Abstract:
An untrasonic technique is presented which permits accurate measurement of the Poisson&#39;s ratio for a specimen which exists as a thin (&lt;10 mil) layer. This technique measures resonance and should prove especially useful in situations where a bulk specimen is either not readily available or would not properly reflect the properties of the material when configured as a thin layer. A detailed discussion of the theory underlying the technique is included. The technique is then used to determine the values of Poisson&#39;s ratio of three thin specimen materials. These values were then contrasted to those of the bulk specimens obtained in a more conventional manner. The technique as presented could be extended for a number of applications, including the cure monitoring of adhesives.

Description:
STATEMENT OF GOVERNMENT INTEREST 
     The invention was made with Government support under Contract No. F04701-88-C-0089 awarded by the Department of the Air Force. The Government has certain rights in the invention. 
    
    
     BACKGROUND OF THE INVENTION 
     1. FIELD OF THE INVENTION 
     The present invention relates to the determination of Poisson&#39;s ratio, particularly for thin layers. 
     2. DESCRIPTION OF THE PRIOR ART 
     It is known that the Poisson&#39;s ratio for a material can be determined from the ratio of the acoustic shear velocity to the longitudinal velocity. These velocities can often be easily measured using conventional ultrasonic time-of-flight techniques. Over the years, however, difficulties associated with making velocity measurements on thin attenuative specimens have forced ultrasonic researchers to become more innovative. For example, in an early effort to measure the velocity of both transverse and longitudinal waves in a buna-N vulcanizate as a function of temperature, A. W. Nolle and P. W. Sieck, J. Appl. Phys., Vol. 23, 888, (1952) employed a method involving the use of solid transmission media to conduct pulses into a thin flat specimen. The error associated with the acoustic measurement, however, was estimated in the report to be as high as 5% and 10% for the longitudinal and shear wave data, respectively. J. R. Cunningham and D. G. Ivey, J. Appl. Phys., Vol. 27,967 (1956) improved upon the aforementioned technique by incorporating a double acoustic path comprised of two separate specimens. Again, however, a number of experimental difficulties (e.g., producing uniform sample bondlines) contributed to significant measurement error. More recently, V. K. Kinra and V. Dayal, Experimental Mechanics, Vol. 28, 289 (1988) report combining standard FFT methods with conventional ultrasonics to measure the longitudinal phase velocity in specimens of submillimeter thicknesses. Their report includes a brief summary and comparison of their technique to those recently developed by others including the resonance method of F. H. Chang, J. C. Couchman and B. G. W. Yee, J. Comp. Mat., Vol. 8, 356 (1974) and the phase insensitive tone-burst spectroscopic method of J. S. Heyman J. Acoust. Soc. Amer., Vol. 64, 243 (1968) [5]. A primary object of the present invention is therefore to develop a schematic for making velocity measurements on thin attenuative specimen. 
     SUMMARY OF THE INVENTION 
     A simple ultrasonic resonance technique which permits measurement of the Poisson&#39;s ratio of thin adhesive material specimens is presented. The acoustic shear and longitudinal velocity are also determined. This technique is characterized by a number of advantages. The Poisson&#39;s ratio measurement accuracy is limited only by that of the resonant frequency; the Poisson&#39;s ratio is independent of the thickness of the specimen. The same specimen and transducer pair are used to determine both the shear and longitudinal response. A fluid medium is employed to couple sound into the specimen, thereby eliminating many of the problems associated with the bonding of transducers. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 illustrates a thin medium of acoustic impedance Z 2 , sandwiched between two semi-infinite media such that the third (left most) medium sees the composite system formed by the first and second mediums as having an input acoustic impedance of Z 21 . 
     FIG. 2 illustrates the coefficient of acoustic power transmission across the second medium as a function of frequency for the three layer system depicted in FIG. 1, where Z 1  =Z 3  =17 Rays (Al) and Z 2  =1.48 Rayls (water). 
     FIG. 3 is a top view of the experimental apparatus. 
     FIGS. 4A and 4B illustrate ray diagrams for the apparatus depicted in FIG. 3 where the specimen thickness has been enlarged for clarity. 
