WAVE PROPAGATION IN A LAYER CONTAINING A RANDOM OR PERIODIC

Transcription

1 WAVE PROPAGATION IN A LAYER CONTAINING A RANDOM OR PERIODIC DISTRIBUTION OF INCLUSIONS Nathan A. Day, Changyi Zhu, and V. K. Kinra Department of Aerospace Engineering Texas A&M University, College Station, Texas INTRODUCTION Wave propagation through particulate composites has received an increased amount of attention in recent years. One of the earlier studies on this subject was presented almost 50 years ago. Albert Wolf modeled the motion of a single rigid spherical inclusion in an elastic medium [1]. Later, Mow investigated scattering by a single elastic or fluid inclusion [2]. In 1970, Moon and Mow calculated a cutoff frequency for a random dispersion of rigid spherical inclusions in an elastic medium [3]. The subject of wave propagation through particulate composites was studied experimentally by Kinra, et al. in a series of papers examining dispersive wave propagation through particulate composites with both random and periodic distributions [4-9]. The results from one of these papers, Kinra and Ker [8], are presented in Figures 1 and 2. Figure 1 shows the phase velocity versus frequency for periodic and random steel/pmma composites. Figure 2 shows the attenuation of the same composites. Notice that phase velocity discontinuities occur at the same frequencies as local peaks in the attenuation. The objective of this work is to study the effect of the in-plane structure of an inclusion layer and resonance of individual particles on the wave propagation phenomena. Experiments were conducted on cylindrical specimens containing a layer with periodic or random distributions of spherical inclusions. By determining the complex-valued transfer functions for both random and periodic distributions of a single layer, one can predict the response of a layered composite with a finite number of layers [10]. SPECIMEN PREPARATION Neat Specimens A polyester casting resin hardened by a curing agent was chosen as the matrix material for this study [11]. Before curing, this material has a low viscosity and is easy to cast. After curing, the material is transparent making any imperfections Review of Progress in Quantitative Nondestructive Evaluation. Vol. 14 Edited by D.O. Thompson and D.E. Chimenti. Plenum Press, New York,

2 n, MHz r- 1.0r- 0.8~ ocnf6p o~\r~.... Steel-PMMA kl a =1.314 n Periodic o Ra~om 0.40L---~----~2----~3----~4----~5 n Figure 1: Phase velocity of periodic and random particulate composites [8]. - - E E "- 1/ ti C A CL ts V n Steel - Periodic k1a=1.314 n MatrlK E E "- I/) c A a: ts V n, MHz Figure 2: Attenuation of periodic and random particulate composites [8]. visible. Neat specimens without inclusions were manufactured for reference purposes. A predetermined amount of resin is mixed with the curing agent at a ratio of four drops of curing agent per ounce of resin. The mixture is mixed thoroughly and poured into a hollow cylinder mm in diameter. The resin is then allowed to cure for 24 hours at room temperature and pressure. The specimen is removed 236

3 from the mold and cut to the desired length perpendicular to the axis of the cylinder. The ends are then polished using 5 micron alumina polishing powder. Periodic Specimens The manufacture of the specimens with a layer of periodically arranged inclusions is more complicated. A steel mold mm in diameter with 0.5 mm diameter holes drilled in a 1.32 mm by 1.32 mm square array was used. Lead balls with a diameter of 1.19 mm were manually placed in lattice sites to produce an area fraction of 64%. The mold was placed into a hollow cylinder and resin poured on top. When the specimen is removed from the mold, it has lead balls protruding from the surface. The specimen is placed back into the hollow cylinder and a thin (approx. 1 cm) layer of polyester resin is poured on top. The thin layer forms a good bond with the lead balls and the already cured polyester. After the thin layer cures, more polyester is added to both ends to produced the desired length. The ends are then prepared in the same manner as the neat specimens. Figure 3 shows an end view of a typical periodic specimen. Random Specimens The manufacturing process for specimens with a random dispersion of inclusions is similar. Instead of a mold with an array of holes, a cylinder of the same dimensions with a flat surface is used. The proper number of inclusions to produce a 64 % area fraction is placed on this flat surface and the mold is shaken to produce a random distribution of inclusions. Resin is then poured over the inclusions and the rest of the procedure is identical to the periodic specimen manufacturing process. Figure 4 shows an end view of a typical random specimen. EXPERIMENTAL PROCEDURES Apparatus A water-immersion through-transmission apparatus was used to conduct the experiments. The transmitters and receivers were matched pairs of Panametrics broadband, piezoelectric transducers with center frequencies ranging from 0.5 MHz to 5.0 MHz and a crystal diameter of mm. A piece of silicon rubber 10 mm thick with a hole 25 mm in diameter is placed in between the transmitter and the specimen to act as an acoustic shield. The silicon rubber has a very high attenuation and a acoustic impedance close to that of water. This shield is designed to counteract the possible effects of beam spreading. A short-duration pulse is applied to the transmitter by a Panametrics Pulser/Receiver (Model 5052UA). The received signal is postamplified by the pulser/receiver and then digitized by a Tektronix DSA 601 at a sampling interval of 10 ns and a record length of 4096 points. To improve the signal to noise ratio, the signal is averaged 64 times. The signal is then sent to a PC with a 486 processor. A FFT routine then transforms the time domain signal to the frequency domain and is recorded on the PC. 237

