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.     

Approximated Analytical Solution of a Problem on Indentation of an Electro-Elastic Half-Space with Inhomogeneous Coating by a Conductive Punch

The paper addresses to the development of the bilateral asymptotic method of solution of contact problems taking into account electro-mechanical fields coupling. Approximated analytical solutions are provided for contact problems on indentation of an electroelastic piezoelectric half-space with inhomogeneous in depth piezoelectric coating by a rigid conductive circular flat, spherical or conical punch.     

Electroelastic field of a sphere located in the vicinity of a plane piezoelectric surface

The electric field generated by a scanning probe microscope is determined. Analytical expressions for the electroelastic field in a piezoelectric sample and the external electric field are derived for a spherical probe. It is demonstrated that the coupling of elastic and electrostatic fields in the piezoelectric material leads to energy redistribution between such fields. This circumstance causes variations in the normal component of the electric field strength at the interface and the capacitance of a probe.     

Quantitative determination of contact stiffness using atomic force acoustic microscopy

Atomic force acoustic microscopy is a near-field technique which combines the ability of ultrasonics to image elastic properties with the high lateral resolution of scanning probe microscopes. We present a technique to measure the contact stiffness and the Young's modulus of sample surfaces quantitatively, with a resolution of approximately 20 nm, exploiting the contact resonance frequencies of standard cantilevers used in atomic force microscopy. The Young's modulus of nanocrystalline ferrite films has been measured as a function of oxidation temperature. Furthermore, images showing the domain structure of piezoelectric lead zirconate titanate ceramics have been taken.     

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.     

Inclusions and inhomogeneities in electroelastic media with hexagonal symmetry

For a long time, the absence of explicit Green's functions (fundamental solutions) for electroelastic media has hindered progress in the modelling of the properties of piezoelectric materials. Michelitsch's recently derived explicit electroelastic Green's function for the infinite medium with hexagonal symmetry (transversely isotropic medium) [4] is used here to obtain compact closed-form expressions for the electroelastic analogue of the Eshelby tensor for spheroidal inclusions. This represents a key quantity for the material properties of piezoelectric solids and analysis of the related electroelastic fields in inclusions. For the limiting case of continuous fibers our results coincide with Levin's expressions [8]. The derived method is useful for an extension to non-spheroidal inclusions or inhomogeneities having an axis of rotational symmetry parallel to the hexagonal c-axis. Received 14 September 1999     

Geometry independence of the normalized relaxation functions of viscoelastic materials in indentation

The normalized relaxation modulus represents a salient feature of viscoelastic materials and its determination is of great significance for various applications. From the normalized relaxation modulus, for instance, one can derive the loss factor of a viscoelastic polymer and judge whether a material is suitable for damping applications or not. By using dimensional analysis and the elastic–viscoelastic correspondence principle, the normalized relaxation function of a linear viscoelastic material obtained from indentation relaxation tests is shown to depend only on the indentation load but not on the indenter geometry and the shape of the indented solid. The result could enable circumvention of the difficulties encountered in the calibration of the indenter geometry and the preparation of indented samples. Numerical simulations are performed on a number of cases of practical interest, including the spherical indentation test of a soft layer lying on a rigid substrate, a flat punch indenter indenting into a soft layer with a rough surface bonded to a rigid substrate, a rigid indenter with irregular shape indenting into a particle, inclined contact of a cylindrical indenter with a cylinder, and indentation of porous substrates. The numerical examples demonstrate that the conclusion from the theoretical analysis is valid for all these situations.     

Acoustic Imaging Frequency Dynamics of Ferroelectric Domains by Atomic Force Microscopy

We report the acoustic imaging frequency dynamics of ferroelectric domains by low-frequency acoustic probe microscopy based on the commercial atomic force microscopy. It is found that ferroelectric domain could be firstly visualized at lower frequency down to 0.5 kHz by AFM-based acoustic microscopy. The frequency-dependent acoustic signal revealed a strong acoustic response in the frequency range from 7kHz to 10 kHz, and reached maximum at 8.1 kHz. The acoustic contrast mechanism can be ascribed to the different elastic response of ferroelectric microstructures to local elastic stress fields, which is induced by the acoustic wave transmitting in the sample when the piezoelectric transducer is vibrating and exciting acoustic wave under ac electric fields due to normal piezoelectric effects.     

Calculation of the thicknesses and elastic properties of thin-film coatings using atomic-force acoustic microscopy data

The feasibility of applying atomic-force acoustic microscopy to measure the elastic properties of thin-film coatings and their thickness in the range from several to several hundreds of nanometers is studied. In practice, this technique can be used to study diamond-like coatings. The key point of our method is application of “flat” tips, which provide a constant tip-surface contact area and, hence, a constant contact stiffness. The reason for using such tips is that experimental data for thin-film structures gained with standard (rounded) tips cannot be given an adequate quantitative interpretation. Numerical simulation is used to evaluate the thickness and indentation modulus of a coating from contact stiffness k _cont measured by atomic-force acoustic microscopy.     

