Homogenization of periodic 1-3 piezocomposite using wave propagation: Toward an experimental method.

1-3 piezocomposites are first choice materials for integration in ultrasonic transducers due to their high electromechanical performance, particularly, in their thickness mode. The determination of a complete set of effective electroelastic parameters through a homogenization scheme is of primary importance for their consideration as homogeneous. This allows for the simplification of the transducer design using numerical methods. The method proposed is based on acoustic wave propagation through an infinite piezocomposite, which is considered to be homogeneous material. Christoffel tensor components for the 2 mm symmetry were expressed to deduce slowness curves in several planes. Simultaneously, slowness curves of a numerical phantom were obtained using a finite element method (FEM). Dispersive curves were initially calculated in the corresponding heterogeneous structure. The subsequent identification of the effective parameters was based on a fitting process between the two sets of slowness curves. Then, homogenized coefficients were compared with reference results from a numerical method based on a fast Fourier transform for heterogeneous periodic piezoelectric materials in the quasi-static regime. A relative error of less than 2% for a very large majority of effective coefficients was obtained. As the aim of this paper is to implement an experimental procedure based on the proposed homogenization scheme to determine the effective parameters of the material in operating conditions, it is shown that simplifications to the procedure can be performed and a careful selection of only seven slowness directions is sufficient to obtain the complete database for a piezocomposite containing square-shaped fibers. Finally, further considerations to adapt the present work to a 1-3 piezocomposite with a fixed thickness are also presented.

Modeling the Electroelastic Moduli of Porous Textured Piezoceramics.

A new model for piezoelectric textured ceramics was developed that considers the presence of porosity, which can appear during heat treatment (ceramic sintering). In the long wavelength approximation, a matrix method, which has already been applied to piezoelectric composites, was extended to textured ceramics for three phases [porosity (air), piezoelectric single-crystal (related to the texturation degree), and ceramic] to calculate the effective electroelastic modulus. This method was first compared and validated with finite-element calculations. A computation was applied to two systems with lead-based (PMN-PT) and lead-free (KNN) compositions. The results showed that the introduction of porosity in the whole material promotes electromechanical performance, particularly the electromechanical coupling factor kt , while limiting the degree of texturation. As an example, for the chosen PMN-PT system, an equivalent kt factor of 60% can be obtained with 1% porosity and an 85% single-crystal volume fraction or with 16% porosity and a 40% single-crystal volume fraction. According to the database used, this tradeoff is different. With the chosen lead-free composition, the degree of texture is less important than in the lead-based composition. Consequently, the porosity content is of primary importance for significantly improving the electromechanical coupling factor kt .

Electrical Evidence of the Tunable Electrical Bragg Bandgaps in Piezoelectric Plates.

A piezoelectric plate, poled along its thickness and supporting on its top and bottom surfaces a periodic grating of electrodes, is considered. An analytical model allowing band structure calculation is derived for the first symmetrical mode propagating along the length of the plate. Analytical results show that an electrical Bragg (EB) bandgap can be observed for this mode, depending on the electrical boundary conditions applied on the electrodes. This "EB bandgap" is associated with a discontinuity of the electric field between two successive unit cells. These results are validated with the numerical simulations based on the finite element method. Analytical and numerical results prove that the EB bandgap is highly tunable and can be optimized by changing the crystallographic orientation of the material. A simple tunable filter exploiting this bandgap is designed and fabricated. Experimental results of electrical impedance and electrical potential at the output together with a scanning laser vibrometer analysis are presented, which confirm the theoretical predictions.

Super-Cell Piezoelectric Composite With 1-3 Connectivity.

The standard fabrication method for 1-3 piezocomposites for ultrasound transducers is the "dice and fill" method (DFM) in which lateral periodicity is introduced. This contributes to the appearance of spurious modes that can drastically affect the performance of the device if they appear near its thickness mode frequency, thus limiting the effective frequency range. A new 1-3 piezocomposite fabricated with a super-cell structure [1-3 super cell (13SC)] was designed in order to overcome these limitations. It consists of the merging of several periodic cells with 47% PZT volume fraction and epoxy resin as the matrix. Two lateral periodicities in one direction are defined as well as two different kerfs. The chosen cell shape is composed of five nonaligned square section rods ( 1 ×1 mm 2 ). For comparison of performance, two regular 1-3 piezocomposites (the same materials and equivalent periodicities) were fabricated by DFM. Electroacoustic responses in water were measured for the three composites being considered as transducers. Successive regular thinnings (from 2.8 to 1.1 mm) were carried out for each sample to increase the operating frequency (from around 0.4 to 1.3 MHz) and study the evolution of the characteristics (bandwidth and sensitivity). The experimental results confirmed the behavior of those obtained with numerical simulations, showing that the 13SC composite can be used in this entire frequency range, unlike regular composites.