     FIG. 5 is a graph of sample data for an Al/H 2  O/Al sandwich, where the circles result from a measurement of the frequency response of a solid aluminum standard of the same dimensions as the sandwich. 
     FIGS. 6A and 6B illustrate longitudinal and shear response data for a thin Black Wax specimen. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The present invention is directed toward making Poisson&#39;s ratio measurements on thin material specimens, where thin is understood to be on the order of several mils. The Poisson&#39;s ratio for a material can be determined through measurement of its acoustic velocities using the relation: ##EQU1## 
     where σ is the Poisson&#39;s ratio and c 1  and c s  are the longitudinal and shear acoustic velocities, respectively. 
     Referring to FIG. 1, consider a first semi-infinite medium 2, with acoustic impedance Z 1 , as depicted in FIG. 1. If a thin layer of a second medium 4, with acoustic impedance Z 2 , is placed in contact with the left side of the first medium, the input impedance of the composite system (first medium+second medium ) is given by the complex relation: ##EQU2## 
     where j is the imaginary index and for the second medium, x 2  is the thickness, β 2  =2π/λ 2  is the propagation constant and λ 2  is the acoustic wavelength, G. S. Kino, Acoustic Waves: Devices, Imaging and Analog Processing, (Prentice-Hall, 1987), p. 12. It should be noted that in the derivation of Eqn. 2, a steady state condition (continuous plane wave stimulation of the system of frequency, f) was assumed. Wavelength is related to frequency according to the relation, c=λ 2  f, where c is the acoustic velocity (either shear or longitudinal) in Medium 2. Next, consider what happens when a third (semi-infinite) medium 6, is placed in contact with the left side of the second medium as is also depicted in FIG. 1. If medium 6 has the same acoustic impedance as that of the first medium, then the stress wave reflection coefficient R, is given by the relation: ##EQU3## 
     The coefficient of acoustic power transfer P T , across medium 2 can now be calculated; ##EQU4## 
     where the substitution β 2  =2πf/c has been applied. 
     In FIG. 2, P T  is plotted as a function of frequency for the case where the first and third media are aluminum (Z 1  =17 Rayls) and the second medium is water (Z 2  =1.48 Rayls). It can be seen that the condition of maximum power transmission occurs when f=f R  =nc/2x 2 , where n is an integer. This behavior can be expected whenever Z 2  is higher than both Z 1  and Z 3  or lower than both Z 1  and Z 3  (as in this case). If Z 2  falls midrange between Z 1  and Z 3 , maximum power transmission occurs when f=f R  =(n+1)c/4x 2 . Note that the second medium was assumed to be lossless in the derivation of Eqn. 4. Taking attenuation into account, one would expect the amplitude of the local maxima to decrease with increasing n. The expression for Poisson&#39;s ratio (Eqn. 1) can now be rewritten as, ##EQU5## 
     where f RS  ∝c s  /2x 2  and F RI  ∝c 1  /2x 2  are determined from the condition of maximum power transmission for shear and longitudinal acoustic waves, respectively. 
     EXPERIMENTAL TECHNIQUE 
     The experimental apparatus is depicted in FIG. 3. Two aluminum blocks 10, 12 are prepared from 1.5 inch square rod stock, each having a θ=74° face and a 90° face with respect to one side of the block. A milled finish was determined to be adequate on the 74° faces. The 90° faces were lapped to insure that the surfaces were flat. A thin uniform layer of the specimen 14 being tested was sandwiched between the 90° faces. The spacing between these faces and hence, the specimen thickness, was determined by two identical, stainless steel wire spacers 16, 18. To hold this Al/specimen/Al sandwich together, small rubber O-rings 20, 22 was stretched over a set of threaded pegs located on each of two opposite sides of the sandwich as shown. The depth of thread for these pegs was less than 0.125 inches. The transducers and O-ring are bisected by a plane parallel to that of the page and 0.75 inches into sandwich. The approximate length 1 of the Al/specimen/Al sandwich is 1.75 inches. The dimension d is 1.5 inches, as the blocks were cut from square rod stock of 1.5 inch side. 