4 [111\111[11'1'1 11'\1 '['I', 1 2 Figure 3: End view of periodic specimen. Figure 4: End view of random specimen. Transmission Technique To determine the transmission transfer function, the technique of Hanneman, Kinra, and Zhu was used [12]. Two measurements were taken for each test. The first measurement was taken through a neat polyester specimen and the second was through a specimen containing a layer of inclusions normal to the direction of wave propagation. The measurements will be referred to as the incident wave, f(t), and the transmitted wave, g(t), respectively. The transmission transfer function of the layer of spheres immersed in polyester, H\(w), is found by dividing the FFT of the transmitted wave, G*(w), by the FFT of the incident wave, F*(w). Ten sets of data were collected for each specimen. The mean value at each frequency was calculated. RESULTS Acoustical Properties of the Constituents To determine the properties of the polyester, reference specimens without inclusions were tested. The specimens were tested with several sets of transducers having center frequencies ranging from 0.5 MHz to 5.0 MHz. The useful frequency range of each transducer is the range for which the amplitude is within 6 db of the amplitude at the center frequency. The technique for analysis of thick specimens detailed by Kinra and Dayal was used to determine the phase velocity and the attenuation [13]. The phase velocity was found to differ by less than 1 % between samples. The density of the polyester was measured using Archimedes' principle and varied less that 1 % in different samples. The attenuation was found to vary by as much as 20% between samples. 238

5 Table 1. Acoustic properties of the constituents CI C 2 p ex Material (mm/p.s) (mm/p.s) (g/cm3) (nepers/mm) Polyester MHz Lead MHz The measured attenuation was found to increase linearly with the frequency with the exception of the 0.5 MHz transducers. This led us to believe that another phenomena may be occurring at long wavelengths. One possibility for this effect is beam spreading. The angle for beam spreading is inversely proportional to the frequency so the effect is magnified at low frequencies. The geometry and the size of the lead spheres made is difficult to measure the acoustic properties of lead. Therefore, the acoustic properties of lead are taken from an earlier work [6]. The acoustic properties of the constituents are presented in Table I. Random Distribution of Inclusions Figures 5 and 6 show the magnitude and phase, respectively, of the transmitted transfer function through a layer with a random distribution of inclusions. A nondimensional frequency, {1 = kia, is introduced where kl is the longitudinal wavenumber in the matrix material and a is the radius of the spheres. We conjecture that the minima in the magnitude of the transfer function are due to the excitation of resonances of the individual spheres. When the particle resonance is excited, the spheres will scatter the energy from the wave in all directions, resulting in a decrease in the energy of transmitted field. This effect translates into a minimum in the transmitted transfer function. The phase of the transfer function, shown in Figure 6, generally follows the pattern of increasing when the magnitude is going through a local minimum. Notice that the phase becomes almost linear at high frequencies. In Figure 1, Kinra and Ker show that the phase velocity for a particulate composite also becomes linear with increasing frequencies. Several random distributions with the same area fraction were tested yielding nearly identical magnitude and phase results. Periodic Distribution of Inclusions The magnitude and phase of the transfer function through a periodic distribution of inclusions is presented in Figures 7 and 8, respectively. The minima in the magnitude of the transfer function for the periodic layer occur at the same frequencies as the minima of the transfer function for the random layer with one exception. The exception is marked with an "ST" on Figure 7. We speculate that this minimum is due to the periodicity of the inclusions and does not occur in a random distribution of inclusions. Aside from the exception, the correlation in the minima frequencies lend support to our conjecture that they are caused by particle resonances. The phase of the transfer function for the periodic layer exhibits similar behavior to that of the random layer. The phase decreases near the frequency of the first minimum in magnitude, but increases near the frequencies of the other minima. 239