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.     

The effect of small-amplitude load oscillations on the nanocontact characteristics of materials in nanoindentation

A continuous stiffness measurement method allowing one to obtain physical-mechamical characteristics of materials in the process of indentation during gradually increasing loading has been scrutinized. The limits of applicability of this approach depend on the testing conditions at which the additional smallamplitude oscillations exert no influence on the mechanics, kinetics, and plastic strain of material being in contact with the indenter. As is shown by taking as examples specimens with amorphous structure, and fcc and bcc lattices, various techniques for determining the nanocontact characteristics of materials are different by their sensitivity to small-amplitude load oscillations. The oscillation amplitudes and the indentation depth ranges where one can neglect the oscillation effects have been evaluated. The impacts of the oscillations on the behavior of the contact (local) characteristics of the studied materials in supercritical modes have been established.     

Effect of initial stress on Love waves in a piezoelectric structure carrying a functionally graded material layer

The effect of initial stress on the propagation behavior of Love waves in a piezoelectric half-space of polarized ceramics carrying a functionally graded material (FGM) layer is analytically investigated in this paper from the three-dimensional equations of linear piezoelectricity. The analytical solutions are obtained for the dispersion relations of Love wave propagating in this kind of structure with initial stress for both electrical open case and electrical short case, respectively. One numerical example is given to graphically illustrate the effect of initial stress on dispersive curve, phase velocity and electromechanical coupling factor of the Love wave propagation. The results reported here are meaningful for the design of surface acoustic wave (SAW) devices with high performance.     

基于2-2型压电复合材料的新型宽频带径向振动超声换能器

本文提出了一种基于2-2型压电复合材料的新型宽频带径向振动超声换能器,它主要由内金属圆环和外压电陶瓷复合材料圆环组成.首先利用Newnham串并联理论和均匀场理论推导了2-2型压电复合材料的等效参数;其次利用解析法得到了金属圆环和径向极化压电复合陶瓷圆环径向振动的机电等效电路;最后得到了换能器的六端机电等效电路,从而得到了换能器的频率方程.接着分析了换能器共振频率和反共振频率以及有效机电耦合系数与几何尺寸、两相体积占比的关系,采用仿真软件对新型换能器的径向振动进行了数值模拟.结果表明,利用解析法得到的共振频率和反共振频率与数值模拟结果吻合较好.此外,对换能器在水下的辐射声场进行了仿真研究,结果表明新型复合材料径向换能器相比传统纯陶瓷径向换能器,发射电压响应幅值更大,工作带宽提高接近一倍,声匹配更佳.     

Characteristics of backing impacts on the performance of the thickness mode piezoelectric transducer

The impacts of the thickness, the specific acoustic impedance and the mechanical loss factor of the backing on the performance of the thickness mode piezoelectric transducer are systemically studied, which are focused on the effective electromechanical coupling coefficient and the mechanical quality factor near the center frequency. The results show that with contin?uous increasing of the backing thickness, the effective electromechanical coupling coefficient and the mechanical quality factor are found rapidly declined by oscillation way. With the increase of the difference value of the acoustic impedance between the backing material and the piezoelec?tric material, the effective electromechanical coupling coefficient deceases and the mechanical quality factor increases. Under condition that the thickness of the piezoelectric material is fixed, the effective electromechanical coupling coefficient is found declined with the increase of the mechanical loss factor by monotonous way. The mechanical quality factor has minimum value and the electric characteristic curve tends to be smooth in a given frequency range. The equivalent circuit theory result is in good agreement with the ones by finite element method and the experimental results. The work mentioned above provides a theoretical guidance for the design and experimental fabrication of the thickness mode piezoelectric transducer.     

Crack extension force and rate of mechanical work of fracture in linear dielectrics and piezoelectrics

The physical significance of the crack extension force produced by mechanical loads and electric fields in linear dielectric and piezoelectric materials is examined using simple thermodynamic arguments. General expressions are derived for the crack extension force in dielectrics and piezoelectrics, under mechanical and electrical loads, in terms of the measurable parameters elastic compliance and electric capacitance . It is shown that the crack extension force produced by an electric field on an impermeable crack is always negative and it is argued that under combined electromechanical loads the total crack extension force in a piezoelectric cannot be separated into a mechanical component and an electrical component. Expressions for the crack extension force in terms of mechanical and electrical intensity factors are also given. Their derivation from available solutions for the electromechanical field at the crack tip (in a transversely isotropic material of crystal class 6 mm) is presented in detail to emphasize the physical significance of the coefficients that appear in the final expressions. In the light of these results, existing experimental observations that appear to be inconsistent with theoretical expectations are re-examined. The suggestion that the crack extension force is not a valid parameter to characterize the fracture behaviour of ferroelectrics is justified on physical grounds. Its importance is discussed and the rate of mechanical work of fracture is proposed as a more suitable parameter for those cases where the electric field does not produce dielectric rupture, nor degradation of the material at the crack tip.     

