from the corresponding load–displacement curves, to show the similar characteristic on the elastic–inelastic transition. When a Berkovich indenter was employed for mechanical response testing, another pop-in event was observed at a smaller indentation depth compared to the one for elastic–inelastic transition, which may indicate a pressure-induced phase transition from rhombohedral (R) to tetragonal (T) of the PMN-PT single crystals.     

Thermal shock fracture of piezoelectric materials

This paper deals with a case study for the piezoelectric materials suddenly exposed to an environmental medium of different temperature. The problem is idealized to a plate containing an edge crack or an embedded crack. The stress and electric displacement histories in an uncracked plate are calculated. These stresses and electric displacements are then added to the crack surface tractions and electric displacements with opposite sign to formulate a mixed boundary value problem. The cracking problem is thus reduced to a singular integral equation of Cauchy type, which is then solved numerically. Both impermeable crack assumption and permeable crack assumption are considered. The results for stress and electric displacement intensity factors are computed as a function of normalized time and crack size. Lower bound solutions are obtained for the maximum thermal shock that the material can sustain without catastrophic failure according to the two distinct criteria: (i) The maximum local tensile stress equals the tensile strength of the medium. (ii) The maximum stress intensity factor for the pre-existing representative crack equals the fracture toughness of the medium. The parameters that control the transient thermal stress and electric displacement are also identified. The method can be used to explore susceptibility to thermal fracture in piezoelectric materials containing pre-cracks.     

Surface waves in a generalised thermopiezoelectric half space

Propagation of longitudinal surface waves on the surface of a rotating monoclinic piezoelectric half space has been considered taking into account generalized thermal coupling as in Green and Lindsay model. Variations of piezoelectric potential, particle displacement, temperature, etc. with depth into the medium and the piezoelectric Poynting vector together with power fiow components have been derived, assuming the existence of a shorting plane at a certain distance above the free surface of the half space.