6 1.00 [] :-.~ 'l--r ,,-..-,-,.--r.--r-'--;---'-'-'-'-'-''''l Frequency (MHz) Figure 5: Magnitude of transfer function of random layer [] 6.00 '" Vl 0..c a Frequency (MHz) Figure 6: Phase of transfer function of random layer. The phase also becomes linear similar to the phase of the random case at higher frequencies. CONCLUSION For a single layer of inclusions, we have shown that the arrangement of the inclusions has a significant effect on a wave propagating normal to the layer. As 240

7 ( ;:. *:::J::: Frequency (MHz) Figure 7: Magnitude of transfer function of periodic layer ( ,-"w...l..lllllllllllllllllllllllllj...lu-l.u-l.llllllw 6.00 Q) VI o.r:: a ,,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-, Frequency (MHz) Figure 8: Phase of transfer function of periodic layer. expected, the resonances of the individual spheres significantly influence the wave propagation behavior. Also, we have identified a resonant frequency attributable to the periodic structure of a layer. 241

8 ACKNOWLEDGEMENTS This work is supported in part by the Texas Advanced Research Program (Advanced Technology Program) under Grant No to Texas A&M University, College Station, Texas REFERENCES 1. A. Wolf, "Motion of a Rigid Sphere in an Acoustic Wave Field", Geophysics 10, (1945). 2. C. C. Mow, "Transient Response of a Rigid Spherical Inclusion in an Elastic Medium", 1. Appl. Mech. 32, (1965). 3. F. C. Moon and C. C. Mow, "Wave Propagation in a Composite Material Containing Dispersed Rigid Spherical Inclusions", Rand Corporation Report, RM-6139-PR, Rand, Santa Monica, California (1970). 4. V. K. Kinra, M. S. Petraitis and S. K. Datta, "Ultrasonic Wave Propagation in a Random Particulate Composite", Int. J. Solids Struct. 16, (1980). 5. v. K. Kinra and A. Anand, "Wave Propagation in a Random Particulate Composite at Long and Short Wavelengths", Int. 1. Solids Struct. 18, (1982). 6. v. K. Kinra, E. L. Ker and S. K. Datta, "Influence of Particle Resonance on Wave Propagation in a Random Particulate Composite", Mech. Res. Commun. 9, (1982). 7. V. K. Kinra and E. L. Ker, "Effective Elastic Moduli of a Thin-Walled Glass Microsphere/PMMA Composite", J. Compos. Mater. 16, (1983). 8. V. K. Kinra and E. L. Ker, "An Experimental Investigation of Pass Bands and Stop Bands in Two Periodic Particulate Composites", Int. J. Solids Struct. 19, (1983). 9. V. K. Kinra and Peining Li, "Resonant Scattering of Elastic Waves by a Random Distribution ofinclusions", Int. J. Solids Struct. 22, 1-11 (1986). 10. N Day, C. Zhu, V. K. Kinra, in Review of Progress in QNDE, Vol. 13, eds. D. O. Thompson and D. E. Chimenti (Plenum, New York, 1993), p Clear Casting Resin, ETI, Fields Landing, CA S. E. Hanneman, V. K. Kinra, and C. Zhu, "A New Technique for Ultrasonic Nondestructive Evaluation of Adhesive Joints: Part II. Experiment", Experimental Mechanics 32(4), (1992). 13. V. K. Kinra, and V. Dayal, "A New Technique for Ultrasonic-Nondestructive Evaluation of Thin Specimens", Experimental Mechanics 28(3), (1988). 242