Stoney formula for piezoelectric film/elastic substrate system

With the trends in miniaturization, and particularly the introduction of micro- and nano-electro-mechanical system,piezoelectric materials used in microelectronic devices are deposited usually in the form of thin film on elastic substrates.In this work, the bending of a bilayer comprising a piezoelectric film deposited on an elastic substrate, due to the mismatch,is investigated. An analytic formula relating the curvature of the bilayer to the mismatch, the electroelastic constants and the film thickness is obtained, and from this formula, a transverse piezoelectric constant d_31 can be estimated. Meanwhile the influence of electromechanical coupling coefficient on the curvature is discussed.     

Study on the electromechanical coupling coefficient of Rayleigh-type surface acoustic waves in semi-infinite piezoelectrics/non-piezoelectrics superlattices

The electromechanical coupling coefficient of Rayleigh-type surface acoustic waves in semi-infinite piezoelectrics/non-piezoelectrics superlattices is investigated by the transfer matrix method. Research results show the high electromechanical coupling coefficient can be obtained in these systems. The optimization design of it is also discussed fully. It is significantly influenced by electrical boundary conditions on interfaces, thickness ratios of piezoelectric and non-piezoelectric layers, and material parameters (such as velocities of pure longitudinal and transversal bulk waves in non-piezoelectric layers). In order to obtain higher electromechanical coupling coefficient, shorted interfaces, non-piezoelectric materials with large velocities of longitudinal and transversal bulk waves, and proper thickness ratios should be chosen.     

Application of piezoelectric transient response in transducer acoustic power measurement

To explore a more convenient method for measuring the focused ultrasound power,a piezoelectric ceramic plate was used to receive ultrasonic signal directly.Due to an acoustic force acts on the surface of piezoelectric ceramic plate,the piezoelectric response was obtained by means of electromechanical analogy,which was composed of voltage response caused by forced vibration and high frequency attenuation response caused by natural vibration.The conversion relationship between output signal of piezoelectric ceramic plate and acoustic power of transducer was analyzed.The envelope of output piezoelectric signal was extracted in twice,and a voltage amplitude curve with sinusoidal distribution that could describe the changes of acoustic power was obtained.Under different drive voltage of transducer,the maximum peak voltage of envelope curve was found respectively.Their squared values were made a linear fitting with acoustic power measured by acoustic power meter,and then the proportional coefficient of theoretical relational expression was calibrated.The experimental results are in good agreement with the theory.The relative error between calibrated theoretical acoustic power and that measured by acoustic power meter was less than 8.7%.The paper can provide a guideline for measuring acoustic power of transducer by using piezoelectric signal.     

Quantitative measurement of local elasticity of SiOx film by atomic force acoustic microscopy

<正>In this paper the elastic properties of SiO_x film are investigated quantitatively for local fixed point and qualitatively for overall area by atomic force acoustic microscopy(AFAM) in which the sample is vibrated at the ultrasonic frequency while the sample surface is touched and scanned with the tip contacting the sample respectively for fixed point and continuous measurements.The SiO_x films on the silicon wafers are prepared by the plasma enhanced chemical vapour deposition(PECVD).The local contact stiffness of the tip-SiO_x film is calculated from the contact resonance spectrum measured with the atomic force acoustic microscopy.Using the reference approach,indentation modulus of SiO_x film for fixed point is obtained.The images of cantilever amplitude are also visualized and analysed when the SiO_x surface is excited at a fixed frequency.The results show that the acoustic amplitude images can reflect the elastic properties of the sample.     

Tunable bulk acoustic wave resonators with the induced piezoelectric effect in a ferroelectric

An electromechanical model of the piezoelectric effect induced in an acoustic resonator based on a ferroelectric film under the action of a dc or weak ac voltage is developed. The basic equation is obtained by expansion of the free energy in a series with respect to the electric induction and the mechanical deformation. The system of electromechanical equations for variable components of the induction and the mechanical deformation involves all linear terms along with the component of the electrostriction nonlinear with respect to the mechanical deformation. These electromechanical equations made it possible to obtain a one-dimensional approximation for the effective parameters of the material: the piezoelectric modulus and the elastic modulus as a function of the strength of the electric field applied to the acoustic layer. Expressions for the controlled electromechanical coupling coefficient and resonance frequencies of the tunable acoustic resonator are found. It is shown that the most significant parameter responsible for the tuning is the nonlinear electros-triction coefficient M, whose magnitude and sign were evaluated from the available experimental data.