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Questions are frequently asked about the techniques used at the Lawrence Livermore National Laboratory to determine the velocity of sound through the various materials examined in the Acoustic Properties of Materials Laboratory, Manufacturing & Materials Engineering Division. Acoustic techniques enable us to measure material properties without harming the materials being tested, an important advantage over other methods. This paper describes some of the techniques and foibles used to measure sonic velocities and how these velocities are used to establish the elastic constants of investigated materials. However, it is not intended, nor should the reader infer that this report is comprehensive, since there is a limitation on how much a report of this size should contain. Moreover, there are a great many books available for the reader who wishes more detail. Books on elastic waves generally fall into two broad categories: applications and theory. The bibliography at the end of this report lists selected material in these two areas of elastic wave research. An example of a test report is given in Appendix A.
TABLE OF SYMBOLS with the exact grafic symboles
a = coefficient of thermal expansion (m/m/°K)
CF = Sound velocity in fluids (m/s)
Cl = longitudinal velocity (m/s)
CM = mean integrated velocity (m/s)
CO = extensional velocity (m/s)
CS = shear velocity (m/s)
CP = specific heat at constant stress (J / kg·°K)
E = Young's modulus (Pa)
EA = adiabatic Young's modulus (Pa)
EI = isothermal Young's modulus (Pa)
f = frequency (Hz)
G = shear modulus (Pa)
h = Planck's constant (662.60755 x 10 -36 J·s)
K = bulk modulus (Pa)
k = Boltzmann's constant (13.80658 x 10 -24 J /°K )
h/k = constant (47.99216 x 10 -12 °K·s )
M = atomic weight _ _
M = mean atomic weight
N = Avogadro's number (602.2167 x 10 21 gmol -1 )
P = number of atoms per molecule
R = reflection coefficient
Z = acoustic impedance (kg/m²s)
e = strain (m/m)
q = Debye temperature (K)
l = wavelength (m/cycle)
l' = Lame modulus (Pa)
n = Poisson's ratio
r = density (kg/m³ )
s = stress (Pa)
The measurement of ultrasonic velocities depends upon generating a dynamic pressure wave (pulse) into a material of known thickness and measuring the transit time of the emerging acoustic pressure wave. The generation and detection of an acoustic wave is usually accomplished by a piezoelectric transducer. The element is cut to a particular frequency, mode of operation (compressional, distortional, etc.), and is then assembled into a transducer case. The element has a measured amount of mechanical damping applied to the back surface. The transducers are purchased from commercial transducer manufacturers for the specific purpose of measuring the ultrasonic velocities with a high degree of resolution. Piezoelectric transducers are frequently used instead of magnetostrictive transducers because they are more adaptable to extreme ranges of frequency, have greater conversion efficiency and provide the greatest sensitivity. There are vast differences in frequency response, selectivity and sensitivity among the various materials used for their piezoelectric properties. Transducer manufacturers have discovered methods to significantly alter many transducer characteristics, such as, using mechanically coupled damping materials to prevent the excessive ringing following the initial excitation of the piezoelectric element. Many aspects of transducer fabrication rely on propriatary factors.
The shape of the pulse, generated by the electronic pulser, has a major influence on the pressure wave induced in the material. The input pulse is generally shaped to form a tail pulse in order to damp ringing following the initial pulse step. The ideal broadbandpressure pulse should:
(a) be a (sin x)/x function, since such a pulse can be shown to have an extremely wide bandwidth, and
(b) have a well defined, characteristic peak.
In practice, the (sin x)/x pulse is approximated by filtering the pulse spectrum at the spectrum extremes. The result of this attenuation is a time domain pulse of 5 to 7 half-cycles, with a well defined peak amplitude at the center. In the spectral domain, this waveform is shown bandpassed, resulting in the attenuation of both the lowest and highest frequencies, with a gradual peaking at the central frequency that, in turn, is proportional to the reciprocal of the period of the central peak in the time history.