Control of elastic wave propagation in one-dimensional piezomagnetic phononic crystals.

Two ways of controlling the acoustic waves propagation by external inductance or capacitance in a one-dimensional (1-D) piezomagnetic phononic crystal are investigated. The structure is made of identical bars, constituted of a piezomagnetic material, surrounded by a coil and connected to an external impedance. A model of propagation of longitudinal elastic waves through the periodic structure is developed and the dispersion equation is obtained. Reflection and transmission coefficients are derived from a 2 × 2 transfer matrix formalism that also allows for the calculation of elastic effective parameters (density, Young modulus, speed of sound, impedance). The effect of shunting impedances is numerically investigated. The results reveal that a connected external inductance tunes the Bragg band gaps of the 1-D phononic crystal. When the elements are connected via a capacitance, a hybridization gap, due to the resonance of the LC circuit made of the piezomagnetic element and the capacitance, coexists with the Bragg band gap. The value of the external capacitance modifies the boundaries of both gaps. Calculation of the effective characteristics of the phononic crystal leads to an analysis of the physical mechanisms involved in the wave propagation. When periodically connected to external capacitances, a homogeneous piezomagnetic stack behaves as a dispersive tunable metamaterial.

Tunable phononic crystals based on piezoelectric composites with 1-3 connectivity.

Phononic crystals made of piezoelectric composites with 1-3 connectivity are studied theoretically and experimentally. It is shown that they present Bragg band gaps that depend on the periodic electrical boundary conditions. These structures have improved properties compared to phononic crystals composed of bulk piezoelectric elements, especially the existence of larger band gaps and the fact that they do not require severe constraints on their aspect ratios. Experimental results present an overall agreement with the theoretical predictions and clearly show that the pass bands and stop bands of the device under study are easily tunable by only changing the electrical boundary conditions applied on each piezocomposite layer.

Theoretical and experimental analyses of tunable Fabry-Perot resonators using piezoelectric phononic crystals.

Theoretical and experimental analyses of piezoelectric stacks submitted to periodical electrical boundary conditions via electrodes are conducted. The presented structures exhibit Bragg band gaps that can be switched on or off by setting electrodes in short or open circuit. The band gap frequency width is determined by the electromechanical coupling coefficient. This property is used to design a Fabry-Perot cavity delimited by a periodic piezoelectric stack. An analytical model based on a transfer matrix formalism is used to model the wave propagation inside the structure. The cavity resonance tunability is obtained by varying the cavity length (i.e., by spatially shifting boundary conditions in the stack). 26% tuning of resonance and antiresonance frequencies with almost constant electromechanical coupling coefficient of 5% are theoretically predicted for an NCE41 resonator. To optimize the device, the influence of various parameters is theoretically investigated. The cavity length, phononic crystal (number and length of unit cells), and transducer position can be adapted to tune the frequency shift and the coupling coefficient. When the transducer is located at a nodal plane of the cavity, the value of the coupling coefficient is 30%. Experimental results are presented and discussed analyzing the effects of damping.

Focusing capability of a phononic crystal based on a hollow metallic structure.

The dispersion curves of a phononic crystal (PC) based on a hollow metallic structure are presented. They exhibit a negative refraction dispersion branch and perfect refractive index matching with the surrounding water, leading to focusing capability. Numerical and experimental results are reported for a flat PC lens. The characteristics of the focal spot (intensity, dimensions, etc.) are numerically and experimentally investigated with the goal of finding the frequency of the optimal imaging performance.

Analysis of signals propagating in a phononic crystal PZT layer deposited on a silicon substrate.

The design of a stop-band filter constituted by a periodically patterned lead zirconate titanate (PZT) layer, polarized along its thickness, deposited on a silicon substrate and sandwiched between interdigitated electrodes for emission/reception of guided elastic waves, is investigated. The filter characteristics are theoretically evaluated by using finite element simulations: dispersion curves of a patterned PZT layer with a specific pattern geometry deposited on a silicon substrate present an absolute stop band. The whole structure is modeled with realistic conditions, including appropriate interdigitated electrodes to propagate a guided mode in the piezoelectric layer. A robust method for signal analysis based on the Gabor transform is applied to treat transmitted signals; extract attenuation, group delays, and wave number variations versus frequency; and identify stop-band filter characteristics.

Experimental and numerical study of evanescent waves in the mini stopband of a 1D phononic crystal.

This paper deals with the analysis of the guided evanescent waves in stopbands of a 1D phononic crystal (PC). A new numerical implementation is shown in order to get the complex values of the wavenumbers in a frequency range where a gap occurs. The considered phononic system is an aluminum plate with a one-dimensional sinusoidal grating. For this structure a mode-gap (mini stopband) occurs at low frequency: it involves the two fundamental Lamb modes A(0) and S(0). The numerical study is performed by using a finite element method (ATILA code). The experiments deal with a finite length grating and evanescent waves are characterized at the vicinity of the mini stopband.

