Resonant-ultrasound spectroscopy for studying annealing effect on elastic constant of thin film

In this paper, we study the elastic property of thin films using resonant-ultrasound spectroscopy (RUS). RUS determines the elastic constants of a solid from its resonance frequencies of free vibration. There were two problems to be solved for applying RUS to thin films: accurate measurement of the resonance frequencies and mode identification of each resonance frequency. We solve these problems using the tripod needle transducers and the laser-Doppler interferometry (LDI). In this paper, we describe the RUS/LDI measurement setup we developed, and show the relationship between the elastic constant and annealing temperature for Cu thin films. Then, we discuss the effects of recrystallization and recovery on the elastic constant referring the X-ray diffraction measurement.     

Elastic constants of natural quartz

The elastic constants of a natural-quartz sphere using resonance-ultrasound spectroscopy (RUS) are measured. The measurements of the near-traction-free vibrational frequencies of the sphere are matched with the predicted frequencies from the dynamic theory of elasticity, with optimized estimates for the elastic constants driving the differences between these sets of frequencies to a minimal value. The present computational model, although based on earlier approaches, is the first application of RUS to trigonal-symmetry spheres. Quartz shows six independent elastic constants, and our estimates of these constants are close to those computed by other means. Except for C14, after a 1% mass-density correction, natural quartz and cultured quartz show the same elastic constants. Natural quartz shows higher internal frictions.     

Resonant ultrasound spectroscopy and homogeneity in polycrystals

Resonant ultrasound spectroscopy (RUS) is capable of determining the bulk elastic properties of a solid from its characteristic vibration frequencies, given the dimensions, density and shape of the sample. The model used for extracting values of the elastic constants assumes perfect homogeneity, which can be approximated by average-isotropic polycrystals. This approximation is excellent in the small grain regime assumed for most averaging procedures, but for real samples with indeterminate grain size distributions, it is not clear where the approximation breaks down. RUS measurements were made on pure copper samples where the grain size distribution was changed by progressive heat treatments in order to find a quantitative limit for the loss of homogeneity. It is found that when a measure of the largest grains is 15% of the sample’s smallest dimension, the deviation in RUS fits indicates elastic inhomogeneity.     

Determination of elastic moduli of rock samples using resonant ultrasound spectroscopy

Resonant ultrasound spectroscopy (RUS) is a method whereby the elastic tensor of a sample is extracted from a set of measured resonance frequencies. RUS has been used successfully to determine the elastic properties of single crystals and homogeneous samples. In this paper, we study the application of RUS to macroscopic samples of mesoscopically inhomogeneous materials, specifically rock. Particular attention is paid to five issues: the scale of mesoscopic inhomogeneity, imprecision in the figure of the sample, the effects of low Q, optimizing the data sets to extract the elastic tensor reliably, and sensitivity to anisotropy. Using modeling and empirical testing, we find that many of the difficulties associated with using RUS on mesoscopically inhomogeneous materials can be mitigated through the judicious choice of sample size and sample aspect ratio.     

Elastic constants of layers in isotropic laminates

The individual laminae elastic constants in multilayer laminates composed of dissimilar isotropic layers were determined using ultrasonic-resonance spectroscopy and the linear theory of elasticity. Ultrasonic resonance allows one to measure the free-vibration response spectrum of a traction-free solid under periodic vibration. These frequencies depend on pointwise density, laminate dimensions, layer thickness, and layer elastic constants. Given a material with known mass but unknown constitution, this method allows one to extract the elastic constants and density of the constituent layers. This is accomplished by measuring the frequencies and then minimizing the differences between these and those calculated using the theory of elasticity for layered media to select the constants that best replicate the frequency-response spectrum. This approach is applied to a three-layer, unsymmetric laminate of WpCu, and very good agreement is found with the elastic constants of the two constituent materials.     