     The resultant AF specimen/Al sandwich was set upon a fixture (not shown) and submerged in a water tank (not shown) so as to be centered between a pair of 5 MHz, plane wave, through-transmission, ultrasonic transducers 26, 28. The 14 with respect to the transducers 26, 28. The apparatus is designed to approximate the conditions leading to the derivation of Eqn. 4. The Al blocks correspond to first and third media, the specimen to the second medium. 
     To perform measurements, one transducer 26 was stimulated to propagate a toneburst (single frequency) longitudinal drive pulse (water will not support a shear wave). The fixture was aligned so that this pulse would strike the first 74° face of the Al/specimen/Al sandwich at a particular angle of incidence θ i  to its normal, giving rise to both a longitudinal and a shear wave pulse within the aluminum, with angles of emergence θ 1  and θ S , respectively. The emergence angles for a particular θ i  can be determined by the well known Snell&#39;s law analog, ##EQU6## 
     where V w  is the acoustic velocity in water, and V s  and V 1  are the acoustic velocities for shear and longitudinal waves in aluminum, respectively. 
     In the first phase of the experiment illustrated in FIG. 4A, θ i  i was set at ˜3.7° so that the longitudinal pulse L transmitted through the first 74° face would strike the Al-specimen interface at normal incidence, and then travel through the remainder of the Al/specimen/Al sandwich to produce a signal at the receive transducer 28 as depicted in FIG. 4a. This signal will be referred to as the longitudinal response. Because the longitudinal velocity in AI exceeds that of the shear, the longitudinal response was the first signal detected after each drive pulse. This first signal was followed by others due to the shear waves transmitted through the first 74° face and various internal reflections within the sandwich. The duration of the drive pulse was chosen to be ˜1.5 μs, which was short enough to permit temporal separation of the various signals, yet long enough to allow reverberations within the thin specimen layer to approximate a steady state (i.e., these reverberations did not significantly affect the lengths of the pulses). The amplitude of the longitudinal response was measured as a function of frequency (between 2 and 8 MHz). To isolate the effect of the specimen, the Al/specimen/Al sandwich was then replaced with a solid Al block of the same dimensions and the measurement repeated. This second set of data was divided into the first. The resultant data set was then normalized and plotted. Peaks in the the amplitude rs. frequency plot, which in accordance with Eqn. 4, occur when f=f R1  =nc 1  /2x 2  were then used to determine c 1 , the longitudinal acoustic velocity of the specimen. 
     In the second phase of the experiment, illustrated in FIG. 4B, θ i  was set to ˜7.5°, so that the direction of the transmitted shear waves was normal to the Al-specimen interface as depicted in FIG. 4B. The first signal resulting from this normal shear pulse will be referred to as the shear response. Following each drive pulse, the shear response was preceded by not only the longitudinal response, but also other signals resulting from internal reflections involving the faster longitudinal pulse L. To positively identify the shear response, one could increase θ i  beyond the critical angle for longitudinal wave production in the aluminum, so that the shear response would be the first remaining signal. One could then track this signal while decreasing θ i  to the appropriate value. The frequency dependence of the shear response amplitude was then measured, normalized and plotted in the same manner as that of the longitudinal response amplitude. Peaks in this plot, occurring when f=f RS  =nc s  /2x 2 , were then be used to determine the shear acoustic velocity of the specimen, c s . 
     RESULTS 
     Before attempting to measure both c s  and c 1  in a material, a simpler system having no shear response was tested. An Al/H 2  O/Al sandwich with wire spacers of diameter x 2  =1.016×10 -4  m (4 mil) was prepared. It should be emphasized that in this case (and all that follow), the 90° faces on the Al blocks were lapped flat with 600 grit garnet. Earlier tests, using blocks which had not been lapped, yielded results which were not satisfactory. In FIG. 5, the frequency response of the Al/H 2  O/Al sandwich and a solid Al standard are plotted as boxes and circles, respectively. The second plot was divided into the first and normalized to produce the solid curve. The importance of this correction (for the transducer response and attenuation in the Al) is obvious as the peak shifts significantly from its position in the raw data. The peak in the solid curve, at f R  =7.3 MHz, was used to calculate c 1  =2f r1  x 2  =1483 m/s for water in excellent agreement with the literature value. 