Reflection of the ultrasonic wave occurs at the interface of two different acoustic impedances. The specific acoustic impedance is given in rayls (kg/m³s). The acoustic impedance is defined for bulk materials where the sound propagation is normal to the transducer to specimen interface as Cl , where is the mass density and Cl is the longitudinal velocity of sound in that medium. However, the impedance is further multiplied by the cross-sectional area for materials that are small compared to the wavelength of the sound. Materials whose lateral dimensions fall between 0.2 and 5 are avoided since the resulting longitudinal velocity is not defined. The transducer-to-specimen interface is a planar surface that frequently involves two or more different acoustic impedances. This difference in impedances causes a phase shift of the reflected ultrasound under specific conditions. The general equation for the sound pressure Reflection Coefficient shows how the phase shift may occur.
This indicates that when the sound is reflected from an interface where the second medium has a lower acoustic impedance than the first medium, a 180° phase shift occurs. The sound transmitted through an interface undergoes no phase shift, however. There may be varying amounts of phase shift from zero to several radians, depending upon the relative complex impedances involved in the reflection. This phase shift causes a discrepancy in the measured sound velocity, as described by McSkimin.3,4 The phase shift is a result of the transit (round-trip) time for sound to penetrate through the couplant and through the piezoelectric element and transducer wear face.
The usual method of measuring the time between reflected ultrasonic echoes shows a disparity in the time interval between one pair of adjacent echoes compared with the time interval between any other pair of adjacent echoes in an echo train. The amount of error may be determined, and the velocity of sound thereby corrected, by a technique known as the Pulse Superposition Method.3-5
This technique was further refined and modified by Papadakis 6,7 to include the correction for ultrasonic diffraction loss and phase change. This later technique replaced the former method of time interval, or echo repetition rate measurements, with a frequency measurement. The horizontal sweep on an oscilloscope is driven by an oscillator operating on a frequency whose period is equal to the time between echoes. The frequency is measured by a frequency counter, increasing the number of significant digits, which in turn increases the precision of the measurement. In addition, a tone burst generator permits a slight shift in the piezoelectric driving frequency that allows the operator to detect, and correct for, the amount of phase shift that has occurred within the specimen, thereby increasing the accuracy of measurement. This technique is amoung those used at LLNL for the determination of the ultrasonic velocities in lightly attenuating specimens that permit the ultrasound to form multiple echoes.
If the specimen is highly attenuating, multiple echoes cannot be detected and a single through-the-specimen pulse is measured for the time interval. The accuracy of the time interval measurement can be enhanced by an acoustic buffer or delay line. The accuracy can be only roughly estimated for velocities where extreme attenuation of sound causes difficulty in defining a separation of the pulse from the noise. In addition, the sound-attenuating materials usually force the use of thin specimens and low frequency transducers, further degrading the accuracy of the measurements.
The derivation of the sonic velocities is a property of the elastic constants of materials. The analysis of the fundamental assumption of elasticity theory, which states that linear stress is proportional to linear strain,
is the origin of all further analysis. In general, it may be shown that for any prescribed direction of propagation three plane waves can exist. The direction of propagation is taken to be perpendicular to the planes of equal phase. The phase velocities (V) for these waves may be obtained from the density r and the elastic stiffness moduli by the use of the Christoffel relations. The number of physical properties may be substantially reduced, depending upon the crystallographic symmetries, according to Neuman's principle 8,12 by a reduction of redundancies. An example of a simple cubic, single crystal is shown below.
The required elements for isotropic materials are adapted from the simple cubic as illustrated. The directional arrows show the particle direction for the shear waves.