Measurements of Young’s and shear moduli of rail steel at elevated temperatures

The design and modelling of the buckling effect of Continuous Welded Rail (CWR) requires accurate material constants, especially at elevated temperatures. However, such material constants have rarely been found in literature. In this article, the Young’s moduli and shear moduli of rail steel at elevated temperatures are determined by a new sonic resonance method developed in our group. A network analyser is used to excite a sample hanged inside a furnace through a simple tweeter type speaker. The vibration signal is picked up by a Polytec OFV-5000 Laser Vibrometer and then transferred back to the network analyser. Resonance frequencies in both the flexural and torsional modes are measured, and the Young’s moduli and shear moduli are determined through the measured resonant frequencies. To validate the measured elastic constants, the measurements have been repeated by using the classic sonic resonance method. The comparisons of obtained moduli from the two methods show an excellent consistency of the results. In addition, the material elastic constants measured are validated by an ultrasound test based on a pulse-echo method and compared with previous published results at room temperature. The measured material data provides an invaluable reference for the design of CWR to avoid detrimental buckling failure.     

RUS study of the elastic constants in silver halide crystals

By using RUS method, the temperature dependence of the elastic constants in the silver halide crystals has been measured above room temperature. The elastic constants decrease linearly with increasing temperature. Numerical calculations are presented for the contribution of the multipole polarization to the elastic constants in the silver halide crystals. The calculated values of the deviation from the Cauchy relation are about a quarter of the experimental values. The multipole polarization significantly affects the elastic constants and contributes to the violation of the Cauchy relation.     

Modal resonant ultrasound spectroscopy for ferroelastics

Recent experimental and theoretical improvements of resonant ultrasound spectroscopy (RUS) are summarized to investigate elastic constants of phases in shape memory alloys. The proposed inversion procedure, described in this work, is particularly suitable to reliable evaluation of the temperature dependence of elastic constants of low-symmetry ferroelastic materials which may be strongly elastically anisotropic and tend to exist in twinned forms. The method is applicable even for the evaluation of single-crystal elastic constants from RUS measurements on microtwinned crystals, since it involves a homogenization algorithm based on the macroscopic deformation response of the layered structure. This potentially allows performing meaningful acoustic studies on samples with a general submicron-size layered structure.     

Resonance ultrasound spectroscopy with laser-Doppler interferometry for studying elastic properties of thin films

We propose an advanced method to determine the elastic-stiffness coefficients Cij of thin films using resonance ultrasound spectroscopy (RUS). It uses free-vibration resonance frequencies of a film/substrate layered solid and derives inversely the film's Cij from the resonance frequencies. We develop a piezoelectric tripod consisting of two pinducers and one support to place the specimen on it and measure the resonance frequencies with high enough accuracy. Furthermore, we achieve mode identification by measuring deformation distributions on the vibrating specimen surface using laser-Doppler interferometry. Accurate measurements of frequencies and correct mode identification are the keys for deducing reliable Cij of the film. We applied this technique to copper thin films deposited of Si substrates. The resulting film's Cij are considerably smaller than the bulk's Cij and show anisotropy between the out-of-plane direction and in-plane direction.     

Resonance frequencies of piezoelectric plates surrounded by solid and fluid half-spaces

The design of ultrasound transducers, resonators and other piezoelectric devices usually requires the calculation of the resonance frequencies of piezoelectric plates. Recent studies have shown that the resonance frequencies for plates in vacuum correspond to frequencies where the waveguide group velocity vanishes (zero-group-velocity points). However, those studies are limited to vacuum boundary conditions. The objective of the present study is to analyze the resonance frequencies of layered piezoelectric plates in contact with solid and fluid half-spaces and their relation to the dispersion behavior of the elastic guided wave propagation. Theoretical analysis using partial-wave approach of leaky Lamb waves is performed to study wave propagation in, and resonance behavior of, multilayered plates in contact with solid and fluid half-spaces. A novel observation resulted from this analysis is that, for plates in contact with solid and fluid half-spaces, the resonance frequencies occur at points where the magnitude of the wavenumber reaches a minimum. This frequency is named as a ‘transition frequency’. Such observations are important because they allow an easy identification of resonance frequencies with high amplitude response directly from the dispersion curves. This study will be helpful for the design of piezoelectric components used for resonators and sensors.     