Analysis of elastic waves transmitted through a 2-D phononic crystal exhibiting negative refraction.

A two-dimensional phononic crystal (PC) made of a square lattice of air holes in an aluminum matrix is studied. The band structure calculated in the irreducible Brillouin zone of the PC exhibits a branch with a negative slope that allows negative refraction. This phenomenon has been numerically verified using a prism-shaped PC for plane waves entering the PC with two different incidences. A detailed study of the waves at the exit of the PC shows that the plane wave is reconstructed after several wavelengths. Finally, the description of the refracted waves is interpreted using a point source array, giving information about the angular spreading and the relative amplitude of each refracted beam.

Attenuation of Lamb waves in the vicinity of a forbidden band in a phononic crystal.

When a Lamb wave propagates on a plate engraved by a periodic grating, it may exhibit attenuation. This attenuation is related to a coupling of this incident mode with other propagating modes. As the propagation takes place in a periodic medium, the dispersion curves of the modes are of interest because they exhibit passbands and stopbands related to the geometry of the waveguide. The goal of this work is to quantitatively establish the relation between the value of the attenuation of the propagating waves and the width of the forbidden bands appearing inside the Brillouin zone. This study is performed by using a finite element method (ATILA code).

Miniature multimode monolithic flextensional transducers.

Traditional flextensional transducers classified in seven groups based on their designs have been used extensively in 1-100 kHz range for mine hunting, fish finding, oil explorations, and biomedical applications. In this study, a new family of small, low cost underwater, and biomedical transducers has been developed. After the fabrication of transducers, finite-elements analysis (FEA) was used extensively in order to optimize these miniature versions of high-power, low-frequency flextensional transducer designs to achieve broad bandwidth for both transmitting and receiving, engineered vibration modes, and optimized acoustic directivity patterns. Transducer topologies with various shapes, cross sections, and symmetries can be fabricated through high-volume, low-cost ceramic and metal extrusion processes. Miniaturized transducers posses resonance frequencies in the range of above 1 MHz to below 10 kHz. Symmetry and design of the transducer, polling patterns, driving and receiving electrode geometries, and driving conditions have a strong effect on the vibration modes, resonance frequencies, and radiation patterns. This paper is devoted to small, multimode flextensional transducers with active shells, which combine the advantages of small size and low-cost manufacturing with control of the shape of the acoustic radiation/receive pattern. The performance of the transducers is emphasized.

Experimental validation of band gaps and localization in a one-dimensional diatomic phononic crystal.

The propagation of compressional ultrasonic pulses through a finite one-dimensional chain of various unit cells is investigated experimentally. The chain, initially compressed by an axially applied constant force, is excited by a periodic force, which acts in line with axis of bead chain. The experimental measurements giving the eigenfrequencies of the specimen are based on a Fourier analysis of the transmitted acoustic pulse. The results are compared with the numerical calculations and it is shown that the two approaches are well correlated. A phononic band structure is observed and under certain conditions, depending on the parity of the number and on the masses of the beads in the chain, it is shown that localized modes propagating in the forbidden band are exhibited. Much attention is devoted to the existence of these localized modes according to the mass ratio between two adjacent beads constituting the unit cell.

Numerical homogenization techniques applied to piezoelectric composites.

With the recent availability of piezoelectric fibers, the design and the analysis of piezoelectric composites needs new modeling tools. Therefore, a numerical homogenization technique has been developed, based on the ATILA finite element code, that combines two techniques: one relying upon the representative volume element (RVE) the other relying upon the wave propagation (WP). The combination of the two methods allows the whole tensor of the homogenized properties of the piezoelectric composite to be found. Considering a fiber embedded in epoxy, the numerical results are compared to the results obtained using previous analytical models, thus validating the models. Even if the method is presented in a particular case, its extension to any piezoelectric composite is straightforward.

On the existence of subresonance generated in a one-dimensional chain of identical spheres.

In this paper, the propagation of short compressional pulses through a one-dimensional chain of identical spherical beads is analyzed. First, a single sphere is studied. Then, an infinite chain of identical spheres is considered. Finally, a finite linear chain can be seen as a particular case of an infinite chain. It shows the existence of allowed and forbidden frequency bands. In the case of a finite chain of spheres, discrete resonance frequencies are excited and may be identified as "subresonance" of modes observed with a single sphere. We also analyze how the coupling between two adjacent spheres and the number of beads in the chain, modify the degeneracy of the stationary states. Finally, the theoretical results are qualitatively verified with experimental data obtained with the "resonant sphere technique."