Isotropic materials provide the greatest reduction in number of elastic moduli since for any direction through the material, the moduli are repeating. Anisotropic materials are frequently sent into the laboratory for ultrasonic velocity measurements, but it is seldom that such materials are sent in for moduli calculation. The primary reason for this is probably one of the following:
1. The crystallographic axis is not known.
2. Only one specimen of a single crystallographic orientation is available
3. Cost or time is prohibitive.
4. The client submitting the specimen is only interested in knowing whether or not it is anisotropic. For this reason, the rigorous analysis of elastic constants for anisotropic materials will not be described here, and the reader is referred to some excellent treatises on the subject.1,8 The subject matter described here is concerned with the unique solutions of the basic matrix equations for isotropic materials. Indeed, the material requirements for these analyses are rigidly limited to both isotropic and homogeneous materials. The principal ultrasonic velocities measured at the laboratory are the infinite medium (bulk specimen) longitudinal and shear velocities, although the extensional velocity is measured, on occasion, on wires, fibers, and thin rods.
The specimen geometry for a particular ultrasonic velocity determination may have a
decided effect on the wave propagation mode and on the measured wave speed. Generally,
velocity determinations are more accurate on relatively thick specimens, for three reasons:
1. As the thickness of the specimen increases, so does the number of significant digits to which one can accurately measure the thickness.
2. The effect on specimen geometry of a short time interval change in ambient temperature decreases with thickness due to the greater specimen volume (thermal inertia).
3. The pulsed ultrasound pressure wave has finite dimensions, generallized as a relatively small diameter cylinder or cone, roughly approximating the diameter of the transducer within the near field and spreading conically in the far field to:
where D is the transducer diameter: Calculated for the -6 dB down amplitude for flat transducers.
4. The first maxima back from the extreme far field (Y + O ) is given as:
where Z is the distance along the ultrasonic beam, CL is the longitudinal velocity of the medium, a is the transducer radius, l is the wavelength, and f is the frequency in Hertz. The start of the far field in the ultrasonic beam profile is approximated as:
where D is the transducer diameter.
The induced pressure wave should involve no more than one-sixth of the smallest dimensions of a specimen, depending upon the desired wave propagation mode. While it can be shown, one may intuitively see that as a sound pressure wave interacts with the specimen boundaries, tension, compression and shear forces contend for position, with the result that during this convolution of waves a severe distortion and interference of the waves occurs, rendering accurate measurement of time of arrival of these waves difficult. Most materials tend to perform as mechanical filters to ultrasound. Most generally, the materials behave like low pass filters, which attenuate the amplitude of the pressure waves in proportion to the increasing frequency and the increasing thickness in the direction of the wave propagation. However, attenuation is also a function of many other phenomena inherent in the materials, see Appendix B. Losses in solids are mainly the result of heat conduction, viscous friction, dislocation motion, and scattering. Losses in high-polymer materials (plastic, rubber, etc.) is mainly of viscous nature. It is therefore neither practical nor possible to set down a fixed specimen geometry beyond stating that the specimen may have flat, parallel faces with a parallelism of better than 25 x 10 -5 (in./in., m/m, etc.) for many polycrystalline materials, and to about 10 -6 for single-crystal specimens.
In addition, for most materials the diameter, and for rectangular specimens the edge dimension, should be three times the thickness (an aspect ratio of ³3:1). The thickness is usually 3 to 25 mm, depending upon the attenuation properties of the material. Sometimes these dimensions are radically altered, as for example, when the material is found only in the shape of rods, or wires, or even a textile yarn. A specialized geometry is required when the acoustic velocities are desired as a function of stress, strain, or temperature. Proper specimen preparation is essential for the determination of specific ultrasonic velocities.