Determination of ultrasonic wave velocities and phase velocity dispersion curves of an Inconel 600 plate using resonant ultrasound spectroscopy and leaky Lamb waves

A plate of Inconel 600 was interrogated using the resonant ultrasound spectroscopy (RUS) and the reflected leaky Lamb waves (LLW). It was found that the plate used in the present work has anisotropy in its material properties by the RUS. The longitudinal and the transverse wave velocities of the Inconel 600 plate were determined by the RUS, ultrasonic pulse-echo method and cut-off frequencies of the LLWs. The wave velocities in the direction of thickness determined by the RUS under the assumption of the orthotropic symmetry were quite similar to those obtained by other methods, the pulse-echo method and from cut-off frequencies. The reflected LLW from the plate was measured with varying the incident angle. The dispersion curves obtained from the reflected LLWs show good agreement with the theoretical calculation in general. The mismatches may be caused by anisotropy of the plate.     

Measuring elastic constants of arbitrarily shaped samples using resonant ultrasound spectroscopy

One of the most important undertakings for materials is the measurement of the elastic behavior. As derivatives of the free energy with respect to atomic displacements, the elastic properties are closely connected to the thermodynamic properties of the material. Elastic behavior is a sensitive probe of the lattice environment in which all solid state phenomena occur, particularly in the vicinity of a phase transition. A useful method for measuring elastic properties is resonant ultrasound spectroscopy (RUS). Some novel materials to which RUS might be applied are often fragile or chemically reactive so that they cannot be polished into the shapes required by conventional RUS; for such cases a finite element method may be used. In this paper a discussion and test of a finite element method for RUS with arbitrarily shaped samples is provided.     

The influence of localized damage in a sample on its resonance spectrum

A nonlinear version of resonance ultrasound spectroscopy (RUS) theory is presented. This is important for NDT-purposes as damage manifests itself more pronounced and in an earlier stage by changes in the nonlinear elastic constants. General equations are derived for the 1-D case, describing the interaction between the modes due to the presence of nonlinearity. An analytical solution of these equations is derived which predicts the shift of the resonance frequency versus amplitude in a bar with localized damage. The damage was modelled as a finite region, having a constant cubic nonlinearity, in an otherwise linear 1-D bar. The analytical expressions for the shifts in resonance frequency at different modes were used to infer information about the position, nonlinearity and width of the damage. Unlike other techniques, the proposed method does not require scanning to locate the defect, as it lets the different modes, each with a different vibration pattern, probe the structure.     

Evaluation of the contact resonance frequencies in atomic force microscopy as a method for surface characterisation (invited)

The combination of ultrasound with atomic force microscopy (AFM) opens the high lateral resolution of scanning probe techniques in the nanometer range to ultrasonics. One possible method is to observe the resonance frequencies of the AFM sensors under different tip-sample interaction conditions. AFM sensors can be regarded as small flexible beams. Their lowest flexural and torsional resonance frequencies are usually found to be in a range between several kHz and several MHz depending on their exact geometrical shape. When the sensor tip is in a repulsive elastic contact with a sample surface, the local indentation modulus can be determined by the contact resonance technique. Contact resonances in the ultrasonic frequency range can also be used to improve the image contrast in other dynamic techniques as, for example, in the so-called piezo-mode. Here, an alternating electric field is applied between a conducting cantilever and a piezoelectric sample. Via the inverse piezoelectric effect, the sample surface is set into vibration. This excitation is localised around the contact area formed by the sensor tip and the sample surface. We show applications of the contact resonance technique to piezoelectric ceramics.     

Determination of elastic, anelastic, and piezoelectric coefficients of piezoelectric materials from a single specimen by acoustic resonance spectroscopy

We describe an advanced methodology to determine all the independent elastic-stiffness coefficients Cijkl, the associated internal frictions Qijkl(-1), and piezoelectric coefficients eijk of piezoelectric materials from a single monocrystal specimen using resonant-ultrasound spectroscopy with laser-Doppler interferometry. The mechanical-resonance frequencies of a piezoelectric solid depend on all of the elastic and piezoelectric coefficients, and their accurate measurement allows one to determine the elastic and piezoelectric coefficients simultaneously. Resonance-peak-width measurements yield the internal-friction tensor. Successful determination requires correct vibration-mode identification for the observed resonance frequencies. This is achieved unambiguously by measuring deformation distributions on the vibrating-specimen surface with laser-Doppler interferometry and comparing them with calculated displacement distributions. The methodology is applied to lithium niobate (LiNbO3) and langasite (La3Ga5SiO14) crystals.     