     Three materials, Black Wax (BW), Five Minute Epoxy (FME) and Crystalbond (CB) adhesive, were selected for measurement of both c s  and c 1 . These materials were easily obtained and could be easily formed into both a thin specimen, as needed for this experiment, and a thick specimen for comparative purposes. The longitudinal response and shear response data for the thin Black Wax specimen are plotted in FIG. 6A and 6B. The raw data was corrected as outlined above to produce these curves. The wire spacers used for the Al/BW/Al sandwich were of diameter 2.39×10 -4  m (9.4 mils). The longitudinal response in FIG. 6A exhibits a peak at 4.8 MHz, corresponding to a longitudinal velocity of 2290 m/s. The shear response curve in FIG. 6B exhibits four equally spaced peaks, corresponding to n=1, 2, 3 and 4 in the relation; f RS  =nc S  /2x 2 . As expected, the amplitude of these peaks decreases with increasing n, due to attenuation within the black wax. It should be noted that the data become less reliable as one pushes the limits on the operating range of the transducer (see the Al Standard plot of FIG. 5). The positions of the shear peaks correspond to a shear velocity of c s  =1120 m/s. As mentioned earlier, a thick specimen of black wax was also prepared of approximate dimensions, 2.5×2.5×0.66 cm. The longitudinal and shear velocities of this specimen were measured using a standard through-transmission, ultrasonic technique. The values thus obtained are compared to those for the thin specimen in Table 1, along with similar results which were obtained for Crystalbond adhesive and Five Minute Epoxy. The entrees for Poisson&#39;s ratio were calculated using Eqn. 5. 
     DISCUSSION 
     It can be observed that for the three materials tested, the resonance technique for thin specimens described above leads to values for c 1 , c s  and Poisson&#39;s Ratio σ which compare well with those determined using thicker specimens and a conventional through-transmission ultrasonic method (Table 1), differing by only a few percent. This difference falls within the uncertainty ascribed to the conventional measurement. When used to measure the acoustic velocity of water, the resonance technique yielded a value essentially equivalent to that reported in the literature. Sources of error for the resonance technique include error in determination of the resonant frequency, wire thickness and sample alignment, or nonuniformities in the specimen thickness which lead to a broadening of the resonance. The resonant frequency can bc determined more accurately by sampling more points. Problems with specimen alignment can bc mitigated through a well designed apparatus and careful experimentation. The wire thickness is important for determination of the velocities, but drops out of the equation for the Poisson&#39;s ratio. 
     The resonance technique presented should prove particularly useful for measurement of the mechanical properties of materials which exist (perhaps only) as thin (subwavelength) layers such as adhesives or highly attenuative materials. The use of a fluid couplant medium provides for uniform and repeatable coupling of the sound into the specimen. The capability for measuring both velocities on a single sample becomes especially useful when multi-component polymer systems are being tested, because of the variations within and between batches. In addition to the Poisson&#39;s ratio and acoustic velocities, if the density, p of the specimen is known, the Young&#39;s, shear and bulk moduli, E, α and K, can also be calculated via the familiar relations, ##EQU7## 
     It is implicit that the properties measured are dynamic. In many instances, however, the temperature of the specimen and/or the frequency of measurement can be arranged so that the measurement occurs above or below the glass transition. 
     
                       TABLE 1______________________________________Comparison of velocities measured for thin and thick specimens.c.sub.1 (10.sup.5 cm/s)          c.sub.s (10.sup.5 cm/s)                       σSample Thin     Thick   Thin   Thick Thin   Thick______________________________________BW    2.29      2.23   1.12   1.11  0.364  0.355CB    2.33     2.3     1.12   1.15  0.350  0.333FME   2.7      2.7     1.25   1.28  0.343  0.335______________________________________ 
    
     In addition to the aforementioned applications, a number of enhancements and/or extensions of the technique can be envisioned. Instead of using a toneburst signal and sweeping the frequency, one might instead use a broadband pulse and Fourier transform the received signal. One could then effectively get all the information from a single pulse. The technique could also be arranged to study or monitor the cure of adhesives. 
     Although the invention has been described in terms of a preferred embodiment, it will be obvious to those skilled in the art that alterations and modifications may be made without departing from the invention. Accordingly, it is intended that all such alterations and modifications be included within the spirit and scope of the invention as defined by the appended claims.