The first velocity usually determined is the longitudinal, often referred to as compressional, nondistortional, L wave, P wave, waves of dilatation, and irrotational waves. The longitudinal velocity is specimen-geometry to wavelength dependent. Thin bars, rods, or wires will propagate a longitudinal wave at a lower velocity than the same material ofinfinite dimension. The first observation of this difference in velocities shows a strong wavelength dependence.When the wavelength (l = C/f) is less than one-fifth of the solid specimen's lateral dimensions along the path of wave propagation, the specimen is often referred to as a bulk specimen, and the velocity is often referred to as a bulk specimen
The first velocity usually determined is the longitudinal, often referred to as compressional, nondistortional, L wave, P wave, waves of dilatation, and irrotational waves. The longitudinal velocity is specimen-geometry to wavelength dependent. Thin bars, rods, or wires will propagate a longitudinal wave at a lower velocity than the same material of infinite dimension. The first observation of this difference in velocities shows a strong wavelength dependence. When the wavelength (l = C/f) is less than one-fifth of the solid specimen's lateral dimensions along the path of wave propagation, the specimen is often referred to as a bulk specimen, and the velocity is often referred to as a bulk specimen longitudinal velocity. This is the velocity most usually measured at the laboratory, and it has a definite relationship to the elastic properties of a solid isotropic homogeneous material:
The shear wave appears to be the most easily propagated waveform in nature and usually has the highest amplitude. Shear waves predominate in seismic activities and in all forms of fracturing. Typically, shear waves exhibit amoung the lowest frequency of vibration of sound within a specimen and amoung the greatest displacement of material. Shear waves are often called transverse waves, distortional waves, rotational waves, torsional waves, and S waves. They are usually the most difficult kind of wave to generate with transducers, and since the shear velocity is approximately one-half the velocity of longitudinal waves, the shear wavelength is on the order of one-half the wavelength of longitudinal waves for the same frequency. For this reason it is frequently necessary to use a shear wave frequency that is one-half the maximum longitudinal wave frequency. Some materials may support a longitudinal wave to the exclusion of shear waves, but any material that supports a shear wave will support a longitudinal wave. All other wave velocities, e.g., Stonely, Lamb, Rayleigh, Love waves, and thin rod or bar longitudinal waves, can exist only because the material can sustain both a shear and longitudinal wave. Materials that cannot sustain a shear wave, or shearing force, must be materials that are perfectly fluid, i.e., have zero viscosity. The shear velocity, CS , is related to the shear modulus in isotropic homogeneous materials by
where G is the shear modulus and r is the density. The particle direction of shear waves generated by shear wave transducers is always normal to the propagation direction. Shear waves are divided by usage into two types, Sv and SH , called shear vertical and shear horizontal, depending upon the particle direction with respect to the propagation direction. SH waves are usually introduced into a material through a wedge such that the particle direction is in the plane of the material. The SH wave will not mode convert to a longitudinal signal, but remain as a pure shear wave. The S v wave may also be introduced into a material through a wedge, but the particle direction is perpendicular to the propagation of the SH wave. As S v the wave just touches the material, part of the wave enters the material at the same moment that it is still partly in the wedge. This S v wave will mode convert according to Snell's law,9 which states
and the primed values refer to the second medium.
This is the same condition that exists when a longitudinal wave is introduced into a material at an angle. Angles are referenced to the axis-normal (perpendicular) to the plane of the specimen. Note that there is no refracted critical angle when the sound velocity in the first medium is greater than the sound velocity in the second medium. When the longitudinal sound velocity in the first medium is less than the sound velocity in the second medium, the incident longitudinal wave angle will be:
and when the refracted longitudinal wave is at the first critical angle, the refracted shear wave (Sv ) angle will be:
The Sv wave also results when a shear-wave transducer at normal incidence is coupled with a viscous, sticky, resinous coupling agent to a material capable of sustaining shearing forces.
The ultrasonic transducer must be coupled to the material in such a manner as to exclude the presence of air voids. With shear waves, coupling is even more critical than with longitudinal waves. Coupling may be accomplished through pressure if the transducer face and material have excellent mating surfaces, similar to those of gage blocks. Transducers are sometimes coupled to specimens with adhesives or low-melting-temperature salts, but easier methods are usually available. Specimens may react to certain coupling agents or otherwise adversely affected by absorption; in these cases, coupling can sometimes be accomplished by a combination of pressure and a sheet of Saran Wrap© sandwiched between the specimen and the transducer. The usual shear wave couplant is a modified resin without a catalyst. This resin has a high viscosity at or below room temperature, but thins rapidly as the temperature increases. It works well for longitudinal waves as well as shear waves.