Circumferential resonance modes of solid elastic cylinders excited by obliquely incident acoustic waves

When an immersed solid elastic cylinder is insonified by an obliquely incident plane acoustic wave, some of the resonance modes of the cylinder are excited. These modes are directly related to the incidence angle of the insonifying wave. In this paper, the circumferential resonance modes of such immersed elastic cylinders are studied over a large range of incidence angles and frequencies and physical explanations are presented for singular features of the frequency-incidence angle plots. These features include the pairing of one axially guided mode with each transverse whispering gallery mode, the appearance of an anomalous pseudo-Rayleigh in the cylinder at incidence angles greater than the Rayleigh angle, and distortional effects of the longitudinal whispering gallery modes on the entire resonance spectrum of the cylinder. The physical explanations are derived from Resonance Scattering Theory (RST), which is employed to determine the interior displacement field of the cylinder and its dependence on insonification angle.     

Determination of elastic modulus of a beam by using electronic speckle pattern interferometry

This paper proposes a sonic resonance test for an elastic modulus measurement which is based on the electronic speckle pattern interferometry. Previous measurement technique of elastic constant has the limitation of application for thin film or polymer material because contact to specimen affects the result. It has been developed as a non-contact optical measurement technique which can visualize resonance vibration mode shapes with whole field. The maximum vibration amplitude at each vibration mode shape is a clue to find the resonance frequencies. The dynamic elastic constant of test material can be easily determined from vibration of a beam equation using the measured resonance frequencies. The proposed technique is able to give high accurate elastic modulus of materials through a simple experimental set-up and analysis. The experimental result also compared to the theoretical result.     

Non-ambiguous recovery of Biot poroelastic parameters of cellular panels using ultrasonicwaves

The inverse problem of the recovery of the poroelastic parameters of open-cell soft plastic foam panels is solved by employing transmitted ultrasonic waves (USW) and the Biot-Johnson-Koplik-Champoux-Allard (BJKCA) model. It is shown by constructing the objective functional given by the total square of the difference between predictions from the BJKCA interaction model and experimental data obtained with transmitted USW that the inverse problem is ill-posed, since the functional exhibits several local minima and maxima. In order to solve this problem, which is beyond the capability of most off-the-shelf iterative nonlinear least squares optimization algorithms (such as the Levenberg Marquadt or Nelder-Mead simplex methods), simple strategies are developed. The recovered acoustic parameters are compared with those obtained using simpler interaction models and a method employing asymptotic phase velocity of the transmitted USW. The retrieved elastic moduli are validated by solving an inverse vibration spectroscopy problem with data obtained from beam-like specimens cut from the panels using an equivalent solid elastodynamic model as estimator. The phase velocities are reconstructed using computed, measured resonance frequencies and a time-frequency decomposition of transient waves induced in the beam specimen. These confirm that the elastic parameters recovered using vibration are valid over the frequency range ofstudy.     

Resonant ultrasound spectroscopy measurement of Young's modulus,shear modulus and Poisson's ratio as a function of porosity for alumina and hydroxyapatite

Modulus–porosity relationships are critical for engineered bone tissue scaffold materials such as hydroxyapatite (HA), where porosity is essential to biological function. Resonant ultrasound spectroscopy (RUS) measurements revealed that the Young's modulus, E, and shear modulus, G, of both alumina and HA decrease monotonically with increasing volume fraction porosity, P, for 0.06 < P < 0.39 (alumina) and 0.05 < P < 0.51 (HA). Although the elastic moduli of porous materials have been measured by a number of different ultrasonic resonance techniques (of which the RUS technique is one example) and over the last decade the elastic moduli of many solids have been measured by the RUS technique, this study is the first systematic RUS study of porous materials. Comparison of E versus P data for alumina (which has been studied extensively) with literature data from several measurement techniques indicates the RUS technique is effective for modulus–porosity measurements. Another key result is that although the HA specimens included in this study have a unimodal pore size distribution, the details of the decrease in E and G with increasing P agree well with literature data for HA with both unimodal and bimodal pore size distributions. In addition, Poisson's ratio exhibits a local minimum in the porosity range of 0.2 < P < 0.25 for both HA and alumina, which may be related to the pore morphology evolution during sintering.